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TITLE: Minimal polynomial of $\sqrt[3]{2} + \omega$ over $\mathbb{Q}.$
QUESTION [0 upvotes]: Is the polynomial $f(x) = x^9 - 9x^6 - 27x^3 - 27$ irreducible over $\mathbb{Q}?$ I think it is because of Eisenstein's applied to the prime $3.$
Is it the minimal polynomial of $x = e^{2 \pi i/3} + \sqrt[3]{2}$ over $\mathbb{Q}$? Wolfram Alpha says the answer is $x^6+3 x^5+6 x^4+3 x^3+9 x+9.$
I am a bit confused.
REPLY [3 votes]: Note that Eisenstein's criterion requires $p^2$ to not divide the constant term. Your polynomial is reducible. Note that the minimal polynomial of $\omega$ is the quadratic $x^2 + x + 1$, so the extension $\Bbb Q(\omega, \sqrt[3]{2})$ is of degree six, and thus $\omega + \sqrt[3]{2}$ has a minimal polynomial of degree dividing six.
(Wolfram alpha factors $ x^9 - 9x^6 - 27x^3 - 27$ as $x^3-3x^2+3x-3$ times the degree six polynomial you cite in your question.) | 176,437 |
TITLE: Random walks on infinite directed regular graphs
QUESTION [6 upvotes]: Let us consider a directed graph $\Gamma=(V,E,s,t)$ ($V$ set of vertices, $E$ set of edges, $s,t: E \rightarrow V$ are the "source" and "target" maps).
Assume that $\Gamma$ is bi-regular, that is there are two integers $d_1 \geq 1$ and $d_2 \geq 1$ such that $|s^{-1}(v)|=d_1$ and $|t^{-1}(v)|=d_2$ for every $v \in D$. Assume that $\Gamma$ weakly connected (there is an undirected path relying any two vertices).
And finally assume that $\Gamma$ has no forward-closed finite subset (i.e. if $S$ is a finite subset of $V$, there is an edge in $E$ with source in $S$ and target not in $S$). In particular $\Gamma$ is infinite.
Consider the standard discrete-time forward random walk on $\Gamma$, starting at $x \in V$: at time $0$ you are at $x$ with probability $1$; for any $n \geq 0$, at time $n+1$, conditional to being at $y$ at time $n$, your odds of being at any of the $d_1$ forward-neighbors of $y$ (i.e. $t(e)$ for $e \in E$, $s(e)=y$) is $1/d_1$.
Let $p^n_{x,y}$ be the probability of being at $y$ at time $n$.
Is it true that for every $y$, $p_n(x,y) \rightarrow 0$ as $n \rightarrow \infty$?
Here are some remarks. This question is related to Fedja's beautiful answer to this question. Fedja proves the result when $\Gamma$ is an undirected regular graph (seen as an undirected graph by replacing each undirected edge by two directed edges going both way). Unfortunately, I have not been able to extend his argument to my directed case.
The hypothesis that $\Gamma$ has no forward-closed finite subset is certainly necessary: If $S$ was such a subset, and $x \in S$, then you would be sure to stay in the finite set $S$ forever, so $\sum_{y \in S} {p^n_{x,y}} = 1$ and one of these $p^n_{x,y}$ at least can not tend to $0$.
The hypothesis that $|s^{-1}(v)|=d_1$ (or at least $s^{-1}(v)$ finite) for all $v$ is necessary to define the random walk, but that $|t^{-1}(v)|=d_2$ (or at least is finite) for all $v$ is also necessary for the theorem to be true. Without it, consider the graph with $V=\mathbb Z$, and for every $a \in \mathbb Z$, there is one directed edge from $a$ to $a+1$ and one directed edge from $a$ to $0$ (called the "speedy return" edge). Thus $|s^{-1}(a)|=2$ for every $a \in \mathbb Z$, but $|t^{-1}(0)|=\infty$. The probability $p^n_{0,0}$ is $\geq 1/2$ since every path that ends up with the "speedy return edge" goes from $0$ to $0$.
REPLY [5 votes]: Here is a counterexample:
Let $G_1$ be the digraph with vertex set $\mathbb N$, two loops at $0$, an edge from $0$ to $1$, and for every $i \geq 1$ an edge from $i$ to $(i+1)$ and two parallel edges from $i$ to $(i-1)$. Let $G_2$ be any countable digraph in which every vertex has $2$ outgoing edges and $4$ incoming edges, and let $f \colon V(G_2) \to V(G_1)$ be such that $|f^{-1}(0)| = 0$ and $|f^{-1}(i)| = 1$ for every $i \geq 1$.
Let $G$ be the digraph obtained from $G_1 \uplus G_2$ by adding all edges from $v$ to $f(v)$. Then $G$ is bi-regular with $d_1 = 3$ and $d_2 = 4$. Moreover, $G$ is weakly connected and has no finite forward closed sets since every vertex has a forward edge connecting it to the forward ray in $G_1$.
The simple random walk on $G$ almost surely enters $G_1$ after finitely many steps (and remains in $G_1$ thereafter since there are no edges from $G_1$ to $G_2$). But the simple random walk on $G_1$ is just a biased random walk on $\mathbb N$ with bias towards $0$. This random walk is irreducible, aperiodic (because of the loops at $0$), and positive recurrent. Thus $\lim_{n \to \infty} p_n(x, i) = \mu(i)$ independently of the starting point $x$, where $\mu \neq 0$ is the invariant probability measure on $\mathbb N$. | 79,473 |
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Pregnancy May Slow Progression of MS
Researchers have made some startling findings: according to a recent study, having a baby may slow down the damaging effects of multiple sclerosis (MS), a serious autoimmune disease.
Multiple Sclerosis is a disorder in which the body's own defense system attacks myelin, the protective fatty substance that surrounds nerve fibers in the central nervous system. This causes a disruption to nerve signals traveling to and from the brain, which causes the numbness, walking problems, blurry vision and fatigue associated with MS. About 85 percent of those with MS start with a relapsing-remitting course, in which attacks are followed by partial or total recovery.
Belgian researchers followed 330 women who had their first MS symptoms between ages 22 and 38 and published their study in the Journal of Neurology, Neurosurgery and Psychiatry. About three-quarters of the women in the study had children. Their research, says women who have given birth to at least one child were 34 percent less likely to have multiple sclerosis progress to a stage where they needed walking assistance (cane or brace) than women without children.
While giving birth either before or after their symptoms started was a benefit, the women who had the most benefits were the ones who had a baby after the onset of symptoms. They were 39 percent less likely to have their disease progress to a point where they had trouble walking. Researchers concluded that women with Multiple Sclerosis who have children seem to have a more benign course than those without children.
Why does childbirth apparently slow the progress of MS? Patricia O'Looney, director of biomedical research for the National Multiple Sclerosis Society and quoted in the journal article says, "Having one or more children does seem to be beneficial, but, we don't know enough about the patient demographics to really draw some major conclusions." The study didn't elaborate on other treatments the women might be receiving.
They developed two theories for why having children may slow MS progression.
- One theory is that pregnancy "downregulated" or suppressed the immune system to prevent the mother's body from rejecting the fetus. Suppressing the immune system with medications like interferon are common treatments for controlling MS.
- A second theory is that pregnancy causes estrogen levels to rise which may help protect women from MS by stimulating the mother's body to make myelin.
Further research is being done to understand the benefits of pregnancy on MS.
Does this mean women with MS should have a baby as a treatment option for Multiple Sclerosis? Not necessarily. MS is a complicated disease and doctors and scientists still don't know as much as they should about why one person's disease is less aggressive than another. Having a baby is a serious life decision and isn't the right choice for many women. Women diagnosed with MS already considering pregnancy should talk to their physicians about how it may affect their health. | 27,109 |
UNZA Vice-Chancellor Professor Stephen Simukanga has said the University of Zambia has re-branded itself strategically to be a 'green campus' starting From 2013. Speaking when he officially launched the 'keep UNZA clean campaign' on 22nd February, 2013, Prof Simukanga said the sustainability emphasizes the point that cleanliness at unza should not just be a one-day activity but that it should be sustained into the future. He said the campaign is also contextualized within the African Union's New Partnership for Africa's Development (NEPAD) whose environmental initiative recognizes that a healthy environment through cleanliness is a pre-requisite for Africa’s Development. “The event we are witnessing today is set within wider policy frameworks beyond that of Zambia. As part of a global community, we are now living within the united nations decade of education for sustainable development, which runs from the year 2005 to 2014. The decade was designated in this manner in order to recognize that change towards a better quality of life through cleanliness starts with education and awareness raising,” he said. Zambians are alive to the reality that a campaign such as that of 'keep unza clean' should be more than just a slogan. The campaign must be a concrete reality for all of us - as individuals, managers, academic or non-academic staff, students as well as organizations and governments - in all our daily decisions and actions. The 'keep UNZA clean' campaign is an initiative by the School of Education which in the recent past has spearheaded various projects and made donations to Kalingalinga police post. The School also has a vibrant degree programme in environmental education meant to produce the needed human resource for various sectors, including for the Ministry of Education, Science, Vocational Training and Early Education. | 50,025 |
tag:blogger.com,1999:blog-4356285099890378613.post1316671020755990646..comments2017-05-25T09:02:39.117+01:00Comments on Pulcinella - Fashion and Lifestyle: Meeting Ms. Bianca [email protected]:blogger.com,1999:blog-4356285099890378613.post-80934794382594620032012-04-06T23:33:42.648+01:002012-04-06T23:33:42.648+01:00So cool that you met her! Amy fashionandbeautyf...So cool that you met her! <br /><br />Amy<br /><br />fashionandbeautyfinds.blogspot.comAmy! That´s nice! You have a really wonderfull blo...Wow! That´s nice!<br />You have a really wonderfull blog and an amazing style :)<br />xoxo from MunichLola Finn can believe You meet her! She is such and amazin...I can believe You meet her! She is such and amazing lady!! Love her she is gorgeous a she was very fashion forward in her times when she got married in a white tuxedo to Mick Jagger wow! she looked amazing and she still has this inner elegance.Victoria're a lucky girl! And you look pretty, as a...You're a lucky girl! And you look pretty, as always!Vanessa. look great! Would you like to follow each othe...You look great! Would you like to follow each other? :) Dea dell'Apparenza. for sharing. I can resist everything ...Great,thanks for sharing.<br />I can resist everything except <a href="" rel="nofollow">Abercrombie Outlet</a>Blaine wow how cool! wow how cool!<br /><br /> Conversations're a very lucky girl :D enjoy every second...You're a very lucky girl :D enjoy every second of this experience!Avi post, she looks amazing :)Great post, she looks amazing :)Psycho Cat necklace!!!great necklace!!!Riyana N´re lucky! great necklace ! :)´re lucky! great necklace ! :) <br /><br /> Z. are very lucky. Congratulations. Bianca Jagger...You are very lucky. Congratulations. Bianca Jagger is always very stylish. <br />xxdimitri, i love your blog too! I'm following ;) Bi...Hey, i love your blog too! I'm following ;)<br />Bianca Jagger really is a diva!<br /><br />XoXo<br /> de Andrade Gomes fab!! Congrats darling :)Just fab!! Congrats darling :)Marta, lucky you, you look great :)Amazing, lucky you, you look great :)Natascha Coester you both look lovely! you both look lovely!<br /><br /> of Mia have a great blog!love bianca jagger,and this ...you have a great blog!love bianca jagger,and this post is amazing! fashion street blog! take a look on mine and it u like it, w...nice blog!<br />take a look on mine and it u like it, we can follow each other. what do you think?<br />xxx<br /><br />thefrontrow-tfr.blogspot.com<br />twitter: @thefrontrow_tfrSILVIA - thefrontrow;];]<br /><br /> Mila gorgeous lady! Love the necklace :)A gorgeous lady! Love the necklace :)lorenabr an amazing experience..she still looks stunni...What an amazing experience..she still looks stunning today. REally enjoyed reading your post :)<br />Happy Thursday hun xoxo<br /> ,great Pictures ,lucky Girl! :) xoxoWow ,great Pictures ,lucky Girl! :)<br /><br />xoxoLoveT. | 234,290 |
Hey there! My name is Lexi and I’m a sophomore com munitions major at Villanova University. I am eternally on the hunt for the perfect macaroon, but when that job gets too stressful I can be found hanging around my campus with friends and sorority sisters, managing PR platforms for various clubs, editing or writing for the school newspaper, and of course, blogging. I’ve been into fashion ever since I was a kid and realized that there are more stores than “Limited Too” out there, and I have since been harnessing a deep appreciation for numerous different styles and looks! Keep up with me here on CollegeFashionista! | 200,338 |
TITLE: It is possible to find at least one function satisfying both $a)$ and $b)$?
QUESTION [2 upvotes]: Let $a<1$ and let $\varepsilon>0$. I am looking for a function $f\in C^1(\mathbb{R}\setminus\{0\})$ satisfying both
$$a)\quad f^{\prime}(x)\ge \frac{1}{x^2} \qquad \mbox{ for } 0<x<\varepsilon;$$
$$b)\quad f(x)\ge \frac{1}{x^a} \qquad \mbox{ for } 0<x<\varepsilon.$$
About me, some examples can be found only when $a<0$. I am not sure, but maybe one example can be constructed in this way: take $a=-1$ and $f(x) = x$. Thus $f(x)$ satisfies $b)$ and $f^{\prime}(x) =1$ satisfies $a)$ for $x>1$,i.e you can take $\varepsilon=2$ and $a)$ is satisfied for $1<x<2$, but not in the range $0<x<1$ (this is the reason why I am not convinced).
Could someone please help me?
Thank you in advance!
REPLY [1 votes]: There is no such a function.
First, point $a$ means $f' \ge 0$ so $f$ increases for $0<x<\varepsilon$ which means that for example as $n \to \infty, f(1/n)$ decreases to some $L$ which can be finite or $-\infty$
But if $L$ would be finite, if we fix such an $n$ large enough so $0<1/n <\varepsilon$, one has $f(1/n)=\int_{1/m}^{1/n}f'(x)dx+f(1/m) \ge \int_{1/m}^{1/n}\frac{1}{x^2}dx+f(1/m) \ge m-n+L$ for all $m >n$ which means that $f(1/n)=\infty$ and that is not possible.
Hence $L$ must be $-\infty$ and since $f$ monotonic, it follows that $f(x) \to -\infty, x \to 0, x >0$ so in particular $f(x)<0$ for $x>0$ small enough contradicting point $b$ | 150,099 |
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\subsection{Preliminary definitions on Markov chains}
In this section, we introduce some of the definitions on Markov chains used throughout this work (see e.g. \cite{LevPerWil09}).
A Markov chain over a finite state space $S$ is specified by a transition matrix $P$, where $P(a,a')$ is the probability of
moving from state $a$ to state $a'$ in one step. The $t^{th}$ power of the transition matrix gives the probability of moving from one state to another state in $t$ steps. The chain studied in this work is ergodic (see below for a proof), meaning that it has a unique \emph{stationary distribution} $\pi$, that is, $\lim_{t\rightarrow \infty} P^t(a,a')=\pi(a')$ for any two states $a$ and $a'$.
\paragraph{Reversible chains.}
We shall use the definition of a \emph{reversible} Markov chain, also called \emph{detailed balanced condition}: If the transition matrix $P$ admits a vector $\pi$ such that $\pi(a) P(a,a') = \pi(a')P(a',a)$ for all $a$ and $a'$, then $\pi$ is the \emph{stationary distribution} of the chain with transitions $P$.
\paragraph{Total Variation Distance and Mixing Time.}
The \emph{total variation} distance of two distributions $\mu$ and $\pi$ is
\[ d_{TV}(\mu,\pi) := \frac{1}{2} \cdot \sum_{a\in S} \Big|\mu(a)-\pi(a)\Big|. \]
The \emph{mixing time} of an ergodic Markov chain with transition matrix $P$ is defined as
\[
t_{mix} (\epsilon) = \min\left\{t : d_{TV}(P^t(a_0,\cdot),\pi)\leq \epsilon
\ \text{ for all } a_0 \in S\right\} .
\]
It is common to also define the quantity $t_{mix}:=t_{mix}(1/4)$, which is justified by the fact that, for any $\epsilon$, $t_{mix}(\epsilon) \leq \lceil\log_2 1/\epsilon\rceil \cdot t_{mix}$. In our experiments, we shall evaluate the total variation distance for $\epsilon =0.05$ to get better estimate.
\subsection{A Markov Chain Sampler}
\label{sec:sampling-markov-chain}
Suppose we have a set of $k$ possible local changes transforming any sequence $a$ into another sequence $a'$ such that all sequences can be obtained
by applying a certain number of such operations. Then the following standard Metropolis chain samples sequences with the desired distribution:
\begin{enumerate}
\item \label{trans1} With probability $\frac{1}{2}$ do nothing. Otherwise,
\item \label{trans2} Select one of the $k$ local operations u.a.r. If this operation cannot be applied to the current sequence $a$ (the new
sequence is unfeasible) do nothing; Otherwise, let $a'$ be the sequence obtained from $a$ by applying this operation;
\item \label{trans3} Accept the operation transforming $a$ to $a'$ with probability
\begin{equation}
\label{eq:accept_probability}
A(a,a') :=\min\left\{1,\frac{P(a')}{P(a)}\right\}=\min\left\{1,\frac{m(a')}{m(a)}\cdot \frac{perm(a')}{perm(a)}\right\},
\end{equation}
and do nothing with remaining probability $1 - A(a,a')$.
\end{enumerate}
\paragraph{Local operations over the sequences}
We define our Metropolis chain $\MCblocks$ through four types of operations: Peak to Flat (PF), Flat to Valley (FV), Flat to Flat (FF), and Peak into Valley (PV). We formally define them as:
\begin{linenomath}
\begin{align*}
PF(i,j) :=& \begin{cases}
p_i & \leftarrow\ p_i - 1 \\
f_{i-1} & \leftarrow\ f_{i-1} + 2 \\
f_j & \leftarrow\ f_j - 1 \\
f_{j+1} & \leftarrow\ f_{j+1} + 1
\end{cases}, &
FV(i,j) :=& \begin{cases}
f_i & \leftarrow\ f_i - 2 \\
p_i & \leftarrow\ p_i + 1 \\
f_j & \leftarrow\ f_j - 1 \\
f_{j+1} & \leftarrow\ f_{j+1} + 1
\end{cases}
\\
FF(i,j) :=& \begin{cases}
f_i & \leftarrow\ f_i - 1 \\
f_{i+1} & \leftarrow\ f_{i+1} + 1 \\
f_j & \leftarrow\ f_j - 1 \\
f_{j-1} & \leftarrow\ f_{j-1} + 1 \\
\end{cases} &
PV(i,j) :=& \begin{cases}
p_i & \leftarrow\ p_i - 1 \\
f_{i-1} & \leftarrow\ f_{i-1} + 2 \\
p_j & \leftarrow\ p_j - 1 \\
f_{j} & \leftarrow\ f_{j} + 2 \\
\end{cases}
\end{align*}
\end{linenomath}
Note that each type of operation applies to two indices $i$ and $j$, and we also implicitly consider the reversed operations which ``undo'' the changes. We now explain step~\ref{trans2} of the chain $\MCblocks$ in more detail: The Markov chain $\MCblocks$ picks two indices $i$ and $j$ at random, then picks one of the four operations above, and decides with probability $1/2$ whether to choose the operation or its reversed version.
As for step~\ref{trans3}, computing the transitional probability $A(a,a')$ can be done in constant time as only a few of the factors in Equations~\eqref{eq:perm-product} and~\eqref{eq:ma-product} change.
\begin{restatable}[]{theorem}{MCstationarytheorem}
\label{th:MC:stationary}
The Markov chain $\MCblocks$ defined above is ergodic and its unique stationary distribution satisfies $\pi(a)\propto P(a)$ for every $a\in S(n,A)$.
\end{restatable}
\begin{proof}
The proof consists of two steps. First, we have to show that the chain is ergodic, that is, it is aperiodic and connected (see e.g. \cite{LevPerWil09}). Then we use the standard detail balance condition to obtain the stationary distribution.
\subsubsection{Connectivity of $\MCblocks$.}
To prove that the chain $\MCblocks$ is connected (from every \emph{building sequence} $a$ we can reach every other building sequence $a'$ in a maximum of $\bigO(A)$ operations) we argue in two steps.
Intuitively, we show that we can transform any two \emph{paths} into each other by some operations depicted in Figure~\ref{fig:path-moves}.
Then it can be seen that every operation in Figure~\ref{fig:path-moves} corresponds to a sequence of operations in the Markov chain $\MCblocks$, given in Figure~\ref{fig:basic-moves}.
Formally:
Every path of width $n$ and area $A\leq \left \lfloor n^2/4 \right \rfloor $ can be turned into any other path of the same area and width by using the operations in Figure~\ref{fig:path-moves}.
To prove this we consider a \emph{canonic path} for a given width $n$ and area $A$.
The canonic path is the uniquely defined path $\mz \in \MZ(n,A)$ for which the following holds:
For every $i$, after $i$ steps (i.e. between $x=0$ and $x=i$) $\mz$ has maximum area among all paths in $\MZ(n,A)$.
The possible forms of the canonic path
are shown in Figure~\ref{fig:connectivity-1}.
Any given path with width $n$ and area $A$ can be transformed into the canonic path of the same area using the steps from Figure~\ref{fig:path-moves}.
We overlay the given path with the canonic path and proceed in steps to the right as is schematically shown in Figure~\ref{fig:connectivity-1} with the black path being the given path and the red path being the canonic path.
There are three possibilities. Either the paths coincide, in which case we proceed to the right, or the given path differs proceeding with a $D$ move or with an $H$ move.
In both cases the given path must intersect the canonic path on the falling part because otherwise the area cannot be the same. Now we use the operations in
Figure~\ref{fig:path-moves} in horizontal sweeps from left
to right to fill-in the missing area of the canonic path. At
the end both paths must coincide because the areas are the
same.
Each of these operations can be simulated by some
operations on the sequences in
Figure~\ref{fig:basic-moves}. This can be seen immediately
because the four cases in Figure~\ref{fig:path-moves}
correspond directly to one or two operations in
Figure~\ref{fig:basic-moves}.
\begin{figure}[t!]
\includegraphics[width=\textwidth]{connected.pdf}
\caption{
(left, middle) The canonic path for $n=12$ and $A=18$, resp. $A=21$.
(right) Building a canonic path (red) from a given path (black) with same area.}
\label{fig:connectivity-1}
\end{figure}
\begin{figure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=.7]{PF-sequence.pdf}
\caption{Peak to Flat.}
\label{fig:PF}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=.7]{FV-sequence.pdf}
\caption{Flat to Valley.}
\label{fig:FV}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=.7]{FF-sequence.pdf}
\caption{Flat to Flat.}
\label{fig:FF}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[scale=.7]{PV-sequence.pdf}
\caption{Peak into Valley.}
\label{fig:PV}
\end{subfigure}
\caption{The basic operations over the sequences.}
\label{fig:basic-moves}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.8]{moves-simulation.pdf}
\caption{Changing \emph{the path}. The shape on one side is transformed to the shape on the other side. In some cases we need two of the operations defined in Figure~\ref{fig:basic-moves}.}
\label{fig:path-moves}
\end{figure}
\subsubsection{Stationary Distribution of $\MCblocks$.}
It is well-known that the Metropolis chains with acceptance probability $A(a,a')=\min\left\{1,\frac{\pi(a')}{\pi(a)}\right\}$ have stationary distribution $\pi$ since the detailed balance condition is obviously satisfied: given that the number of operations is $k$, we have
\begin{linenomath}
\begin{align*}
P(a,a') =& \frac{A(a,a')}{2k} & \text{and}& & P(a',a) =& \frac{A(a',a)}{2k}
\end{align*}
\end{linenomath}
and the definition of $A(\cdot,\cdot)$ yields the detailed balance condition, that is,
$\pi(a) P(a,a')=\pi(a')P(a',a).$
\end{proof} | 89,525 |
Considerations in Athletic Performance Enhancement Training: Athlete Weight Room Preparation
Robert A. Panariello MS, PT, ATC, CSCS
Professional Physical Therapy
Professional Athletic Performance Center
New York, New York
During my 30+ career as a Physical Therapist (PT), Certified Athletic Trainer (ATC) and Strength and Conditioning (S&C) Coach, I have been involved in both the Sports Rehabilitation and Performance Enhancement Training of athletes and have had many valued experiences throughout my years of practice in these two related professions. When confronted with an athlete who presents with a pathology that occurred during the course of S&C or personal training participation, my observations of the athlete, the review of the athlete’s injury and medical history, and my experiences in the sports rehabilitation of athletes, often reveals that the injury is not directly due to a specific exercise performance, but to one of two other training considerations. The first possible cause is the implementation of a poor program design, i.e. inappropriately prescribed exercise weight intensities and exercise performance volumes, which is beyond the subject matter of this dialog, and the second, is the athlete was not properly physically prepared prior to their participation into the formal training program design. Often times, the athlete enters the weight room to initiate their physical training and regardless of their physical condition and/or training experience, they are expected and instructed, along with their peers, to participate in the first day of the identically prescribed formal training program design. This is especially true of the high school athlete. The question then arises, how does the S&C Professional know the athlete will be able to correctly perform and physically tolerate the prescribed program design when implementing this manner of training?
My good friend and one of my mentors, Hall of Fame S&C Coach Al Vermeil has established and imparted upon me his hierarchy of athletic development. This system is utilized as a well-organized progression to assist the S&C Professional in the optimal athletic development of the athlete (Figure 1.)
Figure 1. Vermeil’s Hierarchy of Athletic Development
Coach Vermeil’s system is fostered upon a continuum of the physical qualities necessary for optimal athletic performance. A review of this hierarchy will reveal that strength is the physical quality, the foundation, from where all other physical qualities evolve. Each physical quality is dependent upon the optimal development of the preceding physical quality so that the ideal development of each successive physical quality in the hierarchy may transpire. One should note that although several physical qualities may be trained simultaneously, the emphasis of training is placed upon one specific physical quality until the time where the next ascending physical quality in the pyramid is determined to be developed.
Prior to the initiation of training, a review of Coach Vermeil’s hierarchy will exhibit the necessity for the physical evaluation of the athlete, as well as the development of the athlete’s work capacity, or as some coaches may call this level of the pyramid “general physical preparation (GPP)”. Work capacity or GPP is necessary for the preparation of the athlete for their eventual safe participation in the formal weight training program design.
During my time studying at the Soviet Institute for Physical Culture and Sport in Moscow, prior to the break up of the USSR, the topic of the system of athletic development that ensued at the thousands of Soviet Sports Schools across the USSR was discussed. Included in this lesson was the necessity for the preparation of the young Soviet athlete prior to the progression of applied higher stresses,over time, that would occur during their specific athletic development (specialization). A modification of this concept is presented in figure 2.
Figure 2. The General Physical Preparation and Specialization of the Young Athlete
The successful Soviet structure of training acknowledges the importance and incorporation of a systematic process of general physical preparation prior to the athete’s eventual participation in 100% specialization of training, therefore, shouldn’t we as S&C professionals also heed from this lesson of athletic development?
Javorek’s Exercise Complexes
One method utilized over the years to prepare our athlete’s for the participation into the formal training program design is to incorporate Javorek’s exercise complex system into the training process. These exercise complexes were developed by S&C Coach Istvan “Steve” Javorek as part of the training system utilized with his athletes. These exercise complexes require the athlete to perform a series of specific exercises, employing either barbells or dumbells, with one exercise performance immediately followed by another until an “exercise cycle” or “set” is completed. The athlete then performs the prescribed number of exercise cycles/sets to complete their prescribed daily workout. An example of a Javorek’s exercise complex is as follows:
Barbell Upright Row X 6 Reps
Barbell Snatch High Pull X 6 Reps
Barbell Behind the Head Squat Push Press X 6 Reps
Barbell Behind the Head Good Morning X 6 Reps
Barbell Bent Over Row X 6 Reps
In this example the athlete will have performed a total of 30 successive exercise repetitions while incorporating the entire body during the training in the exercise cycle/set. Exercise weight intensities are initiated with 10% to 15% of the athlete’s body weight and are progressed over time until the athlete is able to perform the exercise complex with 30% – 35% of their body weight. The workouts are performed three days per week and depending upon the individual athlete, may begin with three exercise cycles/sets in their initial workout and progressed over time until the athlete demonstrates the performance of 5-6 cycles/sets at 30% to 35% of their body weight per daily workout.
Some of the advantages for incorporating Javorek’s exercise complexes include but are not limited to:
- Establish the proficiency of exercise technical performance
- Preparation of the neuro-muscular and musculo-tendonous systems of the body for the eventual application of high volume, high weight intensity exercise performance
- Enhance joint mobility and soft tissue compliance
- Enhance strength and power output
- Increase work capacity
Depending upon the specific needs, presentation, and medical history of the athlete, exercises may be substituted and/or modified for the athlete as part of their prescribed exercise performance.
Javorek’s exercise complex systems work well to assist in the preparation of the athlete for the ensuing intergration of the formal training program design. A program design that will include the application of higher exercise volumes and weight intensity performance. We have also implemented Javorek exercise complexes during the “end stage” of the athletes sports rehabilitation prior to their discharge from the clinic and eventual particpation in a formal off-season S&C program.
The preparation of the athlete prior to their initiation into the formal training program design is an important aspect of training that is often overlooked. A properly prepared athlete will not only perform superiorly in the weight room, but likely reduce the incidence of training injuries as well. | 65,922 |
TITLE: Let $B$ be a countable set and let $f$ be a surjective function ($f:A\to B$), then $A$ is also countable
QUESTION [0 upvotes]: Question: Given a countable set $B$ and a function $f:A\to B$ which is surjective, then $A$ is also countable.
I think it's a false affirmation, but I have no idea of a counter example I can use here.
I basically know that: every subset $X \subseteq N$ is a countable set. And the corollary: given a countable set $A$ and a function $f:A\to B$ which is surjective on $B$, then $B$ is also countable.
REPLY [0 votes]: Simple counter:
f: R--->{x} be a function such that
f(y) = x for all y in R
Co domain is a singleton set ,so it is countable.
But domain is set of all real numbers which is uncountable. | 16,942 |
In this section will show that the functors introduced by A. Thomas in \cite{Thomas2006} take graphs of $H$-lattices with a fixed Bass-Serre tree to complexes of $H$-lattices whose development is a ``sufficiently symmetric" right-angled building (we will make this precise later). Finally, we will combine these tools to construct a number of examples. In particular, non-residually finite $(\Isom(\EE^n)\times A)$-lattices where $A$ is the automorphism group of a sufficiently symmetric right-angled building, and non-residually finite algebraically irreducible lattices in products of arbitrarily many isometric and non-isometric sufficiently symmetric right-angled buildings.
\subsection{Right angled buildings} \label{sec.functor.rab}
Let $(W,I)$ be a right-angled Coxeter system. Let $N$ be the finite nerve of $(W,I)$ and $P'$ be the simplicial cone on $N'$ with vertex $x_0$. A \emph{right-angled building} of type $(W,I)$ is a polyhedral complex $X$ equipped with a maximal family of subcomplexes called \emph{apartments}. Such an apartment is isometric to the Davis complex for $(W,I)$ and the copies of $P'$ in $X$ are called \emph{chambers}. Moreover, the apartments and chambers satisfy the axioms for a Bruhat--Tits building.
Let $\cals$ denote the set of $J\subseteq I$ such that $W_J\leq W$ is finite. Note that $W_\emptyset=\{1\}$ so $\emptyset\in\cals$. For each $i\in I$, the vertex $P'$ of \emph{type} $\{i\}$ will be called an \emph{$i$-vertex}, and the union of the simplices of $P'$ which contains the $i$-vertex but not $x_0$ will be called the \emph{$i$-face} There is a one-to-one correspondence between the vertices of $P'$ and the types $J\in\cals$.
Let $X$ be a right-angled building. A vertex of $X$ has a type $J\in\cals$ induced by the types of $P'$. For $i\in I$ an \emph{$\{i\}$-residue} of $X$ is the connected subcomplex consisting of all chambers which meet in a given $i$-face. The cardinality of the $\{i\}$-residue is the number of copies of $P'$ in it.
\begin{thm}[\cite{HaglundPaulin2003}]
Let $(W,I)$ be a right-angled Coxeter system and $\{q_i\colon i\in I\}$ a set of integers such that $q_i\geq 2$. Then up to isometry there exists a unique building $X$ of type $(W,I)$ such that for each $i\in I$ the $\{i\}$-residue of $X$ has cardinality $q_i$.
\end{thm}
If $(W,I)$ is generated by reflections in an $n$-dimensional right-angled hyperbolic polygon $P$, then $P'$ is the barycentric subdivision of $P$. Moreover, the apartments of $X$ are isometric to $\RH^n$. In this case we call $X$ a \emph{hyperbolic building}. We remark that a right-angled building can be expressed as the universal cover of a polyhedral product, however, we will not use this observation elsewhere.
\begin{remark}
Let $(W,I)$ be a right-angled Coxeter system with parameters $\{q_i\}$ and nerve $N$. Let $E_i$ be a set of size $q_i$ and let $CE_i$ denote the simplicial cone on $E_i$, denote the collections of these by $\underline{E}$ and $C\underline{E}$ respectively. The right-angled building of type $(W,I)$ with parameters $\{q_i\}$ is the universal cover of the polyhedral product $(C\underline{E},\underline{E})^N$.
\end{remark}
\subsection{A functor theorem}
In this section we will recap a functorial construction of A.~Thomas which takes graphs of groups with a given universal covering tree to complexes of groups with development a right-angled building. We will then show that this functor takes graphs of lattices to complexes of lattices and deduce some consequences.
Let $X$ be a right-angled building of type $(W,I)$ and parameters $\{q_i\}$ with chamber $P'$. Suppose $m_{i_1,i_2}=\infty$ and define the following two symmetry conditions due to Thomas \cite{Thomas2006}:
\begin{enumerate}[label=(T\arabic*)]
\item There exists a bijection $g$ on $I$ such that $m_{i,j}=m_{g(i),g(j)}$ for all $i,j\in I$, and $g(i_1)=i_2.$ \label{Thomas.1}
\item There exists a bijection $h:\{i\in I: m_{i_1,i}<\infty\}\to \{i\in I:m_{i_2,i}<\infty\}$ such that $m_{i,j}=m_{h(i),h(j)}$ for all $i,j$ in the domain, $h(i_1)=i_2$, and for all $i$ in the domain $q_i=q_{h(i)}$. \label{Thomas.2}
\end{enumerate}
We include the construction adapted from \cite{Thomas2006} for completeness and for utility in the proofs of the new results which will follow. An example of the construction for a graph of groups consisting of a single edge is given in Figure~\ref{fig.pentagon}
\newdimen\R
\R=2.3cm
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\draw[xshift=0.0\R] (0:\R) \foreach \x in {72,144,...,359} {
-- (\x:\R) }
-- cycle (360:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=right:{$\{i_2,i_4\}$}] (c) {}
-- cycle (288:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=below right:{$\{i_2,i_5\}$}] (d) {}
-- cycle (216:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=below left:{$\{i_1,i_5\}$}] (e) {}
-- cycle (144:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=above left:{$\{i_1,i_3\}$}] (a) {}
-- cycle (72:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=above right:{$\{i_3,i_4\}$}] (b) {};
\draw (a) -- (b) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=above left:{$\{i_3\}$}] (ab) {};
\draw (b) -- (c) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=above right:{$\{i_4\}$}] (bc) {};
\draw (c) -- (d) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=below right:{$\{i_2\}$}] (cd) {};
\draw (d) -- (e) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=below:{$\{i_5\}$}] (de) {};
\draw (e) -- (a) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=left:{$\{i_1\}$}] (ea) {};
\node [circle,fill=black,inner sep=0pt,minimum size=5pt,label={[right,xshift=0.34cm,yshift=0.125cm]:{$\emptyset$}}] (m) {};
\draw (m) -- (a);
\draw (m) -- (b);
\draw (m) -- (c);
\draw (m) -- (d);
\draw (m) -- (e);
\draw (m) -- (ab);
\draw (m) -- (bc);
\draw[dashed] (m) -- (cd);
\draw (m) -- (de);
\draw[dashed] (m) -- (ea);
\end{tikzpicture}
\begin{tikzpicture}
\draw[xshift=0.0\R] (0:\R) \foreach \x in {72,144,...,359} {
-- (\x:\R) }
-- cycle (360:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=right:{$G_w\times \ZZ_{q_4}$}] (c) {}
-- cycle (288:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=below right:{$G_w\times \ZZ_{q_5}$}] (d) {}
-- cycle (216:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=below left:{$G_v\times \ZZ_{q_5}$}] (e) {}
-- cycle (144:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label=above left:{$G_v\times\ZZ_{q_3}$}] (a) {}
-- cycle (72:\R) node[circle,fill=black,inner sep=0pt,minimum size=5pt,label={[above,xshift=0.25cm]:{$G_e\times \ZZ_{q_3}\times\ZZ_{q_4}$}}] (b) {};
\draw (a) -- (b) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label={[above,xshift=-0.15cm]:{$G_e\times\ZZ_{q_3}$}}] (ab) {};
\draw (b) -- (c) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=right:{$G_e\times \ZZ_{q_4}$}] (bc) {};
\draw (c) -- (d) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=below right:{$G_w$}] (cd) {};
\draw (d) -- (e) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=below:{$G_e\times \ZZ_{q_5}$}] (de) {};
\draw (e) -- (a) node [midway, circle,fill=black,inner sep=0pt,minimum size=5pt,label=left:{$G_v$}] (ea) {};
\node [circle,fill=black,inner sep=0pt,minimum size=5pt, label={[right,xshift=0.34cm,yshift=0.125cm]:{$G_e$}}] (m) {};
\draw (m) -- (a);
\draw (m) -- (b);
\draw (m) -- (c);
\draw (m) -- (d);
\draw (m) -- (e);
\draw (m) -- (ab);
\draw (m) -- (bc);
\draw[dashed] (m) -- (cd);
\draw (m) -- (de);
\draw[dashed] (m) -- (ea);
\end{tikzpicture}
\caption[A complex of groups over the pentagon.]{The left pentagon shows a labelling of the types $J\in\cals$. The right pentagon shows the local groups after applying Thomas' functor to a graph of groups with a single edge. In both pentagons the dashed line shows the embedding of the graph. If the graph of groups has a single vertex, then $G_v=G_w$, $q_1=q_2$, $q_3=q_4$, the edge $(\{i_1,i_5\},\{i_1\})$ is glued to $(\{i_2,i_5\},\{i_2\})$, and the edge $(\{i_1,i_3\},\{i_1\})$ is glued to $(\{i_2,i_4\},\{i_2\})$.}
\label{fig.pentagon}
\end{figure}
\begin{construction}[Thomas' Functor \cite{Thomas2006}]
\textit{Let $X$ be a right-angled building of type $(W,I)$ and parameters $\{q_i\}$. For each $i_1,i_2\in I$ such that $m_{i_1,i_2}=\infty$ let $\calt$ be the $(q_{i_1},q_{i_2})$-biregular tree. Suppose \ref{Thomas.1} holds and if $q_{i_1}=q_{i_2}$ then \ref{Thomas.2} holds with $g$ an extension of $h$. Then there is functor $F:\calg(\calt)\to\calc(X)$ preserving faithfulness and coverings.}
We will construct $F$ as a composite $F_2\circ F_1$. We first define $F_1:\calg\to\calc_1$. Let $(A,\cala)$ be a graph of groups and $|A|$ the geometric realisation of $A$. We will construct a complex of groups $F_1(A)$ over $|A|$. For the objects we have:
\begin{itemize}
\item The local groups at the vertices of $|A|$ are the vertex groups of $\cala$.
\item For all $e\in EA$ let $\sigma_e=\sigma_{\overline{e}}$ be the vertex of the barycentric subdivision $|A|'$ at the midpoint of $e$.
\item The local group at $\sigma_e$ in $F_1(A)$ is $A_e=A_{\overline{e}}$.
\item A monomorphism $\alpha_e:A_e\to A_{i(e)}$ in $A$ induces the same monomorphism in $F_1(A)$.
\end{itemize}
Let $\phi:A\to B$ be a morphism of graphs of groups over a map of graphs $f$, note that by \cite[Proposition~2.1]{Thomas2006} $F_1$ is not injective on morphisms. We define $F_1(\phi)$ as follows:
\begin{itemize}
\item The map $f$ induces a polyhedral map $f':|A|'\to|B|'$ so we will define $F_1(\phi):F_1(A)\to F_1(B)$ over $f$.
\item Now take the morphisms on the local groups to be the same as for $\phi$.
\end{itemize}
Let $\calc(\calt)=\im(F_1(\calg(\calt)))$ and $G(Y)\in\calc(\calt)$. Now, we will define $F_2:\calc(\calt)\to\calc(X)$ as follows:
\begin{itemize}
\item We first embed $Y'$ into a canonically constructed polyhedral complex $F_2(Y)$. For each $e\in EY$ let $P_e'$ be a copy of $P'$ and identify the midpoint of $e$ with the cone vertex $x_0$ of $P_e'$.
\item If $Y$ is $2$-colourable with colours $i_1$ and $i_2$ (from the valences of the Bass-Serre tree if $q_{i_1}\neq q_{i_2}$), then we identify the vertex of $e$ of type $i_j$ with the $i_j$-vertex of $P_e'$.
\item Suppose $Y$ is not $2$-colourable. If $e\in EY$ is not a loop in $Y$ then identify one vertex of $e$ with the $i_1$-vertex of $P_e'$ and the other with the $i_2$-vertex. If $e$ forms a loop then we attach $P_e'/h$ (where $h$ is the isometry from the assumption) and identify the vertex of $e$ to the image of the $i_1$- and $i_2$-vertices of in $P_e'/h$.
\item Glue together, either by preserving type on the $i_1$- and $i_2$-faces or by the isometry $h$, the faces of the the $P_e'$ and $P_e'/h$ whose centres correspond to the same vertex of $Y$. Let $F_2(Y)$ denote the resulting polyhedral complex.
\item Note that $Y'\rightarrowtail F_2(Y)$ and that each vertex of $F_2(Y)$ has a unique type $J\in\cals$ or two types $J$ and $h(J)$ where $i_1\in J\in\cals$ and $h$ is the isometry from the assumption.
\item Fix the local groups and structure maps induced by the embedding of $Y'$ in $F(Y)$. For each $i\in I$ let $G_i=\ZZ_{q_i}$ and for $J\subseteq I$ let $G_J=\prod_{j\in J}G_j$. For each $e\in EY$ let $G_e$ be the local group at the midpoint of $e$.
\item Let $J\in \cals$ such that neither $i_1$ or $i_2$ are in $J$. The local group at a vertex of type $J$ is $G_e\times G_J$. The structure maps between such local groups are the natural inclusions.
\item Let $J\in\cals$ and suppose $i_k\in J$ for one of $k=1$ or $k=2$. Since $m_{i_1,i_2}=\infty$ both $i_1$ and $i_2$ cannot be in $J$. Let $F_e$ be the $i_k$-face of $P_e'$ or the glued face of $P_e'/h$. The vertex of type $J$ in $P_e'$ or $P_e'/h$ is contained in $F_e$. Let $v$ be the vertex of $Y$ identified with the centre of $F_e$ and let $G_v$ be the local group at $v$ in $G(Y)$
\item The local group at the vertex of type $J$ is $G_v\times G_{J\backslash \{i_k\}}$. For each $J'\subset J$ with $i_k\in J'$ the structure map $G_v\times G_{J'\backslash \{i_k\}}\rightarrowtail G_v\times G_{J\backslash \{i_k\}}$ is the natural inclusion. For each $J'\subset J$ with $i_k\not\in J'$ the structure map $G_e\times G_{J'}\rightarrowtail G_v\times G_{J\backslash \{i_k\}}$ is the product of the structure map $G_e\rightarrowtail G_v$ in $G(Y)$ and the natural inclusion.
\end{itemize}
Now, let $\phi:G(Y)\to H(Z)$ be a morphism in $\calc(\calt)$ over a non-degenerate polyhedral map $f:Y\to Z$. We will define $F_2(\phi)$ as follows:
\begin{itemize}
\item If $Y$ and $Z$ are two colourable $f$ extends to a polyhedral map $F_2(f):F_2(Y)\to F_2(Z)$. Otherwise we use \ref{Thomas.1} to construct $F_2(f)$.
\item If $\tau\in VF(Y)$ then $G_\tau=G_\sigma\times G_J$ where $\sigma$ is a vertex of $Y'$. The homomorphism of local groups $G_\sigma\times G_J\to H_{f(\sigma)}\times G_{J}$ is $\phi_\sigma$ on the first factor and the identity on the other factors.
\item Let $a\in EF(Y)$. If $\psi_a$, the structure map along $a\in F_2(G(Y))$, has a structure map $\psi_b$ from $G(Y)$ as its first factor, put $F_2(\phi)(b)=\phi(a)$. Otherwise set $F_2(\phi)(b)=1$.
\end{itemize}
\end{construction}
We will now show the functor takes graphs of lattices to complexes of lattices and deduce a number of consequences. For locally compact group $H$ let $\rm{Lat}(H)$ denote the set of $H$-lattices and let $\rm{Lat}_u(H)$ denote the set of uniform $H$-lattices.
\begin{thm}\label{thm.functor.new}
Let $Y$ be a right-angled building of type $(W,I)$ and parameters $\{q_i\}$ and let $A=\Aut(Y)$. For each $i_1,i_2\in I$ such that $m_{i_1,i_2}=\infty$ let $\calt$ be the $(q_{i_1},q_{i_2})$-biregular tree and let $T=\Aut(\calt)$. Suppose \ref{Thomas.1} holds and if $q_{i_1}=q_{i_2}$ then \ref{Thomas.2} holds with $g$ an extension of $h$, and let $F:\calg(\calt)\to\calc(Y)$ be Thomas' functor. Let $X$ be a finite dimensional proper $\CAT(0)$ space and assume $H=\Isom(X)$ contains a cocompact lattice. The following conclusions hold:
\begin{enumerate}
\item If $G(\calt)$ is a graph of $H$-lattices, then $F(G(\calt))$ is a complex of $H$-lattices. \label{thm.functor.gtoc}
\item $F$ induces an inclusion of sets $\rm{Lat}_u(H\times T)\rightarrowtail\rm{Lat}_u(H\times A)$. \label{thm.functor.uniform}
\item If $Y$ is a $\CAT(0)$ polyhedral complex then $F$ induces an inclusion of sets $\rm{Lat}(H\times T)\rightarrowtail \rm{Lat}(H\times A)$. \label{thm.functor.nonuni}
\end{enumerate}
Let $\Gamma$ be a uniform $(H\times T)$-lattice and let $F\Gamma$ be the corresponding $(H\times A)$-lattice.
\begin{enumerate}[resume]
\item $\pi_T(\Gamma)$ is discrete if and only if $\pi_A(F\Gamma)$ is discrete. Moreover, $\pi_H(\Gamma)=\pi_H(F\Gamma)$. \label{thm.functor.proj}
\item If $\Gamma$ satisfies any of $\{$algebraically irreducible, non-residually finite, not virtually torsion free$\}$, then so does $F\Gamma$. \label{thm.functor.props}
\end{enumerate}
\end{thm}
\begin{proof}
We first prove \eqref{thm.functor.gtoc}. We will first verify the conditions on the local groups and then construct a morphism to $H$. Let $(B,\calb,\psi)$ be a graph of $H$-lattices and consider the image $L(Z)$ of $\calb$ under $F$. Here $Z=F(B)$. Each local group in $L(Z)$ is of the form $G_\sigma\times G_J$ where $G_\sigma$ is a local group in $\calb$ and $G_J$ is a finite product of finite cyclic groups. We have a morphism $\psi:\calb\to H$ such that the image of each local group $G_\sigma$ is an $H$-lattice and the restriction to $G_\sigma$ has finite kernel. Thus, by construction the local groups in $L(Z)$ are commensurable in $\pi_1(L(Z))$. We define $F(\psi_\sigma)$ to be the composite $\psi|_{G_\sigma}\circ\pi_\sigma:G_\sigma\times G_J\twoheadrightarrow G_\sigma\to\psi(G_\sigma)$, thus commensurability of the images in $H$ is immediate.
We will now deal with the edges. Note the twisting elements in $L(Z)$ are all trivial and the complex of groups $H$ has all structure maps the identity. Let the structure maps in $L(Z)$ be denoted by $\lambda_a$ for $a\in EZ'$ and the structure maps in $\calb$ by $\alpha_e$ for $e\in EB$. The family of elements $(t_e)_{e\in EB}$ in the path group $\pi(\calb)$ are mapped under $\psi$ to elements of $\Comm_H(\psi(G_\sigma))$ where $G_\sigma$ is some local group. Now, let $a\in EZ'$, then by construction $a$ either corresponds to a subdivision of an edge $a$ in $EB$ in which case we define $(F\psi)(a)=\psi(a)$. Or, $a$ corresponds to a inclusion of local groups $G_\sigma\times G_{J'}\to G_\sigma\times G_J$, in which case we define $(F\psi)(a)=1_H$.
It remains to verify the two edge axioms for a morphism. For each $a\in EZ'$ corresponding to the subdivision of an edge $a$ in $EB$ we have \[\Ad((F\psi)(a))\circ F(\psi_{i(a)})=\Ad(\psi(a))\circ \psi_{i(a)}\circ \pi_a=\psi_{t(a)}\circ\alpha_a\circ \pi_a=F(\psi_{t(a)})\circ F(\alpha_a),\]
where $\pi_a$ is the surjection $G_a\times G_J\twoheadrightarrow G_a$. For any other edge $a\in EZ'$ we have
\[\Ad((F\psi)(a))\circ F(\psi_{i(a)})=F(\psi_{i(a)})\text{ and }F(\psi_{t(a)})\circ\lambda_a=F(\psi_{i(a)}).\]
Finally, the other condition that $(F\psi)(ab)=(F\psi)(a)(F\psi)(b)$ for $(a,b)\in E^2Z'$ is verified trivially. Thus, $F(\calb)=L(Z)$ is a complex of $H$-lattices. $\blackdiamond$
We will next prove \eqref{thm.functor.uniform}. Let $\Gamma$ be an $(H\times T)$-lattice. By Theorem~\ref{thm.structure}, $\Gamma$ splits as graph of $H$-lattices $\calb$. Thus, by \eqref{thm.functor.gtoc} we obtain a complex of $H$-lattices $F(\calb)$ with fundamental group $\Lambda$. By Theorem~\ref{thm.col}\eqref{thm.col.uniform} it suffices to show that for each local group $G_\sigma$ in $F(\calb)$ the kernel $K_\sigma=\Ker(\pi_H|_{FG_\sigma})$ acts faithfully on $X$. Now, $K_\sigma$ is a direct product of $L_\sigma=\Ker(\pi_H|_{G_\sigma})$ with a direct product of cyclic groups $G_J$, where $G_\sigma$ is a local group in $\calb$. By construction $G_J$ acts faithfully on $X$ and by Theorem~\ref{thm.structure}, $K_\sigma$ acts faithfully on $\calt$ which embeds into $Y$. In particular, $K_\sigma$ acts faithfully on $X$. $\blackdiamond$
We will next prove \eqref{thm.functor.nonuni}. We construct a complex of lattices as in the previous case. The proof for \eqref{thm.functor.nonuni} is now identical once we have verified that covolume condition in Theorem~\ref{thm.col}\eqref{thm.col.nonuniform}. Let $c$ denote the covolume of an $(H\times T)$-lattice $\Gamma$ with associated graph of lattices $(B,\calb)$, this is given by the formula $c=\sum_{\sigma\in VA}\mu(\Gamma_\sigma)<\infty$. Now, every vertex of the complex $Z=F(B)$ has local group isomorphic to a finite extension of some $\Gamma_\sigma$. In particular we may bound $\sum_{\sigma\in Z}\mu(\Gamma_\sigma)$ by $\ell\times c$ where $\ell$ is the number of vertices in the finite Coxeter nerve of $X$. $\blackdiamond$
The proof of \eqref{thm.functor.proj} follows from the proof of \eqref{thm.functor.gtoc}. $\blackdiamond$
The proof of \eqref{thm.functor.props} follows from either applying Theorem~\ref{thm.CMirrCrit} to \eqref{thm.functor.proj} (algebraically irreducible) or the fact $\Gamma\rightarrowtail F\Gamma$ and the properties of residual finiteness and virtual torsion-freeness are subgroup closed. $\blackdiamond$
\end{proof}
\subsection{Examples and applications}
In this section we will detail some sample examples and applications of the functor theorem.
We can obtain a number of examples by applying Thomas' functor to any irreducible $(\Isom(\EE^n)\times T)$-lattice. This will give a non-biautomatic group acting properly discontinuously cocompactly on $\EE^n\times X$ where $X$ is a sufficiently symmetric right-angled building. More precisely, we have the following corollary:
\begin{corollary}[General version of Corollary~\ref{corx.thomas.notbiaut}]\label{cor.thomas.notbiaut}
Let $Y$ be a right-angled building of type $(W,I)$ and parameters $\{q_i\}$ and let $A=\Aut(Y)$. For each $i_1,i_2\in I$ such that $m_{i_1,i_2}=\infty$ let $\calt$ be the $(q_{i_1},q_{i_2})$-biregular tree and let $T=\Aut(\calt)$. Suppose \ref{Thomas.1} holds and if $q_{i_1}=q_{i_2}$ then \ref{Thomas.2} holds with $g$ an extension of $h$ and let $F:\calg(\calt)\to\calc(Y)$ be Thomas' functor. Let $\Gamma$ be a uniform $(\Isom(\EE^n)\times T)$-lattice and suppose $\pi_{\OO(n)}(\Gamma)$ is infinite. Then, $F\Gamma$ is a uniform $(\Isom(\EE^n)\times A)$-lattice which is not virtually biautomatic nor residually finite. In particular, if $Y$ is irreducible, then the direct product of a uniform $A$-lattice with $\ZZ^2$ is not quasi-isometrically rigid.
\end{corollary}
\begin{proof}
By Theorem~\ref{thm.functor.new} $F\Gamma$ is a uniform $(\Isom(\EE^n)\times A)$-lattice with a non-discrete projection to $\OO(n)$. That $F\Gamma$ is not virtually biautomatic then follows from Theorem~\ref{thm.notbiaut}. The failure of quasi-isometric rigidity follows from the fact that the direct product of a uniform $A$ lattice with $\ZZ^2$ is reducible, whereas, the weakly irreducible lattice is algebraically irreducible by Theorem~\ref{thm.CMirrCrit} and so does not virtually split as a direct product of two infinite groups. In particular, the groups cannot by virtually isomorphic.
\end{proof}
\begin{example} \label{ex.LMX.pres}
Let $\Gamma=\LM(A)$ where $A$ is the matrix corresponding to the Pythagorean triple $(3,4,5)$ . Recall the group acts on $\EE^2\times\calt_{10}$. Let $X$ be the right angled building whose Coxeter nerve is the regular pentagon and whose parameters are given by $q_1=q_2=10$, $q_3=q_4=k$, and $q_5=\ell$. Let $A$ be the automorphism group of $X$ and consider $F\Gamma$ the image of $\Gamma$ under Thomas' functor $F$ as in Figure~\ref{fig.pentagon}. By Theorem~\ref{thm.functor.new}, the group $F\Gamma$ is a non-residually finite $(\Isom(\EE^n)\times A)$-lattice with non-discrete projections to both factors and is irreducible as an abstract group. Moreover, by the previous corollary, $F\Gamma$ is not virtually biautomatic.
We will now construct a presentation for $\Lambda_{k,\ell}:=F\Gamma$. The group has generators $a,b,x_3,x_4,x_5,t$ and relations
\[x_3^{k}=x_4^k=x_5^\ell=1,\ [a,b],\ [a,x_3],\ [a,x_4],\ [a,x_5],\ [b,x_3],\ [b,x_4],\ [b,x_5],\ [x_3,x_4], \]
\[ta^2b^{-1}t^{-1}=a^2b,\ tab^2t^{-1}=a^{-1}b^2,\ tx_3t^{-1}=x_4, [t,x_5]. \]
\end{example}
\begin{prop}
The group $\Lambda_{2,2}$ in Example~\ref{ex.LMX.pres} is virtually torsionfree. This is witnessed by the index $16$ subgroup \[\Delta:=\langle a,\ b,\ x_3tx_4t^{-1},\ x_3x_4t^{-2},\ (x_5x_3)^2,\ (x_5x_4)^2,\ t^{-1}x_3x_4t^{-1},\ (tx_5x_4t^{-1})^2\rangle.\]
\end{prop}
\begin{proof}
The quotient $\Lambda_{2,2}/\Delta$ is isomorphic to $D_4\times \ZZ_2$ which has order $16$. By construction every torsion element of $\Lambda_{2,2}$ is conjugate to some power of $x_3$, $x_4$, $x_5$ or $x_3x_4$. Indeed, every torsion element is contained in a vertex or edge stabiliser of the action on the pentagonal building and acts trivially on $\EE^2$. Each of these elements is mapped to a non-trivial element of $D_4\times\ZZ_2$. In particular, the kernel $\Delta$ is torsion-free.
\end{proof}
\begin{example}
Let $n\geq 2$ and let $\Gamma_n$ be the irreducible lattice constructed in Example~\ref{ex.LM.odddim.lat} acting on $\EE^n\times\calt_{10n}$. Let $X$ be a right angled building satisfying \ref{Thomas.1} and \ref{Thomas.2} with automorphism group $A$ and parameters $\{q_i \}$ all equal to $10n$. Applying Thomas' functor and Theorem~\ref{thm.functor.new} to $\Gamma_n$ we obtain a non-residually finite $(\Isom(\EE^n)\times A)$-lattice with non-discrete projections to both factors. Moreover by Corollary~\ref{cor.thomas.notbiaut}, $\Gamma_n$ is not virtually biautomatic.
\end{example}
We will now show the existence of non-residually finite lattices in arbitrary products of sufficiently symmetric isometric and non-isometric right-angled buildings. We note that Bourdon's ``hyperbolization of Euclidean buildings" \cite[Section~1.5.2]{Bourdon2000} can be used to construct weakly irreducible uniform lattices in products of hyperbolic buildings. We will provide a number of examples to show that the groups we construct here are distinct.
\begin{corollary}\label{thm.treelatstobuildings}
Let $\Gamma$ be a weakly irreducible lattice in product of trees $\calt_1\times\dots\times \calt_n$ such that $\calt_k$ is $(t_{k_1},t_{k_2})$-biregular. Let $X_1\times\dots\times X_n$ be a product of irreducible right angled buildings satisfying \ref{Thomas.1} and \ref{Thomas.2}. Suppose $X_k$ is of type $(W_k,I_k)$, has parameters $\{t_{k_1},t_{k_2},q_{k_3},\dots,q_{k_{n_k}}\}$ where $m_{k_{i_1},k_{i_2}}=\infty$ and $A_k=\Aut(X_k)$. Then, the lattice $\Lambda=F^n\Gamma$ obtained by applying Thomas' functor $n$ times (once for each tree $\calt_k$ corresponding to the building $X_k$) is a lattice in $A_1\times\dots\times A_n$, is weakly and algebraically irreducible, and is non-residually finite.
\end{corollary}
\begin{proof}
Let $T_k=\Aut(\calt_k)$. The result follows from applying Theorem~\ref{thm.functor.new} $n$ times as follows. Consider $\Gamma$ as a graph of $(T_2\times\dots\times T_n)$-lattices and apply $F$ to obtain a $(A_1\times T_2\times\dots\times T_n)$-lattice with the desired properties (non-residual finiteness follows from the fact that the projection to $T_2\times\dots\times T_n$ has a non-trivial kernel). Now, consider $F\Gamma$ as a graph of $(A_1\times T_3\times\dots T_n)$-lattices and proceed by induction on the index $k$.
\end{proof}
\begin{example} In \cite{BurgerMozes2000lat,BurgerMozes1997} Burger and Mozes construct for each pair of sufficiently large even integers $(m,n)$ a finitely presented simple group as a uniform lattices in a product of trees $\calt_m\times\calt_n$ (for more examples see \cite{Rattaggi2007a,Rattaggi2007b,Radu2020}). Applying Theorem~\ref{thm.treelatstobuildings}, we obtain uniform non-residually finite algebraically and weakly irreducible lattices acting on a product of buildings $X_1\times X_2$ each satisfying \ref{Thomas.1} and \ref{Thomas.2} with $X_1$ having some parameters equal to $m$ and $X_2$ having some parameters equal to $n$.
\end{example}
\begin{example}
In \cite{Hughes2021b} the author constructed a non-virtually torsion-free irreducible uniform lattice in a product of two locally finite trees. Applying Theorem~\ref{thm.treelatstobuildings}, we obtain uniform non-residually finite non-virtually torsion-free algebraically and weakly irreducible lattices acting on a product of buildings.
\end{example} | 143,617 |
What do drain flies look like?
1/5 to 1/6 inch long, drain flies have tan-colored bodies with black and white wings. Both their bodies and wings covered with hair. The wings appear larger than their bodies and are blamed for their weak flying ability. Their antennae have 13 segments with the last being a bulb-like shape.
Life cycle of drain flies of drain flies
-.
Are drain flies dangerous?
The drain fly is not known to spread any disease at this time and they do not bite humans. They may carry bacteria on their bodies and could contaminate food in the process.
Where do drain flies nest?
Drain flies can be found in sink drains, especially in the spring.
Helpful hints for drain flies
To prevent a drain fly infestation in your home or business, our Arizona pest control company recommends:
- Eliminating stagnant water especially that which is accumulating near or inside the home or structure
- Adhering to very thorough sanitation practices
- Using a pest control company on a regular basis
- Removing the breeding site
An interesting fact about drain flies
They are possibly the most common insects living in sewage plants.
><< | 385,614 |
Alessandra Thomas
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This is a quick walk-through of how to get started with the Google APIs for Go.
The first thing to understand is that the Google API libraries are auto-generated for each language, including Go, so they may not feel like 100% natural for any language. The Go versions are pretty natural, but please forgive any small non-idiomatic things. (Suggestions welcome, though!)
Pick an API and a version of that API to install. You can find the complete list by looking at the directories here.
For example, let's install the urlshortener's version 1 API:
$ go get -u google.golang.org/api/urlshortener/v1
Now it's ready for use in your code.
Once you've installed a library, you import it like this:
package main import ( "context" "golang.org/x/oauth2" "golang.org/x/oauth2/google" "google.golang.org/api/urlshortener/v1" )
The package name, if you don't override it on your import line, is the name of the API without the version number. In the case above, just
urlshortener.
Each API has a
New function taking an
*http.Client and returning an API-specific
*Service.
You create the service like:
svc, err := urlshortener.New(httpClient)
The HTTP client you pass in to the service must be one that automatically adds Google-supported Authorization information to the requests.
There are several ways to do authentication. They will all involve the package golang.org/x/oauth2 in some way.
For 3-legged OAuth (your application redirecting a user through a website to get a token giving your application access to that user's resources), you will need to create an oauth2.Config,
var config = &oauth2.Config{ ClientID: "", // from<your-project-id>/apiui/credential ClientSecret: "", // from<your-project-id>/apiui/credential Endpoint: google.Endpoint, Scopes: []string{urlshortener.UrlshortenerScope}, }
... and then use the AuthCodeURL, Exchange, and Client methods on it. For an example, see:
For the redirect URL, see
To use a Google service account, or the GCE metadata service, see the golang.org/x/oauth2/google package. In particular, see google.DefaultClient.
Some APIs require passing API keys from your application. To do this, you can use transport.APIKey:
ctx := context.WithValue(context.Background(), oauth2.HTTPClient, &http.Client{ Transport: &transport.APIKey{Key: developerKey}, }) oauthConfig := &oauth2.Config{ .... } var token *oauth2.Token = .... // via cache, or oauthConfig.Exchange httpClient := oauthConfig.Client(ctx, token) svc, err := urlshortener.New(httpClient) ...
Each service contains zero or more methods and zero or more sub-services. The sub-services related to a specific type of “Resource”.
Those sub-services then contain their own methods.
For instance, the urlshortener API has just the “Url” sub-service:
url, err := svc.Url.Get(shortURL).Do() if err != nil { ... } fmt.Printf("The URL %s goes to %s\n", shortURL, url.LongUrl)
For a more complete example, see urlshortener.go in the examples directory. (the examples use some functions in
main.go in the same directory)
Most errors returned by the
Do methods of these clients will be of type
googleapi.Error. Use a type assertion to obtain the HTTP status code and other properties of the error:
url, err := svc.Url.Get(shortURL).Do() if err != nil { if e, ok := err.(*googleapi.Error); ok && e.Code == http.StatusNotFound { ... } } | 416,811 |
\begin{document}
\renewcommand{\refname}{References}
\thispagestyle{empty}
\title{A solvable group isospectral to $\S_4(3)$}
\author{{Andrei V. Zavarnitsine}}
\address{Andrei V. Zavarnitsine
\newline\hphantom{iii} Group Theory Lab.
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} Acad. Koptyug avenue, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia
\newline\hphantom{iii} \sc{and}
\newline\hphantom{iii} Mechanics and Mathematics Dept.
\newline\hphantom{iii} Novosibirsk State University,
\newline\hphantom{iii} Pirogova St., 2,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}
\email{[email protected]}
\thanks{\rm Supported by RFBR, grants 06-01-39001 and 08-01-00322;
by the Council of the President (project NSh-344.2008.1); by the Russian Science Support Foundation (grant of
the year 2008); by SB RAS, Integration Project 2006.1.2; by ADTP ``Development of the Scientific
Potential of Higher School'' of the Russian Federal Agency for Education (Grant 2.1.1.419). }
\maketitle
{\small
\begin{quote}
\noindent{\sc Abstract. }
We construct a solvable group $G$ of order 5\,648\,590\,729\,620 such that the set of element orders of $G$ coincides
with that of the simple group $\S_4(3)$. This completes the determination of finite simple groups isospectral
to solvable groups.
\end{quote}
}
\section{Introduction}
The set of elements orders of a finite group $G$ is denoted by $\om(G)$ and called the {\em spectrum} of $G$.
The spectrum $\om(G)$ contains all divisors of each of its elements, hence is uniquely determined by the set
$\mu(G)$ of its maximal with respect to divisibility elements. Finite groups $G$ and $H$ are {\em isospectral}
if $\om(G)=\om(H)$.
It was shown in \cite{lm} that if a nonabelian simple group $G$ is isospectral to a solvable group then $G\cong
\L_3(3),\U_3(3),\S_4(3),\A_{10}$. The existence of solvable groups isospectral to $\L_3(3)$ and
$\U_3(3)$ was proved in \cite{m02,al}. In \cite{st}, it was shown that there exists no solvable group
isospectral to $\A_{10}$. Thus, the problem remained open only for $\S_4(3)$.
The main result of this paper is as follows:
\begin{thm}\label{main} There exists a solvable group isospectral to the simple group $\S_4(3)$.
\end{thm}
The solvable group from Theorem \ref{main} is constructed as a $17\!\times\! 17$-matrix group over the field
$\FF_3$ of three elements. This group has order
$$5\,648\,590\,729\,620 = 2^2\cdot3^{24}\cdot 5$$
and is believed to be the smallest
solvable group in question. Although the proofs given in this paper do not depend on computer calculations, the
first evidence for the existence of this group was obtained with the aid of the computer algebra system GAP
\cite{gap}.
As a consequence of Theorem \ref{main}, we have
\begin{cor}\label{cm} A finite simple nonabelian group $G$ is isospectral to a solvable group if and only if
$G\cong \L_3(3),\U_3(3),\S_4(3)$.
\end{cor}
\section{Preliminaries}
From \cite{atl}, it follows that
\be
\mu(\S_4(3))=\{5,9,12\}.
\ee
We will require the following auxiliary result:
\begin{lem}\label{x3}
Let
$$
X=
\left(
\begin{array}{ccccc}
1 & x_1 & y_1 & z_1 & t_1 \\
. & 1 & x_2 & y_2 & z_2 \\
. & . & 1 & x_3 & y_3 \\
. & . & . & 1 & x_4 \\
. & . & . & . & 1 \\
\end{array}
\right)
$$
be an upper unitriangular $5\!\times\!5$-matrix over a (not necessarily commutative) ring
of characteristic $3$, where the dots stand for zeros. Then $X^9=1$ and $|X|<9$ if and only if
\be\label{9e}
\begin{array}{lr}
x_1x_2x_3=0,&x_2x_3x_4=0,\\[3pt]
\multicolumn{2}{c}{x_1x_2y_3+x_1y_2x_4+y_1x_3x_4=0.}
\end{array}
\ee
\end{lem}
\begin{proof} A direct calculation shows that
$$
X^3=
\left(
\begin{array}{ccccc}
1 & . & . & z_1' & t_1' \\
. & 1 & . & . & z_2' \\
. & . & 1 & . & . \\
. & . & . & 1 & . \\
. & . & . & . & 1 \\
\end{array}
\right),
$$
where
$$
\begin{array}{lr}
z_1'=x_1x_2x_3,& z_2'=x_2x_3x_4,\\[3pt]
\multicolumn{2}{c}{t_1'=x_1x_2y_3+x_1y_2x_4+y_1x_3x_4.}
\end{array}
$$
The claim follows.
\end{proof}
Let $G$ be a group, $K$ a field, and let $U_1,U_2,U_3$ be right $KG$-modules. Recall that a $K$-bilinear
map $\vf:U_1\times U_2\to U_3$ is called {\em balanced \/} if
$$\vf(u_1g,u_2g)=\vf(u_1,u_2)g$$
for all $u_1\in U_1$, $u_2\in U_2$, $g\in G$. It is a well-known universal property of the tensor
product of $KG$-modules that every balanced map factors through '$\otimes$', i.\,e. there exists a unique module
homomorphism $\tilde\vf: U_1\otimes U_2\to U_3$ such that the following diagram commutes
\be\label{diag}
\xymatrix{
&U_1\otimes U_2\ar@{.>}[dr]^{ \tilde{\vf} } & \\
U_1\times U_2\ar[rr]^{\vf}\ar[ur]^{\otimes}& & U_3 }
\ee
Recall also that the exterior square $\wedge^2 U$ of a $KG$-module $U$ is the quotient of $U\otimes U$ by the
submodule generated by all elements of the form $u\otimes u$, $u\in U$.
Henceforth, all the introduced notation will be fixed throughout.
We define the following $4\!\times\!4$-matrices over $\FF_3$:
\be \label{rep}
a=\left(
\begin{array}{rrrr}
.&1&.&. \\
.&.&1&. \\
.&.&.&1 \\
-1&-1&-1&-1 \\
\end{array}
\right),
\qquad
b=\left(
\begin{array}{rrrr}
1& .& .& . \\
.& .& 1& . \\
-1&-1&-1&-1 \\
.& 1& .& . \\
\end{array}
\right),
\ee
Observe that
\be\label{az}
1+a+a^2+a^3+a^4=0.
\ee
Let $F=\la a, b \ra$. Then $F$ is a Frobenius group of shape $5\!:\!4$ which is isomorphic to the abstract
group
\be
\la a,b\mid a^5=b^4=1, a^b=a^2 \ra,
\ee
and (\ref{rep}) is the matrix representation of
$F$ that corresponds to the simple right $\FF_3F$-module
\be\label{v}
V=\la v\mid vb=v,\ v(1+a+a^2+a^3+a^4)=0\ra
\ee
written in the basis $(v,va,va^2,va^3)$. Observe that $\mu(V\sd F)=\{5,12\}$. In particular,
$a$ acts on $V$ fixed-point freely. Moreover,
\be\label{cb2}
C_V(b^2)=\la v, va^2+va^3\ra.
\ee
The following identities hold in $F$:
\be\label{rels}
\begin{array}{lll}
b^a=ba^4=a^2b, &(b^2)^a=b^2a^2=a^3b^2, & (b^3)^a=b^3a^3=ab^3, \\[3pt]
b^{a^2}=ba^3=a^4b, & (b^2)^{a^2}=b^2a^4=ab^2, & (b^3)^{a^2}=b^3a=a^2b^3, \\[3pt]
b^{a^3}=ba^2=ab, & (b^2)^{a^3}=b^2a=a^4b^2, & (b^3)^{a^3}=b^3a^4=a^3b^3, \\[3pt]
b^{a^4}=ba=a^3b, & (b^2)^{a^4}=b^2a^3=a^2b^2, & (b^3)^{a^4}=b^3a^2=a^4b^3.
\end{array}
\ee
Denote by $M$ the right $\FF_3F$-module $\M_4(\FF_3)$ on which the elements of $F$ act by conjugation, i. e.
$m\circ g=g^{-1}mg$ for $m\in M$, $g\in F$.
\begin{lem}\label{mac} Let $m\in M$. The map $v\mapsto m$ can be extended to an $\FF_3F$-module
homomorphism $V\to M$ if $m\in \la b,b^2,b^3\ra_{\FF_3^{\vphantom{A^A}}}$.
\end{lem}
\begin{proof} Since $V$ is a cyclic module, in view of (\ref{v}), it is
sufficient to show that
$$
m\circ b=m, \qquad m\circ (1+a+a^2+a^3+a^4)=0.
$$
The former holds, because $m$ is a linear combination of powers of $b$, hence commutes with $b$. The latter
need only be checked for $m$ equal to one of $b,b^2,b^3$ due to linearity. By (\ref{rels}), for $m=b$ we have
$$
m\circ (1+a+a^2+a^3+a^4)=b+b^a+b^{a^2}+b^{a^3}+b^{a^4}=b(1+a+a^2+a^3+a^4)=0
$$
due to (\ref{az}). Similarly, the claim follows for $m$ equal to $b^2$ or $b^3$.
\end{proof}
It is not difficult to show that the converse of Lemma \ref{mac} holds as well. However, we will not need this
fact.
\begin{lem}\label{wedge} There is an isomorphism of $\FF_3F$-modules
$$
\wedge^2 V \cong V \oplus U
$$
where $U$ is a $2$-dimensional submodule generated by $v\wedge va+v\wedge va^3+va^2\wedge va^3$.
\end{lem}
\begin{proof}
We will need the following fragment of the ordinary character table of $F$:
$$
\begin{array}{c|crrrr}
& 1a & 4a & 4b & 2a & 5a \\
\hline
\x_1^{\vphantom{A^A}} & 1 & i &-i &-1 & 1 \\
\x_2 & 1 &-i & i &-1 & 1\\
\x_3 & 4 & . & . & . & -1
\end{array}
$$
The module $V$ corresponds to a representation of $F$ whose character is $\x_3$. A direct calculation shows
that $\wedge^2\x_3=\x_3+\x_2+\x_1$. Hence, we need only find a $2$-dimensional submodule of $\wedge^2 V$.
Denote
\begin{align*}
&u_1=v\wedge va+v\wedge va^3+va^2\wedge va^3, \\
&u_2=v\wedge va^2 +va\wedge va^2 +va \wedge va^3.
\end{align*}
Then we have
$$
u_1a=u_1,\quad u_1b=u_2,\quad u_2a=u_2,\quad u_2b=-u_1.
$$
Therefore, the $2$-dimensional $\FF_3$-subspace $\la u_1,u_2 \ra$ is a submodule which is generated by
each of the two vectors $u_1$ and $u_2$.
\end{proof}
\section{The group}
We introduce four $17\!\times\!17$-matrices written in the block form as follows:
\begin{align*} \label{gens}
&A=\diag(1,a,a,a,a),\qquad B=\diag(1,b,b,b,b), \\[5pt]
&C=\left(\begin{array}{c|cccc}
1&.&.&.&. \\
\hline
.&1^{\vphantom{A^A}}&c_1&c_3&. \\
.&.&1&.&c_4 \\
.&.&.&1&c_2 \\
.&.&.&.&1 \\
\end{array}
\right),
\quad
D=\left(
\begin{array}{c|cccc}
1 &d& .& .& . \\
\hline
. &1^{\vphantom{A^A}}& .& .& . \\
. &.& 1& .&. \\
. &.& .& 1& . \\
. &.& .& .& 1 \\
\end{array}
\right),
\end{align*}
where $d=(1,0,0,0)\in \FF_3^4$, and
\be\label{cdef}
c_1=b, \quad c_2=b^3, \quad c_3=b^2, \quad c_4=-b^2.
\ee
\begin{prop}\label{mp} The group $G=\la A,B,C,D \ra$ is a solvable group of order $2^2\cdot 3^{24}\cdot 5 $
isospectral to $\S_4(3)$.
\end{prop}
\begin{proof}
Clearly, $G\cong P\sd F$, where $P=\la C,D \ra^G$ is the largest normal $3$-subgroup of $G$ and $F$ is
the above-defined Frobenius group of shape $5\!:\!4$ which we identify with $\la A,B \ra$.
In order to determine $\om(G)$, we will study more closely the structure of $P$ and the action of $F$ on $P$.
Each element of $P$ has the form
\be\label{form}
\left(
\begin{array}{c|cccc}
1 &d_1& d_2& d_3& d_4 \\
\hline
. &1^{\vphantom{A^A}}& f_1& f_3 & h \\
. &.& 1& .& f_4 \\
. &.& .& 1& f_2 \\
. &.& .& .& 1 \\
\end{array}
\right)
\ee
for some $d_i\in \FF_3^4$, $f_i,h\in \M_4(\FF_3)$, $i=1,\ldots,4$.
First, we observe that $P$ has exponent $9$.
Indeed, the exponent is at most $9$ by Lemma \ref{x3}. \big(In order
to apply Lemma
\ref{x3}, we may naturally embed $P$ into the upper unitriangular $5\!\times\!5$-matrix group over
$\M_4(\FF_3)$.\big) Moreover, Lemma \ref{x3} implies that the element (\ref{form})
has order $9$ if and only if
\be\label{o9}
d_1(f_1f_4+f_3f_2)\ne 0.
\ee
Thus, there is an element of order $9$ of the form $CD_1$, where $D_1\in
\la D\ra^F$. Indeed, such an element has order $9$ if and only if $0\ne
d_1(c_1c_4+c_3c_2)=d_1(b-b^3)$, where $d_1\in
\FF_3^4$ is the block of $D_1$ in the position $(1,2)$. Since $\la D\ra^F$ is isomorphic to $V=\FF_3^4$ as an
$\FF_3F$-module, $d_1$ can be chosen arbitrarily, and we choose it so that it is not annihilated by the nonzero
matrix $b-b^3$.
We now prove that $P$ has an $F$-invariant normal series with factors isomorphic to $V$ as $\FF_3F$-modules.
This will imply that $G$ has no elements of order $15$.
It can be seen from (\ref{form}) that
$P=\la D \ra^G\sd\la C \ra^F$, where
the group $\la D \ra^G$ consists of the matrices (\ref{form}) with $f_i,h=0$
and is isomorphic to $V^{\oplus 4}$ as an $\FF_3F$-module; whereas the group
$\la C \ra^F$ consists of the matrices (\ref{form}) with $d_i=0$.
Observe that the components $f_i$ and $h$ in
(\ref{form}) cannot be chosen arbitrarily. If we identify an element of $\la C \ra^F$ with the tuple
$(f_1,f_2,f_3,f_4,h)$, we have
\be\label{multc}
\begin{array}{r@{}l}
(f_1,f_2,f_3,f_4,h)\cdot(f_1',f_2',f_3',f_4',h')=&\\[3pt]
(f_1+f_1',\ f_2+f_2',\ f_3+f_3',\ f_4+f_4',\ &h+h'+f_1f_4'+f_2f_3').
\end{array}
\ee
By Lemma \ref{mac} the maps $\a_i:v\mapsto c_i$, $i=1,\ldots,4$ can be extended to $\FF_3F$-module
homomorphisms $V\to M$.
Let $T$ be subgroup of the group $\{(u,m)\mid u\in V,m\in M\}$, with the multiplication
$$
(u,m)\cdot (u',m')=(u+u',m+m'+u\a_1\cdot u'\a_4+u\a_3\cdot u'\a_2)
$$
and componentwise action of $F$, generated by the element $(v,0)$ as an $F$-group;
i.\,e. $T=\big\la(v,0)\big\ra^F$. Then there is an $F$-group isomorphism $\a: T\to \la C \ra^F$
(meaning that $\a$ commutes with the action of $F$)
defined by the map
\be\label{isom}
\a:(v,0)\mapsto (v\a_1,v\a_2,v\a_3,v\a_4,0)=C.
\ee
Observe that $\big[(u,m),(u',m')\big]=(0,\vf(u,u'))$, where
$$
\vf(u,u')=u\a_1\cdot u'\a_4 - u'\a_1\cdot u \a_4 +u\a_3\cdot u'\a_2- u'\a_3\cdot u\a_2
$$
defines a map $\vf: V\times V\to M$ whose image generates a nonzero $\FF_3F$-module $W$ isomorphic to the derived
subgroup of $\la C\ra^F$. Clearly, $\vf$ is a balanced map of $\FF_3F$-modules, hence $W$ is a homomorphic image
of $V\otimes V$ under $\tilde\vf$ as in (\ref{diag}). Moreover, $\tilde\vf(u\otimes u)=\vf(u,u)=0$ for all $u\in V$,
thus $W$ is a homomorphic image of $\wedge^2 V$.
Using (\ref{rels}), we also have
\begin{align*}
\tilde\vf(v\wedge va\,+&\,v\wedge va^3+va^2\wedge va^3)=
\vf(v,va)+\vf(v,va^3)+\vf(va^2,va^3)=\\
&\ \ \ \, b\cdot(-b^2)^a-b^a\cdot(-b^2)+b^2\cdot(b^3)^a-(b^2)^a\cdot b^3\\
&+b\cdot(-b^2)^{a^3}-b^{a^3}\cdot(-b^2)+b^2\cdot(b^3)^{a^3}-(b^2)^{a^3}\cdot b^3\\
&+b^{a^2}\cdot(-b^2)^{a^3}-b^{a^3}\cdot(-b^2)^{a^2}+(b^2)^{a^2}\cdot(b^3)^{a^3}-(b^2)^{a^3}\cdot (b^3)^{a^2}=\\
&\hspace{100pt}-b^3a^2+b^3a+ba^3-ba\\
&\hspace{100pt}-b^3a+b^3a^3+ba^4-ba^3\\
&\hspace{100pt}-b^3a^3+b^3a^2+ba-ba^4=0.
\end{align*}
Consequently, $\tilde\vf(U)=0$, where $U$ is the $2$-dimensional submodule of $\wedge^2V$ from Lemma \ref{wedge},
and we thus have $W\cong V$. This proves in particular that $\la C \ra^F$ has nilpotency
class $2$ and order $3^8$. Therefore, $P$ has order $3^{16}\cdot 3^8$ and $A$ acts
fixed-point freely on $P$.
It remains to prove that $G$ has no elements of order $18$. By the Schur--Zassenhaus theorem,
all involutions of $G$ of are conjugate to $B^2$. We will show that $C_P(B^2)$ contains no elements of order $9$.
Every element of $C_P(B^2)$ has the form (\ref{form}) with $d_i\in C_V(b^2)$ and $f_i,h\in C_M(b^2)$.
Moreover, due to the $F$-group isomorphism (\ref{isom}) the components $f_i$ of such an element
must be of the form $u\a_i$ for some $u\in C_V(b^2)$. Thus, in view of the condition (\ref{o9}),
we need only show that
\be\label{lk}
w\psi(u)=0
\ee
for all $w,u\in C_V(b^2)$, where $\psi: C_V(b^2)\to M$ is given by
$$
\psi(u)=u\a_1\cdot u\a_4+u\a_3\cdot u\a_2.
$$
Let $N$ be the $\FF_3\la b\ra$-module generated by $\Im\psi$. Observe that $C_V(b^2)$ is $\la b\ra$-invariant.
Hence, the condition (\ref{lk}) has to be checked only when $\psi(u)$ is a generator of $N$.
Using (\ref{cb2}) it can be shown that $N$ is generated as a module by $\psi(v)$ and $\psi(va^2+va^3)$.
\big(Indeed, the map $\psi$ is quadratic in $u$, hence $N$ is a homomorphic image of the symmetric square
of $C_V(b^2)$ which can be generated by two elements.\big) By (\ref{rels}) and (\ref{az}), we have
$$
\psi(v)=b\cdot(-b^2)+b^2\cdot b^3=(1-b^2)b,
$$
and
\begin{align*}
\psi(va^2&+va^3)=\\
&(b^{a^2}+b^{a^3})((-b^2)^{a^2}+(-b^2)^{a^3})+((b^2)^{a^2}+(b^2)^{a^3})((b^3)^{a^2}+(b^3)^{a^3})=\\
-&(ba^3+ba^2)(b^2a^4+b^2a)+(b^2a^4+b^2a)(b^3a+b^3a^4)=\\
-&b^3(a+a^2+a^3+a^4)+b(a+a^2+a^3+a^4)=-(1-b^2)b.
\end{align*}
Thus, in both cases, $\psi(u)$ is divisible from the left by $1-b^2$. However, if $w\in C_V(b^2)$,
we have $w(1-b^2)=0$; therefore, (\ref{lk}) holds as is required.
\end{proof}
Theorem \ref{main} is now a consequence of Proposition \ref{mp}.
{\em Acknowledgement.} The author is thankful to Prof. V. D. Mazurov for discussing the
problem and for the remarks about this paper. | 106,679 |
Cumberland Gap Convention Center
601 Colwyn Avenue
Cumberland Gap, TN
Arts in the Gap, in collaboration with the Middlesborough Little Theatre, will reprise the Appalachian Youth Performing Arts Camp July 16-21, 2018. This one-week day camp, which includes workshops in acting, singing, dance, stage makeup, costuming, and stage choreography, will culminate in a performance of the work on Saturday, July 21st, at 2:30 p.m. in the Cumberland Gap Convention Center. (Participants must attend the full week of camp in order to perform.)
Thirty slots are available for children rising to grades 3-8.
The cost is $50 per child, including lunch, and some scholarships are available.
For information about the camp or about scholarships, contact Joe Gill @ [email protected].
Director/instructor: Mississippi native Amy Oden Simpson serves as director of the Middlesborough Little Theatre and Professor of English and Theatre Arts at Southeast KY Community & Technical College, where she has taught since 1997. In 2013, she received her MFA in writing with a playwriting emphasis from Spalding University in Louisville. In 2014, Simpson produced her adaptation of Candida Sullivan’s picture book Zippy’s Club, which was subsequently published by Shade Tree Press. She has also written and staged four full-length local history plays, most recently Cakewalk in the Sky: The Story of Ragtime Founder Ben Harney (2015). Simpson was nominated for the UK Library’s 2012 Medallion Award for Intellectual Achievement, and she was nominated by Spalding faculty for the 2014 Kennedy Center Playwrights’ Workshop. Amy Simpson lives in Middlesboro with her husband Bill and two children Corley and Sidney.
co-instructor: Born in New York and moved to Los Angeles at age 19, Joe Gill has worked in theatre all over the country and is now planting roots in Tennessee. Joe is the new Director of the Arts in the Gap as well as LMU’s professor of Theatre. studied Viewpoints with Mary Overlie, Theatre of the Oppressed with Augosto Boal and. | 179,805 |
TITLE: A geometric property of singular matrices
QUESTION [1 upvotes]: Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let $M=Det^{-1}\{0\}-S$ be the codimension one submanifold of $M_{n}(\mathbb{R})$ which has a natural Riemannian metric induced by the standard metric of $M_{n}(\mathbb{R})\simeq\mathbb{R}^{n^{2}}.$
What is a linear algebraic and matrix meaning for a matrix $A\in M$ with the following property:
"The sectional curvature of $M$ at $A$ is independent of choosing a $2$-plane tangent to $M$ at $A$"
What is a precise example of this situation, for $n=2$?
REPLY [7 votes]: Your first question has an easy answer. The differential of $\det$ is
$$\sum_{i,j}\hat a_{ij}{\rm d}a_{ij},$$
where $\hat A$ is the cofactor matrix. Thus a singular point is such that $\hat A=0_n$, in other words, it has rank $\le n-2$.
Actually, ${\bf M}_n({\mathbb R})$ can be stratified by the sets $R_0,\ldots,R_n$ of matrices of rank $k=0,\ldots n$ respectively. Each $R_k$ is a submanifold of dimension $k(2n-k)$. $R_k$ is homogeneous, in the sense that ${\bf GL}_n({\mathbb R})\times{\bf GL}_n({\mathbb R})$ operates transitively on it by $(P,Q)\cdot A=PAQ^{-1}$. The relative boundary of $R_k$ is $R_0\cup\cdots\cup R_{k-1}$. In particular, there is no removable singularity. | 165,972 |
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Spurred by real-world examples of ICU family meetings where physicians and nurses failed to work in an interdisciplinary format to clearly and sensitively convey crucial information to the families of critically ill patients, this project aims to simulate high-stakes ICU family meetings where nursing students role play as ICU nurses, medical students role play as ICU physicians, chaplaincy residents role play as hospital chaplains, and combinations of students role play as family members. Three scenarios are currently utilized:
A patient with metastatic pancreatic cancer, complicated by a bowel perforation, requiring ICU care whose family must be told that, due to his current severe illness, he no longer can participate in a stage 1 drug trial he was to start
A patient with long-standing metastatic colon cancer with declining quality of life due to pain and foul-smelling fistulas, who suffers a bowel perforation complicated by multi-system organ failure and whose family must decide whether he would want to continue on life support\
A patient with severe H1N1 influenza, multi-system organ failure, and severe ventilator dependence, with no option of lung transplant, whose family must discuss potential withdrawal of life-support
The primary goal of the simulations is to teach how, in these nuanced and difficult conversations, nursing, physician, and chaplaincy colleagues can and must work together to ensure clear, sensitive, and honest communication with families. These ICU situations are particularly challenging as the patient is too ill to participate in the meeting and consequently, discussions must be convened with and decisions made by family members; this adds a layer of complexity and causes us to direct these scenarios at advance level, final year medical and nursing students.
Secondary goals are for students to learn and practice: effective and yet sensitive and empathic communication methods to convey bad news and emotionally laden information; conflict resolution techniques; family meeting-related practical skills (such as introducing meeting participants, active listening, and setting meeting goals); and techniques useful in sensitively dealing with distraught and/or dysfunctional patient families and family members.
Simulations at Hopkins Nursing | 191,324 |
Homemade Spaghetti Sauce
By
nellie gragg
1
@appleseed
this recipe is over 40yrs old and we love it i also like it cause it can be done in one pot
Ingredients
- 8 slice
- bacon chopped
- 1 lb
- hambuger
- 8 oz
- mushrooms optional
- 1/4 c
- onion chopped
- 2 clove
- of garlic copped
- 1/2 c
- celery
- 1 small
- bay leave
- 1 tsp
- salt
- 1/2 tsp
- pepper
- 1/2 tsp
- paprika
- 1 large
- can of tomato paste
- 1 large
- 48 oz of tomatoes diced
- 1 tsp
- red cayeane pepper
Step-By-Step
1In large pot cook chopped bacon till cooked but not crisp take out of pot with slotted spoon leaving grease,add hamburger,onion,mushroom,garlic and celery cook till hamburger is done drain,add all other ingredients and simmer for 1 1/2 hrs
2put spaghetti on about 12 mins before sauce is done
we serve this with a salad and cheesy garlic bread
we serve this with a salad and cheesy garlic bread
About this Recipe
Course/Dish: Other Sauces | 249,930 |
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Due to the presence of a high amount of fibre, people who consume sarson ka saag are less susceptible to constipation and colon cancer. This leafy green vegetable ensures adequate bowel movement in the body.
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Contents
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Fha 203K Down Payment Requirements PDF FHA 203(k) Rehabilitation Mortgage Insurance Program – Must meet standard FHA 203(b) down payment requirements. Must pay up-front and annual Mortgage insurance premiums. fha training module. an FHA loan as a 203(k). "Good Neighbor Next Door" and $100 down programs can be used with 203(k).
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\begin{document}
\title[Chaos Round--a--Face]{Many Body Quantum Chaos and Dual Unitarity Round--a--Face}
\author{Toma\v z Prosen}
\affiliation{Faculty of Mathematics and Physics, Universty of Ljubljana, Jadranska 19, Si-1000 Ljubljana, Slovenia}
\email{[email protected]}
\date{\today}
\begin{abstract}
We propose a new type of locally interacting quantum circuits which are generated by {\em unitary} interactions round--a--face (IRF). Specifically, we discuss a set (or manifold) of {\em dual-unitary} IRFs with local Hilbert space dimension $d$ (DUIRF$(d)$) which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite dimensional completely positive trace preserving unital maps, in complete analogy to recently studied circuits made of dual unitary brick gates (DUBG). In fact, we show that the simplest non-trivial (non-vanishing) local correlation functions in dual-unitary IRF circuits involve observables non-trivially supported on at least two sites.
We completely characterise the 10-dimensional manifold of DUIRF$(2)$ for qubits ($d=2$) and provide, for $d=3,4,5,6,7$, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine ${\rm dim}\,{\rm DUBG}(d)$ and show that they are of similar order though systematically larger than ${\rm dim}\,{\rm DUIRF}(d)$ for $d=2,3,4,5,6,7$. It is remarkable that both sets have rather complex topology for $d\ge 3$ in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions $d\neq d'$ residing at even/odd lattice sites.
\end{abstract}
\maketitle
\begin{quotation}
Quantum many-body dynamics of generic interacting systems is essentially intractable and is amenable only to quantum simulation.
One may wonder, whether there exist
non-integrable (generically, quantum chaotic) many-body systems with local interactions which would have exactly solvable spatio-temporal correlation functions of local observables.
These models would be understood as quantum many-body analogs of baker and cat maps, playing a similar role in classical single-particle chaos.
We outline two complementary classes of quantum dynamical systems with exactly solvable dynamical correlations exhibiting a rich ergodic hierarchy of dynamical behaviors: the dual-unitary brickwork circuits and, newly proposed dual-unitary face models (circuits with dual-unitary interactions round--a--face (DUIRF)). Remarkably, dynamical correlation functions of local observables in these families of 1+1 dimensional interacting systems are non-vanishing only along the edges of causal cones, where they are given in terms of dissipative single-particle quantum (Markov) dynamical systems. The latter in turn can be clearly classified as non-ergodic, ergodic and mixing, based on the spectrum of finite-dimensional quantum Markov matrix.
The dynamical and geometric features of such DUIRF dynamical systems are discussed in relation to previously studied dual unitary circuits. We conjecture that recent exact results on random matrix spectral statistics, entanglement dynamics and operator spreading in dual-unitary brickwork circuits can be adapted to dual-unitary IRF circuits.
\end{quotation}
\section{\label{sec:intro}Introduction}
Precise definition of quantum chaos of many-body quantum systems has been elusive for a long time, even in the simplest context of quantum spin-lattice systems with local interactions. For example,
it has been observed a while ago\cite{Mila1993,Poilblanc1993,Hsu1993} that spectral statistics of non-integrable spin-1/2 chain Hamiltonians with nearest-neighbor interactions conform to Random matrix theory\cite{Mehta} (RMT) and the match to RMT statistics on appropriate physical time/energy scales has been considered as a working definition of quantum chaos in condensed matter theory community for decades. Nevertheless, the first analytical
explanations\cite{KLP18,ChalkerPRX,ChalkerPRL,Friedman,Roy} or
proofs\cite{BKP18,BKP20} of this \emph{quantum chaos conjecture} came only very recently, and only in quite restricted contexts. On the other front, people have been trying to identify the quantum analogs of Lyapunov exponents\cite{Maldacena,Swingle} and Kolmogorov-Sinai (dynamical) entropies (characterising algorithmic complexity of dynamics)\cite{Yoshida}. Such quantum dynamical entropies\cite{Alicki}, however, cannot even discriminate between free and interacting evolutions in the thermodynamic limit, as the information (entropy) is generically propagating from an (infinite) bath of degrees of freedom to the subsystem of interest \cite{TP07}, and hence obscuring `dynamical generation' of entropy (the notion which thus cannot be precisely defined in extended systems). On the other hand, quantum Lyapunov exponents have been nevertheless defined through the out-of-time-ordered correlation functions (OTOCs), but these construction is meaningful only in the so-called ``large $N$'' theories (models) which are essentially semiclassical (with an effective Planck constant $\hbar=1/N$). Thus, the genuinely hardest, and arguably the most interesting cases are related to understanding of dynamical complexity in spin lattice models with finite local Hilbert space dimension and with local interactions. The simplest among those are the local quantum circuit models, which can be understood as a discrete-time quantum dynamical systems on a lattice, or quantum cellular automata\cite{Farrelly,QCA}.
Recently, a substantial progress has been achieved in understanding quantum chaos conjecture and dynamical complexity in (Floquet) local quantum circuit models\cite{ChalkerPRX,nahum2017quantum,nahum2018operator,khemani,vonKeyserlingk2018operator} In those models, besides spatio-temporal OTOCs the most fruitful measure of dynamical complexity has been identified as the operator-space entanglement entropy \cite{PP07}. The latter quantifies the so-called operator spreading, or growing bipartite correlations of time-dependent local operators interpreted as elements of tensor products of local Hilbert spaces. Moreover, it has been shown that explicit and exact results on RMT spectral correlations\cite{BKP18,BKP20}, dynamical correlation functions\cite{bertini2019exact,austen_prl},
quantum quenches\cite{Lorenzo}, (operator) entanglement dynamics\cite{PRX19,bertini2020operatorI,bertini2020operatorII,Reid}, information scrambling\cite{BrunoLorenzo}, and OTOCs\cite{austen_PRR}, can be obtained even for local qudit circuits (with fixed local Hilbert space dimension $d$, say $d=2$) provided the circuit, i.e. the local gates, satisfy the so-called dual-unitarity (DU) condition\cite{bertini2019exact}. It has been shown that DU circuits include integrable and (generically) non-integrable (chaotic) systems\cite{bertini2019exact}, in particular the previously studied self-dual kicked Ising model\cite{Sarang}. Studying {\em space-time duality} proved useful also to get important new insights into the behavior of non-DU circuits\cite{Gutkin_JPA,Gutkin_pert,Amos,Garratt,Ippoliti1,Ippoliti2,Grover,Abanin,Pavel}.
DU circuits are thus a representative class of exactly solvable chaotic quantum systems, very much like the baker and cat maps in classical chaos theory\cite{ott}. In analogy to structural stability of hyperbolic flows \cite{Robbin,Robinson} in classical chaos theory we conjectured (and found partial evidence of) \cite{PRX2021} perturbative stability of DU quantum dynamical systems.
In this paper we propose an extension of a class of local quantum circuits in terms of a concept of \emph{unitary} interactions round--a--face (IRF). Unitary IRF circuits can be thought of as a complementary model to brickwork quantum circuits and yet another realization of quantum cellular automata. Specifically, IRF gate is just a controlled (or kinetically constrained) local unitary gate, where the control is placed on the neigbouring two qudits and could hence capture the dynamics of (Floquet) driven Rydberg atom chains\cite{Lukin} or similar manipulated systems. As a deterministic version of unitary IRF dynamics, we should mention a rule 54 reversible cellular automaton \cite{rule54review}. While Yang-Baxter integrable IRF models (also known as RSOS models)\cite{baxter,forrester} can give rise to integrable quantum spin chain Hamiltonians\cite{Pearce}, it is not clear if unitary integrable IRF circuits can be generated beyond the singular case of classical reversible cellular automata mentioned earlier\cite{bobenko1993two,rule54review} (for which Yang-Baxter structure is not clear at the moment anyway). Another related integrable kinetically constrained continuous time (Hamiltonian) dynamics has been studied in Ref.~\cite{Lenart1,Lenart2,Vernier}.
We then extend the concept of IRF circuits to DU IRF circuits of qudits ($d=2,3\ldots$). We show that, similarly as for DU brickwork circuits, the space-time correlation functions of any local observable supported on a pair of neighbouring sites can be shown to be non-vanishing only along two light-rays, where it is evaluated in terms of a pair of completely positive, trace preserving, unital maps acting on pairs of qudits (note that for DU brickwork circuits the correspondig maps act on a single qudit). This map can in fact be interpreted as a classical Markov chain as it acts non-trivially only on a $(d-1)^2 + 1$ dimensional subspace spanned by diagonal operators with vanishing partial traces plus the identity operator.
We show how to completely characterise DUIRF circuits of qubits, $d=2$, and explicitly parametrize the corresponding 10-dimensional manifold ${\rm DUIRF}(2)$. We also empirically estimate dimensions of ${\rm DUIRF}(d)$ and of related DU brick gates ${\rm DUBG}(d,d')$ (where dimensions of local Hilbert spaces on even and odd checkerboard sublattices of the brickwork, $d$ and $d'$ respectively, can be different), for $d (d') = 3,4,5,6,7$.
It is remarkable that both sets ${\rm DUIRF}(d)$, ${\rm DUBG}(d,d')$ have non-uniform dimensions, i.e. the dimensions of tangent space at different generic (random) elements of the set are different for $d,d'\ge 3$.
Nevertheless, we find consistently that
${\rm dim}\,{\rm DUIRF}(d) > {\rm dim}\,{\rm DUBG}(d,d)$, locally everywhere, i.e. for all elements of the sets. We sketch as well some other interesting problems that one could approach using DUIRF circuits, most specifically the problem of spectral statistics and the idea of the proof of RMT spectral form factor for DU IRF circuits.
\section{Unitary IRF circuits}
\begin{figure}
\includegraphics[scale=0.45]{Fig1a}
\hspace{1mm}
\includegraphics[scale=0.6]{Fig1c}
\caption{\label{fig:BWcircuit} Brickwork local circuit composed of brick gates (local gate indicated on the right) for $t=2$ (depth 4). Note that the dimensions of the even/odd local spaces could be different (indicated by thin/thick wires). Evolution time runs bottom-up throughout the paper.}
\end{figure}
Let us consider a chain of even number, $2L$, $L\in\mathbb{N}$, of qudits ($d$-level quantum systems), such that the Hilbert space of the system is given as a $d^{2L}$ dimensional tensor product
${\cal H} = {\cal H}_1^{\otimes 2L}$, ${\cal H}_1 = \mathbb{C}^{d}$.
We may also consider a more general, \emph{chiral} situation, where the pair of neighboring sites have different Hilbert space dimensions
${\cal H}_1 = \mathbb{C}^d$, ${\cal H}'_1 = \mathbb{C}^{d'}$, and two isomorphic system Hilbert spaces (with even/odd sublattices interchanged),
${\cal H} = ({\cal H}_1\otimes {\cal H}'_1)^{\otimes L}$,
${\cal H}'= ({\cal H}'_1\otimes {\cal H}_1)^{\otimes L}$.
In many physical situations, such as when discussing periodically driven (Floquet) spin chains, or Trotterized Hamiltonian evolutions with local one-dimensional interaction (in the latter only the case $d'=d$ makes sense), as well as in protocols for analog quantum simulation\cite{Nori} of local interactions, it is customary to consider brickwork quantum circuits. For simplicity, we assume space-time homogeneity\footnote{The extension of the entire analysis in our paper to arbitrarily spatially and temporally modulated gates is straightforward, so we do not mention it any further}. Hence, considering a single unitary gate $U \in {\rm U}(d d')$ interpreted as a linear map
${\cal H}_1\otimes {\cal H}'_1 \to
{\cal H}'_1\otimes {\cal H}_1$, or in explicit matrix/Dirac notation\footnote{Throughout the paper we use the convention that row/column of a matrix is labeled by a lower/upper index, while multi-indices at the same level label tensor- (Kronecker-) product spaces.}
\begin{equation}
U^\brick=\sum_{j,j'=1}^d \sum_{s,s'=1}^{d'}
U_{j\,s}^{s'j'} \ket{s'}\otimes \ket{j'} \bra{j} \otimes \bra{s}\,,
\label{eq:BG}
\end{equation}
we define a generator (or Floquet propagator) of a brickwork local circuit as
\begin{equation}
\mathcal U =
\mathcal U^{\rm o} \mathcal U^{\rm e} \;:\; {\cal H} \to {\cal H}
\label{eq:UF}
\end{equation} where
\begin{eqnarray}
\mathcal U^{\rm e} &=& \prod_{x=1}^L U^\brick_{2x-1,2x} \;:\; {\cal H} \to {\cal H}',\label{eq:BWC}\\
\mathcal U^{\rm o} &=& \prod_{x=1}^L U^{\brick}_{2x,2x+1} \;:\; {\cal H}' \to {\cal H}',\nonumber
\end{eqnarray}
and where the subscripts in $U^\brick_{x,y}$ denote the positions $x,y$ of two qudits (sites) where the brick gate $U^\brick$ acts non-trivially (see Fig.~\ref{fig:BWcircuit} for an unambiguous graphical definition). Periodic boundaries are assumed throughout: $x+2L\equiv x$.
Although we will use brickwork circuits later for comparison, we make a twist in this paper and propose to study another physics paradigm of \emph{generic local spatiotemporal dynamics on 1+1 dimensional lattice}. Specifically, we propose unitary face circuits where the local interactions are given in terms of nearest-neighbor controlled (e.g., kinetically constrained) local unitary gates or, equivalently, in terms of unitary interactions round--a--face.
Here we assume all local spaces to be isomorphic\footnote{We could as well define more general class of IRF circuits where dimensions of local spaces at even/odd lattice sites would be different, $d\neq d'$, but this would require two different IRF gates at even/odd half-time steps (\ref{eq:BWC}) and will not be considered here.} $d=d'$.
\begin{figure}
\includegraphics[scale=0.57]{Fig1b}
\caption{\label{fig:IRFcircuit}
Face local circuit composed of IRF gates
(local gate indicated on the bottom-right) with duration $t=3$ (depth 6), using two different notations, either in terms of controlled unitary gates (top) or face plaquettes (bottom).}
\end{figure}
Consider a set of $d^2$ arbitrary unitary matrices $\{u_{ik}\in {\rm U}(d)\}_{i,k\in\{1,\ldots,d\}}$ which define a general 2-controlled 3-qudit unitary gate (as a unitary over ${\cal H}_1^{\otimes 3}$)
\begin{equation}
U^\IRF = \sum_{i,j,k,j'=1}^d (u_{ik})_j^{j'} \ket{i}\otimes \ket{j'}\otimes \ket{k} \bra{i}\otimes\bra{j}\otimes\bra{k}\,
\label{eq:UIRF}
\end{equation}
Equivalently, a set of $d^4$ amplitudes $(u_{ik})_{j}^{j'}$ can be understood as defining a (unitary) IRF model (see Fig.~\ref{fig:IRFcircuit}).
Such 3-qudit gates, embedded into the many-body Hilbert space ${\cal H}$ as $U^\IRF_{x-1,x,x+1}$ now define locally interacting unitary circuit with the generator of the form (\ref{eq:UF}), where
\begin{eqnarray}
\mathcal U^{\rm e} &=& \prod_{x=1}^L U^\IRF_{2x-1,2x,2x+1}, \label{eq:IRFC}\\
\mathcal U^{\rm o} &=& \prod_{x=1}^L U^\IRF_{2x,2x+1,2x+2}. \nonumber
\end{eqnarray}
Similarly to brickwork circuits (\ref{eq:BWC}), which behave as quantum cellular automata\cite{Farrelly}, namely they propagate information/correlation by one-site per layer of the gates, one notes the same feature for IRF circuits (\ref{eq:IRFC}).
An example of a unitary IRF circuit is a Trotterization\cite{FloquetPXP1,FloquetPXP2} of the so-called PXP model\cite{PXP,abanin_NP2018} beautifully modelling kinetically constrained Rydberg atom chains\cite{Lukin}. Specifically, the three site Hamiltonian of the PXP model
$h_{x-1,x,x+1}= P_{x-1} X_x P_{x+1}$, where
$$P=\begin{pmatrix} 1 & 0 \cr 0 & 0\end{pmatrix}\,\quad X=\begin{pmatrix} 0 & 1 \cr 1 & 0\end{pmatrix}\,,$$
clearly exponentiates to a unitary IRF gate $U^\IRF_{x-1,x,x+1} = \exp(-{\rm i}\Delta t h_{x-1,x,x+1})$, where $\Delta t$ is the time step.
Other recently studied examples of unitary IRF cicruits are classical reversible cellular automata\cite{bobenko1993two}, like the rule 54\cite{prosen2016integrability,rule54review,katja_PRL} or the rule 201 (`classical PXP')\cite{wilkinson2020exact}.
More broadly, unitary IRF circuits represent a natural language to describe Floquet or driven quantum kinetically constrained models.
It is interesting to note that both manifolds of brick and IRF local gates share the same number of independent real parameters (for $d'=d$), specifically $d^4$, i.e. the number of parameters of ${\rm U}(d^2)$ or the number of parameters for $d^2$ independent elements of ${\rm U}(d)$, respectively. However, we should then also mention different gauge-invariance groups of these parametrizations. While the brick gate can be transformed as
\begin{equation}
U^\brick \leftarrow (h^\dagger \otimes g^\dagger) U^\brick (g\otimes h),
\end{equation}
for arbitrary $g\in{\rm SU}(d), h\in{\rm SU}(d')$, to yield an equivalent circuit, the IRF gate can be gauge-transformed
as
\begin{equation}
U^\IRF \leftarrow
(\Delta^\dagger \otimes g^\dagger \otimes \Delta^\dagger)
U^\IRF (\Delta \otimes g \otimes \Delta),
\end{equation}
where $g\in{\rm SU}(d)$ arbitrary and $\Delta_{j}^{j'} = \delta_{j,j'}
e^{{\rm i}\theta_j}$, $\theta_j \in[0,2\pi)$, is a \emph{diagonal} phase matrix (where one of the phases $\theta_j$ can be fixed without loss of generality). We thus have the following gauge groups for the two classes of circuits
\begin{eqnarray}
G^\brick{}&=&{\rm SU}(d)\otimes {\rm SU}(d'),\qquad
\textrm{ for\;brickwork\;circuits},\\
G^\IRF&=&{\rm SU}(d)\otimes {\rm U}(1)^{\otimes (d-1)},\qquad
\textrm{ for\;IRF\;circuits}.
\label{gaugeIRF}
\end{eqnarray}
One may wish to investigate dynamics, entanglement propagation and operator spreading in IRF circuits and compare to existing results for brickwork circuits. Specifically, it would be desirable to derive analogous results to\cite{nahum2017quantum,nahum2018operator,khemani,vonKeyserlingk2018operator} for \emph{random} IRF circuits where matrices $u_{ik}$ are independent Haar-random ${\rm U}(d)$ matrices for all pairs of components $i,k$ and for each space time point.
In this paper, however, we aim at investigating IRF circuits with an additional structure, namely, the \emph{dual-unitarity}.
\section{Correlation decay in dual-unitary quantum lattice dynamical systems}
\subsection{Spatio-temporal correlation function and folded circuit representation}
Here we set the fundamental problem of quantum dynamics on a space-time lattice, specifically, the computation of space-time correlation function of local observables in the tracial (infinite temperature/maximum entropy) state.
Considering a pair of local traceless observables $a,b$, with $a_x,b_x$ being their embedding into ${\cal H}$ at site $x$, we aim at calculating
\begin{equation}
C_{a,b}(x,y;t) = \lim_{L\to\infty} \frac{1}{{\rm dim}{\cal H}} {\rm tr}(a_x \mathcal U^t b_y \mathcal U^{-t}).
\end{equation}
Explicit, exact or analytical computation of correlation functions, being the fundamental importance in diverse areas of
condensed matter and statistical physics, represent an insurmountable obstacle even in the simplest (say integrable) interacting theories. Nevertheless, we will show below how the correlations can be explicitly treated in a class of generically non-integrable cuircuit models.
\begin{figure}
\includegraphics[scale=0.6]{Fig5}
\caption{\label{fig:5}
Definition of the folded (Heisenberg picture) brick (top) and IRF (bottom) gate. Note that thick wires correspond to doubled Hilbert space (ket$=$left, bra$=$right thin wire).}
\end{figure}
In the so-called folded-circuit representation\cite{Banuls}, one defines a doubled (operator) Hilbert space
$\mathcal H^{\rm op} = \mathcal H\otimes \mathcal H $, which can be considered as composed of doubled local spaces
$\mathcal H^{\rm op}_1 = \mathcal H_1\otimes \mathcal H_1 \simeq \mathbb C^{d^2}$, and possibly different local operator space $\mathcal H^{\rm op'}_1 = \mathcal H'_1\otimes \mathcal H'_1 \simeq \mathbb C^{{d'}^2}$ for even-site sublattice.
Defining doubled local brick gate over $({\mathcal H}^{\rm op}_1)^{\otimes 2}$ (Fig.~\ref{fig:5}-top)
\begin{equation}
W^\brick = U^\brick \otimes (U^\brick)^T\,
\end{equation}
where $^T$ denotes the matrix transposition, and local operator states
\begin{eqnarray}
\kket{a} &=& \frac{1}{\sqrt{d}}\sum_{i,j}a_{i}^{j} \ket{i}\otimes\ket{j},\\
\kket{b} &=& \frac{1}{\sqrt{d}}\sum_{i,j} b_{i}^{j} \ket{i}\otimes\ket{j},\\
\kket{\circ} &=& \frac{1}{\sqrt{d}} \sum_{i} \ket{i}\otimes\ket{i},
\end{eqnarray}
with possibly $d$ replaced by $d'$ for even-labelled sites,
one immediately writes an equivalent expression for the correlation function
\begin{equation}
C_{a,b}(x,y;t) = \lim_{L\to\infty} \bbra{b_y} \mathcal W^t \kket{a_x},
\label{eq:Cab}
\end{equation}
where $\kket{a_x} = \kket{\circ}^{\otimes (x-1)}\otimes\kket{a}\otimes\kket{\circ}^{\otimes(L-x)} $ and
$\bbra{b_y} =\bbra{\circ}^{\otimes(y-1)}\otimes\bbra{b}\otimes\bbra{\circ}^{\otimes(L-y)}$. Here $\mathcal W$ is the
\emph{operator circuit} over $\mathcal H^{\rm op}$ built as in Eqs.(\ref{eq:UF},\ref{eq:BWC}) with $U$'s replaced by $W$'s.
Completely analogous folded circuit construction applies also for IRF circuits, where the (folded) IRF operator gate reads as (Fig.~\ref{fig:5}-bottom)
\begin{equation}
W^\IRF = U^\IRF \otimes (U^\IRF)^T
\label{eq:IRFfold}
\end{equation}
which is a unitary IRF gate as well (over local Hilbert spaces of dimension $d^2$).
Unitarity conditions for, respectively, brick and IRF local gates can be now expressed
as \emph{unitality} (schematically in Figs.~\ref{fig:6},\ref{fig:7}-top) \begin{eqnarray}
W^\brick \kket{\circ}\otimes\kket{\circ} &=& \kket{\circ}\otimes\kket{\circ}, \label{eq:B1}\\
\bbra{\circ}\otimes\bbra{\circ} W^\brick&=& \bbra{\circ}\otimes\bbra{\circ},\label{eq:B2}\\
W^\IRF \kket{\circ}\otimes\kket{\circ}
\otimes\kket{\circ}&=&\kket{\circ}\otimes\kket{\circ}\otimes\kket{\circ},\label{eq:I1}\\
\bbra{\circ}\otimes\bbra{\circ}
\otimes\bbra{\circ}W^\IRF&=&\bbra{\circ}\otimes\bbra{\circ}\otimes\bbra{\circ}.\label{eq:I2}
\end{eqnarray}
These rules, and the fact that the operators are traceless, i.e.
$\bbraket{\circ|a}=\bbraket{\circ|b}=0$, immediately imply strict causality of the correlator, namely that the maximal speed of information propagation equals 1 (one site per circuit layer): $C_{a,b}(x,y;t)=0$ for $|x-y|>2t$. Aside from that, the computation of the correlator $C_{a,b}(x,y;t)$ for a generic local gate circuit is believed to be hard, i.e. to have a positive Kolmogorov algorithmic complexity in $t$.
\subsection{Dual-unitary brickwork circuits: review}
\begin{figure}
\includegraphics[scale=0.55]{Fig3}
\caption{\label{fig:3}
Dual-unitarity: time unitarity (left) and space unitarity (right) condition for the dual-unitary brick gate (element of ${\rm DUBG}(d,d')$). Wires are drawn at $45^\circ$ angles from the gates to stress the space-time symmetry.}
\end{figure}
It has been noted in Ref.~\cite{bertini2019exact} that there exist a rich class of brickwork unitary circuits where computation of arbitrary local correlations can be drastically simplified. These are the so-called dual-unitary brickwork circuits which generate unitary dynamics not only in time (vertical) direction, but also in space (horizontal) direction. In other words, not only the local brick gate $U^\brick$ (\ref{eq:BG}) is unitary, but also the space-time reshuffled gate
\begin{equation}
\tilde{U}^\brick =\sum_{j,j'=1}^d \sum_{s,s'=1}^{d'}
U_{j\,s'}^{s j'} \ket{s'}\otimes \ket{j'} \bra{j} \otimes \bra{s}\,,
\end{equation}
is unitary
\begin{equation}
\tilde{U}^\brick (\tilde{U}^\brick)^\dagger =\one\,.
\label{eq:DU}
\end{equation}
The gate $\tilde{U}^\brick$ is referred to as the space-time dual of $U^\brick$, and the condition (\ref{eq:DU}) (see Fig.~\ref{fig:3}-right) as space unitarity.
The gates which are both, time unitary and space unitary, form a local submanifold (locally smooth subset) ${\rm DUBG}(d,d')$ of the Lie group ${\rm U}(d d')$ and can be completely characterized \cite{bertini2019exact}
for qubits.
Specifically, one can write an arbitrary dual unitary gate
for $d=d'=2$ as
\begin{equation}
{\rm DUBG}(2,2) = \{
(u\otimes v)S e^{{\rm i}(
\beta\one+ \gamma\,\sigma\otimes\sigma)}
(w\otimes r)\},
\end{equation}
where $u,v,w,r\in {\rm SU}(2)$, $\beta,\gamma\in\mathbb R$,
and $\sigma_j^{j'}= (-1)^{j-1} \delta_{j,j'}$ (Pauli-Z matrix),
$S_{i\,j}^{i'j'}\!=\delta_{i,j'}\delta_{j,i'}$
(SWAP, $S\ket{j}\otimes\ket{s} = \ket{s}\otimes \ket{j}$).
Counting the number of independent real parameters, one should note that out of 3 parameters (e.g. Euler angles) determining each local ${\rm SU}(2)$ gate, two can be removed, as Euler rotations around $z-$axis commute with the Ising interaction, so one is left with ${\rm dim}\,{\rm DUBG}(2,2)=12$ independent parameters.
Although large multi-parametric families of DU gates have been proposed\cite{Gutkin_Hadamard,austen_prl,balazs_private} for $d>2$, the complete characterization of ${\rm DUBG}(d,d')$ remains a challenging open problem (see section~\ref{estimates} for some empirical observations). One should note that dual-unitarity condition is equivalent to requiring that $(S U^\brick)^{T_1}$ is unitary, where $^{T_1}$ is a partial transposition. Using the result (Theorem 3.1) of Ref.~\cite{pellegrini} one can show that ${\rm DUBG}(d,d')$ can be identified with the set of {\em unital channels} over $\mathbb C^d\otimes \mathbb C^{d'}$, whose complete characterisation is, however, still open. We note that the \emph{entangling power} of such bi-partite partial-transpose unitaries have been discussed also in Refs.~\cite{PRA07,Karol}.
\begin{figure}
\includegraphics[scale=0.65]{Fig6}
\caption{\label{fig:6}
Compact expressions of unitarity (top) and dual unitarity (bottom) for folded brick gates.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.65]{Fig9}
\caption{\label{fig:8}
The non-vanishing (light-ray) contribution to correlation function between local observables
$a,b$ for the DU brickwork circuit -- using the folded circuit formulation -- (the second term of (\ref{eq:Cablr}) for $t=2$), and the definition of the corresponding transfer matrix $\mathcal M_-$ (right).}
\end{figure}
Computation of local spatiotemporal correlation functions of DU brickwork circuits can be largely simplified, namely it is easy to show that both, the (time) unitarity (\ref{eq:B1},\ref{eq:B2}), as well as the space unitarity (schematically depicted in Fig.~\ref{fig:6})
\begin{eqnarray}
\tilde{W}^\brick \kket{\circ}\otimes\kket{\circ} &=& \kket{\circ}\otimes\kket{\circ}, \label{eq:B3}\\
\bbra{\circ}\otimes\bbra{\circ} \tilde{W}^\brick&=& \bbra{\circ}\otimes\bbra{\circ},\label{eq:B4}
\end{eqnarray}
where $\tilde{W}^\brick = \tilde{U}^\brick\otimes (\tilde{U}^\brick)^T$ imply that the expression (\ref{eq:Cab}) vanishes unless $|x-y|=2t$.
This is a consequence of causality within both, space-like and time-like cones, so the correlator can be non-vanishing only along two light-rays\cite{bertini2019exact}. There, it is expressed as
\begin{eqnarray}
C_{a,b}(x,y;t) &=& \delta_{y,x+2t}\delta_{{\rm mod}(x,2),1}
\,{\rm tr}\,(b \mathcal M_+^{2t}(a)) \nonumber\\
&+& \delta_{y,x-2t}\delta_{{\rm mod}(x,2),0}
\,{\rm tr}\,(b \mathcal M_-^{2t}(a))
\label{eq:Cablr2}
\end{eqnarray}
in terms of completely positive, trace preserving and unital maps over ${\rm End}(\mathcal H_1)$, and
${\rm End}(\mathcal H'_1)$, respectively (see Fig.~\ref{fig:8}),
\begin{eqnarray}
\mathcal M_+ (a) &=&
\frac{1}{d'}({\rm tr}\otimes\One)\left(
(U^\brick)^\dagger
(a\otimes\one) U^\brick\right)\,,\\\
\mathcal M_- (a) &=&
\frac{1}{d}(\One\otimes {\rm tr})\left(
(U^\brick)^\dagger
(\one\otimes a) U^\brick\right)\,.
\end{eqnarray}
$\One$ represents an identify map over the local space $\mathcal H_1^{(')}$, hence $\One\otimes{\rm tr}$ and ${\rm tr}\otimes\One$ denote the partial traces.
As $\mathcal M_\pm$ are linear non-expanding maps, their spectra are confined within the unit disk. Depending on whether there are additional eigenvalues, besides one eigenvalue $1$ corresponding to trivial eigenvector $\one$, which lie on the unit circle (respectively, at $1$), our Floquet circuit system is non-mixing (respectively, non-ergodic), otherwise it is mixing and ergodic. It has been shown in \cite{bertini2019exact} (and elaborated further in other DU models in \cite{austen_prl,arul21}) that one can have all different types of ergodic behavior even in the simplest class of DU brickwork circuits with $d=d'=2$. In the generic case (with probability 1 for a suitably random element of ${\rm DUBG}(d,d')$) the maps $\mathcal M_\pm$ have full rank ($d^2$ or ${d'}^2$) with all eigenvalues, except the trivial one, lying strictly inside the unit disk implying asymptotic \emph{exponential decay of correlations} (\ref{eq:Cab}) (mixing behavior) with the exponent given by the spectral gap of $\mathcal M_\pm$.
\subsection{Dual-unitary IRF circuits}
\begin{figure}
\includegraphics[scale=0.55]{Fig4}
\caption{\label{fig:4}
Time unitarity (left) and space unitarity (right) condition for the dual-unitary IRF gate (element of ${\rm DUIRF}(d)$).}
\end{figure}
In somewhat close analogy to brickwork circuits we define DU IRF circuits, composed of IRF gate (\ref{eq:UIRF}),
for which also the space-time dual $\tilde{U}^\IRF\in {\rm End}(\mathcal H_1^{\otimes 3})$:
\begin{equation}
\tilde{U}^\IRF = \sum_{i,j,k,j'=1}^d (u_{jj'})_i^{k} \ket{i}\otimes \ket{j'}\otimes \ket{k} \bra{i}\otimes\bra{j}\otimes\bra{k}\,
\label{eq:UtIRF}
\end{equation}
is unitary
\begin{equation}
\tilde{U}^\IRF (\tilde{U}^\IRF)^\dagger = \one.
\label{eq:IRFdu}
\end{equation}
This condition is equivalent to a condition that a set of $d^2$ (space-time flipped) matrices $\tilde{u}_{jj'} \in {\rm End}(\mathcal H_1)$, $j,j'=1,2\ldots,d$, defined as
\begin{equation}
(\tilde{u}_{jj'})_{i}^k := (u_{ik})_{j}^{j'},
\label{eq:ut}
\end{equation}
is unitary, $\tilde{u}_{jj'} \tilde{u}^\dagger_{jj'} = \one$, $j,j'=1,\ldots,d$. See Fig.~\ref{fig:4} for a diagrammatic illustration of these properties.
In the next subsection \ref{d2IRF} we provide a complete parametrization of a set ${\rm DUIRF}(d)$ of DU IRF gates for $d=2$, while in section \ref{estimates} we estimate its dimensionality for larger $d$.
\begin{figure}
\includegraphics[scale=0.6]{Fig7}
\caption{\label{fig:7}
Compact expressions of time unitarity (top) and space unitarity (bottom) for the folded IRF gates.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.65]{Fig8}
\caption{\label{fig:9}
Schematic illustration of computation of correlation function between local (2-site) observables in the folded IRF circuit formulation. The yellow-shaded area indicates the intersection of temporal causal cones to which the correlator can be simplified using only unitarity (Fig.~\ref{fig:7}-top). For DU IRF circuit one can apply (all) rules of Fig.~\ref{fig:7} to show that such correlation function identically vanishes (unless the supports of operators $a$ and $b$ are shifted precisely by $2t$, as used in Fig.~\ref{fig:10}.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.65]{Fig10}
\caption{\label{fig:10}
The nonvanishing (light-ray) contribution to correlation function between local observables
$a,b$ for the DU IRF circuit -- using the folded circuit formulation -- and the definition of the corresponding transfer matrix $\mathcal K_-$ (right).
}
\end{figure}
In terms of the folded IRF gate $\tilde{W}^\IRF = \tilde{U}^\IRF\otimes(\tilde{U}^\IRF)^T$, cf. (\ref{eq:IRFfold}), the space unitarity (\ref{eq:IRFdu}) of DU IRF gate is elegantly expressed in terms of the second set of unitality conditions (graphically encoded in Fig.~\ref{fig:7}-bottom)
\begin{eqnarray}
\tilde{W}^\IRF \kket{\circ}\otimes\kket{\circ}
\otimes\kket{\circ}&=&\kket{\circ}\otimes\kket{\circ}\otimes\kket{\circ},\label{eq:I3}\\
\bbra{\circ}\otimes\bbra{\circ}
\otimes\bbra{\circ}\tilde{W}^\IRF&=&\bbra{\circ}\otimes\bbra{\circ}\otimes\bbra{\circ}.\label{eq:I4}
\end{eqnarray}
The complete set of unitality relations (\ref{eq:I1},\ref{eq:I2},\ref{eq:I3},\ref{eq:I4}) is then facilitated to show that the correlator (\ref{eq:Cab}) (Fig.~\ref{fig:9}) vanishes unless $|x-y|=2t$. Without loss of generality we can now assume that local operators are supported on two sites $a,b\in {\rm End}(\mathcal H_1^{\otimes 2})$ (including single-site observables which are trivial on the second site) and write $\kket{a_x} = \kket{\circ}^{\otimes (x-1)}\otimes\kket{a}\otimes\kket{\circ}^{\otimes(L-x-1)} $ and
$\bbra{b_y} =\bbra{\circ}^{\otimes(y-1)}\otimes\bbra{b}\otimes\bbra{\circ}^{\otimes(L-y-1)}$.
Diagrammatically, this is illustrated in Fig.~(\ref{fig:10}), where the resulting light-cone correlators:
\begin{eqnarray}
C_{a,b}(x,y;t) &=& \delta_{y,x+2t}\delta_{{\rm mod}(x,2),1}
\,{\rm tr}\,(b \mathcal K_+^{2t}(a)) \nonumber\\
&+& \delta_{y,x-2t}\delta_{{\rm mod}(x,2),0}
\,{\rm tr}\,(b \mathcal K_-^{2t}(a))\,,
\label{eq:Cablr}
\end{eqnarray}
are expressed in terms of completely positive, trace preserving and unital maps over ${\rm End}(\mathcal H_1^{\otimes 2})$,
\begin{eqnarray}
\mathcal K_+ (a) &=&
\frac{1}{d}({\rm tr}\otimes\One\otimes\One)\left( (U^\IRF)^\dagger
(a\otimes\one) U^\IRF\right)\,,\\\
\mathcal K_- (a) &=&
\frac{1}{d}(\One\otimes \One\otimes {\rm tr})\left( (U^\IRF)^\dagger
(\one\otimes a) U^\IRF \right)\,,
\end{eqnarray}
(see Fig.~\ref{fig:10}-right for graphical defintion of $\mathcal K_-$).
Although the maps $\mathcal K_\pm$ act on a much larger (2-qudit) space as $\mathcal M_\pm$, they also have a large trivial subspace (of eigenvalue 0) and hence can be reduced to a simpler form.
This essentially follows from the trivial action of the IRF gate on the control (left and right) qudits. Let
\begin{equation}
\mathcal D ( \ket{j}\bra{j'}) = \delta_{j,j'} \ket{j}\bra{j'}
\end{equation}
represent a projector to diagonal subspace of ${\rm End}(\mathcal H_1)$.
The correlation maps clearly satisfy the identities (following from diagrammatics of Fig.~\ref{fig:10}):
\begin{eqnarray}
&&\mathcal K_+ (\mathcal D \otimes \One) =
(\One\otimes\mathcal D\mathcal) \mathcal K_+ = \mathcal K_+\,, \label{eq:KD}\\
&&\mathcal K_- (\One\otimes\mathcal D) =
(\mathcal D \otimes \One) \mathcal K_- = \mathcal K_- \,. \nonumber
\end{eqnarray}
Defining the diagonally projected maps
\begin{equation}
\mathcal K'_\pm = (\mathcal D\otimes \mathcal D)\mathcal K_\pm
(\mathcal D\otimes \mathcal D),\,
\end{equation}
and using the projector property $\mathcal D^2=\mathcal D$,
one finds that Eqs. (\ref{eq:KD}) imply, for any $t\in\mathbb Z$:
\begin{equation}
(\mathcal D\otimes \mathcal D)(\mathcal K_\pm)^t (\mathcal D\otimes \mathcal D)
= (\mathcal K'_\pm)^t\,.
\end{equation}
This in turn implies that the correlation functions (\ref{eq:Cablr2}) are given in terms of simple iteration of diagonally projected maps
\begin{equation}
{\rm tr}\left(b \mathcal K^t_\pm (a)\right) =
{\rm tr}\left(b_{\rm d}(\mathcal K'_\pm)^t (a_{\rm d})\right)
\end{equation}
where $a_{\rm d} = \mathcal D\otimes\mathcal D a$, $b_{\rm d} = \mathcal D\otimes\mathcal D b$ are diagonal (projected) 2-site observables.
In fact the maps $\mathcal K'_\pm$ can be identified with the classical Markov chains. By identifying the basis $\{j \leftarrow \ket{j}\bra{j}\}$, the explicit matrix representation of correlation maps reads
\begin{equation}
(\mathcal K'_+)_{i\,j}^{i'j'} = \frac{1}{d}\left| (u_{ij'})_j^{i'} \right|^2\,,\quad
(\mathcal K'_-)_{i\,j}^{i'j'} = \frac{1}{d}\left| (u_{i'j})_i^{j'} \right|^2\,.
\label{Kmarkov}
\end{equation}
These matrices are bistochastic. In fact, they are bistochastic also under the flip of indices
($j\leftrightarrow i'$) which would correspond to space-time flip if one composes from them a brickwork classical Markov circuit like those studied in Ref.~\cite{PRX2021}, hence they may be referred to as {\em dual bistochastic}.\footnote{Note, however, that in spite of some formal similarities these matrices are not unistochastic.}
It follows from the form (\ref{Kmarkov}) and unitarity of
$u_{ik}$ and $\tilde{u}_{jj'}$ that the map $\mathcal K'_\pm$ annihilates the diagonal operators of the form $\one\otimes a_{\rm d}$ or $a_{\rm d} \otimes \one$, where $a_{\rm d}\in {\rm End}(\mathcal H_1)$, ${\rm tr}\, a_{\rm d}=0$.
Hence $\mathcal K'_\pm$ act nontrivially within a subspace spanned by $\one$ and traceless operators supported on no less than 2 neighbouring sites, which yields their maximal rank
\begin{equation}
\max\,{\rm rank}\,\mathcal K_\pm = 1 + (d-1)^2.
\label{maxrank}
\end{equation}
The above observation also implies that all correlation functions between single-site (ultra-local) observables vanish,
while the simplest non-trivial correlations involve two-site observables. In summary, the decay of correlation functions of local observables in DU IRF circuits is thus completely determined by the spectra of dual bistochastic $d^2\times d^2$ matrices $\mathcal K'_\pm$ (in fact, by their $(d-1)^2$ dimensional nontrivial blocks) and the absence of nontrivial eigenvalue $1$ (respectively, unimodular eigenvalue) signals ergodic (respectively, mixing) dynamics.
\subsection{Complete parametrization of dual-unitary IRF qubit gates}
\label{d2IRF}
Let us now consider the case $d=2$ with an attempt to parametrize all DU IRF gates. We start by Euler angle parametrization of
${\rm U}(2)$ matrices $u_{ik}$
\begin{equation}
u_{ik} = e^{{\rm i} \phi_{ik}}
\begin{pmatrix}
e^{{\rm i}\nu_{ik}}\cos\theta_{ik} &
e^{{\rm i}\eta_{ik}}\sin\theta_{ik} \cr
-e^{-{\rm i}\eta_{ik}}\sin\theta_{ik} &
e^{-{\rm i}\nu_{ik}}\cos\theta_{ik}
\end{pmatrix}\,,
\end{equation}
where $\phi_{ik},\nu_{ik},\eta_{ik},\theta_{ik}\in [0,2\pi)$, $i,k=1,2$, are 16 real parameters (note that such parametrization is non-injective).
Solving for unitarity of $\tilde{u}_{jj'}$, defined in (\ref{eq:ut}), separates nicely into two sets of equations: The equations for $\theta_{ik}$
\begin{eqnarray}
&\cos^2\theta_{11}=\sin^2\theta_{12},\quad
\cos^2\theta_{22}=\sin^2\theta_{21}, \nonumber\\
&\cos\theta_{11}\cos\theta_{21}+\cos\theta_{12}\cos\theta_{22}=0, \label{eqtheta}\\ &\sin\theta_{11}\sin\theta_{21}+\sin\theta_{12}\sin\theta_{22}=0, \nonumber
\end{eqnarray}
and a set of linear equations for the other variables which fixes, say $22$-components of the angles $\nu_{ik},\eta_{ik},\phi_{ik}$ in terms of components $11,12,21$:
\begin{eqnarray}
\nu_{22}&=&\nu_{12}+\nu_{21}-\nu_{11},\nonumber\\
\eta_{22}&=&\eta_{12}+\eta_{21}-\eta_{11},\\
\phi_{22}&=&\phi_{12}+\phi_{21}-\phi_{11}.\nonumber
\end{eqnarray}
Eqs.~(\ref{eqtheta}) in turn result in expressing
three $\theta_{ik}$ in terms of the fourth, say $\theta_{22}$. There are two equivalent solutions, while without loss of generality we take:
\begin{equation}
\theta_{11} = \theta_{22} + \pi,\quad
\theta_{12} = \theta_{21} = \theta_{22} + \frac{\pi}{2}.
\end{equation}
We thus parametrized ${\rm DUIRF}(2)$ in terms of 10 independent free parameters
$\{ \theta_{22},\nu_{11},\nu_{12},\nu_{21},\eta_{11},\eta_{12},\eta_{21},\phi_{11},\phi_{12},\phi_{21}\}$, hence ${\rm dim}\,{\rm DUIRF}(2)={\bf 10}$. Considering 4-dimensional gauge symmetry (\ref{gaugeIRF}) and a global (overall) phase, we have in fact $10-4-1={\bf 5}$ parametric set of physically inequivalent IRF gates of qubits.
Expressing the diagonally projected transfer matrices we obtain a simple result
\begin{equation}
\mathcal K'_+ = \mathcal K'_- =
\frac{1}{2}\begin{pmatrix}
\cos^2\theta_{22} & \sin^2\theta_{22} & \sin^2\theta_{22} & \cos^2\theta_{22} \\
\sin^2\theta_{22} & \cos^2\theta_{22} & \cos^2\theta_{22} & \sin^2\theta_{22} \\
\sin^2\theta_{22} & \cos^2\theta_{22} & \cos^2\theta_{22} & \sin^2\theta_{22} \\
\cos^2\theta_{22} & \sin^2\theta_{22} & \sin^2\theta_{22} & \cos^2\theta_{22}
\end{pmatrix}.
\end{equation}
$\mathcal K'_\pm$ have rank 2 and a single nontrivial eigenvalue $\lambda = \cos(2\theta_{22})$ with the corresponding left\&right eigenvector $(1,-1,-1,1)$ corresponding to eigenoperator $a = \sigma\otimes\sigma$.
We have thus shown that the only nontrivial (nonzero) correlation function -- autocorrelation of the Ising interaction -- of all DU IRF circuits with $d=2$ has a universal form
\begin{equation}
C_{\sigma\otimes\sigma,\sigma\otimes\sigma}(x,y;t)
= (\delta_{y,x+2t}\delta_{{\rm mod}(x,2),1}+ \delta_{y,x-2t}\delta_{{\rm mod}(x,2),0}) \lambda^{2t},
\end{equation}
independent of all other parameters (but $\theta_{22}$) of the gate.\footnote{Note that this feature is similar to a kicked Hadamard spin chains studied in Ref.\cite{Gutkin_Hadamard}. It would be interesting to investigate if theres is a link between those sistems and DU IRF circuits.}
Of course, we expect, and find, that DU IRF curcits for higher $d$ have much richer behavior, as reported in the next section.
\section{Estimating the dimension of manifolds of Dual-unitary gates for $d> 2$ ($d'>2$)}
\label{estimates}
This is an experimental section of the paper where we provide some empirical observation which can hopefully guide further progress. Earlier we managed to fully characterize the manifolds of DU brick and IRF gates for qubits $d=2$. It has become clear that obtaining rigorous results in this direction for DU brick gates with $d>2$ or $d'>2$ is notoriously difficult, while this task does not appear to get any easier for DU IRF gates with $d>2$.
Therefore we take a different approach here and try to estimate numerically the number of free real parameters (dimensionality) of ${\rm DUBG}(d,d')$ and ${\rm DUIRF}(d)$.
We do this by determining the dimensions of tangent spaces at {\em random} instances of solutions to dual (time and space) unitarity conditions.
\begin{table}
\label{table}
\begin{ruledtabular}
\begin{tabular}{ccccc}
$d$ & $d'$ & $N=(d d')^2$ &
${\rm dim}\,{\rm DUBG}(d,d')$ &
${\rm dim}\,{\rm DUIRF}(d)$ \\
\hline
2 & 2 & 16 & 12 (12) & 10 (10) \\
3 & 3 & 81 & 45 (45,43,41) & (33,29,25) \\
4 & 4 & 256 & 112 (94) & (49) \\
5 & 5 & 625 & 225 (97) & (81) \\
6 & 6 & 1296 & 396 (141) & (121) \\
7 & 7 & 2401 & 637 (193) & (169) \\
3 & 2 & 36 & 24 (24) & --- \\
4 & 2 & 64 & 40 (40) & --- \\
5 & 2 & 100 & 60 (60) & --- \\
6 & 2 & 144 & 84 (84) & --- \\
7 & 2 & 196 & 112 (112) & --- \\
4 & 3 & 144 & 72 (66,64) & --- \\
5 & 3 & 225 & 105 (77) & --- \\
\end{tabular}
\end{ruledtabular}
\caption{Local dimensions of manifolds ${\rm DUBG}(d,d')$ and ${\rm DUIRF}(d)$ estimated as dimensions of the tangent spaces at randomly sampled solutions of DU constraints (numbers within brackets). For comparison we show for {\rm DUBG} also dimensions of tangent spaces at random instances of explicit parametrization \cite{balazs_private} of DU brick gates of Eq.~(\ref{balazs}) (unbracketed numbers).}
\end{table}
\subsection{Dual unitary brick gate manifolds}
Writing $N=2(d d')^2$ real components of a $dd' \times dd'$ complex matrix $ U^\brick $ in terms of a vector $\vec{z}=(z_1,z_2,\ldots,z_{N})$ we can write the dual unitarity conditions
$ U^\brick (U^\brick)^\dagger = \one$, $\tilde{U}^\brick (\tilde{U}^\brick)^\dagger=\one$, in terms of a zero of a nonlinear (quadratic) vector function $\vec{f}(\vec{z})$. Note that the number $M$ of components of $\vec{f}$ (number of equations) is in general different (larger) than the number of variables $N$.
Considering an instance $\vec{z}_*$ of a solution $\vec{f}(\vec{z}_*) = \vec{0}$, corresponding to an elelement $U^\brick \in {\rm DUBG}(d,d')$ we can estimate a \emph{local dimension} ${\rm dim}(\vec{z}_*)$ of ${\rm DUBG}(d,d')$ as the dimension of the tangent space, i.e. by the rank of the $M\times N$ derivative matrix
$F(\vec{z}_*) = \{\partial f_i (\vec{z}_*)/\partial z_j\}^{i=1\ldots M}_{j=1\ldots N}$,
which is numerically determined by the number of nonvanishing singular values of $F(\vec{z}_*)$:
\begin{equation}
{\rm dim}(\vec{z}_*) = N - {\rm rank}\, F(\vec{z}_*)\,.
\label{dim}
\end{equation}
If ${\rm DUBG}(d,d')$ were a simple manifold the dimension should not depend on the point
$\vec{z}_*\in {\rm DUBG}(d,d')$. This, however, does not seem to be the case when both $d,d'\ge 3$, so different pieces of the set ${\rm DUBG}(d,d')$ may have different topological dimensions.
We made the following numerical experiment. We sampled an ensemble of the order of $10^2 - 10^4$ (depending on values of $d,d'$) random solutions $\vec{z}_{*}$ of $\vec{f}(\vec{z}_*)=\vec{0}$ which were obtained by
running Wolfram's Mathematica routine {\tt FindMinimum} on $|\vec{f}(\vec{z})|^2$ applied to random initial seeds where $z_j$ were i.i.d. Gaussian random with zero mean and variance $1/(dd')$ (reproducing unitarity in the limit $d,d'\to\infty$). We note that random instances of DU gates could also be generated by iteration of a non-linear map proposed in Ref.~\cite{arul_prl2020}.
We have then determined the possible values of
${\rm dim}(\vec{z}_*)$ and collected them in
Table I.
We note that numerical values of $|\vec{f}(\vec{z}_*)|$ were typically between $10^{-13}$ and $10^{-7}$ and there was always a clear cutoff in the singular value spectrum of $F(\vec{z}^*)$, where the `zero' singular values were at least five orders of magnitude smaller than the rest.
For $d=d'=2$ we reproduce the expected analytical result ${\rm dim}=12$. For $d=d'=3$ we obtain three different values ${\rm dim}=45,43,41$ within our data, while for larger $d,d'$ we empirically find only a single dimension (as shown within brackets in Table I). We suspect that for $d,d'>3$ we do not observe other (higher) local dimensions simply because hitting such solutions $\vec{z}_*$ becomes statistically increasingly unlikely. We confirm this speculation by analysing the parametrization that has been proposed by Bal\' azs Pozsgay \cite{balazs_private}.
Specifically, one can write the elements of essentially the largest known subset of ${\rm DUBG}(d,d')$ as
\begin{equation}
U^\brick = S\sum_{s=1}^{d'} u_s \otimes \ket{s}\bra{s},
\label{balazs}
\end{equation}
where $S\ket{j}\otimes\ket{s}=\ket{s}\otimes\ket{j}$.
Clearly, $U^\brick$ is dual-unitary when $u_s$ are arbitrary $d\times d $ unitary matrices. This gives us $d^2 d'$ free parameters, where we can assume $d\ge d'$ without loss of generality. It is striking that empirically found local dimensions in Table I are smaller than $d^3$ for $d=d'\ge 4$. However, determining the dimension (\ref{dim}) for $\vec{z}_*$ parametrizing random DU brick gate of the form (\ref{balazs}) (considering $u_s$ as Haar random) we obtain consistent (non-fluctuating) numbers considerably larger than $d^2 d'$ shown as unbracketed numbers in Table I.
When the smaller of the spaces is a qubit ($d'=2$) we consistenly find that all random solutions of $\vec{f}(\vec{z}_*)=\vec{0}$ give the same dimension (as well as random instances of (\ref{balazs})), so the complexity/topology of the space
${\rm DUBG}(d,d')$ seems fundamentally different when both spaces are non-qubit.
We have also checked that the correlation maps $\mathcal M_\pm$ have full ranks, $d^2$ and ${d'}^2$ respectively, for all randomly generated solution instances $\vec{z}_*$.
\\\\
{\bf\em Conjecture:} Data of Table I suggest a clear conjecture on dimensions of the manifolds of dual unitary brick gates: (i) When one of the spaces is a qubit ($d'=2$) we find a simple quadratic scaling of manifold dimensions
\begin{equation}
{\rm dim}\,{\rm DUBG}(d,2) = 2 (d+1)d\,,
\end{equation}
while (ii) in general we find that, although the local dimensions are fluctuating, the maximal dimension (tangent to (\ref{balazs}) is perfectly fitted by a cubic polynomial in $d,d'$
\begin{eqnarray}
&&{\rm max}\,{\rm dim}\,{\rm DUBG}(d,d') = \\
&&(d^2+{d'}^2)d' +(4d-5d')d' + 6(d'-d)\,,\quad{\rm if}\;\; d\ge d'\,,
\nonumber
\end{eqnarray}
with $d,d'$ swapped if $d\le d'$. For $d=d'$ we have
${\rm max}\,{\rm dim}= (2d-1)d^2$.
\subsection{Dual unitary IRF manifolds}
Completely analogous Mathematica program has been developed for targeting the local dimensions of the manifold ${\rm DUIRF}(d)$ where $\vec{f}(\vec{z})$ now encodes the constraints on dual (space and time) unitarity of the IRF gate, while vector $\vec{z}$ completely encodes the matrices $u_{ik}$, and initial seed variables $z_n$ were taken as Gaussian i.i.d. with zero mean and variance $1/d$. We have found consistently smaller dimensions than for ${\rm DUBG}(d,d)$ (see Table I). We reproduced the correct analytic result ${\rm dim}=10$ for $d=2$, and again, for $d=3$, we have found multiple local dimensions depending on the instance of the solution $\vec{f}(\vec{z}_*)=0$. For $d\ge 4$ the empirical dimensions were again unique, but this might be a statistical effect, like in the case of DU brick gates. Obtaining a systematic (analytic) parametrization (of large subsets) of ${\rm DUIRF}(d)$ for $d\ge 3$ remains an open problem.
We have checked as well that the correlation maps $\mathcal K'_\pm$ have maximal ranks (\ref{maxrank}), $d^2-2(d-1)$, for all randomly generated solution instances $\vec{z}_*$.
\section{Discussion}
Dual unitary circuits, either in brickwork or IRF form, allow for an exact reduction of dynamics of an interacting theory in 1+1 (space-time) dimensions to an open, dissipative (markovian) quantum dynamics of a single particle (qudit) or a pair of particles (qudits). Although this connection has so far been elaborated only for ultra-local (for brickwork circuits) or $2-$local (for IRF circuits)
observables, it is straightforward to generalise it to observables with arbitrary finite range support: the resulting quantum Markov chain process is then defined on a sufficiently large (finite) set of qudits reflecting the range of the observables.
We expect that other recent exact results on dynamics of DU brickwork circuits can be extended to DU IRF circits. Firstly, one may attempt to compute the \emph{spectral form factor} written as\cite{BKP20}
\begin{equation}
K(t) = \mathbb E\left(
\left|{\rm tr}\, \mathcal U^t \right|^2\right)
= \mathbb E\left( {\rm tr}\, \mathcal W^t\right)
\end{equation}
where $\mathbb E$ represents a suitable quench-disorder averaging.
The natural form of disorder is now a random unitary diagonal transformation $\Delta_x={\rm diag}\{ e^{{\rm i}h^{(j)}_x}; j=1,\ldots,d\}$ at each site $x$, after each half-time-step (\ref{eq:UF}), which preserves the DU IRF gate structure, where the fields $h^{(j)}_x$ are i.i.d. random variables (of essentially arbitrary smooth distribution). After a space-time flip we can perform averaging locally (in full analogy to Refs.~\cite{BKP18,BKP20}) and write
\begin{equation}
K(t) = \mathbb E\left( {\rm tr}\, {\mathcal{\tilde{W}}}^L\right) =
{\rm tr} \left(\mathbb E \mathcal{\tilde{W}}\right)^L,\,
\end{equation}
where $\tilde{\mathcal W}$ denotes the folded circuit propagator in space direction, composed as Eqs. (\ref{eq:UF},\ref{eq:IRFC}) with $U^\IRF$ replaced by $\tilde{W}^\IRF$ and $L$ replaced by $t$ (which depends on site-independent random fields $h^{(j)}_x$ at fixed $x$).
Averaging should then again result\cite{BKP18,BKP20} in additional non-expanding piece $\mathcal O$ to the transfer matrix
$\mathcal T =
\mathbb E \tilde{\mathcal{W}} = \mathcal O \tilde{\mathcal{W}}.
$
Translational invariance in time (together with the structure of the generic gate $U^{\rm IRF}$ and absence of time-reversal symmetry) should then result in $\mathcal T$ having exactly $t$-dimensional eigenspace of eigenvalue $1$ and the rest of the spectrum gapped within the unit disk, resulting in exact RMT expression in the thermodynamic limit $\lim_{L\to\infty} K(t) = t$.
Similarly, we expect that the explicit results on solvable initial states in quenched DU circuits \cite{Lorenzo}, as well as on maximal growth rate of entanglement\cite{PRX19}, operator spreading \cite{bertini2020operatorI,bertini2020operatorII,Reid}, and tripartite information \cite{BrunoLorenzo}, should have their close analogs for DU IRF circuits.
Lastly, it is an interesting open question if one can find mappings between brickwork and IRF circuit models on abstract level, similarly as the class of integrable IRF models could be understood in terms of a subalgebra of integrable vertex models\cite{pasquier}.
\begin{acknowledgments}
The author warmly acknowledges Bruno Bertini and Pavel Kos for very fruitful collaboration on closely related topics, and to Bal\' azs Pozsgay and Vedika Khemani for inspiring discussions.
In particular, Bal\' azs Pozsgay is acknowledged for providing the
parametrization (\ref{balazs}).
This work has been supported by the European Research Council (ERC) under the
Advanced Grant No.\ 694544 -- OMNES, and by the Slovenian Research Agency (ARRS)
under the Program P1-0402.
\end{acknowledgments}
\bibliography{bibliography}
\end{document} | 140,587 |
Shirley Collins
~ Person
Wikipedia
Sh.
Discography
Showing official release groups by this artist. Show all release groups instead, or show various artists release groups. | 249,769 |
The coronavirus crisis has catalyzed a drastic surge in homeschooling as parents’ concerns grow about public education.
While homeschooling numbers are slightly lower than last year’s record high, they are still significantly higher than pre-COVID-19 levels. During the 2020-2021 school year, the number of homeschooling students in the U.S. increased by 63%. Even though schools have re-opened, the majority of families that opted for homeschooling over remote learning have continued with it.
The Wisconsin Policy Forum reported historic drops in enrollment for public and private schools in Wisconsin. Nearly 31,900 students were homeschooled during the 2020-21 school year, which is a 47% increase from the previous year.
Private school enrollment took less of a hit than public school enrollment, but still saw the greatest decrease since 2013.
Virtual charter schools also attracted new Wisconsin families, as enrollment increased by 84% from last year.
In Milwaukee, public school enrollment decreased by 4.2% and 151 students switched to homeschooling.
Parents have attributed their decision to homeschool to philosophical differences with their schools, Covid restrictions, and concerns about the quality of education. Generally, children arebetter off learning vital skills and values from their parents than being propagandized from activist teachers.
With so much corruption happening in public schools, homeschooling acts as a beacon of hope. Children benefit greatly, especially mentally and spiritually, when they are primarily guided by their parents rather than the anti-Christian culture that is overrunning schools.
Thankfully, Wisconsin is leading the country in school choice, and it offers homeschooling parents plenty of freedom to direct their children’s education. The homeschool law was passed in 1983 and hasn’t been changed since then, making it one of the best homeschool laws in the country.
The Academy of Excellence Online (AoE) provides an excellent homeschooling opportunity for families in Wisconsin – it is a virtual program available for Wisconsin parents interested in homeschooling and also is part of the Wisconsin Parental Choice Program (“the voucher program”).. The deadline for applying is April 21st! AoE also has brick-and-mortar Christian schools in Milwaukee that also participate in the Milwaukee Parental Choice Program.
Homeschooling is an excellent option for parents who want to protect their children from radical indoctrination. It provides parents the opportunity to train their children for the ideological battle being waged against them. Children need to learn how to recognize and combat the lies they are being fed, meaning all parents must engage in some form of education at home. The very best defense for children is parents being on offense!
This transition away from public school is a necessary and exciting change. Let’s keep the momentum going in Wisconsin and continue to vouch for homeschooling and school choice in order to protect children and parents’ rights. Hopefully next legislative session we can expand educational freedom in Wisconsin. | 268,425 |
labelling to gold.
“Green jewellery jewellery.
The.
Cred’s wedding or engagement rings typically cost about 10 percent more than average prices but are about 15 percent below the top luxury brands such as Tiffany.
A bespoke Cred 18-carat gold wedding ring might cost from 195 to 800 pounds ($390 to $1,600) depending on its size and design — the cheapest engagement ring one could expect to buy in a standard British store would cost about 80 pounds ($160).
| 404,926 |
TITLE: Details in a proof of coarea formula in $\mathbb{R}^n$.
QUESTION [10 upvotes]: Let $\Omega\subseteq\mathbb{R}^n$ be an open set, and let $u:\bar{\Omega}\rightarrow\mathbb{R}$ be a function of class $C^1(\bar{\Omega})$. Given $\lambda\in\mathbb{R}$, let $\Gamma_\lambda=\{x\in\Omega:\,u(x)=\lambda\}$.
Coarea formula: Suppose $|\nabla u|>0$ on $\bar{\Omega}$, and let $f\in L^1(\Omega)$. Then $$\int_\Omega f\,dx=\int_\mathbb{R}\int_{\Gamma_\lambda} \frac{f}{|\nabla u|}\,d\sigma\,d\lambda.$$
I present a proof that I think would work, and I want to ask about a (very important) step inside the proof. I know that coarea formula can be proved in a more general setting, but I am interested on solving this particular proof.
For each $p\in\Omega$, there is $i\in\{1,\ldots,n\}$ with $|u_{x_i}(p)|>0$. By continuity, there exists $r_p>0$ such that for all $x\in B(p,r_p)\subseteq\Omega$ we have $|u_{x_i}(x)|>0$. As $\Omega$ is Lindelöf, $\Omega=\cup_{j=1}^{\infty} B_j$, where $B_j$ is some of such balls. Partition of the unity: there exists $\psi_j\in C_c^{\infty}(B_j)$ with $0\leq\psi_j\leq 1$ and $\sum_{j=1}^{\infty}\psi_j(x)=1$ for all $x\in\Omega$. Define $f_j=f\,\psi_j$ on $\Omega$. From now on, fix one of the balls $B=B_j$ and for simplicity of notation assume that $|u_{x_n}|>0$ on $B$. By the implicit function theorem, $\Gamma_\lambda\cap B=\{x\in B:\,u(x)=\lambda\}=\{(x',\varphi(x',\lambda)):\,x'\in\tilde{B}\}$, where $\tilde{B}$ open in $\mathbb{R}^{n-1}$ and $\varphi(\cdot,\lambda)\in C^1(\tilde{B})$.
Question: can I assume that $\tilde{B}$ is chosen independently of $\lambda$? Do we have $\varphi(x',\cdot)\in C^1$?
On the one hand, $$\int_\mathbb{R}\int_{\Gamma_\lambda}f_j\,d\sigma\,d\lambda=\int_\mathbb{R}\int_{\Gamma_\lambda\cap B}f_j\,d\sigma\,d\lambda=\int_\mathbb{R}\int_{\tilde{B}}f_j(x',\varphi(x',\lambda))\sqrt{1+|\nabla_{x'}\varphi(x',\lambda)|^2}\,dx'\,d\lambda.$$ On the other hand, if the question has a positive answer,
\begin{equation}
\begin{split}
\int_\Omega f_j\,|\nabla u|\,dx &=\int_B f_j\,|\nabla u|\,dx \\
&\stackrel{\substack{\text{change}\\\text{variable}}}{=}\int_\mathbb{R}\int_{\tilde{B}}f_j(x',\varphi(x',\lambda))|\nabla u(x',\varphi(x',\lambda))|\,|\varphi_\lambda (x',\lambda)|\,dx'\,d\lambda.
\end{split}
\end{equation}
Finally use $u(x',\varphi(x',\lambda))=\lambda$ for all $x'\in \tilde{B}$, and derivating with respect to $x'$ and $\lambda$ (if the question has a positive answer), one arrives at the equality between the last two big expressions.
If the answer to the question were negative, are there any alternatives to make this proof work? I don't know, something like changing the balls by rectangles...
REPLY [4 votes]: No, you cannot assume $\tilde B$ to be independent of $\lambda$.
Consider the example $u(x)=x_n$ and $B=B^n(0,1)\subset\mathbb R^n$.
Now $\Gamma_\lambda=\{\lambda\}\times B^{n-1}(0,\sqrt{1-\lambda^2})$.
Your set $\tilde B$ is now the ball $B^{n-1}(0,\sqrt{1-\lambda^2})$, and its radius shrinks to zero as you approach the boundary of $B$ (when $\lambda\to\pm1$).
Here is a possible alternative approach.
Take any point $x\in\Omega$.
Since $\nabla u(x)\neq0$, there is $i\in\{1,\dots,n\}$ so that $u_i(x):=\partial u(x)/\partial x_i\neq0$.
By continuity there is a ball $B^n(x,\epsilon)$ so that $u_i\neq0$ in $B^n(x,\epsilon)$ and $\bar B^n(x,\epsilon)\subset\Omega$.
For notational simplicity assume that $i=n$ and $u_n(x)>0$.
Denote $x'=(x_1,\dots,x_{n-1})\in\mathbb R^{n-1}$.
Suppose $\delta<\epsilon$; we will fix the value later.
Consider the tube $T=B^n(x,\epsilon)\cap (B^{n-1}(x',\delta)\times\mathbb R)$.
The tube has two "caps": $C_\pm=\{x\in\partial B^n(x,\epsilon)\cap\partial T;\pm x_n>0\}$.
Since $u_n>0$, the function is supposed to be bigger on $C_+$ than $C_-$, but this is only true if $\delta$ is small enough.
We need an auxiliary claim:
For sufficiently small $\delta>0$ we have $\inf_{C_+}u>u(x)>\sup_{C_-}u$.
Proof: In the limit $\delta\to0$ the infimum on $C_+$ goes to $u(x+\epsilon e_n)$ and the supremum on $C_-$ to $u(x-\epsilon e_n)$ by continuity, where $e_n=(0,\dots,0,1)$.
Since $u_n>0$, we have $u(x+\epsilon e_n)>u(x)>u(x-\epsilon e_n)$.
Take any such $\delta>0$.
Denote $a_+:=\inf_{C_+}u$ and $a_-:=\sup_{C_-}u$.
Let $U=\{x\in T;a_-<u(x)<a_+\}$.
This $U$ is now a "convenient neighborhood of $x$".
The functions given by the implicit function theorem are1 all defined over $\tilde B:=B^{n-1}(x',\delta)$.
The mental image is that $U$ is foliated by the level sets of $u$, and they all have the same projection $\tilde B$.
In the definition of $U$ we cut the tube $T$ so that all level sets are "full" (their projections are the whole $\tilde B$).
But if $\delta$ is too large, this cut might make $U$ empty.
It would be unfortunate if $x\notin U$.
Every point $x\in\Omega$ has a neighborhood like this, and the rest of the proof works from here.
1
Elaboration: As you wrote in your question, it follows from the implicit function theorem that the level sets of $u$ can be written as graphs in the set $B=B^n(x,\epsilon)$. The problem was that the functions (whose graphs we have) were defined over different sets in $\mathbb R^{n-1}$. For any $x\in U$, consider the level set $L=U\cap u^{-1}(u(x))$. We want to show that the projection of $U$ is $\tilde B$. Now $a_-<u(x)<a_+$. Take any $y'\in\tilde B$ and consider the function $h(t)=u(y',t)$. The function $h$ is strictly increasing. The point $(y',t)$ is in $\bar T$ when $|y'-x'|^2+(t-x_n)^2\leq\epsilon^2$ or $t\in[x_n-\sqrt{\epsilon^2-|y'-x'|^2},x_n+\sqrt{\epsilon^2-|y'-x'|^2}]=:[b_-,b_+]$. By definition $h(b_+)\geq a_+>a_-\geq h(b_-)$. Shrinking $T$ to $U$ corresponds to shrinking $[b_-,b_+]$ to $[c_-,c_+]$, where $h(c_\pm)=a_\pm$. Since $h|_{[c_-,c_+]}$ is strictly increasing and continuous, there is exactly one $t\in(c_-,c_+)$ so that $h(t)=u(x)$ (recall that $a_-<u(x)<a_+$). This means that the point $(y',t)\in U$ projects down to $y'$ and $u$ has the value $u(x)$ on it. | 37,802 |
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One of Super Bowl XLI's most controversial advertisements -- "Life Comes at You Fast," starring Kevin Federline, courtesy of Nationwide Insurance -- conveyed a compelling message for executives, board members, contributors and consumers of not-for-profit services and programming.
Federline, also known as Britney Spears' ex-flame, demonstrated through his riches-to-rags video the way life changes, often in a dramatic and unexpected way.
For years, the not-for-profit sector has gotten by simply by getting by. It was enough for not-for-profits to communicate through heart-rending dramatizations of children suffering the pangs of hunger, the tragedies of homelessness or an elderly person sitting alone in a room.
But the expressive paradigm of need, no matter how worthy, was no longer sufficient once the winds of change swept across the not-for-profit landscape. The Sarbanes-Oxley Act of 2002 demanded increased scrutiny, the result of corporate and accounting scandals.
All the platitudes about doing good, being a good neighbor and rescuing those in distress notwithstanding, this business of health and human services is just that, a business. As such, it demands efficiency, effectiveness and an entrepreneurial zeal. While the stories are compelling, charity's most powerful allies are the business practices that keep the for-profit economy moving.
Each individual who comes into contact with an organization deserves, expects and demands support and services provided at the highest levels one can possibly muster.
As in the corporate sector, employees play a critical role. Not only are they providing hands-on, necessary service to clients, they are ambassadors, telling others about a venture's worth.
Terry Allen Perl is the president and CEO of Chimes International, an organization serving individuals with disabilities in Maryland. He can be reached at tperl@chimes. | 387,258 |
ST3288 : Hafal office, 47 Duckpool Road, Newport
taken 1 year ago, near to Newport/Casnewydd, Great Britain
Hafal office, 47 Duckpool Road, Newport
On the corner of Bristol Street. Hafal Newport helps people
with serious mental illness and their families to work towards recovery.
with serious mental illness and their families to work towards recovery.
TIP: Click the map for Large scale mapping
Change to interactive Map >
Change to interactive Map >
- Grid Square
- ST3288, 57513 88<< | 81,631 |
Potato Chips, the sandwich shop closed last summer by owner Steve Arroyo, is now reopen on Stanley Ave. According to Daily Candy, Arroyo shuttered the original location of his taco shop, Escuela, for the reopening of Chips, which had itself been closed in 2012 to become Otra Escuela, an expanded sister restaurant to the taqueria. The deli has a sunny new coastal cape look, washed in white slats and bearing art of the Pacific surf and California cliffs. The sandwich menu includes roast beef with aged cheddar, eggplant parms, grinders with roast turkey, and meatball subs on locally baked bread, complimented by the chip selection, which features old school crisps like Utz and Zapp's.
The news may also bode well for anyone still hoping for some sort of return of Arroyo's Matadors & Cobras, which shuttered its first and final location in the winter of 2012.
Potato Chips, 308 N. Stanley Ave. Los Angeles; 323-939-8226.
Potato Chips Deli Reopens on Beverly [DC] | 2,873 |
Hi David, I have few comments about the modification you did to the original code. On 1/25/2011 11:01 PM, Lambert, David wrote: > From: Benoit Cousson<b-cousson at ti.com> > > Adds HWMOD entries for the OMAP DMIC driver and creates > a platform device. The HWMOD entires define the system The changelog does not really reflect what the patch is doing. You are not creating a platform device in that patch. > resource requirements for the drvier such as DMA addresses, channels, > and IRQ's. Placing this information in the HWMOD database allows > for more generic drivers to be written and having the specific > implementation details defined in HWMOD. > We already discussed that, but my S-O-B is missing. > Signed-off-by: David Lambert<dlambert at ti.com> > --- > arch/arm/mach-omap2/omap_hwmod_44xx_data.c | 91 ++++++++++++++++++++++++++++ > 1 files changed, 91 insertions(+), 0 deletions(-) > > diff --git a/arch/arm/mach-omap2/omap_hwmod_44xx_data.c b/arch/arm/mach-omap2/omap_hwmod_44xx_data.c > index 7274db4..f9b2ad3 100644 > --- a/arch/arm/mach-omap2/omap_hwmod_44xx_data.c > +++ b/arch/arm/mach-omap2/omap_hwmod_44xx_data.c > @@ -383,6 +383,95 @@ static struct omap_hwmod omap44xx_l4_wkup_hwmod = { > }; > > /* > + * 'dmic' class > + * digital microphone controller > + */ > + > +static struct omap_hwmod_class_sysconfig omap44xx_dmic_sysc = { > + .rev_offs = 0x0000, > + .sysc_offs = 0x0010, > + .sysc_flags = (SYSC_HAS_EMUFREE | SYSC_HAS_RESET_STATUS | > + SYSC_HAS_SIDLEMODE | SYSC_HAS_SOFTRESET), > + .idlemodes = (SIDLE_FORCE | SIDLE_NO | SIDLE_SMART), SIDLE_SMART_WKUP flag is missing. > + .sysc_fields =&omap_hwmod_sysc_type2, > +}; > + > +static struct omap_hwmod_class omap44xx_dmic_hwmod_class = { > + .name = "omap-dmic", This is not the right name. Please stick to the original one from the HW spec. You can rename it the way you want in the device only. > + .sysc =&omap44xx_dmic_sysc, > +}; > + > +/* dmic */ > +static struct omap_hwmod omap44xx_dmic_hwmod; > +static struct omap_hwmod_irq_info omap44xx_dmic_irqs[] = { > + { .irq = 114 + OMAP44XX_IRQ_GIC_START }, > +}; > + > +static struct omap_hwmod_dma_info omap44xx_dmic_sdma_reqs[] = { > + { .dma_req = 66 + OMAP44XX_DMA_REQ_START }, > +}; > + > +static struct omap_hwmod_addr_space omap44xx_dmic_addrs[] = { > + { it might not be useful for your driver, but we should use a name to differentiate the dual memory mapping here. It was introduce by Kishon in his McBSP series. > + .pa_start = 0x4012e000, > + .pa_end = 0x4012e07f, > + .flags = ADDR_TYPE_RT > + }, > +}; > + > +/* l4_abe -> dmic */ > +static struct omap_hwmod_ocp_if omap44xx_l4_abe__dmic = { > + .master =&omap44xx_l4_abe_hwmod, > + .slave =&omap44xx_dmic_hwmod, > + .clk = "ocp_abe_iclk", > + .addr = omap44xx_dmic_addrs, > + .addr_cnt = ARRAY_SIZE(omap44xx_dmic_addrs), > + .user = OCP_USER_MPU, > +}; > + > +static struct omap_hwmod_addr_space omap44xx_dmic_dma_addrs[] = { > + { > + .pa_start = 0x4902e000, > + .pa_end = 0x4902e07f, > + .flags = ADDR_TYPE_RT > + }, > +}; > + > +/* l4_abe -> dmic (dma) */ > +static struct omap_hwmod_ocp_if omap44xx_l4_abe__dmic_dma = { > + .master =&omap44xx_l4_abe_hwmod, > + .slave =&omap44xx_dmic_hwmod, > + .clk = "ocp_abe_iclk", > + .addr = omap44xx_dmic_dma_addrs, > + .addr_cnt = ARRAY_SIZE(omap44xx_dmic_dma_addrs), > + .user = OCP_USER_SDMA, > +}; > + > +/* dmic slave ports */ > +static struct omap_hwmod_ocp_if *omap44xx_dmic_slaves[] = { > + &omap44xx_l4_abe__dmic, > + &omap44xx_l4_abe__dmic_dma, > +}; > + > +static struct omap_hwmod omap44xx_dmic_hwmod = { > + .name = "omap-dmic", > + .class =&omap44xx_dmic_hwmod_class, > + .mpu_irqs = omap44xx_dmic_irqs, > + .mpu_irqs_cnt = ARRAY_SIZE(omap44xx_dmic_irqs), > + .sdma_reqs = omap44xx_dmic_sdma_reqs, > + .sdma_reqs_cnt = ARRAY_SIZE(omap44xx_dmic_sdma_reqs), > + .main_clk = "dmic_fck", > + .prcm = { > + .omap4 = { > + .clkctrl_reg = OMAP4430_CM1_ABE_DMIC_CLKCTRL, > + }, > + }, > + .slaves = omap44xx_dmic_slaves, > + .slaves_cnt = ARRAY_SIZE(omap44xx_dmic_slaves), > + .omap_chip = OMAP_CHIP_INIT(CHIP_IS_OMAP4430), > +}; > + > +/* > * 'mpu_bus' class > * instance(s): mpu_private > */ > @@ -826,6 +915,8 @@ static __initdata struct omap_hwmod *omap44xx_hwmods[] = { > &omap44xx_l4_cfg_hwmod, > &omap44xx_l4_per_hwmod, > &omap44xx_l4_wkup_hwmod, > + /* dmic class */ > + &omap44xx_dmic_hwmod, This is not the right place, and a blank line is missing before and after. Please find the fixed version below. This is based on top of the spinlock + mcspi + mcbsp to avoid conflict and to get the hwmod memory name support. Regards, Benoit --- | 401,173 |
TITLE: What is the probability that the first white ball is seen after the 6th draw?
QUESTION [1 upvotes]: An urn contains $3$ white balls and $7$ red balls. Balls are drawn from the urn one by one and without replacement.What is the probability that the first white ball is seen after the $6$th draw?
My analysis:
The probability of picking the first red ball is :7/10
The probability of picking the second red ball is :6/9
The probability of picking the third red ball is :5/8
The probability of picking the fourth red ball is :4/7
The probability of picking the fifth red ball is :3/6
The probability of picking the first red ball is 2/5
And the probability of picking the white ball after all is 1/4
Multiplying all since the draws are independent gives me:1/120 as an answer whereas the true answer must be: 1/30
REPLY [0 votes]: Let us think of a different situation or Problem. A Plane can be hit by an anti-aircraft gun. The Probability of hitting the plane at the 1st,2nd,3rd,4th,5th, and 6th shots are $[{{P_{1}}}],[{{P_{2}}}][{{P_{3}}}],[{{P_{4}}}],[{{P_{5}}}],[{{P_{6}}}]$ respectively. Then The Probability that the plane is hit in None of the Shots is given by : $[1 -{{P_{1}}}][1 -{{P_{2}}}][1 -{{P_{3}}}][1 -{{P_{4}}}][1 -{{P_{5}}}][1 -{{P_{6}}}]$ equation.
Similarly, the probability that the first white ball is seen after the 6th draw means that All these 6 draws are devoid of any White Balls. Therefore we can use the same equation here also as the situation is the same but the words are different. Hence, The Required Probability that the first white ball is seen after the 6th draw = $[1 -{{P_{1}}}][1 -{{P_{2}}}][1 -{{P_{3}}}][1 -{{P_{4}}}][1 -{{P_{5}}}][1 -{{P_{6}}}]$ = $\frac{1}{30}$ . Here, The Probability of getting a White ball at the 1st,2nd,3rd,4th,5th, and 6th shots are denoted by $[{{P_{1}}}],[{{P_{2}}}][{{P_{3}}}],[{{P_{4}}}],[{{P_{5}}}],[{{P_{6}}}]$ respectively. | 43,469 |
TITLE: What is the negation of each of the following statement?
QUESTION [0 upvotes]: Points R,S and T are collinear and T is not a point on the centered at R with radius RS
Negation:
Points R,S and T are not collinear and T is a point on the center at R with radius RS
is this correct?
Given a line l and a point P that is not on l, there is exactly one line through P that is parallel to l
Negation:
Given a line l and a point p that is on l, the is at least one line through P that is not parallel to l
is this correct? Thank you
REPLY [1 votes]: Re (1), NO. The negation of a proposition of the form $A$ and $B$ is either not-$A$ or not-$B$ (and not not-$A$ and not-$B$).
Re (2), NO. For a start the negation of there is exactly one F is there are either no F's or more than one F.
There are interesting issues lurking in the background in the second case about how to construe the Given ... construction, but we can probably ignore them here.
REPLY [0 votes]: Not correct, to negative and statement negate both sides and use or instead of maintaining and.
($A$ and $B)^{n}$ = ($A^{n}$ or $B^{n}$)
($A$ or $B)^{n}$ = ($A^{n}$ and $B^{n}$) | 173,579 |
TITLE: Is $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$ differentiable?
QUESTION [2 upvotes]: Let $f : AC[0, 1] \to R$ be defined by $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}$, and $F: R^n \to R$ is continuously differentiable function. My question is that, Is $f$ Fréchet differentiable? If not is it at least Gateaux differentiable?
The motivation of this question comes from the question https://math.stackexchange.com/q/1517707/219176 . Note that in my question no growth condition is assumed. However I believe since $[0,1]$ is compact and $F$ is locally Lipschitz, this growth condition is automatically satisfied.
REPLY [3 votes]: From the context, here $AC[0, 1]$ should be the set of all absolutely continuous functions from $[0,1]$ to $\mathbb R^n$ with the norm $W^{1,1}$.
Note that for all $x=(x_1,\dots,x_n)\in AC[0, 1]$ we have $\|x\|_\infty:=\sup\{|x_i(t)|\colon i=1,\dots,n,0\le t\le 1\}\le2\|x\|_{1,1}$; see the lemma at the end of this answer for details. So, for any $x$ and $h$ in $AC[0, 1]$ and any $s\in[0,1]$ we have $\|x+sh\|_\infty\le2\|x\|_{1,1}+2\|h\|_{1,1}$. Because $F$ is continuously differentiable, its derivative $F'$ is uniformly continuous on any bounded set in $\mathbb R^n$. So, for any given $x\in AC[0, 1]$ we have $F'(x(t)+sh(t))-F'(x(t))\to0$ uniformly in $s,t$ in $[0,1]$ as $\|h\|_{1,1}\to0$ (and hence $\|h\|_\infty\to0$). So,
\begin{multline}
f(x+h)-f(x)-\int_0^1 dt\, F'(x(t))\cdot h(t) \\
=\int_0^1 dt\,\int_0^1 ds\,[F'(x(t)+sh(t))-F'(x(t))]\cdot h(t) \\
=o(\|h\|_\infty)=o(\|h\|_{1,1}),
\end{multline}
where $\cdot$ denotes the dot product.
Thus, $f$ is indeed Fréchet-differentiable, with $f'(x)(h)=\int_0^1 dt\, F'(x(t))\cdot h(t)$.
Added:
Lemma: If $\|x\|_{1,1}\le1$ then $\|x\|_\infty\le2$.
Proof. It is enough to prove this lemma when the dimension $n$ is $1$. Let then $m$ and $M$ denote the minimum and maximum of $x$, respectively. Suppose that $\|x\|_\infty>2$. Then without loss of generality $M>2$. On the other hand, the condition $\|x\|_{1,1}\le1$ implies $\|x'\|_1\le1$ and hence $M-m\le1$. So, $m>1$ and hence $1<\|x\|_1\le\|x\|_{1,1}\le1$. This contradiction completes the proof. $\Box$ | 47,876 |
delicious food. Sometimes it was carne con chile, sopes, or when you got really lucky it was a hot pot of espinazo. So what is it? Well its basically a Mexican stew with pork, vegetables, and a delicious broth. Growing up it was rare that my mom made this stew because living in a small town of Crete NE. The Mexican grocery store options were limited. Normally only Mexican stores have this cut of meat. The piece is basically the spine of the pig, and then they cut it up in pieces that are about 2 inches thick. You could use another cut of meat, but it wouldn’t taste the same.
Every time you have this stew it warms you right up. Its healthy and its so flavorful. The trick to making this stew is making you sure you add plenty of bay leaves and the best cut of meat at your local butcher.
Sometimes you get some pieces that don’t have a lot of meat, but the best is when it has a lot of meat on the bone. Another important thing is to sauté the guajillo chiles with the tomato and other spices. That will give the broth the most flavor and a smokiness to it that will make you want to make it again.
Having stew is great by itself, but we always tend to add some additional side items to compliment the stew. For example, we always heat up tortillas in order to wrap them up and eat them with the stew or taking the meat off the bone to make tacos. Usually, my mom cuts up some onion, avocado, limes, and hot peppers to add to the tacos or your stew. Its simple things but it really turns up the dish into something remarkable. Any time that I know that I am going to Crete I drop by my favorite Mexican store in Omaha and get three pounds of espinaso so that my mom and I can make it. It helps us spend some one on one time together since its rare that I get that with her or my dad as we’re both very busy.
If you want to challenge yourself and you’re feeling an authentic Mexican stew get yourself some espinazo and warm your tummy up by making this wonderful dish. I promise you that you won’t regret it and that you might instantly love this new cut of meat. I wonder if you could make some other yummy dishes using it… but that might have to be another time on a different blog post. Hope you enjoyed reading this post! Have a happy Sunday! | 309,015 |
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tag:blogger.com,1999:blog-8470604276410220159.post8158605488866611673..comments2015-09-09T09:41:44.134-04:00Comments on The Vault of Horror: Monster Cereals: Eating What Scares UsB-Sol for the praise, Carol--and for quoting my p...Thanks for the praise, Carol--and for quoting my piece, Gene!<br />Qosmiq, your comment was quite thought-provoking, and I think we are on the same page with a lot of this. I look forward to Freddy-O's with milk in the future...B-Sol the way my kids are attracted to monsters...Watching the way my kids are attracted to monsters, but always on a level that THEY judged acceptable, I'd say that that there is probably a lot of empowerement in eating Frankenberry. <br /.<br /!<br />One of the big plus, the Frankenstein monster and Dracula have over the likes of the 70ies and up horror beings is that they had already been in the public domain for quite a while when they started their movie career though.qosmiq, thoughtful piece. thanks!nice, thoughtful piece. thanks!carol interesting aspect to me is that many of the.<br /><br />I liked your essay enough to quote in the 2nd part of a 2-part essay.<br /><br /> Phillips interview with the actual folks who unleashed F...An interview with the actual folks who unleashed Frankenberry and Count Chocula on our culture?? I think you know what my answer will be! :-)B-Sol read as always B..perhaps you'd like me ...Great read as always B..perhaps you'd like me to get you in touch with the actual family who founded General Mills? See what their take is...you know my life is like 3 degrees of separation--i know someone who knows someone...Captain monsters in milk REALLY kills them. Silve...Drowning monsters in milk REALLY kills them. Silver? Stakes? Dissection? Bah.<br />.<br /><br />But when those elements are completely forgotten, then perhaps they will re-appear...<br /><br />My contribution to this LOTTD round table is here:<br /> the drunken severed head Kruger cereal?! Now that would be ironicall...Freddy Kruger cereal?! Now that would be ironically cool. Couldn't be much more bizarre than the commercialism that followed the film back in its heyday... I mean, a pull-string Freddy doll? Really?Strange Kid, fascinating insight. I don't think we&...Writer, fascinating insight. I don't think we've totally defanged these monsters, meaning they can be resurrected in their original form if need be. However, we've sanitized to the point where this now requires effort and a certain amount of audience cooperation.<br />Ricki, I think it will never cease to amaze me how much the human psyche is governed by fear.<br /.B-Sol i was Bill Gates i`d offer my entire $100 billi...If i was Bill Gates i`d offer my entire $100 billion dollar fortune to anyone who could bring Heather O`Rourke back to life.jervaise brooke [email protected]:blogger.com,1999:blog-8470604276410220159.post-27910733816046735272011-02-23T18:27:21.681-05:002011-02-23T18:27:21.681-05:00I'm somewhat inclined to say that they no long.....Krista post. It's hilarious what people do...Fantastic post. It's hilarious what people do to make themselves feel in control of the things they fear.Ricki don't know much about the de-fanging of the ...I don't know much about the de-fanging of the monsters or the psychology of whether or not kids will be less afraid of them because they float in milk. <br /><br />But I do know they are a damn good way to start my day!<br /><br />Alice<br /> Sweet question, for me, is: have we really de-fanged...The question, for me, is: have we really de-fanged these monsters? Will generations of children who grew up with Count Chocula never be scared by a vampire again?<br /><br />You might be interested in reading the article on the tension between horror and humour, by Noel Carroll:<br /><br />Using numerous examples, Carroll states that the framing of the monster decides whether or not it is perceived as threatening (and thus scary), drawing on examples such as the appearance of Bela Lugosi in <i>Abbott and Costello Meet Frankenstein</i>. Interestingly, Carroll also draws attention to the opposite, namely the creatures of comedy becoming a menace (most notably, clowns).<br /><br />Perhaps this 'warping' of the monster into something fun and familiar will make the opposite even more uncanny. Anyone up for writing a story on the Yummy Mummy coming back from the dead? ;)WriterME | 318,403 |
\begin{document}
\begin{abstract}
This work is motivated from finding the limit of the applicability of G\"{o}del's first incompleteness theorem ($\sf G1$): can we find a minimal theory in some sense for
which $\sf G1$ holds? The answer of this question depends on our definition of minimality. We first show that the Turing degree structure of r.e.~ theories for which $\sf G1$ holds is as complex as the structure of r.e.~ Turing degrees. Then we examine the interpretation degree structure of r.e.~ theories weaker than the theory $\mathbf{R}$ for which $\sf G1$ holds, and answer some open questions about this structure in the literature.
\end{abstract}
\setcounter{page}{0}
\thispagestyle{empty}
\newpage
\maketitle
\section{Preliminaries}
Robinson Arithmetic $\mathbf{Q}$ and the theory $\mathbf{R}$ are both introduced in \cite{undecidable} by Tarski, Mostowski and Robinson as base axiomatic theories for investigating incompleteness and undecidability.
\begin{definition}[Robinson Arithmetic $\mathbf{Q}$]~\label{def of Q}
Robinson Arithmetic $\mathbf{Q}$ is defined in the language $\{\mathbf{0}, \mathbf{S}, +, \times\}$ with the following axioms:
\begin{description}
\item[$\mathbf{Q}_1$] $\forall x \forall y(\mathbf{S}x=\mathbf{S} y\rightarrow x=y)$;
\item[$\mathbf{Q}_2$] $\forall x(\mathbf{S} x\neq \mathbf{0})$;
\item[$\mathbf{Q}_3$] $\forall x(x\neq \mathbf{0}\rightarrow \exists y (x=\mathbf{S} y))$;
\item[$\mathbf{Q}_4$] $\forall x\forall y(x+ \mathbf{0}=x)$;
\item[$\mathbf{Q}_5$] $\forall x\forall y(x+ \mathbf{S} y=\mathbf{S} (x+y))$;
\item[$\mathbf{Q}_6$] $\forall x(x\times \mathbf{0}=\mathbf{0})$;
\item[$\mathbf{Q}_7$] $\forall x\forall y(x\times \mathbf{S} y=x\times y +x)$.
\end{description}
\end{definition}
\begin{definition}[The theory $\mathbf{R}$]~
Let $\mathbf{R}$ be the theory consisting of schemes $\mathbf{Ax1}-\mathbf{Ax5}$ in the language $\{\mathbf{0}, \mathbf{S}, +, \times, \leq\}$.\footnote{For any $n\in \omega$, we define the term $\overline{n}$ as follows: $\overline{0}=\mathbf{0}$, and $\overline{n+1}=\mathbf{S}\overline{n}$.}
\begin{description}
\item[Ax1] $\overline{m}+\overline{n}=\overline{m+n}$;
\item[Ax2] $\overline{m}\times\overline{n}=\overline{m\times n}$;
\item[Ax3] $\overline{m}\neq\overline{n}$ if $m\neq n$;
\item[Ax4] $\forall x(x\leq \overline{n}\rightarrow x=\overline{0}\vee \cdots \vee x=\overline{n})$;
\item[Ax5] $\forall x(x\leq \overline{n}\vee \overline{n}\leq x)$.
\end{description}
\end{definition}
The theory $\mathbf{R}$ contains all key properties of arithmetic for the proof of G\"{o}del's first incompleteness theorem $(\sf G1)$.
Unlike $\mathbf{Q}$, the theory $\mathbf{R}$ is not finitely axiomatizable.\smallskip
\begin{definition}[Translations and interpretations]~
\begin{itemize}
\item Let $T$ be a theory in a language $L(T)$, and $S$ a theory in a language $L(S)$. In its simplest form,
a \emph{translation} $I$ of language $L(T)$ into language $L(S)$ is specified by the following:
\begin{itemize}
\item an $L(S)$-formula $\delta_I(x)$ denoting the domain of $I$;
\item for each relation symbol $R$ of $L(T)$, as well as the equality relation =, an $L(S)$-formula $R_I$ of the same arity;
\item for each function symbol $F$ of $L(T)$ of arity $k$, an $L(S)$-formula $F_I$ of arity $k + 1$.
\end{itemize}
\item If $\phi$ is an $L(T)$-formula, its $I$-translation $\phi^I$ is an $L(S)$-formula constructed as follows: we rewrite the
formula in an equivalent way so that function symbols only occur in atomic subformulas of the
form $F(\overline{x}) = y$, where $x_i, y$ are variables; then we replace each such atomic formula with $F_I(\overline{x}, y)$,
we replace each atomic formula of the form $R(\overline{x})$ with $R_I(\overline{x})$, and we restrict all quantifiers and
free variables to objects satisfying $\delta_I$. We take care to rename bound variables to avoid variable
capture during the process.
\item A translation $I$ of $L(T)$ into $L(S)$ is an \emph{interpretation} of $T$ in $S$ if $S$ proves the following:
\begin{itemize}
\item for each function symbol $F$ of $L(T)$ of arity $k$, the formula expressing that $F_I$ is total on $\delta_I$:
\[\forall x_0, \cdots \forall x_{k-1} (\delta_I(x_0) \wedge \cdots \wedge \delta_I(x_{k-1}) \rightarrow \exists y (\delta_I(y) \wedge F_I(x_0, \cdots, x_{k-1}, y)));\]
\item the $I$-translations of all axioms of $T$, and axioms of equality.
\end{itemize}
\end{itemize}
\end{definition}
The simplified picture of translations and interpretations above actually describes only \emph{one-dimensional}, \emph{parameter-free}, and \emph{one-piece} translations. For precise
definitions of a
\emph{multi-dimensional interpretation}, an
\emph{interpretation with parameters}, and a
\emph{piece-wise interpretation}, we refer to \cite{Visser16} for more details.
\begin{definition}[Interpretations II]~
\begin{itemize}
\item A theory $T$ is \emph{interpretable} in a theory $S$ if there exists an
interpretation of $T$ in $S$.
\item Given theories $S$ and $T$, let `$S\unlhd T$' denote that $S$ is interpretable in $T$ (or $T$ interprets $S$); let `$S\lhd T$' denote that $T$ interprets $S$ but $S$ does not interpret $T$; we say $S$ and $T$ are
\emph{mutually interpretable}, denoted by $S\equiv_I T$, if $S\unlhd T$ and $T\unlhd S$.
\item We say that
\emph{the theory $S$ is weaker than the theory $T$ w.r.t.~ interpretation} if $S\lhd T$.
\end{itemize}
\end{definition}
The notion of interpretation provides us a method to compare different theories in different languages. If $T$ is interpretable in $S$, then all sentences provable (refutable) in $T$ are mapped, by the interpretation function, to sentences provable (refutable) in $S$.\footnote{If theories $S$ and $T$ are mutually interpretable, then $T$ and $S$ are equally strong w.r.t.~ interpretation.}
The interpretation relation among first order theories ($\unlhd$) is reflexive and transitive. The equivalence
classes of theories, under the equivalence relation $\equiv_I$, are
called the interpretation degrees.\footnote{In this paper, we only consider countable theories. There are $2^{\omega}$
countable theories, and $2^{\omega}$ associated interpretation degrees.}
\smallskip
In this paper, we work with first-order theories with finite signature, and always assume the \emph{arithmetization} of the base theory. Under arithmetization, we equate a set of sentences with the set of G\"{o}del's numbers of sentences in it.
\begin{definition}~
Let $\sqsubseteq$ be a binary relation on r.e. theories.
\begin{enumerate}[(1)]
\item For r.e. theories $S$ and $T$, define that $S\sqsubset T$ iff $S\sqsubseteq T$ and $T\sqsubseteq S$ does not hold.
\item We say $S$ is a \emph{minimal theory w.r.t. the relation $\sqsubseteq$} if there is no theory $T$ such that $T\sqsubset S$.
\item We say $S$ is a \emph{maximal theory w.r.t. the relation $\sqsubseteq$} if there is no theory $T$ such that $S\sqsubset T$.
\end{enumerate}
\end{definition}
\begin{definition}[Folklore]~
\begin{enumerate}[(1)]
\item Given two arithmetic theories $U$ and $V$, \emph{$U\leq_T V$} denotes that the theory $U$ is Turing reducible to the theory $V$, and \emph{$U<_T V$} denotes that $U\leq_T V$ but $V\nleq_T U$.
\item We say that \emph{the theory $S$ is weaker than the theory $V$ w.r.t.~ Turing reducibility} if $S<_T V$.
\item We say a set $A$ separates $B$ and $C$ if $B\subseteq A$ and $A\cap C=\emptyset$.
\item We say $\langle S, T\rangle$ is a recursively inseparable pair if $S$ and $T$ are disjoint r.e.~ subsets of $\omega$, and there is no recursive set $X\subseteq\omega$ such that $X$ separates $S$ and $T$.
\item Let $\langle W_e: e\in\omega\rangle$ be the list of all r.e. sets, and $\langle \varphi_e: e\in\omega\rangle$ be the list of all Turing programs.
\item A theory $T$ is \emph{essentially undecidable} if any recursively axiomatizable consistent extension of $T$ in the same language is undecidable.
\item A theory $T$ is \emph{essentially incomplete} if any recursively axiomatizable consistent extension of $T$ in the same language is incomplete.\footnote{The theory of completeness/incompleteness is closely related to the theory of decidability/undecidability (see \cite{undecidable}).}
\end{enumerate}
\end{definition}
Note that $x\in W_e$ if and only if for some $y$, the $e$-th Turing program with input $x$ yields an output in less than $y$ steps. We assume that such $y$ is unique if it exists.
A lattice can be considered as algebraic structures with a signature consisting of two binary operations $\wedge$ and $\vee$.
\begin{definition}\label{}
The theory of lattice consists of the following axioms:
\begin{description}
\item[Commutative laws] $\forall a\forall b (a\vee b=b\vee a); \forall a\forall b (a\wedge b=b\wedge a)$.
\item[Associative laws] $\forall a\forall b\forall c (a\vee (b\vee c)=(a\vee b)\vee c); \forall a\forall b\forall c (a\wedge (b\wedge c)=(a\wedge b)\wedge c)$.
\item[Absorption laws] $\forall a\forall b (a\vee (a\wedge b)=a)$; $\forall a\forall b (a\wedge (a\vee b)=a)$.
\item[Distributive] $\forall x\forall y\forall z (x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z))$; $\forall x\forall y\forall z (x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z))$.
\end{description}
\end{definition}
The following theorem provides us with a method for proving the essentially undecidability of a theory via interpretation.
\begin{theorem}[Theorem 7, Corollary 2, \cite{undecidable}]\label{interpretable theorem}~
Let $T_1$ and $T_2$ be two consistent theories with finite signature such that $T_2$ is interpretable in $T_1$. If $T_2$ is essentially undecidable, then $T_1$ is essentially undecidable.\footnote{For theories with infinite signature, this theorem does not hold.}
\end{theorem}
\begin{lemma}[The fixed point lemma, Folklore]~\label{diagonal lemma}
Let $T$ be a consistent r.e.~ extension of $\mathbf{Q}$. For any formula $\phi(x)$ with exactly one free variable, there exists a sentence $\theta$ such that $T\vdash\theta\leftrightarrow\phi(\ulcorner\theta\urcorner)$.
\end{lemma}
\section{Introduction}
Let $T$ be a consistent r.e. theory. To generalize $\sf G1$ to weaker theories than $\mathbf{PA}$ w.r.t. interpretation, we introduce the notion ``$\sf G1$ holds for $T$".
\begin{definition}[Cheng, \cite{Cheng20}]\label{}
We say that \emph{$\sf G1$ holds for a r.e.~ theory $T$} if any consistent r.e.~ theory that interprets $T$ is incomplete.
\end{definition}
\begin{proposition}[Cheng, \cite{Cheng20}]
$\sf G1$ holds for $T$ iff $T$ is essentially incomplete iff $T$ is essentially undecidable.
\end{proposition}
It is well known that $\sf G1$ holds for Robinson Arithmetic $\mathbf{Q}$ and the theory $\mathbf{R}$ (see \cite{undecidable}). In fact, $\sf G1$ holds for many theories weaker than $\mathbf{PA}$ w.r.t. interpretation.
In summary, we have the following picture:\footnote{For the definition of these weak theories, we refer to \cite{Cheng20}.}
\begin{itemize}
\item $\mathbf{Q}\lhd I\Sigma_0+\mathbf{exp}\lhd I\Sigma_1\lhd I\Sigma_2\lhd\cdots\lhd I\Sigma_n\lhd\cdots\lhd \mathbf{PA}$, and ${\sf G1}$ holds for them.
\item The theories $\mathbf{Q}, I\Sigma_0, I\Sigma_0+\Omega_{1}, \cdots, I\Sigma_0+\Omega_{n}, \cdots, B\Sigma_1, B\Sigma_1+\Omega_{1}, \cdots$, $B\Sigma_1+\Omega_{n}, \cdots$ are all mutually interpretable, and ${\sf G1}$ holds for them.
\item Theories $\mathbf{PA}^{-}, \mathbf{Q}^{+}, \mathbf{Q}^{-}, \mathbf{TC}, \mathbf{AS},\mathbf{S^1_2}$ and $\mathbf{Q}$ are all mutually interpretable, and ${\sf G1}$ holds for them.
\item $\mathbf{R}\lhd\mathbf{Q}\lhd \mathbf{EA}\lhd \mathbf{PRA}\lhd\mathbf{PA}$, and ${\sf G1}$ holds for them.
\end{itemize}
This paper is motivated from finding the limit of the applicability of $\sf G1$: can we find a minimal theory in some sense for which $\sf G1$ holds? The answer of this question depends on our definition of minimality.
If we define minimality as having the minimal number of axiom, then any finitely axiomatized essentially undecidable theory (e.g., Robinson Arithmetic $\mathbf{Q}$) is a minimal r.e. theory for which $\sf G1$ holds. For theories which is not finitely axiomatized, if we define minimality as having the minimal number of axiom schemes, then the following theory $\mathbf{VS}$ is a minimal r.e.~ theory for which $\sf G1$ holds
since it has only one axiom scheme and is essentially undecidable. The Vaught set theory $\mathbf{VS}$, originally introduced by Vaught \cite{Vaught67}, is axiomatized by the schema
\[(V_n)\qquad \forall x_0, \cdots, \forall x_{n-1} \exists y \forall t (t \in y \leftrightarrow \bigvee_{i<n} t = x_i)\]
for all $n \in\omega$, asserting that $\{x_i: i < n\}$ exists.
When we talk about minimality, we should specify the degree structure involved. In \cite{Cheng20}, we examine the following two degree structures that are respectively induced from Turing reducibility and interpretation: $\langle {\sf \overline{D}}, \leq_T\rangle$ and $\langle {\sf D}, \unlhd\rangle$.
\begin{definition}[Cheng, \cite{Cheng20}]~\label{}
\begin{enumerate}[(1)]
\item Let ${\sf \overline{D}}=\{S: S<_{T} \mathbf{R}$, and $\sf G1$ holds for the r.e.~ theory $S$\}.
\item Let ${\sf D}=\{S: S\lhd \mathbf{R}$ and $\sf G1$ holds for the r.e. theory $S$\}.
\end{enumerate}
\end{definition}
In \cite{Cheng20}, we show that there is no minimal r.e. theory w.r.t. Turing reducibility for which $\sf G1$ holds, and prove some results about the structure of $\langle {\sf D}, \unlhd\rangle$. In this paper, we prove more facts about $\langle {\sf \overline{D}}, \leq_T\rangle$, and answer open questions about the structure $\langle {\sf D}, \unlhd\rangle$ in \cite{Cheng20}. Moreover, we prove in Theorem \ref{V complexity} that the index set of r.e. theories for which $\sf G1$ holds is $\Pi^0_3$-complete. As a corollary, we show that for some degree structures satisfying the conditions in Theorem \ref{general important thm}, there is no minimal r.e. theory for which $\sf G1$ holds.
\section{The structure $\langle {\sf \overline{D}}, \leq_T\rangle$}
In this section, we examine the Turing degree structure of r.e.~ theories below $\mathbf{R}$ for which $\sf G1$ holds.
Hanf shows that there is a finitely axiomatizable theory in each recursively enumerable tt-degree (see \cite{Rogers87}).
Feferman shows in \cite{Feferman57} that if $A$ is any recursively enumerable set, then there is a recursively axiomatizable theory $T$ having the same Turing degree as $A$.
In \cite{Shoenfield 58}, Shoenfield improves Feferman's result and shows that if $A$ is not recursive, then there is an essentially undecidable theory with the same Turing degree. To make readers to have a good sense of Shoenfield's theorem, we give a proof of it which is a reconstruction of Shoenfield's proof in \cite{Shoenfield 58} with more details.
\begin{theorem}[Shoenfield, \cite{Shoenfield 58}]\label{Shoenfield first}
If $A$ is recursively enumerable and not recursive, there is a recursively inseparable pair $\langle B, C\rangle$ such that $A$, $B$ and $C$ have the same Turing degree.
\end{theorem}
\begin{theorem}[Shoenfield, \cite{Shoenfield 58}]\label{Shoenfield second}
Let $A$ be recursively enumerable and not recursive. Then there is a consistent axiomatizable theory $T$ having one non-logical symbol which is essentially undecidable and has the same Turing degree as $A$.
\end{theorem}
\begin{proof}\label{}
Pick a recursively inseparable pair $\langle B, C\rangle$ as in Theorem \ref{Shoenfield first} such that $A$, $B$ and $C$ have the same Turing degree.
The theory $T$ we define has only one non-logical symbol: a binary relation symbol $R$.
Let $\Phi_n$ be the statement that there is an
equivalence class of $R$ consisting of $n$ elements.
The theory $T$ contains the following axioms:
\begin{itemize}
\item axioms asserting
that $R$ is an equivalence relation;
\item $\Phi_n$ for all $n\in B$;
\item $\neg\Phi_n$ for all $n\in C$;
\item for each $n$ we
adopt an axiom asserting there is at most one equivalence class of $R$ having $n$
elements.
\end{itemize}
Note that $T$ is consistent and axiomatizable. Since $\Phi_n$ is
provable iff $n\in B$, and $\neg\Phi_n$ is provable iff $n \in C$, we have
$B$ and $C$ are recursive in $T$.
Disjunctions of conjunctions whose terms are $\Phi_n$ or $\neg \Phi_n$ for some $n\in\omega$, are called a disjunctive normal form of $\langle\Phi_n: n\in\omega\rangle$.
\begin{lemma}[Janiczak, Lemma 2 in \cite{Janiczak}]\label{Janiczak's Lemma}
Any sentence $\phi$ of the theory $T$ is equivalent to a disjunctive normal form of $\langle\Phi_n: n\in\omega\rangle$, and this disjunctive normal form can be found explicitly once $\phi$ is explicitly given.\footnote{This is a reformulation of Janiczak's Lemma 2 in \cite{Janiczak} in the context of the theory $T$. Janiczak's Lemma is proved by means of a method known as the elimination of quantifiers.}
\end{lemma}
By Lemma \ref{Janiczak's Lemma},
every sentence $\phi$ of $T$ is equivalent to a disjunctive normal form of $\langle\Phi_n: n\in\omega\rangle$, and
this disjunctive normal form can be calculated from $\phi$. It follows that $T$ is
recursive in $(B, C)$. Hence, $T$ has the same Turing degree as $A$.
Finally, by a standard argument, we can show that $T$ is essentially undecidable (see \cite{Cheng20}).
\end{proof}
\begin{corollary}
The structure $\langle {\sf \overline{D}}, \leq_T\rangle$ is as complex as the Turing degree structure of r.e.~ sets.
\end{corollary}
However, Shoenfield's theory $T$ in Theorem \ref{Shoenfield second} is not unique as can be seen from the following theorem which improves Theorem \ref{Shoenfield first}.
\begin{theorem}\label{}
For any non-recursive r.e. set $A$, there is a countable sequence of r.e. sets $\langle B_n: n\in\omega\rangle$ such that $(B_n, B_m)$ is a recursively inseparable pair for $m\neq n$ and each $B_n$ has same Turing degree as $A$.
\end{theorem}
\begin{proof}\label{}
Suppose $A$ is a non-recursive r.e. set. We construct a sequence of disjoint r.e. sets $\langle B_n: n\in\omega\rangle$ such that for any $m\neq n$, $(B_n, B_m)$ is a recursively inseparable pair and $B_n\equiv_T B_m \equiv_T A$.
Let $f$ be the recursive function that enumerates $A$ without repetitions. Define a partial function $g$ as follows:
$g(\langle x,y\rangle)=n$ iff $\exists s\exists t (f(s)=x, t<s$ and the program $\varphi_s$ with input $\langle x,y\rangle$ yields $n$ as output in $\leq t$ steps).
By the projection theorem, $g$ is partial recursive. Let $B_n=\{x: g(x)=n\}$. For any $n\neq m$, we show that $B_n$ and $B_m$ have the same Turing degree as $A$ and $(B_n, B_m)$ is a recursively inseparable pair.
Note that $B_n$ and $B_m$ are disjoint r.e. sets.
\begin{claim}
$B_n\leq_T A$.
\end{claim}
\begin{proof}\label{}
We test $\langle x,y\rangle\in B_n$ as follows. If $x\notin A$, then $\langle x,y\rangle\notin B_n$. If $x\in A$, let $s=f^{-1}(x)$. We can decide whether the program $\varphi_y$ with input $\langle x,y\rangle$ yields $n$ in $< s$ steps. If yes, then $\langle x,y\rangle\in B_n$; if no, then $\langle x,y\rangle\notin B_n$.
\end{proof}
\begin{claim}
$A\leq_T B_n$.
\end{claim}
\begin{proof}\label{}
We test $x\in A$ as follows. Suppose the index of the program with constant output value $n$ is $e_0$, i.e. $\varphi_{e_0}(x)=n$ for any $x$. If $\langle x, e_0\rangle\in B_n$, then $x\in A$. Suppose $\langle x, e_0\rangle\notin B_n$, let $w$ be the number of computation steps such that $\varphi_{e_0}(\langle x, e_0\rangle)=n$. Decide whether $f(s)=x$ and $s\leq w$ for some $s$. If yes, then $x\in A$; if no, then $x\notin A$.
\end{proof}
Similarly, we can show that $B_m\leq_T A$ and $A\leq_T B_m$.
\begin{claim}
There is no recursive set $X$ that separates $B_n$ and $B_m$.
\end{claim}
\begin{proof}\label{}
Suppose not. Then we can find a recursive function $h$ such that $ran(h)=\{m,n\}, h(B_n)=m$ and $h(B_m)=n$. Let $h=\varphi_{e_1}$.
\begin{claim}
If $x\in A$, then the program $\varphi_{e_1}$ with input $\langle x, e_1\rangle$ yields output in at least $s$ steps where $s=f^{-1}(x)$.
\end{claim}
\begin{proof}\label{}
Suppose not. Then the program $\varphi_{e_1}$ with input $\langle x, e_1\rangle$ halts in less than $s=f^{-1}(x)$ steps. Then by definition, we have $\varphi_{e_1}(\langle x, e_1\rangle)=n \Leftrightarrow \varphi_{e_1}(\langle x, e_1\rangle)=m$, which leads to a contradiction.
\end{proof}
Let $l$ be the recursive function such that $l(x)=$ the number of steps to compute the value of $\langle x, e_1\rangle$ in the program $\varphi_{e_1}$. Then we have:
\[x\in A\Leftrightarrow\exists s (s\leq l(x)\wedge f(s)=x).\]
Thus, $A$ is recursive which leads to a contradiction.
\end{proof}
Thus, for any $n\neq m$, $(B_n, B_m)$ is a recursively inseparable pair and $B_n\equiv_T B_m \equiv_T A$.
\end{proof}
As a corollary, for any no-recursive r.e.~ degree $a$, there are countably many essentially undecidable r.e.~ theories with the same Turing degree $a$. Since there are only countably many r.e.~ theories, this is the best result we can have.
Now, we list some key results about the Turing degree structure of r.e.~ sets.
\begin{fact}[Folklore, many authors]~\label{key fact in recursion theory}
\begin{itemize}
\item The r.e. degrees are dense: for any r.e. sets $A<_T B$, there is a r.e. set $C$ such that $A <_T C <_T B$.
\item No r.e. degree is minimal.
\item For any r.e. set $\mathbf{0}<_T C <_T \mathbf{0}^{\prime}$, there exists an r.e. set $A$ such that $A$ is incomparable with $C$ w.r.t. Turing degree. Furthermore, an index for $A$ can be found uniformly from one for $C$.
\item Given r.e. sets $A<_T B$, there is an infinite r.e. sequence of r.e. sets $C_n$ such that $A<_T C_n <_T B$ and $C_n$'s are incomparable w.r.t. Turing degree.
\item If $a$ and $b$ are r.e. degrees such that $a<b$, then any countably partially ordered set can be embedded in the r.e. degrees between $a$ and $b$.
\end{itemize}
\end{fact}
\begin{corollary}~
\begin{itemize}
\item $\langle {\sf \overline{D}}, \leq_T\rangle$ is dense: for theories $A, B \in {\sf \overline{D}}$ such that $A<_T B$, there is a theory $C\in {\sf \overline{D}}$ such that $A <_T C <_T B$.
\item For any theory $A\in {\sf \overline{D}}$, there exists a theory $B\in {\sf \overline{D}}$ such that $B$ is incomparable with $A$ under $\leq_T$. As a corollary, for any theory $A\in {\sf \overline{D}}$, there exists a theory $B\in {\sf \overline{D}}$ such that $A<_T B$.
\item $\langle {\sf \overline{D}}, \leq_T\rangle$ has no minimal element, and has no maximal element.
\item Given theories $A, B\in {\sf \overline{D}}$ such that $A<_T B$, there is an infinite r.e. sequence of theories $C_n\in {\sf \overline{D}}$ such that $A<_T C_n <_T B$ and $C_n$'s are incomparable w.r.t. Turing degree.
\item Given theories $A, B\in {\sf \overline{D}}$ such that $A<_T B$, any countably partially ordered set can be embedded in $\langle {\sf \overline{D}}, \leq_T\rangle$ between $A$ and $B$.
\end{itemize}
\end{corollary}
Thus, $\langle {\sf \overline{D}}, \leq_T\rangle$ is a dense distributive lattice without endpoints.
\section{Some interpretation degree structures}
In this section, before we examine the structure of $\langle {\sf D}, \unlhd\rangle$, we first review some interpretation degree structures we know in the literature.
Let $\langle D_\mathbf{PA}, \unlhd\rangle$ denote the interpretation degree structure of consistent r.e. extensions of $\mathbf{PA}$. For consistent r.e. extensions of $\mathbf{PA}$, we have some equivalent characterizations of the notion of interpretation.
\begin{fact}[Folklore, \cite{Per 97}]
Suppose theories $S,T$ are consistent r.e. extensions of $\mathbf{PA}$. Then the following are equivalent:
\begin{enumerate}[(1)]
\item $S\unlhd T$;
\item $T\vdash Con(S\upharpoonright k)$ for any $k\in\omega$;\footnote{$Con(S)$ is the canonical arithmetic formula which expresses the consistency of the theory $S$ saying that $\mathbf{0}\neq \mathbf{0}$ is not provable in $S$.}
\item $(S\upharpoonright k)\unlhd T$ for any $k\in\omega$;
\item For any $\Pi^{0}_1$ sentence $\phi$, if $S\vdash \phi$, then $T\vdash \phi$.
\item For every model $M$ of $T$, there is a model $N$ of $S$ such that $M$ is isomorphic to an initial segment of $N$.
\end{enumerate}
\end{fact}
The structure $\langle D_\mathbf{PA}, \unlhd\rangle$ is well known in the literature. In fact, $\langle D_\mathbf{PA}, \unlhd, \downarrow, \uparrow\rangle$ is a dense distributive lattice.\footnote{For consistent r.e. extensions $A$ and $B$ of $\mathbf{PA}$, $A\downarrow B=\mathbf{PA}+\{Con(A\upharpoonright k)\vee Con(B\upharpoonright k): k\in\omega\}$ and
$A\uparrow B= \mathbf{PA} +\{Con(A\upharpoonright k)\wedge Con(B\upharpoonright k): k\in\omega\}$.} For more properties of $\langle D_\mathbf{PA}, \unlhd\rangle$, we refer to \cite{Per 97}.
Now, we examine the interpretation degree structure of general r.e. theories, not restricting to theories interpreting $\mathbf{PA}$.
\begin{definition}[Folklore]\label{}
We introduce two natural operators on r.e. theories.
\begin{itemize}
\item The supremum $A \otimes B$ is defined as follows: $A \otimes B$ is a theory in
the disjoint sum of the signatures of $A$ and $B$ plus two new
predicates $P_0$ and $P_1$. We have axioms that say that $P_0$ and $P_1$
form a partition of the domain and the axioms of $A$ relativized to
$P_0$ and the axioms of $B$ relativized to $P_1$.
\item
The infimum $A \oplus B$ is defined as follows: $A \oplus B$ is a theory in the
disjoint sum of the signatures of $A$ and $B$ plus a fresh $0$-ary
predicate symbol $P$. The theory is axiomatized by all $P \rightarrow\varphi$,
where $\varphi$ is an axiom of $A$ plus $\neg P\rightarrow\psi$, where $\psi$ is an axiom of $B$.
\end{itemize}
\end{definition}
Note that the interpretation degree structure of r.e. theories with the
operators $\oplus$ and $\otimes$ is a distributive lattice.
The interpretation degree structure of all finitely axiomatized theories, denoted by $\langle D_{finite}, \unlhd\rangle$, is studied by
Harvey Friedman in [Fri07]. Note that there are only $\omega$ many interpretation degrees of finitely axiomatized theories. The structure $\langle D_{finite}, \unlhd\rangle$ forms a (reflexive)
partial ordering with a minimum element $\top$ and a maximum element $\bot$ where $\top$ is the equivalence
class of all sentences with a finite model, and $\bot$ is the equivalence class of all sentences
with no models.
\begin{theorem}[Harvey Friedman, \cite{Friedman07}]~
\begin{enumerate}[(1)]
\item The structure $\langle D_{finite}, \unlhd, \oplus, \otimes, \bot, \top\rangle$ forms a distributive
lattice.
\item For any $a\in D_{finite}$ such that $a \lhd \bot$, there exists $b\in D_{finite}$ such that $a \lhd b
\lhd \bot$.
\item The structure $\langle D_{finite}, \unlhd\rangle$ is dense, i.e., $a \lhd b \rightarrow (\exists c)(a \lhd c
\lhd b)$.
\item For any $a, b\in D_{finite}$, if $a \lhd b$, then there
exists an infinite sequence $c_n$ such that $a\lhd c_n\lhd b$ for each $n$ and $c_n$'s are incomparable w.r.t. interpretation.
\end{enumerate}
\end{theorem}
Thus, $\langle D_{finite}, \unlhd, \oplus, \otimes, \bot, \top\rangle$ is a dense distributive lattice without endpoints.
Especially, there is no minimal finitely axiomatized theory w.r.t. interpretation.
The following theorem shows that above any finitely axiomatizable sub-theory of $\mathbf{PA}$, there are continuum many incomparable sub-theories of $\mathbf{PA}$ w.r.t. interpretation.
\begin{theorem}[Montague, \cite{Montague62}]\label{}
Let $T$ be any finitely axiomatizable subtheory of $\mathbf{PA}$. Then there is a set $C$ of cardinality $2^{\omega}$ such that (i) every member of $C$ is a sub-theory of $\mathbf{PA}$ and an extension of $T$, (ii) any two distinct elements of $C$ are incomparable w.r.t. interpretation.
\end{theorem}
The structure $\langle D_{finite}, \unlhd\rangle$ bears a rough resemblance to the Turing degree structure of r.e. sets which is very well studied from recursion
theory, and is very complicated.
It is not fully clear how complicated $\langle D_{finite}, \unlhd\rangle$
is, and how these two structures are related.
Moreover, there are many similarities between the structure $\langle D_{finite}, \unlhd\rangle$ and the structure $\langle {\sf \overline{D}}, \leq_T\rangle$; an interesting question is: what is the difference between these two structures?
\begin{definition}[\cite{Visser14}]\label{}~
\begin{enumerate}[(1)]
\item An interpretation
is direct when it is un-relativized and identity preserving.
\item A theory is sequential if
it directly interprets the theory Adjunctive Set Theory ($\mathbf{AS}$).\footnote{The theory $\mathbf{AS}$ has only one binary relation symbol `$\in$', and the following axioms: (1) $\exists x \forall y (y \notin x)$; (2) $\forall x\forall y \exists z \forall u (u \in z \leftrightarrow (u = x \vee u = y))$.}
\item We say two sentences have the same derivability degree iff they are provably equivalent over $\mathbf{EA}$.
\end{enumerate}
\end{definition}
Let $\langle D_{Seq}, \unlhd\rangle$ denote the interpretation degree structure of finitely axiomatized sequential
theories.
The following theorem gives us a nice characterization of the structure $\langle D_{Seq}, \unlhd\rangle$.
\begin{theorem}[Visser, \cite{Visser14}]\label{}
The structure $\langle D_{Seq}, \unlhd\rangle$ is recursively equivalent to the degrees of derivability of $\Pi^0_1$-sentences over
$\mathbf{EA}$.
\end{theorem}
\section{The structure $\langle {\sf D}, \unlhd\rangle$}
In this section, we examine the interpretation degree structure of r.e.~ theories for which $\sf G1$ holds.
The interpretation degree structure of r.e.~ theories interpreting $\mathbf{PA}$ is well known.
However, the interpretation degree structure of r.e.~ theories weaker than the theory $\mathbf{R}$ is much more complex and we know few about it.
Now, we examine the interpretation degree structure of r.e.~ theories weaker than $\mathbf{R}$ for which $\sf G1$ holds, and answer open questions about the structure $\langle {\sf D}, \unlhd\rangle$ in \cite{Cheng20}.
\begin{definition}[\cite{Cheng20}]
Let $\langle A, B\rangle$ be a recursively inseparable pair. Consider the following r.e.~ theory $U_{\langle A,B\rangle}$ with the signature $\{\mathbf{0}, \mathbf{S}, \mathbf{P}\}$ where $\mathbf{P}$ is a unary relation symbol, and $\overline{n}=\mathbf{S}^n\mathbf{0}$ for $n\in \mathbb{N}$:
\begin{enumerate}[(1)]
\item $\overline{m}\neq\overline{n}$ if $m\neq n$;
\item $\mathbf{P}(\overline{n})$ if $n\in A$;
\item $\neg \mathbf{P}(\overline{n})$ if $n\in B$.
\end{enumerate}
\end{definition}
\begin{theorem}[Cheng, \cite{Cheng20}]\label{main thm}
For any recursively inseparable pair $\langle A,B\rangle$, $\sf G1$ holds for $U_{\langle A, B\rangle}$ and $U_{\langle A, B\rangle}\lhd\mathbf{R}$.
\end{theorem}
Since there are countably many recursively inseparable pairs, there are countably many elements of ${\sf D}$.
\begin{definition}[Visser]\label{}
We say a r.e.~ theory $U$ is Turing persistent if for any consistent r.e. theory $V$, if $U\subseteq V$, then $U \leq_T V$.
\end{definition}
There is no direct relation between the notion of interpretation and the notion of Turing reducibility. Given r.e. theories $U$ and $V$, $U \unlhd V$ does not imply $U \leq_T V$, and $U \leq_T V$ does not imply $U\unlhd V$.
The notion of ``Turing persistent'' establishes the relationship between $U\unlhd V$ and $U \leq_T V$. Note that if $U$ is Turing persistent, then for any r.e.~ theory $V$, if $U \unlhd V$, then $U \leq_T V$. Many essentially undecidable theories we know are Turing persistent. A natural question is: can we find an essentially undecidable theory which is not Turing persistent?
Now, we give some examples of Turing persistent theories.
\begin{proposition}\label{}
For any consistent r.e. theory $T$, if all recursive functions are representable in $T$, then $T$ is Turing persistent.\footnote{Our notion of representability is standard. We refer to \cite{Murawski99} for definitions. }
\end{proposition}
\begin{proof}\label{}
This follows from the fact that $T$ has Turing degree $\mathbf{0}^{\prime}$ since any r.e. set is representable in $T$.
\end{proof}
As a corollary, $\mathbf{R}$ is Turing persistent.
\begin{theorem}[Essentially due to Shoenfield]\label{Visser's thm}
For any r.e.~ set $A$, there are disjoint r.e.~ sets $B$ and $C$ with $B,C \leq_T A$ such that for any r.e.~ $D$ which separates $B$ and $C$, we have $A\leq_T D$.\footnote{The proof of this theorem is based on Shoenfield's construction in \cite{Shoenfield 58}. Albert Visser also discovered this form in his note on Shoenfield's theorem.}
\end{theorem}
\begin{proof}\label{}
Suppose that $A=W_e$, i.e. $x\in A$ if and only if for some $y$, the $e$-th Turing program with input $x$ yields an output in $< y$ steps.
\begin{itemize}
\item Define $x\in B$ iff for some $y$, the $e$-th Turing program with input $(x)_0$ yields an output in $< y$ steps and for all $z\leq y$, the $(x)_1$-th Turing program with input $x$ does not yield an output in $< z$ steps.
\item
Define $x\in C$ iff for some $y$, the $e$-th Turing program with input $(x)_0$ yields an output in $< y$ steps and for some
$z\leq y$, the $(x)_1$-th Turing program with input $x$ yields an output in $< z$ steps.
\end{itemize}
Note that $B$ and $C$ are disjoint r.e.~ sets. If $(x)_0 \notin A$, then $x\notin B$ and $x \notin C$; if $(x)_0 \in A$, then we can decide either $x\in B$ or $x \in C$. Thus, we have $B,C \leq_T A$.
Suppose $D$ is a r.e. set with index $d$, and $B\subseteq D$ and $D\cap C=\emptyset$. We show that $A\leq_T D$.
\begin{claim}~
$x\in A$ if and only if for some $z$, the $d$-th Turing program with input $\langle x,d\rangle$ yields an output in $< z$ steps and for some $y<z$, the $e$-th Turing program with input $x$ yields an output in $< y$ steps.
\end{claim}
\begin{proof}\label{}
The right-to-left direction is obvious. Suppose $x\in A$. Then either $\langle x,d\rangle\in B$ or $\langle x,d\rangle\in C$. Suppose $\langle x,d\rangle\in C$. Let $y$ be the unique witness such that the $e$-th Turing program with input $x$ yields an output in $< y$ steps. Then there exists $z\leq y$ such that the $d$-th Turing program with input $\langle x,d\rangle$ yields an output in $< z$ steps. Then $\langle x,d\rangle\in D$, which contradicts that $D\cap C=\emptyset$. Thus we have $\langle x,d\rangle\in B$.
Let $y$ be the unique witness such that the $e$-th Turing program with input $x$ yields an output in $< y$ steps. Since $\langle x,d\rangle\in D$, we have for some $z$, the $d$-th Turing program with input $\langle x,d\rangle$ yields an output in $< z$ steps. Since for all $z\leq y$, the $d$-th Turing program with input $\langle x,d\rangle$ does not yield an output in $< z$ steps, we have $z>y$. Thus, the left-to-right direction holds.
\end{proof}
Now we show that $A\leq_T D$. If $\langle x,d\rangle\notin D$, then $x\notin A$. If $\langle x,d\rangle\in D$, from the above claim, we can effectively decide whether $x\in A$.
\end{proof}
From Theorem \ref{Visser's thm}, we can give a simpler proof of Theorem \ref{Shoenfield first}.
\begin{proof}\label{}
Suppose $A$ is a non-recursive r.e. set. Pick the disjoint r.e. sets $\langle B, C\rangle$ as in Theorem \ref{Visser's thm}. We show that $\langle B, C\rangle$ is a recursively inseparable pair, and $B$ and $C$ have the same Turing degree as $A$.
Suppose $D$ is a recursive set which separates $B$ and $C$. By Theorem \ref{Visser's thm}, $A\leq_T D$ and thus $A$ is recursive which leads to a contradiction.
From Theorem \ref{Visser's thm}, we have $B, C \leq_T A$. Since $B$ is a r.e. set which separates $B$ and $C$, by Theorem \ref{Visser's thm}, we have $A\leq_T B$. Similarly, we have $A\leq_T C$. Thus, $A$, $B$ and $C$ have the same Turing degree.
\end{proof}
\begin{theorem}[Visser, Theorem 6, \cite{Visser 14}]\label{visser thm on R}
For any r.e.~ theory $T$ with finite signature, $T$ is locally finitely satisfiable iff $T$ is interpretable in $\mathbf{R}$.\footnote{In fact, if $T$ is locally finitely satisfiable, then $T$ is interpretable in $\mathbf{R}$ via a one-piece one-dimensional parameter-free interpretation.}
\end{theorem}
By Theorem \ref{Shoenfield second}, for any r.e.~ Turing degree $\mathbf{0}<\mathbf{d}< \mathbf{0}^{\prime}$, there is an essentially undecidable theory with Turing degree $\mathbf{d}$. We denote this theory by $T_\mathbf{d}$.
\begin{theorem}\label{key thm 1}
For any r.e.~ Turing degree $\mathbf{0}<\mathbf{d}< \mathbf{0}^{\prime}$, the theory $T_\mathbf{d}$ is Turing persistent and $T_\mathbf{d}\lhd\mathbf{R}$.
\end{theorem}
\begin{proof}\label{}
Suppose $A$ is a r.e. set with Turing degree $\mathbf{d}$, and $T_\mathbf{d}$ is constructed as in Theorem \ref{Shoenfield second} via the recursively inseparable pair $\langle B, C\rangle$ constructed as in Theorem \ref{Visser's thm}.
Suppose $V$ is a consistent r.e. extension of $T_\mathbf{d}$. Define $D=\{n: V\vdash\Phi_n\}$. Note that $B\subseteq D$ and $D\cap C=\emptyset$. By Theorem \ref{Visser's thm}, $A\leq_T D$. Since $T_d$ has the same Turing degree as $A$, we have $T_\mathbf{d}\leq_T V$. Thus, $T_\mathbf{d}$ is Turing persistent.
Since $T_\mathbf{d}$ is locally finitely satisfiable, we have $T_\mathbf{d}\unlhd\mathbf{R}$. If $\mathbf{R}\unlhd T_\mathbf{d}$, since $\mathbf{R}$ is Turing persistent, then $\mathbf{R}\leq_T T_\mathbf{d}$ which leads to a contradiction. Thus $T_\mathbf{d}\lhd\mathbf{R}$.
\end{proof}
\begin{theorem}\label{my thm}
For any non-recursive r.e.~ set $A$, we can uniformly find a recursively inseparable pair $\langle B,C\rangle$ such that:
\begin{enumerate}[(1)]
\item $\sf G1$ holds for $U_{\langle B, C\rangle}$;
\item $U_{\langle B, C\rangle}\lhd\mathbf{R}$;
\item $U_{\langle B, C\rangle}$ has the same Turing degree as $A$;
\item $U_{\langle B, C\rangle}$ is Turing persistent.
\end{enumerate}
\end{theorem}
\begin{proof}\label{}
Take the pair of r.e. sets $\langle B,C\rangle$ as in Theorem \ref{Visser's thm} such that $A, B, C$ have the same Turing degree. Since $A$ is not recursive, $\langle B,C\rangle$ is a recursively inseparable pair. From Theorem \ref{Visser's thm}, we can uniformly find such a recursively inseparable pair $\langle B,C\rangle$ from a non-recursive r.e.~ set $A$.
By Theorem \ref{main thm}, it suffices to prove (3) and (4).
(3): Since $\mathbf{P}(\overline{n})$ is in $U_{\langle B, C\rangle}$ iff $n\in B$, and $\neg \mathbf{P}(\overline{n})$ is in $U_{\langle B, C\rangle}$ iff $n \in C$, we have
$B$ and $C$ are recursive in $U_{\langle B, C\rangle}$.
By essentially the same argument as Theorem \ref{Shoenfield second}, we can show that $U_{\langle B, C\rangle}$ is recursive in $(B, C)$. The key point is that the theory $U_{\langle B, C\rangle}$ admits the elimination of quantifiers. Thus, any sentence $\theta$ of the theory $T$ is equivalent to a disjunctive normal form of $\langle \mathbf{P}(\overline{n}): n\in\mathbb{N}\rangle$, and this disjunctive normal form can be found explicitly once $\theta$ is explicitly given.
(4): Suppose $V$ is a consistent r.e. extension of $U_{\langle B, C\rangle}$. Define $D=\{n: V\vdash \mathbf{P}(\overline{n})\}$. Note that $B\subseteq D$ and $D\cap C=\emptyset$. By Theorem \ref{Visser's thm}, $A\leq_T D$. Since $U_{\langle B, C\rangle}$ has the same Turing degree as $A$, we have $U_{\langle B, C\rangle}\leq_T V$. Thus, $U_{\langle B, C\rangle}$ is Turing persistent.
\end{proof}
It is an open question in \cite{Cheng20}: can we show that for any Turing degree $\mathbf{0}<\mathbf{d}\leq\mathbf{0}^{\prime}$, there is a theory $U$ such that $\sf G1$ holds for $U$, $U\lhd \mathbf{R}$ and $U$ has Turing degree $\mathbf{d}$?
As a corollary of Theorem \ref{my thm}, the following theorem answers this question positively and proves a stronger result.
\begin{theorem}\label{main key thm}
For any r.e.~ Turing degree $\mathbf{0}<\mathbf{d}\leq \mathbf{0}^{\prime}$, we can uniformly find a Turing persistent theory $T$ with Turing degree $\mathbf{d}$ such that $\sf G1$ holds for $T$ and $T\lhd \mathbf{R}$.
\end{theorem}
Since there are only countably many r.e. degrees, we have countably many Turing persistent theories in ${\sf D}$.
By default, theories refer to first order theories. Now, we show that Theorem \ref{main key thm} also holds for theories in propositional logic.
We work in propositional logic with countable many variables $\{p_n: n\in\omega\}$. A theory in propositional logic is just a set of formulas in the language of propositional logic. We could view the language of propositional logic as a special instance of first order language: propositional variables can be viewed as constants in first order language. Thus, our notion of interpretation also applies to theories in propositional logic.
\begin{theorem}[Je$\check{r}\acute{a}$bek, Theorem 4.5, \cite{Emil}]~\label{general result}
\begin{enumerate}[(1)]
\item For $\Sigma^0_2$-axiomatized theory $T$, $T$ is interpretable in some consistent existential theory iff $T$ is weakly interpretable in $\mathbf{EC}_L$ for some language $L$.\footnote{The theory of existentially closed $L$-structures ($\mathbf{EC}_L$) is the model completion of the empty theory in the language $L$ (see \cite{Emil}).}
\item
The theory $\mathbf{R}$ is not weakly interpretable in $\mathbf{EC}_{L}$ for any language $L$.
\end{enumerate}
\end{theorem}
By Theorem \ref{general result}, $\mathbf{R}$ is not interpretable in any consistent existential theory.
\begin{theorem}\label{key thm 2}
For any r.e. Turing degree $\mathbf{0}<\mathbf{d}\leq \mathbf{0}^{\prime}$, there exists a Turing persistent theory $U_\mathbf{d}$ in proposition logic with Turing degree $\mathbf{d}$ such that $U_\mathbf{d}\lhd\mathbf{R}$ and $\sf G1$ holds for $U_\mathbf{d}$.
\end{theorem}
\begin{proof}\label{}
Suppose $A$ is a r.e. set with Turing degree $\mathbf{d}$, and $\langle X, Y\rangle$ is the recursively inseparable pair with Turing degree $\mathbf{d}$ constructed as in Theorem \ref{Visser's thm}.
Let $U_\mathbf{d}=\{p_n: n\in X\}\cup\{\neg p_n: n\in Y\}$. Note that $U_\mathbf{d}$ is consistent.
\begin{claim}
$U_\mathbf{d}$ is essentially incomplete.
\end{claim}
\begin{proof}\label{}
Let $S$ be a recursively axiomatized consistent extension of $U_\mathbf{d}$. Define $B=\{n\in \omega: S\vdash p_n\}$ and $C=\{n\in\omega: S\vdash \neg p_n\}$. Note that $B,C$ are r.e.~ sets, $X\subseteq B$, and $Y\subseteq C$. Since $\langle X, Y\rangle$ is a recursively inseparable pair, we have $B\cup C\neq \omega$. Thus, there exists $n\in \omega$ such that $S \nvdash p_n$ and $S \nvdash \neg p_n$. Hence, $S$ is incomplete.
\end{proof}
Now we define a theory $T$ as follows. The language of $T$ consists of the signature of $\mathbf{R}$ and an extra unary predicate symbol $P$. The axioms of $T$ consist of axioms of $\mathbf{R}$ plus the following axioms: $P(\overline{n})$ if $n\in X$; and $\neg P(\overline{n})$ if $n\in Y$.
We can show that $U_\mathbf{d}\unlhd T$ by mapping $p_n$ to $P(\overline{n})$. Since $T$ is locally finitely satisfiable with finite signature, we have $T\unlhd \mathbf{R}$. Thus, $U_\mathbf{d}\unlhd \mathbf{R}$. Since $\mathbf{R}$ is not interpretable in any consistent existential theory, we have $U_\mathbf{d}\lhd \mathbf{R}$.
Now we show that $U_\mathbf{d}$ has Turing degree $\mathbf{d}$. Clearly, $X, Y\leq_T U_\mathbf{d}$. By the normal form theorem in propositional logic, any formula in proposition logic is equivalent to a disjunctive normal form. Thus, $U_\mathbf{d}\leq_T (X, Y)$.
So the theory $U_\mathbf{d}$ has Turing degree $\mathbf{d}$.
\end{proof}
Note that for Turing persistent theories, if they are comparable w.r.t. interpretation, then they are comparable w.r.t. Turing degree. From Theorem \ref{main key thm}, given incomparable r.e. sets w.r.t. Turing degree, we can find incomparable r.e. theories in ${\sf D}$ w.r.t. interpretation. It is an open question in \cite{Cheng20}: are elements of $\langle {\sf D}, \unlhd\rangle$ comparable? The following theorem answers this question negatively and provides much more information.
\begin{theorem}\label{incomparable ele}
Given r.e. sets $A<_T B$, there is a sequence of Turing persistent r.e.~ theories $\langle S_n: n\in\omega\rangle$ such that $S_n\in {\sf D}$, $A<_T S_n <_T B$ and $S_n$ are incomparable w.r.t.~ interpretation.
\end{theorem}
\begin{proof}\label{}
By Fact \ref{key fact in recursion theory}, there exists a sequence of r.e. sets $\langle C_n: n\in\omega\rangle$ such that $A<_T C_n <_T B$ and $C_n$ are incomparable w.r.t. Turing degreee.
By Theorem \ref{main key thm}, for each $n$, we can find a Turing persistent r.e. theory $S_n\in {\sf D}$ with the same Turing degree as $C_n$. Thus, $A<_T S_n <_T B$ for each $n$. Since each $S_n$ is Turing persistent, and $C_n$'s are incomparable w.r.t. Turing degree, we have $S_n$'s are incomparable w.r.t. interpretation.
\end{proof}
\begin{lemma}\label{key lemma}
For r.e. theories $A$ and $B$, if $\sf G1$ holds for both $A$ and $B$, then $\sf G1$ also holds for $A\oplus B$.
\end{lemma}
\begin{proof}
It suffices to show that $A\oplus B$ is essentially undecidable. Suppose $U$ is a consistent decidable extension of $A\oplus B$. Define $X=\{\langle\ulcorner\phi\urcorner, \ulcorner\psi\urcorner\rangle: U\vdash P\rightarrow \phi$ or $U\vdash \neg P\rightarrow \psi\}$. Since $U$ is decidable, $X$ is recursive. Note that $A\subseteq (X)_0$ and $B\subseteq (X)_1$. We claim that at least one of $(X)_0$ and $(X)_1$ is consistent. If both $(X)_0$ and $(X)_1$ are inconsistent, then $U\vdash (P\rightarrow \perp)$ and $U\vdash (\neg P\rightarrow \perp)$. Thus, $U\vdash\perp$ which contradicts that $U$ is consistent. WLOG, we assume that $(X)_0$ is consistent. Then $(X)_0$ is a consistent decidable extension of $A$, which contradicts that $A$ is essentially undecidable.
\end{proof}
As a corollary, the interpretation degree structure of r.e. theories for which $\sf G1$ holds with the
operators $\oplus$ and $\otimes$ is also a distributive lattice.
From the email communication, Albert Visser shows that there is a strictly descending chain of essentially undecidable theories w.r.t.~ interpretation. The following theorem is inspired by Visser's this result.
\begin{theorem}\label{descending chain}
Given r.e. sets $A<_T B$, there is a sequence of r.e.~ theories $\langle C_n: n\in\omega\rangle$ such that $C_n\in {\sf D}$, $A\leq_T C_n \leq_T B$, and $\langle \ldots C_{n+1}\lhd C_{n}\lhd\ldots \lhd C_0\rangle$ is a strictly descending chain of elements of ${\sf D}$.
\end{theorem}
\begin{proof}\label{}
By Theorem \ref{incomparable ele}, we can pick a sequence of Turing persistent r.e.~ theories $\langle S_n: n\in\omega\rangle$ such that $S_n\in {\sf D}$, $A<_T S_n <_T B$ and $S_n$ are incomparable w.r.t.~ interpretation.
Define $C_n=S_n\oplus \ldots \oplus S_0$. By Lemma \ref{key lemma}, $C_n\in {\sf D}$. Note that $A\leq_T C_n \leq_T B$.
Now we show that $C_{n+1}\lhd C_{n}$ for any $n\in\omega$.
We prove by induction on $n$. It is easy to show that $C_1\lhd C_0$ since $S_0$ and $S_1$ are incomparable w.r.t.~ interpretation. Now, we suppose $C_{n+1}\lhd C_{n}$ and show that $C_{n+2}\lhd C_{n+1}$. It suffices to show that $C_{n+1} \ntrianglelefteq C_{n+2}$. Suppose not, i.e. $C_{n+1} \unlhd C_{n+2}$. Then $C_{n+1} \unlhd S_{n+2} \oplus C_{n+1}\unlhd S_{n+2}$.
\begin{claim}\label{compare lemma}
Suppose $T$ is a consistent r.e. theory. For any $n$, if $C_n\unlhd T$, then $S_{i}\leq_T T$ for some $0\leq i\leq n$.
\end{claim}
\begin{proof}\label{}
We prove by induction on $n$. If $n=0$, the conclusion holds since $S_0$ is Turing persistent. Suppose $C_{n+1}= S_{n+1} \oplus C_n \unlhd T$, $\tau$ is the interpretation of $C_{n+1}$ in $T$, and $\mathbf{P}$ is the new predicate used in $S_{n+1} \oplus T_n$. If $T+\mathbf{P}^{\tau}$ is consistent, then $S_{n+1}\unlhd T+\mathbf{P}^{\tau}$. Since $S_{n+1}$ is Turing persistent, $S_{n+1}\leq_T T+\mathbf{P}^{\tau}\leq_T T$. Otherwise, $T+\neg \mathbf{P}^{\tau}$ is consistent. Then $C_n\unlhd T+\neg \mathbf{P}^{\tau}$. By induction, $S_{i}\leq_T T+\neg \mathbf{P}^{\tau}\leq_T T$ for some $0\leq i\leq n$. Thus, we have $S_{i}\leq_T T$ for some $0\leq i\leq n+1$.
\end{proof}
By the above claim, we have $S_{i}\leq_T S_{n+2}$ for some $0\leq i\leq n+1$, which leads to a contradiction. Thus, $\langle \ldots C_{n+1}\lhd C_{n}\lhd\ldots \lhd C_0\rangle$ is a strictly descending chain of elements of ${\sf D}$.
\end{proof}
As a corollary of Theorem \ref{descending chain}, there are many strictly descending chains of elements of ${\sf D}$.
The following theorem shows that the interpretation degree structure of r.e.~ theories for which $\sf G1$ holds has no maximal element.
\begin{theorem}\label{}
For any r.e. theory $A$ for which $\sf G1$ holds, we can uniformly find a r.e. theory $B$ for which $\sf G1$ holds such that $A\lhd B$.
\end{theorem}
\begin{proof}\label{}
Let $A$ be any r.e. theory for which $\sf G1$ holds. By Fact \ref{key fact in recursion theory}, we can uniformly find a r.e. set $C$ such that $A$ is incomparable with $C$ w.r.t. Turing degree. By Theorem \ref{main key thm}, from $C$ we can uniformly find a Turing persistent theory $T$ for which $\sf G1$ holds such that $T$ has the same Turing degree as $C$. Let $B=A\otimes T$. Suppose $B\unlhd A$. Since $T\unlhd B\unlhd A$ and $T$ is Turing persistent, $T$ is Turing comparable with $A$ which leads to a contradiction. Thus, $A\lhd B$.
\end{proof}
\begin{theorem}\label{minimum thm}
If $\langle {\sf D}, \unlhd\rangle$ has a minimal element, then it is also a minimum, and is not Turing persistent.
\end{theorem}
\begin{proof}\label{}
Suppose $A$ is a minimal element of $\langle {\sf D}, \unlhd\rangle$. We show that for any r.e. theory $B\in {\sf D}$, we have $A\unlhd B$. Since $A, B\in {\sf D}$, by Lemma \ref{key lemma}, $A\oplus B\in {\sf D}$. Since $A$ is minimal, we have $A\oplus B$ is mutually interpretable with $A$. Thus, $A\unlhd B$.
Now we show that $A$ is not Turing persistent.
Suppose $A$ is Turing persistent and $\mathbf{d_0}$ is the Turing degree of $A$. Take a r.e. Turing degree $\mathbf{0}<\mathbf{d_1}<\mathbf{0}^{\prime}$ such that $\mathbf{d_0}$ is incomparable with $\mathbf{d_1}$. By Theorem \ref{main key thm}, take a Turing persistent theory $T\in {\sf D}$ with the Turing degree $\mathbf{d_1}$. Since $A$ is the minimum element of $\langle {\sf D}, \unlhd\rangle$, we have $A\unlhd T$. Since $A$ is Turing persistent, we have $A\leq_T T$, which contradicts the fact that $\mathbf{d_0}$ is incomparable with $\mathbf{d_1}$.
\end{proof}
By Theorem \ref{descending chain}, we can effectively find many strictly descending chains of elements of ${\sf D}$.
But it is unknown that whether for any r.e.~ theory $A$ for which $\sf G1$ holds, we can effectively find a r.e.~ theory $B$ for which $\sf G1$ holds such that $B\lhd A$. But for finite axiomatized theories, we can effectively find such a theory $B$. We first introduce some notions.
\begin{definition}[The theory $\mathbf{TN}$, \cite{Visser16}]~\label{}
The theory $\mathbf{TN}$ consists of the following axioms:
\begin{itemize}
\item $\forall x (x\nless 0);\forall x\forall y\forall z((x<y\wedge y<z)\rightarrow x<z)$;
\item $\forall x\forall y\forall z (x<y\vee x=y\vee y<x)$;
\item $\forall x (\mathbf{S} x\nless x)$;
\item $\forall x\forall y(x<y\rightarrow (x< \mathbf{S} x\wedge y\nless \mathbf{S} x))$;
\item $\forall x (x + \mathbf{0} = x)$;
\item $\forall x\forall y(x+\mathbf{S} y=\mathbf{S} (x+y))$;
\item $\forall x (x \times\mathbf{0} = \mathbf{0})$;
\item $\forall x\forall y(x\times \mathbf{S} y=x\times y+x)$.
\end{itemize}
\end{definition}
Note that a model of $\mathbf{TN}$ is a linear
ordering that either represents a finite ordinal or starts with a
copy of $\omega$.
For $\Sigma^0_1$-sentence $\psi=\exists x \phi(x)$ where $\phi$ is $\Delta^0_0$-sentence, define the finitely axiomatizable theory $[\psi]$ as follows:
\[[\psi]=\mathbf{TN}+ \exists x\exists y<x \, \phi(y).\]
\begin{definition}[\cite{Visser16}]\label{}
Suppose $\varphi=\exists x\, A(x)$ and $\psi=\exists x\, B(x)$ are two $\Sigma^0_1$-sentences.
We Define:
\begin{enumerate}[(1)]
\item $\varphi\preceq\psi\triangleq \exists x (A(x)\wedge\forall y< x \,\neg B(y))$;
\item $\varphi\prec\psi\triangleq \exists x (B(x)\wedge\forall y\leq x \, \neg A(y))$;
\item If $\theta$ is $\varphi\preceq\psi$, then $\theta^{\perp}=\psi\prec \varphi$;
\item If $\theta$ is $\varphi\prec\psi$, then $\theta^{\perp}=\psi\preceq \varphi$.
\end{enumerate}
\end{definition}
\begin{fact}[Visser, \cite{Visser16}]~\label{key fact on Q}
Suppose $\varphi, \psi$ are $\Sigma^0_1$-sentences.
\begin{itemize}
\item If $\psi$ is false, then $[\psi] \supseteq \mathbf{R}$.
\item If $\psi$ is true, then $[\psi]\unlhd \top$.
\item If $\varphi\preceq\psi$, then $[\psi]\vdash \varphi$.
\item Let $A=\varphi\preceq\psi$. If $\varphi$ (or $\psi$) holds, then either $A$ holds or $A^{\perp}$ holds.
\end{itemize}
\end{fact}
\begin{theorem}[Harvey Friedman]\label{dense lemma}
For any finitely axiomatized theory $A$, if $\top\lhd A$, then there exists a finitely axiomatized theory $B$ such that
$\top\lhd B\lhd A$.\footnote{This proof is simple than Friedman's proof in \cite{Friedman07}, and the idea of this proof is due to Visser in \cite{Visser16}.}
\end{theorem}
\begin{proof}\label{}
By the fix point lemma, we can find a sentence $\theta$ such that $\mathbf{PA}\vdash \theta\leftrightarrow (A\unlhd A\oplus [\theta])\preceq (A\oplus [\theta]\unlhd \top)$.
\begin{claim}
$A\ntrianglelefteq A\oplus [\theta]$.
\end{claim}
\begin{proof}\label{}
Suppose $A\unlhd A\oplus [\theta]$ holds. By Fact \ref{key fact on Q}, either $\theta$ holds or $\theta^{\perp}$ holds.
Case one: Suppose $\theta$ holds. By Fact \ref{key fact on Q}, $[\theta]\unlhd \top$. Since $A\unlhd A\oplus [\theta]\unlhd [\theta]$, we have $A\unlhd \top$, which leads to a contradiction.
Case two: Suppose $\neg\theta$ holds. Then $\theta^{\perp}$ holds. Thus, $\theta^{\perp}\preceq \theta$ holds. By Fact \ref{key fact on Q}, $[\theta]\vdash \theta^{\perp}$ and thus $[\theta]\vdash \bot$. Since $\neg\theta$ holds, we have $A\oplus [\theta]\unlhd \top$. Thus, $A\unlhd \top$, which leads to a contradiction.
\end{proof}
By the similar argument, we can show that $A\oplus [\theta]\ntrianglelefteq \top$.
Thus, $\top\lhd A\oplus [\theta]\lhd A$.
\end{proof}
\begin{corollary}\label{effective version for finte theory}
If $A$ is a finitely axiomatized theory for which $\sf G1$ holds and $\top\lhd A$, then we can effectively find a finitely axiomatized theory $B$ for which $\sf G1$ holds such that $\top\lhd B\lhd A$
\end{corollary}
\begin{proof}\label{}
This follows from Theorem \ref{dense lemma}. Take the finitely axiomatized theory $B=A\oplus [\theta]$ as in Theorem \ref{dense lemma}. Recall that the proof of Theorem \ref{dense lemma} uses the fixed point lemma, but we can give an effective proof of the fixed point lemma. If $\theta$ is true, then $[\theta]\unlhd \top$ which contradicts that $B\lhd A$. Thus, $\theta$ is false. Since $[\theta]\supseteq \mathbf{R}$, $\sf G1$ holds for $[\theta]$. By Lemma \ref{key lemma}, $\sf G1$ holds for $B$.
\end{proof}
From Corollary \ref{effective version for finte theory}, the interpretation degree structure of finitely axiomatized theories for which $\sf G1$ holds has no minimal element. The following theorem shows that there is no finitely axiomatized theory interpretable in $\mathbf{R}$ for which $\sf G1$ holds.
\begin{theorem}
If $T\in {\sf D}$, then $T$ is not finitely axiomatized: i.e., $D_{finite}\cap {\sf D}=\emptyset$.
\end{theorem}
\begin{proof}\label{}
Suppose $S\in D_{finite}\cap {\sf D}$. Since $S\unlhd \mathbf{R}$, $S$ is locally finitely satisfiable. Since $S$ is finitely axiomatized, then $S$ has a finite model, which contradicts the fact that $S$ is essentially undecidable.
\end{proof}
Moreover, for any Turing persistent r.e.~ theory $A$ for which $\sf G1$ holds, we can effectively find a weaker r.e. theory $B$ than $A$ w.r.t. interpretation such that $\sf G1$ holds for $B$.
\begin{theorem}\label{}
For any Turing persistent r.e. theory $A$ for which $\sf G1$ holds, we can effectively find a r.e. theory $B$ for which $\sf G1$ holds such that $B\lhd A$.
\end{theorem}
\begin{proof}\label{}
Let $A$ be any Turing persistent r.e. theory for which $\sf G1$ holds. By Fact \ref{key fact in recursion theory}, we can effectively find a r.e. set $C$ such that $A$ is incomparable with $C$ w.r.t. Turing degree. By Theorem \ref{main key thm}, from $C$ we can effectively find a Turing persistent theory $T$ for which $\sf G1$ holds such that $T$ has the same Turing degree as $C$. Let $B=A\oplus T$. We show that $B\lhd A$. Suppose $A\unlhd B$. Then $A\unlhd B\unlhd T$. Since $A$ is Turing persistent, we have $A\leq_T C$ which contradicts the fact that $A$ is Turing incomparable with $C$. Thus, $B\lhd A$.
\end{proof}
Recall that we assume by default that the signature of the language is finite.
Finally, we show that whether $\langle {\sf D}, \unlhd\rangle$ has a minimal element (or $\langle {\sf D}, \unlhd\rangle$ is well founded) depends on the signature of the language. If the signature of the language is infinite, then $\langle {\sf D}, \unlhd\rangle$ has minimal elements.
In the following, we assume that the signature of theories is infinite.
Now, we show that for any recursively inseparable pair $\langle X, Y\rangle$, there is a minimum theory $T_{\langle X, Y\rangle}$ w.r.t. interpretation for which $\sf G1$ holds.
\begin{theorem}\label{minimal theory}
For any recursively inseparable pair $\langle X, Y\rangle$, there is a theory $T_{\langle X, Y\rangle}$ with infinite signature such that $\sf G1$ holds for $T_{\langle X, Y\rangle}$ and $T_{\langle X, Y\rangle}$ is interpretable in any first order theory.
\end{theorem}
\begin{proof}\label{}
Let $\langle X, Y\rangle$ be a recursively inseparable pair.
Define the theory $T_{\langle X, Y\rangle}$ as follows.
The language of $T_{\langle X, Y\rangle}$ consists of a countable list of unary predicate symbols $\langle P_n: n\in\omega\rangle$.
The axioms of $T_{\langle X, Y\rangle}$ consist of the following:
\begin{enumerate}[(1)]
\item $\forall x P_n(x)$ if $n\in X$;
\item
$\exists x \neg P_n(x)$ if $n\in Y$.
\end{enumerate}
\begin{lemma}
The theory $T_{\langle X, Y\rangle}$ is essentially incomplete.
\end{lemma}
\begin{proof}\label{}
Let $S$ be a recursively axiomatized consistent extension of $T_{\langle X, Y\rangle}$. Define $A=\{n\in \omega: S\vdash \forall x P_n(x)\}$ and $B=\{n\in \omega: S\vdash \exists x \neg P_n(x)\}$. Note that $A,B$ are r.e.~ sets, $X\subseteq A$, and $Y\subseteq B$. Since $\langle X, Y\rangle$ is a recursively inseparable pair, we have $A\cup B\neq \omega$. Thus, there exists $n\in\omega$ such that $n\notin A\cup B$. Hence, $S$ is incomplete.
\end{proof}
\begin{lemma}
The theory $T_{\langle X, Y\rangle}$ is interpretable in any first order theory.
\end{lemma}
\begin{proof}\label{}
For any $n\in X$, we interpret $P_n(x)$ as $x=x$; and for any $n\in Y$, we interpret $P_n(x)$ as $x\neq x$.
\end{proof}
\end{proof}
Theorem \ref{minimal theory} shows that, interpretation for theories with infinite signature is not a good notion for comparing essentially undecidable theories since an essentially undecidable theory may be interpretable in a decidable theory.
\section{Conclusion}
Finally, we give some concluding remarks.
The existence of non-recursive r.e. set is essential to understand the incompleteness phenomenon. Given any non-recursive r.e. set $A$, we can uniformly construct a r.e. theory for which $\sf G1$ holds with the same Turing degree as $A$. In \cite{Friedman21}, Harvey Friedman proves $\sf G2$ for theories interpreting $I\Sigma_1$ based on the existence of a remarkable set which is equivalent to the existence of non-recursive r.e. set.
We have shown that whether there is a minimal r.e. theory for which $\sf G1$ holds depends on the definition of minimality. The fact that there is no minimal r.e. theory and no maximal r.e. theory for which $\sf G1$ holds w.r.t. some degree structures shows that incompleteness is omnipresent and there is no limit of the first incompleteness theorem.
Both \cite{Cheng20} and this paper are about the limit of $\sf G1$. A natural question is: what is the limit of the second incompleteness theorem ($\sf G2$) if any?
Both mathematically and philosophically, $\sf G2$ is more problematic than $\sf G1$. In the case of $\sf G1$, we are mainly interested in the fact that \emph{some} sentence is independent of the base theory. But in the case of $\sf G2$, we are also interested in the content of the consistency statement.
We can say that ${\sf G1}$ is extensional in the sense that we can construct a concrete independent mathematical statement without referring to arithmetization and provability predicate. However, ${\sf G2}$ is intensional and ``whether the consistency of $T$ is provable in $T$" depends on many factors such as the way of formalization, the base theory we use, the way of coding, the way to express consistency, the provability predicate we use, the way we enumerate axioms of the base theory, etc. For the discussion of the
intensionality of ${\sf G2}$, we refer to \cite{Cheng 19}.
We assume that $\mathbf{Con}(T)$ is the canonical arithmetic formula expressing the consistency of the base theory $T$ defined as $\neg \mathbf{Pr}_T(\mathbf{0}\neq \mathbf{0})$ where both the coding and provability predicate it uses are standard.\footnote{This means that the coding we use is the standard G\"{o}del coding, and the provability predicate we use satisfies the Hilbert-Bernays-L\"{o}b derivability conditions.}
We define that $\sf G2$ holds for a r.e. theory $T$ if for any r.e. theory $S$ interpreting $T$, we have $S\nvdash \mathbf{Con}(S)$.
Pavel Pudl\'{a}k shows that for any r.e. theory $T$ interpreting $\mathbf{Q}, T\nvdash \mathbf{Con}(T)$ (see \cite{Pudlak 85}). Thus, $\sf G2$ holds for $\mathbf{Q}$. But from \cite{Fedor Pakhomov}, $\sf G2$ does not hold for $\mathbf{R}$ since we can find a theory mutually interpretable with $\mathbf{R}$ but it proves its consistency.
The following are two natural questions about the limit of $\sf G2$ worthy of future examination.
\begin{question}~\label{}
\begin{enumerate}[(1)]
\item Is there a r.e. theory $T$ such that $\sf G2$ holds for $T$ and $T$ has Turing degree less than $\mathbf{0}^{\prime}$?
\item Is there a r.e. theory $T$ such that $\sf G2$ holds for $T$ and $T \lhd \mathbf{Q}$?
\end{enumerate}
\end{question} | 192,945 |
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\begin{document}
\begin{abstract}
We consider the following question: if a function of the form $\int_0^{\infty}\varphi(t)\, e^{-xt}dt$ is completely monotonic, is it then $\varphi\ge0$? It turns out that the question is related to a moment problem. In the end we apply those results to answer some questions concerning complete monotonicity of certain functions raised in F.~Qi and R.~Agarwal, \textit{On complete monotonicity for several classes of functions related to ratios of gamma functions}, J Inequal Appl (2019).
\end{abstract}
\maketitle
\section{Introduction}
At the beginning we review basic notions and facts related to \textit{completely monotonic functions}. An infinitely differentiable function $f:(0,\infty)\rightarrow\mathbb{R}$ is called \textit{completely monotonic}, if
$$(-1)^nf^{(n)}\ge0,\quad n=0,1,2,\dots.$$
The crucial fact concerning this class of functions is \textit{Bernstein theorem}: a function $f$ is completely monotonic is and only if there exists a positive Borel measure $\mu$ on $[0,\infty)$, such that
\begin{equation}\label{eq:Bern}
f(x)=\int_{[0,\infty)}e^{-xt}d\mu(t),
\end{equation}
for all $x>0$. Furthermore, the measure $\mu$ is uniquely determined (see \cite{BergForst}, p.\, 61). In many applications one comes up to the situation that a function of the form $\int_0^{\infty}\varphi(t)\, e^{-xt}dt$ is completely monotonic. Usually, in view of Bernstein theorem, it is tacitly assumed that the function $\varphi$ is then necessarily non-negative. Our aim here is to clarify this question: we give a sufficient condition on $\varphi$ which guarantees the claim and provide a complete proof. It turns out that our question has to do with uniqueness of measures in a moment problem which we consider in the next section. In the sequel we apply those results in order to answer the question (in a slightly more general form) raised in \cite{QiAgar} on page 34 whether the functions $\psi'(x+1)-\sinh\frac1{x+1}$ and $\frac12\sinh\frac2x - \psi'(x+1)$ are completely monotonic, where $\psi$ is digamma function.
\section{A moment problem}
As we previously mentioned, our considerations are tightly related to the uniqueness question for measures in a moment problem, which we state in Theorem \ref{moment} (see below). It resembles the \textit{Stieltjes moment problem}: if two non-negative measures $\mu$ and $\nu$ with support on $[0,\infty)$ have the same moments, that is, if
$\int_0^{\infty}t^n\, d\mu(t)=\int_0^{\infty}t^n\, d\nu(t)$, for all $n=0,1,\dots$, is it then $\mu=\nu$? In our case, we use a substitution and reduce it to the \textit{Hausdorff moment problem}, where the support of measures is $[0,1]$.
Now, we turn to our moment problem.
\begin{thm}\label{moment}
Assume $\mu$ and $\nu$ are complex Borel measures on $[0,\infty)$ with the property
\[
\int_{[0,\infty)}e^{-nt}\, d\mu(t)=\int_{[0,\infty)}e^{-nt}\, d\mu(t), \enspace n=0,1,2\dots.
\]
Then, $\mu=\nu$.
\end{thm}
We need the following change of variables formula.
\begin{prop}\label{change}
Let $(X,\mathcal{M},\mu)$ be a measure space, $(Y,\mathcal{N})$ a measurable space and $F:X\to Y$ a measurable map. Then for every measurable function $f:Y\to \mathbb{C}$ and every $E\in\mathcal{N}$ we have
\[
\displaystyle \int_E f(y)\, dF_*\mu(y)=\int_{F^{-1}(E)} f(F(x))\, d\mu(x),
\]
in the case either of two sides is defined. Here $F_*\mu=\mu\circ F^{-1}$ is a measure on $(Y,\mathcal{N})$, the so-called push-forward of $\mu$.
\end{prop}
\noindent For the proof, see \cite[p.\, 30-31]{Durrett}.
\noindent\begin{rem}\label{Remark}
We notice that the change of variable formula also holds for complex Borel measures.
\end{rem}
\textit{Proof of Theorem \ref{moment}}.\\[1ex]
Recall that the complex measures $\mu$ and $\nu$ are of bounded variation, $M_{\mu}:=|\mu|([0,\infty))<\infty$ and $ M_{\nu}:=|\nu|([0,\infty))<\infty$ (see \cite{Rudin}). Let us define a homeomorphism $F:[0,\infty)\rightarrow (0,1]$, $F(t)=e^{-t}$. Applying Proposition \ref{change} (more precisely Remark \ref{Remark}), we obtain
\[
\displaystyle \int_{[0,\infty)}e^{-nt}d\mu(t)=\int_{(0,1]}s^n\, dF_*\mu(s),\quad \int_{[0,\infty)}e^{-nt}d\nu(t)=\int_{(0,1]}s^n\, dF_*\nu(s).
\]
From the assumptions of Theorem \ref{moment}, we have
\[
\int_{(0,1]}s^n\, dF_*\mu(s)=\int_{(0,1]}s^n\, dF_*\nu(s),
\]
for all $n=0,1,2\dots$. Hence
\begin{equation}
\int_{(0,1]}P(s) \, dF_*\mu(s)=\int_{(0,1]}P(s) \, dF_*\nu(s),
\end{equation}
for all polynomials $P$. Notice
\[
|F_*\mu|((0,1])=F_*|\mu|((0,1])=|\mu|([0,\infty))=M_{\mu}
\]
and similarly $|F_*\nu|((0,1])=M_{\nu}$. Therefore, each bounded and measurable (in Borel sense) function on $(0,1]$ is integrable with respect to the both measures $F_*\mu$ and $F_*\nu$. In view of
\[
\left|\int_{(0,1]}g(s)\, dF_*\mu(s)\right|\le M_{\mu}\, \|g\|_{\infty},\quad \left|\int_{(0,1]}g(s)\, dF_*\nu(s)\right|\le M_{\nu}\, \|g\|_{\infty},
\]
for all bounded measurable functions $g:(0,1]\rightarrow\mathbb{R}$, where $\|g\|_{\infty}=\sup\{|g(x)|\, :\, x\in (0,1]\}$, we conclude from Stone - Weierstrass theorem that
\begin{equation}\label{eq:SW}
\int_{(0,1]}g(s) \, dF_*\mu(s)=\int_{(0,1]}g(s) \, dF_*\nu(s),
\end{equation}
for all $g\in C[0,1]$. For small $\delta>0$ introduce a continuous, piecewise linear function $I_{\delta}:(0,1]\to\mathbb{R}$,
\[
I_{\delta}(t)=\left\{
\begin{array}{cc}
0,&t<a-\delta\\
\frac{t-(a-\delta)}\delta,&a-\delta\leq t\leq a\\
1,&a\leq t\leq b\\
\frac{b+\delta-t}\delta,&b\leq t\leq b+\delta\\
0,&b+\delta\leq t,
\end{array}
\right.
\]
where $[a,b]\subset (0,1]$. From \eqref{eq:SW}, we have
\begin{equation}\label{eq:delta}
\int_{(0,1]}I_{\delta}(s) \, dF_*\mu(s)=\int_{(0,1]}I_{\delta}(s) \, dF_*\nu(s).
\end{equation}
Taking into account that $I_{\delta}\to\chi_{[a,b]}$ pointwise as $\delta\to 0+$ (here $\chi$ denotes characteristic function) and $0\le I_{\delta}\le1$, one infers, applying Lebesgue dominant convergence theorem to integrals in \eqref{eq:delta}, that
$\int_{(0,1]}\chi_{[a,b]}(s) \, dF_*\mu(s)=\int_{(0,1]}\chi_{[a,b]}(s) \, dF_*\nu(s)$, or equivalently $F_*\mu([a,b])=F_*\nu([a,b])$, for all $[a,b]\subset (0,1]$. Following a similar procedure one can also deduce $F_*\mu((0,b])=F_*\nu((0,b])$, for all $(0,b]\subset(0,1]$. Therefore $F_*\mu(E)=F_*\nu(E)$ for all Borel sets $E\subset (0,1]$, which implies $F_*\mu=F_*\nu$. Finally, we obtain $\mu=\nu$, since $F$ is a homeomorphism.\ $\Box$
\begin{prop}\label{main}
Let $\varphi:[0,\infty)\rightarrow\mathbb{R}$ be a continuous function with the property
\begin{equation}\label{eq:cond}
\int_0^{\infty}|\varphi(t)|\, dt<\infty.
\end{equation}
If $f(x)=\int_0^{\infty}\varphi(t)\, e^{-xt}\, dt$ is completely monotonic, then $\varphi\ge0$.
\end{prop}
\begin{proof}
Since $f$ is completely monotonic, then according to Bernstein theorem, there exists a non-negative Borel measure $\mu$ on $[0,\infty)$ satisfying \eqref{eq:Bern} for all $x>0$. Due to \eqref{eq:cond}, we have
\[
\mu([0,\infty))=\int_{[0,\infty)}d\mu=f(0)=\int_0^{\infty}\varphi(t)\, dt<\infty,
\]
and consequently, $\mu$ is a finite measure. Again, thanks to \eqref{eq:cond}, we conclude that $\nu(E)=\int_E\varphi(t)\, dt$ is a Borel measure of bounded variation $|\nu|([0,\infty))=\int_0^{\infty}|\varphi(t)|\, dt<\infty$.
Taking into account that
\[
\int_{[0,\infty)}e^{-xt}\, d\nu(t)=\int_0^{\infty}\varphi(t)\, e^{-xt}\, dt=f(x)=\int_{[0,\infty)}e^{-xt}\, d\nu(t),
\]
for all $x\ge0$, we see that the assumptions of Theorem \ref{moment} are fulfilled. Therefore, $\mu=\nu$. This implies
\[
\int_a^b \varphi(t)\, dt=\nu([a,b])=\mu([a,b])\ge0,
\]
for all $[a,b]\subset[0,\infty)$. However, $\varphi$ is continuous, whence $\varphi\ge0$.
\end{proof}
\section{Applications}
We apply the results from the previous section with the aim to answer two questions stated in \cite{QiAgar} on page 34, which concern complete monotonicity of functions $\psi'(x+1)-\sinh\frac1{x+1}$ and $\frac12\sinh\frac2x - \psi'(x+1)$. We will actually prove slightly more general assertions. Here $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma function.
\begin{prop}
For all $m>0$ the function $f(x)=\psi'(x+1)-\frac{1}{m}\sinh\frac{m}{x+1}$ is not completely monotonic.
\end{prop}
\begin{proof}
We employ the following representations
\begin{equation}\label{eq:repr}
\displaystyle \frac1{x^n}=\frac1{(n-1)!}\int_0^{\infty}t^{n-1}\, e^{-xt}\, dt, \quad \psi'(x)=\int_0^{\infty}\frac{t}{1-e^{-t}}\, e^{-xt}\, dt,
\end{equation}
for all $x>0$ and $n\in\mathbb{N}$. The latter one is due to S.\, Ramanujan (see \cite[p.\ 374]{Berndt}). From
\[
\displaystyle \frac{1}{m}\sinh\frac{m}{x+1}=\sum_{n=0}^{\infty}\frac{m^{2n}}{(2n+1)!}\,\frac1{(x+1)^{2n+1}},
\]
we conclude that
\begin{align*}
&\psi'(x+1)-\frac{1}{m}\sinh\frac{m}{x+1}=\\
&=\int_0^{\infty}\left(\frac{t}{1-e^{-t}}-\sum_{n=0}^{\infty}\frac{m^{2n}{t^{2n}}}{(2n)!(2n+1)!}\right)e^{-(x+1)t}\, dt \\
&= \int_0^{\infty}\varphi(t)\, e^{-xt}\, dt,&
\end{align*}
where $\varphi(t)=\left(\frac{t}{1-e^{-t}}-\sum_{n=0}^{\infty}\frac{m^{2n}{t^{2n}}}{(2n)!(2n+1)!}\right)e^{-t}$. Owing to $\frac{t}{1-e^{-t}}\sim t$ as $t\to \infty$ and $\sum_{n=0}^{\infty}\frac{m^{2n}{t^{2n}}}{(2n)!(2n+1)!}\ge \frac{m^2t^2}{4!\, 5!}$, one obtains that $\varphi$ is negative for large $t$. It is easy to see that $\int_0^{\infty}|\varphi(t)|\, dt<\infty$ and using Proposition \ref{main}, we infer that $f$ is not completely monotonic.
\end{proof}
\begin{prop}
For all $m>0$ function the $f(x)=\frac{1}{m}\sinh \frac{m}{x}-\psi'(x+1)$ is completely monotonic.
\end{prop}
\begin{proof}
Using \eqref{eq:repr}, we have
\[
\frac{1}{m}\sinh \frac{m}{x}=\sum_{n=0}^\infty \frac{m^{2n}}{(2n+1)!\, x^{2n+1}}=
\int_0^\infty\sum_{n=0}^\infty\frac{m^{2n}t^{2n}}{(2n)!\,(2n+1)!}\, e^{-tx}\, dt,
\]
and
\[
\frac{1}{m}\sinh\frac{m}{x}-\psi'(x+1)=\int_0^\infty\left(\sum_{n=0}^\infty\frac{m^{2n}t^{2n}}{(2n)!(2n+1)!}-\frac{te^{-t}}{1-e^{-t}}\right)e^{-xt}\, dt,
\]
for all $x>0$. Hence
\begin{equation}\label{eq:final}
f(x)=\int_0^{\infty}\varphi(t)\, e^{-xt}\, dt,
\end{equation}
where $\displaystyle \varphi(t)=\sum_{n=0}^\infty\frac{m^{2n}t^{2n}}{(2n)!(2n+1)!}-\frac{te^{-t}}{1-e^{-t}}$. Further, it is
\begin{align*}
\varphi(t)(e^t-1)=(e^t-1)\sum_{n=0}^\infty\frac{m^{2n}t^{2n}}{(2n)!(2n+1)!}-t\ge t\cdot 1-t=0,
\end{align*}
for all $t\ge0$. Consequently, $\varphi\ge0$ on $[0,\infty)$ and by \eqref{eq:final} one concludes the proof.
\end{proof} | 140,451 |
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“We leveraged the concerns about Y2K to get a diesel generator system, but rolling blackouts weren't a concern back then, ... low-cost solution.”
Related quotes
“Relaxation means releasing all concern and tension and letting the natural order of life flow through one's being”
“Sir, my concern is not whether God is on our side; my greatest concern is to be on God's side, for God is always right”
“There is a great difference between worry and concern. A worried person sees a problem, and a concerned person solves a.” | 283,884 |
Inspiring respect for the environment in the younger generations is a moral obligation of the elderly, of society in general, whether in the form of policies, through teaching in schools or by example in their immediate environment. Not in vain, they are the future and the future of the planet will depend on that transmission of values. But how to generate ecological awareness in young people ? In this AgroCorrn article we explain it to you.
- First years of life
- Set an example
- Recycling is fun
- Know and love nature
First years of life
Generating ecological awareness in young people goes beyond specific campaigns that invite them to recycle or declarations in favor of the environment from their idols. Although everything helps, in reality only an education that begins in the earliest childhood will achieve the objective: that they be informed citizens , with a critical spirit, because only by understanding and loving nature will we contribute to its preservation.
The first years are essential . Within the family or in schools is where the child is socialized, the period in which the personality develops and is oriented for better development. The values, knowledge and attitudes that instill respect for others (which includes Nature) will be transmitted through stories, school activities and, especially, through family life.
A comprehensive development of the child based on positive principles of respect but also love towards himself will be, in short, the basis of a healthy, balanced personality, from which to be able to successfully teach him to take care of the environment in a practical and pleasant, without impositions.
Set an example
Attitude is key to achieving a good predisposition that helps to implement eco-friendly behaviors on a daily basis, so promoting green attitudes is like planting a seed that will end up germinating and bearing fruit: those ecological gestures that end up being infected at the family or family level. in circles of friends. Therefore, if we want the youngest to be environmentally conscious, we must lead by example at home and also on the street.
Through reasoning and suggestion, but especially positive example , recycling, and other desirable behaviors will come naturally. Above all, it is important to take an optimistic approach to eco-responsible everyday actions, such as separating waste to facilitate recycling, not wasting water or electricity, growing an organic garden with them or practicing sensible consumption. Far from being an obligation or a punishment, it must be seen as something desirable , as a contribution that benefits the environment and also us.
Recycling is fun
Creative recycling is a gold mine to entertain young people and open up a world of possibilities that they can take advantage of for their interests. They will discover that they can turn discarded or no longer used objects into other really useful and fun ones, while they will understand from their own experience the problem of waste and the importance of separating it, as well as controlling it by generating the minimum and recycling correctly .
Taking advantage of the holidays to decorate the house with crafts made by the smallest of the house is a nice way to turn those days off into moments that will remain to be remembered and will make the children feel like protagonists. It will be easy to color the eggshells to make Easter decorations that we can hang from the ceiling or use to create original centerpieces.
At Christmas, for example, it would be fun to make the typical wreaths in an alternative way , using candy wrappers, aluminum foil with which they wrap the school sandwich, plastic bottles or any other recycled object.
Know and love nature
You also have to familiarize the youngest with nature . Being in direct contact with her on a regular basis is essential for her good physical and mental development. But not only that, because the environment is not only a place to benefit from, but a space shared with other living beings that deserve respect.
The exits to the field are a wonderful opportunity for the young people aware that it is possible to enjoy the natural environment without adversely affecting the ecosystem. Again, the imposition does not work. Discovering Nature is an exciting adventure that must be enriching in every way.
If you want to read more articles similar to How to generate ecological awareness in young people ,. | 42,453 |
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\section{Spectral mapping theorem}
In this section, we explain the spectral mapping theorem (SMT) for our model, which plays a crucial role in our paper.
Let $\mathcal{K}=\ell^2(\mathbb{Z}^n\setminus\{\bm{0}\})$ and $\tilde{\mathcal{K}}=\ell^2(\mathbb{Z}^n)$.
We define an operator
$\iota:\mathcal{K}\to\tilde{\mathcal{K}}$ as
\begin{align}
\label{def_iota}
(\iota\phi)(\bm{x}) =
\begin{cases}
\phi(\bm{x}), & \bm{x}\in\mathbb{Z}^n\setminus\{\bm{0}\},\\
\bm{0}, & \bm{x}=\bm{0},
\end{cases}
\quad
\text{for $\phi \in \mathcal{K}$.}
\end{align}
For any $\psi\in\tilde{\mathcal{K}}$ and $\phi\in \mathcal{K}$, we have
\begin{align*}
\langle \psi , \iota\phi\rangle_{\tilde{\mathcal{K}}} =
\sum_{\bm{x}\in\mathbb{Z}^n}\overline{\psi(\bm{x})}(\iota\phi)(\bm{x}) =
\sum_{\bm{x}\in\mathbb{Z}^n\setminus\{\bm{0}\}}\overline{\psi(\bm{x})}\phi(\bm{x}).
\end{align*}
Then the conjugate $\iota^{\ast}:\tilde{\mathcal{K}}\to\mathcal{K}$ is given by
\begin{align}
\label{def_iota_conj}
(\iota^{\ast}\psi)(\bm{x}) =
\psi(\bm{x}),\quad \bm{x}\in\mathbb{Z}^n\setminus\{\bm{0}\}.
\end{align}
\begin{lemma}\label{iota}
Let $\iota$ be defined as above.
\begin{enumerate}
\item $\iota^{\ast}\iota = 1\quad (\text{The identity operator on }\mathcal{K})$．
\item $\iota\iota^{\ast} = \mathbbm{1}_{\mathbb{Z}^n\setminus\{\bm{0}\}}\quad (\text{A multiplication operator on }\tilde{\mathcal{K}})$．
\end{enumerate}
\end{lemma}
{\it Proof. }
A direct calculation with (\ref{def_iota}) and (\ref{def_iota_conj}) yields
\begin{align*}
(\iota^{\ast}\iota\phi)(\bm{x}) =
(\iota^{\ast}(\iota\phi))(\bm{x}) =
(\iota\phi)(\bm{x})=\phi(\bm{x})
\end{align*}
for any $\phi\in\mathcal{K}$ and $\bm{x}\in \mathbb{Z}^n\setminus{\{\bm{0}\}}$,
which proves (1).
Similarly, we have
\begin{align*}
(\iota\iota^{\ast}\psi)(\bm{x})=(\iota(\iota^{\ast}\psi))(\bm{x})=(\iota^{\ast}\psi)(\bm{x})=\psi(\bm{x})
\end{align*}
for $\psi\in\tilde{\mathcal{K}}$ and $\bm{x}\in \mathbb{Z}^n\setminus{\{\bm{0}\}}$.
Moreover, if $\bm{x}=\bm{0}$, then
\begin{align*}
(\iota\iota^{\ast}\psi)(\bm{x})=(\iota(\iota^{\ast}\psi))(\bm{x})=\bm{0}.
\end{align*}
Therefore, $(\iota\iota^{\ast}\psi)(\bm{x})=\mathbbm{1}_{\mathbb{Z}^n\setminus\{\bm{0}\}}(\bm{x})\psi(\bm{x})$.
This completes the proof of (2).
\hfill $\square$
We define $d:\mathcal{H}\to\mathcal{K}$ as
\begin{align*}
d &= \iota^{\ast}\tilde{d},
\end{align*}
where $\tilde{d}:\mathcal{H}\to\tilde{\mathcal{K}}$ is given by
\begin{align}
(\tilde{d}\Psi)(\bm{x}) =
\langle \chi(\bm{x}), \Psi(\bm{x})\rangle_{\mathbb{C}^{2n}} \label{d}
\end{align}
for $\bm{x}\in \mathbb{Z}^n$ and $\Psi\in \mathcal{H}$.
Observe that $d$ is a coisometry, i.e., $dd^*=1$, but $\tilde{d}$ is not.
Because for any $\psi\in\tilde{\mathcal{K}}$ and $\Psi\in\mathcal{H}$,
\begin{align*}
\langle \psi, \tilde{d}\Psi\rangle_{\tilde{\mathcal{K}}}
= \sum_{\bm{x}\in\mathbb{Z}^n}\overline{\psi(\bm{x})}\langle \chi(\bm{x}), \Psi(\bm{x})\rangle_{\mathbb{C}^{2n}}
= \sum_{\bm{x}\in\mathbb{Z}^n}\langle \psi(\bm{x})\chi(\bm{x}), \Psi(\bm{x})\rangle_{\mathbb{C}^{2n}}.
\end{align*}
Then the conjugate of $\tilde{d}$ is given by
\begin{align}
(\tilde{d}^{\ast}\psi)(\bm{x})
=
\chi(\bm{x})\psi(\bm{x}),\quad
\bm{x}\in\mathbb{Z}^n. \label{dast}
\end{align}
\begin{lemma}
The coin operator $C$ is expressed as follows：
\begin{align}
C = 2\tilde{d}^{\ast}\tilde{d} - 1 = 2d^{\ast}d - 1.\label{cd}
\end{align}
\end{lemma}
{\it Proof.}
An argument similar to \cite{FFS1, FFS2} shows
$\tilde{d}^{\ast}\tilde{d} = \bigoplus_{\bm{x}\in\mathbb{Z}^n}|\chi(\bm{x})\rangle\langle \chi(\bm{x})|$, which gives the first equality of (\ref{cd}).
The second equality of (\ref{cd}) is proven by $\tilde{d}^{\ast}\tilde{d} =d^{\ast}\iota\iota^{\ast}d = d^{\ast}\mathbbm{1}_{\mathbb{Z}^n\setminus \{\bm{0}\}}d = d^{\ast}d$.
\hfill $\square$
We define two operators
\begin{align}
\label{def_T}
T = dSd^{\ast}\ \text{and}\ \ \tilde{T} = \tilde{d}^{\ast}S\tilde{d}.
\end{align}
Because $T$ and $\tilde{T}$ are bounded self-adjoint operators whose norms are less than $1$, both $\sigma(T)$ and $\sigma(\tilde T)$ are closed sets contained in the interval $[-1,1]$.
The relation among $S,T,$ and $\tilde{T}$ is illustrated in the following figure.
\[
\xymatrix{
\mathcal{H} \ar[d]_{S} & \tilde{\mathcal{K}} \ar[l]_{\tilde{d}^{\ast}} \ar[d]_{\tilde{T}} & \mathcal{K} \ar[d]_{T}\ar@/_18pt/[ll]_{d^{\ast}} \ar[l]_{\iota}\\
\mathcal{H} \ar@/_18pt/[rr]^{d} \ar[r]^{\tilde{d}} & \tilde{\mathcal{K}} \ar[r]^{\iota^{\ast}}& \mathcal{K}
}
\]
We recall $M_{\pm} = \dim\mathcal{B}_{\pm}$ is defined in Theorem \ref{Thm_cont}.
\begin{theorem}[Spectral mapping theorem]\label{SMT}
The following holds：
\begin{align*}
&\sigma_{\sharp}(U) = \varphi^{-1}(\sigma_{\sharp}(T))\cup \{1\}^{\dim \mathcal{B}_{+}}\cup \{-1\}^{\dim \mathcal{B}_{-}} \quad (\sigma_{\sharp}
= \sigma \ {\rm or} \ \sigma_{\rm p}),\\
&\dim\ker (U\mp 1) = M_{\pm} + \dim\ker(T\mp 1).
\end{align*}
\end{theorem}
{\it Proof. } See \cite{SMTSS,SMTSS2}. \hfill $\square$ | 123,562 |
Yes, it has been almost a week. A week in which not much has happened. Last Thursday the headache had come back, on Friday it was still lingering and during the weekend I unpacked box after box after box. So I did get a bit of training in. Ahem...
Last night however, it was back to training. Swimming training! And not just any swimming either: a huge bunch of people swimming. Like a pack of piranhas eating a zebra. I got my own personal co-swimmer though. Since I am not that strong yet. And half way through the first lap, I got a personal buoy as well, since I was doing pretty badly at that point. To hang on to just in case.
But, eventually I did better than last week. I managed to get my whole head under water (instead of just my nose) and I noticed it immediately in my back and neck. Because my body was flatter in the water, the friction was less and I was able to swim a bit faster. Still nowhere near the speed everybody else has, but I can practice.
Tomorrow will be another swim training, and possibly Thursday as well. The rest of the week I have to figure out for myself. Which I must admit I find quite hard. The telly is very tempting!! I am however already starting to see differences. Or rather feeling them. It's easier to stay in a higher gear. My clothes are a bit looser (even though not a single gram has been lost so far) and the only thing left to clear up now is the wheezing and whistling when I am short of breath.
Hari om
Every stroke, a win! YAMxx (in Cincinnati now).
Your clothes are looser!!!!! Yaaaaaaaaay! Doesn't matter if no weight is lost. You've lost inches and that's even better. The weight loss will come later. AND you put your whole head in the water ... You're going soooooo well!! | 67,752 |
\begin{document}
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\nc{\cE}{{\cal E}}\nc{\cG}{{\cal G}}\nc{\cH}{{\cal H}}
\nc{\cX}{{\cal X}}\nc{\cR}{{\cal R}}\nc{\cL}{{\cal L}}
\nc{\cK}{{\cal K}}\nc{\cV}{{\cal V}}\nc{\cI}{{\cal I}}
\nc{\cU}{{\cal U}}
\nc{\sh}{\on{sh}}\nc{\Id}{\on{Id}}\nc{\Diff}{\on{Diff}}
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\nc{\CC}{{\mathbb C}}\nc{\ZZ}{{\mathbb Z}}
\nc{\NN}{{\mathbb N}}\nc{\zz}{{\mathbf z}}\nc{\kk}{{{\mathbf k}}}
\nc{\bl}{{{\mathbf l}}}\nc{\pp}{{{\mathbf p}}}
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\nc{\cO}{{\cal O}} \nc{\cF}{{\cal F}}
\nc{\cS}{{\cal S}}\nc{\cW}{{\cal W}}\nc{\cJ}{{\cal J}}
\nc{\cZ}{{\cal Z}}
\nc{\la}{{\lambda}}\nc{\G}{{\mathfrak g}}\nc{\mm}{{\mathfrak m}}
\nc{\J}{{\mathfrak j}}
\nc{\A}{{\mathfrak a}}\nc{\gotS}{{\mathfrak S}}
\nc{\HH}{{\mathfrak h}} \nc{\T}{{\mathfrak t}}
\nc{\N}{{\mathfrak n}}\nc{\B}{{\mathfrak b}}\nc{\LL}{{\mathfrak l}}
\nc{\La}{\Lambda}
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\nc{\wh}{\widehat}
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\title[PBW and duality for quantum groups and quantum current algebras]
{PBW and duality theorems for quantum groups and quantum current algebras}
\author{B. Enriquez}
\address{Centre de Math\'ematiques, Ecole Polytechnique,
UMR 7640 du CNRS, 91128 Palaiseau, France}
\date{April 1999}
\begin{abstract}
We give proofs of the PBW and duality theorems for the quantum
Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra
duality. We also show that the classical limit of the quantum current
algebras associated with an untwisted affine Cartan matrix is the enveloping
algebra of a quotient of the corresponding toroidal algebra; this quotient is
trivial in all cases except the $A_1^{(1)}$ case.
\end{abstract}
\maketitle
\section{Outline of results}
\subsection{Quantum Kac-Moody algebras} \label{KM}
Let $A = (a_{ij})_{1\leq i,j\leq n}$ be a symmetrizable Cartan matrix.
Let $(d_i)_{1\leq i \leq n}$ be the coprime positive integers such that
the matrix $(d_i a_{ij})_{1\leq i,j\leq n}$ is symmetric. Let $r$ be
the rank of $A$; we assume that the matrix $(a_{ij})_{n-r+1\leq i,j\leq
n}$ is nondegenerate.
Let $\G$ be the Kac-Moody Lie algebra associated with $A$; let $\N_+$ be
its positive pro-nilpotent subalgebra and $(\bar e_i)_{1\leq i\leq n}$
be the generators of $\N_+$ corresponding to the simple roots of $\G$.
Let $\CC[[\hbar]]$ be the formal series ring in $\hbar$. Let
$U_\hbar\N_+$ be the quotient of the free algebra with $n$ generators
$\CC[[\hbar]]\langle e_i, i = 1,\ldots,n\rangle$ by the two-sided ideal
generated by the quantum Serre relations \begin{equation}
\label{quantum:serre} \sum_{k=0}^{1 - a_{ij}}
(-1)^k \bmatrix 1 - a_{ij} \\ k
\endbmatrix_{q^{d_i}} e_i^k e_j e_i^{1 - a_{ij} - k} = 0,
\end{equation} where $\bmatrix m \\ p \endbmatrix_{q} = {{ [m]^!_q
}\over{ [p]^!_q [m-p]^!_q }}$, $[k]^!_q = [1]_q\cdots[k]_q$, $[k]_q =
{{q^k - q^{-k}}\over{q - q^{-1}}}$, and $q = e^\hbar$
(\cite{Drinf:def,Jimbo}).
We will show:
\begin{thm} \label{thm:first}
$U_\hbar\N_+$ is a free $\CC[[\hbar]]$-module, and the map
$e_i\mapsto \bar e_i$ defines an algebra isomorphism of $U_\hbar\N_+
/\hbar U_\hbar\N_+$ with $U\N_+$.
\end{thm}
This Theorem may be derived from the Poincar\'e-Birkhoff-Witt (PBW)
results of Lusztig's book \cite{Lusztig}; in the case $\G = \SL_n$, it can
also be derived from those of Rosso (\cite{Rosso:PBW}), and in the cases when
$\G$ is semisimple or untwisted affine, from those of \cite{Kh:T}.
The proof presented here is based on the comparison of $U_\hbar\N_+$
with a quantum shuffle algebra, Lie bialgebra duality and the
Deodhar-Gabber-Kac theorem.
As a corollary of this proof, we show
\begin{cor} \label{cor:comparison}
The map $p_\hbar$ defined in Lemma \ref{pourim} is an algebra
isomorphism from $U_\hbar\N_+$ to the subalgebra $\langle V \rangle$
of the shuffle algebra $\Sh(V)$ defined in sect.\ \ref{sect:shuffles}.
\end{cor}
This result was proved in \cite{Rosso:shuffle}; it can also be derived from the
results of \cite{Schau}.
Rosso's proof uses the nondegeneracy of the pairing between opposite
Borel quantum algebras (\cite{Lusztig}, Cor.\ 33.1.5; see also Thm.\
\ref{thm:second}). Schauenburg shows that $\langle V \rangle$ is
isomorphic to the quotient of the free algebra generated by the $e_i$
by the radical of a braided Hopf pairing. Together with
\cite{Lusztig}, Cor.\ 33.1.5, this implies Cor.\ \ref{cor:comparison}.
Define $U_\hbar\N_-$ as the algebra with generators $f_i$, $i = 1,\ldots,n$,
and the same defining relations as $U_\hbar\N_+$ (with $e_i$ replaced by $f_i$).
Define a grading on $U_\hbar\N_\pm$ by $(\pm\NN)^n$ by $\deg(e_i) = \eps_i$,
$\deg(f_i) = -\eps_i$, where $\eps_i$ is the $i$th basis vector of $\NN^n$, and
define the braided tensor products $U_\hbar\N_\pm \bar\otimes U_\hbar\N_\pm$
as the algebras isomorphic to $U_\hbar\N_\pm \otimes_{\CC[[\hbar]]} U_\hbar\N_\pm$
as $\CC[[\hbar]]$-modules, with mutiplication rule
\begin{equation} \label{braided:tensor:pdt}
(x\otimes y)(x'\otimes y')
= q^{-\langle \deg(x'), \deg(y)\rangle}(xx'\otimes yy');
\end{equation}
we set $\langle \eps_i,\eps_j\rangle = d_i a_{ij}$.
Then $U_\hbar\N_\pm$ are endowed with braided Hopf algebra structures,
defined by $\Delta_+(e_i) = e_i\otimes 1 + 1\otimes e_i$, and
$\Delta_-(f_i) = f_i\otimes 1 + 1\otimes f_i$.
In \cite{Drinf:ICM}, Drinfeld showed that there exists a unique pairing
$\langle , \rangle_{U_\hbar\N_\pm}$ of $U_\hbar\N_+$ and $U_\hbar\N_-$
with values in $\CC((\hbar)) = \CC[[\hbar]][\hbar^{-1}]$, defined by
\begin{equation} \label{pair:1}
\langle e_i, f_{i'} \rangle_{U_\hbar\N_\pm} = {1\over\hbar}
d_i^{-1}\delta_{ii'},
\end{equation}
\begin{equation} \label{pair:2}
\langle x,yy' \rangle_{U_\hbar\N_\pm} = \sum \langle x^{(1)},
y\rangle_{U_\hbar\N_\pm} \langle x^{(2)}, y'\rangle_{U_\hbar\N_\pm},
\end{equation}
and
\begin{equation} \label{pair:3}
\langle xx',y \rangle_{U_\hbar\N_\pm} = \sum \langle x,
y^{(1)}\rangle_{U_\hbar\N_\pm} \langle x',
y^{(2)}\rangle_{U_\hbar\N_\pm}
\end{equation}
for $x,x'$ in $U_\hbar\N_+$ and $y,y'$ in $U_\hbar\N_-$, and
$\Delta_\pm(z) = \sum z^{(1)} \otimes z^{(2)}$ (braided Hopf pairing
axioms).
As a direct consequence of Cor.\ \ref{cor:comparison}, we show:
\begin{thm} \label{thm:second}
The pairing $\langle , \rangle_{U_\hbar\N_\pm}$ between $U_\hbar\N_+$
and $U_\hbar\N_-$ is nondegenerate.
\end{thm}
This result can be found in Lusztig's book (\cite{Lusztig}, Cor.\
33.1.5, Def.\ 3.1.1 and Prop.\ 3.2.4); it relies on the construction of
dual PBW bases. Another argument using Lusztig' results on integrable modules
is in \cite{Tanisaki}, and an argument using irreducible Verma modules is in
\cite{Rosso:CRAS}.
We also show:
\begin{prop} \label{R:mat}
For any $\al$ in $\NN^n$, let $U_\hbar\N_\pm[\pm\al]$ be the part of
$U_\hbar\N_\pm$ of degree $\pm\al$, and let $P[\al]$ be the element of
$U_\hbar\N_+[\al] \otimes U_\hbar\N_-[-\al]$ induced by $\langle ,
\rangle_{U_\hbar\N_\pm}$. Let $\Delta_+$ be the set of positive roots
of $\G$ (the $\eps_i$ are the simple roots). Let $(\bar
e_{\al,i})_{\al\in\Delta_+}$ and $(\bar f_{\al,i})_{\al\in\Delta_+}$ be dual
Cartan-Weyl bases of $\N_+$ and $\N_-$, and let $e_{\al,i},f_{\al,i}$ be lifts
of the $\bar e_{\al,i},\bar f_{\al,i}$ to $U_\hbar\N_\pm$. Then, if $k$
is the integer such that $\al$ belongs to $k\Delta_+ \setminus
(k-1)\Delta_+$, we have
$$
P[\al] = {{\hbar^k}\over{k!}} \sum_{\al_1,\ldots,\al_k\in\Delta_+ | \sum_i\al_i = \al; i_j}
e_{\al_1,i_1} \cdots e_{\al_k,i_k} \otimes f_{\al_1,i_1}\cdots f_{\al_k,i_k} + o(\hbar^k).
$$
\end{prop}
The fact that $P[\al]$ has $\hbar$-adic valuation equal to $k$ was
stated by Drinfeld in \cite{Drinf:ICM}.
\subsubsection{The case of a generic deformation parameter}
\label{generic}
It is easy to derive from the above results, PBW and nondegeneracy
results in the case where the parameter $q = e^\hbar$ is
generic.
\begin{cor} \label{generic:1}
Let $q'$ be an indeterminate, and let $U_{q'}\N_+$ be the algebra over
$\CC(q')$ with generators $e'_i$, $i = 1,\ldots,r$, and relations
(\ref{quantum:serre}), with $e_i$ and $q = e^\hbar$ replaced by $e'_i$
and $q'$. We have for any $\al$ in $\NN^n$, $\dim_{\CC(q')}
U_{q'}\N_+[\al] = \dim_{\CC(q')} U_{q'}\N_+[\al]$.
Let $(\bar e_\nu)_{\nu\in I}$ be a basis of homogeneous elements of $\N_+$.
Let
$$
\bar e_\nu = \sum_{i_j\in \{1,\ldots,n\}} C_{\nu; i_1,\ldots,i_k}
\bar e_{i_1} \cdots \bar e_{i_k}
$$
be expressions of the $\bar e_\nu$ in terms
of the generators $\bar e_1,\ldots,\bar e_n$.
Let $C_{\nu; i_1,\ldots,i_k}(q')$ be rational functions of $q'$, such that
$C_{\nu; i_1,\ldots,i_k}(1) = C_{\nu; i_1,\ldots,i_k}$, and set
$e'_\nu = \sum_{i_j\in \{1,\ldots,n\}} C_{\nu; i_1,\ldots,i_k}(q')
e_{i_1} \cdots e_{i_k}$. Then the family
$(\prod_{\nu} e_\nu^{n_\nu})$, where the $n_\nu$ are in $\NN$ and
vanish except for a finite number of them, forms a basis of
$U_{q'}\N_+$ (over $\CC(q')$).
\end{cor}
\begin{cor} \label{generic:2}
Let $U_{q'}\N_-$ be the $\CC(q')$-algebra with generators $f'_i,1\leq i \leq n$,
and relations (\ref{quantum:serre}), with $e_i$ replaced by $f'_i$. Define
the braided tensor squares $U_{q'}\N_\pm\bar\otimes U_{q'}\N_\pm$ using
(\ref{braided:tensor:pdt}). We have a braided Hopf pairing
$\langle , \rangle_{\CC(q')}$ between $U_{q'}\N_+$ and $U_{q'}\N_-$,
defined by (\ref{pair:1}), (\ref{pair:2}) and (\ref{pair:3}).
The pairing $\langle , \rangle_{\CC(q')}$ is nondegenerate.
\end{cor}
\subsection{Quantum current and Feigin-Odesskii algebras} \label{QC}
Our next results deal with quantum current algebras. Assume that
the Cartan matrix $A$ is of finite type. Let $L\N_+$ be
the current Lie algebra $\N_+\otimes \CC[t,t^{-1}]$, endowed with the
bracket $[x\otimes t^n,y\otimes t^m] = [x,y]\otimes t^{n+m}$.
\subsubsection{Quantum affine algebras}
Let $\cA$ be the quotient of the free algebra
$\CC[[\hbar]]\langle e_i[k], i = 1, \ldots,n, k\in\ZZ\rangle$ by the
two-sided ideal generated by the coefficients of monomials in the
formal series identities
\begin{equation} \label{crossed:vertex}
(q^{d_i a_{ij}}z - w)e_i(z) e_j(w) = (z - q^{d_i a_{ij}}w) e_j(w)
e_i(z) ,
\end{equation}
\begin{equation} \label{q:serre:nr}
\Sym_{z_1,\ldots,z_{1-a_{ij}}} \sum_{k=0}^{1-a_{ij}} (-1)^k \bmatrix 1 -
a_{ij} \\ k \endbmatrix_{q^{d_i}} e_i(z_1) \ldots e_i(z_{k}) e_j(w)
e_i(z_{k+1}) \ldots e_i(z_{1-a_{ij}}) =0,
\end{equation}
where $e_i(z)$ is the generating series $e_i(z) = \sum_{k\in z}
e_i[k]z^{-k}$, and $q = e^\hbar$.
Let $U_\hbar L\N_+$ be the quotient $\cA / (\cap_{N>0}\hbar^N \cA)$.
Define $\wt A$ as the quotient of the free algebra $\CC[[\hbar]]\langle
e_i[k]^{\wt \cA}, i = 1, \ldots, n k\in\ZZ\rangle$ by the two-sided ideal
geberated by the coefficients of monimials of (\ref{crossed:vertex})
and
\begin{equation} \label{variant:serre}
\sum_{k = 0}^{1 - a_{ij}} (-1)^k \bmatrix 1 - a_{ij} \\ k
\endbmatrix_{q^{d_{i}}} (e_i[0]^{\wt \cA})^k e_j[l]^{\wt \cA}
(e_i[0]^{\wt \cA})^{1 - a_{ij} - k} = 0 ,
\end{equation}
for any $i,j = 1,\ldots,n$ and $l$ integer. Define
$\wt U_\hbar L\N_+$ as the quotient
$\wt\cA / (\cap_{N>0}\hbar^N \wt\cA)$.
\begin{thm} \label{thm:third}
1) $U_\hbar L\N_+$ is a free $\CC[[\hbar]]$-module, and the map
$e_i[n]\mapsto \bar e_i\otimes t^n$ defines an algebra isomorphism from
$U_\hbar L\N_+ / \hbar U_\hbar L\N_+$ to $U L\N_+$.
2) Let $U_\hbar L\N_+^{top}$ be the quotient of $\CC\langle e_i[k], i =
1,\ldots,n,k\in\ZZ\rangle[[\hbar]]$ by the $\hbar$-adically closed
two-sided ideal generated by the coefficients of monomials in relations
(\ref{crossed:vertex}) and (\ref{q:serre:nr}). Then $U_\hbar
L\N_+^{top}$ is a topologically free $\CC[[\hbar]]$-module; it is
naturally the $\hbar$-adic completion of $U_\hbar L\N_+$, and the map
$e_i[n]\mapsto \bar e_i\otimes t^n$ defines an algebra isomorphism
from $U_\hbar L\N_+^{top} / \hbar U_\hbar L\N_+^{top}$ to $U L\N_+$.
3) There is a unique algebra map from $\wt U_\hbar L\N_+$ to $U_\hbar L\N_+$,
sending each $e_i[k]^{\wt \cA}$ to $e_i[k]$; it is an algebra isomorphism.
4) Let $\wt U_\hbar L\N_+^{top}$ be the quotient of $\CC\langle
e_i[k]^{\wt \cA}, i = 1,\ldots,n,k\in\ZZ\rangle[[\hbar]]$ by the $\hbar$-adically closed
two-sided ideal generated by the coefficients of monomials in
relations (\ref{crossed:vertex}) and (\ref{variant:serre}). Then
$e_i[k]^{\wt \cA} \mapsto e_i[k]$ defines an algebra isomorphism between
$\wt U_\hbar L\N_+^{top}$ and $U_\hbar L\N_+^{top}$.
\end{thm}
The statements 1) and 2) of this Theorem can be derived from the results
of \cite{Beck}.
In \cite{FO1,FO2}, Feigin and Odesskii defined the algebra $FO$, which may be
viewed as a functional version of the shuffle algebra. $FO$ is defined
as
\begin{equation} \label{grading}
FO = \oplus_{\kk\in\NN^n} FO_\kk,
\end{equation}
where if $\kk = (k_i)_{1\leq i\leq n}$, we set
$$
FO_\kk =
{1\over{\prod_{i<j} \prod_{1\leq\al\leq k_i, 1\leq\beta\leq k_j}
( t^{(i)}_\al - t^{(j)}_\beta ) }}\CC[[\hbar]][(t^{(i)}_j)^{\pm 1}
, i = 1,\ldots,n, j = 1,\ldots,k_i]^{\gotS_{k_1}\times\cdots\times
\gotS_{k_n}} ,
$$
where the product of symmetric groups acts by permutation of variables
of each group of variables $(t^{(i)}_j)_{1\leq j \leq k_i}$. $FO_\kk$ therefore
consists of rational functions in the $t^{(i)}_j$, symmetric in each
group $(t^{(i)}_j)_{1\leq j \leq k_i}$, regular except for poles when the variables go to
$0$ or infinity, and simple poles when variables of different ``colors'' collide.
(\ref{grading}) defines a grading of $FO$ by $\NN^n$.
The product on $FO$ is also graded, and we have, for $f$ in $FO_\kk$ and $g$ in
$FO_\bl$ ($\bl = (l_i)_{1\leq i\leq n}$) ,
\begin{align} \label{pdt:FO}
& (f*g)(t^{(i)}_j)_{1\leq i\leq n, 1\leq j \leq k_i + l_i} \\ & \nonumber
= \Sym_{\{ t^{(1)}_j \}}
\ldots \Sym_{\{ t^{(n)}_j \}} [\prod_{1\leq i\leq N} \prod_{N+1\leq
j\leq N+M} {{q^{ \langle \eps(i), \eps(j)\rangle} t_i - t_j}\over{t_i - t_j}}
f(t_1,\ldots,t_N) g(t_{N+1},\ldots,t_{N+M})] ,
\end{align}
where $N = \sum_{i=1}^n k_i$ and $M = \sum_{i=1}^n l_i$; we set
for any $s$,
$$
t_{k_1 + \cdots + k_{s-1} + 1} = t^{(s)}_1 ,
\ldots ,
t_{k_1 + \cdots + k_{s}} = t^{(s)}_{k_s} ,
\quad
t_{N+l_1 + \cdots + l_{s-1} + 1} = t^{(s)}_{k_s + 1} , \ldots ,
t_{N+l_1 + \cdots + l_s} = t^{(s)}_{k_s + l_s} ;
$$
we also define
$\eps(i) = \eps_k$ if $t_i = t^{(k)}_l$ for some $l$;
as before, $\langle \eps_i, \eps_j\rangle = d_i a_{ij}$ for $i,j =
1,\ldots,n$.
In the right side of (\ref{pdt:FO}), each symmetrization can be
replaced by a sum over shuffles, since the argument in symmetric in
each group of variables $(t^{(s)}_1,\ldots, t^{(s)}_{k_s})$ and
$(t^{(s)}_{k_s + 1},\ldots, t^{(s)}_{k_s + l_s})$.
In Prop.\ \ref{Hopf:FO}, we define a topological braided Hopf structure on
$FO$.
In \cite{Enr}, we showed:
\begin{prop}
There is a unique algebra morphism $i_\hbar$ from $U_\hbar L\N_+$ to $FO$,
such that $i_\hbar(e_i[n])$ is the element $(t^{(i)}_1)^n$ of
$FO_{\eps_i}$.
\end{prop}
Let us denote by $LV$ the direct sum $\oplus_{i=1}^n FO_{\eps_i}$ and let
$\langle LV\rangle$ be the sub-$\CC[[\hbar]]$-algebra of $FO$ generated by
$LV$. As a corollary of the proof of Thm.\ \ref{thm:third}, we prove:
\begin{cor} \label{cor:second}
$i_\hbar$ is an algebra isomorphism between $U_\hbar L\N_+$ and
$\langle LV\rangle$.
\end{cor}
$U_\hbar L\N_+$ is also endowed with a topological braided Hopf structure
(the Drinfeld comultiplication); it is then easy to see that $i_\hbar$ is
compatible with both Hopf structures.
Define $T(LV_\pm)$ as the free algebras
$\CC[[\hbar]]\langle e_i[k]^{(T)}, 1\leq i\leq n, k\in\ZZ \rangle$ and
$\CC[[\hbar]]\langle f_i[k]^{(T)}, 1\leq i\leq n, k\in\ZZ \rangle$.
We
have a pairing $\langle , \rangle_{ T(LV_\pm) }$ between
$T(LV_+)$ and $T(LV_-)$ defined by
\begin{align} \label{pairing:introd}
& \langle e_{i_1}[k_1]^{(T)} \cdots e_{i_p}[k_p]^{(T)} ,
f_{j_1}[l_1]^{(T)} \cdots f_{j_{p'}}[l_{p'}]^{(T)}
\rangle_{T(LV_\pm)} = \delta_{pp'} {1\over{\hbar^p}} \cdot \\ & \nonumber
\cdot \sum_{\sigma\in \gotS_p} \res_{z_1 = 0} \cdots \res_{z_p = 0}
(\prod_{s>t,\sigma^{-1}(s) < \sigma^{-1}(t)} { { q^{(\eps_{i_s}, \eps_{i_t})}
z_s - z_t} \over{ z_s - q^{(\eps_{i_s}, \eps_{i_t})} z_t}}
\prod_{s=1}^p {1\over {d_{j_s}}}\delta_{i_{\sigma(s),j_s}} \prod_{s=1}^p
z_i^{k_s + l_s} \prod_{s=1}^p {{dz_s}\over{z_s}})
\end{align}
where the ratios ${ { q^{(\eps_{i_s}, \eps_{i_t})}
z_s - z_t}\over{ z_s - q^{(\eps_{i_s}, \eps_{i_t})} z_t }}$ are expanded
for $z_t << z_s$.
Let $U_\hbar L\N_-$ be the quotient of $T(LV_-)$ by the homomorphic image of the
ideal defining $U_{-\hbar} L\N_+$ by the map $e_i[k]^{(T)} \mapsto f_i[k]^{(T)}$.
\begin{prop} \label{prop:morphism} (see \cite{Enr})
This pairing induces a pairing $\langle , \rangle_{U_\hbar L\N_\pm}$ between
$U_\hbar L\N_+$ and $U_\hbar L\N_-$.
\end{prop}
We then prove:
\begin{thm} \label{nondeg:L}
The pairing $\langle, \rangle_{U_\hbar L\N_\pm}$ is nondegenerate.
\end{thm}
\subsubsection{The form of the $R$-matrix}
Let us set $A_\pm = U_\hbar L\N_\pm$. Let $a$ and $b$ be two integers.
Define $I^+_{\geq a}$ and $I^+_{\leq a}$, as the right, resp.\ left
ideals of $A_+$ generated by the $e_i[k],k\geq a$ , resp.\
the $e_i[k],k\leq a$. The ideals $I^+_{\geq a}$ and $I^+_{\leq a}$
are graded; for $\al$ in $(\pm\NN)^n$, denote by $I^+_{\geq a}[\al]$
and $I^+_{\leq a}[\al]$ their component of degree $\al$.
\begin{prop} \label{R:mat:QC}
For any $\al$ in $\NN^n$, for any integers $a$ and $b$, $(I^+_{\leq a} +
I^+_{\geq b})^\perp[-\al]$ and $[A_+ / (I^+_{\leq a} + I^+_{\geq
b})^{\perp\perp}][\al]$ are free finite-dimensional
$\CC[[\hbar]]$-modules. The pairing between $A_+$ and $A_-$ induces a
nondegenerate pairing between them. Moreover, the intersection
$\cap_{a,b} (I^+_{\leq a} + I^+_{\geq b})^{\perp\perp}$ is zero.
Denote by $P_{a,b}[\al]$ the corresponding element of $[A_+ /
(I^+_{\leq a} + I^+_{\geq b})^{\perp\perp}][\al] \otimes (I^+_{\leq a}
+ I^+_{\geq b})^\perp[-\al][\hbar^{-1}]$. $P_{a,b}[\al]$ defines an
element of $\limm_{\leftarrow a,b} A_+ / (I^+_{\leq a} + I^+_{\geq
b})^{\perp\perp} \otimes_{\CC[[\hbar]]} A_-[\hbar^{-1}]$.
Let $(\bar e_{\beta,i})_{\beta\in\Delta_+}$ and $(\bar
f_{\beta,i})_{\beta\in\Delta_+}$ be dual Cartan-Weyl bases of $\N_+$ and
$\N_-$. Let $e_{\beta,i}[p]$ and $f_{\beta,i}[p]$ be lifts to $U_\hbar L\N_\pm$
of $\bar e_{\beta,i}\otimes t^p$ and $\bar f_{\beta,i}\otimes t^p$.
Then if $\al$ belongs to
$k\Delta_+ - (k-1)\Delta_+$, $P[\al]$ has the form
\begin{align*}
& P[\al] = {{\hbar^k}\over{k!}}
\sum_{\al_1,\ldots,\al_k\in\Delta_+,\sum_i\al_i = \al; i_j}
\sum_{p_1,\ldots,p_k\in \ZZ} e_{\al_1,i_1}[p_1] \ldots e_{\al_k,i_k}[p_k]
\otimes f_{\al_1,i_1}[-p_1] \ldots f_{\al_k,i_1}[-p_k] \\ & + o(\hbar^k)
\end{align*}
(all but a finite number of elements of this sum belong to $(I^+_{\leq
a} + I^+_{\geq b})^{\perp\perp} \otimes_{\CC[[\hbar]]}
A_-[\hbar^{-1}]$.)
\end{prop}
Let $(h'_i)_{i = 1,\ldots,n}$ be the basis of $\HH$, dual to
$(h_i)_{i = 1,\ldots,n}$. Set $\cK = \exp(\sum_{i = 1}^n
h_i[0] \otimes h'_i[0] + \sum_{p>0}
h_i[p] \otimes h'_i[-p]) $. Then the elements $\cR[\al] = \cK P[\al]$
of $\limm_{\leftarrow N}
(U_\hbar L\B_+\otimes U_\hbar L\B_-) / I_N^{\B_\pm,(2)}$,
where $I_N^{\B_\pm,(2)}$ is the ideal generated by the $h_i[p]\otimes 1,
e_i[p]\otimes 1, p \geq N$, and the $1\otimes f_i[p], p\geq N$,
satisfy the $R$-matrix identity
$$
\sum_{\gamma \in\NN^n, \beta \in (\pm \NN)^n, \beta + \gamma = \la }
\cR[\gamma] \Delta(x)_{(\beta,\al - \beta)}
=
\sum_{\gamma \in\NN^n, \beta \in (\pm \NN)^n, \beta + \gamma = \la }
\Delta'(x)_{(\beta,\al - \beta)} \cR[\gamma] ,
$$
for any $\la\in \ZZ^r$ and $x$ in the double $U_\hbar L\G$ of $U_\hbar
L\B_+$ of degree $\al$ (the sums over the root lattice are obviously
finite, and each product makes sense in $\limm_{\leftarrow N}
(U_\hbar L\G \otimes U_\hbar L\G) / I_N^{\G,(2)}
$, where $I_N^{\G,(2)}$ is the left ideal generated by the $x[p]\otimes 1$ and
$1\otimes x[p], p\geq N$, $x = e_i,h_i,f_i$).
\subsubsection{Yangians}
Let us describe how the above results are modified in the case of Yangians.
Let $\cA^{rat}$ be the quotient of the free algebra
$\CC[[\hbar]]
\langle e_i[k]^{rat}
, i = 1,\ldots,n,k\in\ZZ \rangle$ by the
two-sided ideal generated by the coefficients of the relations
\begin{equation} \label{crossed:vx:rat}
(z-w + \hbar a_{ij}) e_i(z)^{rat} e_j(w)^{rat} =
(z-w - \hbar a_{ij}) e_j(w)^{rat} e_i(z)^{rat},
\end{equation}
\begin{equation} \label{yg:serre}
\Sym_{z_1,\ldots,z_{1 - a_{ij}}} \ad(e_i(z_1)^{rat}) \cdots
\ad(e_i(z_{1 - a_{ij}})^{rat}) (e_j(w)^{rat}) = 0,
\end{equation}
where we set $e_i(z)^{rat} = \sum_{k\in\ZZ} e_i[k]^{rat} z^{-k-1}$, and let us set
$U_\hbar^{rat}L\N_+ = \cA^{rat} / \cap_{N>0} \hbar^N \cA^{rat}$.
Define also $\wt \cA^{rat}$ as the quotient of the free algebra
$\CC[[\hbar]]
\langle e_i[k]^{\wt\cA^{rat}}
, i = 1,\ldots,n,k\in\ZZ \rangle$ by the
two-sided ideal generated by the coefficients of the relations
(\ref{crossed:vx:rat}) and
\begin{equation} \label{yg:variant:serre}
\ad (e_i[0]^{\wt\cA^{rat}})^{1 - a_{ij}} e_j[l]^{\wt\cA^{rat}} = 0,
\end{equation}
for any $i,j = 1,\ldots, n$ and integer $l$.
\begin{thm}
1) $U_\hbar L\N_+$ is a free $\CC[[\hbar]]$-module. There is a unique
algebra isomorphism from $U_\hbar^{rat} L\N_+ /\hbar U_\hbar^{rat} L\N_+$ to
$U L\N_+$, sending the class of $e_i[k]^{rat}$ to $\bar e_i\otimes t^k$.
2) There is a unique algebra morphism from $\wt U_\hbar L\N_+$ to
$U_\hbar L\N_+$, sending $e_i[k]^{rat}$ to $e_i[k]^{\wt\cA^{rat}}$;
it is an isomorphism between these algebras.
3) Let $U_\hbar L\N_+^{rat,top}$ and $\wt U_\hbar L\N_+^{rat,top}$ be
the quotients of $\CC\langle e_i[k]^{rat}, i =
1,\ldots,n,k\in\ZZ\rangle[[\hbar]]$ and $\CC\langle
e_i[k]^{\wt\cA^{rat}}, i = 1,\ldots,n,k\in\ZZ\rangle[[\hbar]]$ by the
$\hbar$-adically closed two-sided ideals generated by the coefficients
of monomials in relations (\ref{crossed:vx:rat}) and
(\ref{yg:serre}), resp.\ (\ref{crossed:vx:rat}) and
(\ref{yg:variant:serre}). Then $e_i[k]^{\wt \cA^{rat}} \mapsto
e_i[k]^{rat}$ defines an algebra isomorphism between $U_\hbar
L\N_+^{rat,top}$ and $\wt U_\hbar L\N_+^{rat,top}$. $U_\hbar
L\N_+^{rat,top}$ is the $\hbar$-adic completion of $U_\hbar
L\N_+^{rat}$; it is a topologically free $\CC[[\hbar]]$-module.
\end{thm}
Define $FO^{rat}$ as the graded space $FO$, endowed with the product
obtained from (\ref{pdt:FO}) by the replacement of each $q^{\la}z -
q^{\mu}w$ by $z - w+\hbar(\la - \mu)$. $FO^{rat}$ is an associative algebra
and we have
\begin{thm}
There is a unique algebra map $i_\hbar^{rat}$ from $U_\hbar L\N_+^{rat}$ to
$FO^{rat}$, sending each $e_i[k]^{rat}$ to $e_i\otimes t^k$. $i_\hbar$
is an isomorphism between $U_\hbar L\N_+^{rat}$ and its subalgebra
$\langle LV\rangle^{rat}$ generated by the degree one elements.
\end{thm}
Define a pairing $\langle , \rangle_{T(LV_\pm), rat}$ between $T(LV_+)$ and
$T(LV_-)$ by the formula (\ref{pairing:introd}), where each
$q^{\la} z - q^{\mu}w$ is replaced by $z - w + \hbar(\la - \mu)$.
Let $U_\hbar L\N_-^{rat}$ be the quotient $T(LV_-)$ by the homomorphic
image of the ideal defining $U_\hbar L\N_+^{rat}$ by the map $e_i[k]^{(T)}
\mapsto f_i[k]^{(T)}$.
\begin{thm}
$\langle , \rangle_{T(LV_\pm), rat}$ induces a nondegenerate pairing
$\langle , \rangle_{U_\hbar L\N_\pm, rat}$ between $U_\hbar L\N_\pm^{rat}$.
Define $I_a^{\pm, rat}$ as the right, resp.\ left ideals of $U_\hbar L\N_\pm^{rat}$
generated by the $e_i[k]^{rat}$, $k\geq a$, resp.\ by the $f_i[k]^{rat}$, $k\geq a$.
Then for any $\al$ in $\NN^n$, $I^{+,rat}_a[\al] \cap (I^{-,rat}_b[-\al])^\perp$ is a
space of finite codimension in $(I^{-,rat}_b[-\al])^\perp$.
Let $P[\al]^{rat}$ be the corresponding element of
$\limm_{\leftarrow a,b} (U_\hbar L\N_+^{rat} / I_a^{+,rat})[\al]
\otimes (U_\hbar L\N_-^{rat} / I_b^{-,rat})[-\al][\hbar^{-1}]$.
Let $e_{\beta,i}[p]^{rat}$ be lifts to $U_\hbar L\N_\pm$ of the
$\bar e_{\beta,i}[p]$.
Then if $\al$ belongs to
$k\Delta_+ - (k-1)\Delta_+$, $P[\al]^{rat}$ has the form
\begin{align*}
& P[\al]^{rat} = {{\hbar^k}\over{k!}}
\sum_{\al_1,\ldots,\al_k\in\Delta_+,\sum_i\al_i = \al; i_j}
\sum_{p_1,\ldots,p_k\in \ZZ} \\ & e_{\al_1,i_1}[p_1]^{rat} \ldots
e_{\al_k,i_k}[p_k]^{rat} \otimes f_{\al_1,i_1}[-p_1-1]^{rat} \ldots
f_{\al_k,i_1}[-p_k - 1]^{rat} + o(\hbar^k).
\end{align*}
\end{thm}
The proofs of the statements of this section are analogous to those of
the quantum affine case and will be omitted.
\subsection{Quantum current algebras of affine type (toroidal
algebras) } \label{QC:affine}
Assume that $A$ is an arbitrary symmetrizable Cartan matrix. Define
$U_\hbar L\N_+$ and $\wt U_\hbar L\N_+$ as in sect.\ \ref{QC}.
\begin{prop} \label{oz}
1) Let $\wt F_+$ be the Lie algebra with generators $\wt x_i^+[k], 1\leq i \leq n,
k\in\ZZ$, and relations given by the coefficients of monomials in
$$
(z - w) [\wt x_i^+(z), \wt x_j^+(w)] = 0, \quad
\ad (\wt x_i^+(z_1)) \cdots \ad (\wt x_i^+(z_{1 - a_{ij}}))
(\wt x_j^+(w)) = 0,
$$
where for $x$ any $\wt x_i^+$, $x(z)$ is the generating series
$\sum_{k\in\ZZ} x[k]z^{-k}$. If we give degree $\eps_i$ to $\wt x_i^+[k]$,
$\wt F_+$ is graded by set $\Delta_+$ of the roots of $\N_+$.
Then $U_\hbar L\N_+ / \hbar U_\hbar L\N_+$ and $\wt U_\hbar L\N_+ / \hbar \wt
U_\hbar L\N_+$ are both isomorphic to the enveloping algebra $U(\wt
F_+)$.
2) There is a unique Lie algebra morphism $j_+: \wt F_+ \to L\N_+$, such
that $j_+(\wt e_i^+[k]) = \bar e_i\otimes t^k$. $j_+$ is graded and
surjective. The kernel of $j_+$ is contained in $\oplus_{\al\in \Delta_+,
\al\ \on{imaginary}} \wt F_+[\al]$.
\end{prop}
Let us assume now that $A$ is untwisted affine. $\N_+$ is the isomorphic
to a subalgebra of the loop algebra $\bar\G[\la,\la^{-1}]$, with $\bar\G$ semisimple.
Define $\T_+$ as the direct sum $L\N_+ \oplus (\oplus_{k>0,l\in\ZZ}
\CC K_{k\delta}[l])$, and endow it with the bracket such that the
$K_{k\delta}[l]$ are central, and
$$
[(x \otimes t^l,0), (y \otimes t^m,0)] = ([x \otimes t^l, y\otimes t^m],
\langle \bar x, \bar y\rangle_{\bar\G} (lk'' - m k') K_{(k' + k'')\delta}[l+m] )
$$
for $x \mapsto \bar x \otimes \la^{k'}, y \mapsto \bar y \otimes \la^{k''}$
by the
inclusion $\N_+ \subset \bar\G[\la,\la^{-1}]$, where
$\langle , \rangle_{\bar\G}$ is an invariant scalar product on
$\bar\G$.
Then $\T_+$ is a Lie subalgebra of the
toroidal algebra $\T$, which is the universal central
extension of $L\G$ (\cite{Kassel,Moody}). In what follows, we
will set $x[k]^{\T} = (x\otimes t^k,0)$.
\begin{prop} \label{nagila}
1) When $A$ is of affine Kac-Moody type, the kernel of $j_+$ is equal to
the center of $\wt F_+$, so that $\wt F_+$ is a central extension of
$L\N_+$.
2) We have a unique Lie algebra map $j'$ from $\T_+$ to $\wt F_+$ such
that $j'(\bar e_i \otimes t^n) = \wt e_i[n]$. This map is an isomorphism
iff $A$ is not of type $A_1^{(1)}$. If $A$ is of type $A_1^{(1)}$, $j'$
is surjective, and its kernel is $\oplus_{n\in\ZZ} \CC K_{\delta}[n]$.
\end{prop}
In Remark \ref{rem:generalizations}, we discuss possible generalizations
of Thm.\ \ref{thm:third} to the case of affine quantum current
algebras, and the connection Prop.\ \ref{oz} with the results of
\cite{FO:new}.
\subsection{}
The basic idea of the constructions of the two first parts of this work
is to compare the quantized algebras defined by generators and
relations with quantum shuffles algebras. The idea to use shuffle
algebras to provide examples of Hopf algebras dates back to Nichols
(\cite{Nichols}). Later, Schauenburg (\cite{Schau}) and Rosso
(\cite{Rosso:shuffle}) showed that the positive part $U_\hbar\N_+$ of
the Drinfeld-Jimbo quantized enveloping algebras are isomorphic to the
subalgebra $\Sh(V)$ of quantum shuffle Hopf algebras generated in degree
$1$. Their results rely on Lusztig's PBW or duality (nondegeneracy of
Drinfeld's pairing) results. A nonabelian generalization of Schauenburg's result can be found
in \cite{AG}.
In sect.\ \ref{proof:KM}, we show that applying Drinfeld's theory of Lie
bialgebras to $\Sh(V)$ yields at the same time
proofs of these results (PBW for $U_\hbar\N_+$ and isomorphism of
$U_\hbar\N_+$ with $\Sh(V)$, and nondegeneracy of the pairing as a simple
consequence), when the deformation parameter is formal or generic.
In sect.\ \ref{proof:QC}, we apply the same idea to quantum current
algebras. These algebras, also know as ``new realizations''
algebras, depend on the datum of a Cartan matrix.
In that situation, the proper replacement of shuffle algebras
are the functional shuffle algebras introduced by Feigin and Odesskii
(\cite{FO1,FO2}). We show that when the Cartan matrix is of
finite type, the ideas of sect.\ \ref{proof:KM} allow
to complete the results of \cite{Enr} on comparison of the quantum
current algebras and the Feigin-Odesskii algebras. However, there
are still some open problems in this direction, see Rem.\ \ref{open}.
We hope that the ideas of this section will help generalize
the results of \cite{Enr:Rub} from $\SL_2$ to arbitrary semisimple Lie
algebras. For this one should, in particular, find analogues of the
quantum Serre relations for the algebras in genus $\geq 1$.
In sect.\ \ref{toroidal}, we consider the classical limit of the
quantum current algebras in the case of an affine Kac-Moody Cartan
matrix. We show that this classical limit is the enveloping algebra of
a Lie algebra $\wt F_+$, which is a central extension of the Lie
algebra $L\N_+$ of loops with values in the positive subalgebra $\N_+$
of the affine Kac-Moody algebra. $\wt F_+$ is graded by the roots of
$\N_+$, and its center is contained in the part of imaginary degrees.
We show that in all affine untwisted cases, except the $A_1^{(1)}$ case,
$\wt F_+$ is isomorphic to a subalgebra $\T_+$ of the toroidal
algebra $\T$ introduced and studied in
\cite{Moody,GKV,Vass}. In the
$A_1^{(1)}$ case, we identify $\wt F+$ with a quotient of $\T_+$.
In the quantum case, the center of $U_\hbar L\N_+$ seems closely
connected with the central part of the affine elliptic algebras
constructed in the recent work of Feigin and Odesskii (\cite{FO:new}).
We hope that a better understanding of this center will enable to
extend to toroidal algebras the results of sect.\ \ref{proof:QC}.
This paper grew from the notes of the DEA course I taught at univ.\ Paris
6 in february-april 1999. I would like to thank P.\ Schapira for giving
me the opportunity to give this course and its participants, notably C.\
Grunspan, O.\ Schiffmann and V.\ Toledano Laredo, for their patience and
attention.
I also would like to thank N.\ Andruskiewitsch and B.\ Feigin for
valuable discussions. In particular, the idea that quantum shuffle algebras
could be a tool to construct quantizations of Lie bialgebras is due to
N.\ Andruskiewitsch; it is clear that this idea plays an important role
in the present work. I also would like to thank N.\ Andruskiewitsch
for his kind invitation to the univ.\ of C\'ordoba in August 1998, where these
discussions took place.
\section{Quantum Kac-Moody algebras (proofs of the results of sect.\
\ref{KM})} \label{proof:KM}
\subsection{PBW theorem and comparison with shuffle algebra
(proofs of Thm.\ \ref{thm:first} and Cor.\ \ref{cor:comparison})}
\subsubsection{Definition of $\Sh(V)$ and $\langle V \rangle$} \label{sect:shuffles}
Let us set $V = \oplus_{i=1}^n \CC v_i$. Let $\eps_i$ be the $i$th basis
vector of $\NN^n$. Define the grading of $V$ by $\NN^n$ by $\deg(v_i) =
\eps_i$. Let $\Sh(V)$ be the quantum shuffle algebra constructed from
$V$ and the braiding $V\otimes V \to V\otimes V[[\hbar]]$, $v_i \otimes
v_j \mapsto q^{ - d_i a_{ij}} v_j \otimes v_i$. That is, $\Sh(V)$ is
isomorphic, as $\CC[[\hbar]]$-module, to $\oplus_{i\geq 0} V^{\otimes
i}[[\hbar]]$. Denote the element $z_1\otimes\cdots\otimes z_k$ as
$[z_1|\cdots|z_k]$. The product is defined on $\Sh(V)$ as follows:
\begin{align*}
[z_1|\cdots|z_k]
& [z_{n+1}|\cdots|z_{k+l}] \\ & = \sum_{\si\in \Sigma_{k,l}}
q^{ - \sum_{1\leq i < j\leq k+l, \si(i) > \si(j)} \langle \deg(z_i)
, \deg(z_j) \rangle} [z_{\si(1)}|\cdots|z_{\si(k+l)}] ,
\end{align*}
if the $z_i$ are homogeneous elements of $V$, and where $\Sigma_{k,l}$ is
the subset of the symmetric group $\gotS_{k+l}$ consisting of shuffle
permutations $\sigma$ such that $\si(i)< \si(j)$ if $1\leq i < j \leq k$
or $k+1\leq i < j \leq k+l$; the bilinear form on $\NN^n$ is defined by
form $\langle \eps_i,\eps_j \rangle = d_i a_{ij}$.
\begin{lemma} \label{<V>:free}
$\langle V \rangle$ is the direct sum of its graded components, which
are free $\CC[[\hbar]]$-modules. It follows that $\langle V \rangle$ is a free
$\CC[[\hbar]]$-module.
\end{lemma}
{\em Proof.} That $\langle V \rangle$ is the direct sum of its graded
components follows from its definition. These graded components are
$\CC[[\hbar]]$-submodules of finite-dimensional free
$\CC[[\hbar]]$-modules (the graded components of $\Sh(V)$). Each graded
component is therefore a finite-dimensional free module over
$\CC[[\hbar]]$. The Lemma follows. \hfill \qed \medskip
\subsubsection{Crossed product algebras $\cV$ and $\cS$}
Define linear endomorphisms $\wt h_i,i= 1,\ldots,n$ and
$\wt D_j, j = 1,\ldots, n-r$ of $V$ by the formulas
$$
\wt h_i(v_j) = a_{ij}v_j, \quad \wt D_j(v_i) = \delta_{ij}v_i.
$$
Extend the $\wt x$, $x\in \{h_i,D_j\}$ to linear endomorphisms of $\Sh(V)$ by the formulas
$$
\wt x([x_1|\cdots|x_n]) = \sum_{k=1}^n [x_1|\cdots|\wt x(x_k)|\cdots |x_n].
$$
It is clear that the $\wt x$ define derivations of $\Sh(V)$. These
derivations preserve $\langle V \rangle$.
Define $\cV$ and $\cS$ as the crossed product algebras of $\langle V
\rangle$ and $\Sh(V)$ with the derivations $\wt h_i,\wt D_j$. More precisely,
$\cV$ and $\cS$ are isomorphic, as $\CC[[\hbar]]$-modules, to their
tensor products $\langle V\rangle
\otimes_{\CC[[\hbar]]}\CC[h^{\cV}_i,D_j^{\cV}][[\hbar]]$ and $\Sh(V)
\otimes_{\CC[[\hbar]]}\CC[h^{\cV}_i,D_j^{\cV}][[\hbar]]$ with $\hbar$-adically
completed polynomial
algebras in $n+r$ variables. The products on $\cV$ and $\cS$ are then
defined by the rules
$$
(x\otimes \prod_{s = 1}^{2n-r} (X_s^{\cV})^{\al_s})
(y\otimes \prod_{s = 1}^{2n-r} (X_s^{\cV})^{\beta_s})
=
\sum_{(i_s)} \prod_{s = 1}^{2n-r}
\pmatrix \al_s \\ i_s \endpmatrix
\left( x \prod_{s = 1}^{2n-r} \wt X_s^{i_s} (y) \right) \otimes \left(
\prod_{s = 1}^{2n-r}
(X_s^{\cV})^{\al_s + \beta_s - i_s} \right) ,
$$
where we set $X_s = h_s$ for $s = 1,\ldots , n$, and
$X_s = D_{s-n}$ for $s = n+1,\ldots, 2n-r$.
In what follows, we will denote $x\otimes 1$ and $1\otimes X_s$ simply by
$x$ and $X_s$, so that $x\otimes \prod_s (X_s^\cV)^{\al_s}$ will be
$x\prod_s (X_s^\cV)^{\al_s}$.
$\cS$ is then endowed with a Hopf $\CC[[\hbar]]$-algebra structure (that is,
all maps of Hopf algebra axioms are $\CC[[\hbar]]$-module maps, and the tensor
products are completed in the $\hbar$-adic topology),
defined by
$$
\Delta_\cV(h^{\cV} ) = h^{\cV} \otimes 1 + 1 \otimes h^{\cV} \on{\ for\ }
h\in \{h_i,D_j\},
$$
$$
\Delta_\cV([v_{i_1} | \cdots | v_{i_m} ]) = \sum_{k=0}^m [v_{i_1} |
\cdots | v_{i_k} ] \otimes \exp( \hbar \sum_{j=1}^{k} d_{i_j}
h_{i_j}^{\cV})[v_{i_{k+1}} | \cdots | v_{i_m} ]) .
$$
$\cV$ is then a Hopf subalgebra of $\cS$.
Assign degrees $0$ to the elements $h_i^{\cV},D_j^{\cV}$, and $\eps_i$
to $v_i$. $\cV$ is then the direct sum of its homogeneous components,
which are free finite-dimensional modules over
$\CC[h_i^{\cV},D_j^{\cV}][[\hbar]]$; the grading of $\cV$ is compatible
with its algebra structure.
\subsubsection{Hopf co-Poisson and Lie bialgebra structures} \label{adama}
Define $\cV_0$ as $\cV / \hbar\cV$.
\begin{lemma}
$\cV_0$ is a cocommutative Hopf
algebra.
\end{lemma}
{\em Proof.} Define $\Delta'_{\cV}$ as $\Delta_{\cV}$ composed
with the permutation of factors.
We have to show that for $x$ in $\cV$, we have
\begin{equation} \label{balus}
(\Delta_{\cV} - \Delta'_{\cV})(x)
\subset \hbar(\cV\otimes_{\CC[[\hbar]]}\cV).
\end{equation}
For $x$ one of the $\bar h_i,\bar D_j$, (\ref{balus}) is clearly
satisfied. On the other hand, if (\ref{balus}) is satisfied for $x$ and $y$
in $\cV$, then $(\Delta_{\cV} - \Delta'_{\cV})(xy)$ is equal to
$(\Delta_{\cV} - \Delta'_{\cV})(x)\Delta_{\cV}(y) +
\Delta'_{\cV}(x) (\Delta_{\cV} - \Delta'_{\cV})(y)$ and therefore belongs to
$\hbar(\cV\otimes_{\CC[[\hbar]]}\cV)$. It follows that (\ref{balus})
holds for any $x$ in $\cV$. \hfill\qed\medskip
\begin{lemma} \label{pourim}
1) There exists a unique surjective Hopf algebra morphism $p_\hbar$
from $U_\hbar\B_+$ to $\cV$, such that $p_\hbar(h_i) = h_i^{\cV}$ and
$p_\hbar(x_i^+) = [v_i]$.
2) The map $D_j\mapsto \bar D_j,h_i\mapsto \bar h_i,x_i^+\mapsto \bar x_i^+$
extends to an isomorphism from $U_\hbar\B_+ / \hbar U_\hbar\B_+$ to
$U\B_+$.
3) $p_\hbar$ induces a surjective cocommutative Hopf algebra morphism
$p$ from $U_\hbar\B_+ = U\B_+$ to $\cV / \hbar\cV = \cV_0$.
\end{lemma}
{\em Proof.} That the quantum Serre relations are satisfied in $\Sh(V)$
by the $[v_i]$ follows from \cite{Rosso:shuffle}, Lemma 14 (the proof
relies on $q$-binomial coefficients identities, which are proved by
induction); this proves the first part of the Lemma.
Let us show that $U_\hbar\B_+ / \hbar U_\hbar\B_+$ is isomorphic
$U\B_+$. $U_\hbar\B_+ / \hbar U_\hbar\B_+$ is equal to the quotient of
$\CC\langle h_i,D_j,x_i^+, i = 1,\ldots,n, j = 1,\ldots,2n-r
\rangle[[\hbar]]$ by the sum of $\hbar \CC\langle
h_i,D_j,x_i^+\rangle[[\hbar]]$ and the closed ideal generated by the relations
$[h_i,e_j] = a_{ij}e_j$, $[D_i,e_j] = \delta_{ij}e_j$
and the quantum Serre relations (\ref{quantum:serre}). This sum is the
same as that of $\hbar \CC\langle h_i,D_j,x_i^+\rangle[[\hbar]]$ and the
closed ideal generated by $[h_i,e_j] = a_{ij}e_j$, $[D_i,e_j] =
\delta_{ij}e_j$ and the classical Serre relations. The quotient of
$\CC\langle h_i,D_j,x_i^+ \rangle[[\hbar]]$ by this last space is equal
to $U\B_+$. This proves the second part of the Lemma.
The third part is immediate.
\hfill \qed \medskip
\begin{prop} \label{ratiu}
Let $\LL$ be a Lie algebra and let $J$ be a two-sided ideal of $U\L$
such that $\Delta_{U\LL}(J)\subset U\LL \otimes J + J \otimes U\LL$.
Then $\J = J\cap \LL$ is an ideal of the Lie algebra $\LL$ and we have
$J = (U\LL)\J = \J(U\LL)$.
\end{prop}
{\em Proof.}
We first show:
\begin{lemma} \label{lausanne}
Let $\LL$ be a Lie algebra and let $J$ be a left ideal of $U\LL$
such that $\Delta_{U\LL}(J) \subset J \otimes U\LL + U\LL \otimes J$.
Let $\J$ be the intersection $\LL\cap J$. Then $J$ is equal to
($U\LL)\J$.
\end{lemma}
{\em Proof of Lemma.} Denote by $(U\LL)_n$ the subspace of $U\LL$ spanned
by the monomials in elements of $\LL$ of degree $\leq n$. Let us set
$\bar\Delta_{U\LL}(x) = \Delta_{U\LL}(x) - x\otimes 1 - 1\otimes x$. We
have $\bar\Delta_{U\LL}((U\L)_n) \subset \sum_{p,q>0, p+ q = n} (U\LL)_p
\otimes (U\LL)_q$. Denote by $J_n$ the intersection $J\cap (U\LL)_n$.
Then we have $\bar\Delta_{U\LL}(J_n) \subset \sum_{p,q>0, p+q = n}
J_p\otimes (U\LL)_q + (U\LL)_p \otimes J_q$.
Let us show by induction that $J_n$ is contained in $(U\LL)_{n-1}\J$.
This is clear if $n= 1$; assume it is true at order $n-1$.
Let $x$ be an element of $J_n$. Then
$\bar\Delta_{U\LL}(x)$ is contained in $\sum_{p,q>0, p+q = n}
(U\LL)_{p-1}\J \otimes (U\LL)_q + (U\LL)_p \otimes (U\LL)_{q-1}\J$.
Let $\bar x$ be the image of $x$ in $(U\LL)_n / (U\LL)_{n-1}$.
$(U\LL)_n / (U\LL)_{n-1}$ is isomorphic to the $n$th symmetric
power $S^n\LL$. Let $\Delta_{S\LL}$ be the coproduct of the
symmetric algebra $S\LL$, for which elements of degree $1$ are
primitive, and set $\bar\Delta_{S\LL}(\bar x) = \Delta_{S\LL}(\bar x)
- \bar x \otimes 1 - 1\otimes \bar x$. Then $\bar \Delta(\bar x)$
is contained in $\sum_{p,q>0,p+q = n} (S^{p-1}\LL) \J \otimes S^q \LL
+ S^p \LL \otimes (S^{q-1}\LL) \J$. It follows that $\bar x$ belongs to
$(S^{n-1}\LL) \J$. The difference of $x$ with some element of
$(U\LL)_{n-1}\J$ therefore belongs to $(U\LL)_{n-1}$, so that it belongs to
$J_{n-1}$ and by hypothesis to $(U\LL)_{n-2}\J$. Therefore, $x$ belongs to
$(U\LL)_{n-1}\J$. This proves the Lemma.
\hfill \qed\medskip
Let us prove Prop.\ \ref{ratiu}. Lemma \ref{lausanne} implies that $J =
(U\LL) \J$ and its analogue for right Hopf ideals implies that $J = \J
(U\LL)$. Therefore, we have $(U\LL)\J = \J (U\LL)$. Let us fix $x$ in
$\LL$ and $j$ in $\J$, then $[x,j]$ belongs $J$; since it also belongs
to $\LL$, $[x,j]$ belongs to $\J$. Therefore $\J$ is an ideal of $\LL$.
\hfill \qed\medskip
Let $J$ be the kernel of the cocommutative Hopf algebras morphism $p$
defined in Lemma \ref{pourim}, 3). Let us set $\J = J\cap \B_+$ and
$\A = \B_+ / \J$. prop.\ \ref{ratiu} then implies
\begin{lemma} The Lie algebra structure on $\B_+$ induces a Lie algebra
structure on $\A$. Moreover, $\cV_0$ is isomorphic with $U\A$, and $p$
can be identified with the quotient map $U\B_+ \to U\A$.
\end{lemma}
Define $\delta_{\cV_0}$ as
${{\Delta_\cV - \Delta_\cV'}\over\hbar}$ mod $\hbar$. $\delta_{\cV_0}$
is a linear map from $\cV_0$ to the antisymmetric part of its tensor
square $\wedge^2\cV_0$. It obeys the rules
\begin{equation} \label{co-leib}
(\Delta_{\cV_0}\otimes id)\circ \delta_{\cV_0} =
(\delta_{\cV_0}^{2\to 23} + \delta_{\cV_0}^{2\to 13}) \circ
\Delta_{\cV_0},
\end{equation}
$$
\Alt ( \delta_{\cV_0}\otimes
id)\circ\delta_{\cV_0} ) = 0,
$$
$$ \delta_{\cV_0}(xy) = \delta_{\cV_0}(x) \Delta_{\cV_0}(y) + \Delta_{\cV_0}(x)
\delta_{\cV_0}(y) \on{\ for\ } x,y \on{\ in\ } \cV_0,
$$ if
$\delta_{\cV_0}(y) = \sum_i y'_i \otimes y''_i$, we set $\delta^{2\to
23}_{\cV_0}(x\otimes y) = \sum_i x\otimes y'_i \otimes y''_i$, and
$\delta^{2\to 13}_{\cV_0}(x\otimes y) = \sum_i y'_i \otimes x\otimes
y''_i$. These rules are the co-Leibnitz, co-Jacobi and Hopf
compatibility conditions; they mean that $(\cV_0,\delta_{\cV_0})$ is a
Hopf co-Poisson algebra (see \cite{Drinf:ICM}).
\begin{lemma} \label{A:bialg}
$\delta_{\cV_0}$ maps $\A$ to $\wedge^2 \A$.
\end{lemma}
{\em Proof.} Let $a$ be an element of $\A$ and set $\delta_{\cV_0}(a) = \sum_i
x_i \otimes y_i$, where $(y_i)$ is a free family. Then (\ref{co-leib})
implies that $\Delta_{\cV_0}(x_i) \otimes y_i = \sum_i x_i \otimes
1\otimes y_i + 1 \otimes x_i\otimes y_i$, so that each $x_i$ is
primitive and therefore belongs to $\A$. So $\delta_{\cV_0}(a)$
belongs to $\A\otimes \cV_0$. Since $\delta_{\cV_0}(a)$ is also antisymmetric,
it belongs to $\wedge^2\A$.
\hfill \qed\medskip
Call $\delta_\A$ the map from $\A$ to $\wedge^2\A$ defined as the
restriction of $\delta_{\cV_0}$ to $\A$. $(\A,\delta_\A)$ is then a Lie
bialgebra, which means that $\delta_\A$ is a $1$-cocyle of $\A$ with
values in the antisymmetric part of the tensor square of its
adjoint representation, satisfying the co-Jacobi identity
$\Alt(\delta_\A\otimes id)\delta_\A = 0$.
\begin{remark}
The Hopf co-Poisson algebra and Lie bialgebra axioms were introduced by
Drinfeld in \cite{Drinf:ICM}. Drinfeld showed that the quantization of
a cocommutative Hopf algebra lead to such structures. He also stated
that there is an equivalence of categories between the category of Hopf
co-Poisson algebras and that of Lie bialgebras. Lemma \ref{A:bialg} can
therefore be viewed as the proof of one part of this statement (from
Hopf co-Poisson to Lie bialgebras). It is also not difficult to prove the
other part (from Lie bialgebras to Hopf co-Poisson).
\end{remark}
\subsubsection{Kac-Moody Lie algebras} \label{KM:rappels}
Let $\G$ be the Kac-Moody Lie algebra associated with $A$.
$\G$ has generators $\bar x_i^\pm,\bar h_i$, $i = 1,\ldots, n$ and
$\bar D_j, j = 1, \ldots, n-r$, and relations
\begin{equation} \label{comm:cartan:class}
[h,h'] = 0 \on{\ if\ } h,h'\in \{\bar h_i, \bar D_j\},
\end{equation}
\begin{equation} \label{cartan:n+:class}
[\bar h_i, \bar x_{i'}^\pm] = \pm a_{ii'} \bar x_{i'}^\pm,
\quad [\bar D_j, \bar x_{i}^\pm] = \pm \delta_{ij} \bar x_{i}^\pm,
\end{equation}
\begin{equation} \label{serre:class}
\ad(\bar x_i^\pm)^{1 -a_{ij}} (\bar x_j^\pm) = 0,
\end{equation}
$$
[\bar x_i^+, \bar x_{i'}^-] = \delta_{ii'} \bar h_i, \on{\ for \ all\ }
i,i' = 1, \ldots,n, j = 1,\ldots,n-r.
$$
Let $\HH$ and $\N_\pm$ be the subalgebras of $\G$ generated by $\{\bar
h_i,\bar D_j\}$ and $\{\bar x_i^\pm\}$. Then $\G$ has the Cartan decomposition
$\G = \N_+ \oplus \HH \oplus \N_-$. Let us set $\B_\pm =
\HH\oplus\N_\pm$. $\N_\pm$ and $\B_\pm$ are the Lie algebras with
generators $\{\bar x_i^\pm\}$ and $\{\bar h_i, \bar D_j,\bar x_i^\pm\}$ and
relations (\ref{serre:class}) for $\N_\pm$ and
(\ref{comm:cartan:class}), (\ref{cartan:n+:class}) and
(\ref{serre:class}) for $\B_\pm$. $\G$ is endowed with a nondegenerate
bilinear form $\langle , \rangle_\G$, which is determined by
$\langle \bar h_i, \bar h_{i'} \rangle_\G = d_{i'}^{-1} a_{ii'}$,
$\langle \bar x_i^+, \bar x_{i'}^- \rangle_\G = d_i^{-1}\delta_{ii'}$,
$\langle \bar h_i, \bar D_j \rangle_\G = d_i^{-1}\delta_{ij} $,
and that its values for all other pairs of generators
is zero (see \cite{DGK,Kac}).
\subsubsection{Hopf algebras $U_\hbar\B_\pm$}
Define $U_\hbar\B_\pm$ as the algebras with generators $h_i^\pm,D_j^\pm$ and
$x_i^\pm$, and relations
$$
[h_i^\pm,x^\pm_{i'}] = \pm a_{ii'} x^\pm_{i'}, \quad
[D_j^\pm,x^\pm_{i}] = \pm \delta_{ij} x^\pm_{i},
$$
and relations (\ref{quantum:serre}), with $e_i$ replaced by $x^\pm_i$.
It is easy to see that the maps $e_i \mapsto x^+_i$ and $f_i\mapsto x^-_i$ define
algebra inclusions of $U_\hbar\N_\pm$ in $U_\hbar\B_\pm$.
We have Hopf algebra structures on $U_\hbar\B_+$ and
$U_\hbar\B_-$, defined by
$$
\Delta_\pm(h^\pm) = h^\pm \otimes 1 + 1\otimes h^\pm \on{\ for\ }
h\in \{\bar h_i,\bar D_j\},
\quad
\Delta_\pm(x_i^\pm) = x_i^\pm \otimes e^{\pm\hbar d_i h_i^\pm}
+ 1\otimes x_i^\pm.
$$
\subsubsection{Comparison lemmas}
Recall that $\cV$ is the direct sum of its graded components.
Its component of degree zero is $\CC[h_i^{\cV}, D_j^{\cV}][[\hbar]]$.
Let $h_i^{\A}, D_j^{\A}$ be the images of $h_i^{\cV}, D_j^{\cV}$
in $\cV_0$. We have an inclusion of $\CC[h_i^{\A},D_j^{\A}]$
in $\cV_0$. It follows that the $h_i^{\A}$ and $D_j^{\A}$ are
linearly independent and commute to each other.
On the other hand, as the elements $h_i^{\cV}$ and $D_j^{\cV}$ are
primitive in $\cV$, the $h_i^{\A}$ and $D_j^{\A}$ are also primitive; it
follows that they belong to $\A$, and we have
\begin{equation} \label{shoshan}
p(\bar h_i) = h_i^\A, p(\bar D_j) = D_j^\A, \quad \delta_\A(h_i^\A) =
\delta_\A(D_j^\A) = 0.
\end{equation}
The degree $\eps_i$ component of $\cV$ is
$\CC[h_i,D_j][[\hbar]] \cdot[v_i]$, and the map $x\mapsto x[v_i]$ is a
$\CC[[\hbar]]$-module isomorphism from $\CC[h_i,D_j][[\hbar]]$ to this
component. Therefore, $[v_i]$ has a nonzero
image in $\cV / \hbar\cV = \cV_0$. Since we have
$$
\Delta_\cV([v_i]) = [v_i]\otimes e^{\hbar d_i h_i^{\cV}} +
1\otimes [v_i],
$$
$[v_i]\ mod\ \hbar$ is primitive in $\cV_0$. Therefore, $[v_i]\
mod\ \hbar$ belongs to $\A$; call $v_i^\A$ this element of $\A$.
It is clear that
\begin{equation} \label{habira}
\delta_\A(v_i^\A) = d_i v_i^\A \wedge h_i^{\A}.
\end{equation}
On the other hand, $p_\hbar(\bar x_i^+) = [v_i]$ implies that
\begin{equation} \label{megila}
p(\bar x_i^+)=v_i^\A.
\end{equation}
Recall that we have a Lie bialgebra structure on $\B_+$; it consists in
a map $\delta_{\B_+}$ from $\B_+$ to $\wedge^2\B_+$, which is uniquely
determined by the conditions $\delta_{\B_+}(\bar x_i^+) = d_i \bar x_i^+
\wedge \bar h_i$, $\delta_{\B_+}(\bar h_i) = \delta_{\B_+}(\bar D_j)
= 0$ and that it satisfies the $1$-cocycle identity.
\begin{lemma} \label{amo}
$p$ is also a Lie bialgebra morphism from $(\B_+,\delta_{\B_+})$ to
$(\A,\delta_\A)$.
\end{lemma}
{\em Proof.} This means that
\begin{equation} \label{esther}
(\wedge^2 p) \circ \delta_{\B_+}(x) = \delta_{\A}
\circ p(x), \on{\ for\ }x\on{\ in \ }\B_+.
\end{equation}
For $x$ equal to $\bar h_i$ and $\bar D_j$, (\ref{esther}) follows from
(\ref{shoshan}). It follows from (\ref{habira}) and (\ref{megila}) that
$$
\delta_\A\circ p(\bar x_i^+) = \delta_\A(v_i^\A) = d_i v_i^\A \wedge h_i^\A
= (\wedge^2 p)(d_i \bar x_i^+\wedge \bar h_i) =\wedge^2 p(\delta_\A(\bar x_i^+)),
$$
so that (\ref{esther}) is also true for $x = \bar x_i^+$.
Since $p$ is a Lie algebra morphism, both sides of (\ref{esther}) are
$1$-cocycles of $\B_+$ with values in the antisymmetric part of the
tensor square of $(\A,\ad_\A\circ p)$. Therefore, (\ref{esther}) holds
on the subalgebra of $\B_+$ generated by the $\bar h_i,\bar D_j$ and
$\bar x_i^+$, which is $\B_+$ itself.
\hfill \qed \medskip
Denote by $\HH_\A$ the subspace of $\A$ spanned by the $h_i^\A$ and
$D_j^\A$; it forms an abelian Lie subalgebra of $\A$.
Since $p_\hbar(h_i) = h_i^{\cV}$ and $p_\hbar(D_j) = D_j^{\cV}$,
the restriction of $p$ to the Cartan subalgebra $\HH$ of $\B_+$
is a Lie algebra isomorphism from $\HH$ to $\HH_\A$.
Define, for $\al$ in $\HH^*$, the root subspace $\A[\al]$ associated
with $\al$ by
$$
\A[\al] =\{x\in \A | [p(h),x] = \al(h) x, \on{\ for\ all\ }
h \on{\ in\ } \HH\},
$$
and as usual
$$
\B_+[\al] =\{x\in \B_+ | [h,x] = \al(h) x, \on{\ for\ all\ }
h \on{\ in\ } \HH\}.
$$
\begin{lemma}
$\A$ is the direct sum of its root subspaces $\A[\al]$, where $\al$ belongs to
the set $\Delta_+\cup\{0\}$ of roots of $\B_+$. Each $\A[\al]$
is finite dimensional. $\delta_{\A}$ is a graded map from $\A$ to $\wedge^2\A$,
therefore the graded dual $\A^*$ of $\A$, defined as $\oplus_{\al}\A[\al]^*$,
has a Lie bialgebra structure.
\end{lemma}
{\em Proof.} For any $\al$ in $\HH^*$, $p$ maps $\B_+[\al]$ to $\A[\al]$. It
follows that $\A$ is the sum of the root subspaces $\A[\al]$, where $\al$ belongs to
the set of roots of $\B_+$. That this
sum is direct is proved as in the case of $\B_+$: let $\al_i$ be the
root such that $\bar x_i^+$ belongs to $\B_+[\al_i]$. Then $\al_1,\ldots,\al_n$
form a basis of $\HH^*$ (they are the simple roots of $\G$). Let $(\bar
H_1,\ldots,\bar H_n)$ be the basis of $\HH$ dual to
$(\al_1,\ldots,\al_n)$. Then for any family $x_{\al}$ of $\A[\al]$ such
that $\sum_\al x_\al = 0$, we have, by applying $\ad(p(\bar H_1))^k$ to
this equality, $\sum_{(n_i)\in \NN^n} n_1^k x_{\sum_i n_i \al_i} = 0$,
which gives for any integer $a_1$, $\sum_{(n_i)\in \NN^n | n_1 = a_1}
x_{\sum_i n_i\al_i} = 0$; applying $\ad(p(\bar H_2))^k$, we get
$\sum_{(n_i)\in \NN^n | n_1 = a_1, n_2 = a_2 } x_{\sum_i n_i\al_i} = 0$;
finally each $x_\al$ vanishes.
That $\delta_\A$ is graded follows from the fact that its restriction to
$\HH_\A$ vanishes and from the cocycle identity.
\hfill \qed\medskip
\begin{lemma} \label{isom:Lie}
$p$ is a Lie bialgebra isomorphism.
\end{lemma}
{\em Proof.} It follows from Lemma \ref{amo} that $p^*$ is an injective
Lie bialgebra morphism from $\A^*$ in the graded dual $\B_+^*$ of $\B_+$.
Recall that $\B_+^*$ is
isomorphic, as a Lie algebra, to $\B_- = \HH \oplus \N_-$; this relies
on the nondegeneracy of the invariant pairing between $\B_+$ and
$\B_-$, itself a consequence of \cite{DGK} (in what follows, we will
denote by $\HH_-$ the Cartan subalgebra $\HH$ of $\B_-$). We will show that
the image of $p^*$ contains a generating family of $\B_-$.
Let us denote by $\HH_{\A^*}$ the space of forms on $\A$, which vanish
on all the $\A[\al],\al\neq 0$. The duality beween $\B_+$ and $\B_-$
identifies $\HH_-$ with the space of the forms on $\B_+$ which vanish on
$\N_+ = \oplus_{\al\neq 0}\B_+[\al]$. We have $p(\B_+[\al])\subset
\A[\al]$ for any $\al$, therefore
$$
p^*[(\oplus_{\al\neq 0}\A[\al])^{\perp}] \subset
(\oplus_{\al\neq 0}\B_+[\al])^{\perp},
$$
which means that $p^*(\HH_{\A^*})\subset \HH_-$. Since $p^*$ is
injective and $\HH_{\A^*}$ and $\HH_-$ have the same dimension, $p^*$
induces an isomorphism between $\HH_{\A^*}$ and $\HH_-$. It follows
that the image of $p^*$ contains $\HH_-$.
Since $\bar x_i^+$ belongs to $\B_+[\al_i]$, the element $v_i^\A$ of $\A$
defined before Lemma \ref{amo} belongs to $\A[\al_i]$. We have seen that
$v_i^\A$ is nonzero.
Let $\xi_i$ be the element of $\A^*$ which is $1$ on $v_i^\A$ and zero
on each $\A[\al]$, $\al\neq\al_i$. For $x$ in $\B_+[\al]$,
$\al\neq\al_i$, $\langle p^*(\xi_i), x \rangle_{\B_+\times\B_-} =
\langle \xi_i, p(x) \rangle_{\A^*\times\A} = 0$ because $\xi_i$ vanishes
on $\A[\al]$. It follows that $p^*(\xi_i)$ has weight $-\al_i$ in
$\B_-$. On the other hand, $p^*(\xi_i)$ is nonzero, because $p^*$ is
injective, so it is a nonzero constant times $\bar x_i^-$.
Since the image of $p^*$ contains $\HH_-$ and the
$\bar x_i^-$, $p^*$ is an isomorphism. \hfill \qed \medskip
\begin{lemma} \label{isom:n+}
$p_\hbar$ mod $\hbar$ restricts to an isomorphism of $\NN^n$-graded
algebras from $U\N_+$ to $\langle V \rangle / \hbar \langle V \rangle$.
\end{lemma}
{\em Proof.} It follows from Lemma \ref{isom:Lie} that $p_\hbar$ mod
$\hbar$ induces an isomorphism from $U\B_+ = U_\hbar\B_+ / \hbar
U_\hbar\B_+$ to $\cV_0 = \cV / \hbar\cV$. Therefore it induces an
isomorphism from $U\N_+$ to its image in $\cV_0$. Since $U\N_+$
coincides with the image of $U_\hbar\N_+$ by the projection
$U_\hbar\B_+ \to U_\hbar\B_+ / \hbar U_\hbar\B_+ = U\B_+$, this
image coincides with that of the composed map
\begin{equation} \label{composed}
U_\hbar\N_+\to U_\hbar\B_+\to \cV\to \cV_0.
\end{equation} The image
of the composed map $U_\hbar\N_+\to U_\hbar\B_+\to \cV$ is equal to
$\langle V \rangle$. We have a $\CC[[\hbar]]$-module isomorphism of
$\cV$ with $\langle V \rangle \otimes_{\CC[[\hbar]]}
\CC[h_i^{\cV},D_j^{\cV}][[\hbar]]$, so that $\hbar\cV\cap \langle V
\rangle = \hbar\langle V \rangle$. It follows that the image of $\langle
V \rangle$ by $\cV\to \cV_0$ is $\langle V \rangle / \hbar\langle V
\rangle$. Therefore the image of (\ref{composed}) is
$\langle V \rangle / \hbar\langle V \rangle$.
\hfill \qed \medskip
{\em Proof of Thm.\ \ref{thm:first}.} Assign degree $\eps_i$ to the
generator $e_i$ of $U_\hbar\N_+$. Then $U_\hbar\N_+$ is the direct sum
of its homogeneous components $(U_\hbar\N_+)[\al]$, $\al\in\NN^n$,
which are finitely generated $\CC[[\hbar]]$-modules. As a
$\CC[[\hbar]]$-module, $(U_\hbar\N_+)[\al]$ is therefore isomorphic to
the direct sum $\oplus_{i=1}^{q_\al} \CC[[\hbar]] /
(\hbar^{n_i^{(\al)}}) \oplus \CC[[\hbar]]^{p_\al} $ of its torsion part
with a free module (see Lemma \ref{str:modules}).
Lemma \ref{pourim}, 2) implies that
$U_\hbar\N_+[\al] / \hbar U_\hbar\N_+[\al] = U\N_+[\al]$ so
\begin{equation} \label{pal}
p_\al+q_\al = \dimm U\N_+[\al].
\end{equation}
On the other hand, $\langle V \rangle[\al]$ is a free finite dimensional
module over $\CC[[\hbar]]$, by Lemma \ref{<V>:free} above, so it is
isomorphic, as a $\CC[[\hbar]]$-module, to $\CC[[\hbar]]^{p'_\al}$.
$p_\hbar$ restricts to a surjective $\CC[[\hbar]]$-module morphism from
$U_\hbar\N_+[\al]$ to $\langle V \rangle[\al]$, therefore $p_\hbar$ maps
the torsion part of $U_\hbar\N_+[\al]$ to zero and
\begin{equation} \label{ma}
p_\al\geq p'_\al.
\end{equation}
Moreover,
\begin{equation} \label{kh}
p'_\al = \dimm U\N_+[\al]
\end{equation}
by Lemma \ref{isom:n+}.
It follows from (\ref{pal}), (\ref{ma}) and (\ref{kh}) that
$p_\al = p'_\al$ and $q_\al=0$.
This means that $U_\hbar\N_+$ has no torsion, and is isomorphic to
$\langle V\rangle$. In view of Lemma \ref{pourim}, 2), this proves
Thm.\ \ref{thm:first}. This also proves Cor.\ \ref{cor:comparison}.
\hfill \qed \medskip
\subsection{Nondegeneracy of Hopf pairing (proof of
Thm.\ \ref{thm:second})}
Let $(v_i^*)$ be the basis of $V^*$ such that $\langle v_i^*,v_j\rangle
= d_i^{-1}\delta_{ij}$. Assign to $v_i^*$ the degree $-\eps_i$. Let
$T(V^*)$ be the tensor algebra $\oplus_i (V^*)^{\otimes i}[[\hbar]]$.
Define the braided tensor product strucutre on the tensor square of
$T(V^*)$ according to (\ref{braided:tensor:pdt}). $T(V^*)$ is endowed
with the braided Hopf structure defined by $\Delta_{T(V^*)}(v_i^*) =
v_i^*\otimes 1 + 1\otimes v_i^*$, for any $i = 1,\ldots,n$. We have a
surjective braided Hopf algebra morphism from $T(V^*)$ to
$U_\hbar\N_-$, defined by $v_i^*\mapsto f_i$, for $i = 1,\ldots,n$.
Then we have a braided Hopf pairing
$$
\langle , \rangle_{\Sh(V) \times T(V^*)}
: \Sh(V) \times T(V^*)
\to \CC((\hbar)),
$$
defined by the rules
\begin{equation} \label{formula:crossed:pdt}
\langle [v_{i_1} | \cdots | v_{i_k}],\xi_{i'_1}\cdots
\xi_{i'_{k'}}
\rangle_{\cS \times U_\hbar\wt \B_- } =
{1\over\hbar} \delta_{kk'} \prod_{j=1}^k\langle v_{i_j},
\xi_{i'_j}\rangle_{V \times V^*} .
\end{equation}
The ideal of $T(V^*)$ generated by the quantum Serre
relations is in the radical of this pairing (see e.g.\ \cite{Lusztig},
chap.\ 1; this is a consequence of $q$-binomial identities).
It follows that
$\langle , \rangle_{\Sh(V) \times T(V^*)}$ induces a braided Hopf
pairing
$$
\langle , \rangle_{\Sh(V) \times U_\hbar\N_- }:
\Sh(V) \times U_\hbar\N_- \to \CC((\hbar)).
$$
By Thm.\ \ref{thm:first}, $U_\hbar\N_+$ is a braided Hopf subalgebra of
$\Sh(V)$. The restriction of
$\langle , \rangle_{\Sh(V) \times U_\hbar\N_- }$
to $U_\hbar\N_+\times
U_\hbar\N_-$ therefore induces a braided Hopf pairing between $U_\hbar\N_+$
and $U_\hbar\N_-$; since it coincides on generators with $\langle
, \rangle_{U_\hbar\N_+\times U_\hbar\N_-}$, it is equal to
$\langle , \rangle_{U_\hbar\N_+\times U_\hbar\N_-}$.
View $V^{\otimes k}$ as a subspace of $\Sh(V)$. Assign degree $1$ to each
element of $V^*$ in $T(V^*)$; then
$T(V^*)$ is a graded algebra; we denote by
$T(V^*)^{(k)}
$ is homogeneous component of degree $k$. The
restriction of $\langle , \rangle_{\Sh(V) \times T(V^*) }$
to
$V^{\otimes k} \times T(V^*)^{(k)}$ can be identified with the
natural pairing of $V^{\otimes k}$ with $(V^*)^{\otimes k}$, which is
nondegenerate. Therefore the annihilator of $T(V^*)$ in
$\Sh(V)$ for $\langle , \rangle_{\Sh(V) \times T(V^*) }$
is zero. By Thm.\ \ref{thm:first}, it follows that the annihilator
of $U_\hbar\N_-$ in $U_\hbar\N_+$ for $\langle , \rangle_{U_\hbar\N_\pm}$
is zero.
Since the pairing $\langle , \rangle_{U_\hbar\N_+\times U_\hbar\N_-}$ is
graded and the graded components of $U_\hbar\N_+$ and $U_\hbar\N_-$
have the same dimensions (as $\CC[[\hbar]]$-modules), the pairing
$\langle , \rangle_{U_\hbar\N_+\times U_\hbar\N_-}$ is nondegenerate.
\hfill\qed \medskip
\subsection{The form of the $R$-matrix (proof of Prop.\ \ref{R:mat})}
Let us endow $U_\hbar\G = U_\hbar \B_+ \otimes U_\hbar \N_-$ with the double
algebra structure such that $U_\hbar \B_+ \to U_\hbar\G$, $x_+ \mapsto
x_+ \otimes 1$, and $U_\hbar \N_-\to U_\hbar\G$, $x_- \mapsto 1\otimes x_-$
are algebra morphisms and, if $e_i^{\G} = e_i\otimes 1$,
$f_i^{\G} = 1\otimes f_i$, $h_i^\G = h_i \otimes 1$ and $D_j^\G = D_j \otimes 1$,
$$
[e^\G_i , f^\G_j] = \delta_{ij}{{q^{d_ih^\G_i}
- q^{ - d_ih_i^\G}}\over{q^{d_i} - q^{-d_i}}} ,
$$
and
$$
[h^\G_i,f^\G_j] = - a_{ij} f^\G_j, \quad [D^\G_j, f^\G_j] = -\delta_{ij} f^\G_j.
$$
$U_\hbar\G$ is endowed with a topological Hopf algebra structure
$\Delta: U_\hbar \G\mapsto U_\hbar \G \hat\otimes U_\hbar\G =
\limm_{\leftarrow N} ( U_\hbar \G \otimes U_\hbar \G ) / \hbar^N (
U_\hbar \G \otimes U_\hbar \G )$, extending $\Delta_+$ and $\Delta_-$
(\cite{Drinf:ICM}).
Let $t_0$ be the element of $\HH \otimes \HH$ corresponding to the
restriction of the invariant pairing of $\G$ to $\HH$ and let
$\cR[\al]$ be the element of $[\limm_{\leftarrow N} ( U_\hbar \G
\otimes U_\hbar\G) /
\hbar^N (U_\hbar\G\otimes U_\hbar\G)][\hbar^{-1}]$
$$
\cR[\al] = \exp(\hbar t_0)P[\al].
$$
Then we have the equalities
\begin{equation} \label{kolia}
\cR[\al - \al_i] (e^\G_i \otimes q^{d_i h^\G_i }) + \cR[\al] (1
\otimes e^\G_i) = (q^{d_i h^\G_i } \otimes e^\G_i ) \cR[\al] +
(e^\G_i \otimes 1) \cR[\al - \al_i] ,
\end{equation}
for any $i = 1,\ldots, n$.
\begin{lemma}
For any nonzero $\al$ in $\NN^n$, $P[\al]$ belongs to $\hbar U_\hbar \N_+ \otimes
U_\hbar\N_-$.
\end{lemma}
{\em Proof.} Let us show this by induction on the height of $\al$ (we
say that the height of $\al = (\al_i)_{1\leq i \leq n}$ is $\sum_{i=
1}^n \al_i$). If $\al$ is a simple root $\al_i$, $P[\al] = \hbar
e_i^{\G} \otimes f_i^{\G}$, so that the statement holds when $\deg(\al)
= 1$.
Assume that we know that $P[\al]$ belongs to $\hbar U_\hbar\N_+ \otimes
U_\hbar \N_-$ for any $\al$ of height $<\nu$. Let $\al$
be of height $\nu$. Let $v$ be the $\hbar$-adic valuation of
$P[\al]$, and assume that $v\leq 0$. $\cR[\al]$ belongs to $\hbar^{v}
(U_\hbar\G \hat\otimes U_\hbar\G)$, and since $v\leq 0$ the equality
(\ref{kolia}) takes place in $\hbar^{v} (U_\hbar\G \hat\otimes
U_\hbar\G)$. Let us set $R_\al = \hbar^{-v} \cR[\al]$ mod $\hbar$;
$R_\al$ is an element of $U\B_+ \otimes U\B_-$. Since $\hbar^{-v}
\cR[\al - \al_i]$ is zero mod $\hbar$, (\ref{kolia}) implies that
$R_\al$ commutes with each $1\otimes \bar e_i$.
Lemma 1.5 of \cite{Kac} says that if $a$ belongs to $\N_-$ and commutes
with each $e_i$, then $a$ is zero. It follows that if $x$ belongs to
$U\N_-$ and commutes with each $\bar e_i$, $x$ is scalar; and if in
addition $x$ has nonzero degree, $x$ is zero. Therefore $R_\al$ is zero.
It follows that $v\geq 1$, which proves the induction.
\hfill \qed\medskip
It follows from \cite{Drinf:ICM} that the $\cR[\al]$ satisfy
the quasi-triangular identities
\begin{equation} \label{QT1}
(\Delta\otimes id)\cR[\al] = \sum_{\beta,\gamma\in\NN^n,
\beta + \gamma = \al} \cR[\beta]^{(13)} \cR[\gamma]^{(23)} ,
\end{equation}
\begin{equation} \label{QT2}
(id \otimes \Delta)\cR[\al] = \sum_{\beta,\gamma\in\NN^n,
\beta + \gamma = \al} \cR[\beta]^{(13)} \cR[\gamma]^{(12)} .
\end{equation}
Let us set, for $\al\neq 0$,
$r[\al] = R[\al] / \hbar$ mod $\hbar$; $r[\al]$ belongs
to $U\B_+ \otimes U\B_-$. Dividing the equalities (\ref{QT1}) and
(\ref{QT2}) by
$\hbar$, we get $(\Delta_{U\B_+} \otimes id) r[\al] =
r[\al]^{(13)} + r[\al]^{(23)}$ and
$(id \otimes \Delta_{U\B_+}) r[\al] =
r[\al]^{(12)} + r[\al]^{(13)}$. Therefore $r[\al]$ belongs to
$\B_+ \otimes \B_-$.
Moreover, (\ref{kolia}) implies the identity
$$
\delta(x)_{(\beta,\al - \beta)} = [r[\beta - \al], x\otimes 1] +
+ [r[\beta], 1\otimes x]
$$
for $x$ in $\G[\al]$, where we set
$(\sum_i x_i\otimes y_i)_{(\al,\beta)} = \sum_i (x_i)_{(\al)} \otimes
(y_i)_{(\beta)}$ and we denote by $x_{(\al)}$ the degree $\al$ component
of an element $x$ of $U\G$.
It follows that $r[\al]$ is the element of $\N_+[\al] \otimes
\N_-[-\al]$ corresponding to the invariant pairing of $\G$.
Let us now prove by induction on $k$ that it $\al$ belongs to
$k\Delta_+ - (k-1)\Delta_+$, $P[\al]$ belongs to $\hbar^k U_\hbar
\N_+ \otimes U_\hbar \N_-$ and
$$
P[\al] = {{\hbar^k}\over{k!}}
\sum_{\al_1,\ldots, \al_k \in \Delta_+,
\sum_{i = 1}^k \al_i = \al } r[\al_1] \cdots r[\al_k] + o(\hbar^k).
$$
Assume that the statement is proved up to order $k-1$ and let $\la$
belong to $k\Delta_+ - (k-1)\Delta_+$. Then (\ref{QT1}) and the
induction hypothesis implies that
\begin{align} \label{hannibal}
& (\wt\Delta \otimes id) (P[\la]) =
\sum_{\al_1, \ldots,\al_k\in\Delta_+, \sum_{i = 1}^k \al_i = \la; i_j}
\sum_{l,l' >0, l + l' = k} {{\hbar^k}\over{l! l'! }}
\\ & e_{\al_1,i_1} \cdots e_{\al_l,i_l} \otimes
e_{\al_{l+1},i_{l+1}} \cdots e_{\al_k,i_k} \otimes
f_{\al_1,i_1} \cdots f_{\al_k,i_k} ,
\end{align}
where $\wt\Delta(x) = \Delta (x) - x\otimes 1 - 1 \otimes x$. Let $\sigma$ be any
permutation of $\{1,\ldots,k\}$.
For any $\al_1, \ldots,\al_k$ in $\Delta_+$, such that $\sum_{i = 1}^k \al_i = \la$,
we have
$$
f_{\al_1,i_1} \cdots f_{\al_k,i_k} = f_{\al_{\sigma(1)},i_{\sigma(1)}}
\cdots f_{\al_{\sigma(k)},i_{\sigma(k)}} + o(\hbar).
$$
Indeed, the difference of both sides is a sum of products of the
$[f_{\al_s,i_s}, f_{\al_t,i_t}]$ with elements of $\N_+$; but $\al_s +
\al_t$ does not belong to $\Delta_+$ by hypothesis on $\la$, so
$[f_{\al_s,i_s}, f_{\al_t,i_t}] = o(\hbar)$.
The right side of (\ref{hannibal}) can the be rewritten as
\begin{align*}
& \sum_{\al_1, \ldots,\al_k\in\Delta_+, \sum_{i = 1}^k \al_i = \la; i_j}
\sum_{l,l' >0, l + l' = k}
{{\hbar^k}\over{l! l'! }}
{1\over{\on{card} \Sigma_{l,l'}}}
\\ &
\sum_{\sigma\in\Sigma_{l,l'}}
e_{\al_{\sigma(1)},i_{\sigma(1)}} \cdots e_{\al_{\sigma(l)},i_{\sigma(l)}} \otimes
e_{\al_{\sigma(l+1)},i_{\sigma(l+1)}} \cdots e_{\al_{\sigma(k)},i_{\sigma(k)}} \otimes
f_{\al_1,i_1} \cdots f_{\al_k,i_k} ,
\end{align*}
where $\Sigma_{l,l'}$ is the set of shuffle transformations of $((1, \ldots,l),
(l+1,\ldots, l+l'))$.
Therefore the right side of (\ref{hannibal}) is equal to
$$
{{\hbar^k}\over{k!}}\wt\Delta
\left( \sum_{\al_1, \ldots,\al_k\in\Delta_+, \sum_{i = 1}^k \al_i = \la; i_j}
e_{\al_1;i_1} \cdots e_{\al_k;i_k} \otimes f_{\al_1;i_1} \cdots f_{\al_k;i_k}
\right) + o(\hbar^k).
$$
Let $v$ be the $\hbar$-adic valuation of $P[\la]$. Assume that $v<k$.
Set $\bar P[\al] = \hbar^{-v} P[\al]$ mod $\hbar$. Then if we call
$\Delta_0$ the coproduct of $U\N_+$, and we set $\wt \Delta_0(x) =
\Delta_0(x) - x \otimes 1 - 1 \otimes x$, (\ref{hannibal}) gives $(\wt
\Delta_0 \otimes id) (\bar P[\al]) = 0$, so that $\bar P[\al]$ belongs
to $\N_+ \otimes U \N_-$; since $\bar P[\al]$ also belongs to $U\N_+[\al]\otimes
U\N_-[-\al]$ and $\al$ does not belong to $\Delta_+$, $\bar P[\al]$ is zero,
contradiction. Therefore $v\geq k$. Let us set $P'[\al] = \hbar^{-k}P[\al]$
mod $\hbar$; we find that
$$
(\wt\Delta_0\otimes id) \left( P'[\al] - {1\over {k!}}
\sum_{\al_1, \ldots,\al_k\in\Delta_+, \sum_{i = 1}^k \al_i = \la; i_j}
\bar e_{\al_1;i_1} \cdots \bar e_{\al_k;i_k} \otimes
\bar f_{\al_1;i_1} \cdots \bar f_{\al_k;i_k}
\right) = 0,
$$
so that $P'[\al]$ belongs to ${1\over {k!}}
\sum_{\al_1, \ldots,\al_k\in\Delta_+, \sum_{i = 1}^k \al_i = \la; i_j}
\bar e_{\al_1;i_1} \cdots \bar e_{\al_k;i_k} \otimes
\bar f_{\al_1;i_1} \cdots \bar f_{\al_k;i_k}
+ \N_+ \otimes U\N_-$; as $\N_+[\al]$ is zero, $P'[\al]$ is equal to
${1\over {k!}}
\sum_{\al_1, \ldots,\al_k\in\Delta_+, \sum_{i = 1}^k \al_i = \la; i_j}
\bar e_{\al_1;i_1} \cdots \bar e_{\al_k;i_k} \otimes
\bar f_{\al_1;i_1} \cdots \bar f_{\al_k;i_k}$,
which proves the induction.
\hfill \qed\medskip
\subsection{The generic case (proof of Cors.\ \ref{generic:1} and
\ref{generic:2})}
We have the equality $U_\hbar \N_+ \otimes_{\CC[[\hbar]]} \CC((\hbar))
= U_{q'}\N_+ \otimes_{\CC(q')} \CC((\hbar))$, therefore the graded
components of $U_{q'}\N_+$ have the same dimension as those of $U_\hbar
\N_+$. Cors.\ \ref{generic:1} and \ref{generic:2}) follow.
\section{Quantum current algebras of finite type
(proofs of results of sect.\ \ref{QC})} \label{proof:QC}
\subsection{PBW theorem and comparison with Feigin-Odesskii
algebra (proofs of Thm.\ \ref{thm:third} and Cor.\ \ref{cor:second})}
\subsubsection{Identification of algebras generated by the classical limits
of quantum currents relations}
Recall that $A$ is now assumed of finite type. Define $L\B_+$ as the Lie
subalgebra $(\HH\otimes\CC[t^{-1}]) \oplus (\N_+\otimes\CC[t,t^{-1}])$
of $\G\otimes\CC[t,t^{-1}]$.
\begin{prop} \label{fargo}
Define $U_\hbar L\B_+$ and $\wt U_\hbar L\B_+$
as the algebra with generators $h_i[k],
i = 1,\ldots, n, k\leq 0$ and $x^+_i[k], i = 1,\ldots, n, k\in\ZZ$,
and relations
$$
[h_i[k], h_j[l]] = 0, \quad [h_i[k], x^+_j[l]] = {{ q^{k d_i a_{ij}}
- q^{ - k d_i a_{ij}} }\over{2\hbar k d_i}} x^+_j[k+l],
$$
and relations (\ref{crossed:vertex}) and (\ref{q:serre:nr})
among the $x^+_j[k]$ (with $e_i$ replaced by $x_i^+$),
resp.\ (\ref{crossed:vertex}) and (\ref{variant:serre}).
There are algebra
isomorphisms from $U_\hbar L\B_+ / \hbar U_\hbar L\B_+$ and
$\wt U_\hbar L\B_+ / \hbar \wt U_\hbar L\B_+$ to $U L\B_+$,
sending $h_i[k]$ to $\bar h_i\otimes t^k$ and $x^+_i[k]$ to
$\bar x_i^+\otimes t^k$.
\end{prop}
{\em Proof.} $U_\hbar L\B_+ / \hbar U_\hbar L\B_+$ is the algebra
with generators $\bar h_i[k],\bar e_i[l],
1\leq i,j\leq n, k\leq 0,l\in\ZZ$ and relations
$$
[\bar h_i[k], \bar e_i[l]] = a_{ij} \bar e_i[k+l],
$$
and
\begin{equation} \label{ceb:1}
(z-w)[\bar e_i(z),\bar e_j(w)] = 0,
\end{equation}
$$
\Sym_{z_1,\ldots,z_{1 - a_{ij}}}
\left( \ad(\bar e_i(z_1)) \cdots \ad(\bar e_i(z_{1 - a_{ij}}))
(\bar e_j(w)) \right) = 0 ,
$$
where $\bar e_i(z) = \sum_{k\in \ZZ} \bar e_i[k] z^{-k}$.
It follows from (\ref{ceb:1}) with $i = j$ that we have $[\bar
e_i[n],\bar e_j[m]] = 0$ for all $n,m$. Therefore, $\ad(\bar e_i(z_1))
\cdots \ad(\bar e_i(z_{1 - a_{ij}})) (\bar e_j(w))$ is symmetric in the
$z_i$, so that the last equation is equivalent to
\begin{equation} \label{ceb:2}
\ad(\bar e_i(z_1)) \cdots \ad(\bar e_i(z_{1 - a_{ij}}))
(\bar e_j(w)) = 0.
\end{equation}
On the other hand, $\wt U_\hbar L\B_+ / \hbar \wt U_\hbar L\B_+$ is the algebra
with generators $\bar h_i[k]',\bar e_i[l]',
1\leq i,j\leq n, k\leq 0,l\in\ZZ$ and relations
$$
[\bar h_i[k]', \bar e_i[l]'] = a_{ij} \bar e_i[k+l]',
$$
and
\begin{equation} \label{cef:1}
(z-w)[\bar e_i(z)',\bar e_j(w)'] = 0,
\end{equation}
and
\begin{equation} \label{cef:2}
(\ad e_i[0]')^{1 - a_{ij}} e_j[k]' = 0.
\end{equation}
The algebras presentad by the pairs of relations (\ref{ceb:1}) and (\ref{ceb:2}) on one hand,
and (\ref{cef:1}) and (\ref{cef:2}) on the other, are isomorphic.
Indeed, (\ref{ceb:1}) and (\ref{cef:1}) are equivalent,
and (\ref{ceb:2}) implies (\ref{cef:2}); on the other hand,
(\ref{ceb:1}) implies that
$[e_i[0]', e_j[k+l]'] = [e_i[k]', e_j[l]']$,
so that
$[e_i[0]',[e_i[0]', e_j[k+k'+l]']] = [e_i[0]',[e_i[k]', e_j[k'+l]']]
= [e_i[k]',[e_i[0]', e_j[k'+l]']]$, because the (\ref{ceb:1})
implies that $[e_i[0]',e_i[k]'] = 0$, therefore
$[e_i[0]',[e_i[0]', e_j[k+k'+l]']]
= [e_i[k]',[e_i[k']', e_j[l]']]$; one then proves by
induction that
$(\ad e_i[0]')^p ( e_j[k + k_1 + \cdots + k_p])
= \ad e_i[k_1]' \cdots \ad e_i[k_p]'( e_j[k])$.
With $p = 1 - a_{ij}$, this relation shows that the $e_i[k]'$
satisfy (\ref{ceb:2}).
If follows that if $\wt F_+$ is the Lie algebra
defined in Prop.\ \ref{oz}, both quotient algebras $U_\hbar L\B_+ / \hbar U_\hbar L\B_+$ and
$\wt U_\hbar L\B_+ / \hbar \wt U_\hbar L\B_+$ are isomorphic to
the crossed product of $U\wt F_+$ with the derivations
$\tilde h_i[k]'$, defined by $\wt h_i[k]'(\bar e_i[l]) = a_{ij}
\bar e_i[k+l]$.
It is clear that there is a unique Lie algebra morphism $j_+$ from the
Lie algebra $\wt F_+$ defined in Prop.\ \ref{oz} to $\N_+\otimes
\CC[t,t^{-1}]$, sending $\bar e_i[k]$ to $\bar x_i^+ \otimes t^k$. Let
us prove that it is an isomorphism.
For this, let us define $\wt F$ as the Lie algebra with generators
$\wt x_i^\pm[k],\wt h_i^\pm[k]$, $1\leq i\leq n, k\in\ZZ$,
and relations given by the coefficients of the monomials in
$$
(z-w)[\wt x_i^\pm(z), \wt x_j^\pm(w)] = 0, \quad\on{if}\quad
x,y\in\{\wt x_i^\pm\},
$$
$$
[\wt h_i(z), \wt h_j(w) ] = 0,
$$
$$
[\wt h_i(z), \wt x_j^\pm(w)] = \pm a_{ij} \delta(z/w) \wt x_j^\pm(w),
$$
$$
[\wt x_i^+(z), \wt x_j^-(w) ] = \delta_{ij}\delta(z/w)
\wt h_i(z),
$$
$$
\ad(\wt x_i^\pm(z_1)) \cdots \ad(\wt x_i^\pm ( z_{1- a_{ij}}) )
(\wt x_j^\pm(w)) = 0,
$$
where we set $\wt x(z) = \sum_{k\in\ZZ} x[k]z^{-k}$ for $x$ in
$\{\wt x_i^\pm,\wt h_i\}$.
\begin{lemma} \label{cite:U}
(In this Lemma, $\G$ may be an arbitrary Kac-Moody Lie algebra.)
Let $W$ be the Weyl group of $\G$, and $s_i$ be its elementary
reflection associated to the root $\al_i$. Then there is a unique
action of $W$ on $\wt F$ such that
$$
s_i(\wt x_i^\pm[k]) = \wt x_i^\mp[k],
$$
$$
s_i(\wt x_j^\pm[k]) = \ad(\wt x_i^\pm[0])^{-a_{ij}}
(\wt x_j^\mp[k]), \quad \on{if}\quad j\neq i,
$$
$$
s_i(\wt h_j^\pm[k]) = \wt h_j^\pm[k] - a_{ij}\wt h_i^\pm[k].
$$
\end{lemma}
{\em Proof of Lemma.} The proof follows the usual proof for Kac-Moody Lie
algebras. For example, if $j,k$ are different from $i$, we have
\begin{align*}
(z-w) [s_i(\wt x_j^\pm(z)
) , s_i( \wt x_k^\pm(w) ) ] & =
(z-w) [ \ad(\wt x_i^\pm[0])^{-a_{ij}}
(\wt x_j^\pm(z))
, \ad(\wt x_i^\pm[0])^{-a_{ik}}(\wt x_k^\pm(w)) ]
\\ & =
\ad (\wt x_i^\pm[0])^{-a_{ij} -a_{ik} } \left(
(z-w) [\wt x_j^\pm(z) , \wt x_k^\pm(w) ] \right )
\end{align*}
because the Serre relations imply that $\ad(\wt x_i^\pm[0])^{1
- a_{il}} (\wt x_l(u)) = 0$ for $j = k,l$ and $u = z,w$; therefore
$(z-w) [s_i(\wt x_j^\pm(z)
) , s_i( \wt x_k^\pm(w) ) ]$ is zero.
\hfill \qed\medskip
\begin{lemma} \label{india}
There is a unique Lie algebra isomorphism $j$ from $\wt F$ to
$\G\otimes \CC[t,t^{-1}]$, such that $j(\wt x[k]) = \bar x\otimes t^k$,
for any $x$ in $\{x_i^\pm, h_i\}$ and $k$ in $\ZZ$.
\end{lemma}
{\em Proof of Lemma.} Let $\wt F_-$ be the Lie algebra with generators
$\bar x_i^-[k], 1\leq i\leq n, k\in\ZZ$ and relations (\ref{ceb:1}) and (\ref{ceb:2}),
with $\bar x_i^+[k]$ replaced by $\bar x_i^-[k]$, and let $\wt H$ be the
abelian Lie algebra with generators $\bar h_i[k], 1\leq i \leq n, k\in\ZZ$.
There are unique Lie algebra morphisms from $\wt F_\pm$ and $\wt H$ to $\wt F$,
sending the $\bar x_i^\pm[k]$ to $\wt x_i^\pm[k]$ and the $\bar h_i[k]$ to
$\wt h_i[k]$. These morphisms are injections, so that we will indentify
$\wt F_\pm$ and $\wt H$ with their images in $\wt F$.
Moreover, let $F_\pm$ be the free Lie algebras with generators
$x_i^\pm[k]^{F}$, $i = 1,\ldots,n$, $k$ integer. Endow $F_\pm \oplus \wt H$
with the Lie algebra structure such that $\wt H$ is abelian, $F_\pm$ is a
Lie subalgebra of $F_\pm \oplus \wt H$, and $[\wt h_i[k], x_i^\pm[l]^F]
= \pm a_{ij} x_i^\pm[k+l]^F$
There are unique derivations $\Phi^{\mp}_{i,k}$ from $F_\pm$ to
$F_\pm \oplus \wt H$ such that
$$
\Phi^\mp_{i,k}(x_{i'}^\pm[l]) = \delta_{ii'} \wt h_i[k+l].
$$
Let $I^F_\pm$ be the ideals of $F_\pm$ generated by relations (\ref{ceb:1})
and (\ref{ceb:2}); then computation shows that $I^F_\pm$ are
preserved by the $\Phi^\mp_{i,k}$. It follows that $\wt F$ is the
direct sum of its subspaces $\wt F_\pm$ and $\wt H$.
The rules $\deg(\wt x_i^\pm[k]) = (\pm\eps_i,k)$ and $\deg(\wt h_i[k]) =
(0,k)$ define a Lie algebra grading of $\wt F$ by $\ZZ^n\times \ZZ$,
because the relations of $\wt F$ are homogeneous for this grading.
Clearly, $\dimm \wt F_\pm[(\pm\eps_i,k)] = 1$ for any $i$ and $k$, so
that $\dimm \wt F[(\pm\eps_i,k)] = 1$.
Let $\al$ be any root on $\G$. Then there is some simple root
$\pm\eps_i$ an element $w$ of $W$ such that $\al = w(\pm\eps_i)$.
Then $\wt F[(\pm\eps_i,k)] = \wt F[(\al,k)]$ so that $\dimm\wt F[(\al,k)] = 1$.
It is clear that the map $j$ defined in the statement of the Lemma
defines a Lie algebra morphism. Define a grading by $\ZZ^r\times\ZZ$ on
$\G\otimes\CC[t,t^{-1}]$, by the rules $\deg(x\otimes t^k) =
(\deg(x),k)$, for $x$ a homogeneous (for the root grading) element of
$\G$. Then $j$ is a graded map. Moreover, if $\al$ is in $\pm\Delta_+$
and $x$ is a nonzero element of $\G$ od degree $\al$, then $x$ can be
written as a $\sum \la_{i_1,\cdots,i_p} [x^\pm_{i_1}, [\ldots,
x^\pm_{i_p}]]$; then the image by $j$ of $\sum \la_{i_1,\cdots,i_p}
[x^\pm_{i_1}[0], [\ldots, x^\pm_{i_p}[k]]]$ is equal to $x\otimes t^k$;
therefore the map induced by $j$ from $\wt F[(\al,k)]$ to
$\G\otimes\CC[t,t^{-1}][(\al,k)]$ is nonzero and therefore an
isomorphism.
It follows that $\Ker j$ is equal to $\sum_{\al\in\ZZ^n \setminus
(\Delta_+\cup \{0\}\cup (-\Delta_+)), k\in\ZZ} \wt F[(\al,k)]$. Any element of
$\wt F[(\al,k)]$ is a linear combination of brackets
$[x^\pm_{i_l}[k_l], [\ldots, x^\pm_{i_1}[k_1]]]$, with $\sum_{s=1}^l \pm
\eps_{i_s} = \al$. Assume $\al$ is not a root of $\G$ and let $l'$ be
the smallest integer such that $\sum_{s=1}^{l' + 1} \pm \eps_{i_s}$ is
not a root of $\G$. Let us show that each $[x^\pm_{i_{l' + 1}}[k_{l' +
1}], [\ldots, x^\pm_{i_1}[k_1]]]$ vanishes. It follows from the
fact that $j$ is an isomorphism when restricted to the parts of
degree in $\Delta_+ \cup (-\Delta_+)$
that we may write each $[x^\pm_{i_{l'}}[k_{l'}], [\ldots,
x^\pm_{i_1}[k_1]]]$ as a linear combination $\sum_s \la_s
[x^\pm_{i_{l'}}[k^{(s)}(i_{l'})] \cdots x^\pm_{i_{1}}[k^{(s)}(i_{1})]]$,
where for each $s$, $i\mapsto k^{(s)}(i)$ is a map from
$\{1,\ldots,n\}$ to $\ZZ$, such that $k^{(s)}(i_{l' + 1}) = k_{l' +
1}$. The defining relations for $\N_\pm$ hold among the
$x^\pm_1[k^{(s)}_1], \ldots, x^\pm_n[k^{(s)}_n]$, therefore we have Lie
algebra maps from $\N_\pm$ to $\wt F^\pm$ sending each $x_i^\pm$ to
$x^\pm_i[k^{(s)}_i]$. $[x^\pm_{i_{l' + 1}}[k^{(s)}(i_{l' + 1})] \cdots
x^\pm_{i_{1}}[k^{(s)}(i_{1})]]$ is the image of zero by one of these
maps, and is therefore zero. It follows that $[x^\pm_{i_{l'}}[k_{l'}], [\ldots,
x^\pm_{i_1}[k_1]]]$ vanishes, so that $\Ker j$ is zero.
\hfill \qed \medskip
{\em End of proof of Proposition.} The restriction of $j$ to
$\wt F_+$ coincides with the map $j_+$ define before Lemma
\ref{cite:U}, therefore $j_+$ induces an isomorphism between
$\wt F_+$ and $\N_+\otimes\CC[t,t^{-1}]$.
\hfill \qed \medskip
\subsubsection{Crossed product algebras $\cV^L$ and $\cS^L$}
For $k$ an integer $\leq 0$ and $1\leq i \leq n$,
define endomorphisms $\wt{h_i[k]}$ of $FO$ by
\begin{equation} \label{def:hik}
(\wt{h_i[k]} f) (t^{(i)}_l) = [\sum_{j=1}^n \sum_{l=1}^{k_j}
{{q^{kd_i a_{ij}} - q^{- kd_i a_{ij}}}\over{2\hbar kd_i}}
(t^{(j)}_l)^k ] f(t^{(i)}_l)
\end{equation}
if $f\in FO_\kk$. The $\wt{h_i[k]}$ are
derivations of $FO$. These derivations preserve $LV$, therefore
they preserve $\langle LV\rangle$.
Define $\cV^L$ and $\cS^L$ as the crossed product algebras of $\langle
LV \rangle $ and $FO$ with the derivations $\wt{x[k]}$: $\cV^L$, resp.\
$\cS^L$ is equal to $\langle LV\rangle \otimes \CC[h_i[k]^{\cV^L}, k\leq
0]$, resp.\ $FO \otimes \CC[h_i[k]^{\cV^L}, k\leq 0]$; both spaces are
endowed with the products given by formula (\ref{formula:crossed:pdt}),
where $x$ now belongs to $\langle LV\rangle$, resp.\ $FO$ and the $X_s$
are replaced by $h_i[k]$, $k\leq 0$.
Define $\hat\cV^L$ and $\cS^L$ as the partial $\hbar$-adic completions
$\CC[h_i[k]^{\cV^L}, k<0] \otimes \langle LV\rangle
\otimes_{\CC[[\hbar]]} \CC[h_i[0]^{\cV^L}][[\hbar]]$ and
$\CC[h_i[k]^{\cV^L}, k<0] \otimes FO \otimes_{\CC[[\hbar]]}
\CC[h_i[0]^{\cV^L}][[\hbar]]$. .
\begin{lemma} \label{LV:free}
1) The rules $\deg(h_i[k]) = 0,\deg(t_i^k) = \eps_i$ define gradings of
$\langle LV\rangle$, $FO$, $\cV^L$, $\cS^L$, $\hat\cV^L$ and $\hat\cS^L$ by
$\NN^n$, which are compatible with the inclusions. For $X$ any of
these algebras, we denote by $X_\kk$ its homogeneous component of degree
$\kk$. $X$ is therefore the direct sum of the $X_\kk$.
2) For any $\kk$, $\langle LV\rangle_\kk$ is a free
$\CC[[\hbar]]$-modules with a countable basis.
3) For any $\kk$, $\cV^L_\kk$ and $\cS^L$ are free
$\CC[[\hbar]][h_i[k]^{\cV^L}]$-modules; and $\hat\cV^L_\kk$ and
$\hat\cS^L$ are free $\CC[[\hbar]][h_i[k]^{\cV^L},k<0]
\otimes_{\CC[[\hbar]]} \CC[h_i[0]^{\cV^L}][[\hbar]]$-modules.
\end{lemma}
{\em Proof.} 1) is clear. $\langle LV\rangle_\kk$ is a
$\CC[[\hbar]]$-submodule of $FO_\kk$, and by Lemma \ref{free:inf:dim},
it is a free $\CC[[\hbar]]$-module with a countable basis. This shows
2). 3) is a direct consequence of 2). \hfill \qed \medskip
\subsubsection{Ideals and completions}
Define for $N$ positive integer, $I_N$ as the left ideal of $\langle
LV\rangle$ generated by the elements $(t_i^k)$ of $FO_{\eps_i}$, $k\geq
N$, $i = 1,\ldots,n$. Define $\cI_N$ and $\hat \cI_N$ as the left
ideals of $\cV^L$ and $\hat\cV^L$ generated by the same family. For
$s\geq 0$, set $I_N^{(s)} = \hbar^{-s} (I_N \cap \hbar^s \langle
LV\rangle)$, and $I_N^{(\infty)} = \cup_{s\geq 0} I_N^{(s)}$; define
$\cI_N^{(s)}$, $\cI_N^{(\infty)}$ and $\hat\cI_N^{(s)}$,
$\hat\cI_N^{(\infty)}$ in the same way.
For any integer $a$, define $LV^{\geq a}$ as the subspace of $FO$ equal to
the direct sum $\oplus_{i= 1}^n t_i^a \CC[[\hbar]][t_i]$ and let
$\langle LV^{\geq a}\rangle$ be the subalgebra of $FO$ generated by
$LV^{\geq a}$. Define $I_N^{(0),\geq a}$ as the left ideal of $\langle LV^{\geq a}\rangle$
generated by the $t_i^{k},k\geq N$ and
$I_N^{\geq a}$ as the ideal of $\langle LV^{\geq a}\rangle$ formed of the elements
$x$ such that for some $k\geq 0$, $\hbar^k x$ belongs to $I_N^{(0),\geq a}$.
For any integer $a$ and $\kk$ in $\NN^r$, define $FO^{\geq a}$ as the subspace of
$FO_\kk$
consisting of the rational functions
$$
g(t^{(i)}_\al) = {1\over{\prod_{i = 1}^n} \prod_{1\leq \al \leq k_i, 1\leq \beta\leq k_j (t^{(i)}_\al
- t^{(j)}_\beta)}} f(t^{(i)}_\al),
$$
where the $f(t^{(i)}_\al)$ have degree $\geq a$ in each variable $t^{(i)}_\al$
and the total degree of $g$ is $\geq (\sum_i k_i)a$.
Set $FO^{\geq a} = \oplus_{\kk\in\NN^n} FO^{\geq a}_\kk$. Then
$FO^{\geq a}$ is a subalgebra of $FO$.
Define $\cI_{(N)}^{\geq a}$ as the set of elements of $FO^{(\geq a)}$, where
$f(t^{(i)}_\al)$ has total degree $N$ in the variables $t^{(i)}_\al$, and let
$\cI_N^{\geq a}$ be the direct sum $\oplus_{k\geq N} \cI_N^{\geq a}$.
\begin{lemma} \label{ladino}
For $(J_N)_{N>0}$ a family of left ideals of some algebra $A$, say that
$(J_N)_{N>0}$ has property $(*)$ if for any integer $N>0$ and element
$a$ in $A$, there is an integer $k(a,N)>0$ such that $J_N a\subset
J_{k(a,N)}$ for any $N$ large enough, and $k(N,a)$ tends to infinity
with $N$, $a$ being fixed. Then the inverse limit $\limm_{\leftarrow N}
A/J_N$ has an algebra structure.
Say that $J_N$ has property $(**)$ if for any integer $N>0$ and element
$a$ in $A$, there is are integer $k'(a,N)$ and $k''(a,N)>0$ such that $J_N a\subset
J_{k'(a,N)}$ and $a J_N\subset
J_{k''(a,N)}$ for any $N$ large enough, and $k(N,a)$ tends to infinity
with $N$, $a$ being fixed. In that case also, the inverse limit $\limm_{\leftarrow N}
A/J_N$ has an algebra structure.
1) The family $(I^{(\infty)}_N)_{N>0}$ of ideals of $\langle LV\rangle$
has property $(*)$;
2) the family $(\cI^{(\infty)}_N)_{N>0}$ of ideals of $\cV^L$ has property $(*)$;
3) the family $(\hat\cI^{(\infty)}_N)_{N>0}$ of ideals of $\hat\cV^L$ has
property $(*)$;
4) the family $(I_N^{\geq a})_{N>0}$ of ideals of $\langle LV^{\geq a} \rangle$
has property $(*)$;
5) the family $(\cI^{\geq a}_N)_{N>0}$ of ideals of $FO^{\geq a}$ has property $(**)$.
\end{lemma}
{\em Proof.} Set for any $a$ in $A$ and $N>0$, $k'(a,N) =$ inf
$\{k| J_N a \subset J_k\}$; then $k'(N,a)$ tends to infinity with $N$,
$a$ being fixed and we have $k'(N,a')\geq$ inf $(k(N,a), p)$ if
$a'$ belongs to $a + J_p$.
An element of $\limm_{\leftarrow N} A / J_N$ is a family $(a_N)_{N>0}$,
$a_N\in A / J_N$, such that $a_{N+1} + J_N = a_N$. For $a = (a_N)_{N>0}$
and $b = (b_N)_{N>0}$ in $\limm_{\leftarrow N} A / J_N$, choose
$\beta_N$ in $b_N$ and let $N'(N,\beta_N)$ be the smallest integer $N'$
such that $k'(N',\beta_N)\geq N$; $N'(N,\beta_N)$ is independent of the
choice of $\beta_N$, we denote it $N'(N,b)$. Choose then $\al_N$ in
$a_{N'(N,b)}$; then $\al_N\beta_N + J_N$ is independent of the choice of
$\al_N$ and $\beta_N$; one checks that $\al_{N+1}\beta_{N+1} + J_N =
\al_N\beta_N + J_N$, so that $(\al_N\beta_N + J_N)_{N>0}$ defines an
element of $\limm_{\leftarrow N} A / J_N$. The product $ab$ is defined
to be this element.
The construction is similar in the case of property $(**)$.
1) The equality
\begin{align*}
& t_i^k * t_j^l = q^{-n d_i a_{ij}} t_i^{k-n} * t_j^{l+n}
+ q^{- d_i a_{ij}} t_j^l * t_i^k \\ & + \sum_{n' = 1}^{n-1}
(q^{ - (n' + 1) d_i a_{ij}} - q^{ - (n' - 1) d_i a_{ij}} )
t_j^{l+n'} * t_i^{k-n'} - q^{- (n-1)d_i a_{ij}} t_j^{l+n} *
t_i^{k-n} ,
\end{align*}
where $n = N - l$ and $k\geq 2N - l$, implies that if $k\geq 2N - l$,
$t_i^k * t_j^l$ belongs to $I_N$. It follows that $I_{2N - l} * t_j^l
\subset I_N$. Set then $k(t_j^l,N) = [{1\over 2}(N + l/2)] + 2$; for
$a$ in $\langle LV \rangle$, and any decomposition $dec$ of $a$ as a
sum $\sum_{(j_i),(l_i)} \la_{(j_i),(l_i)} t_{j_1}^{l_1} * \cdots *
t_{j_p}^{l_p}$, define $k(a,N,dec)$ as the smallest of integers
$k(t_{j_p}^{l_p}, \cdots, k(t_{j_1}^{l_1},N))$; finally, define
$k(a,N)$ as the largest of all $k(a,N,dec)$.
The family $(I_N)_{N>0}$ has property $(*)$, with this function
$k(a,N)$. Then for any $a$ in $\langle LV\rangle$, $(I_N \cap \hbar^s
\cV^L) a \subset \hbar^s \cV^L \cap I_{k(a,N)}$, so that $I_N^{(s)} a
\subset I^{(s)}_{k(a,N)}$. It follows that we have also
$I_N^{(\infty)} a \subset I^{(\infty)}_{k(a,N)}$, so that the families
$(I_N^{(s)})_{N>0}$ and $(I_N^{(\infty)})_{N>0}$ have property $(*)$.
2) follows from the fact that $I_{N} h_j[l]^{\cV^L}
\subset I_{N + l}$.
3) follows from the fact that for $a$ any element of $\CC[h_i[0]^{\cV^L}][[\hbar]]$,
we have $\hat\cI_N^{(\infty)}a \subset \hat\cI_N^{(\infty)}$.
4) is proved in the same way as 1) ans 2).
5) For $\kk$ in $\NN^n$, set $|\kk| = \sum_{i = 1}^n k_i$. Then if $f$ belongs
to $FO^{\geq a}_{\kk}$, we have $f * \cI_N^{\geq a} \subset \cI^{\geq a}_{N + a|\kk|}$
and $\cI_N^{\geq a} * f\subset \cI^{\geq a}_{N + a|\kk|}$. Therefore the family
$(\cI_N^{\geq a})_N$ has property $(**)$.
\hfill \qed \medskip
It follows that the inverse limits
$\limm_{\leftarrow N} \langle LV\rangle / I_N^{(\infty)}$,
$\limm_{\leftarrow N} \cV^L / \cI_N^{(\infty)}$,
$\limm_{\leftarrow N} \hat\cV^L / \hat\cI_N^{(\infty)}$,
$\langle LV^{\geq a}\rangle / I_N^{\geq a}$ and
$\limm_{\leftarrow N}FO^{\geq a} / \cI_N^{\geq a}$
have algebra structures.
Moreover, as we have $\cI_N^{(\infty)}\cap \langle LV\rangle = I_N^{(\infty)}$ and
$\hat\cI_N^{(\infty)}\cap \cV^L = \cI_N^{(\infty)}$, we have natural algebra inclusions
$$
\limm_{\leftarrow N} \langle LV\rangle / I_N^{(\infty)} \subset
\limm_{\leftarrow N} \cV^L / \cI_N^{(\infty)} \subset
\limm_{\leftarrow N} \hat\cV^L / \hat\cI_N^{(\infty)}.
$$
Moreover, there exists a function $\phi(\kk,N)$, tending to infinity with $N$,
such that $\cI_{N,\kk}^{\geq a} \cap \langle LV^{\geq a}
\rangle \subset I_{\phi(\kk,n)}$. Indeed, if the $k_i$ are $\geq a$ and
$t_{i_1}^{k_1} * \cdots * t_{i_{l}}^{k_l}$ belongs to $\cI^{\geq a}_{N,\kk}$
($l = |\kk|$), then $k_1 + \cdots + k_l \geq N$ so that one of the $k_i$ is
$\geq N/l$. The statement then follows from the proof of the above Lemma, 1).
It follows that we have an algebra inclusion
$$\limm_{\leftarrow N}
\langle LV^{\geq a} \rangle / I^{\geq a}_N \subset \limm_{\leftarrow N}
FO^{\geq a} / \cI_N^{\geq a}.
$$
If a family $(J_N)_{N>0}$ of left ideals of the algebra $A$ has
property $(*)$, and $B$ is any algebra, the family $(J_N \otimes B)_{N>0}$
also satisfies $(*)$; therefore the inverse limit $\limm_{\leftarrow N}
(A\otimes B) / (J_N \otimes B)$ has an algebra structure. It follows
that we have algebra structures on $\limm_{\leftarrow N} \langle
LV\rangle \otimes A/ I_N^{(\infty)}\otimes A$, $\limm_{\leftarrow N} \cV^L
\otimes A/ \cI_N^{(\infty)}\otimes A$ and $\limm_{\leftarrow N} \hat\cV^L \otimes
A/ \hat\cI_N^{(\infty)}\otimes A$ for any algebra $A$.
\subsubsection{Topological Hopf structures on $\cV^L$ and $\cS^L$}
For $i = 1,\ldots,n$ and $l\geq 0$, define $K_i[-l]$ as the element of $\cV^L$
(or $\cS^L$)
$$
K_i[-l] = e^{ - \hbar d_i h_i[0]^{\cV^L}} S_l(-2\hbar d_ih_i[k]^{\cV^L},k<0),
$$
where
$S_l(z_1,z_2,\ldots)$ are the Schur polynomials in variables $(z_i)_{i<0}$,
which are determined by the relation $\exp(\sum_{i<0} z_it^{-i})
= \sum_{l\leq 0} S_l(z_k) t^{-l}$.
\begin{prop} \label{Hopf:FO}
There is unique graded algebra morphism $\Delta_{\cS^L}$ from $\cS^L$
to $\limm_{\leftarrow N}(\cS^L \otimes_{\CC[[\hbar]]}\hat\cS^L) /
(\cI_N^{\cS} \otimes_{\CC[[\hbar]]} \hat\cS^L )$, such that
$$
\Delta_{\cS^L}(h_i[k]) = h_i[k] \otimes 1 + 1\otimes h_i[k]
$$
for $1\leq i \leq n$ and $k\leq 0$, and
its restriction to $\cS^L_\kk$ is the direct sum of the maps
$\Delta_{\cS^L}^{\kk',\kk''}: FO^{\geq a}_\kk \to
\limm_{\leftarrow N}(\cI^{\geq a}_{\kk'} \otimes_{\CC[[\hbar]]} \hat\cS^L_{\kk''})
/ [FO_{\kk',N}^{\geq a}\otimes_{\CC[[\hbar]]} \hat \cS^L_{\kk''} ] $,
where $\kk = \kk' + \kk''$, defined by
$$
\Delta_{\cS^L}^{\kk',\kk''}(P) = \sum_{p_1,\ldots,p_{N'}\geq 0}
\left( \prod_{i = 1}^{N'} u_i^{p_i} P'_{\al}(u_1,\ldots,u_{N'})
\right) \otimes \left(
P''_\al(u_{N' + 1}, \ldots, u_N)
\prod_{i = 1}^{N'} K_{\eps(i)}[-p_i]
\right),
$$
where $N' = \sum_{i=1}^n k'_i$, $N'' = \sum_{i=1}^n k''_i$,
$N = N' + N''$, the arguments of the functions in $FO_{\kk'}$
and $FO_{\kk''}$ are respectively $(t^{(i)}_j)_{1\leq i\leq n,
1\leq j\leq k'_i}$ and $(t^{\prime (i)}_j)_{1\leq i\leq n,
1\leq j\leq k''_i}$; we set
$$
(u_1,\ldots, u_{k'_1}) = (t^{(1)}_1, \ldots, t^{(1)}_{k'_1}), \ldots,
(u_{k'_1 + \ldots + k'_{n-1} + 1}, \ldots, u_{N'}) =
(t^{(n)}_1, \ldots, t^{(n)}_{k'_n}),
$$
$$
(u_{N' + 1},\ldots, u_{N' + k''_1}) = (t^{\prime (1)}_1, \ldots, t^{\prime
(1)}_{k''_1}), \ldots,
(u_{N' + k''_1 + \ldots + k''_{n-1} + 1}, \ldots, u_{N}) =
(t^{\prime (n)}_1, \ldots, t^{\prime (n)}_{k''_n}),
$$
$$
t^{(i)}_{k'_i + j} = t^{\prime (i)}_j, \quad j = 1,\ldots, k''_i,
$$
$$
(t_1,\ldots, t_{k_1}) = (t^{(1)}_1, \ldots, t^{(1)}_{k_1}), \ldots,
(t_{k_1 + \ldots + k_{n-1} + 1}, \ldots, t_{N}) =
(t^{(n)}_1, \ldots, t^{(n)}_{k_n}),
$$
and
\begin{equation} \label{weil}
\sum_{\al} P'_\al(u_1,\ldots,u_{N'}) P''_\al(u_{N'+1},\ldots,u_{N})
= P(t_1,\ldots,t_N) \prod_{1\leq l\leq N', N' + 1\leq l'\leq N}
{{u_{l'} - u_l}\over{q^{\langle \eps(l),\eps(l') \rangle} u_{l'} -
u_l}},
\end{equation}
for $l$ in $\{1,\ldots,N'\}$, resp.\ $\{N' + 1,\ldots,N\}$,
$\eps(l)$ is the element of $\{1,\ldots,n\}$
such that $u_l = t^{(\eps(l))}_j$ , resp.\ $u_l = t^{\prime(\eps(l))}_j$
for some $j$; in (\ref{weil}), the ratios are expanded for $u_l << u_{l'}$.
\end{prop}
{\em Proof.} Define $FO^{(2)}$ as the $(\NN^r)^2$-graded $\CC[[\hbar]]$-module
$FO^{(2)} = \oplus_{\kk,\kk'\in\NN^r} FO^{(2)}_{\kk,\kk'}$, where
\begin{align*}
& FO^{(2)}_{\kk,\kk'} =
\\ & {1\over{
\prod_{i<j, 1\leq \al \leq k_i,
1\leq \beta \leq k_j} ( t^{(i)}_\al - t^{(j)}_{\beta} )
\prod_{i<j, 1\leq \al \leq k'_i, 1\leq \beta \leq k'_j
} ( u^{(i)}_\al - u^{(j)}_{\beta} )
\prod_{i,j, 1\leq \al \leq k_i, 1\leq \beta \leq k'_j} (t^{(i)}_\al - u^{(j)}_\beta) }}
\cdot \\ & \cdot \CC[[\hbar]][(t^{(i)}_j)^{\pm1}, (u^{(i)}_{j'})^{\pm1}, j = 1,\ldots, k_i ,
j' = 1,\ldots, k'_i ]^{\prod_{i=1}^n\gotS_{k_i} \times \gotS_{k'_i}} ,
\end{align*}
where the groups $\gotS_{k_i}$ and $\gotS_{k'_i}$ act by permutation of the
variables $t^{(i)}_j$ and $u^{(i)}_j$. Define on $FO^{(2)}$, the graded composition
map $*$ as follows: for $f$ in $FO^{(2)}_{\kk,\kk'}$ and $g$ in $FO^{(2)}_{\bl,\bl'}$,
\begin{align*}
& (f*g)(t_1,\ldots, t_P, u_1, \ldots, u_{P'}) =
\Sym_{t^{(1)}_j} \cdots \Sym_{t^{(n)}_j}
\Sym_{u^{(1)}_j} \cdots \Sym_{u^{(n)}_j}
\\ & (
\prod_{1\leq i \leq N, N+1\leq j \leq P} { {q^{ \langle \eps_t(i), \eps_t(j) \rangle }
t_i - t_j }\over{t_i - t_j } }
\prod_{1\leq i \leq N', N+1\leq j \leq P} { {q^{ \langle \eps_u(i), \eps_t(j) \rangle }
u_i - t_j }\over{u_i - t_j }}
\\ & \prod_{1\leq i \leq N, N'+1\leq j \leq P'} { {q^{ \langle \eps_t(i), \eps_u(j) \rangle }
t_i - u_j }\over{t_i - u_j }}
\prod_{1\leq i \leq N', N'+1\leq j \leq P'} {{q^{ \langle \eps_u(i), \eps_u(j) \rangle }
u_i - u_j }\over{u_i - u_j }}
\\ &
f(t_1,\ldots, t_N, u_1,\ldots, u_M)
g(t_{N+1},\ldots, t_P, u_{M+1},\ldots, u_P) ) ,
\end{align*}
where $N = \sum_i k_i, N' = \sum_i k'_i, M = \sum_i l_i, M' = \sum_i l'_i,
P = N + M, P' = N' + M'$, and $(t_1,\ldots, t_{k_1 + k'_1}) = (t^{(1)}_1, \ldots,
t^{(1)}_{k_1 + k'_1})$, ...,
$(t_{k_1 + \cdots + k'_{n-1} + 1 },\ldots, t_{P}) = (t^{(n)}_1, \ldots,
t^{(n)}_{k_n + k'_n})$, and $(u_1,\ldots, u_{l_1 + l'_1}) = (u^{(1)}_1, \ldots,
u^{(1)}_{l_1 + l'_1})$, ...,
$(u_{l_1 + \cdots + l'_{n-1} + 1 },\ldots, u_{P'}) = (u^{(n)}_1, \ldots,
u^{(n)}_{l_n + l'_n})$. We set $\eps_x(\al) = i$ if $x_\al = x^{(i)}_j$ for some $j$
($x$ is $t$ or $u$). It is easy to check that $µ$ defines an algebra structure on
$FO^{(2)}$.
Let $\Delta_{FO}$ be the linear map from $FO$ to $FO^{(2)}$, which maps $FO_{\kk}$
to $\oplus_{\kk' + \kk'' = \kk} FO^{(2)}_{\kk',\kk''}$ as follows
$$
\Delta_{FO}(P)(t^{(1)}_1,\ldots, t^{(n)}_{k'_n},
u^{(1)}_1,\ldots, u^{(n)}_{k''_n} ) =
P(t^{(1)}_1,\ldots, t^{(1)}_{k'_1}, u^{(1)}_1,\ldots, u^{(1)}_{k''_1},
t^{(2)}_1, \ldots, u^{(n)}_{k''_n}) .
$$
Then it is immediate that $\Delta_{FO}$ defines an algebra morphism.
Consider the $(\NN^n)^2$-graded map $\mu : FO^{(2)} \to
\cup_a \limm_{\leftarrow N}
(FO^{\geq a} \otimes_{\CC[[\hbar]]} \hat\cS^L)
/ [FO_N^{\geq a}\otimes_{\CC[[\hbar]]} \hat
\cS^L]$
defined by
$$
\mu(P) = \sum_{p_1,\ldots,p_{N'}\geq 0} (t_1^{p_1}\cdots t_{N'}^{p_{N'}}
\bar P_\al) \otimes (\bar P'_\al K_{\eps_t(1)}[-p_1] \cdots K_{\eps_t(N')}[-p_{N'}])
$$
if $P(t_1,\ldots, t_N, u_1,\ldots, u_{N'})$ belongs to $FO^{(2)}_{\kk,\kk'}$, and we set
$$
P(t_1,\ldots, u_{N'}) \prod_{1\leq l \leq N, 1\leq l'\leq N'}
{{t_l - u_{l'}}\over{q^{\langle \eps_t(l), \eps_u(l')\rangle} t_l - u_{l'}}}
= \sum_{\al} \bar P_\al(t_1,\ldots,t_{N}) \bar P'_\al(u_1,\ldots,u_{N'})
$$
(the expansion is for $u_{l'} << t_l$).
Let $P,Q$ belong to $FO_{\kk,\bl}$ and $FO_{\kk',\bl'}$. $\mu(P)\mu(Q)$ is
equal to
\begin{align} \label{pioche}
& \sum_{\al,\beta} \left( t_1^{p_1} \cdots t_N^{p_N}
P_\al(t_1,\cdots,t_N) \otimes P'_\al(u_1,\ldots,u_P)
K_{\eps_t(1)}[-p_1]\cdots K_{\eps_t(N)}[-p_N]\right) \cdot \\ &
\nonumber \cdot \left( t_1^{\prime p'_1} \cdots t_{N'}^{\prime p'_{N'}}
Q_\al(t'_1,\cdots,t'_{N'}) \otimes Q'_\al(u'_1,\ldots,u'_{P'})
K_{\eps_{t'}(1)}[-p'_1]\cdots K_{\eps_{t'}(1)}[-p'_{N'}]\right) ;
\end{align}
since
$$
(\sum_{i\geq 0} t^i K_\al[-i])
Q(u_1,\ldots,u_P) = \prod_{i=1}^P {{u_i -
q^{\langle \eps_u(i), \al \rangle }t}\over
{ q^{\langle \eps_u(i), \al \rangle } u_i - t}}
Q(u_1,\ldots,u_P) (\sum_{i\geq 0} t^i K_\al[-i]) ,
$$
(\ref{pioche}) is equal to
\begin{align} \label{pioche'}
& \sum_{\gamma}
\sum_{\al,\beta} \left( f_\gamma(t_1,\ldots, t_N) t_1^{p_1} \cdots t_N^{p_N}
P_\al(t_1,\cdots,t_N) \otimes
P'_\al(u_1,\ldots,u_P)
\right) \cdot
\\ & \nonumber \cdot
\left( t_1^{\prime p'_1} \cdots t_{N'}^{\prime p'_{N'}} Q_\al(t'_1,\cdots,t'_{N'}) \otimes
g_\gamma(u'_1,\ldots,u'_{P'})
Q'_\al(u'_1,\ldots,u'_{P'}) K_{\eps_t(1)}[-p_1]\cdots
\right. \\ & \nonumber
K_{\eps_t(N)}[-p_N] K_{\eps_{t'}(1)}[-p'_1]\cdots K_{\eps_{t'}(1)}[-p'_{N'}] ) ,
\end{align}
with
$$
\sum_\gamma f_\gamma(t_1,\ldots, t_N) g_\gamma(u'_1,\ldots, u'_{P'}) =
\prod_{1\leq i \leq N, 1\leq j\leq P'}
{{u_j - q^{\langle \eps_u(j), \al \rangle }t_i}\over
{q^{\langle \eps_u(j), \al \rangle } u_j - t_i}}.
$$
After some computation, one finds that (\ref{pioche'}) coincides with
$\mu(PQ)$. Therefore $\mu$ is an algebra morphism.
It follows that the composition $\mu\circ \Delta_{FO}$ is an algebra morphism.
This composition coincides with $\Delta_{\cS^L}$, which is therefore an algebra
morphism.
\hfill\qed\medskip
\begin{remark}
$\Delta_{\cS^L}^{\kk',\kk''}$ may also be expressed by the formula
$$
\Delta_{\cS^L}^{\kk',\kk''}(P) = \sum_{p_1,\ldots,p_{N'}\geq 0}
\left( \prod_{i = 1}^{N'} u_i^{p_i} P'''_{\al}(u_1,\ldots,u_{N'})
\right) \otimes \left( \prod_{i = 1}^{N'} K_{\eps(i)}[-p_i]
P''''_\al(u_{N' + 1}, \ldots, u_N)
\right),
$$
where
$$
\sum_{\al} P'''_\al(u_1,\ldots,u_{N'}) P''''_\al(u_{N'+1},\ldots,u_{N})
= P(t_1,\ldots,t_N) \prod_{1\leq l\leq N', N' + 1\leq l'\leq N}
{{u_{l'} - u_l}\over{u_{l'} - q^{\langle \eps(l),\eps(l') \rangle}
u_l}} .
$$
\hfill \qed\medskip
\end{remark}
\begin{cor}
There is a unique algebra morphism $\Delta_{\cV^L}$
from $\cV^L$
to $\limm_{\leftarrow N} (\cV^L \otimes_{\CC[[\hbar]]} \hat \cV^L) /
(\cI_N^{(\infty)}\otimes_{\CC[[\hbar]]} \hat\cV^L)$, such that
$$
\Delta_{\cV^L}(h_i[k]^{\cV^L}) = h_i[k]^{\cV^L}\otimes 1 + 1 \otimes h_i[k]^{\cV^L},
\quad 1\leq i \leq n, \quad k\leq 0,
$$
$$
\Delta_{\cV^L}(t_i^k) = \sum_{l\geq 0}t_i^{k+l}\otimes K_i[-l]
+ 1\otimes t_i^{k}
\quad 1\leq i \leq n, \quad k\in\ZZ.
$$
\end{cor}
{\em Proof.} For each $i$, $\Delta_{\cS^L}$ maps $FO_{\eps_i}$ to
$\limm_{\leftarrow N} (\cV^L \otimes_{\CC[[\hbar]]} \hat \cV^L) /
(\cI_N^{(\infty)}\otimes_{\CC[[\hbar]]} \hat\cV^L)$, because
$\cI_{N,\kk}^{\geq a} \cap \langle LV^{\geq a}
\rangle \subset I_{\phi(\kk,n)}$. It follows that
$\Delta_{\cS^L}$ maps $\cV^L$ to the same space. Call $\Delta_{\cV^L}$
the restriction of $\Delta_{\cS^L}$ to $\cV^L$. This restriction is
clearly characterized by its values on $h_i[k]$ and $t_i^k$. \hfill
\qed\medskip
\subsubsection{Construction of a Hopf algebra structure}
Recall that we showed in Lemma \ref{LV:free} that $\cV^L$ is a free
$\CC[[\hbar]]$-module. Let us set $\cV^L_0 = \cV^L / \hbar\cV^L$.
Let $\overline{\cI}_N$ be the image of the ideal $\cI_N^{(\infty)}$ of $\cV^L$ by the
projection from $\cV^L$ to $\cV^L_0$. By Lemma \ref{ladino}, 2), and since
the map from $\cV^L$ to $\cV^L_0$ is surjective, the ideals
$\ol{\cI}_N$ have property $(*)$, so that $\cap_{N>0}\ol{\cI}_N$ is
a two-sided ideal of $\cV^L_0$. Define $\cW_0$ as the quotient algebra
$\cW_0 = \cV^L_0 / \cap_{N>0}\ol{\cI}_N$. We are going to define a
Hopf algebra structure on $\cW_0$.
Since the $\cV^L_0 \otimes\ol{\cI}_N$ have property $(*)$,
$\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0)
/ (\cV^L_0 \otimes \ol{\cI}_N)$ has an algebra structure.
Moreover, the projection
\begin{align*}
& [\limm_{\leftarrow N} (\cV^L \otimes_{\CC[[\hbar]]} \cV^L)
/ (\cV^L \otimes_{\CC[[\hbar]]} \cI_N^{(\infty)})]
/ \hbar [\limm_{\leftarrow N} (\cV^L \otimes_{\CC[[\hbar]]} \cV^L)
/ (\cV^L \otimes_{\CC[[\hbar]]} \cI_N^{(\infty)})]
\\ & \to
\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0)
/ (\cV^L_0 \otimes \ol{\cI}_N) ,
\end{align*}
is an algebra isomorphism.
$\Delta_{\cV^L}$
induces therefore an algebra morphism $\Delta_{\cV^L_0}$ from $\cV^L_0$
to $\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0)
/ (\cV^L_0 \otimes \ol{\cI}_N ) $.
On the other hand, the $\ol{\cI}_N \otimes \cV^L_0 +
\cV^L_0 \otimes \ol{\cI}_N$ have property $(*)$, so that
$\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0)
/ (\ol{\cI}_N \otimes \cV^L_0 + \cV^L_0 \otimes \ol{\cI}_N)$
has an algebra structure. The composition of $\Delta_{\cV^L_0}$
with the projection
$$
\lim_{\leftarrow N} (\cV_0^L \otimes \cV_0^L) /
(\cV_0^L \otimes \ol{\cI}_N) \to
\lim_{\leftarrow N} (\cV_0^L \otimes \cV_0^L) /
(\ol{\cI}_N \otimes \cV_0^L + \cV_0^L \otimes \ol{\cI}_N )
$$
then yields an algebra morphism $p'\circ \Delta_{\cV_L^0}$ from $\cV_L^{(0)}$ to
$\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0)
/ (\ol{\cI}_N \otimes \cV^L_0 + \cV^L_0 \otimes \ol{\cI}_N)$.
We have for any $k\geq N$, $\Delta_{\cV}(t_i^k)\in \cI_N\otimes\hat\cV +
\cV\otimes \hat\cI_N$, therefore
$\Delta_{\cV^L}(\cI_N) \subset \cI_N \otimes_{\CC[[\hbar]]} \hat\cV
+ \cV\otimes_{\CC[[\hbar]]} \hat\cI_N$, therefore
$\Delta_{\cV^L_0}(\ol{\cI}_N ) \subset \ol{\cI}_N \otimes \cV_0 +
\cV_0\otimes \ol{\cI}_N$. It follows that $p'\circ \Delta_{\cV_L^0}$
maps the intersection $\cap_{N> 0}\ol{\cI}_N$ to the kernel of
the projection $\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0) /
(\cV^L_0 \otimes \ol{\cI}_N) \to \limm_{\leftarrow N} (\cV^L_0
\otimes \cV^L_0) / (\ol{\cI}_N \otimes \cV^L_0 + \cV^L_0 \otimes
\ol{\cI}_N)$.
We have then an algebra morphism $\tilde\Delta_{\cW_0}$
from $\cW_0$ to $\limm_{\leftarrow N}
(\cV^L_0 \otimes \cV^L_0) / (\ol{\cI}_N \otimes \cV^L_0 + \cV^L_0
\otimes \ol{\cI}_N)$.
In Prop.\ \ref{prop:morphism}, we defined a surjective algebra morphism
$i_\hbar$ from $U_\hbar L\B_+$ to $\cV^L$. It induces an algebra
morphism $i$ from $U L\B_+$ to $\cV^L_0$, which is also surjective.
Let $T$ be the free algebra generated by the $h_i[k]^{(T)},
k\leq 0, i = 1,\ldots,n$ and $e_i[k]^{(T)}, i =
1,\ldots,n, k\in \ZZ$. We have a natural projection of $T$ on $U
L\B_+$, sending each $x[k]^{(T)}$ to $x\otimes t^k$; composing it with
$\iota$, we get a surjective algebra morphism $\pi$ from $T$ to
$\cV^L_0$.
We have a unique algebra morphism $\Delta_T:T\to T\otimes T$,
such that $\Delta_T(x_i[k]^{(T)}) = x_i[k]^{(T)} \otimes 1 +
1\otimes x_i[k]^{(T)}$.
\begin{lemma}
Let $\pi_{\cV_0\to \cW_0}$ be the natural projection from $\cV_0$ to $\cW_0$.
We have the identity
\begin{equation} \label{basic}
\tilde\Delta_{\cW_0} \circ (\pi_{\cV_0\to\cW_0}\circ\pi)
= (\nu\circ(\pi_{\cV_0\to \cW_0}\otimes\pi_{\cV_0\to \cW_0}))
\circ\Delta_T,
\end{equation}
where $\nu$ is the natural projection from $(\cV^L_0 \otimes \cV^L_0)$ to
$\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0) / (\ol{\cI}_N \otimes
\cV^L_0 + \cV^L_0 \otimes \ol{\cI}_N)$.
\end{lemma}
{\em Proof.} The two sides are algebra morphisms from $\cW_0$ to
$\limm_{\leftarrow N} (\cV^L_0 \otimes \cV^L_0) / (\ol{\cI}_N \otimes
\cV^L_0 + \cV^L_0 \otimes \ol{\cI}_N)$. The identity is satisfied on
generators of $\cW_0$, therefore it is true. \hfill \qed\medskip
Let $J$ be the kernel of the projection $\pi_{\cV_0\to\cW_0}\circ \pi$
from $T$ to $\cW_0$. It follows from (\ref{basic}) that $\Delta_T(J)$
in contained in the kernel of $\nu\circ(\pi\otimes\pi)$, which is the
preimage by $\pi\otimes\pi$ of $\Ker\nu$; $\Ker\nu$ is equal to $\cap_{N>0}
(\ol{\cI}_N \otimes \cV^L_0 + \cV^L_0 \otimes \ol{\cI}_N)$, which is
$(\cap_{N>0} \ol{\cI}_N)
\otimes \cV^L_0 + \cV^L_0 \otimes (\cap_{N>0} \ol{\cI}_N)$. Therefore
$(\pi\otimes\pi)^{-1}(\Ker\nu)$ is $J\otimes T + T\otimes J$.
We have shown that $\Delta_T(J)\subset J\otimes T + T \otimes J$.
We have shown:
\begin{prop}
$\Delta_T$ induces a cocommutative Hopf algebra structure on
$T / J = \cW_0$.
\end{prop}
We will denote by $\Delta_{\cW_0}$ the coproduct induced
by $\Delta_T$ on $\cW_0$.
\subsubsection{Compatibility of $\Delta_{\cW_0}$ with
$\Delta_{\cV^L_0}$}
Recall that $\Delta_{\cV^L_0}$ is an algebra
morphism from $\cV^L_0$ to $\lim_{\leftarrow N}
(\cV^L_0\otimes\cV^L_0) / (\ol{\cI}_N \otimes \cV^L_0)$.
Let us denote by $\Delta_{\cV_0^L;N}$
the induced map from $\cV_0^L$ to $(\cV_0^L\otimes\cV^L_0)
/ (\ol{\cI}_N \otimes \cV^L_0)$.
We have seen that for any
integer $p>0$, $\Delta_{\cV^L_0;N}(\ol{\cI}_p)$ is contained in
the image of $\ol{\cI}_p \otimes \cV^L_0 + \cV^L_0 \otimes\ol{\cI}_p$
by the projection map $\cV^L_0\otimes \cV^L_0 \to (\cV^L_0\otimes\cV^L_0)
/ (\ol{\cI}_N \otimes \cV^L_0)$. This image is
$$
[(\ol{\cI}_p + \ol{\cI}_N) \otimes \cV^L_0 + \cV^L_0 \otimes \ol{\cI}_p]
/ (\ol{\cI}_N \otimes \cV^L_0) .
$$
Therefore, $\Delta_{\cV^L_0;N}(\cap_{p>0}\ol{\cI}_p)$ is contained in
the intersection of these spaces, which is
$$
[ \ol{\cI}_N \otimes \cV^L_0 + \cV^L_0 \otimes (\cap_{p>0}\ol{\cI}_p) ]
/ (\ol{\cI}_N \otimes \cV^L_0) .
$$
It follows that $\Delta_{\cV_0^L;N}$ induces a linear map from
$\cW_0$ to $(\cW_0 \otimes \cW_0)/(\ol{\cJ}_N \otimes \cW_0)$,
where $\ol{\cJ}_N$ is the image of $\ol{\cI}_N$ by the projection map
$\cV_0 \to \cW_0$.
It also induces an algebra morphism from $\cW_0$ to $\limm_{\leftarrow N}
(\cW_0\otimes\cW_0) / (\ol{\cJ}_N \otimes \cW_0)$.
Then this algebra morphism factors through the coproduct map $\Delta_{\cW_0}$
defined above. To check this, it is enough to check it on generators $x[k]$
of $\cW_0$.
It follows that
\begin{lemma} \label{kably}
1) $i_\hbar$ induces a map $i : UL\B_+ \to \cW_0$, which is a surjective Hopf algebra
morphism.
2) Let $\A_L$ be the Lie algebra of primitive elements of $\cW_0$.
The restriction $\iota_{|L\B_+}$ of $\iota$ to $L\B_+$ to $\A_L$ induces a surjective
Lie algebra morphism.
\end{lemma}
{\em Proof.} $\Delta_{\cW_0}\circ i$ and $(i \otimes i)\circ
\Delta_{U\B_+}$ are both algebra morphisms from $U\B_+$ to $\cW_0
\otimes\cW_0$. Their values on the $x\otimes t^k$ coincide, therefore
they are equal. This shows 1).
2) follows directly from 1) and from Prop.\ \ref{ratiu}. \hfill
\qed\medskip
\subsubsection{Construction of $\delta_{\cW_0}$}
Define $\hat\cV^L \hat\otimes \hat\cV^L$ as the tensor product
$$
\CC[X_s[0]^{\cV^L(1)} ,X_s[0]^{\cV^L(2)}][[\hbar]] \otimes
\CC[X_s[k]^{\cV^L(1)} ,X_s[k]^{\cV^L(2)},k>0] \otimes \langle LV
\rangle \otimes_{\CC[[\hbar]]} \langle LV \rangle;
$$
endow $\hat\cV^L\hat\otimes \hat\cV^L$ with the unique $\hbar$-adically
continuous algebra structure such that $\cV^L \otimes_{\CC[[\hbar]]}
\cV^L \to \cV^L \hat\otimes \cV^L$, $X_s[k]^{\cV^L(1)}\mapsto
X_s[k]^{\cV^L} \otimes 1$, $X_s[k]^{\cV^L(2)}\mapsto 1\otimes
X_s[k]^{\cV^L}$, $1\otimes x\otimes 1\mapsto x\otimes 1$, $1\otimes
1\otimes x\mapsto 1\otimes x$ (where $x$ is in $\langle LV \rangle$) is
an algebra morphism.
Define in the same way $\hat\cV^L \hat\otimes \hat\cI_N^{(\infty)}$ as
the tensor product $ \CC[X_s[k]^{\cV^L(1)},X_s[k]^{\cV^L(2)}][[\hbar]]
\otimes \langle LV \rangle \otimes I_N^{(\infty)}$, where the tensor products are
over $\CC[[\hbar]]$. Each $\hat\cV^L \hat\otimes \hat\cI_N^{(\infty)}$ is then a
left ideal of $\hat\cV^L \hat\otimes\cV^L$.
Clearly, we have $(\hat\cV^L \hat\otimes\hat\cV^L) /
\hbar(\hat\cV^L \hat\otimes\hat\cV^L) = \cV_0^L\otimes\cV_0^L$.
Moreover,
$$
[(\hat\cV^L \hat\otimes\hat\cV^L) / (\hat\cV^L \hat\otimes \hat\cI_N^{(\infty)})]
/ \hbar [(\hat\cV^L \hat\otimes\hat\cV^L) / (\hat\cV^L \hat\otimes \hat\cI_N^{(\infty)})]
= (\cV_0^L\otimes\cV_0^L) / (\cV_0^L\otimes\ol{\cI}_N) .
$$
Then $\Delta_{\cV^L}$ is an algebra morphism from $\cV^L$ to
$\limm_{\leftarrow N} (\hat \cV^L\hat\otimes \hat\cV^L) / (\hat
\cI_N^{(\infty)} \hat\otimes \hat\cV^L)$. We again denote by $\Delta_{\cV^L}$ the
composition of this map with the projection on $\lim_{\leftarrow N} (
\hat\cV^L \hat\otimes \hat\cV^L) / (\hat\cI_N^{(\infty)} \hat\otimes\hat\cV^L +
\hat\cV^L \hat\otimes \hat\cI_N^{(\infty)})$. Define $\Delta'_{\cV^L}$ as
$\Delta_{\cV}^L$ composed with the exchange of factors.
We have then $(\Delta_{\cV^L} - \Delta'_{\cV^L})(\cV^L)
\subset \lim_{\leftarrow N} ( \hbar \hat\cV^L \hat\otimes
\hat\cV^L) / [(\hat\cI_N^{(\infty)} \hat\otimes\hat\cV^L
+ \hat\cV^L \hat\otimes \hat\cI_N^{(\infty)})\cap \hbar
\hat\cV^L \hat\otimes \hat\cV^L]$.
Since $\cI_N^{(\infty)}$ is divisible in $\cV^L$, we have
$$
(\hat\cI_N^{(\infty)} \hat\otimes \hat\cV^L + \hat\cV^L \hat\otimes
\hat\cI_N^{(\infty)})
\cap \hbar (\hat\cV^L \hat\otimes \hat\cV^L) =
\hbar (\hat\cI_N^{(\infty)} \hat\otimes \hat\cV^L
+ \hat\cV^L \hat\otimes \hat\cI_N^{(\infty)} ) ,
$$
so
${{\Delta_{\cV^L} - \Delta'_{\cV^L}
}\over{\hbar}}$ is a linear map from $\cV^L$ to
$$
\lim_{\leftarrow N} ( \hat\cV^L \hat\otimes
\hat\cV^L) / (\hat\cI_N^{(\infty)} \hat \otimes\hat\cV^L
+ \hat\cV^L \hat\otimes \hat\cI_N^{(\infty)}) .
$$
Define $\cI_N^{(1)}$ as the image of $\hat \cI_N^{(1)}$ in $\cV_0^L$ by the
projection $\cV^L\to\cV_0^L$.
Let us set $\delta_{\cV_0^L} = {{\Delta_{\cV^L} - \Delta'_{\cV^L}
}\over{\hbar}}$ mod $\hbar$. Then $\delta_{\cV_0^L}$ is a linear map from
$\cV^L_0$ to
$$
\lim_{\leftarrow N} (\cV_0^L \otimes
\cV_0^L) / (\ol{\cI}_N \otimes \cV_0^L
+ \cV_0^L \otimes \ol{\cI}_N) .
$$
Moreover, ${{\Delta_{\cV^L} - \Delta'_{\cV^L} }\over{\hbar}}$ maps
$\cI_N^{(\infty)}$ to the inverse limit
$$
\lim_{\leftarrow M}
(\hat\cI_N^{(\infty)} \hat\otimes \hat\cV^L + \hat\cV^L\hat\otimes
\hat\cI_N^{(\infty)} ) / (\hat\cI_M^{(\infty)} \hat\otimes \hat\cV^L +
\hat\cV^L\hat\otimes \hat\cI_M^{(\infty)} ).
$$
Therefore, $\delta_{\cV_0^L}$
maps $\ol{\cI}_N$ to $\lim_{\leftarrow M}
(\ol{\cI}_N \otimes \cV^L_0 + \cV^L_0\otimes \ol{\cI}_N )
/ (\ol{\cI}_M \otimes \cV^L_0 + \cV^L_0\otimes \ol{\cI}_M )$.
Therefore, $\delta_{\cV_0^L} (\cap_N \ol{\cI}_N)$ is zero. It follows that
$\delta_{\cV_0^L}$ induces a map $\delta_{\cW_0}$ from $\cW_0$ to
$\lim_{\leftarrow N} (\cW_0 \otimes\cW_0) / (\ol{\cJ}_N \otimes \cW_0 +
\cW_0 \otimes \ol{\cJ}_N)$.
\subsubsection{Identities satisfied by $\delta_{\cW_0}$}
\begin{lemma} \label{peisakh}
$\delta_{\cW_0}$ satisfies
\begin{equation} \label{co-leib:top}
(\Delta_{\cW_0}\otimes id)\circ \delta_{\cW_0} = (\delta_{\cW_0}^{2\to 23}
+ \delta_{\cW_0}^{2\to 13}) \circ \Delta_{\cW_0},
\end{equation}
\begin{equation} \label{co-jacobi:QC}
\Alt ( \delta_{\cW_0}\otimes id)\circ\delta_{\cW_0} ) = 0,
\end{equation}
\begin{equation} \label{comp:Hopf:QC}
\delta_{\cW_0}(xy) = \delta_{\cW_0}(x) \Delta_{\cW_0}(y)
+ \Delta_{\cW_0}(x) \delta_{\cW_0}(y) \on{\ for\ }
x,y \on{\ in\ } \cW_0,
\end{equation}
where we use the notation of sect.\ \ref{adama}.
The two first equalities are identities of maps from $\cW_0$ to
$\limm_{\leftarrow N} \cW_0^{\otimes 3} / (\ol{\cJ}_N \otimes \cW_0^{\otimes 2}
+ \cW_0 \otimes \ol{\cJ}_N \otimes \cW_0 + \cW_0^{\otimes 2} \otimes
\ol{\cJ}_N)$.
\end{lemma}
{\em Proof.}
$\Delta_{\cV^L}$ maps $\cI_N^{(\infty)}$ to
$$
\lim_{\leftarrow M}
(\hat\cI_N^{(\infty)} \hat\otimes \hat\cV^L + \hat\cV^L\hat\otimes
\hat\cI_N^{(\infty)} ) / (\hat\cI_M^{(\infty)} \hat\otimes \hat\cV^L +
\hat\cV^L\hat\otimes \hat\cI_M^{(\infty)} ).
$$
Therefore, $(\Delta_{\cV^L}\otimes id ) \otimes \Delta_{\cV^L}$
and $(id \otimes \Delta_{\cV^L}) \otimes \Delta_{\cV^L}$ both
define algebra morphisms from $\cV^L$ to
$\lim_{\leftarrow N} (\hat\cV^L)^{\hat\otimes 3}
/ [\hat \cI_N^{(\infty)} \hat\otimes (\hat\cV^L)^{\hat\otimes 2}
+ \hat\cV^L \hat\otimes \hat \cI_N^{(\infty)} \hat\otimes \hat\cV^L
+ (\hat\cV^L)^{\hat\otimes 2} \hat\otimes
\hat \cI_N^{(\infty)}]$. These morphisms are the restrictions
to $\cV^L$ of $(\Delta_{\cS^L}\otimes id ) \otimes \Delta_{\cS^L}$
and $(id \otimes \Delta_{\cS^L}) \otimes \Delta_{\cS^L}$, which
coincide, therefore they coincide.
The intersection $\cap_{N>0}\cI_N^{(\infty)}$ is a two-sided
ideal of $\cV^L$. Define $\cW$ as the quotient $\cV^L /
\cap_{N>0}\cI_N^{(\infty)}$. Let $\cJ_N$ be the image of
$\cI_N^{(\infty)}$ by the projection of $\cV^L$ on $\cW$. Define
$\hat\cW$ and $\hat \cJ_N$ in the same way, replacing $\cV^L$
and $\cI_N^{(\infty)}$ by $\hat\cV^L$
and $\hat\cI_N^{(\infty)}$. Then $\Delta_{\cV^L}$ induces an
algebra morphism $\Delta_{\cW}$ from $\cW$ to $\limm_{\leftarrow N}
(\hat\cW\hat\otimes\hat\cW) / (\hat\cJ_N\hat\otimes\hat\cW +
\hat\cW \hat\otimes\cJ_N)$. Moreover,
$(\Delta_{\cW}\otimes id ) \circ \Delta_{\cW}$
and $(id \otimes \Delta_{\cW}) \circ \Delta_{\cW}$
define coinciding algebra morphisms from $\cW$ to
$\limm_{\leftarrow N}
(\hat\cW^{\hat\otimes 3}
) / [\hat\cJ_N \hat\otimes \hat\cW^{\hat\otimes 2}
+ \hat\cW \hat\otimes \cJ_N \hat\otimes \hat\cW
+ \hat\cW^{\hat\otimes 2} \hat\otimes \hat\cJ_N]$.
Moreover, $\cW$ is a free $\CC[[\hbar]]$-module, and we have a
topological Hopf algebra isomorphism of $\cW / \hbar\cW$ with $\cW_0$.
The usual manipulations then imply the statements of the Lemma.
\hfill \qed\medskip
The identities of Lemma \ref{peisakh} are the topological versions of
the co-Leibnitz, co-Jacobi and Hopf compatibility rules.
\subsubsection{Topological Lie bialgebra structure on $\A_L$}
Define $\A_L^{(N)}$ as the intersection $\A_L \cap \ol{\cJ}_N$.
\begin{lemma} \label{coen}
$\ol{\cJ}_N$ is the left ideal $(U\A_L) \A_L^{(N)}$ of
$\cW_0 = U\A_L$. Moreover, $\A_L^{(N)}$ is a Lie subalgebra of
$\A_L$.
\end{lemma}
{\em Proof.} $\cJ_N$ is a left ideal of $\cW$, therefore $\ol{\cJ}_N$
is a left ideal of $\cW_0$. Moreover, $\Delta_{\cW}(\cJ_N)$
is contained in the inverse limit
$\limm_{\leftarrow M} (\hat\cJ_N \hat\otimes \hat\cW + \hat\cW
\hat\otimes \hat\cJ_N) / (\hat\cJ_M \hat\otimes \hat\cW + \hat\cW
\hat\otimes \hat\cJ_M)$. It follows that $\Delta_{\cW_0}(\ol{\cJ}_N)$
is contained in $\ol{\cJ}_N \otimes \cW_0 + \cW_0 \otimes \ol{\cJ}_N$.
The first statement of Lemma \ref{coen} now follows from Lemma \ref{lausanne}.
For $x,y$ in $\A_L^{(N)}$, $[x,y] = xy - yx$ belongs to
$\A_L$ and also to $(U\A_L)\A_L^{(N)}$, so it belongs to
$\A_L^{(N)}$. Therefore $\A_L^{(N)}$ is a Lie subalgebra of $\A_L$.
\hfill \qed\medskip
\begin{lemma} \label{basic:delta}
1) The restriction of $\delta_{\cW_0}$ to $\A_L$ defines a map
$\delta_{\A_L} : \A_L \to \limm_{\leftarrow N} (\A_L \otimes\A_L) /
(\A_L^{(N)} \otimes\A_L + \A_L \otimes \A_L^{(N)})$.
2) For any element $x$ of $\A_L$, $\ad(x)(\A_L^{(N)})$ in contained in
$\A_L^{(N - k(x))}$. The tensor square of the adjoint action therefore induces
a $\A_L$-module structure on $\limm_{\leftarrow N} (\A_L \otimes\A_L) /
(\A_L^{(N)} \otimes\A_L + \A_L \otimes \A_L^{(N)})$. $\delta_{\A_L}$ is
a $1$-cocycle of $\A_L$ with values in this module.
3) We have $\delta_{\A_L}(\A_L^{(N)}) \subset \limm_{\leftarrow M}
(\A_L^{(N)} \otimes\A_L + \A_L \otimes\A_L^{(N)})
/ (\A_L^{(M)} \otimes\A_L + \A_L \otimes\A_L^{(M)})$.
$(\delta_{\A_L}\circ id) \circ \delta_{\A_L}$ therefore defines a map from
$\A_L$ to $\limm_{\leftarrow N} \A_L^{\otimes 3} / (\A_L^{(N)} \otimes
\A_L^{\otimes 2} + \A_L\otimes\A_L^{(N)} \otimes\A_L + \A_L^{\otimes 2}
\otimes\A_L^{(N)})$. It satisfies the rule
\begin{equation} \label{coass}
\Alt(\delta_{\A_L}\otimes id)\circ\delta_{\A_L} = 0.
\end{equation}
\end{lemma}
{\em Proof.} Let us show 1). $\delta_{\cW_0}$
induces an map $\delta_{\cW_0;N}$ from $\cW_0$ to $\cW_0^{\otimes 2} /
(\ol{\cJ}_N \otimes \cW_0 + \cW_0 \otimes\ol{\cJ}_N)$. Let $a$ belong to $\A_L$.
Let us write
$\delta_{\cW_0;N}(a) = \sum_i a_i\otimes b_i$ mod
$\ol{\cJ}_N \otimes \cW_0 + \cW_0 \otimes\ol{\cJ}_N$, with
$(a_i)_i$ and $(b_i)_i$ finite families of $\cW_0$ such that
$(b_i \on{\ mod\ }\ol{\cJ}_N)_i$
is a free family of $\cW_0 / \ol{\cJ}_N$. It follows from
(\ref{co-leib:top}) that
$$
\sum_i (\Delta_{\cW_0}(a_i) - a_i\otimes 1 - 1 \otimes a_i) \otimes b_i
$$
belongs to $\ol{\cJ}_N \otimes\cW_0^{\otimes 2} + \cW_0\otimes
\ol{\cJ}_N \otimes \cW_0 + \cW_0^{\otimes 2} \otimes \ol{\cJ}_N$.
Its image by the projection $\cW_0^{\otimes 3} \to [\cW_0^{\otimes 2} /
(\ol{\cJ}_N \otimes \cW_0 + \cW_0 \otimes \ol{\cJ}_N ) ]\otimes
[\cW_0 / \ol{\cJ}_N]$ its therefore zero.
It follows that each $a_i$ is such that $\Delta_{\cW_0}(a_i)
- a_i\otimes 1 - 1 \otimes a_i$ belongs to $\ol{\cJ}_N \otimes\cW_0
+ \cW_0\otimes \ol{\cJ}_N
$. Reasoning by induction on the degree of $a_i$
(for the enveloping algebra filtration of $\cW_0$), we find that $a_i$
belongs to $\A_L + \ol{\cJ}_N$. Therefore, $\delta_{\cW_0;n}(a)$ belongs to
the image of $\A_L \otimes\cW_0$ in $\cW_0^{\otimes 2} /
(\ol{\cJ}_N \otimes \cW_0 + \cW_0 \otimes \ol{\cJ}_N )$.
Since $\delta_{\cW_0;n}(a)$ is also antisymmetric, it belongs to the
image of $\A_L \otimes\A_L$ in this space. This shows 1).
Let us show 2). For $x$ en element of $\A_L$ and $y$ an element of
$\A_L^{(N)}$, $[x,y] = xy - yx$ belongs to $\ol{\cJ}_N
+ \ol{\cJ}_{N - k(x)} = \ol{\cJ}_{N - k(x)}$; since it also belongs to
$\A_L$, $[x,y]$ belongs to $\A_L^{(N)}$. That $\delta_{\A_L}$ is a $1$-cocycle
then follows from (\ref{comp:Hopf:QC}).
Let us show 3). $\Delta_{\cW;M}(\cJ_N)$ is contained in $(\cJ_N \otimes
\cW + \cW \otimes \cJ_N) / (\cJ_M \otimes \cW + \cW \otimes \cJ_M)$.
It follows that $\delta_{\cW_0;M}(\ol{\cJ}_N)$ is contained in
$(\ol{\cJ}_N \otimes \cW_0 + \cW_0 \otimes \o{\cJ}_N) / (\ol{\cJ}_M
\otimes \cW_0 + \cW_0 \otimes \o{\cJ}_M)$. Therefore,
$\delta_{\A_L}(\A_L^{(N)})$ is contained in $\limm_{\leftarrow M}
(\A_L^{(N)} \otimes\A_L + \A_L \otimes\A_L^{(N)}) /
(\A_L^{(M)} \otimes\A_L + \A_L \otimes\A_L^{(M)})$. (\ref{coass})
in then a consequence of (\ref{co-jacobi:QC}).
\hfill \qed\medskip
Define the restricted dual $\A_L^\star$ of $\A_L$ as the subspace of
$\A_L^*$ composed of the forms $\phi$ on $\A_L$, such for some $N$,
$\phi$ vanishes on $\A_L^{(N)}$.
\begin{lemma} \label{szenes}
The dual map to $\delta_{\A_L}$ defines a Lie algebra structure on $\A_L^\star$.
\end{lemma}
{\em Proof.} Let $\phi,\psi$ belong to $\A_L^\star$. Let $N$ be an
integer such that $\phi,\psi$ vanish on $\A_L^{(N)}$. For any integer $M$
let $\bar\delta_{\A_L;M}$ be a lift to $\A_L^{\otimes 2}$ of the
map $\delta_{\A_L;M}$ from $\A_L$ to $\A_L^{\otimes 2} / (\A_L^{(M)})
\otimes\A_L + \A_L \otimes\A_L^{(M)}$ induced by $\delta_{\A_L}$.
Let $x$ belong to $\A_L$. Then $M\geq N$, the number $\langle
\phi\otimes\psi, \bar\delta_{\A_L;M}(x) \rangle$ is independent of
the lift $\bar\delta_{\A_L;M}$ and of $M$; it defines
a linear form $[\phi,\psi]$ on $\A_L$. The first statement of Lemma
\ref{basic:delta}, 3), implies that $[\phi,\psi]$ actually belongs to
$\A_L^\star$. It is clear that $(\phi,\psi)\mapsto [\phi,\psi]$ is linear
and antisymmetric in $\phi$ and $\psi$. (\ref{coass}) implies that it
satisfies the Jacobi identity. \hfill \qed \medskip
\subsubsection{Topological Lie bialgebra structure on $L\B_+$}
Define for any integer $N$, $(L\B_+)^{(N)}$ as the Lie subalgebra of
$L\B_+$ generated by the $\bar x_i^+\otimes t^k$, $k\geq N, i = 1,\ldots,n$.
For any $x$ in $L\B_+$, there exists an integer $l(x)$ such that
$\ad(x)((L\B_+)^{(N)})$ is contained in $(L\B_+)^{(N - l(x))}$. It follows that
$\limm_{\leftarrow N} (L\B_+)^{\otimes 2} / [(L\B_+)^{(N)} \otimes L\B_+
+ L\B_+ \otimes (L\B_+)^{(N)}]$
and $\limm_{\leftarrow N} (L\B_+)^{\otimes 3} / [(L\B_+)^{(N)} \otimes L\B_+^{\otimes 2}
+ L\B_+ \otimes (L\B_+)^{(N)} \otimes L \B_+ + L\B_+^{\otimes 2} \otimes (L\B_+)^{(N)}]$
have $L\B_+$-module structures.
\begin{lemma} There is a unique map $\delta_{L\B_+}$ from $L\B_+$ to
$\limm_{\leftarrow N} (L\B_+)^{\otimes 2} / [(L\B_+)^{(N)} \otimes L\B_+
+ L\B_+ \otimes (L\B_+)^{(N)}]$, such that
$\delta_{L\B_+} (\bar h_i\otimes t^k) = 0$ and
$$
\delta_{L\B_+}(\bar x_i^+\otimes
t^k) = d_i \Alt [ {1\over 2 }(\bar h_i\otimes 1) \otimes (\bar x_i^+ \otimes 1)
+ \sum_{l>0} (\bar h_i\otimes t^{-l}) \otimes (\bar x_i^+ \otimes t^{k+l}) ]
$$
and $\delta_{L\B_+}$ is a $1$-cocycle. Moreover, $\delta_{L\B_+}$ maps
$L\B_+^{(N)}$ to $\limm_{\leftarrow M} (L\B_+^{(N)} \otimes L\B_+
+ L\B_+ \otimes L\B_+^{(N)}) / (L\B_+^{(M)} \otimes L\B_+
+ L\B_+ \otimes L\B_+^{(M)})$, and it satisfies the co-Jacobi
identity $\Alt (\delta_{L\B_+}\otimes id) \circ \delta_{L\B_+} = 0$.
\end{lemma}
Define the restricted dual $(L\B_+)^\star$ to $L\B_+$ as the subspace of
$(L\B_+)^*$ consisting of the forms on $L\B_+$, which vanish on some
$(L\B_+)^{(N)}$. The argument of Lemma \ref{szenes} implies that
$\delta_{L\B_+}$ induces a Lie algebra structure on $(L\B_+)^\star$.
\begin{lemma}
Define on $\G\otimes\CC((t))$, the pairing $\langle , \rangle_{\G\otimes \CC((t))}$
as the tensor product of the invariant pairing on $\G$ and $\langle f,g \rangle
= \res_0 (fg {{dt}\over t})$. $\langle , \rangle_{\G\otimes \CC((t))}$
an isomorphism of $(L\B_+)^\star$ with the subalgebra $L\B_-$ of $L\G$
defined as $\HH \otimes \CC[[t]] \oplus \N_- \otimes \CC((t))$. This isomorphism
is a Lie algebra antiisomorphism (that is, it is an isomorphism after we
change the bracket of $L\B_-$ into its opposite).
\end{lemma}
The map $\iota_{|L\B_+}$ defined in Lemma \ref{kably}, 2), maps the
generators of $L\B_+^{(N)} $ to $\A_L^{(N)}$; since $\A_L^{(N)}$ is a
Lie subalgebra of $\A_L$ (Lemma \ref{coen}), we have $\iota_{|
L\B_+}(L\B_+^{(N)}) \subset \A_L^{(N)}$.
It follows that $\iota_{|L\B_+}$ induces a linear map $\iota^\star$
from $\A_L^\star$ to $(L\B_+)^\star = L\B_-$.
Moreover, we have
\begin{equation} \label{compat}
\delta_{\A_L} \circ \iota_{|L\B_+} =
(\iota_{L\B_+}^{\otimes 2}) \circ \delta_{L\B_+},
\end{equation}
because both maps are $1$-cocycles of $L\B_+$ with values in
$\limm_{\leftarrow N} \A_L^{\otimes 2} / (\A_L^{(N)} \otimes\A_L
+ \A_L \otimes \A_L^{(N)})$, and coincide on the generators of
$L\B_+$.
(\ref{compat}) then implies that $\iota^\star : \A_L^\star \to
L\B_-$ is a Lie algebra morphism.
Let us set $(L\B_-)_{pol} = \HH\otimes \CC[t^{-1}] \oplus
\N_- \otimes\CC[t,t^{-1}]$; $(L\B_-)_{pol}$ is the polynomial
part of $L\B_-$.
\begin{lemma} \label{surj}
The image of $\iota^\star$ contains $(L\B_-)_{pol}$.
\end{lemma}
{\em Proof.}
As we have seen, $\cV^L$ is graded by $\NN^n$. Each ideal
$\cI_N^{(\infty)}$is a graded ideal, so that $\cW = \cV /
\cap_N \cI_N^{(\infty)}$ is also graded by $\NN^n$. Moreover,
the degree $0$ and $\eps_i$ components of $\cI_N^{(\infty)}$
are respectively $0$ and $\oplus_{k\geq N}
\CC[[\hbar]][h_i[k]^{\cV^L},k\leq 0] t_i^k$. Therefore,
the components of $\cap_N \cI_N^{(\infty)}$ of degree $0$ and
$\eps_i$ are zero. The components
of $\cW$ of degrees $0$ and $\eps_i$ are therefore respectively
$\CC[[\hbar]] [h_i[k], k\leq 0]$ and $\oplus_{l\in\ZZ}
\CC[[\hbar]][h_i[k]^{\cV^L},k\leq 0] t_i^l$.
$\cW_0$ is also graded by $\NN^n$, and its components of degrees
$0$ and $\eps_i$ are $\CC[h_i[k], k\leq 0]$ and $\oplus_{l\in\ZZ}
\CC[h_i[k]^{\cV^L},k\leq 0] t_i^l$.
The primitive part $\A_L$ of $\cW_0$ is therefore also graded by
$\NN^n$, and the computation of $\Delta_{\cW_0}$ on $\cW_0[0]$ and
$\cW_0[\eps_i]$ shows that $\A_L[0] = \oplus_{1\leq i\leq n, k\leq 0}
\CC h_i[k]^{\cV^L}$ and $\A_L[\eps_i] = \oplus_{k\in\ZZ} \CC t_i^k$.
Define linear forms $h_{i,k}^*$ and $e_{i,k}^*$ on $\A_L$ by the
rules that $h_{i,k}^*$ vanishes on $\oplus_{\al\neq 0}\A_L[\al]$,
and the restriction of $h_{i,k}^*$ to $\A_L[0]$ maps
$h_j[l]^{\cV^L}$ to $\delta_{ij}\delta_{kl}$; and
$e_{i,k}^*$ vanishes on $\oplus_{\al\neq \eps_i}\A_L[\al]$,
and the restriction of $e_{i,k}^*$ to $\A_L[\eps_i]$ maps
$t_i^l$ to $\delta_{kl}$.
It follows from the computation of $\cI_N^{(\infty)}[0]$ and
$\cI_N^{(\infty)}[\eps_i]$ that
the $h_{i,k}^*$ and $e_{i,k}^*$ vanish on all the $\ol{\cJ}_N$,
resp.\ on the $\ol{\cJ}_N, N\geq k$, adn therefore on all the
$\A_L^{(N)}$, resp.\ on the $\A_L^{(N)}, N\geq k$. It follows that
the $h_{i,k}^*$ and $e_{i,k}^*$ actully belong to $\A_L^\star$.
Since the images of $\bar h_i\otimes t^k$ and
$\bar x_i^+\otimes t^k$ by $\iota_{|L\B_+}$ are
$h_i[k]^{\cV^L}$ and $t_i^k$, the images of
$h_{i,k}^*$ and $e_{i,k}^*$ by $\iota^\star$ are the generators
$\bar h_i\otimes t^k, 1\leq i\leq n, k\geq 0$ and
$\bar x_i^-\otimes t^k, 1\leq i \leq n, k\in\ZZ$, of
$(L\B_-)_{pol}$.
The statement follows because $\iota^\star$ is a Lie algebra
morphism.
\hfill \qed\medskip
Lemma \ref{surj} implies that the kernel of $\iota_{|L\B_+}$ is
contained in contains the annihilator of $(L\B_-)_{pol}$ in $L\B_+$.
Since this annihilator is zero, $\iota_{|L\B_+}$ is injective.
It follows that $\iota_{|L\B_+}$ is an isomorphism.
Therefore, $\iota: UL\B_+\to \cW_0$ is also an isomorphism.
Recall that $\iota$ was obtained from the surjective
$\CC[[\hbar]]$-modules morphism $\iota_\hbar = p\circ i_\hbar$, where
$p$ is the projection of $\cV^L$ on $\cW$.
We now use:
\begin{lemma} \label{3.15}
Let $E$ and $F$ be $\CC[[\hbar]]$-modules, such that $F$ is
torsion-free and $E$ is separated (i.e. $\cap_{N>0} \hbar^N E = 0$).
Let $\pi : E\to F$ be a surjective morphism of $\CC[[\hbar]]$-modules,
such that the induced morphism $\pi_0: E / \hbar E \to F / \hbar F$
is an isomorphism of vector spaces. Then $\pi$ is an isomorphism.
\end{lemma}
{\em Proof.} Let $x$ belong to $\Ker p$. $\pi_0(x \on{\ mod\ }\hbar)$
is zero, therefore $x$ belongs to $\hbar E$. Set $x = \hbar x_1$.
$\hbar \pi(x_1)$ is zero; since $F$ is torsion-free, $x_1$ belongs to
$\Ker p$. Therefore, $\Ker p\subset \hbar \Ker p$. It follows that
$\Ker p\subset \cap_{N>0} \hbar^N E$, so that $\Ker p = 0$. It
follows that $\pi$ is an isomorphism.
\hfill \qed\medskip
Recall that $U_\hbar L\N_+$ was defined as the quotient $\cA /
(\cap_{N>0}\hbar^N \cA)$. It follows that $U_\hbar L\N_+$ is separated.
The above Lemma therefore shows that $p\circ i_\hbar$ is an isomorphism.
Since $p$ and $i_\hbar$ are both surjective, they are both
isomorphisms. Cor.\ \ref{cor:second} follows, together with $\cap_{N>0}
\cI_N^{(\infty)} = 0$ (from where also follows that $\cap_{N>0} \cI_N =
0$), and also, by Lemma \ref{LV:free}, Thm.\ \ref{thm:third}, 1).
It is then clear that the map $U_\hbar L\N_+ \to
U_\hbar L\N_+^{top}$ is injective and that
$U_\hbar L\N_+^{top}$ is the $\hbar$-adic completion of
$U_\hbar L\N_+$. This proves Thm.\ \ref{thm:third}, 2).
There is a unique algebra morphism $\varsigma$ from $\wt U_\hbar L\N_+$
to $U_\hbar L\N_+$, which sends each $e_i[k]^{\wt\cA}$ to $e_i[k]$. As
we have seen in Prop.\ \ref{fargo}, $\varsigma$ induces an isomorphism
between $\wt U_\hbar L\N_+ / \hbar\wt U_\hbar L\N_+$ and
$U_\hbar L\N_+ / \hbar U_\hbar L\N_+$. Moreover, $\wt U_\hbar L\N_+$
is separated and by Thm.\ \ref{thm:third}, 1), $U_\hbar L\N_+$ is free.
Lemma \ref{3.15} then implies that $\varsigma$ is an isomorphism.
\hfill\qed\medskip
\begin{remark} We have not been able to prove directly that $\cA$ ot
$\wt \cA$ are themselves $\hbar$-adically separated. Since the
homogeneous components of $\cA$ are infinitely generated
$\CC[[\hbar]]$-modules, it might happen that $\cA$ has components of
the type $\CC((\hbar))$ or $\CC((\hbar)) / \CC[[\hbar]]$. We cannot
exclude the existence of these components by the same argument as in the
proof of Thm.\ \ref{thm:first}, because their images by the map $\cA
\to \cA / \hbar\cA$ are zero (whereas in the finitely generated case,
the torsion submodules had a nonzero image by the same map).
\end{remark}
\begin{remark} \label{open}
Let $FO^{(0)}$ be the subspace of $FO$ formed of the
functions satisfying $f(^{(i)}_\al) = 0$ when $t^{(i)}_1
= q_{d_i}^2 t^{(i)}_2 = \cdots = q_{d_i}^{-2a_{ij}} t^{(i)}_{a_{ij}}
= q_{d_i}^{-a_{ij}}t^{(j)}_1$ for any $i,j$. We showed in \cite{Enr} that the image of
$U_\hbar L\N_+$ in $FO$ is contained in $FO^{(0)}[\hbar^{-1}]$. It is natural
to expect that this image is actually the subspace of $FO^{(0)}$ consisting
in the functions such that $f(t_1,\cdots,t_N) = O(\hbar^k)$ whenever $k$
out of the $N$ variables $t_i$ coincide.
\end{remark}
\subsection{Nondegeneracy of the pairing $\langle ,
\rangle_{U_\hbar L\N_\pm}$ (proof of Thm.\ \ref{nondeg:L})}
Let us define $T(LV)$ as the tensor algebra $\oplus_{k\geq 0}
(LV)^{\otimes_{\CC[[\hbar]]}k}$, where $LV = \oplus_{i=1}^{n}
\CC[[\hbar]][t_i,t_i^{-1}]$. Denote in this algebra, the element
$t_i^l$ of $LV$ as $f_i[l]^{(T)}$.
Define a pairing
$$
\langle , \rangle_{FO \times T(LV)} : FO \times T(LV) \to \CC((\hbar))
$$
as follows: if $P$ belongs to $FO_\kk$,
\begin{align} \label{pairing:NR}
& \langle P, f_{i_1}[l_1]^{(T)} \cdots f_{i_{N'}}[l_{N}]^{(T)}
\rangle_{FO \times T(LV)}
\\ & \nonumber
= \delta_{\kk,\sum_{j=1}^{N} \eps_{i_j}}
\res_{u_N = 0} \cdots \res_{u_1 = 0}
\left( P(t_1,\cdots,t_N) \prod_{l < l'} {{u_{l'} - u_l}\over
{q^{\langle \eps_{i_{l'}}, \eps_{i_l}\rangle} u_{l'} - u_l}}
u_1^{l_1} \cdots u_N^{l_N} {{du_1}\over{u_1}}\cdots
{{du_N}\over{u_N}} \right) ,
\end{align}
where we set as usual
$(t_1, \ldots, t_{k_1}) = (t^{(1)}_1, \ldots, t^{(1)}_{k_1})$, etc.,
$(t_{k_1+ \cdots + k_{n-1} + 1}, \ldots, t_{k_1+ \cdots + k_n})
= (t^{(n)}_1, \ldots, t^{(n)}_{k_n})$, and
$u_1 = t^{(i_1)}_1$, $u_2 = t^{(i_2)}_1$ if $i_2\neq i_1$
and in general
$u_s = t^{(i_s)}_{\nu_s + 1}$, where $\nu_s$ is the number of indices $t$
such that $t<s$ and $i_t = i_s$.
\begin{lemma} \label{volod}
The pairing $\langle , \rangle_{FO \times T(LV)}$ verifies $(T(LV))^{\perp} = 0$.
\end{lemma}
{\em Proof.} Assume that the polynomial $P$ of $FO_\kk$ is such that (\ref{pairing:NR})
vanishes for any families of indices $(i_k)$ and $(l_k)$. Fix a family of indices
$(i_k)$ such that $\kk = \sum_{j=1}^{N} \eps_{i_j}$. Since (\ref{pairing:NR})
vanishes for any family $(l_k)$, the rational function
$P(t_1,\cdots,t_N) \prod_{l < l'} {{u_{l'} - u_l}\over
{q^{\langle \eps_{i_{l'}}, \eps_{i_l}\rangle} u_{l'} - u_l}}$ vanishes,
therefore $P$ is zero. \hfill \qed\medskip
Let $\langle , \rangle_{\langle LV \rangle \times T(LV)}$ be the
restriction of $\langle , \rangle_{FO \times T(LV)}$ to $\langle
LV\rangle \times T(LV)$. Lemma \ref{volod} implies that $T(LV)^\perp =
0$ for this pairing. Using the isomorphism of Thm.\ \ref{thm:third}
between $\langle LV \rangle$ and $U_\hbar L\N_+$, we may view
$\langle , \rangle_{\langle LV \rangle \times T(LV)}$ as a pairing
$\langle , \rangle_{U_\hbar L\N_+ \times T(LV)}$ between
$U_\hbar L\N_+$ and $T(LV)$. So again $T(LV)^\perp = 0$ for
$\langle , \rangle_{U_\hbar L\N_+ \times T(LV)}$.
Let $p$ be the quotient map from $T(LV)$ to $U_\hbar L\N_+$. Composing
$\langle , \rangle_{U_\hbar L\N_+ \times T(LV)}$ with $p\otimes id$, we
get a pairing $\langle , \rangle_{T(LV) \times T(LV)}$ between $T(LV)$
and itself. It follows from (\ref{pairing:NR}) and (\ref{pdt:FO}) that
$\langle , \rangle_{T(LV) \times T(LV)}$ is given by formula
(\ref{pairing:introd}). Moreover, it follows from \cite{Enr}, Prop.\
4.1 (relying on an identity of \cite{Jing}) that $\langle ,
\rangle_{T(LV) \times T(LV)}$ induces a pairing $\langle ,
\rangle_{U_\hbar L\N_+ \times U_\hbar L\N_-}$ between $U_\hbar L\N_+$
and $U_\hbar L\N_-$.
Since $\langle , \rangle_{U_\hbar L\N_+ \times U_\hbar L\N_-}$ is
induced by the pairing $\langle , \rangle_{U_\hbar L\N_+ \times
T(LV)}$, and $T(LV)^\perp = 0$ for this pairing, we get that $(U_\hbar
L\N_+)^\perp = 0$ for $\langle , \rangle_{U_\hbar L\N_+ \times U_\hbar
L\N_-}$. Exchanging the roles of $U_\hbar L\N_+$ and $U_\hbar L\N_-$,
we find that $(U_\hbar L\N_-)^\perp = 0$.
Thm.\ \ref{nondeg:L} follows. \hfill \qed \medskip
\begin{remark} This argument is completely similar to the proof of
Thm.\ \ref{thm:second}, the pairing between $U_\hbar L\N_+$ and $FO$
playing the role of the pairing beween $U_\hbar\N_+$ and $\Sh(V)$.
\end{remark}
\subsection{The form of the $R$-matrix (proof of Prop.\
\ref{R:mat:QC})}
Let us define $A_+^{a,b}$ as the subalgebra of $U_\hbar L\N_+$ generated by
the $e_i[k]$, $i = 1, \ldots,n$, $a\leq k \leq b$.
\begin{lemma} \label{bar:mitswah}
$A_+^{a,b}$ is a graded subalgebra of $U_\hbar L\N_+$. We have
$A_+^{a,b} + I^+_{\leq a} + I^+_{\geq b} = U_\hbar L\N_+$. Moreover,
the graded components of $A_+^{a,b}$ are finite $\CC[[\hbar]]$-modules.
\end{lemma}
{\em Proof.} Let us define $A_+^{\leq a}$ and $A_+^{\geq b}$ as the
subalgebras of $A_+$ generated by the $e_i[k]$, $k\leq a$ (resp.\ $k\geq
b$). It follows from Thm.\ \ref{thm:third} that the product defines a
surjective morphism from $A_+^{\leq a}\otimes A_+^{a,b} \otimes
A_+^{\geq b}$ to $A_+$. The Lemma follows. \hfill \qed\medskip
Since $I^+_{\geq a}[\al] + I^+_{\leq b}[\al] \subset (I^+_{\geq a}[\al] +
I^+_{\leq b}[\al])^{\perp\perp}$, it follows from Lemma
\ref{bar:mitswah} that $(I^+_{\geq a}[\al] + I^+_{\leq
b}[\al])^{\perp\perp}$ is a submodule of $A_+$ with a complement of
finite type. Moreover, this module is also divisible, so that $A_+
[\al]/ (I^+_{\geq a}[\al] + I^+_{\leq b}[\al])^{\perp\perp}$ is
torsion-free. Since it is finitely generated, it follows that
$A_+[\al] / (I^+_{\geq a} [\al]+ I^+_{\leq b}[\al])^{\perp\perp}$ is
a free, finite-dimensional $\CC[[\hbar]]$-module.
On the other hand, $(I^+_{\geq a}[\al] + I^+_{\leq b}[\al] )^{\perp}$ is a
submodule of $\Hom_{\CC[[\hbar]]} (A_+[\al] / (I^+_{\geq a}[\al] + I^+_{\leq b}[\al]) ,
\CC[[\hbar]])$, and is therefore a $\CC[[\hbar]]$-module of finite type.
It is a submodule of $A_-[-\al]$, so it is torsion-free. It follows that
$(I^+_{\geq a} [\al]+ I^+_{\leq b}[\al] )^{\perp}$ is also a free,
finite-dimensional $\CC[[\hbar]]$-module.
By construction, the pairing induced by $\langle , \rangle_{U_\hbar
L\N_\pm }$ between $(I^+_{\geq a}[\al] + I^+_{\leq b}[\al] )^{\perp}$
and $A_+ [\al]/ (I^+_{\geq a}[\al] + I^+_{\leq b}[\al])^{\perp\perp}$
is nondegenerate.
The fact that $P_{a,b}[\al]$ defines an element of
$\limm_{\leftarrow a,b} A_+ / (I^+_{\leq a} + I^+_{\geq
b})^{\perp\perp} \otimes_{\CC[[\hbar]]} A_-[\hbar^{-1}]$ follows
from the following fact: if $F\subset G$ is an inclusion of finite
dimensional vector spaces, and $id_F$ and $id_G$ are the identity
elements of $F\otimes F^*$ and $G\otimes G^*$, then their images
in $G\otimes F^*$ by the natural maps coincide.
Let $x$ be any product of the $f_i[k]$, with $c\leq k\leq d$. Then $x$
is orthogonal to $I^+_{\leq -c} + I^+_{\geq -d}$. It follows that
$\cup_{a,b} (I^+_{\leq a} + I^+_{\geq b})^{\perp} = 0$, therefore
$\cap_{a,b}(I^+_{\leq a} + I^+_{\geq b})^{\perp\perp} = 0$.
The proof of Prop.\ \ref{R:mat:QC} follows then the proof of Prop.\ \ref{R:mat}.
\hfill \qed \medskip
\begin{remark} We have in the $\SL_{2}$ case
$$
P = \sum_r \sum_{i_1<\cdots<i_r, n_r\geq 0} {{\hbar^{n_1 + \cdots + n_r}}
\over{[n_1]^!_q \cdots [n_r]^!_q }
}e_{i_1}^{n_1} \cdots e_{i_r}^{n_r} \otimes
f_{-i_1}^{n_1} \cdots f_{-i_r}^{n_r} ;
$$
for $r\leq 2$, this formula is shown in \cite{Kh:D}, App.\ B.
It would be interesting to obtain analogous explicit formulas in Yangian or
elliptic cases.
\end{remark}
\section{Toroidal algebras (proofs of Props.\ \ref{oz}, \ref{nagila})}
\label{toroidal}
\subsection{Proof Prop.\ \ref{oz}}
1) follows from the argument of the beginning of the proof
of Prop.\ \ref{fargo}. The first statements of 2) are obvious.
The proof of Prop.\ \ref{fargo} then implies that $j_+$ induces a surjective
Lie algebra morphism from $\wt F$ to $\G\otimes\CC[t,t^{-1}]$, which
restricts to an isomorphim between $\oplus_{\al\in\pm\Delta_+, \al\
\on{real}; k\in\ZZ} \wt F[(\al,k)]$ and $(\oplus_{\al\in\pm\Delta_+,
\al\ \on{real}} \G[\al]) \otimes\CC[t,t^{-1}]$, which are the real roots
part of both Lie algebras, and that $\wt F_+[(\al,k)] = 0$ if $\al$ does not
belong to $\Delta_+$.
It follows that $\Ker j_+$ is a graded subalgebra of $\wt F_+$, contained in
$$
\oplus_{\al\in\Delta_+, \al\on{\ imaginary}, k\in\ZZ} \wt F_+[(\al,k)].
$$
\hfill \qed\medskip
\subsection{Proof of Prop.\ \ref{nagila}}
Let us prove 1). Let us denote by $Z(\wt F_+)$ the center of $\wt F_+$.
Let us first prove that $\Ker j_+\subset Z(\wt F_+)$. Let $x$ belong to
$\Ker j_+$. We may assume that $x$ is homogeneous of degree $n\delta$.
Then any $n\delta + \al_i$, which is a real root. $[\wt e_i[k], x]$ is
homogeneous of degree $n\delta + \al_i$, which is a real root. Since
the restriction of $j_+$ on the subspace of $\wt F$ of degree $n\delta
+ \al_i$ is injective, $j_+([\wt e_i[k], x])$ is nonzero unless $[\wt
e_i[k], x]$ is itself zero. But $j_+([\wt e_i[k], x])$ is equal to
$[j_+(\wt e_i[k]), j_+(x)]$, which is zero because $j_+(x) = 0$.
Therefore, $[\wt e_i[k], x]$ is equal to zero.
On the other hand, since $j_+$ is surjective and the center of $L\N_+$
is zero, $\Ker j_+ = Z(\wt F_+)$. This proves 1).
Let us prove 2). The argument used in the proof of 1) implies that $\Ker
j_+$ is contained in the center $Z(\wt F)$ of $\wt F$. In the same way,
one proves that $\Ker j_-$ is contained in $Z(\wt F)$, therefore $\Ker
j\subset Z(\wt F)$. On the other hand, $\wt F$ is perfect.
It follows that we have a surjective Lie algebra morphism $j' :
\T \to \wt F$, such that the composition $\T\to \wt F \to L\G$
is the natural projection of $\T$ on $L\G$. Let $j'_+$ be the restriction of
$j'$ to $\T_+$. For any $i,k$, we have
$j'_+(e_i[k]^{\T}) = \wt e_i[k] + k_{i,k}$, with $k_{i,k}$ in $\Ker(j)$.
Let $\la$ be any linear map from $\T_+$ to $\Ker(j)$, such that
$\la(e_i[k]^{\T}) = k_{i,k}$. Set $\wt j'_+ = j'_+ - \la$. Then
$\wt j'_+$ is a Lie algebra map from $\T_+$ to $\wt F$. Since
$\wt j'_+(e_i[k]^{\T}) = \wt e_i[k]$ and the $e_i[k]^{\T}$ generate
$\T_+$, the image of $\wt j'_+$ is contained in $\wt F_+$.
Moreover, $\wt j'_+$ is graded, and it coincides with $j'_+$ on the
nonsimple roots subspace $[\T_+,\T_+] = \oplus_{\al\in\Delta_+ \setminus
\{\eps_i\}} \T_+[\al]$ of $\T_+$. It follows that the restrictions of
$j'$ on $[\T_+,\T_+]$ and $[\T_-,\T_-]$ are graded.
Let us show that $\wt j'_+$ is surjective.
Since the
composition of $\wt j'_+$ with the projection $j:\wt F_+ \to L\N_+$ is
the natural projection, it suffices to show that any element $x$ of
$\Ker j_+$ is contained
in $\wt j'_+(\T_+)$. $x$ belongs to the image of $j'$, so let us set
$x = j'(y)$, with $y = y_+ + y_- + y_0$, $y_\pm$ in $[\T_\pm,\T_\pm]$
and $y_0$ in $\HH_\T \oplus \oplus_{i = 1}^n\T_+[\eps_i] \oplus
\oplus_{i = 1}^n\T_-[-\eps_i]$, where $\HH_\T$ is the Cartan subalgebra
of $\T$ (defined as $\HH[\la^{\pm 1}] \oplus Z_0$, see
Rem.\ \ref{rem:generalizations}).
Then $j'(y_\pm)$ belong to $[\wt F_\pm,\wt F_\pm]$ and
$j'(y_0)$ belongs to $\wt H \oplus \oplus_{i = 1}^n \wt F_+[\eps_i]
\oplus \oplus_{i = 1}^n \wt F_-[-\eps_i] \oplus \Ker(j)$. Moreover, the
map from $\HH_\T \oplus \oplus_{i = 1}^n\T_+[\eps_i] \oplus \oplus_{i =
1}^n\T_-[-\eps_i]$ to $\wt H \oplus \oplus_{i = 1}^n \wt F_+[\eps_i]
\oplus \oplus_{i = 1}^n \wt F_-[-\eps_i]$ induced by $j'$ is injective,
therefore $y_0 = 0$. It follows that $y_- = 0$ and $x = j'(y_+) =
\wt j'_+(y_+)$, because $\wt j'_+$ coincides with $j'$ on $[\T_+,\T_+]$.
\begin{lemma}
1) Assume that $A$ is not of type $A_1^{(1)}$.
There is a unique Lie algebra map $j''$ from $\wt F_+$ to $\T_+$ such that
$j''(\wt e_i[k]) = e_i[k]^\T$, for any $i = 0,\ldots, n$ and $k$ integer.
2) Assume that $A$ is the Cartan matrix of type $A_1^{(1)}$.
There is a unique Lie algebra map $j''$ from $\wt F_+$ to $\T_+ /
\oplus_{l\in\ZZ} \CC K_\delta[l] $ such that
$j''(\wt e_i[k]) = e_i[k]^\T$, for any $i = 0,1$ and $k$ integer.
\end{lemma}
{\em Proof.} One should just check that the defining relations of $\wt
F_+$ are satisfied by the $e_i[k]^{\T}$ (in the $A_1^{(1)}$ case, by
the images of $e_i[k]^{\T}$ in $\T_+ / \oplus_{l\in\ZZ} \CC
K_\delta[l]$). This is the case when $A$ is not of type
$A_1^{(1)}$, because in that case we set $e_i = \bar x_i \otimes
\la^{\delta_{i0}}$ and we always have $\langle \bar x_i, \bar
x_j\rangle_{\bar\G} = 0$ for $i\neq j$.
If $A$ is of type $A_1^{(1)}$, we have $x_0 = \bar f \otimes \la$,
$x_1 = \bar e$, therefore
$$
[x_0[l], x_1[m]] = ( - \bar h\la\otimes t^{l+m}, - m K_\delta[l+m]),
$$
so that $[x_0[l+1], x_1[m]] = [x_0[l], x_1[m+1]]$ holds in
$\T_+ / \oplus_{l\in\ZZ} \CC K_\delta[l]$.
\hfill \qed\medskip
Let us now prove Prop.\ \ref{nagila}, 2). The composition $\wt j'_+
\circ j''$ are Lie algebra maps from $\wt F_+ \to \T_+ \to \wt F_+$
($\wt F_+ \to \T_+ / \oplus_{k\in\ZZ} \CC K_\delta[k] \to \wt F_+$ in
the $A_1^{(1)}$ case), which map the generators $\wt e_i[k]$ to
themselves. Therefore, $\wt F_+$ can be viewed as a subalgebra of
$\T_+$ (of $\T_+ / \oplus_{k\in\ZZ} \CC K_\delta[k]$ in the
$A_1^{(1)}$ case). This subalgebra contains the elements $e_i[k]^\T$ of
$\T_+$ (resp.\ of $\T_+ / \oplus_{k\in\ZZ} \CC K_\delta[k]$). Since the
Lie subalgebra of $\T_+$ generated by the $e_i[k]^\T$ is $\T_+$ itself,
the image of $\wt F_+$ is equal to $\T_+$ (resp.\ to $\T_+ /
\oplus_{k\in\ZZ} \CC K_\delta[k]$). This proves Prop.\ \ref{nagila},
2). \hfill \qed\medskip
\begin{remark} \label{rem:india} Prop.\ \ref{nagila}, 1), can also be
obtained using the presentation given in \cite{Moody} of $\T$. In this
paper, one shows that $\T$ is isomorphic to the algebra $\dot F$ with
generators $\dot e_i^\pm[k],\dot h_i[k]$ and $\dot c$ and relations
$\ad(\dot e_i^\pm[0])^{1 - a_{ij}}(\dot e_j^\pm[k]) = 0$, $[\dot
e^\pm_i[k] , \dot e^\pm_i[l]] = 0$, $[\dot h_i[k], \dot e^\pm_j[l]] =
\pm a_{ij} \dot e^\pm_j[k+l]$, $[\dot e_i^+[k], \dot e_j^-[l]] =
\delta_{ij} \dot h_i[k+l] + i \delta_{i+j,0} \langle e_i, f_i
\rangle_{\bar\G} c$, $[\dot h_i[k],\dot h_j[k]] = k \delta_{k+l,0} \langle h_i,
h_j\rangle_{\bar\G} c$, $c$ central. It is then clear that there is a
Lie algebra map from $\T$ to $\wt F$. On the other hand, the system of
relations $[\dot e_i^+[k], \dot e_i^+[l]] = \ad(\dot e_i^\pm[0])^{1 -
a_{ij}}(\dot e_j^\pm[k]) = 0$ is {\it not} a presentation of $\T_+$,
because the ideal generated by these relations is not preserved by the
analogues of the $\Phi_{i,k}^\pm$ of the proof of Lemma \ref{india}.
\hfill \qed\medskip
\end{remark}
\begin{remark} \label{rem:generalizations}
{\em Toroidal Manin triples.}
It is easy to define an extension of the Lie algebra $\T$ with an
invariant scalar product. Recall first (\cite{Moody,Kassel})
that if $\G$ is the central
extension of the Lie algebra $\bar\G[\la,\la^{-1}]$, $\T$
is the universal central extension of
$\bar\G[\la^{\pm 1},\mu^{\pm 1}]
$. We have therefore
$$
\T = \bar\G[\la^{\pm 1},\mu^{\pm 1}] \oplus Z(\T).
$$
$Z(\T)$ is isomorphic to $\Omega^1_\AAA / d\AAA$, where $\AAA =
\CC[\la^{\pm 1},\mu^{\pm 1}]$. We have
$$
Z(\T) = \oplus_{k,l\in\ZZ} K_{k\delta}[l] \oplus \CC c,
$$
with
$K_{k\delta}[l] = $ the class of ${1\over k} \la^k \mu^{l-1} d\mu$
if $k\neq 0$ , $K_0[l] = $ the class of $\mu^l {{d\la}\over{\la}}$,
$c = $ the class of ${{d\mu}\over\mu}$.
Define for $k,l$ in $\ZZ$, $\wt D_{k\delta}[l]$ as the
derivations of $\bar\G[\la^{\pm 1},\mu^{\pm 1}]$
equal to $\la^k \mu^l (l \la \pa_\la - k \mu \pa_\mu)$ if $k\neq 0$
and to $\mu^l \la \pa_\la$ if $k = 0$, and $\wt d$ as the
derivation $\mu \pa_\mu$.
Endow $\CC^{\times 2}$ with the coordinates $(\la,\mu)$ and consider on
this space the Poisson structure defined by $\{\la,\mu\} = \la\mu$. Let
$\Ham(\CC^{\times 2})$ be the Lie algebra of Hamiltonian vector fields
on $\CC^{\times 2}$ generated by the functions $\la^k\mu^l$,
$k,l\in\ZZ^2$, $\log\la$ and $\log\mu$. For any function $f$ on
$\CC^{\times 2}$, denote by $V_f$ the corresponding Hamiltonian vector
field. Then $\Ham(\CC^{\times 2})$ is a Lie algebra, and the map
$V_{\la^k\mu^l} \mapsto \wt D_{k\delta}[l]$, for $(k,l)\neq (0,0)$,
$V_1 \mapsto 0$, $V_{\log \la} \mapsto \wt D_{0}[0]$, $V_{\log \mu}
\mapsto \wt d$, defines a Lie algebra map from $\Ham(\CC^{\times 2})$
to $\Der(\bar\G[\la^{\pm 1},\mu^{\pm 1}])$.
The formula $V_f (\sum_i a_i db_i) = \sum_i \{f,a_i\} db_i + a_i
d\{f,b_i\}$ defines an action of $\Ham(\CC^{\times 2})$ on
$\Omega^1_{\AAA} / d\AAA$, that is on $Z(\T)$. Define $\bar
D_{k\delta}[l]$ and $\bar d$ as the following endomorphisms of $\T$:
$\bar D_{k\delta}[l](x,0) = (\wt D_{k\delta}[l](x),0), \bar d(x,0) =
(\wt d(x),0)$, and $\bar D_{k\delta}[l](0,\omega) = (0,V_{\la^k
\mu^l}(\omega))$ for $(k,l)\neq (0,0)$, $\bar D_{0}[0](0,\omega) =
(0,V_{\log \la}(\omega))$, $\bar d(0,\omega) =
(0,V_{\log\mu}(\omega))$. These endomorphisms again define derivations
of $\T$, and we have now a Lie algebra map from $\Ham(\CC^{\times 2})$
to $\Der(\T)$. Let $\wt\T$ be the corresponding crossed product Lie
algebra of $\T$ with $\Ham(\CC^{\times 2})$. We denote by
$D_{k\delta}[l]$ and $d$ the elements of $\wt t$ implementing the
extensions of the derivations $\bar D_{k\delta}[l]$ and $\bar d$ to
$\T$.
Define for $a,b$ integers, $x[a,b]$ as the element $(x\otimes \la^a \mu^b)$
of $\bar\G[\la^{\pm 1}, \mu^{\pm 1}]$. Define the bilinear form
$\langle , \rangle_{\wt\T}$ by
$$
\langle x[a,b], x'[a',b'] \rangle_{\wt\T} = \langle x,x'\rangle_{\bar\G}
\delta_{a+a',0} \delta_{b+b',0},
\langle D_{k\delta}[l] , K_{k'\delta}[l']
\rangle_{\wt\T} = \delta_{k+k',0} \delta_{l+l',0},
\langle d, c \rangle_{\wt\T} = 1,
$$
and all other pairings of elements $x[a,b],K_{k\delta}[l],D_{k'\delta}[l'],c$
and $d$ are zero.
Then $\langle , \rangle_{\wt\T}$ is an invariant nondegenerate bilinear form
on $\wt\T$.
Let us define $D$ as the image of $\Ham(\CC^{\times 2})$ in $\wt\T$. Let
us set $D_>, D_<$ and $D_0$ as its subspaces $\oplus_{k>0,l\in\ZZ} \CC
D_{k\delta}[l]$, $\oplus_{k<0,l\in\ZZ} \CC D_{k\delta}[l]$ and
$\oplus_{l\in\ZZ} \CC D_{0}[l] \oplus \CC d$. We have then $D = D_> \oplus
D_0 \oplus D_<$. In the same way, define $Z_>, Z_<$ and $Z_0$ as the subspaces
$\oplus_{k>0,l\in\ZZ} \CC
K_{k\delta}[l]$, $\oplus_{k<0,l\in\ZZ} \CC K_{k\delta}[l]$ and
$\oplus_{l\in\ZZ} \CC K_{0}[l] \oplus \CC c$ of $Z(\T)$. We have then
$Z(\T) = Z_> \oplus Z_< \oplus Z_0$.
Recall we defined $\HH_\T$ as the subalgebra $\bar\HH[\la^{\pm 1}]
\oplus Z_0$ of $\T$. $\wt \HH_\T = \HH_\T\oplus D_0$ is then a Lie
subalgebra of $\wt\T$. In the spirit of the new realizations, we split
$\wt \HH_\T$ in two parts.
Let us set $\HH_+ = \bar\HH[\la] \oplus Z_0$,
$\HH_- = \bar\HH[\la^{-1}] \oplus D_0$; then $\HH_+ + \HH_-
= \wt\HH_\T$, and $\HH_+ \cap \HH_-$ is $\bar\HH$.
Define $L\N_+$ and $L\N_-$ as the linear spans of the $x[a,b]$,
$a\in\ZZ, b>0$ ($b\geq 0$ if $x\in\bar\N_+$), resp.\ of the $x[a,b]$,
$a\in\ZZ, b<0$ ($b\leq 0$ if $x\in\bar\N_-$). $L\N_\pm$ are Lie
subalgebras of $\bar\G[\la^{\pm 1},\mu^{\pm 1}]$. $L\N_\pm \oplus
Z_\pm$ and $\wt{L\N}_\pm = L\N_\pm \oplus D_\pm \oplus Z_\pm$ are also Lie
subalgebras of $\wt\T$. Set $\wt\T_\pm = \wt{L\N}_\pm \oplus \HH_\pm$.
Endow $\wt\T\times \bar\HH$ with the scalar product
$\langle , \rangle_{\wt\T\times \bar\HH}$ defined by
$$
\langle (x,h) , (x',h') \rangle_{\wt\T\times \bar\HH} =
\langle x, x' \rangle_{\wt\T} - \langle h, h' \rangle_{\bar\HH}.
$$
Let $p_\pm$ be the natural projection of $\wt \T_\pm$
on $\bar \HH$. Identify $\wt\T_\pm$ as the Lie subalgebras
of $\{(x, \pm p_\pm(x)), x\in \wt\T_\pm\}$ of
$\wt\T\times \bar\HH[\la,\la^{-1}]$.
$\wt\T_\pm$ are supplementary isotropic subspaces of
$\wt\T\times \bar\HH[\la,\la^{-1}]$ and define therefore a
Manin triple. This Manin triple is a central and cocentral
extension (by $Z(\T)$ and $D$) of the Manin triple
$$
(\bar\G[\la^{\pm1},\mu^{\pm1}] \times \bar\HH,
L\N_+ \oplus \bar\HH[\la], L\N_- \oplus \bar\HH[\la^{-1}])
$$
which is a part of the new realizations Manin triple
$(\G[\mu^{\pm1}]\times \HH, L\B_+,L\B_-)$.
One may also consider ``intermediate'' Manin triples, for example
$$
\left( \{\bar\G[\la^{\pm1},\mu^{\pm1}] \oplus Z_> \oplus D_< \} \times \bar\HH,
L\N_+ \oplus \bar\HH[\la] \oplus Z_>, L\N_- \oplus \bar\HH[\la^{-1}] \oplus D_<
\right).
$$
It is a natural problem to quantize the corresponding Lie
bialgebra structures on $\wt\T_\pm$. For this, one can think of the
following program:
1) to compute the centers of $U_\hbar L\N_+$ and (following
\cite{FO:new}) the center of $FO$. By duality, these central elements
should provide derivations of $U_\hbar L\N_-$ (and $FO$) of imaginary
degree. Compute these derivations and relations between them. One could
expect that the algebra generated by the derivations is some difference
analogue of the Lie algebra $\Ham(\CC^{\times 2})$.
2) it should then be easy, following Thm.\ \ref{thm:third}, to prove
that the analogue of $i_\hbar$ is an isomorphism, and to derive from
there the quantization of the Lie bialgebra $L\B_+$.
We hope to return to these questions elsewhere.
\end{remark}
\appendix
\section{Lemmas on $\CC[[\hbar]]$-modules}
\begin{lemma} \label{str:modules}
Let $E$ be a finitely generated $\CC[[\hbar]]$-module. Let $E_{tors} =
\{x\in E | \hbar^k x = 0 $ for some $k >0\}$ be the torsion part of $E$.
Then $E_{tors}$ is isomorphic to a direct sum $\oplus_{i=1}^p \CC[[\hbar]] /
(\hbar^{n_i})$, where $n_i$ are positive integers, and $E$ is isomorphic to the
direct sum of $E_{tors}$ and a free module $\CC[[\hbar]]^{p'}$.
\end{lemma}
{\em Proof.} As $E$ is finitely generated, we have a surjective
$\CC[[\hbar]]$-modules morphism $\CC[[\hbar]]^N \to E$. Let $K$ be the
kernel of this morphism. Then $E$ is isomorphic to $\CC[[\hbar]]^N /
K$.
Let us determine the form of $K$. Let us set $\bar K_i = K \cap \hbar^i
\CC[[\hbar]]^N$. Then we have $\hbar\bar K_i\subset \bar K_{i+1}$. Let
us set $E_0 = \CC^N$, and $F_i = \hbar^{-i}\bar K_i$ mod $\hbar$. Then
we have $F_0 \subset F_1 \subset\cdots \subset E_0$. Let $p$ the integer
such that $F_k = F_p$ for $k\geq p$. We can then find a basis
$(v_i)_{1\leq i\leq N}$ of $E_0$ such that $(v_1,\ldots,v_{\dimm F_0})$
is a basis of $F_0$, $(v_1,\ldots,v_{\dimm F_1})$ is a basis of
$F_1$, etc., $(v_1,\ldots,v_{\dimm F_p})$ is a basis of $F_p$. Then $K$
is the submodule $ \oplus_{i} (\oplus_{k = \dimm F_{i-1} + 1}^{\dimm
F_i}\hbar^i \CC[[\hbar]] v_k)$ of $E_0[[\hbar]]$.
It follows that the quotient $E_0[[\hbar]] / K$ is isomorphic to a direct
sum $\oplus_{i=1}^p \CC[[\hbar]] / (\hbar^{n_i}) \oplus \CC[[\hbar]]^{p'}$.
The statement of the Lemma follows.
\hfill \qed \medskip
\begin{cor} Any $\CC[[\hbar]]$-submodule of a finite-dimensional free
$\CC[[\hbar]]$-module is free.
\end{cor}
{\em Proof.} This follows from the fact that such a submodule has no torsion
and from the above Lemma. \hfill \qed\medskip
We have also
\begin{lemma} \label{free:inf:dim}
Let $E$ be a free $\CC[[\hbar]]$-module with countable
basis $(v_i)_{i\geq 0}$. Any countably generated $\CC[[\hbar]]$-submodule of
$E$ is free and has a countable basis.
\end{lemma}
{\em Proof.} We repeat the reasoning of the proof of Lemma
\ref{str:modules}. Let $(w_i)_{i\geq 0}$ be a countable family of $E$
and let $F$ be the sub-$\CC[[\hbar]]$-module of $E$ generated by the
$w_i$.
Set $\bar F_i = F\cap \hbar^i E$ and $F_i = \hbar^{-i}F$ mod $\hbar$.
Generating families and bases for the $F_i$ can be constructed inductively
as follows.
A generating family for $F_0$ is $(w_i$ mod $\hbar)_{i\geq 0}$. We can
then construct by induction a partition of $\NN$ in subsets $(i_k)$ and
$(j_k)$ such that $(w_{i_k}$ mod $\hbar)_{k\geq 0}$ is a basis of
Span$(w_i$ mod $\hbar)_{i\geq 0}$.
Let $\la_{kk'}$ be the scalars such that $w_{j_k} - \sum_{k'} \la_{kk'}
w_{i_{k'}}$ belongs to $\hbar E$. Set $w_k^{(1)} = \hbar^{-1}
[w_{j_k} - \sum_{k'} \la_{kk'} w_{i_{k'}}]$. Then a generating family of
$F_1$ is $(w_{i_k},w^{(1)}_k$ mod $\hbar)$. We then construct by
induction a partition of $\NN$ in subsets $(i_k^{(1)})$ and $(j_k^{(1)})$
such that $(w_{i_k},w^{(1)}_{i_k^{(1)}}$ mod $\hbar)$ is a basis of $F_1$.
It is clear how to continue this procedure. Then
$(w_{i_k},w^{(1)}_{i_k^{(1)}},w^{(2)}_{i_k^{(2)}} \ldots)$ forms a basis
of $F$. \hfill \qed \medskip
\frenchspacing | 71,667 |
Virtual Flipbooks
West Jordan, Utah
Print Case Study
Owner:
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Nolte Consultant Engineers
Contractor:
Ames Construction
Installation:
June 2000
Technical Description:
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The 40-foot pipe lengths weighed only 1600 lbs each. The manifolds and manhole risers were prefabricated. The installation was quick and economic.
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800-338-1122 |
© 2020 Contech Engineered Solutions LLC, a QUIKRETE Company | 362,633 |
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TITLE: Proving a polynomial splits over a certain field extension
QUESTION [1 upvotes]: If $K$ is the splitting field of $f\in F[x]$, and $g\in F[x]$ is irreducible and has a root in $K$, prove that $g$ splits over $K$.
My proof (which I don't think is correct) is as follows:
Let $a\in K$ be a root of $g$. As $g(a)=0$, by the division theorem for polynomials $\exists h\in F[x]$ such that $g(x) = (x-a)h(x)$, where $\deg(h)=\deg(g) - 1$. We will prove by induction.
If $\deg(g) = 2$, then $g(x) = (x-a)(x-b)$ where $a,b\in K$. So the base case holds.
Now assume that the proposition is true for $\forall g\in F[x]$ such that $\deg(g) = n$.
If $\deg(g) = n+1$, then $g(x) = (x-a)h(x)$ where $\deg(h) = n$. As $\deg(h) = n$ we see that $h$ splits over $K$ and therefore so does $g$. So by induction we are done.
I feel like even though $\deg(h) = n$, I can't assume that $h$ automatically splits over $K$ because $a$ is not necessarily a root of $h$. Any help or tips appreciated!
REPLY [1 votes]: Try doing it using the normality condition, because automorphisms of a field over its algebraic closure permute the roots of a generator, you can see that if $K$ is the splitting field for $f$ then $K= F(\alpha_1,\ldots, \alpha_n)$ with $\alpha_i$ the roots of $f$. But then as $F(\beta)$ with $\beta$ a root of $g$ is a subfield of $K$, we see that there is an injection
$$E = F[x]/(g(x))\to K$$
so we may consider $F\subseteq E\subseteq K$. But then any automorphism of $E/F$ extends to an automorphism of $K/F$, and vice versa for restriction. But then as all automorphisms of $E/F$ are realized by automorphisms of $K/F$ (when restricted to $E$) and these automorphisms are exactly those that permute the roots (transitively since $g$ is irreducible) all roots of $g$ are just $\varphi(\beta)$ for some $\varphi: K\to K$ fixing $F$, i.e. an automorphism of $K/F$. Since all the roots are in $K$, $g$ splits in $K[x]$ by definition. | 68,006 |
TITLE: Will the magnetic field still remain if there is no current in the electromagnet? Why
QUESTION [2 upvotes]: Will the magnetic field still remain if there is no current in the electromagnet
REPLY [2 votes]: Faraday's laws of induction states that for a change in current in one conductor there is a proportional change in magnetic flux induced by it.
In your case as the conductor has no current for long time hence leading to inference that change in current is zero so there will be no induced magnetic field.
Hence the electromagnetic loses it's magnetic strength.
But coming to a realistic world does it always happen?
No
Your magnetic flux carrier material is made of ferrous materials hence it has prominent dipoles. In case the carrier has the magnetic field for a long time the dipoles will align and hence give the material a permanent/temporary magnetic characteristic, making it essentially a powerful magnet.
And this is also a major reason why choice of material is necessary while building electromagnets.
Higher retentivity leads to higher residual magnetic character in the material even after the current is cut off. | 41,595 |
TITLE: Prove uniqueness of linear spline
QUESTION [2 upvotes]: Let $f: \mathbb{R} \to \mathbb{R}$ be linear spline (continuous, piecewise polynomial of degree $\le 1 $) at knots $x_0 < x_1 < … < x_n$. Prove $f$ can be uniquely represented at form:
$$\displaystyle f(x) = a+bx+ \sum_{j=0}^n c_j|x-x_j|$$ for some $a,b,c_j$.
I've tried to prove it by contradiction assuming there are two representation but I haven't figured out anything so far. I'll be grateful for some hints, since I've got stuck at this problem and don't have any idea
REPLY [1 votes]: Given a set $S=\{x_0,\ldots,x_n\}\subset{\mathbb R}$ of $n+1$ knots denote by $V_S$ the vector space of linear splines $f\colon\>{\mathbb R}\to{\mathbb R}$ with knots in $S$. I claim that ${\rm dim}(V_S)=n+3$.
Proof. In order to fix an $f\in V_S$ we can freely choose $n+2$ slopes $m_i$ in the $n+2$ intervals
$$(-\infty,x_0), \ (x_0,x_1), \ \ldots, \ (x_{n-1},x_n), \ (x_n,\infty)\ ,$$
which then will determine $f'$ wherever defined. In this way $f$ is defined up to an additive constant, which is finally determined by fixing $f(0)$.$\qquad\square$
On the other hand, the $n+3$ functions
$$1,\quad x,\qquad\phi_j(x):=|x-x_j|\quad(0\leq i\leq n)\tag{1}$$
belong to $V_S$. I claim that they are linearly independent.
Proof. If
$$a+b x+\sum_{j=0}^n c_j \phi_j(x)\equiv0$$
then the $c_j$ have to vanish individually, because each $\phi_j$ has a property that cannot be annihilated by a linear combination of the other involved functions: It has a corner at $x_j$. From $ax+b\equiv0$ it then follows that $a=b=0$.$\qquad\square$
It follows that the functions $(1)$ form a basis of $V_S$. This implies that any $f\in V_S$ has a unique representation as a linear combination of the functions $(1)$. | 97,975 |
TITLE: Computing the first $n$ values of the Liouville function in linear time
QUESTION [7 upvotes]: Is it possible to compute the first $n$ values of the Liouville function in linear time? Since we need to output $n$ values we clearly cannot do better than linear time, but the best I can figure out is something like $O(n \cdot \log{\log{n}})$: fill an array of size $n$ with ones, and for each prime power $p^a$, negate the value at the index of each of its multiples. I think it is possible to identify the prime powers as we count up using another table of $O(n)$ bits, essentially the sieve of Eratosthenes counting powers too, but there are still $\sum_{p^a \le n}{\frac{n}{p^a}} = n \cdot \log{\log{n}} + O(n)$ negation operations. Is it possible to do better than this?
REPLY [0 votes]: I know this question was posed 5 months ago but I thought I would add something.
Not sure if this will increase computing time but,
$\lambda(n)=i^{\tau(n^{2})-1}$,
where $\tau(n)$ is the divisor function and $i$ is the imaginary unit. From this of course the summatory Liouville function is $L(n)=\sum_{j=1}^{n}i^{\tau(j^{2})-1}$. Perhaps the divisor function can be calculated in less time. | 144,878 |
Welcome, dear reader, to the fourth and likely final “season” of our play-through of Castle Xyntillan.
The Company:
- Hendrik (MU6)
- Jürg (F6/T1)
- Noel & Göpf (porter)
- Florin (bowmen)
- Francesca, Nathalie & Liv (F1 retainers)
Loot: Oh boy. No mundane treasure was recovered, but the company did acquire a Horn of Blasting, a blue book titled Castle Xyntillan, a black book titled The Compendium of Champions, a crusader’s two-handed axe +1, and a plain gold cup containing powerful evil magics.
Casualties: Göpf and Florin, struck dead while trying to grab treasure; Nathalie, polymorphed into a snail by the eye beams of a giant snail statue; Francesca & Liv, slaughtered by undead crusaders.
It is Monday, September 2, 1527. Five weeks have passed since the last expedition by our nameless company into Castle Xyntillan. Having recovered the Scepter of the Merovings, the company set their sights on the mysterious stuccoed secret door in the castle dungeons, which they suspect may be opened by it.
Upon arrival, they make for the grand entrance. They keep an eye on the walls to see if any of their beans have sprouted. Most appear to have been weeded out by the skeleton gardening crew. But one sturdy stalk is growing along the south wall of the gatehouse.
They enter through the grand entrance, make their way down into the wine cellar without trouble, and head up to the tomb where once the company had an ill-fated encounter with a pack of ghouls, and lost many of their most powerful members and equipment. They go up the the stuccoed secret door depicting a dark-skinned man leading crusaders up a hill lined with palm trees. They hold the scepter up to it, and lo and behold, it opens!
At that very moment, however, the rear ranks are attacked by a lone ghoul. There is a moment of panic while Jürg pushes through to the melee, and cleaves the thing in two with one mighty strike of his chaotic magic zweihander, the Blade of Rel. With that out of the way, they turn back to the newly opened corridor, and cautiously head inside.
Jürg asks his blade if any traps are nearby, and it trembles with confirmation. They spot a suspicious-looking crack half-way down the corridor, and notice an elevated floor tile just in front. They drag the ghoul corpse from the tomb and toss it on the tile. It presses down, making a loud clicking noise, and the next moment, a huge blade slams down along the crack. Next, it slowly ascends again to the sounds of some hidden ratcheting mechanism. When they test the switch again, it does not trigger anymore. Gingerly, the company step over the tile one by one and make their way into the room beyond.
This room’s walls are decorated with portraits of various warrior saints with stern looks on their faces. Their eyes follow the company as they make their way to a door leading east. They enter a large room with a pool and several doors. They first check the door north-west. It opens onto a corridor that stretches into the darkness beyond their torchlight. They try the next door, leading east.
At this point, the players were joking about how it would be nice to come across a door with a sign reading “treasure” for a change. You won’t believe what happened next.
The center piece of this room is a circle of statues of dwarves in chains holding up a massive plateau. On it is piled a massive amount of treasure. Gold, gems, jewelry, magic items, weapons, scrolls and books. Oh my! The plateau’s edge is inscribed with: “To each hero, one treasure of far lands shall be the prize.”
The company agonize over what to do. Is it wise to grab this treasure? Might it be cursed? Etcetera. Finally, Jürg has had enough, and grabs the horn. The very next instance, they hear a loud bang of stone on stone, reverberating throughout the dungeon. They look at each other worriedly. Swallow, take a deep breath, and then Hendrik steps up and grabs one of the books…
Nothing happens. Puzzled, they next ask one of the retainers to grab something. Some resist, worried that it might be a trap or something. Finally, one of the fighting-women retainers is courageous enough to grab the other book. And nothing happens again. So then they convince bowman Florin to grab something. The moment he lays a hand on the treasure, his breath chokes in his throat, his eyes glaze over, and he drops to the dungeon floor, dead.
The company manage to hold it together, and speculate about why this might be. Hendrik and Jürg pressure porter Göpf into grabbing something despite what happened to Florin, and for some reason, the man is foolish enough to go for it. Alas for poor Göpf, his fate is identical to that of the bowman. Soon enough he lies dead on the floor as well.
They decide they have had enough of this room and will not push their luck any further. They return to the pool room. They next check the south door. They find themselves in another large room containing a massive statue of a monstrous snail. They make their way to the next door north, all the while wearily eyeing the snail.
The next room contains dazzling mosaics. From here they take another door north, and find themselves in a room with an altar on which stands a plain gold cup. Hendrik detects strong evil magical energies emanating from it. Jürg (who is aligned to Chaos) goes “great,” steps up to the pedestal and with one smooth motion tips the cup into a sack. Nothing happens.
They backtrack to the room with the snail statue and search it for secret doors. Sure enough, they find one in the south-east corner. They push it open, but something blocks it. Through a crack they see a massive stone block that was clearly dropped very recently.
That very moment, the snail statue animates and starts to crawl their way. They decide to make a run for the exit as fast as they can. They escape the room before the thing can get to them, but it does manage to zap one of the fighting women in their rear rank with a beam from its eye stalks. The poor retainer is polymorphed into a snail. The door slams shut.
They catch their breath and briefly mourn the loss of yet another companion. They decide they should check the door leading to that dark corridor which they abandoned earlier. As they cautiously move down it they pass gated alcoves holding sarcophagi left and right. At the end, a small circular room holds a glass case containing a huge double-headed axe. Before going for the axe they decide to check the sarcophagi. As they open the first one, a skeletal hand reaches out and an undead crusader knight starts to climb out. Meanwhile, the remaining sarcophagi are pushed open from inside as well, and more crusaders appear.
The company decide to face the undead, but make their way to the door they came from, which should serve as a convenient choke point. When they get there they turn around, and Hendrik flings a lightning bolt from his wand back down the hallway. The crusaders are harmed but still going. The distance is closed, and Jürg and a retainer make a stand in the doorway.
A lot of hacking and slashing ensues. Hendrik uses many spells to increase the company’s powers, and reduce that of the undead. When that seems to be insufficient, he pulls out a scroll of protection from undead, and manages to turn away about half of the crusader fighting force. Still, the remaining fighting women are killed, one by one.
At the very end, Hendrik is standing next to Jürg in the doorway. They blast the crusaders with their wand of cold, Jürg hacks away with his Blade of Rel. And finally, the battle is won.
Very cautiously, they had back down the hallway. They know several undead crusaders have fled down it. When they get to the end they see several have returned to their sarcophagi. A lone crusader has ended up in the room with the axe in the glass case. They manage to flush it out, and turn their attention to the axe.
They smash the glass case, and grab the axe. They decide this is the time to leave. They move back the way they came until they get to a door behind which they know is the cellar of the donjon. A ghost once told them it lost her ring in there somewhere. They hear a lot of rodent noises from behind the door. Hendrik uses clairvoyance to see what is behind the door, and detects magic to see if he can find the ring. He sees a large circular cellar lined with alcoves. A lot of human remains and debris on the floor. But no ring.
They decide to definitively end the expedition there, and head back out of the castle and back to town without further issue.
Referee Commentary:
We’re back in that damn castle after a welcome Christmas break. I polled the players if they had enough, but some were keen on going for one more season. It’s clear we are nearing a climax, now that we have several very powerful characters in the company (recall that level 6 is the highest level in my homebrew version of D&D), and they have acquired many, many powerful items. Not to mention a ludicrous amount of wealth.
I keep urging them to think bigger than simply continuing to clear the dungeon room by room. Let’s see if they catch on. This session at least immediately kick-started things with a more or less complete looting of the crusader tomb. However, readers familiar with the module will know they made a very fateful decision in the treasure room.
Without entirely spoiling what the module states, as written it closes off a particular direction for the players entirely if they do something, but they have no real way of knowing this. (At least, mine did not.) So I decided to make it a little less definitive, but still hugely significant. That was the point of the thing with the secret door and the stone blocking it. Sorry for being so vague about it.
As I continue to tinker with our combat sequence, we are now trying full-on phases. So we revolve actions in the order of missile, movement, melee, and magic for both sides. Initiative decides who goes first within each phase. It worked okay, yielding some interesting dilemmas, and it nicely reinforces this idea that everything more or less happens at the same time during combat.
The fight with the undead turned into a bit of a slog. I guess I should have had them change tactics after the second round of simply exchanging blows in the doorway. But my mental bandwidth was too limited to come up with something, I guess. In hindsight, I think I should have had one crusader try and push their way past the front rank. Maybe using an opposed attack roll to resolve it or something? Would have been more dramatic.
The two books recovered in the treasure room are neat. The Champions book in particular is crazy powerful. As written it allows for summoning living or dead characters listed in the module, that were played by the playtesters. I decided to change this into any character lost by the players in our own campaign. They loved that, and have already decided to bring back good old Claus, who died after failing his roll on the table of terror.
And that plain gold cup. Oh boy. Readers who know about the module, know that it was particularly significant for the one chaotic character in the company to grab it. I am still puzzling over what the effects for Jürg will be. They have had it identified and know about its powers, they also know it contains a powerful malevolent spirit. But they don’t know who or what specifically is in there…
2 replies on “Castle Xyntillan – Session #37 – Good and Bad Destiny”
Boy oh boy! That’s quite interesting indeed.
(And a nice collection of prizes, too. This is definitely the high-end stuff.)
Some of those items are positively bonkers. And you’ll see in the next report they were immediately put to good use… | 36,918 |
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When comparing Torchlight II vs AudioSurf, the Slant community recommends Torchlight II for most people. In the question“What are the best Colorful games on Steam?” Torchlight II is ranked 5th while AudioSurf is ranked 29 Can be casual or hardcore.
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\begin{document}
\title{
Bayesian inversion for electromyography using low-rank tensor formats
}
\author{
Anna R{\"o}rich\thanks{
Institute~of~Applied~Analysis~and~Numerical~Simulation, University~of~Stuttgart, Allmandring~5b, 70569~Stuttgart, Germany\newline
\href{mailto:[email protected]}{[email protected]}, \href{mailto: [email protected]}{[email protected]}
}
\and
Tim A.\@ Werthmann\thanks{
Institut~f{\"u}r~Geometrie~und~Praktische~Mathematik, RWTH~Aachen~University, Templergraben~55, 52056~Aachen, Germany\newline
\href{mailto:[email protected]}{[email protected]}, \href{mailto:[email protected]}{[email protected]}
}
\and
Dominik G{\"o}ddeke\footnotemark[1]
\thanks{Stuttgart~Center~for~Simulation~Science, University~of~Stuttgart, Pfaffenwaldring~5a, 70569~Stuttgart, Germany}
\and
Lars Grasedyck\footnotemark[2]
}
\date{\today}
\maketitle
\begin{abstract}
The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications.
Mathematically, this process is an inverse problem.
Furthermore, electromyographic data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue.
Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach.
Solving the discretized Bayesian inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data.
Using, e.g., a Metropolis-Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort.
Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution.
The application of Bayes' theorem proves the well-posedness of the Bayesian inverse problem.
The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm.
Numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters.
The Metropolis-Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods.
\par\vspace\baselineskip\noindent
\emph{Keywords}: inverse problem, parameter-dependent problem, Metropolis-Hastings algorithm, hierarchical Tucker format, EMG
\end{abstract}
\input{tex/introduction}
\input{tex/electromyographic_model}
\input{tex/problem}
\input{tex/representation_parameter_dependent_problems}
\input{tex/cp_format}
\input{tex/ht_format}
\input{tex/Bayes_inverse_EMG}
\input{tex/linear_system}
\input{tex/finite_difference_operator}
\input{tex/finite_difference_rhs}
\input{tex/parameter_dependent_problems}
\input{tex/operator}
\input{tex/algorithms}
\input{tex/numerical_experiments}
\input{tex/related_work}
\input{tex/conclusion}
\section*{Acknowledgments}
We thank Maren Klever for her critical reading of and suggestions for this article.
We thank the anonymous referees for helping to improve the article by their suggestions.
This research was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -- EXC-2075 -- 390740016.
L.\@ Grasedyck and T. \@A.\@ Werthmann have been supported by the DFG within the DFG priority program 1886 (SPPPoly) under Grant No.\@ GR-3179/5-1.
\printbibliography{}
\end{document} | 148,185 |
Check out some available loans that are similar to this one!
Anonymous
About Uganda
- $1,500Average annual income
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A loan of $725 helped this borrower to purchase crops like passion fruit and also poultry to sell.
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Little Lamb Boy Gift Box Bundle
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The Orange led 7-3 at half and 10-6 after three quarters before holding on to defeat Hopkins for the seventh time in the last eight meetings. Read More...Continue Reading
The Johns Hopkins men's lacrosse team got four goals each from Ryan Brown, John Crawley and Brandon Benn and used a 10-1 run to pull away from visiting Towson Saturday afternoon. Continue Reading
CHAPEL HILL, N.C. - North Carolina junior face-off man R.G. Keenan will always remember the first ever game he played in Kenan Stadium. The Perry Hall, Md., native won the overtime face-off and scored six seconds Continue Reading | 165,569 |
Spring Fling: Centenary Square, Bradford, UK3 5 Share Tweet
Bradford has recently gone through a bit of much-needed regeneration to attract more tourists into its city center. For me, this has been a huge success and I intend to photograph it lots and lots over the coming months.
Bradford is not always given the best advertising and is rarely shown for its merits. Despite living in Leeds (a neighboring city just 10 miles away) for almost 8 years, I think I have visited central Bradford three times in total as there has never been loads to do in the center and of course, the shops are not as good as those in Leeds. On a recent visit to a friend’s house however, we found ourselves wandering down to the new fountains in Centenary Square right in the heart of the city and we were all really, really impressed.
There is a really nice article about the new fountains here which I enjoyed because it suggests that the bursts of water represent Yorkshire Pride – what a lovely concept that is. The area itself has had a lot done to it, with a new building near the town hall full of cafes, bars and restaurants. Then there is a huge area (bigger than a football pitch at the very least, probably more) where the “puddle in the park” fountains now stand. They start in the early morning, spraying just a bit of mist around, and get more powerful through the day. There are periods when you can stroll between them all, but as the water gets more forceful as the day goes by, the pathways disappear. Then in the evening it all lights up and looks lovely. This is a great place to take your other half for a wander round in the spring time sunshine (if we ever get any, that is!)
written by kneehigh85 on 2012-03-12 #places #requested-water-fountain-bradford-yorkshire-centre-city-regeneration #location #urban-adventures
3 Comments | 382,291 |
Do you want to know the best podcasts about food? Look no further. Foodgrads has compiled a list of the most amazing podcasts about the food industry.
We’ve got careers in food, food businesses, food topics, start-ups and more…..
Did we miss anyone? Email [email protected] let us know if they should be included on this page.
My Food Job Rocks!
My Food Job Rocks! is the first podcast of it’s kind. We interview food professionals, both fresh and experienced, so that they can tell the world why working in the food industry is awesome and why their Food Job Rocks!
Want to find an episode? You can check out our website for all of our information:
Or we are currently available on Itunes and Stitcher.
Peas On Moss
PeasOnMoss brings chefs, entrepreneurs, and food professionals together to tell their career stories, give some R&D advice, and discuss their passion for food. Whether you’re a chef considering corporate work, a foodie curious about product development, or a seasoned food business owner, we invite you to join our conversations about food careers and the curious world of R&D.
The Food Heros Podcast
The Food Heroes Podcast focuses on ways to change the world through food. They interview people who are having a positive impact on the environment, making healthy food accessible, and empowering us to eat ethically. We learn how food can be a powerful and delicious tool for positive change.
There are so many more episodes to listen to! Check them all out here!
Gastropods
Gastrop.
The Barron Report
The restaurant industry isn’t just the business of food. It’s a cross-functioning, multivariate entity that touches almost every industry — after all, everyone needs to eat. So, why not learn from the greats from all businesses? Join award-winning journalist, host, and author Paul Barron as he connects the dots and discusses innovative strategies with thought leaders around the restaurant industry. Get the inside scoop on trends and open your eyes to the full vision of restaurant and hospitality with The Barron Report.
Food Startups Podcast
If you are a food startup, the chips are stacked against you. It is expensive and complicated. It can be stressful. Many ups + downs.
The food business, unlike the tech world, is not so “user-friendly” for startups. Finding packaging, getting in front of Whole Foods buyers, figuring out what paperwork is required for import/export, how to start, how to scale, etc..
We want to help you share your project with the world and turn it into a sustainable business. You are not alone. Every week, we deliver a new show with a guest that could be in the beginning stage of a startup, a food author/advisor, or head of a 8/9 figure revenue company.
Take a look at our podcasts. To start you may like our interview with Seth Goldman of Honest Tea, acquired by Coca Cola and in over 100,000 outlets in the United States.
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The Slow Melt
Enjoy this 23-minute program hosted by Simran Sethi. The Slow Melt uses chocolate as the thick, delicious lens through which to explore the world—from flavor and physiology to chemistry and conservation, from global markets and gender to climate change, social justice and beyond—highlighting the people, places and processes behind this $100 billion industry! | 344,055 |
Motorola Phones Tmobile
HTC, in cooperation with Google created a new cell phone with an operating system Android. The cell phone is called "T-Mobile G1" and is characterized by a special design with a large screen that can be pushed aside to reveal a full QWERTY keyboard, 5 line.
The phone is said to be a great competitor to Apple's iPhone. The mobile phone is built around an operating system open source, like the iPhone, so both developers and manufacturers mobile content can lead users and support applications. In practice, this means that everyone can build applications for the mobile phone to anyone buying or just download it free via the Internet. Using the built-in keyboard, it became easy to navigate the phone, network and make tasks such as writing e-mails or other documents.
The G1 comes with:
- 528MHz processor
- WLAN function
- 3.2-inch touchscreen
- Full QWERTY keyboard line 5
- 3.2 megapixel color camera with auto focus
- Full HTML Browser
- Google Talk is a chat program
- GPS navigation capability with Google Maps
- Street View, which means you can view maps in real images of where you are and, in view of the street
- Gmail
- YouTube
And many other features of Google.
The phone weights 158 grams, which also include battery. The G1 also has a built-in microphone and speaker and supports multiple audio formats such as video and MP3, AAC, ACC + and Mp4
The G1 has a box search on the desktop that is used to search both the phone and the Internet. To reach the search box, you just have to slip right through your fingers and slide to a desktop screen with a search box only. The search box also has some interesting features. For example, he tries to guess what you want to search for listings the most common searches on the Internet while you are typing.
One of the features that makes the G1 stands out the iPhone is the background image.
On the iPhone, you can not put background images on your desktop, while on a G1 phone you're just a panoramic image when "sliding" around desktop.
The G1 looks virtually the same as the previous T-Mobile Sidekick series.
I do not like the design much. I did not expect a little more with Google and HTC working together, but I also think that the software works very well and has some very nice features. The G1 is very simple to use and you can take some shortcuts and add them directly to your desktop.
The absolute biggest advantage is the ability to accept the open-source software, and provides this concept is the future..
Motorola Razr Camera/ Video Phone and $55 T Mobile…
This entry was posted by admin on October 3, 2009 at 11:06 pm, and is filed under Motorola Phones. Follow any responses to this post through RSS 2.0. You can leave a response or trackback from your own site. | 210,577 |
Contents
Real Estate Analysis is providing this data as advisory and solely as a point of reference. The data is an amalgamation of data reported to TDHCA in the Part D Owner’s Financial Certifications. For that reason, REA utilizes cleansed data for comparative purposes only.
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\begin{document}
\preprint{APS/123-QED}
\title{Analysis of coherent quantum cryptography protocol \\ vulnerability to an active beam-splitting attack}
\author{D.A. Kronberg}
\affiliation{Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia}
\affiliation{\mbox{Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow 119899, Russia}}
\author{E.O. Kiktenko}
\affiliation{Theoretical Department, DEPHAN, Skolkovo, Moscow 143025, Russia}
\affiliation{Bauman Moscow State Technical University, Moscow 105005, Russia}
\author{A.K. Fedorov}
\affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\affiliation{Acronis Ltd, Moscow 127566, Russia}
\affiliation{LPTMS, CNRS, Univ. Paris-Sud, Universit\'e Paris-Saclay, Orsay 91405, France}
\author{Y.V. Kurochkin}
\affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\date{\today}
\begin{abstract}
We consider a new type of attack on a coherent quantum key distribution protocol [coherent one-way (COW) protocol].
The main idea of the attack consists in measuring individually the intercepted states and sending the rest of them unchanged.
We have calculated the optimum values of the attack parameters for an arbitrary length of a channel length and compared this novel attack with a standard beam-splitting attack.
\end{abstract}
\maketitle
\section{Introduction}
Breakthrough in methods of manipulation for individual quantum systems plays a crucial role for the implementation of quantum technologies.
Quantum technologies have a significant potential for design of computers~\cite{Feynman,Feynman2,Deutsch} and communications devices~\cite{BB84}.\,The
use of quantum systems in the role of basic structural elements in computer allows one to achieve considerable improvement in a number of tasks \cite{Deutsch}
such as search in unstructured databases~\cite{Grover}, and in integer factoring and discrete logarithm problems~\cite{Shor}.
The last problems are of significant importance for public-key cryptography~\cite{DiffieHellman,Rivest},
which is based on the complexity of these tasks for classical computers.
But these tasks can be solved faster using quantum computers.
This endangers the existing methods of information security using cryptographic tools.
Furthermore, the absence of an efficient non-quantum algorithm for solving these problems still remains unproved.
In fact, appearance of quantum computing device limits possible tools for cryptography by two methods.
The first is to use problems without both classical and quantum efficient algorithms as novel public-key cryptography schemes.
These methods are in foundations of post-quantum cryptography~\cite{Bernstein}.
However, it is an open problem whether it is possible to design an effective algorithm that solves these problem, so the post-quantum cryptography keeps to be potentially vulnerable.
Another possible solution is to use cryptography with a private key.
On the one hand, these systems (at certain conditions) are absolutely secure~\cite{Shannon}.
If legitimate users (Alice and Bob) have identical random private keys, which are used only once, with the size being equal or greater than the size of the message,
then according to the Shannon theorem messages encrypted using such key via the one-time pad scheme~\cite{Vernam} cannot be decrypted by an eavesdropper (Eve).
On the other hand, however, distribution of keys satisfying these demands is challenging.
\newpage
Towards the solution of this problem one can use the resource of quantum systems~\cite{BB84}.\,By transmitting information using individual quantum objects ({\it e.g.}, single photons)
security is guaranteed by the principles of quantum physics~\cite{Gisin,Scarani}.
First is that the arbitrary quantum state cannot be cloned according to the no-cloning theorem~\cite{Wootters}.
Second reason is that the pair of non-orthogonal states cannot be discriminated with unit probability.
Furthermore, any measurement leads to distortion or annihilation of a quantum state.
Thus, by distribution of a key with the use of single photons one can obtain a setup, which allows observing of inception in the key generation process.
Consequently, this opens a possibility to design a novel architecture for telecommunications systems, in which information security is guaranteed by the fundamentals laws of physics.
Nevertheless, imperfections of technical realizations of key distribution systems such as absorption of photons in an optical fiber,
efficiency of single-photon detectors, and possible actions of an eavesdropper exploiting these vulnerabilities lead to possible attacks.
In particular, if the length of a quantum channel exceeds a certain value then it is impossible to ensure confidentiality of key distribution~\cite{Scarani}.
It should be noted that attacks on quantum key distribution systems can be divided into two classes.
The first class is attacks on quantum key distribution protocols.
It is commonly understood that the quantum key distribution protocol is a general scheme of preparation and measurement of quantum states,
and procedure of obtaining of a key on Alice and Bob sides from results of measurements of quantum states.
The first quantum key distribution protocol is BB84.
When considering the attacks of this class, it is traditionally assumed~\cite{Gisin,Scarani} that an eavesdropper has all technological resources,
which do not contradict the physic laws,
such as quantum computer, quantum memory, and an ideal channel for transmitting of quantum states.
Of course, the practical realization of the considered attack essentially depends on the required technological resources.
The second class presumes attacks on hardware realization of quantum-cryptographic systems (so-called ``quantum hacking'')~\cite{Makarov,Makarov2,Makarov3}.
For example, this is attacks on particular type of single-photon detectors~\cite{Makarov3}.
Quantum key distribution systems are available on the market.
Nevertheless, the way to their implementation encounters with a number of technological challenges.
One of the most promising methods of operating with quantum systems used in commercial devices consists of using coherent states of light for quantum key distributions
(such as coherent one-way protocol, COW)~\cite{Stucki,Stucki2,Stucki3}.
These protocols are inspired by classical communications with optical fiber systems~\cite{Stucki3}.
The most significant advantage of the COW protocol is simplicity of its realization~\cite{Scarani,Stucki,Stucki2,Stucki3},
which is related to its simple optical scheme.
This protocol is easy to implement in experiment, and it has been used in the project on the design of European network for quantum key distribution SECOQC~\cite{SECOQC}.
However, despite prevalence of this method and its high practical importance,
an analysis of the possibility of attacks, which give to an eavesdropper an opportunity to obtain information about a quantum key,
is an important task~\cite{Stucki,Stucki2,Stucki3,SECOQC,Branciard,Branciard2,Curty,Kronberg,Molotkov}.
We note that lower bounds on the key generation rate of a variant of the coherent-one-way quantum key distribution protocol have been obtained in Ref.~\cite{Moroder}.
We note that quantum key distribution systems are not communication systems in the full sense.
The quantum resource in the form of single photons is used not for information transmitting, but for generation of a random identical for legitimate users sequence of bits (key).
The typical generation rate is such systems is about 10 kbit/s on the distances 50--80 km.
It is a limitation of such systems that for generation of keys for legitimate users the direct channel is needed ({\it i.e.}, ``point-to-point'' network topology).
After distribution of a keys, it can be used for encryption on the one-time-pad regime or as a source of entropy.
The final speed of information transmitting in such a case depends on the speed of the communication system transmitting encrypted information.
In this work, we consider a novel attack on the coherent quantum key distribution protocol.
The key idea of the proposed attack is using of individual measurements of intercepted states and transfer of the remaining part in an unmodified form.
The suggested attack belongs to the first class (attacks on the protocol).
One of its advantages is the fact that realization of this attack does not require using of the quantum memory or sophisticated elements,
but it is limited to a common assumption that an eavesdropper has a channel without losses.
In the rest, the suggested attack has rather simple optical scheme for its realization,
allowing one to gain an advantage in comparison with the known beam-splitting attack under certain restrictions on the parameters of a key distribution system.
\newpage
\section{COW protocol}
In the COW protocol for quantum key distribution Alice and Bob
use a coherent state $|{\alpha}\rangle$ in a one of two time slots, whereas the vacuum state is in another window, to encode two information states with bit values $0$ and $1$.
Therefore, the corresponding states can be presented in the following form:
\begin{equation}\label{eq:inf_states}
\begin{split}
|\psi_0\rangle=|\alpha\rangle\otimes|0\rangle, \quad
|\psi_1\rangle=|0\rangle\otimes|\alpha\rangle,
\end{split}
\end{equation}
where $\mu = |\alpha|^2$ stands for the intensity of the coherent state $|{\alpha}\rangle$.
A straightforward scenario of attack in quantum cryptography is the ``intercept and resend'' attack.
In this attack, Eve tries to measure the state in each time slot and resent of the resulting state (with amplification compensating losses).
In order to detect an eavesdropper, using this attack, in addition to the information states Alice and Bob employ the decoy states of the following form:
\begin{equation}
|\psi_c\rangle=|\alpha\rangle\otimes|\alpha\rangle.
\end{equation}
The decoy states are used to detect attempts to distinguish between information states.
During the attack an eavesdropper occasionally sends the information states instead the decoy states that would allow to detect it.
The fraction of the decoy states is denoted as $f$, it is typically about $10\%$.
\section{Beam-splitting attack \\ on the COW protocol}
In real optical fibers there is attenuation of signals leading to the fact that Bob obtains states of lower intensity.
The intensity of states obtained by Bob is as follows:
\begin{equation}\label{eq:Bob_intensity}
\mu_B=\mu 10^{-\frac{\delta l}{10}},
\end{equation}
where $l$ is the length (in km), and $\delta$ is the attenuation coefficient (typical value of the attenuation coefficient for optical fibers is about $0.2$ dB/km).
In an analysis of attacks on the quantum key distribution protocol one can assume that Eve has unlimited technological resources.
That is why one of the possible scenario is to use a beam splitter together with ideal identical channel (\emph{e.g.} by means of quantum teleportation).
Since states transform in a self-similar way both on a beamsplitter and in optical channel with attenuation,
Eve can take a part of a state using beam-splitting with sending of the remaining part to Bob via channel replaced on the ideal one.
Maximal value of the intensity that Eve can take has the following form:
\begin{equation}\label{eq:Max_Eve_intensity}
\mu_E^{\max}=\mu-\mu_B=\mu(1-10^{-\frac{\delta l}{10}}).
\end{equation}
From viewpoint of Bob, this modification of transferred states using a beamsplitter does not differs from losses, and it cannot be detected.
Further strategy of Eve is different for various attacks.
We consider the scenario of a beam-splitting attack, in which Eve stores the withdrawn states in order to then extract information~\cite{Kronberg}.
The set of states and measurements on the receiver side is such that if there is no Eve
(as well as in the case of its withdrawal of a fraction of state and sending using a lossless channel)
the Bob has zero quantum bit error rate (QBER), and the mutual information between Alice and Bob is equal to 1 after discarding inconclusive outcomes.
Since Eva has less information, then in this attacks she introduces additional errors in order to make her information equal to the Bob information.
Thus, information states of Eve in this attack are as follows:
\begin{equation}
|\psi_0^E\rangle = |\sqrt{\mu_E^{\max}}\rangle\otimes|0\rangle, \quad
|\psi_1^E\rangle = |0\rangle\otimes|\sqrt{\mu_E^{\max}}\rangle.
\end{equation}
The information, which can be extracted from such states by Eve at a collective measurement, is given by the Holevo quantity ($\chi$-value)~\cite{Holevo}:
\begin{equation}
I_{AE}=\chi(\{|\psi_0^E\rangle, |\psi_1^E\rangle\}),
\end{equation}
It is easy to find a critical value of QBER, which Eve should add in order to make her information equal to Bob's information.
The critical value of the QBER can be expressed via the binary entropy function as follows:
\begin{equation}
I_{AB}=1-h_2({\rm QBER}),
\end{equation}
where $h_2({\rm QBER})$ is the binary entropy function.
The strong side of the beam-splitting attack is using of the collective measurements over the entire transmitted sequence,
that allows one to achieve the $\chi$-value due the quantum superadditivity and obtain the theoretical maximum of information from these states.
However the disadvantage of such an attack is that the critical value of the QBER does not tend to zero at large lengths of the channel since the Eve information
is limited by the $\chi$-value of initial states.
At the same time, the longer is the channel, the higher are the losses,
and one can use this fact by blocking some of the states, whose information had not been extracted.
\section{Active beam-splitting attack on the COW protocol}
\begin{figure*}
\includegraphics[width=1\linewidth]{fig1.pdf}
\vskip -3mm
\caption
{
The critical value of the QBER for the standard beam-splitting attack (dashed line) and the considered active beam-splitting attack (solid line) for three different values of intensity
$\mu=0.1$ (a),
$\mu=0.2$ (b),
and $\mu=0.5$ (c) are shows as function of the channel length.
}
\label{fig:1}
\end{figure*}
In the present work, we suggest an alternative scenario to attack the COW protocol.
The difference from the beam-splitting attack is as follows.
First, in the considered attack Eve withdraws smaller part of states rather than in the beam-splitting attack, but still needs channel with lower losses.
Second, the eavesdropper performs individual measurements.
In the case of the inconclusive results of measurements ({\it i.e.}, at the detection of vacuum states in both positions) Eve is able to block state since the information is inaccessible.
At the same time, in a general case Eve is unable to block all such states due to the fact that in this case the intensity on the receiver side is lower than it is expected,
which discovers Eve.
Therefore, in order to estimate the efficiency of this attack we consider the idea similar to that in the beam-splitting attack, {\it i.e.},
in the cases where the length of the channel makes it impossible for Eve to block all states, of which information is not extracted,
she introduces errors to make the Bob information equal to the Eve information.
We refer this attack as the active beam-splitting attack since its using depends on ability of extracting information from withdrawn states by Eve.
We describe details of actions of Eve with taking into account all the parameters.
Eve withdraws a fraction of states with intensity $\mu_E$ and implements measurements of each time slot of the state.
The measurement in each time slot is described by the following observable:
\begin{equation}
M_0=|0\rangle\langle 0|,
\quad
M_1=\sum_{i=1}^\infty |i\rangle\langle i|.
\end{equation}
The probability to obtain the result 1 at the measurement of the state with intensity $\mu_E$ reads
\begin{equation}\label{eq:Eve_conclusive_probability_inf}
p_{\mathrm{conc}}^{\mathrm{inf}} = 1 - e^{-\mu_E}.
\end{equation}
This expression gives the probability to obtain the conclusive result at the measurement of the information state (\ref{eq:inf_states}) but with lower intensity.
At the transmitting of the decoy state, the probability of the conclusive result is as follows:
\begin{equation}\label{eq:pcont}
p_{\mathrm{conc}}^{\mathrm{cont}}=2e^{-\mu_E}(1 - e^{-\mu_E}) + (1 - e^{-\mu_E})^2.
\end{equation}
Thus, the total probability of the conclusive result by Eve taking into account the fraction of the decoy states is the following:
\begin{equation}
\label{Eve_conclusive_probability}
p_{\mathrm{conc}}^E=(1-f)\times{p_{\mathrm{conc}}^{\mathrm{inf}}}+f\times{p^{\mathrm{cont}}_{\mathrm{conc}}}.
\end{equation}
It should be noted that the use of the decoy states does not allow to detect Eve since she still sends a fraction of states without measurements.
Possible errors due to wrong distinctions of the information and the decoy states does not lead do decrease of the Eve information,
because the recordings about decoy states do not participate in the generation of the sifted key.
In the case of inconclusive result of Eve measurement, the probability of which is equal to $1-p_{\mathrm{conc}}^E$, Eve seeks to block states transmitting to Bob.
In a general case, Eve is able to implement this not universally but for a small fraction of states to make this undetectable on the receiver side due to the large attenuation.
We calculate this fraction of states, which can be blocked.
Bob expects states with intensity (\ref{eq:Bob_intensity}).
For these states the probability to obtain the conclusive result for information states is equal to $1-e^{-\mu_B}$.
In the reality, by using a beamsplitter Eve can keep for Bob states of higher intensity in order to block useless states, then the intensity of Bob is $\mu'_B=\mu-\mu_E$.
Consequently, permissible fraction of states $b$ blocked by Eve can be obtained from the relation
\begin{equation}\label{eq:block_probability}
(1-b)(1-e^{-\mu_B'}) = 1 - e^{-\mu_B}.
\end{equation}
The fraction of blocked states can be determined for a given channel length and parameters of the Eve attack, in particular, the intensity of withdraw states.
This intensity varies from zero to the maximal intensity $\mu_E^{\max}$ given by expression (\ref{eq:Max_Eve_intensity}).
For each channel length Eve is able to chose the part of withdraw states maximizing her information.
The Eve's information can be calculated as follows.
The information is equal to unity for the cases, where Eve has conclusive results.
At the same time, for an arbitrary channel length Eve cannot block all states, where she extracted no information, then she still should send them to Bob.
For the probability of the conclusive result and the probability of state block $b$, determined be Eq.~(\ref{eq:block_probability}), we have
\begin{equation}\label{eq:Eve_information}
I_{AE}=\frac{p_{\mathrm{conc}}^{\mathrm{inf}}}{1 - b}.
\end{equation}
One can see that if Eve is able to block all states with the inconclusive result of the measurement ({\it i.e.}, if $b=1-p_{\mathrm{conc}}^{\mathrm{inf}}$), then her information is equal to unity.
It is useless to consider the case, where the value $b$ is greater that the probability of the inconclusive result of the measurement,
whereas at lower values Eve information decreases with the minimum at the case of absence of states block, which is equal to the probability of the conclusive measurement result,
and it is the capacity of the channel with the attenuation.
We consider a question about choice of the optimal value of the intensity of withdraw signal $\mu_E$ maximizing the withdraw information $I_{AE}$.
By using expression (\ref{eq:Eve_conclusive_probability_inf}) and expression (\ref{eq:block_probability}) defining the value of the fraction of blocked by Eve states $b$,
we then rewrite (\ref{eq:Eve_information}) as follows:
\begin{equation}\label{eq:Eve_information2}
I_{AE}=\frac{\left(1-e^{-\mu+\mu_E}\right)\left(1-e^{-\mu_E}\right)}{1-e^{\mu_B}}.
\end{equation}
It is clear that $I_{AE}$ in a convex function of the argument $\mu_E$, which has a maximum at $\mu_E=\mu/2$.
However, since the intensity $\mu_E$ is bounded by the value $\mu_E^{\max}$ the maximum value of the withdraw information is achieved at
\begin{equation}
\mu_E=\min\left(\mu_E^{\max},\mu/2\right)
\end{equation}
We note that in the case $\mu_E=\mu_E^{\max}$ we have $\mu_E^{\max}=\mu_E'$ and $b=1$.
Thus, at $\mu_E<\mu/2$ the optimal strategy for Eve is to do not block states at all and send all messages to Bob.
The critical length $l_\mathrm{crit}$at which it is reasonable for Eve to start block states is defined by the expression,
\begin{equation}
1-10^{-\frac{\delta{l_\mathrm{crit}}}{10}}=1/2,
\end{equation}
and it can be calculated as follows:
\begin{equation}
l_\mathrm{crit}=10\log_{10}2/\delta=3/\delta.
\end{equation}
For a typical value $\delta=0.2$ dB/km, we then have $l_\mathrm{crit}=15$ km.
\section{Discussions of results}
In conclusion, we one more time briefly describe a scenario of the considered attack and the method of finding of the critical QBER in the protocol.
For a given value of the channel length, Eve calculates the maximal intensity of states, which can be withdrawn.
Then, she considers possibility of withdraw of states with different intensity from zero to the maximal value.
For each state one can calculate the fraction of states, which Eve can block, and the Eve information.
The greater is the intensity of the state received by Bob, the larger fraction can be blocked.
Eve chooses the intensity maximizing her information.
If she can block all states with inconclusive results of measurements, Eve is able to attack without introducing of additional error, and the protocol is totally insecure.
However, if the fraction of states, which can be blocked, is lower,
then Eve incorporates errors in the channel between Alice and Bob in order to offset the difference between Bob's and her information.
The minimum value of errors, whereby their information are equal to each other, is a critical value of the QBER of the protocol against this attack.
The critical value of QBER for the standard beam-splitting attack and the considered active beam-splitting attack for three different values of intensity ($\mu=$ 0.1, 0.2, and 0.5)
are shows as function of the channel length in Fig.~\ref{fig:1}.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{fig2.pdf}
\end{center}
\vskip -7mm
\caption{The optimal intensity in a presence of active beam-splitting attack for different channel lengths (dotted line),
the critical QBER at the optimal choice of the intensity by Eve (solid line), and the critical QBER of the corresponding beam-splitting attack at the same intensity (dashed line).}
\label{fig:2}
\end{figure}
The question may arise as to which the intensity of the initial state is optimal for legitimate users under the assumption that the eavesdropper uses this particular attack.
On the one hand, it is clear that in order to increase the critical QBER they have to take the lowest intensity of the initial states.
On the other hand, low intensity of the initial states leads to low key generation rate because of too large proportion of too many states are lost.
We can find the optimal intensity as follows.
Assume that Eve uses withdraw of the maximum fraction of states, however without incorporation of errors in the communication channel between Alice and Bob
(since the value of errors is not the mark).
The length of secret key of Alice and Bob recalculated on a single state
is given by difference between the information between Alice and Bob $I_{AB}$ and the information between Alice and Eve $I_{AE}$.
The first is equal to the capacity of the erasure channel with erasure probability equal to $e^{-\mu_B}$, the second can be calculated by using Eq.~(\ref{eq:Eve_information})
and then multiplying on the probability of the conclusive result of Bob.
Therefore, the optimal value of the intensity for the given value of the channel is those, which maximizes the difference
\begin{equation}\label{optimal_intencity}
I_{AB}-I_{AE}=1-e^{-\mu_B}-(1-e^{-\mu_B})\frac{p_{\mathrm{conc}}^{\mathrm{inf}}}{1-b}.
\end{equation}
Fig.~\ref{fig:2} shows the optimal intensity for different channel lengths and the critical value at the optimal choice of the intensity if Eve uses this attack.
For a comparison, we add the critical value of QBER for the beam-splitting attack at the fixed length and given choice of intensity,
however, we note that the initial intensity is optimized on the basis of the assumption that the eavesdropper uses the active beam-splitting attack.
The disadvantage of this attack is that Eve is unable to amplify the signal transmitting to Bob for the states, where she has the conclusive results.
Another weak side of the attack is that Eve has to measure at once, which it eliminates the possibility of achieving of the superadditive information.
On the other hand, in contrast to the usual attacks with a beam splitter, such the attack, starting with the critical channel length, is possible without introducing of additional errors.
However, the attack with unambiguous measurements~\cite{Molotkov} is more effective since it is possible to enhance the intensity of the premises from which all the information can be extracted.
The advantages of the considered attack can also be attributed relatively simple technical implementation.
Thus, the discussed active beam-splitting attack is interesting primarily as implemented in the current technological level than the optimum in terms of the eavesdropper,
limited only by the laws of physics.
The development of the idea of unchanged forwarding states seems to be topical in the context of other quantum-cryptography protocols that are based on the use of coherent states.
We note also that despite the fact that the considered attack does not lead to a change of the status type (decoy state can not be transformed into a signal, and vice versa),
it leads to a distortion of the statistics getting decoy or information states.
This phenomenon is due to both the change in the intensity of the pulses sent to Bob (by $\mu_B$ or $\mu_B'$),
and that the blocking probability able to determine the probability of getting conclusive result by Eve.
In turn, in the case of this decoy state likelihood is higher than in the case of the signal, and the total probability of obtaining the decoy state increases.
As a result, the attack can be considered potentially registered by taking into account statistical registration of decoy states in the classical post-processing key.
Usually, however, the protocol is considered a requirement of the absence of conditions such as change,
and such registration statistics require a significant change in the protocol in terms of evaluation of the interceptor information.
An interesting and relevant question is about required modification of the COW protocol for taking into account the proposed attack.
However, this question further analysis, and it is beyond the scope of this work.
\section{Conclusion}
In this paper, we considered a new type of attack on a coherent quantum key distribution protocol COW with the use of an active beam splitter.
The optimum values of the parameters of attack for an arbitrary length of the channel were calculated, and the comparison with a standard beam-splitting attack was performed .
The advantage of the considered attack is rather simple technical implementation.
It should be noted that the suggested attack is actual, in fact, for channels of arbitrary length.
However, it is of special interest for quantum-cryptographic systems operating in an urban environment and using short (30--50 km) urban fiber-optic communication lines with fairly high losses.
In recent experiments on quantum key distribution in urban conditions,
losses in the channel of 30 km length have been on the level of 11 dB at the key generation rate on the level of 0.5 kbit/s~\cite{Miller} after post processing~\cite{Kiktenko}.
\newpage
We are grateful to A.S. Trushechkin and O.V. Lychkovskiy for helpful discussions, and to the reviewers for valuable comments.
The financial support from Ministry of Education and Science of the Russian Federation in the framework of the Federal Program
(Agreement 14.579.21.0105, ID RFMEFI57915X0105) is acknowledged. | 99,562 |
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"I."
What the heck goes on in our country's middle and high schools that can make a child who formerly loved learning such a wreck? I have a guess or two...
The above quote is from this interview in a Delaware education blog called,
Exceptional Deleware: Helping Parents of Special Needs Children, Eliminating Disability Discrimination, & Understanding Special Education Rights. The post reports, "College student, Melissa Katz, currently studying to be a teacher, is a public education activist, and writes a blog of her own. It is brave to be planning a career in education at this time. I am not sure I would allow my kids to do so, which is quite sad to me. I have already told my youngest daughter that if she wants to be a teacher, she needs a second degree as a fall back. That said, if the next generation of young adults is like Melissa Katz, perhaps we have a chance at saving this important American institution.
As a teacher in a wonderful school district (but one that is still impacted by these politically charged, anti-public education times), the following comment by Ms. Katz is what helps motivate me to "teach right":
"...."
This speaks to me: " ... I had one of the most unconventional yet enlightening teachers I’ve ever had, who has acted as a mentor to me throughout this entire journey." I want my own children to have teachers like this. I want to be a teacher like this. Why else would anyone go into the education field? I don't want "common" anything for my kids, and I don't want to be common myself.
It is the individual gifts that each teacher, instructional assistant, principal, and assistant principal brings to the school that are true goldmines of any school. Standardization slowly and painfully sucks the life out of the living, breathing humans, the very ones who bring passion into our schools and our world: the students and the teachers.
I received an email over the weekend from a former student. I get these every so often, and they are my kind of "merit pay." They make me know that what I am doing makes a difference, and remind me why I love being a teacher. This student remembered a small token I gave him when his grandfather died years ago. He shared how much it meant to him and how it helped him through that time. He shared that he has a friend going through a difficult time, and wondered if I could tell him where I bought the token. He wrote me at 3 a.m. I was so moved. On top of the great literature, current event discussions, and writing assignments, this student remembered (years later) that his 8th grade English teacher cared.
Ms. Katz wants more citizens to understand what is being done to teachers to purposely burn them out, distract them from focusing on our kids with meaningless paperwork and meetings, evaluate them by the junk science of using student test scores as a percentage of their effectiveness (which has been debunked by the American Statistical Association). She wants people to care enough to speak out - even if this doesn't impact one's own kids. She states, ."
If you or someone you know has grown kids or kids in private schools, you & they can be some of the strongest allies we have. Please speak up and urge others to do the same. All kids deserve teachers who are free to bring their strengths to the table, like Katz's unconventional and enlightening high school teacher.
No students or teachers should be having breakdowns from the way our schools are run. Our dedicated school board members need to focus on the very real issues that are impacting the children in our schools, and the teachers one the front lines with them. School leaders, from the superintendent to building principals have the power to set the tone and encourage real creativity and the unconventional. It is time for every citizen to make up his or her own mind about the future schools of our grandchildren and great grandchildren. Like our fingerprints, we are all unique, and no matter how much our politicians and the business people who buy their loyalty want us to believe otherwise, human beings cannot and should not be standardized. We are unique and our schools need an uncommon core that is locally developed and approved. No one ever aspired to be common in our great country... indeed being common is not at all what makes us American, now is it?
Dawn Sweeney has left a new comment on your post ""I just can't handle the pressure..."":
Your blog is excellent. I am a parent from Owen J. Roberts SD down the road from LM. It's disheartening to see so much lost instructional time to test prep and testing schedule. I opt my kids out. I hope this testing madness ends soon! My blog is [email protected] | 161,855 |
\begin{document}
\maketitle \renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnotetext[1]{A preliminary version of this manuscript was
presented as~\cite{DMN-JC:13-acc} at the 2013 American Control
Conference, Washington, D.C. This manuscript is a revision of the version submitted to
SIAM Journal on Control and Optimization 52 (4) (2014), 2399-2421.}
\footnotetext[2]{The authors are with the Department of Mechanical and
Aerospace Engineering, University of California, San Diego, 9500
Gilman Dr, La Jolla, CA 92093, USA, {\tt \small
\{dmateosn,cortes\}@ucsd.edu}.}
\renewcommand{\thefootnote}{\arabic{footnote}}
\begin{abstract}
This paper studies the stability properties of stochastic
differential equations subject to persistent noise (including the
case of additive noise), which is noise that is present even at the
equilibria of the underlying differential equation and does not
decay with time. The class of systems we consider exhibit
disturbance attenuation outside a closed, not necessarily bounded,
set. We identify conditions, based on the existence of Lyapunov
functions, to establish the noise-to-state stability in probability
and in \textit{p}th~moment of the system with respect to a closed
set. As part of our analysis, we study the concept of two functions
being proper with respect to each other formalized via pair of
inequalities with comparison functions. We show that such
inequalities define several equivalence relations for increasingly
strong refinements on the comparison functions. We also provide a
complete characterization of the properties that a pair of functions
must satisfy to belong to the same equivalence class. This
characterization allows us to provide checkable conditions to
determine whether a function satisfies the requirements to be a
strong NSS-Lyapunov function in probability or a $p$th~moment
NSS-Lyapunov function. Several examples illustrate our results.
\end{abstract}
\section{Introduction}\label{sec:intro}
Stochastic differential equations (SDEs) go beyond ordinary
differential equations (ODEs) to deal with systems subject to
stochastic perturbations of a particular type, known as white noise.
Applications are numerous and include option pricing in the stock
market, networked systems with noisy communication channels, and, in
general, scenarios whose complexity cannot be captured by
deterministic models. In this paper, we study SDEs subject to
\emph{persistent noise} (including the case of additive noise), i.e.,
systems for which the noise is present even at the equilibria of the
underlying ODE and does not decay with time. Such scenarios arise, for
instance, in control-affine systems when the input is corrupted by
persistent noise. For these systems, the presence of persistent noise
makes it impossible to establish in general a stochastic notion of
asymptotic stability for the (possibly unbounded) set of equilibria of
the underlying ODE. Our aim here is to develop notions and tools to
study the stability properties of these systems and provide
probabilistic guarantees on the size of the state of the system.
\emph{Literature review:} In general, it is not possible to obtain
explicit descriptions of the solutions of SDEs. Fortunately, the
Lyapunov techniques used to study the qualitative behavior of
ODEs~\cite{HKK:02,RFB:91} can be adapted to study the stability
properties of SDEs as well~\cite{RK:12,UHT:97,XM:99}. Depending on
the notion of stochastic convergence used, there are several types of
stability results in SDEs. These include stochastic stability (or
stability in probability), stochastic asymptotic stability, almost
sure exponential stability, and \textit{p}th moment asymptotic
stability, see e.g.,~\cite{UHT:97,XM:99,XM:11,TT:03}.
However, these notions are not appropriate in the presence of
persistent noise because they require the effect of the noise on the
set of equilibria to either vanish or decay with time.
To deal with persistent noise, as well as other system properties like
delays, a concept of ultimate boundedness is required that generalizes
the notion of convergence. As an example, for stochastic delay
differential equations,~\cite{FW-PEK:13} considers a notion of
ultimate bound in $p$th moment~\cite{HS:01} and employs Lyapunov techniques to
establish it.
More generally, for mean-square random dynamical systems, the concept
of forward attractor~\cite{PEK-TL:12} describes a notion of
convergence to a dynamic neighborhood and employs contraction analysis to establish it.
Similar notions of ultimate boundedness for the state of a system, now
in terms of the size of the disturbance, are also used for
differential equations, and
many of these notions are inspired by dissipativity properties of the
system that are captured via \emph{dissipation inequalities} of a
suitable Lyapunov function:
such inequalities encode the fact that the Lyapunov function decreases
along the trajectories of the system as long as the state is ``big
enough'' with regards to the disturbance. As an example, the concept
of input-to-state stability (ISS) goes hand in hand with the concept
of ISS-Lyapunov function, since the existence of the second implies
the former (and, in many cases, a converse result is also
true~\cite{EDS-YW:95}). Along these lines, the notion of practical
stochastic input-to-state stability (SISS)
in~\cite{SL-JZ-ZJ:08,ZJW-XJX-SYZ:07} generalizes the concept of ISS to
SDEs where the disturbance or input affects both the deterministic
term of the dynamics and the diffusion term modeling the role of the
noise. Under this notion, the state bound is guaranteed in
probability, and also depends, as in the case of ISS, on a decaying
effect of the initial condition plus an increasing function of the sum
of the size of the input and a positive constant related to the
persistent noise.
For systems where the input modulates the covariance of the noise,
SISS corresponds to noise-to-state-stability (NSS)~\cite{HD-MK:00},
which guarantees, in probability, an ultimate bound for the state that
depends on the magnitude of the noise covariance. That is, the noise
in this case plays the main role, since the covariance can be
modulated arbitrarily and can be unknown. This is the appropriate
notion of stability for the class of SDEs with persistent
noise considered in this paper, which are nonlinear systems affine in
the input, where the input corresponds to white noise with locally
bounded covariance. Such systems cannot be studied under the ISS
umbrella, because the stochastic integral against Brownian motion has
infinite variation, whereas the integral of a legitimate input for~ISS
must have finite variation.
\emph{Statement of contributions:} The contributions of this paper are
twofold. Our first contribution concerns the noise-to-state stability
of systems described by SDEs with persistent noise. We
generalize the notion of noise-dissipative Lyapunov function, which is
a positive semidefinite function that satisfies a dissipation
inequality that can be nonexponential (by this we mean that the
inequality admits a convex~$\classkinfty$ gain instead of the linear
gain characteristic of exponential dissipativity). We also introduce
the concept of $p$thNSS-Lyapunov function with respect to a closed
set, which is a noise-dissipative Lyapunov function that in addition
is proper with respect to the set with a convex lower-bound gain
function. Using this framework, we show that noise-dissipative
Lyapunov functions have NSS~dynamics and we characterize the overshoot
gain. More importantly, we show that the existence of a
\textit{p}thNSS-Lyapunov function with respect to a closed set implies
that the system is NSS in \textit{p}th moment with respect to the set.
Our second contribution is driven by the aim of providing alternative,
structured ways to check the hypotheses of the above results. We
introduce the notion of two functions being proper with respect to
each other as a generalization of the notion of properness with
respect to a set. We then develop a methodology to verify whether two
functions are proper with respect to each other by analyzing the
associated pair of inequalities with increasingly strong refinements
that involve the classes~$\classk$, $\classkinfty$, and $\classkinfty$
plus a convexity property. We show that these refinements define
equivalence relations between pairs of functions, thereby producing
nested partitions on the space of functions. This provides a useful
way to deal with these inequalities because the construction of the
gains is explicit when the transitivity property is exploited. This
formalism motivates our characterization of positive semidefinite
functions that are proper, in the various refinements, with respect to
the Euclidean distance to their nullset. This characterization is
technically challenging because we allow the set to be noncompact, and
thus the pre-comparison functions can be discontinuous. We devote
special attention to the case when the set is a subspace and examine
the connection with seminorms. Finally, we show how this framework
allows us to develop an alternative formulation of our stability
results.
\emph{Organization:} The paper is organized as
follows. Section~\ref{sec:preliminaries} introduces preliminaries on
seminorms, comparison functions, and SDEs.
Section~\ref{sec:problem-statement} presents the NSS stability results
and Section~\ref{sec:positive semidefinite-functions} develops the
methodology to help verify their hypotheses. Finally,
Section~\ref{sec:conclusions-future} discusses our conclusions and
ideas for future work.
\section{Preliminary notions}\label{sec:preliminaries}
This section reviews some notions on comparison functions and
stochastic differential equations that are used throughout the paper.
\subsection{Notational conventions}\label{sec:notation}
Let $\real$ and $\realnonnegative$ be the sets of real and nonnegative
real numbers, respectively. We denote by~$\real^n$ the
$n$-dimensional Euclidean space. A subspace $\Uset\subseteq\real^n$
is a subset which is also a vector space. Given a matrix
$A\in\realmatricesrectangulararg{m}{n}$, its nullspace
$\kernel(A)\triangleq\{x\in\real^n:Ax=0\}$ is a subspace. Given
$\dom\subseteq\real^n$, we denote by~$\psddomain$
and~$\psddomaintwice$ the set of positive semidefinite functions
defined on $\dom$ that are continuous and continuously twice
differentiable (if $\dom$ is open), respectively. Given
$\lyap\in\psdtwice$, we denote its gradient by~$\grad\lyap$ and its
Hessian by~$\hessian\lyap$. A seminorm is a function
$S:\real^n\to\real$ that is positively homogeneous, i.e., $S(\lambda
x) = \absolute{\lambda} S(x)$ for any $\lambda\in \real$, and
satisfies the triangular inequality, i.e., $S(x+y)\le S(x)+S(y)$ for
any $x,y\in \real^n$. From these properties it can be deduced that
$S\in\psd$ and its nullset is always a subspace. If, moreover, the
function $S$ is positive definite, i.e., $S(x)=0$ implies $x=0$, then
$S$ is a norm. The Euclidean norm of $x\in\real^n$ is denoted
by~$\norm{x}$, and the Frobenius norm of the matrix
$A\in\realmatricesrectangulararg{m}{n}$ is
$\normfrob{A}\triangleq\sqrt{\trace{(A\tp A)}} = \sqrt{\trace{(A
A\tp)}}$. For any matrix $A\in\realmatricesrectangulararg{m}{n}$,
the function $\seminorm{x}{A}\triangleq\norm{Ax}$ is a seminorm and
can be viewed as a distance to $\kernel(A)$. For a symmetric positive
semidefinite real matrix $A\in\realmatrices$, we order its eigenvalues
as $\lambdamax(A)\triangleq\lambda_1(A)\ge
\dots\ge\lambda_n(A)\triangleq\lambdamin(A)$, so if the dimension of
$\kernel(A)$ verifies $\dim(\kernel(A))=k\le n$, then
$\lambda_{n-k}(A)$ is the minimum nonzero eigenvalue of~$A$.
The Euclidean distance from $x$ to a set $\Uset \subseteq\real^n $ is
defined by
$\distU{x}\triangleq\inf\setdef{\norm{x-u}}{u\in\Uset}$. The function
$\distU{.}$ is continuous when~$\Uset$ is closed. Given
$\map{f,g}{\realnonnegative}{\realnonnegative}$, we say that $f(s)$ is
in $\Obig(g(s))$ as $s\to\infty$ if there exist constants $\kappa,
s_0>0$ such that $f(s)<\kappa g(s)$ for all $s>s_0$.
\subsection{Comparison, convex, and concave
functions}\label{sec:pre-classks-convexity}
Here we introduce some classes of comparison functions
following~\cite{HKK:02} that are useful in our technical treatment. A
continuous function $\alpha:[0,b)\to\realnonnegative$, for $b>0$ or
$b=\infty$, is class $\classk$ if $\alpha(0)=0$ and is strictly
increasing. A function
$\map{\alpha}{\realnonnegative}{\realnonnegative}$ is
class~$\classkinfty$ if $\alpha\in\classk$ and is unbounded. A
continuous function
$\map{\mu}{\realnonnegative\times\realnonnegative}{\realnonnegative}$
is class~$\classkl$ if, for each fixed $s\ge 0$, the function $r
\mapsto \mu(r,s)$ is class $\classk$, and, for each fixed $r\ge 0$,
the function $s\mapsto \mu(r,s)$ is decreasing and
$\lim_{s\to\infty}\mu(r,s)=0$. If $\alpha_1$, $\alpha_2$ are
class~$\classk$ and the domain of $\alpha_1$ contains the range of
$\alpha_2$, then their composition $\alpha_1\circ\alpha_2$ is
class~$\classk$ too. If $\alpha_3$, $\alpha_4$ are
class~$\classkinfty$, then both the inverse function $\alpha_3^{-1}$
and their composition $\alpha_3\circ\alpha_4$ are
class~$\classkinfty$. In our technical treatment, it is sometimes
convenient to require comparison functions to satisfy additional
convexity properties. A real-valued function $f$ defined in a convex
set $X$ in a vector space is convex if $f(\lambda x+(1-\lambda)y)\le
\lambda f(x) + (1-\lambda)f(y)$ for each $x,y\in X$ and any
$\lambda\in[0,1]$, and is concave if $-f$ is
convex. By~\cite[Ex.~3.3]{SB-LV:09}, if $f:[a,b]\to[f(a),f(b)]$ is a
strictly increasing convex (respectively, concave) function, then the
inverse function $f^{-1}:[f(a),f(b)]\to [a,b]$ is strictly increasing
and concave (respectively, convex). Also, following~\cite[Section
3]{SB-LV:09}, if $f,g:\real\to\real$ are convex (respectively,
concave) and $f$ is nondecreasing, then the composition $f\circ g$ is
also convex (respectively, concave).
\subsection{Brownian motion}\label{sec:brownian}
We review some basic facts on probability and introduce the notion of
Brownian motion following~\cite{XM:11}. Throughout the paper, we
assume that $(\Omega, \sigmaalgebra, \filtration, \probability)$ is a
complete probability space, where $\probability$ is a probability
measure defined on
the $\sigma$-algebra $\sigmaalgebra$, which contains all the subsets
of $\Omega$ of probability $0$. The filtration $\filtration$ is a
family of sub-$\sigma$-algebras of $\sigmaalgebra$ satisfying
$\filtrationt\subseteq\filtrations\subseteq\sigmaalgebra$ for any
$0\le t < s < \infty$; we assume it is right continuous, i.e.,
$\filtrationt=\cap_{s>t}\filtrations$ for any $t\ge0$, and
$\mathcal{F}_0$ contains all the subsets of $\Omega$ of
probability~$0$. The Borel $\sigma$-algebra in $\real^n$, denoted by
$\borel^n$, or in $[t_0,\infty)$, denoted by $\borel([t_0,\infty))$,
are the smallest $\sigma$-algebras that contain all the open sets in
$\real^n$ or $[t_0,\infty)$, respectively. A function
$X:\Omega\to\real^n$ is $\sigmaalgebra$-measurable if the set
$\{\omega\in\Omega: X(\omega)\in A\}$ belongs to $\sigmaalgebra$ for
any $A\in\borel^n$. We call such function a
($\sigmaalgebra$-measurable) $\real^n$-valued random variable. If $X$
is a real-valued random variable that is integrable with respect to
$\probability$, its expectation is $\expectation [X]=\int_{\Omega}
X(\omega) \differential\probability(\omega)$. A function
$f:\Omega\times[t_0,\infty)\to\real^n$ is
$\sigmaalgebra\times\borel$-measurable (or just measurable) if the set
$\{(\omega, t)\in\Omega\times[t_0,\infty): f(\omega, t)\in A\}$
belongs to $\sigmaalgebra\times\borel([t_0,\infty))$ for any
$A\in\borel^n$. We call such function an $\filtrationadapted$-adapted
process if $\map{f(.,t)}{\Omega}{\real^n}$ is
$\filtrationt$-measurable for every $t\ge t_0$. At times, we omit the
dependence on ``$\omega$'', in the sense that we refer to the indexed
family of random variables, and refer to the random process
$f=\fprocess$. We define $\lone$ as the set of all $\real^n$-valued
measurable $\filtrationadapted$-adapted processes $f$ such that
$\probability (\setdef{\omega\in\Omega}{\int_{t_0}^T \norm{f(\omega,
s)}\, \ds<\infty})=1$ for every~$T>t_0$. Similarly,
$\ltwomatrices$ denotes the set of all $\realnm$-matrix-valued
measurable $\filtrationadapted$-adapted processes $\galone$ such that
$\probability (\setdef{\omega\in\Omega} {\int_{t_0}^T
\normfrob{\galone(\omega, s)}^2\, \ds<\infty})=1$ for every~$T>t_0$.
A one-dimensional Brownian motion
$\map{\brownian}{\Omega\times[t_0,\infty)}{\real}$ defined in the
probability space $(\Omega, \sigmaalgebra, \filtration, \probability)$
is an $\filtrationadapted$-adapted process such that
\begin{itemize}
\item
$\probability (\setdef{\omega\in\Omega}{\brownian(\omega,
t_0)=0})=1$;
\item the mapping $\map{\brownian(\omega, .)}{[t_0,\infty)}{\real}$,
called sample path, is continuous also with probability~$1$;
\item the increment $\brownian(.,t)-\brownian(.,s):\Omega\to\real$ is
independent of $\filtrations$ for $t_0 \le s < t < \infty$ (i.e., if
$S_b\triangleq\{\omega\in\Omega: \brownian(\omega,
t)-\brownian(\omega, s)\in(-\infty, b)\}$, for $b\in\real$, then
$\probability(A\cap S_b)= \probability(A)\probability(S_b)$ for all
$A\in\filtrations$ and all $b \in \real$). In addition, this
increment is normally distributed with zero mean and variance $t-s$.
\end{itemize}
An $m$-dimensional Brownian motion
$\map{\brownian}{\Omega\times[t_0,\infty)}{\real^m}$ is given by
$\brownian(\omega, t)= [\brownian_1(\omega,
t),\dots,\brownian_m(\omega, t)]\tp$, where each $\brownian_i$ is a
one-dimensional Brownian motion and, for each $t\ge t_0$, the random variables
$\brownian_1(t),...,\brownian_m(t)$ are independent.
\subsection{Stochastic differential equations}\label{sec:prel-sdes}
Here we review some basic notions on stochastic differential equations
(SDEs) following~\cite{XM:11}; other useful references
are~\cite{RK:12,BO:10,JRM:11}. Consider the $n$-dimensional SDE
\begin{align}\label{eq:nonlinear-SDE-preliminaries}
\differential x(\omega, t)= f\big(x(\omega, t),t\big)\dt
+G\big(x(\omega, t),t\big)\Sigma(t)\,\db(\omega, t),
\end{align}
where $x(\omega, t)\in\real^n$ is a realization at time $t$ of the
random variable $\map{x(.,t)}{\Omega}{\real^n}$, for
$t\in[t_0,\infty)$.
The initial condition is given by $x(\omega, t_0)=x_0$ with
probability~$1$ for some $x_0\in\real^n$. The functions
$\map{f}{\real^n\times[t_0,\infty)}{\real^n}$,
$\galone:\real^n\times[t_0,\infty)\to\realmatricesrectangulararg{n}{q}$,
and $\map{\Sigma}{[t_0,\infty)}{\realmatricesrectangulararg{q}{m}}$
are measurable. The functions $f$ and $\galone$ are regarded as a
model for the architecture of the system and, instead, $\Sigma$ is
part of the model for the stochastic disturbance; at any given time
$\Sigma$ determines a linear transformation of the $m$-dimensional
Brownian motion $\{\brownian(t)\}_{t\ge t_0}$, so that at time $t\ge
t_0$ the input to the system is
the process $\{\Sigma(t)\brownian(t)\}_{t\ge t_0}$, with covariance
$\int_{t_0}^t \Sigma(t)\Sigma(t)\tp \ds$. The distinction between the
roles of~$\galone$ and~$\Sigma$ is irrelevant for the SDE; both
together determine the effect of the Brownian motion.
The integral form of~\eqref{eq:nonlinear-SDE-preliminaries} is
given by
\begin{align*}
x(\omega, t)=x_0+\int_{t_0}^t f\big(x(\omega, s),s\big) \ds +
\int_{t_0}^t G\big(x(\omega, s),s\big)\Sigma(s)\,\db(\omega, s),
\end{align*}
where the second integral is an stochastic
integral~\cite[p. 18]{XM:11}.
A $\real^n$-valued random process $\xprocess$ is a solution
of~\eqref{eq:nonlinear-SDE-preliminaries} with initial value $x_0$ if
\begin{enumerate}
\item is continuous with probability~$1$,
$\filtrationadapted$-adapted, and satisfies $x(\omega, t_0)=x_0$
with probability~$1$,
\item the processes $\fprocesssolution$ and $\gprocesssolution$ belong
to $\lone$ and $\ltwomatrices$ respectively, and
\item equation~\eqref{eq:nonlinear-SDE-preliminaries} holds for every
$t\ge t_0$ with probability~$1$.
\end{enumerate}
A solution $\xprocess$ of~\eqref{eq:nonlinear-SDE-preliminaries} is
unique if any other solution $\xbarprocess$ with $\bar{x}(t_0)=x_0$
differs from it only in a set of probability~$0$, that is,
$\probability (\big\{x(t)=\bar{x}(t)$\; $\forall\, t\ge t_0\big\})=1$.
We make the following assumptions on the objects
defining~\eqref{eq:nonlinear-SDE-preliminaries} to guarantee existence
and uniqueness of solutions.
\begin{assumption}\label{ass:assumptions-SDE}
We assume $\Sigma$ is essentially locally bounded. Furthermore, for
any $T>t_0$ and $n\ge 1$, we assume there exists $K_{T,n}>0$ such
that, for almost every $t\in[t_0,T]$ and all $x,y\in\real^n$ with
$\max\big\{\norm{x} ,\norm{y}\big\}\le n$,
\begin{align*}
\max&\big\{\,\norm{f(x,t)-f(y,t)}^2\,,\,\normfrob{\gtx-\gty}^2\,\big\}
\le K_{T,n}\norm{x-y}^2.
\end{align*}
Finally, we assume that for any $T>t_0$, there exists $K_T>0$ such
that, for almost every $t\in[t_0,T]$ and all $x\in\real^n$, $
x\tp f(x,t)+\tfrac{1}{2}\normfrob{\gtx}^2\le K_T(1+\norm{x}^2)$.
\end{assumption}
According to~\cite[Th. 3.6, p. 58]{XM:11},
Assumption~\ref{ass:assumptions-SDE} is sufficient to guarantee global
existence and uniqueness of solutions
of~\eqref{eq:nonlinear-SDE-preliminaries} for each initial condition
$x_0\in\real^n$.
We conclude this section by presenting a useful operator in the
stability analysis of SDEs. Given a function $\lyap\in\psdtwice$, we
define the generator of~\eqref{eq:nonlinear-SDE-preliminaries} acting
on the function $\lyap$ as the mapping
$\map{\itowoarguments{\lyap}}{\real^n\times[t_0,\infty)}{\real}$ given
by
\begin{align}\label{eq:Ito-operator}
\ito{\lyap}{x,t}\triangleq\grad\lyap(x)\tp f(x,t)
+\tfrac{1}{2}\trace\Big(\Sigma(t)\tp\gtx\tp
\hessian\lyap(x)\gtx\Sigma(t)\Big).
\end{align}
It can be shown that $\ito{\lyap}{x,t}$ gives the expected rate of
change of $\lyap$ along a solution
of~\eqref{eq:nonlinear-SDE-preliminaries} that passes through the
point~$x$ at time~$t$, so it is a generalization of the Lie
derivative. According to~\cite[Th. 6.4, p. 36]{XM:11}, if we evaluate $\lyap$
along the solution $\xprocess$
of~\eqref{eq:nonlinear-SDE-preliminaries}, then the process
$\{\lyap(x(t))\}_{t\ge t_0}$ satisfies the new SDE
\begin{align}\label{eq:sde-lyapunov-function}
\lyap(x(t)) & = \lyap(x_0) +\int_{t_0}^t\ito{\lyap}{x(s),s}\ds +
\int_{t_0}^t\grad\lyap(x(s))\tp\gs\Sigma(s)\db(s).
\end{align}
Equation~\eqref{eq:sde-lyapunov-function} is known as It\^o's formula
and corresponds to the stochastic version of the chain rule.
\section{Noise-to-state stability via noise-dissipative Lyapunov
functions}\label{sec:problem-statement}
In this section, we study the stability of stochastic differential
equations subject to persistent noise. Our first step is the
introduction of a novel notion of stability. This captures the
behavior of the $p$th moment of the distance (of the state) to a
given closed set, as a function of two objects: the initial condition
and the maximum size of the covariance. After this, our next step is
to derive several Lyapunov-type stability results that help determine
whether a stochastic differential equation enjoys these stability
properties.
The following definition generalizes the concept of noise-to-state
stability given in~\cite{HD-MK:00}.
\begin{definition}\longthmtitle{Noise-to-state stability with respect
to a set}\label{def:NSS-expectation-subspace}
The system~\eqref{eq:nonlinear-SDE-preliminaries} is
\emph{noise-to-state stable (NSS) in probability with respect to the
set}~$\Uset\subseteq\real^n$ if for any $\epsilon>0$ there exist
$\mu\in\classkl$ and~$\theta\in\classk$ (that might depend
on~$\eps$), such that
\begin{align}\label{eq:def-ISS-probability}
\probability \Big\{\,\distU{x(t)}^p > \mu\big(\distU{x_0},
t-t_0\big) + \theta\Big(\esssup_{t_0\le s\le
t}\normfrob{\Sigma(s)}\Big)\,\Big\} \le\eps,
\end{align}
for all $t\ge t_0$ and any $x_0\in\real^n$. And the
system~\eqref{eq:nonlinear-SDE-preliminaries} is \emph{$p$th moment
noise-to-state stable ($p$thNSS) with respect to}~$\Uset$ if there
exist $\mu\in\classkl$ and $\theta\in\classk$, such that
\begin{align}\label{eq:def-ISS-expectation}
\expectationarg{\distU{x(t)}^p} & \le
\mu\big(\distU{x_0}, t-t_0\big) + \theta\Big(\esssup_{t_0\le
s\le t}\normfrob{\Sigma(s)}\Big) ,
\end{align}
for all $t\ge t_0$ and any $x_0\in\real^n$. The gain functions
$\mu$ and $\theta$ are the \emph{overshoot gain} and the
\emph{noise gain}, respectively.
\end{definition}
The quantity $\normfrob{\Sigma(t)} = \sqrt{ \trace\big(\Sigma(t)
\Sigma(t)\tp\big)}$ is a measure of the size of the noise because it
is related to the infinitesimal covariance
$\Sigma(t)\Sigma(t)\tp$. The choice of the $p$th power is irrelevant
in the statement in probability since one could take any
$\classkinfty$ function evaluated at $\distU{x(t)}$. However, this
would make a difference in the statement in expectation. (Also, we use the
same power for convenience.) When the set $\Uset$ is a subspace, we can
substitute~$\distU{.}$ by $\seminorm{.}{A}$, for some matrix
$A\in\realmatricesrectangulararg{m}{n}$ with $\kernel(A)=\Uset$. In
such a case, the definition above does not depend on the choice of the
matrix~$A$.
\begin{remark}\longthmtitle{NSS is not a particular case of ISS}
{\rm The concept of NSS is not a particular case of input-to-state
stability (ISS)~\cite{EDS:08} for systems that are affine in the
input, namely,
\begin{align*}
\dot{y} = f(y,t) + \galone(y,t) u(t) \;\:\Leftrightarrow\;\:
y(t)=y(t_0)+ \int_{t_0}^t f(y(s),s)\,\ds +\int_{t_0}^t G(y(s),s)
u(s)\,\ds,
\end{align*}
where $\map{u}{[t_0, \infty)}{\real^q}$ is measurable and
essentially locally bounded~\cite[Sec. C.2]{EDS:98}. The reason is
the following: the components of the vector-valued function
$\int_{t_0}^t G(y(s),s) u(s)\,\ds$ are differentiable almost
everywhere by the Lebesgue fundamental theorem of
calculus~\cite[p. 289]{JNM-NAW:99}, and thus absolutely
continuous~\cite[p. 292]{JNM-NAW:99} and with bounded
variation~\cite[Prop. 8.5]{JNM-NAW:99}. On the other hand, at
any time previous to $t_k(t)\triangleq\min\{t,\inf{\setdef{s\ge t_0}{\norm{x(s)}\ge k}}\}$,
the
driving disturbance of~\eqref{eq:nonlinear-SDE-preliminaries} is
the vector-valued function $\int_{t_0}^{t_k(t)}
\gs\Sigma(s)\db(s)$, whose $i$th component has quadratic
variation~\cite[Th. 5.14, p. 25]{XM:11} equal to
\begin{align*}
\int_{t_0}^{t_k(t)}\sum\limits_{j=1}^{m}
\absolute{\sum\limits_{l=1}^{q} \gs_{il}\Sigma(s)_{lj}}^2\ds > 0.
\end{align*}
Since a continuous process that has positive quadratic variation
must have infinite variation~\cite[Th. 1.10]{FCK:05}, we conclude
that the driving disturbance in this case is not allowed in the
ISS framework.} \oprocend
\end{remark}
Our first goal now is to provide tools to establish whether a
stochastic differential equation enjoys the noise-to-state stability
properties given in Definition~\ref{def:NSS-expectation-subspace}. To
achieve this, we look at the dissipativity properties of a special
kind of energy functions along the solutions
of~\eqref{eq:nonlinear-SDE-preliminaries}.
\begin{definition}\longthmtitle{Noise-dissipative Lyapunov
function}\label{def:noise-dissipative-Lyapunov}
A function $\lyap \in \psdtwice$ is a \emph{noise-dissipative
Lyapunov function} for~\eqref{eq:nonlinear-SDE-preliminaries} if
there exist $\lyapw\in\psd$, $\gainnoise\in\classk$, and concave
$\increasingw\in\classkinfty$ such that
\begin{align}\label{eq:theorem-second-hypothesis-VandW}
\lyap(x)\le\increasingw(\lyapw(x)),
\end{align}
for all $x\in\real^n$, and the following dissipation inequality holds:
\begin{align}\label{eq:theorem-hypothesis-ito}
\itowoarguments{\lyap}(x,t)\le-\lyapw(x) +
\gainnoise\big(\normfrob{\Sigma(t)}\big),
\end{align}
for all $(x,t)\in\real^n\times[t_0,\infty)$.
\end{definition}
\begin{remark}\longthmtitle{It\^o formula and exponential
dissipativity}\label{re:convex-dissipativity-vs-exponential-dissipativity}
{\rm Interestingly, the
conditions~\eqref{eq:theorem-second-hypothesis-VandW}
and~\eqref{eq:theorem-hypothesis-ito} are equivalent to
\begin{align}\label{eq:theorem-dissipative-condition}
\itowoarguments{\lyap}(x,t)\le-\increasingw^{-1}(\lyap(x))+
\gainnoise\big(\normfrob{\Sigma(t)}\big),
\end{align}
for all $x\in\real^n$, where $\increasingw^{-1}\in\classkinfty$ is
convex. Note that, since $\itowoarguments{\lyap}$ is not the Lie
derivative of $\lyap$ (as it contains the Hessian of $\lyap$), one
cannot directly deduce
from~\eqref{eq:theorem-dissipative-condition} the existence of a
continuously twice differentiable function $\tilde{\lyap}$ such that
\begin{align}\label{eq:exponential-dissipativity}
\itowoarguments{\tilde{\lyap}}(x,t)\le-c\tilde{\lyap}(x)+
\tilde{\gainnoise}\big(\normfrob{\Sigma(t)}\big),
\end{align}
as instead can be done in the context of~ISS, see
e.g.~\cite{LP-YW:96}.} \oprocend
\end{remark}
\begin{example}\longthmtitle{A noise-dissipative Lyapunov
function}\label{ex:example}
\rm{Assume that $\map{\convexf}{\real^n}{\real}$ is continuously
differentiable and verifies
\begin{align}\label{eq:convex-uniformly-convex}
\gamma(\norm{x-x'}^2)\le(x-x')\tp(\grad
\convexf(x)-\grad\convexf(x'))
\end{align}
for some convex function~$\gamma\in\classkinfty$ for all $x,
x'\in\real^n$. In particular, this implies that~$h$ is
strictly convex. (Incidentally, any strongly convex function
verifies~\eqref{eq:convex-uniformly-convex} for some choice
of~$\gamma$ linear and strictly increasing.) Consider now the
dynamics
\begin{align}\label{eq:inexact-distributed-opt}
\differential x(\omega, t)=-\big(\delta\lap x(\omega, t)
+\grad\convexf(x(\omega, t))\big)\dt +\Sigma(t)\,\db(\omega, t),
\end{align}
for all $t\in[t_0,\infty)$, where $x(\omega, t_0)=x_0$ with
probability~$1$ for some $x_0\in\real^n$, and $\delta>0$. Here,
the matrix $\lap\in\realmatrices$ is symmetric and positive
semidefinite, and the matrix-valued function
$\map{\Sigma}{[t_0,\infty)}{\realmatricesrectangulararg{n}{m}}$ is
continuous.
This dynamics
corresponds to the SDE~\eqref{eq:nonlinear-SDE-preliminaries} with
$f(x,t)\triangleq-\delta\lap x -\grad\convexf(x)$ and
$\galone(x,t)\triangleq\identity_n$ for all
$(x,t)\in\real^n\times[t_0,\infty)$.
Let $x^*\in\real^n$ be the unique solution of the
Karush-Kuhn-Tucker~\cite{SB-LV:09} condition $\delta\lap
x^*=-\grad\convexf(x^*)$, corresponding to the unconstrained
minimization of $F(x)\triangleq\tfrac{\delta}{2}x\tp\lap
x+\convexf(x)$. Consider then the candidate Lyapunov function
$\lyap\in\psdtwice$ given by $\lyap(x)\triangleq
\tfrac{1}{2}(x-x^*)\tp(x-x^*)$. Using~\eqref{eq:Ito-operator}, we
obtain that, for all $x\in\real^n$,
\begin{align*}
\ito{\lyap}{x,t} & = -(x-x^*)\tp\Big(\delta\lap x
+\grad\convexf(x)\Big)
+\tfrac{1}{2}\trace\Big(\Sigma(t)\tp\Sigma(t)\Big)\nonumber
\\
&=- \delta(x-x^*)\tp\lap
(x-x^*)-(x-x^*)\tp\Big(\grad\convexf(x)-\grad\convexf(x^*)\Big)
+\tfrac{1}{2}\normfrob{\Sigma(t)}^2\nonumber
\\
&\le -\gamma(\norm{x-x^*}^2)+\tfrac{1}{2}\normfrob{\Sigma(t)}^2.
\end{align*}
We note that $\lyapw\in\psd$ defined by $\lyapw(x)\triangleq
\gamma(\norm{x-x^*}^2)$ verifies
\begin{align*}
\lyap(x)=\tfrac{1}{2}\gamma^{-1}\big(\lyapw(x)\big)\;\;\quad\forall
x\in\real^n,
\end{align*}
where~$\gamma^{-1}$ is concave and belongs to the
class~$\classkinfty$ as explained in
Section~\ref{sec:pre-classks-convexity}. Therefore, $\lyap$ is a
noise-dissipative Lyapunov function
for~\eqref{eq:inexact-distributed-opt}, with concave
$\increasingw\in\classkinfty$ given by
$\increasingw(r)=1/2\gamma^{-1}(r)$ and $\gainnoise\in\classk$
given by $\gainnoise(r)\triangleq 1/2\,r^2$. \oprocend}
\end{example}
The next result generalizes~\cite[Th. 4.1]{HD-MK-RJW:01} to positive
semidefinite Lyapunov functions that satisfy weaker dissipativity
properties (cf.~\eqref{eq:theorem-dissipative-condition}) than the
typical exponential-like
inequality~\eqref{eq:exponential-dissipativity}, and characterizes the
overshoot gain.
\begin{theorem}\longthmtitle{Noise-dissipative Lyapunov functions have
an NSS dynamics}\label{th:Stability-Non-Linear-Systems}
Under Assumption~\ref{ass:assumptions-SDE},
and further assuming that~$\Sigma$ is continuous,
suppose that~$\lyap$ is
a noise-dissipative Lyapunov function
for~\eqref{eq:nonlinear-SDE-preliminaries}. Then,
\begin{align}\label{eq:ISS-theorem}
\expectationarg{\lyap(x(t))} &
\:\le\muISStilde\big(\lyap(x_0),t-t_0\big)+\increasingw
\Big(2\,\gainnoise\big(\max_{t_0\le s\le
t}\normfrob{\Sigma(s)}\big)\Big),
\end{align}
for all $t\ge t_0$, where the class~$\classkl$ function $(r,s)
\mapsto \muISStilde(r,s)$ is well defined as the solution $y(s)$ to the
initial value problem
\begin{align}\label{eq:class-kl-beta-satisfies}
\dot{y}(s) = -\tfrac{1}{2}{\increasingw}^{-1}(y(s)),\quad y(0)=r.
\end{align}
\end{theorem}
\begin{proof}
Recall that Assumption~\ref{ass:assumptions-SDE} guarantees the
global existence and uniqueness of solutions
of~\eqref{eq:nonlinear-SDE-preliminaries}. Given the process
$\{\lyap(x(t))\}_{t\ge t_0}$, the proof strategy is to obtain a differential inequality
for $\expectationarg{\lyap(x(t))}$ using It\^o formula~\eqref{eq:sde-lyapunov-function},
and then use a
comparison principle to translate the problem into one of standard
input-to-state stability for an appropriate choice of the input.
To carry out this strategy, we consider It\^o formula~\eqref{eq:sde-lyapunov-function} with respect to an arbitrary reference time instant $t'\ge t_0$,
\begin{align}\label{eq:sde-lyapunov-function-arbitrary-reference}
\lyap(x(t)) & = \lyap(x(t')) +\int_{t'}^t\ito{\lyap}{x(s),s}\ds +
\int_{t'}^t\grad\lyap(x(s))\tp\gs\Sigma(s)\db(s),
\end{align}
and we first ensure that the expectation
of the integral against Brownian motion is~$0$.
Let $S_k=\setdef{x\in\real^n}{\norm{x}\le k}$ be the ball of
radius~$k$ centered at the origin. Fix $x_0\in\real^n$ and denote by
$\tau_k$ the first exit time of~$x(t)$ from $S_k$ for integer
values of $k$ greater than~$\norm{x(t_0)}$, namely,
$\tau_k\triangleq\inf{\setdef{s\ge t_0}{\norm{x(s)}\ge k}}$,
for $k>\lceil \norm{x(t_0)}\rceil$.
Since the event
$\setdef{\omega\in\Omega}{\tau_k\le t}$ belongs to~$\filtrationt$ for each $t\ge t_0$, it
follows that
$\tau_k$ is an $\filtrationadapted$-stopping time for each $t\ge t_0$.
Now, for each~$k$
fixed, if we consider the random variable
$t_k(t)\triangleq\min\{t,\tau_k\}$ and
define~$I(t',t)$ as the stochastic
integral in~\eqref{eq:sde-lyapunov-function-arbitrary-reference}
for any fixed $t'\in [t_0,t_k(t)]$,
then the process
$I(t', t_k(t))$
has zero expectation
as we show next. The function $\map{X}{S_k\times[t',t]}{\real}$
given by $X(x,s)\triangleq\grad\lyap(x)\tp \galone(x,s)\Sigma(s)$
is essentially bounded (in its domain),
and thus
$\expectationarg{\int_{t'}^t \indicator{[t', t_k(t)]}(s)\, X(x(s),t)^2 \,\ds}<\infty$,
where
$\indicator{[t', t_k(t)]}(s)$ is the indicator function of the set
$[t', t_k(t)]$.
Therefore,
$\expectationarg{I(t', t_k(t))}=0$
by~\cite[Th. 5.16, p. 26]{XM:11}. Define now
$\lyapbar(t)\triangleq\expectationarg{\lyap(x(t))}$ and
$\lyapwbar(t)\triangleq\expectationarg{\lyapw(x(t))}$ in
$\Gamma(t_0)\triangleq\setdef{t\ge t_0}{\lyapbar(t)<\infty}$.
By the above,
taking expectations in~\eqref{eq:sde-lyapunov-function-arbitrary-reference}
and
using~\eqref{eq:theorem-hypothesis-ito}, we obtain that
\begin{align}\label{eq:differential-ineq-r}
\lyapbar(t_k(t)) & = \lyapbar(t') + \expectation
\Big[\,\int_{t'}^{t_k(t)}\ito{\lyap}{x(s),s}\ds\, \Big]\nonumber
\\
& \le \:\lyapbar(t') -
\expectation\Big[\,\int_{t'}^{t_k(t)}\lyapw(x(s))\ds\, \Big] +
\expectation\Big[\int_{t'}^{t_k(t)} \gainnoise(\normfrob{\Sigma(s)})\,\ds\,\Big]
\end{align}
for all $t\in\Gamma(t_0)$ and any $t'\in [t_0, t_k(t)]$. Next we use the fact that~$\lyap$ is
continuous and $\{x(t)\}_{t\ge t_0}$ is also continuous with
probability~$1$. In addition, according to Fatou's
lemma~\cite[p. 123]{JNM-NAW:99} for convergence in the probability
measure, we get that
\begin{align}\label{eq:fatou-lyapbar}
\lyapbar(t)=\,&\expectationarg{\,\lyap (x(\liminf_{k\to\infty}\,
t_k(t)))\,} =\expectationarg{\,\liminf_{k\to\infty}\lyap (x(
t_k(t)))\,}
\\
\le\,& \liminf_{k\to\infty}\,
\expectationarg{\,\lyap(x(t_k(t)))\,}
=\liminf_{k\to\infty}\lyapbar(t_k(t))\nonumber
\end{align}
for all $t\in\Gamma(t_0)$.
Moreover, using the monotone convergence~\cite[p. 176]{JNM-NAW:99}
when $k\to\infty$ in both Lebesgue integrals
in~\eqref{eq:differential-ineq-r} (because both integrands are
nonnegative and $\indicator{[t', t_k(t)]}$ converges monotonically
to $\indicator{[t', t]}$ as $k\to\infty$ for any $t'\in [t_0, t_k(t)]$), we obtain
from~\eqref{eq:fatou-lyapbar} that
\begin{align}\label{eq:differential-ineq-beforeTonelli}
\lyapbar(t)\le \lyapbar(t') -
\expectation\Big[\,\int_{t'}^{t} \lyapw(x(s))\ds \,\Big]+
\int_{t'}^{t}\gainnoise(\normfrob{\Sigma(s)})\,\ds
\end{align}
for all $t\in\Gamma(t_0)$ and any $t'\in[t_0,t]$.
Before resuming the argument we make two observations. First,
applying Tonelli's theorem~\cite[p. 212]{JNM-NAW:99} to the
nonnegative process $\{\lyapw(x(s))\}_{s\ge t'}$, it follows that
\begin{align}\label{eq:Tonelli_W}
\expectation\big[\,\int_{t'}^{t} \lyapw(x(s))\ds\,\big]
=\int_{t'}^{t} \lyapwbar(x(s))\ds.
\end{align}
Second, using~\eqref{eq:theorem-second-hypothesis-VandW} and
Jensen's inequality~\cite[Ch. 3]{VSB:95},
we get that
\begin{align}\label{eq:vbar-wbar-kinfcc}
\lyapbar(t) = \expectationarg{\lyap(x(t))}
\le\expectation\big[\increasingw(\lyapw(x(t)))\big]
\le\increasingw\big(\expectationarg{\lyapw(x(t))}\big)
=\increasingw\big(\lyapwbar(t)\big),
\end{align}
because~$\increasingw$ is concave, so
$\lyapwbar(t)\ge{\increasingw}^{-1}(\lyapbar(t))$.
Hence,~\eqref{eq:differential-ineq-beforeTonelli}
and~\eqref{eq:Tonelli_W} yield
\begin{align}\label{eq:integral-ineq-lyapbar}
\lyapbar(t) &\:\le \lyapbar(t') -
\int_{t'}^{t}\lyapwbar(s)\,\ds+
\int_{t'}^{t}\gainnoise(\normfrob{\Sigma(s)})\,\ds
\nonumber
\\&\:
\le
\lyapbar(t') +
\int_{t'}^{t} \Big(-{\increasingw}^{-1}(\lyapbar(s))
+\gainnoise(\normfrob{\Sigma(s)})\Big)\,\ds
\end{align}
for all $t\in\Gamma(t_0)$ and any $t'\in [t_0, t]$,
which in particular shows that $\Gamma(t_0)$ can be taken equal to $[t_0,\infty)$.
Now the strategy is to compare~$\lyapbar$ with the unique solution of an ordinary differential
equation that represents an input-to-state stable (ISS) system. First we leverage the integral inequality~\eqref{eq:integral-ineq-lyapbar} to show that
$\lyapbar$ is continuous in $[t_0,\infty)$, which allows us then to rewrite~\eqref{eq:integral-ineq-lyapbar} as a differential inequality at~$t'$. To
to show that~$\lyapbar$ is continuous, we use the dominated
convergence theorem~\cite[Thm. 2.3, P. 6]{XM:11} applied to
$V_k(\hat{t})\triangleq\lyap(x(\hat{t}))-\lyap(x(\hat{t}+1/k))$, for $\hat{t}\in[t_0,t]$, and similarly taking $\hat{t}-1/k$ (excluding, respectively, the cases when $\hat{t}=t\,$ or $\,\hat{t}=t_0$).
The hypotheses are satisfied because $V_k$ can be majorized using~\eqref{eq:integral-ineq-lyapbar} as
\begin{align}\label{eq:majorizing-function-lyapbar}
\absolute{V_k(\hat{t})}
\;\le\;\lyap(x(\hat{t}))+\lyap(x(\hat{t}+1/k))
\; \le\;
2\big( \lyap(x_0) +
\int_{t_0}^{t} \gainnoise(\normfrob{\Sigma(s)})\,\ds\big),
\end{align}
where the term on the right is not a random variable and thus coincides with its expectation.
Therefore, for every $\hat{t}\in[t_0, t]$,
\begin{align*}
\lim_{s\to \hat{t}}
\expectationarg{\lyap(x(s))}
=\expectationarg{ \lim_{s\to \hat{t}} \lyap(x(s)) }
=\expectationarg{\lyap(x(\hat{t}))},
\end{align*}
so $\lyapbar$ is continuous on~$[t_0, t]$, for any $t\ge t_0$.
Now, using again~\eqref{eq:integral-ineq-lyapbar} and the continuity of the integrand,
we can bound the upper right-hand derivative~\cite[Appendix C.2]{HKK:02}
(also called upper Dini derivative), as
\begin{align*}
D^+\lyapbar(t')\triangleq &\, \limsup_{t\to t',\, t> t'}\frac{\lyapbar(t)-\lyapbar(t')}{t-t'}
\\
\le &\,
\limsup_{t\to t',\, t> t'}\frac{1}{t-t'}\int_{t'}^t \big(-{\increasingw}^{-1}(\lyapbar(s))+\gainnoise(\normfrob{\Sigma(s)})\big)\,\ds
\,=\,
h(\lyapbar(t'),\Input(t')),
\end{align*}
for any $t'\in [t_0,\infty)$, where the function
$\map{h}{\realnonnegative\times\realnonnegative}{\real}$ is given by
\begin{align*}
h(y,\Input)\triangleq-{\increasingw}^{-1}(y)+\Input,
\end{align*}
and
$\Input(t)\triangleq\gainnoise(\normfrob{\Sigma(t)})$, which is continuous in $[t_0,\infty)$.
Therefore, according to the
comparison principle~\cite[Lemma 3.4, P. 102]{HKK:02},
using that $\lyapbar$ is continuous in $[t_0,\infty)$ and
$D^+\lyapbar(t')\le h(\lyapbar(t'),\Input(t'))$, for any $t'\in [t_0,\infty)$,
the solutions~\cite[Sec. C.2]{EDS:98} of the initial value problem
\begin{align} \label{eq:comparison-principle-lemma-autoproper-implies-bounded-V}
\lyapudot(t) = h(\lyapu(t),\Input(t)),
\qquad
\lyapu_0\triangleq\lyapu(t_0)=\lyapbar(t_0)
\end{align}
(where~$h$ is locally Lipschitz in the first argument as we show next),
satisfy that $\lyapu(t)\ge \lyapbar(t)\,(\ge 0)$ in the common interval of existence.
We argue
the global existence and uniqueness of
solutions
of~\eqref{eq:comparison-principle-lemma-autoproper-implies-bounded-V}
as follows.
Since
$\alpha\triangleq{\increasingw}^{-1}$ is convex and
class~$\classkinfty$ (see Section~\ref{sec:pre-classks-convexity}),
it holds that
\begin{align*}
\alpha(s')\le\alpha(s) \le
\alpha(s')+\tfrac{\alpha(s'')-\alpha(s')}{s''-s'} (s-s')
\end{align*}
for all $s\in[s', s'']$, for any $s''>s'\ge 0$. Thus,
$\absolute{\alpha(s)-\alpha(s')}=\alpha(s)-\alpha(s')\le L (s-s')$,
for any $s''\ge s\ge s'\ge 0$, where $L\triangleq
(\alpha(s'')-\alpha(s'))/(s''-s')$, so ${\increasingw}^{-1}$ is
locally Lipschitz. Hence,~$h$ is locally Lipschitz in
$\realnonnegative\times\realnonnegative$.
Therefore, given the
input function~$\Input$ and any $\lyapu_0\ge 0$, there is a unique maximal
solution
of~\eqref{eq:comparison-principle-lemma-autoproper-implies-bounded-V}, denoted by
$\lyapu(\lyapu_0, t_0; t)$, defined in a maximal interval $[t_0,
t_{\text{max}}(\lyapu_0,t_0))$. (As a by-product, the initial value
problem~\eqref{eq:class-kl-beta-satisfies}, which can be written as
$\dot{y}(s) = \tfrac{1}{2} h(y(s),0)$, $y(0)=r$, has a unique and
strictly decreasing solution in~$[0,\infty)$, so~$\muISStilde$ in
the statement is well defined and in class~$\classkl$.)
To show
that~\eqref{eq:comparison-principle-lemma-autoproper-implies-bounded-V}
is ISS we follow a similar argument as in
the proof of~\cite[Th. 5]{EDS:08} (and note that, as a consequence, we obtain that
$t_{\text{max}}(\lyapu_0,t_0)=\infty$). Firstly, if
${\increasingw}^{-1}(\lyapu) \ge 2\Input$, then $\lyapudot(t) =
-\tfrac{1}{2}{\increasingw}^{-1}(\lyapu(t))$, which implies that~$\lyapu$ is nonincreasing outside the set
$S\triangleq\setdef{t\ge t_0}{\lyapu(t)\le
\increasingw(2\Input(t))}$. Thus, if some $t^*\ge t_0$ belongs to $S$, then so
does every $t\in [t^*, t_{\text{max}}(\lyapu_0,t_0))$ implying that $\lyapu$ is
locally bounded because~$\Input$ is locally bounded (in fact, continuous).
(Note that $\lyapu(t)\ge 0$ because $\lyapudot(t)\ge 0$ whenever $\lyapu(t)=0$.)
Therefore, for all $t\ge t_0$, and
for~$\muISStilde$ as in the statement (which we have shown is well
defined), we have that
\begin{align*}
\lyapbar(t)\,
\le
\lyapu(t)
\le
\max\Big\{\muISStilde\big(\lyapbar(t_0),t-t_0\big)\,,\,
\increasingw\Big(2\max_{t_0\le s\le t}\Input(s)\Big)\Big\}.
\end{align*}
Since the maximum of two quantities is upper bounded by the sum, and
using the definition of~$\Input$ together with the monotonicity
of~$\gainnoise$, it follows that
\begin{align}\label{eq:ISS-lyapu-to-make-comparison}
\lyapbar(t)\,
\le
\lyapu(t)
\le\muISStilde\big(\lyap(x_0),t-t_0\big)
+ \increasingw\Big(2\gainnoise\Big(\max_{t_0\le s\le t}\normfrob{\Sigma(s)}\Big)\Big),
\end{align}
for all $t\ge t_0$,
where we also used that~$\lyapbar(t_0)=\lyap(x_0)$, and the proof is complete.
\end{proof}
Of particular interest to us is the case when the function $\lyap$ is
lower and upper bounded by class~$\classkinfty$ functions of the
distance to a closed, not necessarily bounded, set.
\begin{definition}\longthmtitle{NSS-Lyapunov
functions}\label{def:Noise-to-state-Lyapunov}
A function $\lyap \in \psdtwice$ is a \emph{strong NSS-Lyapunov
function in probability with respect to}~$\Uset\subseteq\real^n$
for~\eqref{eq:nonlinear-SDE-preliminaries} if $\lyap$ is a
noise-dissipative Lyapunov function and, in addition, there exist
$p>0$ and class~$\classkinfty$ functions $\gainNSSone$ and
$\gainNSStwo$ such that
\begin{align}\label{eq:theorem-third-hypothesis-V}
\gainNSSone(\distU{x}^p) \le \lyap(x)\le\gainNSStwo(\distU{x}^p),
\quad\forall x\in\real^n.
\end{align}
If, moreover, $\gainNSSone$ is convex, then $\lyap$ is a \emph{$p$th
moment NSS-Lyapunov function with respect to}~$\Uset$.
\end{definition}
Note that a strong NSS-Lyapunov function in probability with respect
to a set satisfies an inequality of the
type~\eqref{eq:theorem-third-hypothesis-V} for any $p>0$, whereas the
choice of~$p$ is relevant when $\gainNSSone$ is required to be convex.
The reason for the `strong' terminology is that we
require~\eqref{eq:theorem-dissipative-condition} to be satisfied with
convex $\increasingw^{-1}\in\classkinfty$. Instead, a standard
NSS-Lyapunov function in probability satisfies the same inequality
with a class~$\classkinfty$ function which is not necessarily convex.
We also note that~\eqref{eq:theorem-third-hypothesis-V} implies that
$\Uset=\setdef{x\in\real^n}{\lyap(x)=0}$, which is closed because
$\lyap$ is continuous.
\begin{example}\longthmtitle{Example~\ref{ex:example}--revisited:
an NSS-Lyapunov function}\label{ex:example-two}
{\rm Consider the function $\lyap$ introduced in
Example~\ref{ex:example}. For each $p\in(0,2]$, note that
\begin{align*}
{\gainNSSone}_p(\norm{x-x^*}^p) \le
\lyap(x)\le{\gainNSStwo}_p(\norm{x-x^*}^p)\;\;\quad\forall
x\in\real^n,
\end{align*}
for the convex
functions~${\gainNSSone}_p(r)={\gainNSStwo}_p(r)\triangleq
r^{2/p}$, which are in the class~$\classkinfty$. (Recall that
$\gainNSStwo$ in Definition~\ref{def:Noise-to-state-Lyapunov} is
only required to be~$\classkinfty$.) Thus, the function~$\lyap$ is
a $p$th moment NSS-Lyapunov function
for~\eqref{eq:inexact-distributed-opt} with respect to~$x^*$ for
$p\in(0,2]$. \oprocend}
\end{example}
The notion of NSS-Lyapunov function plays a key role in establishing
our main result on the stability of SDEs with persistent
noise.
\begin{corollary}\longthmtitle{The existence of an NSS-Lyapunov
function implies the corresponding
NSS property}\label{co:of-the-main-theorem}
Under Assumption~\ref{ass:assumptions-SDE},
and further assuming that~$\Sigma$ is continuous,
given a closed
set~$\Uset \subset \real^n$,
\begin{enumerate}
\item if $\lyap \in \psdtwice$ is a strong NSS-Lyapunov function in
probability with respect to~$\Uset$
for~\eqref{eq:nonlinear-SDE-preliminaries}, then the system is NSS
in probability with respect to~$\Uset$ with gain functions
\begin{align}\label{eq:gains-NSS}
\mu(r,s) \:\triangleq\gainNSSone^{-1}\big(\tfrac{2}{\eps}
\muISStilde(\gainNSStwo(r^p),s ) \big),
\quad\theta(r)\: \triangleq\gainNSSone^{-1}
\big(\tfrac{2}{\eps}\increasingw(2\gainnoise(r))\big);
\end{align}
\item if $\lyap \in \psdtwice$ is a $p$thNSS-Lyapunov function with
respect to $\Uset$ for~\eqref{eq:nonlinear-SDE-preliminaries},
then the system is $p$th~moment NSS with respect to~$\Uset$ with
gain functions $\mu$ and $\theta$ as in~\eqref{eq:gains-NSS}
setting $\epsilon=1$.
\end{enumerate}
\end{corollary}
\begin{proof}
To show \emph{(i)}, note that, since $\gainNSSone(\distU{x}^p) \le
\lyap(x)$ for all $x\in\real^n$, with $\gainNSSone\in\classkinfty$,
it follows that for any $\hat{\rho}>0$ and $t\ge t_0$,
\begin{align}\label{eq:corollary-bounded-probability}
\probability\Big\{\distU{x(t)}^p > \hat{\rho}\Big\} & =
\probability\Big\{\gainNSSone(\distU{x(t)}^p) >
\gainNSSone(\hat{\rho}) \Big\}
\le\probability\Big\{\lyap(x(t)) > \gainNSSone(\hat{\rho})\Big\}
\le
\frac{\expectationarg{\lyap(x(t))}}{\gainNSSone(\hat{\rho})}\nonumber
\\
& \le \frac{1}{\gainNSSone(\hat{\rho})}
\bigg(\muISStilde\Big(\gainNSStwo(\distU{x_0}^p),\,t-t_0\Big)
+\increasingw\Big(2\gainnoise \big(\max_{t_0\le s\le
t}\normfrob{\Sigma(s)}\big)\Big)\bigg),
\end{align}
where we have used the strict monotonicity of $\gainNSSone$ in the
first equation, Chebyshev's inequality~\cite[Ch. 3]{VSB:95} in the
second inequality, and the upper bound for
$\expectationarg{\lyap(x(t))}$ obtained in
Theorem~\ref{th:Stability-Non-Linear-Systems},
cf.~\eqref{eq:ISS-theorem}, in the last inequality (leveraging the
monotonicity of $\muISStilde$ in the first argument and the fact
that $\lyap(x)\le\gainNSStwo(\distU{x}^p)$ for all
$x\in\real^n$). Also, for any function $\alpha\in\classk$, we have
that $\alpha(2r)+\alpha(2s)\ge\alpha(r+s)$ for all $r,s\ge 0$. Thus,
\begin{align}\label{eq:rhos-for-probability}
\rho(\eps,x_0,t) & \triangleq\mu\big(\distU{x_0},t-t_0\big)
+\theta\Big(\max_{t_0\le s\le t}\normfrob{\Sigma(s)}\Big)
\\
& \ge\gainNSSone^{-1}
\Bigg(\frac{1}{\eps}\muISStilde\Big(\gainNSStwo(\distU{x_0}^p) ,
t-t_0\Big) + \frac{1}{\eps}\increasingw\Big(2\gainnoise
\big(\max_{t_0\le s\le t}\normfrob{\Sigma(s)}\big)\Big)\Bigg)
\triangleq\hat{\rho}(\eps). \nonumber
\end{align}
Substituting now $\hat{\rho}\triangleq\hat{\rho}(\eps)$
in~\eqref{eq:corollary-bounded-probability}, and
using that $\rho(\eps,x_0,t)\ge \hat{\rho}(\eps)$,
we get that
$\probability\big\{\distU{x(t)}^p >
\rho(\eps,x_0,t)\big\}\le\probability\big\{\distU{x(t)}^p >
\hat{\rho}(\eps)\big\}\le\eps$.
To show \emph{(ii)}, since $\gainNSSone^{-1}$ is concave, applying Jensen's
inequality~\cite[Ch. 3]{VSB:95}, we get
\begin{align*}
\expectationarg{\distU{x(t)}^p}
\:\le\expectation\big[\gainNSSone^{-1}
\big(\lyap(x(t))\big)\big]\le\gainNSSone^{-1}
\Big(\expectationarg{\lyap(x(t))} \Big)
\le\hat{\rho}(1)\le\rho(1,x_0,t),
\end{align*}
where in the last two inequalities we have used the bound for
$\expectationarg{\lyap(x(t))}$
in~\eqref{eq:corollary-bounded-probability} and the definition of
$\hat{\rho}(\eps)$ in~\eqref{eq:rhos-for-probability}.
\end{proof}
\begin{figure*}[bth]
\centering
\subfigure[]{\includegraphics[width=.496\linewidth]{figure-a.eps}}
\subfigure[]{\includegraphics[width=.496\linewidth]{figure-b.eps}}
\caption{Evolution of the
dynamics~\eqref{eq:inexact-distributed-opt} with $\lap=0$,
\,$\convexf(x_1,x_2)=\log\big(e^{(x_1-2)} + e^{(x_2+1)}\big) + 0.5
(x_1+x_2-1)^2 + (x_1-x_2)^2$, and initial condition $[x_1(0),
x_2(0)]=(1,-0.5)$. Since $\convexf$ is a sum of convex functions,
and the Hessian of the quadratic part of $\convexf$ has
eigenvalues $\{2, 4\}$, we can take $\gamma$ given by
$\gamma(r)=2\, r$, for $r\ge 0$. Plot (a) shows the evolution of
the first and second coordinates with $\Sigma=0.1\,\identity_2$.
Plot (b) illustrates the noise-to-state stability property in
second moment with respect to~$x^*=(0.36, 0.14)$, where the matrix
$\Sigma(t)$ is a constant multiple of the identity. (The
expectation is computed averaging over $500$ realizations of the
noise.)}\label{fig:simulation}
\end{figure*}
\begin{example}\longthmtitle{Example~\ref{ex:example}--revisited:
illustration of
Corollary~\ref{co:of-the-main-theorem}}
{\rm Consider again Example~\ref{ex:example}. Since $\lyap$ is a
$p$th moment NSS-Lyapunov function
for~\eqref{eq:inexact-distributed-opt} with respect to the
point~$x^*$ for $p\in(0,2]$, as shown in
Example~\ref{ex:example-two},
Corollary~\ref{co:of-the-main-theorem} implies that
\begin{align}\label{eq:example-NSS-expectation}
\expectationarg{\norm{x-x^*}^p} & \le \mu\big(\norm{x_0-x^*},
t-t_0\big) + \theta\Big(\max_{t_0\le s\le
t}\normfrob{\Sigma(s)}\Big) ,
\end{align}
for all $t\ge t_0$, $x_0\in\real^n$, and $p\in(0,2]$,
where
\begin{align*}
\mu(r,s) \:=\big(2 \muISStilde(r^2,s ) \big)^{p/2},
\quad\theta(r)\: = \big(\gamma^{-1}(r^2)\big)^{p/2},
\end{align*}
and the class~$\classkl$ function $\muISStilde$ is defined as the
solution to the initial value
problem~\eqref{eq:class-kl-beta-satisfies} with
$\increasingw(r)=\tfrac{1}{2}\gamma^{-1}(r)$.
Figure~\ref{fig:simulation} illustrates this noise-to-state
stability property. We note that if the function $\convexf$ is
strongly convex, i.e., if $\gamma(r)=c_{\gamma}\,r$ for some
constant $c_{\gamma}>0$, then $\map{\muISStilde}{\realnonnegative
\times \realnonnegative}{\realnonnegative}$ becomes
$\muISStilde(r, s)=r e^{-c_{\gamma}s}$, and $\muISS(r,
s)=2^{p/2}\,r^p e^{-c_{\gamma} p/2\, s}$, so the bound for
$\expectationarg{\norm{x-x^*}^p}$
in~\eqref{eq:example-NSS-expectation} decays exponentially with
time to~$\theta\big(\max_{t_0\le s\le
t}\normfrob{\Sigma(s)}\big)$.\oprocend}
\end{example}
\section{Refinements of the notion of proper
functions}\label{sec:positive semidefinite-functions}
In this section, we analyze in detail the inequalities between
functions that appear in the definition of noise-dissipative Lyapunov
function, strong NSS-Lyapunov function in probability, and
$p$th~moment NSS-Lyapunov function.
In Section~\ref{sec:equivalent-classes}, we establish that these
inequalities can be regarded as equivalence relations. In
Section~\ref{sec:characterizations}, we make a complete
characterization of the properties of two functions related by these
equivalence relations. Finally, in Section~\ref{sec:alternative},
these results lead us to obtain an alternative formulation of
Corollary~\ref{co:of-the-main-theorem}.
\subsection{Proper functions and equivalence
relations}\label{sec:equivalent-classes}
Here, we provide a refinement of the notion of proper functions with
respect to each other. Proper functions play an important role in
stability analysis, see e.g.,~\cite{HKK:02,EDS:08}.
\begin{definition}\longthmtitle{Refinements of the notion of proper
functions with respect to each
other}\label{def:kinf-proper-wrt-each-other}
Let $\dom\subseteq\real^n$ and the functions $\map{\lyap,
\lyapw}{\dom}{\realnonnegative}$ be such that
\begin{align*}
\alpha_1 (\lyapw(x)) \le \lyap(x) \le \alpha_2 (\lyapw(x)), \quad
\forall x \in \dom,
\end{align*}
for some functions $\map{\alpha_1,\,
\alpha_2}{\realnonnegative}{\realnonnegative}$. Then,
\begin{enumerate}
\item if $\alpha_1,\alpha_2\in\classk$, we say that $\lyap$ is
$\classk$\,-\,\emph{dominated by}~$\lyapw$ in $\dom$, and write
$\linebreak\lyap\proper\lyapw\;\;\text{in}\;\;\dom$;
\item if $\alpha_1,\alpha_2\in\classkinfty$, we say that $\lyap$ and
$\lyapw$ are $\classkinfty$\,-\,\emph{proper with respect to each
other in} $\dom$, and write $
\lyap\properinfty\lyapw\;\;\text{in}\;\;\dom$;
\item if $\alpha_1,\alpha_2\in\classkinfty$ are convex and concave,
respectively, we say that $\lyap$ and $\lyapw$ are
$\classkinftyccp$\,-\,\emph{(convex-concave) proper with respect
to each other in} $\dom$, and write $ \lyap
\properinftyccp\lyapw\;\;\text{in}\;\;\dom$;
\item if $\alpha_1(r)\triangleq c_{\alpha_1} r$ and
$\alpha_2(r)\triangleq c_{\alpha_2} r$,
for some constants
$c_{\alpha_1}, c_{\alpha_2} > 0$, we say that $\lyap$ and $\lyapw$
are \emph{equivalent in $\dom$}, and write
$\lyap\relationconstants\lyapw\;\;\text{in}\;\;\dom$.
\end{enumerate}
\end{definition}
Note that the relations in
Definition~\ref{def:kinf-proper-wrt-each-other} are nested, i.e.,
given $\map{\lyap, \lyapw}{\dom}{\realnonnegative}$, the following
chain of implications hold in~$\dom$\,:
\begin{align}\label{eq:chain-relations}
\lyap\relationconstants\lyapw \,\Rightarrow\,
\lyap\properinftyccp\lyapw\,\Rightarrow\,\lyap
\properinfty\lyapw\,\Rightarrow\,\lyap\proper\lyapw.
\end{align}
Also, note that if $\lyapw(x)=\norm{x}$,
$\dom$ is a neighborhood of~$0$, and $\alpha_1, \alpha_2$ are
class~$\classk$, then we recover the notion of $\lyap$ being a
\emph{proper} function~\cite{HKK:02}. If $\dom=\real^n$, and $\lyap$
and $\lyapw$ are seminorms, then the relation~$\relationconstants$
corresponds to the concept of equivalent seminorms.
The relation~$\properinfty$ is relevant for ISS and NSS in
probability, whereas the relation~$\properinftyccp$ is important for
$p$th moment NSS. The latter is because the inequalities in
$\properinftyccp$ are still valid, thanks to Jensen inequality, if we
substitute $\lyap$ and $\lyapw$ by their expectations along a
stochastic process. Another fact about the relation~$\properinftyccp$
is that $\alpha_1,\alpha_2\in\classkinfty$, convex and concave,
respectively, must be asymptotically linear if
$\lyap(\dom)\supseteq[s_0,\infty)$, for some $s_0\ge 0$, so that
$\alpha_1(s)\le\alpha_2(s)$ for all $s\ge s_0$. This follows from
Lemma~\ref{le:convex-concave}.
\begin{remark}\longthmtitle{Quadratic forms in a constrained
domain}\label{re:another-set-assumptions}
{\rm It is sometimes convenient to view the functions $\map{\lyap,
\lyapw}{\dom}{\realnonnegative}$ as defined in a
domain where their
functional expression becomes simpler. To make this idea precise,
assume there exist $\map{i}{\dom \subset \real^n}{\real^m}$, with
$m \ge n$, and
$\map{\lyaphat, \lyapwhat}{\hat{\dom}}{\realnonnegative}$, where
$\hat{\dom}=i(\dom)$, such that $\lyap =\lyaphat \circ i $ and
$\lyapw =\lyapwhat \circ i$. If this is the case, then the
existence of $\alpha_1,\,
\alpha_2:\realnonnegative\to\realnonnegative$ such that
$\alpha_1\big(\lyapwhat(\hat{x})\big) \le
\lyaphat(\hat{x})\le\alpha_2\big(\lyapwhat(\hat{x})\big)$, for all
$\hat{x}\in\hat{\dom}$, implies that
$\alpha_1\big(\lyapw({x})\big)\le\lyap({x})
\le\alpha_2\big(\lyapw({x})\big)$, for all $x\in\dom$. The reason
is that for any $x\in\dom$ there exists $\hat{x}\in\hat{\dom}$,
given by $\hat{x}=i(x)$, such that $\lyap(x)=\lyaphat(\hat{x})$
and $\lyapw(x)=\lyapwhat(\hat{x})$, so
\begin{align*}
\alpha_1\big(\lyapw(x)\big) =
\alpha_1\big(\lyapwhat(\hat{x})\big)\le\lyap(x) =
\lyaphat(\hat{x})\le\alpha_2\big(\lyapwhat(\hat{x})\big) =
\alpha_2\big(\lyapw(x)\big).
\end{align*}
Consequently, if any of the relations given in
Definition~\ref{def:kinf-proper-wrt-each-other} is satisfied by
$\lyaphat$, $\lyapwhat$ in $\hat{\dom}$, then the corresponding
relation is satisfied by $\lyap$, $\lyapw$ in $\dom$. For
instance, in some scenarios this procedure can allow us to rewrite
the original functions $\lyap$, $\lyapw$ as quadratic forms
$\lyaphat$, $\lyapwhat$ in a constrained set of an extended
Euclidean space, where it is easier to establish the appropriate
relation between the functions. We make use of this observation
in Section~\ref{sec:alternative} below.}\oprocend
\end{remark}
\begin{lemma}\longthmtitle{Powers
of seminorms with the same
nullspace}\label{le:kinf-proper-semidefinite-quadratic-forms}
Let $A$ and $B$ in $\realmatricesrectangulararg{m}{n}$ be nonzero
matrices with the same nullspace, $\kernel(A) = \kernel(B)$. Then,
for any $p,q>0$, the inequalities
$\alpha_1\big(\seminorm{x}{A}^p\big) \le \seminorm{x}{B}^q \le
\alpha_2\big(\seminorm{x}{A}^p\big)$ are verified with
\begin{align*}
\alpha_1(r)\triangleq\Big(\frac{\lambda_{n-k}(\Bt
B)}{\lambdamax(\At A)}\Big)^{\frac{q}{2}}\,
r^{q/p};\quad\alpha_2(r)\triangleq \Big(\frac{\lambdamax(\Bt
B)}{\lambda_{n-k}(\At A)}\Big)^{\frac{q}{2}}\, r^{q/p},
\end{align*}
where $k\triangleq\dim (\kernel(A))$. In particular,
$\seminorm{.}{A}^p\relationconstants\seminorm{.}{B}^p$\, and\,
$\seminorm{.}{A}^p\properinfty\seminorm{.}{B}^q$\, in\, $\real^n$
for any real numbers $p,q>0$.
\end{lemma}
\begin{proof}
For $\Uset\triangleq\kernel(A)$,
write any $x\in\real^n$ as $x=\xu+\xuperpendicular$, where
$\xu\in\Uset$ and $\xuperpendicular\in\setdef{x\in\real^n}{x\tp
u=0\;, \forall u\in\Uset}$, so that
$Ax=A(\xu+\xuperpendicular)=A\xuperpendicular$ and
$Bx=B\xuperpendicular$ because $\kernel(A)=\kernel(B)=\Uset$. Using
the formulas for the eigenvalues in~\cite[p. 178]{RAH-CRJ:85}, we
see that the next chain of inequalities hold:
\begin{align*}
&\alpha_1\big(\seminorm{x}{A}^p\big)
=\alpha_1\Big(\big(\xuperpendicular\tp \At
A\xuperpendicular\big)^{\frac{p}{2}}\Big) \le \alpha_1
\Big(\big(\lambdamax(\At A)\xuperpendicular\tp
\xuperpendicular\big)^{\frac{p}{2}}\Big)
\\
&\le \big(\lambda_{n-k}(\Bt B) \xuperpendicular\tp
\xuperpendicular\big)^{\frac{q}{2}} \le\big(\xuperpendicular\tp
\Bt B\xuperpendicular\big)^{\frac{q}{2}} \le\big(\lambdamax(\Bt B)
\xuperpendicular\tp \xuperpendicular\big)^{\frac{q}{2}}
\\
&\le \alpha_2 \Big(\big(\lambda_{n-k}(\At A) \xuperpendicular\tp
\xuperpendicular\big)^{\frac{p}{2}}\Big)
\le\alpha_2\Big(\big(\xuperpendicular\tp \At
A\xuperpendicular\big)^{\frac{p}{2}}\Big)
=\alpha_2\big(\seminorm{x}{A}^p\big),
\end{align*}
where $\seminorm{x}{B}^q=\big(\xuperpendicular\tp \Bt
B\xuperpendicular\big)^{\frac{q}{2}}$. From this we conclude that
$\seminorm{.}{A}^p\properinfty\seminorm{.}{B}^q$ in $\real^n$. Finally, when
$p=q$, the class~$\classkinfty$ functions $\alpha_1$, $\alpha_2$
in the statement are linear, so we obtain that
$\seminorm{.}{A}^p\relationconstants\seminorm{.}{B}^p$ in $\real^n$.
\end{proof}
Next we show that~$\properinfty$ and~$\properinftyccp$ are reflexive,
symmetric, and transitive, and hence define equivalence relations.
\begin{lemma}\longthmtitle{The $\classkinfty$- and
$\classkinftyccp$-proper relations are equivalence
relations}\label{le:kinf-equivalence-relation}
The relations $\properinfty$ and $\properinftyccp$ in any set
$\dom\subseteq\real^n$ are both equivalence relations.
\end{lemma}
\begin{proof}
For convenience, we represent both relations by~$\relation$. Both
are {reflexive}, i.e., $\lyap\relation\lyap$, because one can take
$\alpha_1(r)=\alpha_2(r)=r$ noting that a linear function is both
convex and concave. Both are {symmetric}, i.e.,
$\lyap\relation\lyapw$ if and only if $\lyapw\relation\lyap$,
because if $\alpha_1 \circ \lyapw \le \lyap \le \alpha_2 \circ
\lyapw$ in $\dom$, then $\alpha_2^{-1} \circ \lyap \le \lyapw \le
\alpha_1^{-1} \circ \lyap$ in $\dom$. In the case of~$\properinfty$,
the inverse of a class~$\classkinfty$ function is
class~$\classkinfty$. Additionally, in the case
of~$\properinftyccp$, if $\alpha\in\classkinfty$ is convex
(respectively, concave), then $\alpha^{-1}\in\classkinfty$ is
concave (respectively, convex). Finally, both are {transitive},
i.e., $\lyapu\relation\lyap$ and $\lyap\relation\lyapw$ imply $
\lyapu\relation\lyapw$, because if $\alpha_1 \circ \lyap \le \lyapu
\le \alpha_2 \circ \lyap$ and $\tilde{\alpha}_1 \circ \lyapw \le
\lyap \le \tilde{\alpha}_2 \circ \lyapw$ in $\dom$, then $\alpha_1
\circ \tilde{\alpha}_1 \circ \lyapw \le \lyapu \le \alpha_2 \circ
\tilde{\alpha}_2 \circ \lyapw$ in $\dom$. In the case
of~$\properinfty$, the composition of two class $\classkinfty$
functions is class~$\classkinfty$. Additionally, in the case
of~$\properinftyccp$, if $\alpha_1,\alpha_2\in\classkinfty$ are both
convex (respectively, concave), then the compositions
$\alpha_1\circ\alpha_2$ and $\alpha_2\circ\alpha_1$ belong
to~$\classkinfty$ and are convex (respectively, concave), as
explained in Section~\ref{sec:pre-classks-convexity}.
\end{proof}
\begin{remark}\longthmtitle{The relation~$\proper$ is not an
equivalence relation}
{\rm The proof above also shows that the relation~$\proper$ is
reflexive and transitive. However, it is not symmetric: consider
$\lyap, \lyapw\in\psd$ given by $\lyap(x)=1-e^{-\norm{x}}$ and
$\lyapw(x)=\norm{x}$. Clearly, $\lyap\proper\lyapw$ in $\real^n$
by taking $\alpha_1=\alpha_2=\alpha \in\classk$, with
$\alpha(s)=1-e^{-s}$. On the other hand, if there exist
$\tilde{\alpha}_1, \tilde{\alpha}_2\in\classk$ such that
$\tilde{\alpha}_1(\lyap(x)) \le \lyapw(x) \le
\tilde{\alpha}_2(\lyap(x))$ for all $x\in\real^n$, then we reach
the contradiction, by continuity of $\tilde{\alpha}_2$, that
$\lim_{\norm{x}\to\infty}\norm{x} \le\tilde{\alpha}_2
\big(\lim_{\norm{x}\to\infty} \big(
1-e^{-\norm{x}}\big)\big)=\tilde{\alpha}_2(1)<\infty$.
} \oprocend
\end{remark}
\subsection{Characterization of proper functions with respect to each
other}\label{sec:characterizations}
In this section, we provide a complete characterization of the
properties that two functions must satisfy to be related by the
equivalence relations defined in Section~\ref{sec:equivalent-classes}.
For $\dom\subseteq \real^n$, consider
$\map{\lyapone, \psdfuncalone}{\dom}{\realnonnegative}$.
Given a real number~$p>0$, define
\begin{align*}
\phip(s)&\triangleq\sup_{\{y\in\dom\,:\, \psdfunc{y}\,\le \sqrt[p]{s}\}}
\lyapone(y),
\\
\psip(s)&\triangleq\inf_{\{y\in\dom\,:\, \psdfunc{y}\,\ge\sqrt[p]{s}\}}
\lyapone(y),
\end{align*}
for $s \ge 0$. The value $\phip(s)$ gives the supremum of the function
$\lyapone$ in the $\sqrt[p]{s}$\,-\,sublevel set of $\psdfuncalone$,
and $\psip(s)$ is the infimum of $\lyapone$ in the
$\sqrt[p]{s}$\,-\,superlevel set of $\psdfuncalone$. Thus, the
functions $\phip$ and $\psip$ satisfy
\begin{align}\label{eq:psi-phi-sandwich}
\psip\big(\psdfunc{x}^p\big)\,
=\,\!\!\!\inf_{\substack{\{y\in\dom\,:\\ \psdfunc{y}\,\ge\,
\psdfunc{x}\}}}\!\!\!\lyapone(y)\, \le\,\lyapone(x)\,
\le\,\!\!\!\sup_{\substack{\{y\in\dom\,:\\ \psdfunc{y}\,\le\,
\psdfunc{x}\}}}\!\!\!\lyapone(y)
\,=\,\phip\big(\psdfunc{x}^p\big),
\end{align}
for all $x\in\dom$, which suggests $\phip$ and $\psip$ as
pre-comparison functions to construct $\alpha_1$ and $\alpha_2$ in
Definition~\ref{def:kinf-proper-wrt-each-other}. To this end, we find
it useful to formulate the following properties of the function
$\lyapone$ with respect to $\psdfuncalone$:
$\hzero$: The set
$\{x\in\dom:\psdfunc{x} = s\}$ is nonempty for all $s\ge0$.
$\hone$: The nullsets of $\lyapone$ and $\psdfuncalone$ are the same,
i.e.,
$\setdef{x\in\dom}{\lyapone(x)=0}=\setdef{x\in\dom}{\psdfunc{x}=0}$.
$\htwo$: The function $\phione$ is locally bounded in
$\realnonnegative$ and right continuous at~$0$, and $\psione$ is
positive definite.
$\hthree$: The next limit holds: $\lim_{s\to\infty}\psione(s)=\infty$.
$\hfour$ (as a function of $p>0$): The asymptotic behavior of
$\phip$ and $\psip$ is such that $\phip(s)$ and $s^2/\psip(s)$ are
both in $\Obig(s)$ as $s\to\infty$.
The next result shows that these properties completely characterize
whether the functions $\lyapone$ and $\lyaptwo$ are related through the
equivalence relations defined in
Section~\ref{sec:equivalent-classes}. This result
generalizes~\cite[Lemma 4.3]{HKK:02} in several ways: the notions of
proper functions considered here are more general and are not necessarily
restricted to a relationship between an arbitrary function and the
distance to a compact set.
\begin{theorem}\longthmtitle{Characterizations of proper functions
with respect to each other}\label{prop:V-is-kinf-proper-seminorm}
Let $\map{\lyapone, \psdfuncalone}{\dom}{\realnonnegative}$, and
assume $\psdfuncalone$ satisfies $\hzero$. Then
\begin{enumerate}
\item $\lyapone$ satisfies $\hipartialset$ with respect to
$\psdfuncalone$ \;$\Leftrightarrow$\;
$\lyapone\,\proper\,\psdfuncalone$ in $\dom$\,;
\item $\lyapone$ satisfies $\hiset$ with respect to $\psdfuncalone$
\;$\Leftrightarrow$\; $\lyapone\properinfty\psdfuncalone$ in
$\dom$\,;
\item $\lyapone$ satisfies $\hisetfull$ with respect to
$\psdfuncalone$ for $p>0$ \;$\Leftrightarrow$\;
$\lyapone\properinftyccp\psdfuncalone^p$ in $\dom$.
\end{enumerate}
\end{theorem}
\begin{proof}
We begin by establishing a few basic facts about the pre-comparison
functions $\psip$ and $\phip$.
By definition and by $\hzero$, it follows that $0\le\psione(s)\le
\phione(s)$ for all $s\ge0$. Since $\phione$ is locally bounded by
$\htwo$, then so is $\psione$. In particular, $\phione$ and
$\psione$ are well defined in $\realnonnegative$. Moreover, both
$\phione$ and $\psione$ are nondecreasing because if $s_2\ge s_1$,
then the supremum is taken in a larger set,
$\{x\in\dom:\psdfunc{x}\le s_2\}\supseteq\{x\in\dom:\psdfunc{x}\le
s_1\}$, and the infimum is taken in a smaller set,
$\{x\in\dom:\psdfunc{x}\ge s_2\}\subseteq\{x\in\dom:\psdfunc{x}\ge
s_1\}$.
Furthermore, for any $q>0$, the functions $\phiq$ and $\psiq$ are
also monotonic and positive definite because
$\phiq(s)=\phione(\sqrt[q]{s})$ and $\psiq(s)=\psione(\sqrt[q]{s})$
for all $s\ge0$. We now use these properties of the pre-comparison
functions to construct $\alpha_1$, $\alpha_2$ in
Definition~\ref{def:kinf-proper-wrt-each-other} required by the
implications from left to right in each statement.
Proof of \emph{(i)} ($\Rightarrow$). To show the existence of
$\alpha_2\in\classk$ such that $\alpha_2(s)\ge\phione(s)$ for all
$s\in\realnonnegative$, we proceed as follows. Since $\phione$ is
locally bounded and nondecreasing, given a strictly increasing
sequence $\{b_k\}_{k\ge1}\subseteq\realnonnegative$ with
$\lim_{k\to\infty}b_k=\infty$, we choose the sequence
$\{M_k\}_{k\ge1}\subseteq\realnonnegative$, setting $M_{0}=0$, in
the following way:
\begin{align}\label{eq:def+Mk}
M_k\triangleq\max\Big\{\sup_{s\in[0, b_k]} \phione(s)\,,\:
M_{k-1}+1/k^2\Big\} =\max\big\{\phione(b_k)\,,\:
M_{k-1}+1/k^2\big\}.
\end{align}
This choice guarantees that $\{M_k\}_{k\ge1}$ is strictly increasing
and, for each~$k\ge 1$,
\begin{align}\label{eq:pi_bound}
0\le M_k-\phione(b_k)\le\sum_{i=1}^k \frac{1}{i^2}\le \pi^2/6.
\end{align}
Also, since~$\phione$ is right continuous at~$0$, we can choose
$b_1>0$ such that there exists
$\map{\alpha_2}{[0,b_1]}{\realnonnegative}$ continuous, positive
definite and strictly increasing, satisfying that
$\alpha_2(s)\ge\phione(s)$ for all $s\in[0, b_1]$ and with
$\alpha_2(b_1)=M_2$. (This is possible because the only function
that cannot be upper bounded by an arbitrary continuous function in
some arbitrarily small interval $[0,b_1]$ is the function that has a
jump at~$0$.)
The rest of the construction is explicit. We
define $\alpha_2$ as a piecewise linear function in $(b_1, \infty)$ in
the following way: for each $k\ge2$, we define
\begin{align*}
\alpha_2(s)\triangleq\alpha_2(b_{k-1}) +
\frac{M_{k+1}-\alpha_2(b_{k-1})}{b_{k}-b_{k-1}}\,(s-b_{k-1}),
\qquad\forall s\in(b_{k-1}, b_{k}].
\end{align*}
The resulting $\alpha_2$ is continuous by construction. Also,
$\alpha_2(b_1)=M_2$, so that, inductively, $\alpha_2(b_{k-1})=M_{k}$
for $k\ge2$. Two facts now follow: first,
$M_{k+1}-\alpha_2(b_{k-1})=M_{k+1}-M_{k}\ge 1/(k+1)^2$ for $k\ge2$,
so $\alpha_2$ has positive slope in each interval $(b_{k-1}, b_{k}]$
and thus is strictly increasing in $(b_1,\infty)$; second,
$\alpha_2(s)>\alpha_2(b_{k-1})= M_k\ge \phione(b_{k})\ge\phione(s)$
for all $s\in(b_{k-1}, b_{k}]$, for each $k\ge2$, so
$\alpha_2(s)\ge\phione(s)$ for all $s\in(b_1,\infty)$.
We have left to show the existence of $\alpha_1\in\classk$ such that
$\alpha_1(s)\le\psione(s)$ for all $s\in\realnonnegative$. First,
since $0\le\psione(s)\le \phione(s)$ for all $s\ge0$ by definition
and by $\hzero$, using the sandwich theorem~\cite[p. 107]{JL-ML:88},
we derive that $\psione$ is right continuous at~$0$ the same as
$\phione$. In addition, since $\psione$ is nondecreasing, it can
only have a countable number of jump discontinuities (none of them
at~$0$). Therefore, we can pick $c_1>0$ such that a continuous and
nondecreasing function $\psionetilde$ can be constructed in $[0,
c_1)$ by removing the jumps of $\psione$, so that
$\psionetilde(s)\le\psione(s)$. Moreover, since $\psione$ is
positive definite and right continuous at~$0$, then $\psionetilde$
is also positive definite. Thus, there exists $\alpha_1$ in $[0,
c_1)$ continuous, positive definite, and strictly increasing, such
that, for some $r<1$,
\begin{align}\label{eq:def-alpha1-before-c1}
\alpha_1(s)\le r\psionetilde(s)\le
r\psione(s)
\end{align}
for all $s\in [0, c_1)$. To extend $\alpha_1$ to a function in
class~$\classk$ in~$\realnonnegative$, we follow a similar strategy
as for $\alpha_2$. Given a strictly increasing sequence
$\{c_k\}_{k\ge 2}\subseteq\realnonnegative$ with $\lim_{k\to\infty}
c_k=\infty$, we define a sequence
$\{m_k\}_{k\ge1}\subseteq\realnonnegative$ in the following way:
\begin{align}\label{eq:def+mk}
m_k\triangleq\inf_{s\in [c_{k}, c_{k+1})} \psione(s)
-\tfrac{\psione(c_1)-\alpha_1(c_1)}{1+k^2}
=\psione(c_{k})-\tfrac{\psione(c_1)-\alpha_1(c_1)}{1+k^2}.
\end{align}
Next we define $\alpha_1$ in
$[c_1, \infty)$ as the piecewise linear function
\begin{align*}
\alpha_1(s)\triangleq\alpha_1(c_{k}) +
\frac{m_{k}-\alpha_1(c_{k})}{c_{k+1}-c_{k}}\,(s-c_{k}), \qquad\forall
s\in[c_{k}, c_{k+1}),
\end{align*}
for all $k\ge 1$, so $\alpha_1$ is continuous by construction. It is
also strictly increasing because
$\alpha_1(c_2)=m_1=(\psione(c_1)+\alpha_1(c_1))/2>\alpha_1(c_1)$
by~\eqref{eq:def-alpha1-before-c1}, and also, for each $k\ge 2$, the
slopes are positive because $m_{k}-\alpha_1(c_{k})=m_{k}-m_{k-1}>0$
(due to the fact that $\{m_k\}_{k\ge 1}$ in~\eqref{eq:def+mk} is
strictly increasing because $\psione$ is nondecreasing). Finally,
$\alpha_1(s)<\alpha_1(c_{k+1})= m_k<\psione(c_k)\le \psione(s)$ for
all $s\in[c_{k}, c_{k+1})$, for all $k\ge 1$ by~\eqref{eq:def+mk}.
Equipped with $\alpha_1$, $\alpha_2$ as defined above, and as a
consequence of~\eqref{eq:psi-phi-sandwich}, we have that
\begin{align}\label{eq:application-psi-phi-sandwich}
\alpha_1(\psdfunc{x})&\,\le\psione(\psdfunc{x}) \le \lyapone(x)
\le\phione(\psdfunc{x})\le\alpha_2(\psdfunc{x}),\quad\forall
x\in\dom.
\end{align}
This concludes the proof of~\emph{(i)}~($\Rightarrow$).
As a preparation for \emph{(ii)-(iii)}~($\Rightarrow$), and
assuming~$\hthree$, we derive two facts regarding the functions
$\alpha_1$ and $\alpha_2$ constructed above.
First, we establish that
\begin{align}\label{eq:asymptotic-phione-alpha}
\alpha_2(s)\in\Obig(\phione(s))\;\,\text{as}\;\, s\to\infty.
\end{align}
To show this, we argue that
\begin{align}\label{eq:ingredient-slackness}
\lim_{k\to\infty}\,\sup_{s\in(b_{k-1}, b_{k}]}
\big(\alpha_2(s)-\phione(s)\big) \le\lim_{k\to\infty}
\big(\phione(b_{k+1})-\phione(b_{k-1})\big)+\pi^2/6,
\end{align}
so that there exist $C, s_1>0$ such that $\alpha_2(s)\le
3\phione(s)+C$, for all $s\ge s_1$. Thus, noting that
$\lim_{s\to\infty} \phione(s)=\infty$ as a consequence of $\hthree$,
the expression~\eqref{eq:asymptotic-phione-alpha} follows.
To establish~\eqref{eq:ingredient-slackness}, we use the
monotonicity of $\alpha_2$ and $\phione$,~\eqref{eq:def+Mk}
and~\eqref{eq:pi_bound}. For $k\ge 2$,
\begin{align*}
&\sup_{s\in(b_{k-1}, b_{k}]}
\big(\alpha_2(s)-\phione(s)\big)\le\alpha_2(b_k)
-\phione(b_{k-1})=M_{k+1}-\phione(b_{k-1})
\\
=\,&\max\big\{\phione(b_{k+1})-\phione(b_{k-1})\,,\:
M_{k}+1/(k+1)^2-\phione(b_{k-1})\big\}
\\
\le\, &\max\big\{\phione(b_{k+1})-\phione(b_{k-1})\,,\:
\phione(b_k)+\pi^2/6+1/(k+1)^2-\phione(b_{k-1})\big\}.
\end{align*}
Second,
the construction of $\alpha_1$ guarantees that
\begin{align}\label{eq:asymptotic-psione-alpha}
\psione(s)\in\Obig(\alpha_1(s))\;\, \text{as}\;\, s\to\infty,
\end{align}
because, as we show next,
\begin{align} \label{eq:ingredient-slackness-psi}
\lim_{k\to\infty}\,\sup_{s\in[c_{k}, c_{k+1})}
\big(\psione(s)-\alpha_1(s)\big) \le\lim_{k\to\infty}
\big(\alpha_1(c_{k+2})-\alpha_1(c_{k})\big),
\end{align}
so there exists $s_2>0$ such that $\psione(s)\le 3\alpha_1(s)$ for
all $s\ge s_2$. To obtain~\eqref{eq:ingredient-slackness-psi}, we
leverage the monotonicity of $\psione$ and $\alpha_1$,
and~\eqref{eq:def+mk}; namely, for $k\ge2$,
\begin{align*}
\,&\sup_{s\in[c_{k}, c_{k+1})} \big(\psione(s)-\alpha_1(s)\big)
\le\psione(c_{k+1})-\alpha_1(c_{k}) \\=\,&
m_{k+1}+\tfrac{\psione(c_1)-\alpha_1(c_1)}{1+(k+1)^2}-\alpha_1(c_{k})
= \alpha_1(c_{k+2})+\tfrac{\psione(c_1)-\alpha_1(c_1)}{1+(k+1)^2} -
\alpha_1(c_{k}).
\end{align*}
Equipped with~\eqref{eq:asymptotic-phione-alpha}
and~\eqref{eq:asymptotic-psione-alpha}, we prove next
\emph{(ii)-(iii)}~($\Rightarrow$).
Proof of \emph{(ii)}~($\Rightarrow$): If, in addition,~$\hthree$ holds,
then $\lim_{s\to\infty} \phione(s)\ge\lim_{s\to\infty}
\psione(s)=\infty$. This guarantees that
$\alpha_2\in\classkinfty$. Also, according
to~\eqref{eq:asymptotic-psione-alpha}, $\hthree$ implies that
$\alpha_1$ is unbounded, and thus in $\classkinfty$ as well. The
result now follows by~\eqref{eq:application-psi-phi-sandwich}.
Proof of \emph{(iii)}~($\Rightarrow$): Finally, assume that $\hfour$
also holds for some $p>0$. We show next the existence of the
required convex and concave functions involved in the
relation~$\properinftyccp$.
Let $\alpha_{1,p}(s)\triangleq\alpha_1(\sqrt[p]{s})$ and
$\alpha_{2,p}(s)\triangleq\alpha_2(\sqrt[p]{s})$ for $s\ge 0$, so
that
\begin{align*}
\alpha_{1,p}(s)=\alpha_1(\sqrt[p]{s})\le\psione(\sqrt[p]{s})=\psip(s)
\quad\text{and}\quad
\phip(s)=\phione(\sqrt[p]{s})\le\alpha_2(\sqrt[p]{s})=\alpha_{2,p}(s).
\end{align*}
From~\eqref{eq:asymptotic-phione-alpha} and~$\hfour$, it follows
that there exist $s'$, $c_1$, $c_2>0$ such that $\alpha_2(s)\le c_1
\phione(s)$ and $\phip(s)\le c_2 s$ for all $s\ge s'$. Thus,
\begin{align*}
\alpha_{2,p}(s)=\alpha_2(\sqrt[p]{s})\le c_1
\phione(\sqrt[p]{s})=c_1 \phip(s)\le c_1 c_2 s,
\end{align*}
for all $s\ge s'$, so $\alpha_{2,p}(s)$ is in $\Obig(s)$ as
$s\to\infty$. Similarly, according to~\eqref{eq:asymptotic-psione-alpha}
and~$\hfour$, there are constants $s''$, $c_3$, $c_4>0$ such
that $\psione(s)\le c_3\, \alpha_1(s)$ and $s^2\le c_4 s\, \psip(s)$
for all $s\ge s''$. Thus,
\begin{align*}
s\, \alpha_{1,p}(s)=s\, \alpha_1(\sqrt[p]{s})\ge s\,\tfrac{1}{c_3}
\psione(\sqrt[p]{s}) =s\,\tfrac{1}{c_3} \psip(s)\ge \tfrac{1}{c_3
c_4} s^2,
\end{align*}
for all $s\ge s''$, so $s^2/\alpha_{1,p}(s)$ is in $\Obig(s)$ as
$s\to\infty$. Summarizing, the construction of $\alpha_1$,
$\alpha_2$ guarantees, under $\hfour$, that $\alpha_{1,p}$,
$\alpha_{2,p}$ satisfy that $s^2/\alpha_{1,p}(s)$ and
$\alpha_{2,p}(s)$ are in $\Obig(s)$ as $s\to\infty$ (and, as a
consequence, so are $s^2/\alpha_{2,p}(s)$ and
$\alpha_{1,p}(s)$). Therefore, according to
Lemma~\ref{le:convex-concave}, we can
leverage~\eqref{eq:application-psi-phi-sandwich} by taking
$\tilde{\alpha}_1$, $\tilde{\alpha}_2\in\classkinfty$, convex
and concave, respectively, such that, for
all~$x\in\dom$,
\begin{align*}
\tilde{\alpha}_1\big(\psdfunc{x}^p\big)
\le&\,\alpha_{1,p}(\psdfunc{x}^p) =\alpha_1(\psdfunc{x})
\le
\psione\big(\psdfunc{x}\big)\le\lyapone(x)
\\
\le&\,\phione\big(\psdfunc{x}\big)
\le\alpha_2(\psdfunc{x}) =\alpha_{2,p}(\psdfunc{x}^p)
\le\tilde{\alpha}_2\big(\psdfunc{x}^p\big).
\end{align*}
Proof of \emph{(i)}~($\Leftarrow$): If there exist class~$\classk$
functions $\alpha_1$, $\alpha_2$ such that $\alpha_1(\psdfunc{x})\le
\lyapone(x)\le\alpha_2(\psdfunc{x})$ for all $x\in\dom$, then the
nullsets of $\lyapone$ and $\psdfuncalone$ are the same, which is
the property~$\hone$. In addition,
$0\le\phione(s)\le\alpha_2(s)$ for all $s\ge 0$, so $\phione$ is
locally bounded and, moreover, the sandwich theorem guarantees that
$\phione$ is right continuous at~$0$. Also, since
$\alpha_1(s)\le\psione(s)$, for all $s\ge 0$, and $\psione(0)=0$, it
follows that $\psione$ is positive definite. Therefore, $\htwo$ also
holds.
Proof of \emph{(ii)}~($\Leftarrow$): Since
$\psione(s)\ge\alpha_1(s)$ for all $s\ge 0$, the property~$\hthree$
follows because
\begin{align*}
\lim_{s\to\infty}\psione(s)\ge\lim_{s\to\infty}\alpha_1(s)=\infty.
\end{align*}
Proof of \emph{(iii)}~($\Leftarrow$): If
$\lyapone\properinftyccp\psdfuncalone^p$, then
$\lyapone\properinfty\psdfuncalone^p$
by~\eqref{eq:chain-relations}. Also, we have trivially that
$\psdfuncalone^p\properinfty\psdfuncalone$. Since~$\properinfty$ is
an equivalence relation by Lemma~\ref{le:kinf-equivalence-relation},
it follows that $\lyapone\properinfty\psdfuncalone$, so the
properties $\hiset$ hold as in \emph{(ii)}~($\Leftarrow$). We have
left to derive $\hfour$. If
$\lyapone\properinftyccp\psdfuncalone^p$, then there exist
$\alpha_1,\alpha_2\in\classkinfty$ convex and concave, respectively,
such that $\alpha_1\big(\psdfunc{x}^p\big)\le\lyapone(x) \le
\alpha_2\big(\psdfunc{x}^p\big)$ for all $x\in\dom$. Hence, by the
definition of $\psip$ and $\phip$, and~$\hzero$, and by the
monotonicity of $\alpha_1$ and $\alpha_2$, we have that, for all
$s\ge 0$,
\begin{align}\label{eq:psis-and-alphas}
\alpha_1(s) & \le \inf_{\{x\in\dom\,:\,\psdfunc{x}^p\ge s\}}
\alpha_1\big(\psdfunc{x}^p\big)
\le\inf_{\{x\in\dom\,:\,\psdfunc{x}^p\ge s\}} \lyapone(x)
=\psip(s)\nonumber
\\
& \le\, \phip(s) =\sup_{\{x\in\dom\,:\, \psdfunc{x}^p\le
s\}}\lyapone(x) \le\sup_{\{x\in\dom\,:\, \psdfunc{x}^p\le
s\}}\alpha_2\big(\psdfunc{x}^p\big) \le\alpha_2(s).
\end{align}
Now, since $\alpha_1$,$\alpha_2\in\classkinfty$ are convex and
concave, respectively, it follows by Lemma~\ref{le:convex-concave}
that $s^2/\alpha_1(s)$ and $\alpha_2(s)$ are in $\Obig(s)$ as
$s\to\infty$. Knowing from~\eqref{eq:psis-and-alphas} that
$\alpha_1(s)\le \psip(s)\le\phip(s)\le\alpha_2(s)$ for all $s\ge 0$,
we conclude that the functions $s^2/\psip(s)$ and $\phip(s)$ are
also in $\Obig(s)$ as $s\to\infty$, which is the property~$\hfour$.
\end{proof}
The following example shows ways in which the conditions of
Theorem~\ref{prop:V-is-kinf-proper-seminorm} might fail.
\begin{example}\longthmtitle{Illustration
of~Theorem~\ref{prop:V-is-kinf-proper-seminorm}}
{\rm Let $\map{\lyaptwo}{\real^2}{\realnonnegative}$ be the distance
to the set $\setdef{(x_1,x_2)\in\real^2}{x_1=0}$, i.e.,
$\lyaptwo(x_1,x_2)=\absolute{x_1}$. Consider the following cases:
\emph{P2 fails ($\psione$ is not positive definite):} Let
$\lyapone(x_1,x_2)=\absolute{x_1}e^{-\absolute{x_2}}$ for
$(x_1,x_2)\in\real^2$. Note that $\lyapone$ is not
$\classk$\,-\,dominated by $\psdfuncalone$ because, given any
$\alpha_1\in\classk$, for every $x_1\in\real$ with
$\absolute{x_1}>0$ there exists $x_2\in\real$ such that the
inequality $\alpha_1(\absolute{x_1})\le
\absolute{x_1}e^{-\absolute{x_2}}$ does not hold (just choose
$x_2$ satisfying $\absolute{x_2} >
\log\big(\tfrac{\absolute{x_1}}{\alpha_1(\absolute{x_1})}\big)$). Thus,
there must be some of the hypotheses on
Theorem~\ref{prop:V-is-kinf-proper-seminorm} that fail to be
true. In this case, we observe that
\begin{align*}
\psione(s)=\inf_{\{(x_1,x_2)\in\real^2\,:\, \absolute{x_1}\,\ge
s\}} \absolute{x_1}e^{-\absolute{x_2}}
\end{align*}
is identically~$0$ for all $s\ge 0$, so it is not positive
definite as required in~$\htwo$.
\emph{P2 fails ($\phione$ is not locally bounded):} Let
$\lyapone(x_1,x_2)=\absolute{x_1}e^{\absolute{x_2}}$ for
$(x_1,x_2)\in\real^2$. As above, one can show that $\alpha_2$ does
not exist in the required class; in this case, the
hypothesis~$\htwo$ is not satisfied because $\phione$ is not
locally bounded in $(0,\infty)$:
\begin{align*}
\phione(s)=\sup_{\{(x_1,x_2)\in\real^2\,:\, \absolute{x_1}\,\le
s\}} \absolute{x_1}e^{\absolute{x_2}}=\infty, \quad \forall\;
s>0.
\end{align*}
\emph{P2 fails ($\phione$ is not right continuous):} Let
$\lyapone(x_1,x_2)=\absolute{x_1}^4 +\absolute{\sin{(x_1
x_2)}}$ for $(x_1,x_2)\in\real^2$.
For every $p>0$, we have that
\begin{align*}
\phip(s)=\sup_{\{(x_1,x_2)\in\real^2\,:\, \absolute{x_1}^p\,\le
s\}} \absolute{x_1}^4 +\absolute{\sin{(x_1 x_2)}}\le
s^{4/p}+1,
\end{align*}
so $\phip$ is locally bounded in $\realnonnegative$, and, again
for every $p>0$,
\begin{align*}
\psip(s)=\inf_{\{(x_1,x_2)\in\real^2\,:\, \absolute{x_1}^p\,\ge
s\}} \absolute{x_1}^4 +\absolute{\sin{(x_1 x_2)}}\ge s^{4/p},
\end{align*}
so $\psip$ is positive definite. However, $\phip$ is not right
continuous at~$0$ because $\sin{(x_1 x_2)}=0$ when $x_1=0$, but
$\sup_{\{(x_1,x_2)\in\real^2\,:\, \absolute{x_1}^p\,\le s_0\}}
\sin{(x_1 x_2)}=1$ for any $s_0>0$, so by
Theorem~\ref{prop:V-is-kinf-proper-seminorm}~\emph{(i)}, it
follows that $\lyapone$ is not $\classk$\,-\,dominated by
$\psdfuncalone$.
\emph{P4 fails (non-compliant asymptotic behavior):} Let
$\lyapone(x_1,x_2)=\absolute{x_1}^4$ for $(x_1,x_2)\in\real^2$. Then $\htwo$ is satisfied
and $\hthree$ also holds because
$\lim_{s\to\infty}\psione(s)=\lim_{s\to\infty} s^4=\infty$, so
Theorem~\ref{prop:V-is-kinf-proper-seminorm}~\emph{(ii)} implies that
$\lyapone$ and $\lyaptwo$ are $\classkinfty$-proper with respect
to each other. However, in this case $\phip(s)=\psip(s)=s^{4/p}$,
which implies that $\phip$ is not in $\Obig(s)$ as $s\to\infty$
when $p\in(0,4)$, and $s^2/\psip(s)$ is not in $\Obig(s)$ as
$s\to\infty$ when~$p>4$. Thus $\hfour$ is satisfied only for
$p=4$, so Theorem~\ref{prop:V-is-kinf-proper-seminorm}~\emph{(iii)} implies
that only in this case $\lyapone$ and $\lyaptwo^p$ are
$\classkinftyccp$- proper with respect to each other. Namely,
for $p>4$, one cannot choose a convex
$\alpha_1\in\classkinfty$ such that $\alpha_1(\absolute{x_1}^p)\le
\absolute{x_1}^4$ for all $x_1\in\real$ and, if
$p<4$, one cannot choose a concave $\alpha_2 \in
\classkinfty$ such that $\absolute{x_1}^4 \le
\alpha_2(\absolute{x_1}^p)$ for all $x_1\in\real$.
\oprocend}
\end{example}
\subsection{Application to noise-to-state
stability}\label{sec:alternative}
In this section we use the results of
Sections~\ref{sec:equivalent-classes} and~\ref{sec:characterizations}
to study the noise-to-state stability properties of stochastic
differential equations of the
form~\eqref{eq:nonlinear-SDE-preliminaries}. Our first result
provides a way to check whether a candidate function that satisfies a
dissipation inequality of the
type~\eqref{eq:theorem-second-hypothesis-VandW} is in fact a
noise-dissipative Lyapunov function, a strong NSS-Lyapunov function in
probability, or a $p$th~moment NSS-Lyapunov function.
\begin{corollary}\longthmtitle{Establishing proper relations between
pairs of functions through
seminorms}\label{co:kinf-proper-wrt-each-other}
Consider $\map{\lyap_1, \lyap_2}{\dom}{\realnonnegative}$ such that
their nullset is a subspace~$\Uset$. Let $A,\Atilde \in
\realmatricesrectangulararg{m}{n}$ be such that $\kernel(A) = \Uset
= \kernel(\Atilde)$. Assume that $\lyap_1$ and $\lyap_2$ satisfy
$\{\operatorname{P}$i$\}_{i=0}^3$ with respect to $\seminorm{.}{A}$
and $\seminorm{.}{\Atilde}$, respectively. Then, for any $q>0$,
\begin{align*}
\lyap_1\properinfty\lyap_2, \quad \lyap_1\properinfty
\seminorm{.}{A}^q,
\quad
\lyap_2\properinfty\seminorm{.}{\Atilde}^q\quad\text{in}\quad\dom.
\end{align*}
If, in addition, $\lyap_1$ and $\lyap_2$ satisfy $\hfour$ with
respect to $\seminorm{.}{A}$ and $\seminorm{.}{\Atilde}$,
respectively, for some $p>0$, then
\begin{align*}
\lyap_1\properinftyccp\lyap_2,
\quad\lyap_1\properinftyccp\seminorm{.}{A}^p,
\quad
\lyap_2\properinftyccp\seminorm{.}{\Atilde}^p
\quad\text{in}\quad\dom.
\end{align*}
\end{corollary}
\begin{proof}
The statements follow from the characterizations in
Theorem~\ref{prop:V-is-kinf-proper-seminorm}~\emph{(ii)}~and~\emph{(iii)},
and from the fact that the relations $\properinfty$ and
$\properinftyccp$ are equivalence relations as shown in
Lemma~\ref{le:kinf-equivalence-relation}. That is, under the
hypothesis~$\hzero$,
\begin{align*}
\!\!\left.\begin{array}{c}
\!\!\lyap_1\:\text{satisfies}\:\hiset\;\text{w/ respect to}\;
\seminorm{.}{A}\; (\Leftrightarrow\;
\lyap_1\properinfty\seminorm{.}{A}\;\text{in}\;\dom)
\\
\!\!\lyap_2\:\text{satisfies}\:\hisettilde\;\text{w/ respect to}\;
\seminorm{.}{\Atilde}\; (\Leftrightarrow\;
\lyap_2\properinfty\seminorm{.}{\Atilde}\;\text{in}\;\dom)
\end{array}
\!\!\right\}\Rightarrow
\lyap_1\properinfty\lyap_2\;\text{in}\;\dom,
\\
\!\!\left.\begin{array}{c}
\!\!\lyap_1\:\text{satisfies}\:\hisetfull\;\text{w/ respect to}\;
\seminorm{.}{A}\; (\Leftrightarrow\;
\lyap_1\properinftyccp\seminorm{.}{A}^p\;\text{in}\;\dom)
\\
\!\!\lyap_2\:\text{satisfies}\:\hisetfulltilde\;\text{w/
respect to}\;
\seminorm{.}{\Atilde}\; (\Leftrightarrow\;
\lyap_2\properinftyccp\seminorm{.}{\Atilde}^p\;\text{in}\;\dom)
\end{array}
\!\!\right\}\Rightarrow
\lyap_1\properinftyccp\lyap_2\;\text{in}\;\dom.
\end{align*}
Note that, by
Lemma~\ref{le:kinf-proper-semidefinite-quadratic-forms}
and~\eqref{eq:chain-relations}, the equivalences
\begin{align*}
\seminorm{.}{A}\properinfty\seminorm{.}{\Atilde}^q\;\;\;\text{in}\;\;\dom,
\qquad
\seminorm{.}{A}^p\properinftyccp\seminorm{.}{\Atilde}^p\;\;\;\text{in}\;\;
\dom
\end{align*}
hold for any $p,q>0$ and any matrices
$A,\,\Atilde\in\realmatricesrectangulararg{m}{n}$ with
$\kernel(A)=\kernel(\Atilde)$.
\end{proof}
We next build on this result to provide an alternative formulation of
Corollary~\ref{co:of-the-main-theorem}. To do so, we employ the
observation made in Remark~\ref{re:another-set-assumptions} about the
possibility of interpreting the candidate functions as defined on a
constrained domain of an extended Euclidean space.
\begin{corollary}\longthmtitle{The existence of a $p$thNSS-Lyapunov
function implies $p$th~moment NSS --revisited}\label{co:alternative}
Under Assumption~\ref{ass:assumptions-SDE}, let $\lyap\in\psdtwice$,
$\lyapw\in\psd$ and $\gainnoise\in\classk$ be such that the
dissipation inequality~\eqref{eq:theorem-hypothesis-ito} holds. Let
$\map{R}{\real^n}{\real^{(m-n)}}$, with $m\ge n$, $\dom \subset
\real^m$, $\lyaphat\in\psddomaintwice$ and $\lyapwhat\in\psddomain$
be such that, for $i(x) = [x\tp,R(x)\tp]\tp$, one has
\begin{align*}
\dom = i(\real^n), \quad \lyap = \lyaphat \circ i, \quad
\text{and} \quad \lyapw = \lyapwhat \circ i .
\end{align*}
Let $A = \diag(A_1, A_2)$ and $\Atilde = \diag(\Atilde_1,\Atilde_2)$
be block-diagonal matrices, with $A_1, \Atilde_1\in\realmatrices$
and $A_2, \Atilde_2\in\realmatricesrectangulararg{(m-n)}{(m-n)}$,
such that $\kernel(A)=\kernel(\Atilde)$ and
\begin{align}\label{eq:bound-R}
\seminorm{R(x)}{A_2}^2\le \kappa\seminorm{x}{A_1}^2
\end{align}
for some $\kappa>0$, for all $x\in\real^n$. Assume that $\lyaphat$
and $\lyapwhat$ satisfy the properties
$\{\operatorname{P}$i$\}_{i=0}^4$ with respect to $\seminorm{.}{A}$
and $\seminorm{.}{\Atilde}$, respectively, for some $p>0$. Then the
system~\eqref{eq:nonlinear-SDE-preliminaries} is NSS in probability
and in $p$th~moment with respect to~$\kernel(A_1)$.
\end{corollary}
\begin{proof}
By Corollary~\ref{co:kinf-proper-wrt-each-other}, we have that
\begin{align}\label{eq:co-other-assumptions}
\lyaphat\properinftyccp\lyapwhat, \quad \text{and} \quad
\lyaphat\properinftyccp\seminorm{.}{\diag(A_1,
A_2)}^p\quad\text{in}\quad\dom.
\end{align}
As explained in Remark~\ref{re:another-set-assumptions}, the first
relation implies that $\lyap\properinftyccp\lyapw$ in
$\real^n$. This, together with the fact
that~\eqref{eq:theorem-hypothesis-ito} holds, implies that $\lyap$
is a noise-dissipative Lyapunov function
for~\eqref{eq:nonlinear-SDE-preliminaries}. Also, setting
$\hat{x}=i(x)$ and using~\eqref{eq:bound-R}, we obtain that
\begin{align*}
\seminorm{x}{A_1}^2\le\seminorm{\hat{x}}{\diag(A_1,
A_2)}^2=\seminorm{x}{A_1}^2+\seminorm{R(x)}{A_2}^2\le
(1+\kappa)\seminorm{x}{A_1}^2,
\end{align*}
so, in particular, $\seminorm{[\,. ,R(.)]}{\diag(A_1,
A_2)}^p\relationconstants\seminorm{.}{A_1}^p$ in~$\real^n$.
Now, from the second relation in~\eqref{eq:co-other-assumptions}, by
Remark~\ref{re:another-set-assumptions}, it follows that $\lyaphat
\circ\, i\properinftyccp\seminorm{[\,.,R(.)]}{\diag(A_1, A_2)}^p$ in
$\real^n$. Thus, using~\eqref{eq:chain-relations} and
Lemma~\ref{le:kinf-equivalence-relation}, we conclude that
$\lyap\properinftyccp\seminorm{.}{A_1}^p$ in $\real^n$. In addition,
the Euclidean distance to the set $\kernel(A_1)$ is equivalent to
$\seminorm{.}{A_1}$, i.e.,
$\distset{.}{\kernel(A_1)}\relationconstants\seminorm{.}{A_1}$. This
can be justified as follows: choose
$B\in\realmatricesrectangulararg{n}{k}$, with
$k=\dim(\kernel(A_1))$, such that the columns of $B$ form an
orthonormal basis of~$\kernel(A_1)$. Then,
\begin{align}
\distset{x}{\kernel(A_1)}=\norm{(\identity -B
B\tp)x}=\seminorm{x}{\identity -B
B\tp}\relationconstants\seminorm{.}{A_1},
\end{align}
where the last relation follows from
Lemma~\ref{le:kinf-proper-semidefinite-quadratic-forms} because
$\kernel(\identity -B B\tp)=\kernel(A_1)$. Summarizing,
$\lyap\properinftyccp\seminorm{.}{A_1}^p$ and
$\seminorm{.}{A_1}^p\relationconstants \distset{x}{\kernel(A_1)}^p$
in $\real^n$ (because the $p$th power is irrelevant for the
relation~$\relationconstants$). As a consequence,
\begin{align}\label{eq:proper-alternative-hypotheses}
\lyap\properinftyccp\distset{.}{\kernel(A_1)}^p\quad
\text{in}\quad\real^n,
\end{align}
which implies condition~\eqref{eq:theorem-third-hypothesis-V} with
convex $\alpha_1\in\classkinfty$, concave $\alpha_2\in\classkinfty$,
and $\Uset=\kernel(A_1)$. Therefore, $\lyap$ is a $p$th moment
NSS-Lyapunov function with respect to the set~$\kernel(A_1)$, and
the result follows from Corollary~\ref{co:of-the-main-theorem}.
\end{proof}
\section{Conclusions}\label{sec:conclusions-future}
We have studied the stability properties of SDEs subject to persistent
noise (including the case of additive noise). We have generalized the
concept of noise-dissipative Lyapunov function and introduced the
concepts of strong NSS-Lyapunov function in probability and
$p$th~moment NSS-Lyapunov function, both with respect to a closed set.
We have shown that noise-dissipative Lyapunov functions have NSS
dynamics and established that the existence of an NSS-Lyapunov
function, of either type, with respect to a closed set, implies the
corresponding NSS property of the system with respect to the set. In
particular, $p$th moment NSS with respect to a set provides a bound,
at each time, for the $p$th~power of the distance from the state to
the set, and this bound is the sum of an increasing function of the
size of the noise covariance and a decaying effect of the initial
conditions. This bound can be achieved regardless of the possibility
that inside the set some combination of the states accumulates the
variance of the noise. This is a meaningful stability property for the
aforementioned class of systems because the presence of persistent
noise makes it impossible to establish in general a stochastic notion
of asymptotic stability for the set of equilibria of the underlying
differential equation. We have also studied in depth the inequalities
between pairs of functions that appear in the various notions of
Lyapunov functions mentioned above. We have shown that these
inequalities define equivalence relations and have developed a
complete characterization of the properties that two functions must
satisfy to be related by them. Finally, building on this
characterization, we have provided an alternative statement of our
stochastic stability results.
Future work will include the study of the effect of delays and
impulsive right-hand sides in the class of SDEs considered in this
paper.
\section*{Acknowledgments}
The first author would like to thank Dean Richert for useful
discussions. In addition,
the authors would like to thank Dr. Fengzhong Li for his kind observations that have made possible an important correction of the proof of Theorem~\ref{th:Stability-Non-Linear-Systems}.
The research was supported by NSF award CMMI-1300272. | 79,326 |
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TITLE: If two subgroups have a complete set of left coset representatives in common, then
QUESTION [5 upvotes]: Let $H,K$ be proper subgroups of a group $G$ having a complete set $S$ of representatives of left cosets in common, that is,
$$
G = \bigsqcup_{s \in S} sH = \bigsqcup_{s \in S} sK
$$
It seems in general one cannot expect any serious relation on $H,K.$ But I am afraid of overlooking some general result here. Any information on the subject will be warmly accepted. Regards, Olod
REPLY [3 votes]: |S| = [G:H] = [G:K], so at least their indexes are equal. I don't think much else is true, because of the following example:
If G = 2 × 2 is the Klein four group, then H = 2 × 1 and K = 1 × 2 are two subgroups with a common set of coset representatives: S = { (0,0), (1,1) }. H and K are not conjugate.
In general if G is a semi-direct product H⋉N, then H has S=N as a set of coset representatives. It is very possible for N to have more than one "complement" H, that is, another subgroup K such that G=K⋉N. In nice situations like G nonabelian of order 6, all complements are conjugate, but in general they need not be as the dihedral 2-groups (including the Klein four group) show.
It would be nice to have an example where H and K are not even isomorphic, but semi-direct products won't do that. I don't believe H, K need to be complemented to have a common transversal, but I guess that is another reasonable guess to rule out.
Edit: Well, the dihedral group G of order 8 has a cyclic normal subgroup H and Klein four normal subgroup K, the union of which is not all of G. Since a set of coset representatives S has only 2 elements, we just need to take the identity and an element neither in H nor K. In particular, H need not be isomorphic to K.
Also the dihedral group of order 16 has a similar pair (H cyclic order 4, K a four-group), and so neither H nor K need be complemented.
One can also use Abelian examples as Arturo points out, and one can extend the D8 example to S4 as Steve points out. A fun problem.
REPLY [2 votes]: If $S$ is a complete set of (left) coset representatives for $H$, then for every $x\in G$ there exists $s\in S$ such that $xH=sH$, and moreover, if $s_1,s_2\in S$ are such that $s_1H=s_2H$, then $s_1=s_2$. That is: there is one and exactly one representative from each coset of $H$ in $G$ in the set $S$. As such, your condition is trivially satisfied under the assumption that $S$ is a complete set of coset representatives for both $H$ and $K$, since for $s_1,s_2\in S$, you have
$$s_1H=s_2H \Longleftrightarrow s_1=s_2\Longleftrightarrow s_1K=s_2K.$$
So either you meant something else, or you are just asking for condition under which two subgroups $H$ and $K$ can have the same complete set of coset representatives.
Jack Schmidt already gave an example with $H$ and $K$ not conjugate, and asked if there is an example in which $H$ and $K$ are not isomorphic. I think this does it: take $G=\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_4$, the product of two cyclic groups of order two and one of order four. Take $H={0}\times{0}\times\mathbb{Z}_4$, and $K={0}\times\mathbb{Z}_2\times\langle 2\rangle$ (so $H$ is cyclic of order $4$, and $K$ is the Klein $4$-group). Let $S={(0,0,1), (1,0,1), (1,1,0), (0,1,0)}$. If I did not make some silly mistake, then this is a complete set of coset representatives for both $H$ and $K$. | 41,654 |
\begin{document}
\begin{abstract}
The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem
by Edmonds and Fulkerson. A packing for a family $ (M_i: i\in\Theta) $ of matroids on the common
edge set $ E $ is a system $ (S_i: i\in\Theta ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is panning in $ M_i $. Similarly,
a covering is a system $ (I_i: i\in\Theta ) $ with $\bigcup_{i\in\Theta} I_i=E $ where $ I_i $ is independent in $ M_i $. The
conjecture states that for every matroid family on $ E $ there is a partition $E=E_p \sqcup E_c$ such that
$ (M_i \upharpoonright E_p: i\in \Theta) $ admits a packing and $ (M_i. E_c: i\in \Theta) $ admits a covering. We prove the
special case where $ E $ is countable and each $ M_i $ is either finitary or cofinitary. The connection between packing/covering
and matroid intersection problems discovered by Bowler and Carmesin can
be established for every well-behaved matroid class. This makes possible to approach the problem from the direction of matroid
intersection. We show that the generalized version of Nash-Williams' Matroid Intersection Conjecture holds for
countable matroids having only finitary and cofinitary
components.
\end{abstract}
\maketitle
\section{Introduction}
Rado asked in 1966 (see Problem P531 in \cite{rado1966abstract}) if it is possible to extend the concept of matroids to infinite
without loosing
duality and minors. Based on the works of Higgs (see \cite{higgs1969matroids}) and
Oxley (see \cite{oxley1978infinite} and \cite{oxley1992infinite}) Bruhn, Diestel, Kriesell, Pendavingh and Wollan
settled Rado's problem affirmatively in \cite{bruhn2013axioms} by finding a
set of cryptomorphic axioms for infinite matroids, generalising the usual independent set-, bases-, circuit-, closure- and
rank-axioms for finite mastoids. Higgs named originally these structures B-matroids to distinguish from the original concept.
Later this terminology vanished and in the context of infinite combinatorics B-matroids are referred as matroids and the term
`finite matroid' is used to
differentiate.
An $ M=(E, \mathcal{I}) $ is a matroid if $ \mathcal{I}\subseteq \mathcal{P}(E) $ with
\begin{enumerate}
[label=(\arabic*)]
\item\label{item axiom1} $ \varnothing\in \mathcal{I} $;
\item\label{item axiom2} $ \mathcal{I} $ is downward closed;
\item\label{item axiom3} For every $ I,J\in \mathcal{I} $ where $J $ is $ \subseteq $-maximal in $ \mathcal{I} $ but $ I $ is
not, there exists an $ e\in J\setminus I $ such that
$ I+e\in \mathcal{I} $;
\item\label{item axiom4} For every $ X\subseteq E $, any $ I\in \mathcal{I}\cap
\mathcal{P}(X) $ can be extended to a $ \subseteq $-maximal element of
$ \mathcal{I}\cap \mathcal{P}(X) $.
\end{enumerate}
If $ E $ is finite, then \ref{item axiom4} is redundant and \ref{item axiom1}-\ref{item axiom3} is one of the usual
axiomatizations
of finite matroids. One can show that every dependent set in an infinite matroid contains a minimal dependent set which is called
a circuit. Before Rado's program
was settled, a more restrictive axiom was used as a replacement of \ref{item axiom4}:
\begin{enumerate}[label={(4')}]
\item\label{item axiom4'} If all the finite subsets of an $ X\subseteq E $ are in $ \mathcal{I} $, then $ X\in \mathcal{I} $.
\end{enumerate}
The implication \ref{item axiom4'}$ \Longrightarrow $\ref{item axiom4} follows directly from Zorn's lemma thus axioms
\ref{item axiom1}, \ref{item axiom2}, \ref{item axiom3} and \ref{item axiom4'} describe a subclass $ \mathfrak{F} $ of the
the matroids. This $ \mathfrak{F} $ consists of the matroids having only finite circuits and called the class of \emph{finitary}
matroids. Class $ \mathfrak{F} $ is closed under several important operations like direct sums and taking minors but not under
taking
duals which was the main motivation of Rado's program for looking for a more general matroid concept. The class $
\mathfrak{F}^{*} $ of the duals of the matroids in $ \mathfrak{F} $
consists of the \emph{cofinitary} matroids, i.e. matroid whose all cocircuits are finite. In order to being closed under all
matroid operations we need, we work with the class $
\mathfrak{F}\oplus \mathfrak{F}^{*} $ of matroids having only finitary and cofinitary components, equivalently that are the
direct sum of a finitary and cofinitary matroid.
Matroid union is a fundamental concept in the theory of finite matroids. For a finite family $ (M_i: i\leq n) $ of matroids on a
common finite edge set $ E $ one can define a matroid $ \bigvee_{i\leq n}M_i $ on $ E $ by letting $ I\subseteq E $ be
independent in $ \bigvee_{i\leq n}M_i $ if $ I=\bigcup_{i\leq n}I_i $ for suitable $ I_i\in \mathcal{I}_{M_i} $ (see
\cite{edmonds1968matroid}). This
phenomenon fails for infinite families of finitary matroids. Indeed, let $ E $ be uncountable and let $ M_i $ be the $ 1
$-uniform
matroid on $ E $ for $ i\in \mathbb{N} $. Then exactly the countable subsets of $ E $ would be independent in
$ \bigvee_{i\in \mathbb{N}}M_i $ and hence there would be no maximal independent set contradicting
\ref{item axiom4}.\footnote{For a finite family of finitary matroids the union operation results in a matroid (Proposition 4.1
in \cite{aigner2018intersection}).} Even so,
Bowler and Carmesin observed (see section 3 in \cite{bowler2015matroid}) that the rank formula in the Matroid Partition
Theorem by Edmonds and Fulkerson (Theorem 13.3.1 in \cite{frank2011connections}), namely:
\[ r\left( \bigvee_{i\leq n}M_i \right) =\max_{I_i\in \mathcal{I}_{M_i}}\left|\bigcup_{i\leq n}I_i\right|=\min_{E=E_p\sqcup
E_c}\left|E_c\right|+\sum_{i\leq n}r_{M_i}(E_p),\]
can be interpreted in infinite setting via the complementary slackness conditions. In the minimax formula above there is equality
for the family $ (I_i: i\leq n) $ and partition $ E=E_p\sqcup E_c $ iff
\begin{itemize}
\item $ I_i $ is independent in $ M_i $,
\item $\bigcup_{i\leq n} I_i\supseteq E_c $,
\item $ I_i\cap E_p $ spans $ E_p $ in $ M_i $ for every $ i $,
\item $ I_i\cap I_j\cap E_p=\varnothing $ for $ i\neq j$.
\end{itemize}
Bowler and Carmesin conjectured that for
every family $\mathcal{M}:= (M_i: i\in\Theta) $ of matroids on a common edge set $ E $ there is a family $ (I_i: i\in \Theta) $
and partition $
E=E_p\sqcup E_c $ satisfying
the conditions above. To explain the name ``Packing/Covering Conjecture'' let us provide an alternative formulation. A
\emph{packing} for $ \mathcal{M} $ is a system $ (S_i: i\in\Theta ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is
spanning in $ M_i $. Similarly, a \emph{covering} for $ \mathcal{M} $ is a system $ (I_i: i\in\Theta ) $ with
$\bigcup_{i\in\Theta} I_i=E $ where $ I_i $ is independent in $ M_i $.
\begin{conj}[Packing/Covering, Conjecture 1.3 in \cite{bowler2015matroid} ]\label{conj: Pack/Cov}
For every family $ (M_i: i\in \Theta) $ of matroids on a common edge set $ E $ there is a partition $E=E_p \sqcup E_c$ in such a
way that $ (M_i \upharpoonright E_p: i\in \Theta) $ admits a packing and $ (M_i. E_c: i\in \Theta) $ admits a covering.
\end{conj}
We shall prove the following special case of the Pacing/Covering Conjecture \ref{conj: Pack/Cov}:
\begin{restatable}{thm}{PC}\label{t: main result0}
For every family $ (M_i: i\in \Theta) $ of matroids on a common countable edge set $ E $ where $ M_i \in
\mathfrak{F}\oplus \mathfrak{F}^{*} $, there is a partition $E=E_p \sqcup E_c$ such
that $ (M_i \upharpoonright E_p: i\in \Theta) $ admits a packing and $ (M_i. E_c: i\in \Theta) $ admits a covering.
\end{restatable}
It is worth to mention that packings and coverings have a crucial role in other problems as well. For example if $ (M_i: i\in\Theta)
$ is as
in Theorem
\ref{t: main result0} and admits
both a packing and a covering, then there is a partition
$ E=\bigsqcup_{i\in \Theta}B_i $ where $ B_i $ is a base of $ M_i $ (see \cite{erde2019base}). Maybe surprisingly, the failure
of the analogous statement
for arbitrary
matroids is consistent with set theory ZFC (Theorem 1.5 of \cite{erde2019base}) which might raise some scepticism about the
provability of Conjecture
\ref{conj: Pack/Cov} for general matroids.
The Packing/Covering Conjecture \ref{conj: Pack/Cov} is closely related to the Matroid Intersection Conjecture which has
been one of the central open problems in the theory of infinite matroids:
\begin{conj}[Matroid Intersection Conjecture by Nash-Williams, \cite{aharoni1998intersection}]\label{MIC}
If $ M $ and $ N $ are finitary matroids on the same edge set $ E $, then they admit a common
independent set $ I $ for which there is a partition $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans $ E_M $ in $
M $ and $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $.
\end{conj}
Aharoni proved in \cite{aharoni1984konig} based on his earlier works with Nash-Williams and Shelah (see
\cite{aharoni1983general} and \cite{aharoni1984another})
that the special case of Conjecture \ref{MIC} where $ M $ and $ N $ are partition matroids holds. The conjecture
is also known
to be true if we assume that $ E $ is countable but $ M $ and
$ N $ can be otherwise arbitrary (see \cite{joo2020MIC}).
Let us call Generalized Matroid Intersection Conjecture what we obtain from \ref{MIC} by extending it to arbitrary matroids
(i.e. omitting the word ``finitary''). Several partial results has been obtained for this generalization but only for well-behaved
matroid classes. The positive answer is known for example if: $ M $ is finitary and $ N $ is cofinatory
\cite{aigner2018intersection} or both
matroids are
singular\footnote{A matroid is singular if it is the direct sum of $ 1 $-uniform matroids and duals of $ 1
$-uniform matroids.} and countable \cite{ghaderi2017} or $ M $ is arbitrary and $ N $ is the direct sum of
finitely many uniform matroids \cite{joó2020intersection}.
Bowler and Carmesin showed that their Pacing/Covering Conjecture \ref{conj: Pack/Cov} and the
Generalized Matroid Intersection Conjecture are equivalent and they also found an important reduction for both (see Corollary
3.9 in \cite{bowler2015matroid}). By analysing their proof it is clear that the equivalence can be established if we restrict both
conjectures to a class of matroids closed under certain operations. It allows us to prove Theorem \ref{t: main result} by showing
the following instance of the Generalized Matroid Intersection Conjecture which itself is a common extension of the singular case
by Ghaderi \cite{ghaderi2017} and our previous work \cite{joo2020MIC}:
\begin{restatable}{thm}{MI}\label{t: main result}
If $ M $ and $ N $ are matroids in $ \mathfrak{F}\oplus \mathfrak{F}^{*} $ on the same countable edge set $ E $, then they
admit a common independent set $ I $ for which there is a partition $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans
$ E_M $ in $ M $ and $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $.
\end{restatable}
The paper is organized as follows. In the following section we introduce some notation and fundamental facts about matroids that
are mostly well-know for finite ones. In Section \ref{s: premil} we collect some previous results and relatively easy technical
lemmas in order be able the discuss later the proof of the main results without any distraction. Then in Section
\ref{s: reduction} we reduce the main results to a key-lemma. After these preparations the actual proof begins with
Section \ref{s: augP} by developing and analysing an `augmenting path' type of technique. Our main principle from this point is
to
handle the
finitary and the
cofinitary parts of matroid $ N $ differently in order to exploit the advantage
of the finiteness of the circuits and cocircuits respectively. Equipped with these ``mixed'' augmenting paths we discuss the proof
of our key-lemma in Section
\ref{s: proof of key-lemma}. Finally, we introduce an application in Section \ref{s: application} about orientations of a graph with
in-degree requirements.
\section{Notation and basic facts}\label{sec notation}
In this section we introduce the notation and recall some basic facts about
matroids that we will use later without further explanation. For more details we refer to \cite{nathanhabil}.
An enumeration of a countable set $ X $ is an $ \mathbb{N}\rightarrow X $ surjection that we write as $ \{x_n: n\in \mathbb{N}
\} $. We denote the symmetric difference $ (X\setminus Y)\cup (Y\setminus X) $ of $ X $ and $ Y $ by $
\boldsymbol{X\vartriangle Y} $.
A pair ${M=(E,\mathcal{I})}$ is a \emph{matroid} if ${\mathcal{I} \subseteq \mathcal{P}(E)}$ satisfies the axioms
\ref{item axiom1}-\ref{item axiom4}.
The sets in~$\mathcal{I}$ are called \emph{independent} while the sets in ${\mathcal{P}(E) \setminus \mathcal{I}}$ are
\emph{dependent}. An $ e\in E $ is a \emph{loop} if $ \{ e \} $ is dependent.
If~$E$ is finite, then \ref{item axiom1}-\ref{item axiom3} is one of the the usual axiomization of matroids in terms of
independent sets (while \ref{item axiom4} is redundant).
The maximal independent sets are called \emph{bases}. If $ M $ admits a finite base, then all the bases have the same size which
is the rank $ \boldsymbol{r(M)} $ of $ M $ otherwise we let $ r(M):=\infty $.\footnote{It is independent of ZFC that the bases of
a fixed matroid have the same
size (see \cite{higgs1969equicardinality} and \cite{bowler2016self}).} The minimal dependent sets are called \emph{circuits}.
Every dependent set contains a circuit. The \emph{components} of a matroid are the
components of the hypergraph of its circuits. The
\emph{dual} of a matroid~${M}$ is the
matroid~${M^*}$ with $ E(M^*)=E(M) $ whose bases are the complements of
the bases of~$M$. For an ${X \subseteq E}$, ${\boldsymbol{M \upharpoonright X} :=(X,\mathcal{I}
\cap \mathcal{P}(X))}$ is a matroid and it is called the \emph{restriction} of~$M$ to~$X$.
We write ${\boldsymbol{M - X}}$ for $ M \upharpoonright (E\setminus X) $ and call it the minor obtained by the
\emph{deletion} of~$X$.
The \emph{contraction} of $ X $ in $ M $ and the contraction of $ M $ onto $ X $ are
${\boldsymbol{M/X}:=(M^* - X)^*}$ and $\boldsymbol{M.X}:= M/(E\setminus X) $ respectively.
Contraction and deletion commute, i.e., for
disjoint
$ X,Y\subseteq E $, we have $ (M/X)-Y=(M-Y)/X $. Matroids of this form are the \emph{minors} of~$M$. The
independence of an $ I\subseteq X $ in $ M.X $ is equivalent with $ I\subseteq \mathsf{span}_{M^{*}}(X\setminus I) $.
If $ I $ is independent in $ M $ but $ I+e $ is dependent for some $ e\in E\setminus I $ then there is a unique
circuit $ \boldsymbol{C_M(e,I)} $ of $ M $ through $ e $ contained in $ I+e $. We say~${X
\subseteq E}$ \emph{spans}~${e \in E}$ in matroid~$M$ if either~${e \in X}$ or there exists a circuit~${C
\ni e}$ with~${C-e \subseteq X}$.
By letting $\boldsymbol{\mathsf{span}_{M}(X)}$ be the set of edges spanned by~$X$ in~$M$, we obtain a closure
operation
$ \mathsf{span}_{M}: \mathcal{P}(E)\rightarrow \mathcal{P}(E) $.
An ${S \subseteq E}$ is \emph{spanning} in~$M$ if~${\mathsf{span}_{M}(S) = E}$. An $ S\subseteq X $ spans $ X $ in $
M.X $ iff $ X\setminus S $ is independent in $ M^{*} $. If $ M_i=(E_i,
\mathcal{I}_i)$ is a matroid for $ i\in
\Theta $ and the sets $ E_i $ are pairwise disjoint, then their direct sum is $ \boldsymbol{\bigoplus_{i\in
\Theta}M_i}=(E,\mathcal{I}) $ where
$ E=\bigsqcup_{i\in \Theta}E_i $ and $ \mathcal{I}=\{ \bigsqcup_{i\in \Theta}I_i : I_i\in \mathcal{I}_i\} $. For a class $
\mathfrak{C} $ of matroids $ \boldsymbol{\mathfrak{C}(E)} $ denotes the subclass $ \{ M\in \mathfrak{C}: E(M)=E \} $.
A matroid is called uniform if for ever base $ B $ and every edges $ e\in B $ and $ f\in E\setminus B $ the set $ B-e+f $ is also a
base. Let
$ \boldsymbol{U_{E,n}}$ be the $ n $-uniform matroid on $ E $, formally $ U_{E,n}:=(E , [E]^{\leq n})$.
We need some further more subject-specific definitions. From now on let $ M
$ and $ N $ be matroids on a common edge set $ E $. We call a $ W\subseteq E $ an
$ (M,N) $-\emph{wave} if $ M\upharpoonright W $ admits an $ N.W $-independent base. Waves in the matroidal context were
introduced by
Aharoni and Ziv in \cite{aharoni1998intersection} but it was also an important tool in the proof of the infinite version of Menger's
theorem \cite{aharoni2009menger} by Aharoni and Berger. We write $ \boldsymbol{\mathsf{cond}(M,N)} $ for
the condition: `For every $ (M,N) $-wave $ W $ there is an $ M $-independent base of $ N.W $.' A set $ I\in\mathcal{I}_M\cap
\mathcal{I}_N $ is \emph{feasible} if $
\mathsf{cond}(M/I,N/I) $ holds.
It is known (see Proposition \ref{wave union}) that there exists a $ \subseteq $-largest $ (M,N) $-wave which we denote by
$ \boldsymbol{W(M,N)} $. Let $ \boldsymbol{\mathsf{cond}^{+}(M,N) }$ be the statement that $ W(M,N) $ consists of $ M
$-loops and $
{r(N.W(M,N))=0} $. As the notation indicates it is a strengthening of $ \mathsf{cond}(M,N) $. Indeed, under the assumption $
\mathsf{cond}^{+}(M,N) $, $ \varnothing $ is an $ M $-independent base of $ N.W $ for every wave $ W $. A feasible $ I $ is
called \emph{nice} if $ \mathsf{cond}^{+}(M/I,N/I) $ holds. For $ X\subseteq E $ let $ \boldsymbol{B(M,N,X)} $ be the
(possibly empty) set of common bases of $ M \upharpoonright
X $ and $ N.X $.
\section{Preliminary lemmas and preparation}\label{s: premil}
We collect those necessary lemmas in this section that are either known from previous papers or follow more or less directly
from definitions.
\subsection{Classical results}
The following two statements were proved by Edmonds' in \cite{edmonds2003submodular}:
\begin{prop}\label{prop: simult change}
Assume that~$I$ is independent,
${e_1, \dots, e_{m} \in \mathsf{span}(I) \setminus I}$
and~${f_1, \dots, f_{m} \in I}$
with ${f_j \in C(e_j, I)}$
but ${f_j\notin C(e_k, I)}$ for~${k < j}$.
Then
\[
{\left( I \cup \{e_1,\dots, e_{m}\} \right) \setminus \{f_1,\dots, f_{m}\}}
\]
is independent and spans the same set as~$I$.
\end{prop}
\begin{proof}
We use induction on~$m$.
The case~${m = 0}$ is trivial.
Suppose that~${m > 0}$.
On the one hand, the set ${I-f_m+e_m}$ is independent and spans the same set as~$I$.
On the other hand, ${C(e_j, I-f_m+e_m) = C(e_j, I)}$ for~${j < m}$ because ${f_m \notin C(e_j, I)}$ for~${j < m}$.
Hence by using the induction hypothesis for~${I-f_m+e_m}$ and ${e_1,\dots, e_{m-1}, f_1,\dots, f_{m-1}}$ we are done.
\end{proof}
\begin{lem}[Edmonds' augmenting path method]\label{l: augP Edmonds}
For $ I\in \mathcal{I}_M\cap \mathcal{I}_N $, exactly one of the following statements holds:
\begin{enumerate}
\item There is a partition $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans $ E_M $ in $ M $ and $ I_N:=I\cap E_N $
spans $ E_N $ in $ N $.
\item There is a
$ P=\{ x_1,\dots, x_{2n+1} \}\subseteq E $ with $ x_{1}\notin \mathsf{span}_N(I) $ and $
x_{2n+1}\notin
\mathsf{span}_M(I) $ such that $ I\vartriangle P\in
\mathcal{I}_M\cap \mathcal{I}_N $ with $ \mathsf{span}_{M}(I\vartriangle P) =\mathsf{span}_{M}(I+x_{2n+1}) $ and
$ \mathsf{span}_{N}(I\vartriangle P) =\mathsf{span}_{M}(I+x_{1}) $.
\end{enumerate}
\end{lem}
\noindent We will develop in Section
\ref{s: augP} a ``mixed'' augmenting path method which operates differently on the finitary
and on the cofinitary part of an $ N\in (\mathfrak{F}\oplus \mathfrak{F}^{*})(E) $. The phrase `augmenting path' refers always
to our mixed method except in the proof of Lemma \ref{one more edge}.
Note that $ E_M $ is an $ (M,N) $-wave witnessed by $ I_M $ and $ E_N $ is an $ (N,M) $-wave witnessed by $
I_N $.
One can define matroids in the language of circuits (see \cite{bruhn2013axioms}). The following claim is one of the axioms in
that case.
\begin{claim}[Circuit elimination axiom]\label{Circuit elim}
Assume that $ C\ni e $ is a circuit and $ \{ C_x: x\in X \} $ is a family of circuits where $ X\subseteq C-e $ and $ C_x $ is a
circuit with $C\cap X=\{ x \} $ avoiding $ e $. Then there is a circuit through $ e $ contained in
\[ \left( C\cup \bigcup_{x\in X}C_x \right) \setminus X =:Y \]
\end{claim}
\begin{proof}
Since $ C_x-x$ spans $ x $ we have $ C-e\subseteq\mathsf{span}(Y-e) $ and therefore $e\in \mathsf{span}(\mathsf{span}(Y-e)
)$. But
then $ e\in \mathsf{span}(Y-e) $ because $ \mathsf{span} $ is a closure operator.
\end{proof}
For finite matroids the axiom above is demanded only in the special case where $ X $ is a singleton (known as ``Strong circuit
elimination``) from which the case of arbitrary $ X $ can be derived by repeated application.
\begin{cor}\label{cor: Noutgoing arc}
Let $ I $ be an independent and suppose that there is a circuit $ C\subseteq \mathsf{span}(I) $
with $ e\in I\cap C $. Then there is an $ f\in C\setminus I $ with $e\in C(f,I) $.
\end{cor}
\begin{proof}
For every $ x\in C\setminus I $ we pick a circuit $ C_x $ with $ C_x\setminus I=\{ x \} $. If $
e\in C_x $
for some $ x $,
then $ f:=x $ is as desired. Suppose for a contradiction that there is no such an $ x $. Then by Circuit elimination (Claim
\ref{Circuit elim}) we obtain a circuit through $ e $ which is
contained entirely in $ I $ contradicting the independence of $ I $.
\end{proof}
The following statement was shown by Aharoni and Ziv in \cite{aharoni1998intersection} using a slightly
different terminology.
\begin{prop}\label{wave union}
The union of arbitrary many waves is a wave.
\end{prop}
\begin{proof}
Suppose that $ W_{\beta} $ is a wave for $ \beta<\kappa $ and let
$W_{<\alpha}:=\bigcup_{\beta<\alpha}W_{\beta} $ for $ \alpha\leq \kappa $. We fix a base
$ B_{\beta} \subseteq W_{\beta} $
of $ M\upharpoonright W_{\beta} $ which is independent in $ N.W_{\beta} $. Let us define
$ B_{<\alpha} $ by transfinite recursion for $ \alpha\leq \kappa $ as follows.
\[B_{<\alpha}:= \begin{cases} \varnothing &\mbox{if } \alpha=0 \\
B_{<\beta}\cup (B_\beta \setminus W_{<\beta}) & \mbox{if } \alpha=\beta+1\\
\bigcup_{\beta<\alpha}B_{<\beta} & \mbox{if } \alpha \text{ is limit ordinal}.
\end{cases} \]
First we show by transfinite induction
that $ B_{<\alpha} $ is spanning in $ M\upharpoonright W_{<\alpha} $. For $ \alpha=0 $ it is trivial.
For a limit $ \alpha $ it follows directly from the induction hypothesis. If $ \alpha=\beta+1 $, then
by the choice of $ B_\beta $, the set $ B_\beta \setminus W_{<\beta} $ spans $ W_{\beta}\setminus W_{<\beta} $
in $ M/W_{<\beta} $. Since $ W_{<\beta} $ is spanned by $ B_{<\beta} $ in $ M $ by induction, it follows that
$ W_{<\beta+1} $ is spanned by $ B_{<\beta+1} $ in $ M $.
The independence of $ B_{<\alpha} $ in
$ N.W_{<\alpha} $ can be reformulated as ``$W_{<\alpha}\setminus B_{<\alpha}$ is spanning in $
N^{*} \upharpoonright
W_{<\alpha} $'',
which can be proved the same way as above.
\end{proof}
\subsection{Some more recent results and basic facts}
\begin{thm}[Aigner-Horev, Carmesin and Frölich; Theorem 1.5 in \cite{aigner2018intersection}]\label{t: mixed}
If $ M\in \mathfrak{F}(E) $ and $ N\in
\mathfrak{F}^{*}(E) $, then there is an $ I\in \mathcal{I}_M\cap \mathcal{I}_N $ and a partition $ E=E_M\sqcup E_N $ such
that $ I_M:=I\cap E_M $ spans $ E_M $ in $ M $ and $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $.
\end{thm}
\begin{cor}\label{cor: applyMixed}
If $ M\in \mathfrak{F}(E) $ and $ N\in
\mathfrak{F}^{*}(E) $ satisfy $ \mathsf{cond}^{+}(M,N) $, then there is an $ M $-independent $ N $-base.
\end{cor}
\begin{proof}
Let $ E_M, E_N,
I_M $ and $ I_N $ be as in Theorem \ref{t: mixed}. Then $ E_M $ is a wave witnessed by $ I_M $ thus by
$ \mathsf{cond}^{+}(M,N) $ we know that $ E_M $ consists of $ M $-loops and $ r(N.E_M)=0 $. But then $ I_M=\varnothing $
and $ I_N $ is a base of $ N $ which is independent in $ M $.
\end{proof}
\begin{obs}\label{o: Mloop}
If $ \mathsf{cond}(M,N) $ holds and $ L $ is a set of the $ M $-loops, then $ r(N.L)=0 $ which means $
L\subseteq\mathsf{span}_N(E\setminus L) $.
\end{obs}
\begin{cor}\label{cor: Mloop}
If $ I $ is feasible, then $ r(N.(\mathsf{span}_{M}(I)\setminus I)=0$.
\end{cor}
\begin{obs}
If $ W_0 $ is an $ (M,N) $-wave and $ W_1 $ is an $ (M/W_0, N-W_0) $-wave, then $ W_0\cup W_1 $ is an $ (M,N)
$-wave.
\end{obs}
\begin{cor}\label{cor: empty wave}
For $ W:=W(M,N) $, the largest $ (M/W, N-W) $-wave is $ \varnothing $. In particular, $ \mathsf{cond}^{+}(M/W,N-W) $
holds.
\end{cor}
\begin{obs}\label{o: condRestrict}
$ \mathsf{cond}^{+}(M,N) $ implies ${\mathsf{cond}^{+}(M\upharpoonright X,N.X) }$ for every $ X\subseteq E $.
\end{obs}
\begin{prop}\label{p: iterate feasible}
If $ I_0\in \mathcal{I}_N\cap \mathcal{I}_M $ and $ I_1 $ is feasible with respect to $ (M/I_0,
N/I_0) $, then $ I_0\cup I_1 $ is
feasible with respect to $ (M,N) $. If in addition $ I_1 $ is a nice feasible in regards to $ (M/I_0, N/I_0) $, then so is
$ I_0\cup I_1 $ to $ (M,N) $.
\end{prop}
\begin{proof}
By definition the feasibility of $ I_1 $ w.r.t. $ (M/I_0, N/I_0) $ means that the condition $ \mathsf{cond}(M/(I_0\cup
I_1),N/(I_0\cup I_1)) $
holds. The feasibility of $ I_0\cup I_1 $ w.r.t. $ (M, N) $ means the same also by definition. For `nice feasible' the argument is
similar, only $ \mathsf{cond} $ must be replaced by $ \mathsf{cond}^{+} $.
\end{proof}
The following lemma was introduced in \cite{joo2020MIC}.
\begin{lem}\label{one more edge}
Condition $ \mathsf{cond}^{+}(M, N) $ implies that
whenever $ W $ is an $ (M/e, N/e) $-wave for some $ e\in E $ witnessed by $ B\subseteq W $, then
$B\in B(M/e,N/e,W) $, i.e. $ B $ is spanning in $ N.W $.
\end{lem}
\begin{proof}
Let $ W $ be an $ (M/e, N/e) $-wave. Note that
$(N/e).W=N.W $ by definition. Pick a $ B\subseteq W $ which is an $ N.W $-independent base of $ (M/e)\upharpoonright
W $. We may assume
that
$ e\in \mathsf{span}_M(W) $ and $ e $ is not an $ M $-loop. Indeed, otherwise $ (M/e) \upharpoonright
W=M\upharpoonright
W$ holds
and hence $ W $ is also an $ (M,N) $-wave. Thus by $ \mathsf{cond}^{+}(M, N) $ we may conclude that $ W $ consists of $ M
$-loops and hence $ B=\varnothing $, moreover, $ r(N.W)=0 $ and therefore $
\varnothing $ is a base of $ N.W $.
Then $ B $ is not a base of
$ M\upharpoonright W $ but ``almost'', namely $r(M/B \upharpoonright W) =1 $. We apply the
augmenting path Lemma \ref{l: augP Edmonds} by Edmonds with $ B$ in regards to $M\upharpoonright W$ and $ N.W $. An
augmenting
path $ P $ cannot exist. Indeed, if $ P $ were an augmenting path then $ B\vartriangle P $ would show that $ W $ is an
$ (M,N) $-wave which does not consist of $ M $-loops, contradiction. Thus we get a partition
$ W=W_{0}\sqcup W_{1} $ instead where $ W_0 $ is an $ (M \upharpoonright W,N.W) $-wave witnessed by $ B\cap W_0 $,
and therefore also an $ (M,N)
$-wave, and $ W_1 $ is an
$ (N.W, M\upharpoonright W) $-wave showed by $ B\cap W_1 $.
Then $ W_0 $ must consist of $ M $-loops by $ \mathsf{cond}^{+}(M, N) $ and
therefore $ B\subseteq W_1 $ by the $ M $-independence of $ B $. We need to show that $ B $ is spanning not just in $ N.W_1 $
but also in $ N.W $. To do so let $ B' $ be a base of $ N-W$.
Then $ B\cup B' $ spans $ E\setminus W_0 $ in $ N $ because $ B=B\cap W_1 $ is a base $ N.W_1 $. But
then $ B\cup B' $ is spanning in $ N $ because $ r(N.W_0)=0 $ by Observation \ref{o: Mloop}. We conclude that $ B $ is
spanning in $ N.W $ as desired.
\end{proof}
\subsection{Technical lemmas in regards to \texorpdfstring{$ \boldsymbol{B(M,N,W)} $}{BMNW}}
\begin{prop}\label{prop: WB nice feasible}
For $ W:=W(M,N) $, the elements of $ B(M,N,W) $ are nice feasible sets.
\end{prop}
\begin{proof}
Let $ B\in B(M,N,W) $. Clearly $ B\in \mathcal{I}_M\cap \mathcal{I}_N $ because $ B\in \mathcal{I}_M\cap
\mathcal{I}_{N.W} $ by definition and $ \mathcal{I}_{N.W} \subseteq \mathcal{I}_N $. We know that $ W\setminus B $ is
an $
(M/B, N/B) $-wave consisting of $ M/B $ loops and $ r(N.(W\setminus B))=0 $ because $ B $ is a
base of $ N.W $. In order to show $ \mathsf{cond}^{+}(M/B, N/B)
$, it is enough to prove that $ W\setminus B $ is actually $ W(M/B, N/B)=:W' $. Suppose for a contradiction that $ W'\supsetneq
W\setminus B$. Let $ B' $ be an $ N.W' $-independent base of $ M/B \upharpoonright W' $. Note that $ B\cup B' $ is a base of $
M \upharpoonright (W\cup
W') $ and $ B'\cap (W\setminus B)=\varnothing $. Since $ B $ is $ N^{*} $-spanned by $ W\setminus B\subseteq (W\cup
W')\setminus (B\cup B') $ and $ B' $ is $ N^{*} $-spanned by $ W'\setminus B'\subseteq (W\cup W')\setminus (B\cup B') $, we
may conclude that $ B\cup B' $ is independent in $ N.(W\cup W') $. Thus $ W\cup W' $ is an $ (M,N) $-wave witnessed by $
B\cup B' $ which contradicts the maximality of $ W $.
\end{proof}
\begin{cor}\label{cor: extNice}
Assume that $ I\in \mathcal{I}_M\cap \mathcal{I}_M $ and $ B\in B(M/I,N/I, W) $ where $ W:=W(M/I,N/I) $. Then $ I\cup B
$ is a nice feasible set.
\end{cor}
\begin{proof}
Combine Propositions \ref{p: iterate feasible} and \ref{prop: WB nice feasible}.
\end{proof}
\begin{obs}\label{obs: cond+ loop delete}
If $ \mathsf{cond}^{+}(M,N) $ holds and $ L $ is a set of $ M $-loops, then $ \mathsf{cond}^{+}(M-L,N-L) $ also holds.
\end{obs}
\begin{obs}\label{obs: common loops remove}
Let $ W $ be a wave and let $ L $ be a set of common loops of $ M $ and $ N $. Then $ W\setminus L $ is also a wave and $
B(M,N,W)=B(M,N,W\setminus L) $.
\end{obs}
\begin{lem}\label{l: wave modify}
Let $ W $ be a wave and $ L\subseteq W $ such that $ L $ consists of $ M $-loops with $ r(N.L)=0 $. Then $ W\setminus L $ is
an $ (M-L, N-L, W\setminus L) $-wave with \[
B(M,N,W)=B(M-L, N-L, W\setminus L). \]
\end{lem}
\begin{proof}
A set $ B $ is a base of $ M\upharpoonright W $ iff it is a base of
$ M\upharpoonright (W\setminus L) $ because $ L $ consists of $ M $-loops. A $ B\subseteq W\setminus L $ is $ N.W
$-independent iff $ B\subseteq\mathsf{span}_{N^{*}}(W\setminus B) $. This holds if and only if $
B\subseteq\mathsf{span}_{N^{*}/L}(W\setminus
(B\cup L)) $ i.e. $ B $ is independent in $ (N-L).(W\setminus L) $. Note that $ r(N.L)=0 $ is equivalent with the $
N^{*} $-independence of $ L $. Thus for $ B\subseteq W\setminus L $, $ W\setminus B $ is $ N^{*} $-independent iff
$ W\setminus (B\cup L) $ is $ N^{*}/L $-independent. It means that $ B $ is spanning in $ N.W $ iff it is spanning in $
(N-L).(W\setminus L) $.
Thus the sets that are witnessing that $ W\setminus L $ is
an $ (M-L, N-L, W\setminus L) $-wave are exactly those that are witnessing that $ W $ is an $ (M,N) $-wave, moreover, $
B(M,N,W)=B(M-L, N-L, W\setminus L) $ holds.
\end{proof}
\begin{lem}\label{l: minorsChanged}
Assume that $ X_j, Y_j \subseteq E $ for $ j\in \{ 0,1 \} $ where $ X_j\sqcup Y_j=Z $ for $ j\in \{ 0,1 \} $, furthermore
$ \mathsf{span}_M(X_0)=\mathsf{span}_M(X_1) $ and $ \mathsf{span}_{N^{*}}(Y_0) =\mathsf{span}_{N^{*}}(Y_1) $.
Then for every $ X\subseteq E\setminus Z $ we have\[ B(M/X_0-Y_0, N/X_0-Y_0, X)=B(M/X_1-Y_1, N/X_1-Y_1, X). \]
\end{lem}
\begin{proof}
The matroids $ M/X_0-Y_0 $ and $ M/X_1-Y_1$ are the same as well as the matroids $ N/X_0-Y_0$ and $ N/X_1-Y_1 $.
\end{proof}
\section{Reductions}\label{s: reduction}
We repeat here our main results for convenience:
\PC*
\MI*
\subsection{Matroid Intersection with a finitary \texorpdfstring{$\boldsymbol{M} $}{M}}
As we mentioned, the method by Bowler and Carmesin used to prove Corollary 3.9 in \cite{bowler2015matroid} works not only
for the class of all matroids but can be adapted for every class closed under certain operations. We apply their
technique to obtain the following reduction:
\begin{lem}\label{l: M finitary}
Theorems \ref{t: main result0} and \ref{t: main result} are implied by the special case of Theorem \ref{t: main result} where $
M\in \mathfrak{F} $.
\end{lem}
\begin{proof}
First we show that one can assume without loss of generality in the proof of Theorem \ref{t: main result0} that $ \Theta $ is
countable. To
do so let \[ E':=\{ e\in E: \left|\{i\in
\Theta: \{ e \}\in \mathcal{I}_{M_i} \}\right|\leq \aleph_0 \} \] and
\[ \Theta':= \{ i\in \Theta: (\exists e\in E' ) (\{ e \}\in \mathcal{I}_{M_i}) \}. \]
We apply Theorem \ref{t: main result0} with $ E' $ and with the countable family $(M_i\upharpoonright E': i\in \Theta') $.
Then we obtain a partition
$ E'=E'_p\sqcup E'_c $ such that $ (M_i \upharpoonright E'_p: i\in \Theta') $ admits a packing $ (S_i: i\in \Theta') $ and
$ (M_i\upharpoonright E'. E_c: i\in \Theta') $ admits a covering $ (I_i: i\in \Theta') $. Let $ E_p:=E'_p $ and $ E_c:= E\setminus
E_p=E'_c\cup (E\setminus E') $. By construction $ r_{M_i}(E')=0 $ for $ i\in
\Theta \setminus \Theta' $. Thus by letting $ S_i:=\varnothing $ for $ i\in \Theta \setminus \Theta' $ the
family $ (S_i: i\in \Theta) $ is a packing w.r.t. $ (M_i \upharpoonright E_p: i\in \Theta) $.
Let $ g: E\setminus E'\rightarrow
\Theta\setminus \Theta' $ be injective. For $ i\in \Theta \setminus \Theta' $, we take $ I_i:=\{ g^{-1}(i) \}$ if $ i\in
\mathsf{ran}(g) $ and $ I_i:=\varnothing $ otherwise. Then $ (I_i: i\in \Theta) $ is a covering for $ (M_i. E_c: i\in \Theta) $ and
we are done.
We proceed with the proof of Theorem \ref{t: main result0} assuming that $ \Theta $ is countable. For $ i\in \Theta $, let $ M'_i
$ be the matroid on $
E\times \{ i \} $ that we
obtain by
``copying'' $ M_i $ via the bijection $ e\mapsto (e,i) $. Then for
\[M:= \bigoplus_{i\in \Theta}M_i',\ \text{ and } N:= \bigoplus_{e\in E}U_{ \{ e \}\times \Theta, 1 } \]
we have $M\in ( \mathfrak{F}\oplus \mathfrak{F}^{*})(E\times \Theta) $ and $ N\in \mathfrak{F}(E\times \Theta) $ where $
E\times \Theta $ is countable. Thus by assumption there is a partition $
E\times \Theta=E_M\sqcup E_N $ and an $ I\in \mathcal{I}_{M}\cap \mathcal{I}_{N} $ such that $ I_M:=I\cap E_M $
spans $ E_M $ in $ M$ and $ I_N:=I\cap E_N $
spans $ E_N$ in $ N$. The $ M $-independence of $ I $ ensures that $ J_i:=\{ e\in E: (e,i)\in I \} $ is $ M_i $-independent.
The $ N $-independence of $ I $ guarantees that the sets $ J_i $ are pairwise disjoint. Let
$ E_c:=\{ e\in E: (\exists i\in \Theta) (e,i)\in E_N \} $. Then for each $ e\in E_c $ there must be some $ i\in \Theta $ with $
(e,i)\in
I_N $ because $ E_N\subseteq \mathsf{span}_N(I_N) $. Thus the sets $ J_i $ cover $ E_c $ and so do the sets $
I_i:=J_i\cap E_c $. It is enough to show that $ S_i:=J_i\setminus I_i $ spans $ E_p:=E\setminus E_c $ in $ M_i $ for every $
i\in \Theta $. Let $ f\in E_p $ and $ i\in \Theta $ be given. Then $ \{ f \}\times \Theta \subseteq E_M $ follows directly from the
definition of $ E_p $, in particular $ (f,i)\in E_M $. We know that $(f,i)\in
\mathsf{span}_{M}(I_M)$ and hence $ f\in \mathsf{span}_{M_i}(\{ e\in E: (e,i)\in I_M \}) $. Suppose for a contradiction that
$h\in E_c\cap \{ e\in E: (e,i)\in I_M \} $. Then for some $ j\in
\Theta $ we have $ (h,j)\in E_N $. Since $ (h,j)\in \mathsf{span}_N(I_N) $, we have $ (h,k)\in I_N $ for some $ k\in \Theta $.
But then $ (h,i), (h,k)\in I $ are distinct elements thus $ i\neq k $ which contradict the $ N $-independence of $ I $. Therefore $
E_c\cap \{ e\in E: (e,i)\in I_M \}=\varnothing $. Since $ \{ e\in E: (e,i)\in I_M \}\subseteq J_i $ by the definition of $ J_i
$ we conclude $ \{ e\in E: (e,i)\in I_M \}=S_i $. Therefore $ (S_i: i\in \Theta) $ is a packing for $ (M_i \upharpoonright E_p:
i\in \Theta) $ and $ (I_i: i\in \Theta) $ is a covering for $ (M_i. E_c: i\in \Theta) $ as desired.
Now we derive Theorem \ref{t: main result} from Theorem \ref{t: main result0}. To do so, we take a partition $E=E_p\sqcup
E_c$ such that $ (S_M, S_N) $ is a packing for
$ (M\upharpoonright E_p, N^{*}\upharpoonright E_p) $ and $ (R_M, R_N) $ is a covering for $ (M.E_c, N^{*}.E_c ) $. Let $
I_M\subseteq S_M $ be a base of $ M\upharpoonright E_p $ and we define $ I_N:= R_M $. By construction
$E_p\subseteq \mathsf{span}_M(I_M) $ and $ I_N\in \mathcal{I}_{M.E_c} $. We also know that
\[ I_M\subseteq \mathsf{span}_{N^{*}}(S_N) \subseteq \mathsf{span}_{N^{*}}(E_p\setminus I_M) \]
which means $ I_M\in \mathcal{I}_{N.E_{p}} $.
Finally, $R_N\in \mathcal{I}_{N^{*}.E_c} $ means that $ E_c\setminus R_N $ spans $ E_c $ in $ N $ and therefore so does $
I_N=R_M\supseteq E_c\setminus R_N $.
\end{proof}
\subsection{Finding an \texorpdfstring{$ \boldsymbol{M} $}{M}-independent base of \texorpdfstring{$ \boldsymbol{N}
$}{N}}
The following reformulation of the matroid intersection problem was introduced by Aharoni and Ziv in
\cite{aharoni1998intersection} but its analogue by Aharoni was already an important tool to attack (and eventually solve in
\cite{aharoni2009menger}) the Erdős-Menger Conjecture.
\begin{restatable}{claim}{indepB}\label{c: M-indep N-base}
Assume that $ M\in \mathfrak{F}(E) $ and $ N\in (\mathfrak{F}\oplus\mathfrak{F}^{*})(E) $ such that $ E $ is countable and
$ \mathsf{cond}^{+}(M,N) $ holds. Then there is an $ M $-independent base of $ N $.
\end{restatable}
\begin{lem}\label{l: reduc2}
Claim \ref{c: M-indep N-base} implies our main results Theorems \ref{t: main result0} and \ref{t: main result}.
\end{lem}
\begin{proof}
By Lemma \ref{l: M finitary} it is enough to show that the special case of Theorem \ref{t: main result} where $ M\in
\mathfrak{F} $ follows from Claim \ref{c: M-indep N-base}.
To do so, let $ E_M:=W(M,N) $ and let $ I_M\subseteq E_M $ be a witness that $ E_M $ is a wave. For $ E_N:=E\setminus
E_M $, we have $
M/E_M\in \mathfrak{F}(E_N) $ and
$ N-E_M\in (\mathfrak{F}\oplus \mathfrak{F}^{*})(E_N) $, furthermore, $ \mathsf{cond}^{+}(M/W,N-W) $ holds (see
Corollary \ref{cor: empty wave}). By Claim \ref{c: M-indep N-base}, there is an $ M/E_M $-independent base $ I_N $ of $
N-E_M $. Then $ I\in \mathcal{I}_M\cap \mathcal{I}_{N}$ and $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans
$ E_M $ in $ M $ and $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $ as desired.
\end{proof}
\subsection{Reduction to a key-lemma}
From now on we assume that $ M\in \mathfrak{F}(E) $ and $ N\in (\mathfrak{F}\oplus\mathfrak{F}^{*})(E) $ where $ E $ is
countable. Let $
\boldsymbol{E_0} $ be
the union of the finitary components of $ N $ and let
$ \boldsymbol{E_1}:=E\setminus E_0 $. Note that $ N\upharpoonright E_0 $ is finitary,
$ N\upharpoonright E_1 $ is cofinitary and no $ N $-circuit meets both $ E_0 $ and $ E_1 $.
\begin{restatable}[key-lemma]{lem}{keylemma}\label{l: key-lemma}
If $ \mathsf{cond}^{+}(M,N) $ holds, then for every $ e\in E_0 $ there is a nice
feasible $ I $ with $ e\in \mathsf{span}_{N}(I) $.
\end{restatable}
\begin{lem}
Lemma \ref{l: key-lemma} implies our main results Theorems \ref{t: main result0} and \ref{t: main result}.
\end{lem}
\begin{proof}
It is enough to show that Lemma \ref{l: key-lemma} implies Claim \ref{c: M-indep N-base} because of Lemma \ref{l: reduc2}.
Let us
fix an enumeration $ \{ e_n: n\in
\mathbb{N} \} $ of $ E_0 $.
We build an $ \subseteq $-increasing sequence $ (I_n) $ of nice feasible sets starting
with $ I_0:=\varnothing $ (who is nice feasible by $ \mathsf{cond}^{+}(M,N) $) in such a way that $ {e_n\in
\mathsf{span}_N(I_{n+1})} $. Suppose that $ I_n $ is already defined.
If $ e_n\notin \mathsf{span}_N(I_n) $, then we
apply Lemma \ref{l: key-lemma} with $ (M/I_n, N/I_n) $ and $ e_n $ and take the union of the resulting $ I $ with $ I_{n} $ to
obtain $ I_{n+1} $ (see Observation \ref{p: iterate feasible}), otherwise let $ I_{n+1}:=I_n $. The recursion is done. Now we
construct an $ M $-independent
$ I^{+}_n \supseteq I_n $ with $ E_1\subseteq \mathsf{span}_N(I^{+}_n) $ for $ n\in \mathbb{N} $. The matroid $ M/I_n
\upharpoonright
(E_1\setminus I_n) $ is finitary and $ N.(E_1\setminus I_n)=(N\upharpoonright E_1)/(I_n\cap E_1)$ is cofinitary, moreover, by
Observation \ref{o: condRestrict}
\[ \mathsf{cond}^{+}(M/I_n, N/I_n) \Longrightarrow \mathsf{cond}^{+}(M/I_n \upharpoonright (E_1\setminus I_n),
N.(E_1\setminus I_n )). \] Thus by Corollary \ref{cor: applyMixed} there is an $ M/I_n $-independent base $ B_n $ of
$ (N\upharpoonright E_1)/(E_1\cap I_n) $ and $
I^{+}_n:=I_n\cup B_n $ is as desired.
Let $ \mathcal{U} $ be a free ultrafilter on $ \mathcal{P}(\mathbb{N}) $ ad we define
\[ S:=\{ e\in E: \{ n\in \mathbb{N}: e\in I^{+}_n \}\in \mathcal{U} \}. \] Then $ S $ is $ M
$-independent and $ N $-spanning and therefore we are done. Indeed, suppose for a contradiction that $ S $ contains an $ M
$-circuit $ C $. For $ e\in C $, we pick a $ U_e\in
\mathcal{U} $ with $ e\in I^{+}_n $ for $ n\in U_e $. Since $ M $ is finitary, $ C $ is finite, thus $U:= \cap \{ U_e: e\in C \}\in
\mathcal{U} $.
But then for $ n\in U $ we have $ C\subseteq I^{+}_n $ which contradicts the $ M $-independence of $ I^{+}_n $. Clearly, $
I_{n+1}\subseteq S $ for every $ n\in \mathbb{N} $ and therefore $ E_0\subseteq \mathsf{span}_N(S) $. Finally, suppose for a
contradiction that there is some
$ N^{*}\upharpoonright E_1 $-circuit $ C' $ with $ S\cap C'=\varnothing $. For $ e\in C' $, we can pick a $ U_e'\in
\mathcal{U} $ with $ e\notin I^{+}_n $ for $ n\in U_e' $. Since $ N^{*}\upharpoonright E_1 $ is finitary, $ C' $ is finite, thus
$U':= \cap \{ U_e': e\in
C' \}\in
\mathcal{U} $.
But then for $ n\in U' $ we have $ I^{+}_n\cap C=\varnothing $ which contradicts $E_1\subseteq
\mathsf{span}_N(I^{+}_n) $.
\end{proof}
\section{Mixed augmenting paths}\label{s: augP}
In the section we introduce an `augmenting path' type of method and analyse it in order to show some properties we need later.
On $ E_0 $ the definition will be
the same as in the proof of
the Matroid Intersection Theorem by Edmonds \cite{edmonds2003submodular} but on $ E_1 $ we need to define it in a different
way considering that $ N\upharpoonright E_1 $ is cofinitary. For brevity we write
$ \boldsymbol{\overset{\circ}{\mathsf{span}}_{M}(F)} $ for $ \mathsf{span}_{M}(F)\setminus F $
and $ \boldsymbol{F^{j}} $ for $F\cap E_j $
where $ F\subseteq E $ and $ j\in \{ 0,1 \} $.
We call an $F\subseteq E $ \emph{dually safe} if $ F^{1} $ is spanned by $
\overset{\circ}{\mathsf{span}}_M(F) $ in $ N^{*} $.
\begin{lem}\label{l: enoughAddB}
If $ I\in \mathcal{I}_M\cap \mathcal{I}_n $ is dually safe and $ B\in B(M/I, N/I, W) $ for $ W:=W(M/I, N/I) $, then
$ I\cup B $ is a nice dually safe feasible set.
\end{lem}
\begin{proof}
We already know by Corollary \ref{cor: extNice} that $ I\cup B $ is a nice feasible set. By using that $ I $ is dually safe and $
\overset{\circ}{\mathsf{span}}_M(I)\subseteq\overset{\circ}{\mathsf{span}}_M(I\cup B) $ we get
\[ I^{1}\subseteq \mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_M(I))\subseteq
\mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_M(I\cup B)). \]
Since $ B\in B(M/I, N/I, W) $ we have $ W\setminus B\subseteq \overset{\circ}{\mathsf{span}}_M(I\cup B) $ and $ B $ is a
base of $ N.W $. The latter can be rephrased as `$ W\setminus B $ is a base of $ N^{*}\upharpoonright W $'. By combining
these $ \overset{\circ}{\mathsf{span}}_M(I\cup B) $ spans $ B $ in $ N^{*} $. Therefore
$(I\cup B)\cap E_1 \subseteq \mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_M(I\cup B)) $, which means that $ I\cup
B $ is dually safe.
\end{proof}
\begin{prop}\label{prop: IE_1 common base}
For a dually safe feasible $ I $, $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1} $ is
a base of $ {N^{*}\upharpoonright \mathsf{span}_{M}(I)^{1} } $.
\end{prop}
\begin{proof}
By the definition of `dually safe', $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1} $ spans $ N^{*}\upharpoonright
\mathsf{span}_{M}(I)^{1} $. Furthermore, $ r(N. \overset{\circ}{\mathsf{span}}_{M}(I))=0 $ by Corollary \ref{cor: Mloop},
which is equivalent with the $ N^{*} $-independence of $
\overset{\circ}{\mathsf{span}}_{M}(I) $.
\end{proof}
For a dually safe feasible $ I $,
we define an auxiliary digraph $ D(I) $ on $ E $.
Let $ xy $ be an arc of $ D(I) $ iff
one of the following possibilities occurs:
\begin{enumerate}
\item\label{item: D(I) 1} $ x\in E\setminus I $ and $ I+x $ is $ M $-dependent with $ y\in C_M(x, I)-x $,
\item\label{item: D(I) 2} $x\in I^{0} $ and $ C_{N}(y,I) $ is well-defined and contains $ x $,
\item\label{item: D(I) 3} $x\in I^{1} $ and
$ y\in C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ) -x $ (see
Proposition \ref{prop: IE_1 common base}).
\end{enumerate}
An augmenting path for a nice dually safe feasible $ I $ is a $ P=\{ x_1,\dots, x_{2n+1} \} $ where
\begin{enumerate}
[label=(\roman*)]
\item $ x_1\in E_0\setminus \mathsf{span}_N(I) $,
\item $ x_{2n+1}\in E_0\setminus \mathsf{span}_M(I) $,
\item $ x_kx_{k+1}\in D(I) $ for $ 1\leq k\leq 2n $,
\item\label{item: no jumping} $ x_kx_{\ell}\notin D(I) $ if $ k+1<\ell $.
\end{enumerate}
\begin{prop}\label{p: augpath}
If $ I $ is a dually safe feasible set and $ P=\{ x_1,\dots, x_{2n+1} \} $ is an augmenting path for $ I $, then
$ I\vartriangle P$ is a dually safe element of $ \mathcal{I}_M\cap \mathcal{I}_N $ with
\begin{enumerate}
[label=(\Alph*)]
\item\label{item: A} $\mathsf{span}_{M}(I\vartriangle P)=\mathsf{span}_{M}(I+x_{2n+1}) $,
\item\label{item: B} $\mathsf{span}_{N}(I\vartriangle P)\cap E_0=\mathsf{span}_{N}(I+x_1)\cap E_0 $,
\item\label{item: C} $\mathsf{span}_{N^{*}} (\overset{\circ}{\mathsf{span}}_{M}(I)^{1})=\mathsf{span}_{N^{*}}
(\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1}) $ and $
\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1}\in \mathcal{I}_{N^{*}} $.
\end{enumerate}
\end{prop}
\begin{proof}
The set $ I+x_{2n+1} $ is $ M $-independent by the definition of $ P $. Property \ref{item: no jumping} ensures that we can
apply Proposition \ref{prop: simult change} with $I+x_{2n+1},\ e_j=x_{2j-1},\ f_j=x_{2j}\ (1\leq j\leq n) $ and $ M $ and
conclude that
$ I\vartriangle P\in \mathcal{I}_M $ and \ref{item: A} holds.
To prove $ (I\vartriangle P)\cap E_0\in \mathcal{I}_N $ and
\ref{item: B} we
proceed similarly. We start with the $ N $-independent set $ (I+x_{1})\cap E_0 $. In order to satisfy the premisses of
Proposition \ref{prop: simult change} via property
\ref{item: no jumping}, we need to enumerate the relevant edge
pairs backwards.
Namely, for $j\leq \left| I^{0}\cap P \right|$ let $ e_j:=x_{i_j+1} $ where $ i_j $ is the $ j $th largest index with $ x_{i_j}\in
I^{0} $ and $ f_j:=x_{i_j} $. We conclude that $ (I\vartriangle P)\cap E_0\in \mathcal{I}_N
$ and \ref{item: B}
holds.
Finally, we let $ e_j:=x_{i_j} $ for $ j\leq \left|I^{1}\cap P\right| $ where $ i_j $ is the $j$th smallest index with $
x_{i_j}\in I^{1} $ and $
f_j:=x_{i_j+1} $.
Recall that $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1} $ is $ N^{*} $-independent
(see Proposition \ref{prop: IE_1 common base}).
We apply Proposition \ref{prop: simult change} with
$\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ,\ e_j,\ f_j $ and $ N^{*} $ to conclude \ref{item: C}. This means that $
\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1} $ is a base of $ N^{*}\upharpoonright
\mathsf{span}_{M}(I)^{1}$ because $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1} $ was a base
of it by Proposition \ref{prop: IE_1 common base}. By \ref{item: A} and by the definition of $ P $ we
know that \[ (I\vartriangle P)\cap
E_1\subseteq
I^{1}\cup P^{1}\subseteq \mathsf{span}^{1}_{M}(I).
\] By combining these we obtain \[ (I\vartriangle P)\cap
E_1\subseteq \mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1}). \] The set $
I\vartriangle
P $ is disjoint from
$ \overset{\circ}{\mathsf{span}}_{M}(I)\vartriangle P$ because $ I $ is disjoint
from $
\overset{\circ}{\mathsf{span}}_{M}(I) $, moreover, $ \overset{\circ}{\mathsf{span}}_{M}(I)\vartriangle P $ contained in $
\mathsf{span}_{M}(I\vartriangle P) $. Hence $ \overset{\circ}{\mathsf{span}}_{M}(I\vartriangle P) $
contains $
\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1} $ and therefore $ N^{*} $-spans
$ (I\vartriangle P) \cap E_1 $, i.e. $ I\vartriangle P $ is dually safe. It means that $ (I\vartriangle P)\cap E_1 $ is independent in $
N.\mathsf{span}_{M}(I\vartriangle P)^{1} $.
Thus uniting $ (I\vartriangle P)\cap E_0\in \mathcal{I}_N $
with $ (I\vartriangle P)\cap E_1 $ preserves $ N
$-independence, i.e. $ I\vartriangle P\in \mathcal{I}_N $.
\end{proof}
\begin{lem}\label{l: aug extend}
If $ I $ is a nice dually safe feasible set and $ P=\{ x_1,\dots, x_{2n+1} \} $ is an augmenting path for $ I $, then
$ I \vartriangle P $ can be extended to a nice dually safe feasible set.
\end{lem}
\begin{proof}
Let $ M':= M/(I\vartriangle P)$ and $ N':=N/(I\vartriangle P) $. By Lemma \ref{l: enoughAddB} it is enough to show that
$ B(M', N', W)\neq \varnothing $ for $
W:=W(M', N') $. Statements \ref{item: A} and \ref{item: B} of Proposition \ref{p: augpath} ensure that the elements of $ L:=
\{
x_1, x_3,\dots,
x_{2n-1} \}\cap E_0 $ are common
loops of $ M'$ and $ N' $. Statement \ref{item: C} tells that $ Y_0:=\overset{\circ}{\mathsf{span}}_{M}(I)^{1}
$ and $
Y_1:=\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1} $ have the same $ N^{*}
$-span, furthermore, $ Y_1 $ is $ N^{*} $-independent, i.e. $ r(N.Y_1)=0 $. Note that $ Y_1 $ consists of $ M' $-loops by
\ref{item: A}.
Thus by applying Observation \ref{obs: common loops remove} with $ W $ and $ L $ and then Lemma \ref{l: wave modify}
with $ W\setminus L $ and $ Y_1 $ we can conclude that $ W\setminus (L\cup Y_1) $ is an
$ (M'-Y_1, N'-Y_1) $-wave, furthermore,
\[ B(M',N', W)=B(M'-Y_1, N'-Y_1, W\setminus (L\cup Y_1)). \]
The sets $X_0:= I+x_{2n+1} $ and $X_1:= I\vartriangle P $ have the same $ M $-span (see
Proposition \ref{p: augpath}/ \ref{item: A}). Recall
that $ M'=M/X_1 $ and
$
N'=N/X_1 $ by definition. Hence Lemma \ref{l: minorsChanged}
ensures that $ W\setminus (Y_1\cup L) $ is also an $ (M/X_0-Y_0, M/X_0-Y_0) $-wave with
\[ B(M/X_1-Y_1, N/X_1-Y_1, W\setminus (Y_1\cup L))=B(M/X_0-Y_0, M/X_0-Y_0, W\setminus (Y_1\cup L)). \]
We have $ \mathsf{cond}^{+}(M/I, N/I) $ because $ I $ is a nice feasible set by assumption. Then by
Observation \ref{obs: cond+ loop delete}
$ \mathsf{cond}^{+}(M/I-Y_0, N/I-Y_0) $ also holds. Applying Lemma \ref{one more edge} with $ M/I-Y_0,\ N/I-Y_0,\
W\setminus(Y_0\cup L) $ and $ x_{2n+1}$ tells
$ B(M/X_0-Y_0, M/X_0-Y_0, W\setminus (Y_1\cup L))\neq \varnothing $ which completes the proof.
\end{proof}
\begin{lem}\label{l: arc remain lemma}
If $ P=\{ x_1, \dots, x_{2n+1} \} $ is an augmenting path for $ I $ which contains neither $x$ nor any of its out-neighbours in $
D(I) $, then $xy\in D(I) $ implies $ xy \in D(I \vartriangle P) $.
\end{lem}
\begin{proof}
Suppose that $ xy\in D(I) $. First we assume that $ x\notin I $. Then the set of the out-neighbours of $ x $ is
$ C_M(x,I)-x $. By
assumption $ P\cap C_M(x,I)=\varnothing $ and
therefore $C_M(x,I)\subseteq I \vartriangle P$ thus $ C_M(x,I)=C_M(x,I \vartriangle P) $. This means by definition that $ x $
has the same out-neighbours in
$ D(I) $ and $ D(I \vartriangle P) $.
If $ x\in I^{1} $, then we can argue similarly. The set of the out-neighbours of $ x $ in $ D(I) $ is
$ C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ) -x $. By
assumption $ P\cap C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} )=\varnothing $ and
therefore $ C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ) \cap (I \vartriangle P)=\{ x\} $. Since $
\mathsf{span}^{1}_{M}(I\vartriangle P) \supseteq
\mathsf{span}^{1}_{M}(I)$ because of Proposition \ref{p: augpath}/\ref{item: A}, we also have $
C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} )\subseteq
\mathsf{span}^{1}_{M}(I\vartriangle P) $. By combining these we conclude
\[ C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I\vartriangle P)^{1} )
=C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ). \] This means
by definition that $ x $ has the same out-neighbours in $ D(I) $ and $ D(I \vartriangle P) $.
We turn to the case where $ x\in I^{0} $. By definition $ C_N(y,I) $ is well-defined and
contains $ x $, in particular $ y\in E_0 $.
For $ k\leq n $, let us denote
$ I+x_1-x_2+x_3-\hdots -x_{2k}+x_{2k+1} $ by $ I_k $. Note that
$ {I_n=I \vartriangle P} $. We show by induction on $ k $ that $ I_k $ is $ N $-independent and
$ {x\in C_N(y, I_k)} $. Since
$ I+x_1 $ is $ N $-independent by definition and $ x_1\neq y $ because $ y\notin P$ by assumption, we obtain $
C_N(y,I)=C_N(y,I_0) $, thus for $ k=0 $ it holds. Suppose that $ n>0 $ and
we already know the statement for some $ k<n $. We have $ C_N(x_{2k+3},I_k)=C_N(x_{2k+3},I)\ni x_{2k+2} $ because
there is no ``jumping arc'' in the augmenting path by property \ref{item: no jumping}. It follows via the $ N $-independence of
$ I_k $ that $ I_{k+1} $ is also $ N
$-independent.
If $ x_{2k+2}\notin C_N(y, I_k) $ then $ C_N(y, I_k)=C_N(y, I_{k+1}) $
and the induction step is done. Suppose that $ x_{2k+2}\in C_N(y, I_k) $. Then $x_{2k+2}, x_{2k+3}\in E_0 $, moreover,
$x\notin C_N(x_{2k+3},I) $ since
otherwise $P$ would contain the out-neighbour $x_{2k+3} $ of $ x $ in $ D(I) $. We apply circuit elimination (Claim
\ref{Circuit elim})
with $C= C_N(y, I_k),\ e=x,\ X=\{ x_{2k+2} \},\ C_{x_{2k+2}}=C_N(x_{2k+3},I_k) $. The resulting circuit $ C'\ni x $ can
have at most one element out of $ I_{k+1} $, namely $ y $. Since $ I_{k+1} $ is $ N $-independent, there must be at least one
such an element and therefore $ C'=C_N(y,I_{k+1}) $.
\end{proof}
\begin{obs}\label{arc remain fact}
If $ xy \in D(I) $ and $ J\supseteq I $ is a dually safe feasible set with
$ \{ x,y \}\cap J=\{ x, y \}\cap I $, then $ xy \in D(J) $ (the same circuit is the witness).
\end{obs}
\section{Proof of the key-lemma}\label{s: proof of key-lemma}
\keylemma*
\begin{proof}
It is enough to build a sequence $(I_n)$ of nice dually safe feasible sets such that
$ (\mathsf{span}_N(I_n)\cap E_0) $ is an ascending sequence exhausting $ E_0 $. We fix a well-order $ \boldsymbol{\prec} $
of type $ \left|E_0\right| $ on $ E_0 $. Let $ I_0=\varnothing $, which is a nice dually safe feasible set by $
\mathsf{cond}^{+}(M,N) $.
Suppose that $ I_n $ is already defined. If there is no augmenting path for $ I_n $, then we let $ I_m:=I_n $ for $ m>n $.
Otherwise we pick an augmenting path $ P_n $ for $ I_n $ in such a way that its first element is as $ \prec $-small as possible.
Then we apply Lemma \ref{p: augpath} to extend $ I\vartriangle P $ to a nice dually safe feasible set which we define to be $
I_{n+1} $. The recursion is done.
Let $ \boldsymbol{X}:=E\setminus \bigcup_{n\in \mathbb{N}}\mathsf{span}_{N}(I_n) $ and for $ x\in X $, let $
\boldsymbol{E(x,n)} $ be the set of elements that are
reachable from $ x $ in $ D(I_n) $ by a directed path. We define $\boldsymbol{n_x} $ to be the smallest
natural number such that for every $ y\in E\setminus X $ with $ y \prec x $ we have $ y\in \mathsf{span}_N(I_{n_x}) $.
We shall prove that \[\boldsymbol{W}:= \bigcup_{x\in X}\bigcup_{n\geq n_x}E(x,n) \] is a wave.
\begin{lem}\label{l: stabilazing stuff}
For every $ x\in X $ and $ \ell\geq m\geq n_x $,
\begin{enumerate}
\item\label{item stabilize} $ I_m\cap E(x,m)=I_{\ell}\cap E(x,m) $,
\item\label{item same circuit} $ C_M(y,I_\ell)=C_M(y,I_{m})\subseteq E(x,m)$ for every $y\in E(x,m)\setminus I_m $,
\item \label{item same cocircuit} $C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_\ell)^{1})=
C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_{m})^{1}) $ for every $y\in E(x,m)\cap I_m^{1} $,
\item\label{item subdigraph} If $ yz\in D(I_m)$ with $y,z\in E(x,m)$, then $yz\in D(I_\ell) $,
\item\label{item increasing} $ E(x,m)\subseteq E(x,\ell) $.
\end{enumerate}
\end{lem}
\begin{proof}
Suppose that there is an $ n\geq n_x $ such that we know already the statement whenever $m,\ell \leq n $. For the induction step
it is
enough to
show that the claim holds for $ n $ and $ n+1 $. We may assume that $ P_n $ exists, i.e. $ I_n\neq I_{n+1} $, since otherwise
there is nothing to prove.
\begin{prop}\label{p: nx no aug}
$ P_n \cap E(x,n)=\varnothing $.
\end{prop}
\begin{proof}
A common element of $ P_n $ and $ E(x,n) $ would show that there is also an augmenting path in $ D(I_n) $ starting at $ x $
which is impossible since $ x\in X $ and $ n\geq n_x $.
\end{proof}
\begin{cor}
$ I_{n}\cap E(x,n)=(I_n \vartriangle P_n)\cap E(x,n) $.
\end{cor}
\begin{prop}
$ (I_n \vartriangle P_n)\cap E(x,n)=I_{n+1}\cap E(x,n) $.
\end{prop}
\begin{proof}
If $y\in E(x,n)\setminus I_n $, then its out-neighbours in $ D(I_n) $ are in $ E(x,n)\cap I_n $ and span $ y $ in $ M $.
Thus $I_{n+1}\setminus (I_n \vartriangle P_n)$ cannot contain any edge from $ E(x,n) $.
\end{proof}
\begin{cor}\label{circ subset}
$ I_n\cap E(x,n)=I_{n+1}\cap E(x,n) $ and for every $y\in E(x,n)\setminus I_n $ we have
$ C_M(y,I_n)=C_M(y,I_{n+1})\subseteq E(x,n)$.
\end{cor}
\begin{cor}\label{cor: cocircuit stabil}
For $y\in E(x,n)\setminus I_n $, $ y $ has the same out-neighbours in $ D(I_n) $ and in $ D(I_{n+1}) $ and they span $ y $ in
$ N^{*} $. More concretely:
\[ C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_{n+1})^{1})=
C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_{n})^{1}). \]
\end{cor}
Finally, for $ y\in E(x,n)$, $ P_n $ does not contain $ y $ or any of its out-neighbours with respect
to $ D(I_n) $ because $ P_n\cap E(x,n)=\varnothing $. Hence by applying Lemma \ref{l: arc remain lemma} with $P_n, y$ and
$I_n $ (and then Observation \ref{arc remain fact}) we may conclude that $ yz\in D(I_{n+1}) $ whenever $yz\in D(I_n) $.
This implies $ E(x,n)\subseteq E(x,n+1) $ since reachability from $ x $ is witnessed by the same directed paths.
\end{proof}
Let \[ B:=\bigcup_{m\in \mathbb{N}} \bigcap_{n>m}W\cap I_n. \] We are going to show that $ B $ witnesses that $ W $ is a
wave. Since $ M $ is
finitary the $ M $-independence of the sets $ I_n\cap W $ implies the $ M $-independence of $ B $. Similarly $ B^{0} $ is
independent in $ N $ because $ N \upharpoonright E_0 $ is finitary. Statements (\ref{item stabilize}) and
(\ref{item same circuit}) of Claim \ref{l: stabilazing stuff} ensure $W\subseteq
\mathsf{span}_{M}(B) $, while (\ref{item stabilize}) and (\ref{item same cocircuit}) guarantee $ B^{1} \subseteq
\mathsf{span}_{N^{*}}(W\setminus B) $. The latter means that $ B^{1} $ is independent in $ N.(W^{1}\setminus B^{1}) $.
Suppose for
a contradiction that $ B^{0} $ is not independent in $ N.W^{0} $. Then there exists an $ N $-circuit $ C\subseteq E_0 $ that
meets $ B $
but avoids $ W\setminus B $. We already know that $ B^{0} $ is $ N $-independent thus $ C $ is not contained in $ B $. Hence
$C\setminus B= C\setminus W\neq\varnothing $. Let us pick some $ e\in C\cap B$.
Since $ C $ is finite, for every large enough $ n $ we have $ C\cap B\subseteq C\cap I_n $ and $ I_n $ spans $ C $ in $ N $ (for
the latter we use
$ X\subseteq W\setminus B $). Applying Corollary \ref{cor: Noutgoing arc}
with $I_n, N, C $ and $ e $ tells that $e\in C_N(f,I_n) $
for
some $ f\in C\setminus W $ whenever $ n $ is large enough.
Then by (\ref{item increasing}) of Claim \ref{l: stabilazing stuff} we can take an $ x\in X $ and an $ n\geq n_x $ such that $ e\in
E(x,n) \cap C_N(f,I_n) $
for some $ f\in C\setminus W $. Then by definition $ f \in E(x,n) \subseteq W $ which contradicts $ f\in C\setminus W $. Thus
$ B^{0} $ is indeed
independent in $ N.W^{0} $ and hence $ B $ in $ N.W $ as well therefore $ W $ is a wave.
By $ \mathsf{cond}^{+}(M,N) $ we know that $ W $ consists of $ M $-loops and $ r(N.W)=0 $. It implies $ r(N.X)=0 $
because $ X\subseteq W $ by definition. This means $ X\subseteq \mathsf{span}_N(E\setminus X) $. Since $X\subseteq E_0 $
and $ E_0 $ is the union of the finitary $ N $-components, $ X\subseteq \mathsf{span}_N(E_0\setminus X) $ follows. Thus for
every $ x\in X $
there is a finite $ N $-circuit $ C\subseteq E_0 $ with $ C\cap X=\{ x \} $. The sequence $ (\mathsf{span}_N(I_n)\cap E_0 )$ is
ascending by construction and exhausts $ E_0\setminus X $ by the definition of $ X $. As $ C-x\subseteq E_0\setminus X $ is
finite, this implies that for every large enough $ n $, $ I_n $
spans $ C-x $ in $ N $ and hence spans $ x $ itself as well. But then by the definition of $ X $, we must have
$ X=\varnothing $. Therefore $ (\mathsf{span}_N(I_n)\cap E_0) $ exhausts $ E_0 $ and the proof of
Lemma \ref{l: key-lemma} is complete.
\end{proof}
\section{An application: Degree-constrained orientations of infinite graphs}\label{s: application}
Matroid intersection is a powerful tool in graph theory and in combinatorial optimization. Our generalization
Theorem \ref{t: main result} extends the scope of its applicability to infinite
graphs. To illustrate this, let us consider a classical problem in combinatorial optimization. A graph is given with
degree-constrains and we are looking for either an orientation
that satisfies it or a certain substructure witnessing the non-existence of such an orientation (see \cite{hakimi1965degrees}).
Let a (possibly infinite) graph $ G=(V,E) $ be fixed through this section. We denote the set of edges incident with $ v $ by $
\boldsymbol{\delta(v)} $.
Let $ o: V\rightarrow \mathbb{Z}$ with $ \left|o(n)\right|\leq
d(v) $ for $
v\in V $ which we will threat as `lower bounds' for in-degrees in orientations in the following sense. We say that the orientation $
D $ of $ G $ is
\emph{above} $
o $ at $ v $ if either $
o(v)\geq 0 $ and $ v $ has at least $ o(v) $ ingoing edges in $ D $ or $ o(v)< 0 $ and all but at most $ -o(v) $ edges in $ \delta(v)
$
are oriented towards $ v $ by $ D $. We say \emph{strictly above} if we forbid equality in the definition. Orientation $ D $ is
above $ o $ if it is above $ o $ at every $ v\in V $. We say that $ D $ is (strictly) bellow $ o $ at $ v $ if the reverse of $ D $ is
(strictly) above $ -o(v) $. Finally, $ D $ is (strictly) bellow $ o $ if the reverse of $ D $ is strictly above $ -o $.
\begin{thm}\label{t: indegree demand}
Let $ G=(V,E) $ be a countable graph and let $ o: V\rightarrow \mathbb{Z} $. If there is no
orientation of $ G $ above $ o $, then there is a $ V'\subseteq V $ and an orientation $ D $ of $ G $ such that
\begin{itemize}
\item $ D $ is bellow $ o $ at every $ v\in V' $;
\item There exists a $ v\in V' $ such that $ D $ is strictly bellow $ o $ at $ v $;
\item Every edge between $ V' $ and $ V\setminus V' $ is oriented by $ D $ towards $ V' $.
\end{itemize}
\end{thm}
\begin{proof}
Without loss of generality we may assume that $ G $ is loopless. We define the digraph $
\overset{\leftrightarrow}{G}=(V, A) $ by
replacing each $ e\in E
$ by back and forth arcs $ a_{e}, a'_e $ between the
end-vertices of $ e $. Let $ \delta^{+}(v) $ be the set of the ingoing edges of $ v $ in $ \overset{\leftrightarrow}{G} $. For
$ v\in V $, let $ M_v $ be $
U_{\delta^{+}(v), o(v)} $ if $
o(v)\geq 0 $ and $
U_{\delta^{+}(v),-o(v)}^{*} $ if $ o(v)<0 $. We define $ N_e $ to be $ U_{\{a_e, a'_e \}, 1} $ for $ e\in E $. Let \[
M:=\bigoplus_{v\in V}M_v\text{ and }N:=\bigoplus_{e\in E} N_e. \] Since $ M, N\in
(\mathfrak{F}\oplus \mathfrak{F}^{*})(A) $, Theorem \ref{t: main result} guarantees that there exists an $ I\in
\mathcal{I}_M\cap
\mathcal{I}_N $ and a partition $ A=A_M\sqcup A_N $ such that $I_M:=I\cap A_M $ spans $ A_M $ in $ M $ and $
I_N:=I\cap A_N $ spans $ A_N $ in $ N $. Note that $ \left|I\cap \{ a_e, a'_e \} \right|\leq 1$ by the $ N $-independence of $ I
$. We define $ D $ by taking the orientation $ a_e $ of $ e $ if $ a_e\in I $ and $ a'_e $ otherwise. Let $ V'' $ consists of those
vertices $ v $ for which $ I_M $ contains a base of $ M_v $ and let $ V':=V\setminus V'' $.
We claim that whenever an edge $ e\in E $ is incident with some $ v\in V' $, then $ I $ contains one of $ a_e $ and $ a'_e $.
Indeed, if $ I $ contains none of them then they cannot be $ N $-spanned by $ I_N $ thus they are $ M $-spanned by $ M $ which
implies that both end-vertices of $ e $ belong to $ V'' $, contradiction. Thus if $ e $ is incident some $ v\in V' $, then all ingoing
arcs of $ v $ in $ D $ must be in $ I $. Then the $ M $-independence of $ I $ ensures that $ D $ is bellow $ o $ at $ v $.
Suppose for a contradiction that $ a_e $ is an arc in $ D $ from a $ v\in V' $ to a $ w\in V'' $. As we have already shown, we must
have $ a_e\in I $ . By $ w\in V''
$ we know that $ I_M $ contains a base of
$ M_w $ thus $ a_e\notin I_N $ by the $ M $-independence of $ I $ and therefore $ a_e\in I_M $. But then $ a'_e $ cannot be
spanned by $ I_N $ in $ N $ hence $ a'_e\in \mathsf{span}_M(I_M) $, which means that $ I_M $ contains a base of $ M_v $
contradicting $ v\in V' $. We conclude that all the edges between $ V'' $ and $ V' $ are oriented towards $ V' $ in $ D $. By the
definition of $ V'' $, $ D $ is above $ o $ at every $ w\in V'' $. If $ D $ is also above $ o $ for every $ v\in V' $, then $ D $
is above $ o $. Otherwise there exists a $v\in V' $ such that $ D $ is strictly bellow $ o $ at $ v $, but then $ V' $ is as desired.
\end{proof}
Easy calculation shows that if $ G $ is finite, then the existence of a $ V' $ described in Theorem \ref{t: indegree demand}
implies the non-existence of an orientation above $ o $. Indeed, the total demand by $ o $ on $ V' $ is more than the number of all
the edges that are incident with a vertex in $ V' $. That is why for finite $ G $, ``if'' can be replaced by ``if and only if'' in
Theorem \ref{t: indegree demand}. For an infinite $ G $ it is not always the case. Indeed, let $ G $ be the one-way infinite path $
v_0, v_1,\dots $ and let $ o(v_n)=1 $ for $ n\in \mathbb{N} $. Then orienting edge $ \{ v_{n}, v_{n+1} \} $ towards $ v_{n+1}
$ for each $ n\in \mathbb{N} $ and taking $ V':=V $ satisfies the three points in Theorem \ref{t: indegree demand}. However,
taking the opposite orientation is above $ o $.
A natural next step is to introduce upper bounds $ p:V\rightarrow \mathbb{Z} $ beside the lower bounds
$ o:V\rightarrow \mathbb{Z} $. To avoid trivial obstructions we assume that $ o $ and $ p $ are \emph{consistent} which means
that for every $ v\in V $ there is an orientation $ D_v $
which is above $ o $ at bellow $ p $ at $ v $.
\begin{quest}
Let $ G $ be a countable graph and let $ o, p: V\rightarrow \mathbb{Z} $ be a consistent pair of bounding functions.
Suppose that there are orientations $ D_o $ and $ D_p $ that are above $ o $ and bellow $ p $ respectively. Is there always a
single
orientation $ D $ which is above $ o $ and bellow $ p $?
\end{quest}
The positive answer for finite graphs is not too hard to prove, as far we know its first appearance in the literature is
\cite{frank1978orient}.
\end{document} | 167,955 |
\begin{document}
\RUNAUTHOR{Ma and Qian}
\RUNTITLE{PDODE on computational graphs}
\TITLE{Estimating probabilistic dynamic origin-destination demands using multi-day traffic data on computational graphs}
\ARTICLEAUTHORS{
\AUTHOR{Wei Ma}
\AFF{Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong SAR\\
The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, Guangdong, China\\
\EMAIL{[email protected]}}
\AUTHOR{Sean Qian}
\AFF{Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, USA\\
H. John Heinz III Heinz College, Carnegie Mellon University, Pittsburgh, PA, USA\\
\EMAIL{[email protected]}}
}
\ABSTRACT{
System-level decision making in transportation needs to understand day-to-day variation of network flows, which calls for accurate modeling and estimation of probabilistic dynamic travel demand on networks. Most existing studies estimate deterministic dynamic origin-destination (OD) demand, while the day-to-day variation of demand and flow is overlooked.
Estimating probabilistic distributions of dynamic OD demand is challenging due to the complexity of the spatio-temporal networks and the computational intensity of the high-dimensional problems.
With the availability of massive traffic data and the emergence of advanced computational methods, this paper develops a data-driven framework that solves the probabilistic dynamic origin-destination demand estimation (PDODE) problem using multi-day data.
Different statistical distances ({\em e.g.}, $\ell_p$-norm, Wasserstein distance, KL divergence, Bhattacharyya distance) are used and compared to measure the gap between the estimated and the observed traffic conditions, and it is found that 2-Wasserstein distance achieves a balanced accuracy in estimating both mean and standard deviation.
The proposed framework is cast into the computational graph and a reparametrization trick is developed to estimate the mean and standard deviation of the probabilistic dynamic OD demand simultaneously.
We demonstrate the effectiveness and efficiency of the proposed PDODE framework on both small and real-world networks.
In particular, it is demonstrated that the proposed PDODE framework can mitigate the overfitting issues by considering the demand variation.
Overall, the developed PDODE framework provides a practical tool for public agencies to understand the sources of demand stochasticity, evaluate day-to-day variation of network flow, and make reliable decisions for intelligent transportation systems.
}
\KEYWORDS{Network Modeling, Dynamic Origin Destination Demand, OD Estimation, Computational Graphs}
\HISTORY{Submitted: \today}
\maketitle
\section{Introduction}
The spatio-temporal traffic origin-destination demand is a critical component to dynamic system modeling for transport operation and management. For decades, dynamic traffic demand is deterministically modeled as it is translated into deterministic link/path flow and travel cost. Recent studies on transportation network uncertainty and reliability indicate that the variation of traffic demand also has equally large economic and environmental regional impacts \citep{mahmassani2014incorporating}. However, traffic demand and flow variation from time to time ({\em e.g.}, morning versus afternoon, today versus yesterday) cannot be captured in those deterministic models. In addition, the multi-day large-scale data cannot be characterized by deterministic traffic models. It is essential to consider the flow variation and understand its causes for system-wide decision making in the real-world.
Therefore, modeling and estimating the stochasticity of traffic demand, namely its spatial-temporal correlation/variation, is a real need for public agencies and decision-makers.
In view of this, this paper addresses a fundamental problem to estimate the probabilistic dynamic origin-destination demand (PDOD) on general road networks. The reasons for estimating the PDOD instead of the deterministic dynamic OD demand (DDOD) are four-fold: 1) PDOD enables the modeling of system variation \citep{han2018stochastic}, and hence the corresponding traffic model is more reliable; 2) later we will show that there is a theoretical bias when using the deterministic dynamic OD demand estimation (DDODE) framework with stochastic traffic flow; 2) the probabilistic dynamic OD estimation (PDODE) framework makes full use of multi-day traffic data, and the confidence level of the estimated PDOD can be quantified. In particular, the confidence in estimation accuracy increases when the number of data increases in the PDODE framework; 4) the estimated PDOD facilitates public agencies to operate and manage the stochastic complex road networks more robustly \citep{jin2019behavior}.
Before focusing on the PDODE problem, we first review the large body of literature for DDODE problems, then their extensions to PDODE problems are discussed.
The DDODE problem is originally proposed and solved through a generalized least square (GLS) formulation by assuming the networks are not congested and travelers' behaviors ({\em e.g.} route choice, departure time choice) are exogenous \citep{cascetta1993dynamic}. On congested networks, travelers' behaviors need to be considered endogenously. A bi-level formulation is then proposed on top of the GLS formulation, in which the upper-level problem solves for the GLS formulation with fixed travelers' behaviors and the lower-level problem updates the travelers' behaviors \citep{tavana2001internally}. Readers are referred to more details on the bi-level formulation from \citet{nguyen1977estimating, leblanc1982selection, fisk1989trip, yang1992estimation, florian1995coordinate, jha2004development,nie2008variational}. The DDODE problem can also be solved with real-time data feeds from and for ATIS/ATMS applications, and state-space models are usually adopted to estimate the OD demand on a rolling basis \citep{bierlaire2004efficient, zhou2007structural, ashok2000alternative}. Another interesting trend is that emerging data sources are becoming available to estimate OD demand directly, which include automatic vehicle identification data \citep{cao2021day}, mobile phone data \citep{bachir2019inferring}, Bluetooth data \citep{cipriani2021traffic}, GPS trajectories \citep{ros2022practical}, and satellite images \citep{kaack2019truck}. Unlike static networks \citep{wu2018hierarchical, waller2021rapidex}, an universal framework that can integrate multi-source data is still lacking for dynamic networks.
Solution algorithms to the DDODE problem can be categorized into two types: 1) meta-heuristic methods; 2) gradient-based methods.
Though meta-heuristics methods might be able to search for the global optimal, most studies only handle small networks with low-dimensional OD demand \citep{patil2022methods}. In contrast, gradient-based methods can be applied to large-scale networks without exploiting computational resources.
The performance of gradient-based methods depends on how to accurately evaluate the gradient of the GLS formulation. \citet{balakrishna2008time, cipriani2011gradient} adopt the stochastic perturbation simultaneous approximation (SPSA) framework to approximate the gradients. \citet{lee2009new, vaze2009calibration, ben2012dynamic, lu2015enhanced, tympakianaki2015c, antoniou2015w, oh2019demand, qurashi2019pc} further enhance the SPSA-based methods. \citet{lu2013dynamic} discuss to evaluate the gradients of dynamic OD under congested networks. \citet{flotterod2011bayesian, yu2021bayesian} derives the gradient of OD demand in a Bayesian inference framework. \citet{osorio2019dynamic, osorio2019high, patwary2021metamodel, dantsuji2022novel} develop a meta-model to approximate the gradients of dynamic OD demand through linear models. Recently, \citet{wu2018hierarchical, ma2019estimating} propose a novel approach to evaluate the gradient of OD demand efficiently through the computational graph approach.
A few studies have explored the possibilities of estimating PDOD, and this problem turns out to be much more challenging than the DDODE problem. As far as we know, all the existing studies related to the probabilistic OD demand focus on static networks. For example, a statistical inference framework with Markov Chain Monte Carlo (MCMC) algorithm is proposed to estimate the probabilistic OD demand \citep{hazelton2008statistical}. The GLS formulation is also extended to consider the variance/covariance matrices in order to estimate the probabilistic OD demand \citep{shao2014estimation, shao2015estimation}. \citet{ma2018statistical} estimate the probabilistic OD demand under statistical traffic equilibrium using Maximum Likelihood Estimators (MLE). Recently, \citet{yang2019estimating} adopt the Generalized Method of Moment (GMM) methods to estimate the parameters of probability distributions of OD demand.
Estimating the probabilistic dynamic OD demand (PDOD) is challenging, and the reasons are three-fold: 1) PDODE problem requires modeling the dynamic traffic networks in the probabilistic space, hence a number of existing models need to be adapted or re-formulated \citep{shao2006reliability, nakayama2014consistent, watling2015stochastic, ma2017variance}; 2) estimating the probabilistic OD demand is an under-determined problem, and the problem dimension of PDODE is much higher than that for DDODE \citep{shao2015estimation,ma2018statistical,yang2019estimating}; 3) solving PDODE problem is more computationally intensive than solving the DDODE problem, and hence new approaches need to be developed to improve the efficiency of the solution algorithm \citep{flotterod2017search, ma2018estimating, shen2019spatial}.
In both PDODE and DDODE formulations, travelers' behaviors are modeled through the dynamic traffic assignment (DTA) models. Two major types of DTA models are Dynamic User Equilibrium (DUE) models and Dynamic System Optimal (DSO) models. The DUE models search for the user optimal traffic conditions such that all travelers in the same OD pair have the same utilities \citep{mahmassani1984dynamic, nie2010solving}; DSO models solve the system optimal in which the total disutility is minimized \citep{shen2007path, qian2012system, ma2014continuous}. Most DTA models rely on the Dynamic Network Loading (DNL) models on general networks \citep{ma2008polymorphic}, and the DNL models simulate all the vehicle trajectories and spatio-temporal traffic conditions given the origin-destination (OD) demand and fixed travelers' behaviors.
One noteworthy observation is that many studies have shown great potential in improving the solution efficiency by casting network modeling formulations into computational graphs \citep{wu2018hierarchical, ma2019estimating, sun2019analyzing, zhang2021network, kim2021computational}. The advantages of using computational graphs for PDODE problem lies in that, the computational graph shares similarities with deep neural networks from many perspectives. Hence a number of state-of-art techniques, which are previously developed to enhance the efficiency for solving neural networks, can be directly used for solving the PDODE problem. These techniques include, but are not limited to, adaptive gradient-based methods, dropout \citep{srivastava2014dropout}, GPU acceleration, multi-processing \citep{zinkevich2009slow}. Some of the techniques have been examined in our previous study, and the experimental results demonstrate great potential on large-scale networks \citep{ma2019estimating}. Additionally, multi-source data can be seamlessly integrated into the computational graph to estimate the OD demand.
The success of computational graphs advocates the development of the end-to-end framework, and this paper inherits this idea to estimate the mean and standard deviation of PDOD simultaneously. Ideally, the computational graph involves the variables that will be estimated (decision variables, {\em e.g.}, mean and standard deviation of PDOD), intermediate variables ({\em e.g.}, path/link flow distributions), and observed data ({\em e.g.}, observed traffic volumes), and it finally computes the objective function as a scalar. Among different variables, all the neural network operations can be employed. The chain rule and back-propagation can be adopted to update all the variables on the computational graph. This process is also known as differential programming \citep{jax2018github}. As some of the variables in the computational graph contain physical meaning, we view the computational graph as a powerful tool to integrate data-driven approaches and domain-oriented knowledge. An overview of a computation graph is presented in Figure~\ref{fig:cg}.
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth]{cg}
\caption{\footnotesize{An illustrative figure for computation graphs.}}
\label{fig:cg}
\end{figure}
In this paper, we develop a data-driven framework that solves the probabilistic dynamic OD demand estimation (PDODE) problem using multi-day traffic data on general networks. The proposed framework rigorously formulates the PDODE problem on computational graphs, and different statistical distances ({\em e.g.}, $\ell_p$-norm, Wasserstein distance, KL divergence, Bhattacharyya distance) are used and compared for the objective function.
The closest studies to this paper are those of \citet{wu2018hierarchical, ma2019estimating}, which construct the computational graphs for static and dynamic OD demand estimation, respectively. This paper extends the usage of computation graphs to solve PDODE problems. The main contributions of this paper are summarized as follows:
\begin{enumerate}[label=\arabic*)]
\item We illustrate the potential bias in the DDODE framework when dynamic OD demands are stochastic.
\item We rigorously formulate the probabilistic dynamic OD estimation (PDODE) problem, and different statistical distances are compared for the objective function. It is found that $\ell_1$ and $\ell_2$ norms have advantages in estimating the mean and standard deviation of the PDOD, respectively, and the 2-Wasserstein distance achieves a balanced accuracy in estimating both mean and standard deviation.
\item The PDODE formulation is vectorized and cast into a computational graph, and a reparameterization trick is first time developed to estimate the mean and standard deviation of the PDOD simultaneously using adaptive gradient-based methods.
\item We examine the proposed PDODE framework on a large-scale network to demonstrate its effectiveness and computational efficiency.
\end{enumerate}
The remainder of this paper as organized as follows. Section~\ref{sec:example} illustrates the necessity of the PDODE, and section~\ref{sec:model} presents the proposed model formulation and casts the formulation into computational graphs. Section~\ref{sec:solution} proposes a novel solution algorithm with a reparameterization trick. Numerical experiments on both small and large networks are conducted in section~\ref{sec:experiment}. Finally, conclusions and future research are summarized in section \ref{sec:con}.
\section{An Illustrative Example}
\label{sec:example}
To illustrate the necessity of considering the demand variation when the traffic flow is stochastic, we show that the DDODE framework can under-estimate the DDOD (mean of PDOD) when traffic flow is stochastic. Considering a simple 2-link network with a bottleneck, as shown in Figure~\ref{fig:bottle}, the capacity of the bottleneck is 2000vehicles/hour, and the incoming flow follows a Gaussian distribution with mean 2000vehicles/hour. Due to the limited bottleneck capacity, we can compute the probability density functions (PDFs) of the queue accumulative rate and flow rate on the downstream link, as shown in Figure~\ref{fig:bottle}.
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth]{example.png}
\caption{\footnotesize{A simple network with a bottleneck.}}
\label{fig:bottle}
\end{figure}
Suppose link 2 is installed with a loop detector, we aim to estimate the OD demand from Origin to Destination. If the DDODE method is used, we ignore the variation in the traffic flow observation on link 2, and the mean traffic flow is used, which is below 2000vehicles/hour. Therefore, the estimated OD demand will be less than 2000vehicles/hour. One can see the demand is under-estimated, and the bias is due to the ignorance of the flow variation. In contrast, the flow variation is considered in our proposed model. By matching the PDF of the observed traffic flow, the distribution of the OD demand can be estimated in an unbiased manner if the model specifications of the OD demand is correct. Overall, considering flow variation could improve the estimation accuracy of traffic demand, which motivates the development of the PDODE framework.
\section{Model}
\label{sec:model}
In this section, we first present model assumptions. Then important components of the probabilistic traffic dynamics on general networks, which include PDOD, route choice models, and network loading models, are discussed. The PDODE problem is formulated and cast to a vectorized representation using random vectors. Lastly, we propose different statistical distances as the objective function.
\subsection{Assumptions}
Let $Q_{rs}^h$ represent the dynamic traffic demand (number of travelers departing to travel) from OD pair $r$ to $s$ in time interval $h$, where $r \in R, s\in S$ and $h \in H$. $R$ is the set of origins, $S$ is the set of destinations and $H$ is the set of all time intervals.
\begin{assumption}
\label{as:mvn}
The probabilistic dynamic OD demand (PDOD) follows multivariate Gaussian distribution with diagonal covariance matrix. We also assume that $Q_{rs}^h$ is bounded such that $Q_{rs}^h \geq 0$ for the sake of simplification. Readers are referred to \citet{nakayama2016effect} for more discussions about the assumptions of bounded Gaussian distribution of OD demand.
\end{assumption}
\begin{assumption}
The dynamic traffic flows, including OD demand, path flow, and link flow, are infinitesimal. Therefore, the variation of travelers' choice is not considered \citep{ma2017variance}.
\end{assumption}
\subsection{Modeling the probabilistic network dynamics}
We present different components and their relationship on a probabilistic and dynamic network.
\subsubsection{Probabilistic dynamic OD demand}
The dynamic OD demand $Q_{rs}^h$ is an univariate random variable, and it can be decomposed into two parts, as shown in Equation~\ref{eq:od}.
\begin{eqnarray}
\label{eq:od}
Q_{rs}^h = q_{rs}^h + \varepsilon_{rs}^h
\end{eqnarray}
where $q_{rs}^h$ is the mean OD demand for OD pair $rs$ in time interval $h$ and it is a deterministic scalar; while $\varepsilon_{rs}^h$ represents the randomness of OD demand. Based on Assumption~\ref{as:mvn}, $\varepsilon_{rs}^h$ follows the zero-mean Gaussian distribution, as presented in Equation~\ref{eq:random}.
\begin{eqnarray}
\label{eq:random}
\varepsilon_{rs}^h \sim \mathcal{N}\left(0, \left(\sigma_{rs}^h\right)^2 \right)
\end{eqnarray}
where $\sigma_{rs}^h$ is the standard deviation of $Q_{rs}^h$, and $\mathcal{N}(\cdot, \cdot)$ represents the Gaussian distribution.
\subsubsection{Travelers' Route Choice}
To model the travelers' route choice behaviors, we define the time-dependent route choice portion $p_{rs}^{kh}$ such that it distributes OD demand $Q_{rs}^{h}$ to path flow $F_{rs}^{kh}$ by Equation~\ref{eq:ODpath}.
\begin{eqnarray}
\label{eq:ODpath}
F_{rs}^{kh} = p_{rs}^{kh} Q_{rs}^h
\end{eqnarray}
where $F_{rs}^{kh}$ is the path flow (number of travelers departing to travel along a path) for $k$th path in OD pair $rs$ in time interval $h$. The route choice portion $p_{rs}^{kh}$ can be determined through a generalized route choice model, as presented in Equation~\ref{eq:gen_choice}.
\begin{eqnarray}
\label{eq:gen_choice}
p_{rs}^{kh} = \Psi_{rs}^{kh}\left( \D\left(\{C_{rs}^{kh}\}_{rskh}\right), \D\left(\{T_{a}^h\}_{ah}\right)\right)
\end{eqnarray}
where $\Psi_{rs}^{kh}$ is the generalized route choice model and the operator $\D(\cdot)$ extracts all the parameters in a certain distribution. For example, if $Y\sim \N(\mu, \sigma^2)$, then $\D(Y) = (\mu, \sigma)$. $T_{a}^{h}$ represents the link travel time for link $a$ in time interval $h$, and $C_{rs}^{kh}$ represents the path travel time for $k$th path in OD pair $rs$ departing in time interval $h$. Equation~\ref{eq:gen_choice} indicates that the route choice portions are based on the distributions of link travel time and path travel time. In this paper we use travel time as the disutility function, while any form of disutility can be used as long as it can be simulated by $\Lambda$. The generalized travel time can include roads tolls, left turn penalty, delay at intersessions, travelers' preferences and so on.
\subsubsection{Dynamic network loading} For a dynamic network, the network conditions ({\em i.e.} path travel time, link travel time, delays) are governed by the link/node flow dynamics, which can be modeled through dynamic network loading (DNL) models \citep{ma2008polymorphic}. Let $\Lambda(\cdot)$ represent the DNL model, as presented in Equation~\ref{eq:dnl}.
\begin{eqnarray}
\label{eq:dnl}
\left\{T_{a}^{h}, C_{rs}^{kh}, {\rho}_{rs}^{ka}(h, h') \right\} _{r,s,k,a,h, h'} = \Lambda(\{F_{rs}^{kh}\}_{r,s,k,h})
\end{eqnarray}
where $\rho_{rs}^{ka}(h, h')$ is the dynamic assignment ratio (DAR) which represents the portion of the $k$th path flow departing within time interval $h$ for OD pair $rs$ which arrives at link $a$ within time interval $h'$ \citep{ma2018estimating}. Link $a$ is chosen from the link set $A$, and path $k$ is chosen from the set of all paths for OD $rs$, as represented by $a \in A, k \in K_{rs}$. We remark that $T_{a}^{h}, C_{rs}^{kh}, \rho_{rs}^{ka}(h, h')$ are random variables as they are the function outputs of the random variable $F_{rs}^{kh}$.
The DNL model $\Lambda$ depicts the network dynamics through traffic flow theory \citep{zhang2013modelling,jin2012link}. Essentially, many existing traffic simulation packages, which include but are not limited to, MATSIM \citep{balmer2009matsim}, Polaris \citep{stolte2002polaris}, BEAM \citep{sheppard2017modeling}, DynaMIT \citep{ben1998dynamit}, DYNASMART \citep{mahmassani1992dynamic}, DTALite \citep{zhou2014dtalite} and MAC-POSTS \citep{qian2016dynamic, CARTRUCK}, can be used as function $\Lambda$. In this paper, MAC-POSTS is used as $\Lambda$.
Furthermore, link flow can be modeled by Equation~\ref{eq:link}.
\begin{eqnarray}
\label{eq:link}
X_a^{h'} = \sum_{rs \in K_q} \sum_{k \in K_{rs}} \sum_{h \in H}{\rho}_{rs}^{ka}(h, h') F_{rs}^{kh}
\end{eqnarray}
where $X_a^{h'}$ represents the flow of link $a$ that arrives in time intervel $h'$, and $K_q$ is the set of all OD pairs.
\subsubsection{Statistical equilibrium}
The route choice proportion $p_{rs}^{kh}$ is a deterministic variable rather than a random variable, and the reason is that we assume travelers' behaviors are based on the statistical equilibrium originally defined in \citet{ma2017variance}, as presented in Definition~\ref{def:equi}.
\begin{definition}
\label{def:equi}
A road network is under statistical equilibrium, if all travelers practice the following behavior: on each day, each traveler from origin $r$ to destination $s$ independently chooses route $k$ with a deterministic probability $p_{rs}^k$. For a sufficient number of days, this choice behavior yields a stabilized distribution of travel costs with parameters $\D\left(\{C_{rs}^{kh}\}_{rskh}\right), \D\left(\{T_{a}^h\}_{ah}\right)$. This stabilized distribution, in turn, results in the deterministic probabilities $p_{rs}^{kh} = \Psi_{rs}^k\left( \D\left(\{C_{rs}^{kh}\}_{rskh}\right), \D\left(\{T_{a}^h\}_{ah}\right)\right)$ where $\psi_{rs}^k(\cdot)$ is a general route choice function. Mathematically, we say the network is under statistical equilibrium when the random variables $(\{Q_{rs}^h \}_{rsh},\{F_{rs}^{kh}\}_{rskh}, \{X_a^h\}_{ah}, \{C_{rs}^{kh}\}_{rskh}, \{T_a^h\}_{ah})$ are consistent with the Formulation \ref{eq:ODpath}, \ref{eq:gen_choice} and \ref{eq:dnl}.
\end{definition}
The statistical equilibrium indicates that the multi-day traffic conditions are independent and identically distributed, which differs from the assumptions in day-to-day traffic models. Readers are referred to \citet{ma2017variance} for more details. The assumption of statistical equilibrium allows us to estimate the distribution of link/path/OD flow from the observed data, and the empirical covariance matrix of link/path/OD flow can approximate the corresponding true covariance matrix when there is a large number of data.
\subsubsection{Vectorization}
To simplify notations, all the related variables are vectorized. We set $N = |H|$ and denote the total number of paths as $\Pi = \sum_{rs} |K_{rs}|$, $K=|K_q|$. The vectorized variables are presented in Table~\ref{tab:mcvec}.
\begin{table*}[h]
\begin{center}
\caption{Variable vectorization table (R.V.: random variable).}
\label{tab:mcvec}
\begin{tabular}{p{3cm}cccccp{4.5cm}}
\hline
Variable & R.V. &Scalar & Vector& Dimension & Type & Description\\
\hline\hline \rule{0pt}{3ex}
Mean OD flow & No &$q_{rs}^h$ & $\vec{q}$ &$\mathbb{R}^{NK}$ & Dense & $q_{rs}^h$ is placed at entry $(h-1)K + k$\\
\hline \rule{0pt}{3ex}
Standard deviation of OD flow & No &$\sigma_{rs}^h$ & $\boldsymbol \sigma$ &$\mathbb{R}^{NK}$ & Dense & $\sigma_{rs}^h$ is placed at entry $(h-1)K + k$\\
\hline \rule{0pt}{3ex}
Randomness of OD flow & Yes &$\varepsilon_{rs}^h$ & $\boldsymbol \varepsilon$ &$\mathbb{R}^{NK}$ & Dense & $\varepsilon_{rs}^h$ is placed at entry $(h-1)K + k$\\
\hline \rule{0pt}{3ex}
OD flow & Yes&$Q_{rs}^h$ & $\vec{Q}$ &$\mathbb{R}^{NK}$ & Dense & $Q_{rs}^h$ is placed at entry $(h-1)K + k$\\
\hline \rule{0pt}{3ex}
Path flow & Yes &$F_{rs}^{kh}$ &$\mathbf{F}$ & $\mathbb{R}^{N\Pi}$ & Dense & $F_{rs}^{kh}$ is placed at entry $(h-1)\Pi + k$\\
\hline \rule{0pt}{3ex}
Link flow & Yes &$X_{a}^h$ & $\mathbf{X}$ &$\mathbb{R}^{N|A|}$ & Dense & $X_{a}^h$ is placed at entry $(N-1)|A| + k$\\
\hline \rule{0pt}{3ex}
Link travel time & Yes&$T_{a}^h$ & $\mathbf{T}$ &$\mathbb{R}^{N|A|}$ & Dense & $T_{a}^h$ is placed at entry $(N-1)|A| + k$\\
\hline \rule{0pt}{3ex}
Path travel time & Yes&$C_{rs}^{kh}$ & $\mathbf{C}$ &$\mathbb{R}^{N\Pi}$ & Dense & $C_{rs}^{kh}$ is placed at entry $(N-1)\Pi + k$\\
\hline \rule{0pt}{3ex}
DAR matrix & Yes&$\rho_{rs}^{ka}(h, h')$ &$\boldsymbol \rho$ & $\mathbb{R}^{N|A| \times N\Pi}$ & Sparse & $\rho_{rs}^{ka}(h, h')$ is placed at entry $[(h'-1)|A| + a, (h-1)\Pi + k]$\\
\hline \rule{0pt}{3ex}
Route choice matrix & No&$p_{rs}^{kh}$ &$\mathbf{p}$ & $\mathbb{R}^{N\Pi \times NK}$ & Sparse & $p_{rs}^{kh}$ is placed at entry $[(h-1)|\Pi| + k, (h-1)K + rs]$\\
\hline
\end{tabular}
\end{center}
\end{table*}
Using the notations presented in Table~\ref{tab:mcvec}, we can rewrite Equation~\ref{eq:od}, \ref{eq:random}, \ref{eq:ODpath}, \ref{eq:gen_choice}, \ref{eq:dnl}, and \ref{eq:link} in Equation~\ref{eq:vec}.
\begin{equation}
\label{eq:vec}
\begin{array}{llllll}
\vspace{5pt}
\vec{Q} &=& \vec{q} + \boldsymbol \varepsilon\\
{\boldsymbol \varepsilon} &\sim& \N\left(\vec{0}, {\boldsymbol \sigma}^2\right)\\
\vec{F} &=& \vec{p}\vec{Q}\\
\vec{p} &= & \Psi \left( \D \left(\vec{C} \right), \D \left(\vec{T}\right) \right)& \\
\left \{ \vec{C}, \vec{T}, {\boldsymbol\rho} \right\} &= & \Lambda(\vec{F}) & \\
\vec{X} &=&{\boldsymbol \rho} \vec{F}
\end{array}
\end{equation}
where ${\boldsymbol \sigma}^2$ denotes the element-wise square for matrix ${\boldsymbol \sigma}$. In the rest of this paper, we will use the vectorized notations for simplicity.
\subsection{Formulating the PDODE problem}
The PDODE problem is formulated in this section. In particular, different objective functions are discussed.
\subsubsection{Objective function}
\label{sec:obj}
To formulate the PDODE problem, we first define the objective function in the optimization problem. DDODE problem minimizes the gap between the estimated (reproduced) and the observed traffic conditions. The gap is usually measured through $\ell^2$-norm, which is commonly used to measure the distance between two deterministic variables. However, the PDODE problem is formulated in the probabilistic space, and we need to measure the distance between the distributions of the observed traffic conditions and the estimated (reproduced) traffic conditions.
To this end, we define a generalized form to measure the observed and estimated distribution of traffic conditions, as presented by Equation~\ref{eq:ere}.
\begin{eqnarray}
\label{eq:ere}
\mathcal{L}_0 = \mathcal{M} \left(\tilde{\mathbf{X}}, \mathbf{X}(\mathbf{Q})\right)
\end{eqnarray}
where $\mathcal{M}$ measures the statistical distance between two distributions, which is defined in Definition~\ref{def:stat}.
\begin{definition}
\label{def:stat}
The statistical distance $\mathcal{M}(\mathbf{X}_1, \mathbf{X}_2)$ is defined as the distance between two random vectors ({\em i.e.}, two probabilistic distributions) $\mathbf{X}_1$ and $\mathbf{X}_2$, and it should satisfy two properties: 1) $\mathcal{M}(\mathbf{X}_1, \mathbf{X}_2) \geq 0, \forall~\mathbf{X}_1, \mathbf{X}_2 $; 2) $\mathcal{M}(\mathbf{X}_1, \mathbf{X}_2) = 0 \iff \mathbf{X}_1 = \mathbf{X}_2$.
\end{definition}
The statistical distance may not be symmetric with respect to $\mathbf{X}_1$ and $\mathbf{X}_2$, and hence it may not be viewed as a metric. Various statistical distances can be used for $\mathcal{M}$, and we review existing literature to list out some commonly used distances that have explicit form for Gaussian distributions. We further simplify the notation $\mathbf{X}(\mathbf{Q})$ to $\mathbf{X}$, and we assume $\tilde{\mathbf{X}} \sim \mathcal{N}\left( \tilde{\mathbf{x}}, \Sigma_{\tilde{\mathbf{X}}} \right)$ and $\mathbf{X} \sim \mathcal{N}\left( \mathbf{x}, \Sigma_{\mathbf{X}} \right)$, then different statistical distances can be computed as follows.
\begin{itemize}
\item $\ell_p$-norm on distribution parameters: this metric directly compare the $\ell_p$-norm of the mean vector and covariance matrix, which can be written as:
$$\|\tilde{\mathbf{x}} - \mathbf{x}\|_p + \| \Sigma_{\tilde{\mathbf{X}}} - \Sigma_{\mathbf{X}}\|_p $$
\item Wasserstein distance: the 2-Wasserstein distance has close-form for Gaussian distributions, and $\mathcal{M}\left(\tilde{\mathbf{X}}, \mathbf{X}\right)$ can be written as:
$$\|\tilde{\mathbf{x}} - \mathbf{x}\|_2^2 + \text{Tr}\left( \Sigma_{\tilde{\mathbf{X}}} + \Sigma_{\mathbf{X}} -2 \left(\Sigma_{\tilde{\mathbf{X}}}^{1/2} \Sigma_{\mathbf{X}} \Sigma_{\tilde{\mathbf{X}}}^{1/2} \right)^{1/2} \right)$$
\item Kullback–Leibler (KL) divergence: also known as relative entropy. KL divergence is not symmetric, and we choose the forward KL divergence to avoid taking inverse of $\Sigma_{\mathbf{X}}$, which can be written as
$$\frac{1}{2} \left[ \log \frac{|\Sigma_{\tilde{\mathbf{X}}}|}{|\Sigma_{\mathbf{X}}|} + (\tilde{\mathbf{x}} - \mathbf{x})^T\Sigma_{\tilde{\mathbf{X}}}^{-1} (\tilde{\mathbf{x}} - \mathbf{x}) + \text{Tr}\left(\Sigma_{\tilde{\mathbf{X}}}^{-1} \Sigma_{\mathbf{X}}\right) \right]. $$
We note that KL divergence can be further extended to Jensen–Shannon (JS) divergence, while it requires to take the inverse of $\Sigma_{\mathbf{X}}$, so we will not consider it in this study.
\item Bhattacharyya distance: we set $\Sigma = \frac{\Sigma_{\tilde{\mathbf{X}}} + \Sigma_{\mathbf{X}}}{2}$, then $\mathcal{M}\left(\tilde{\mathbf{X}}, \mathbf{X}\right)$ can be written as:
$$\frac{1}{8}(\tilde{\mathbf{x}} - \mathbf{x})^T\Sigma^{-1} (\tilde{\mathbf{x}} - \mathbf{x}) + \frac{1}{2}\ln \frac{|\Sigma|}{\sqrt{|\Sigma_{\tilde{\mathbf{X}}}| |\Sigma_{\mathbf{X}}|}}$$
\end{itemize}
All the above statistical distances satisfy Definition~\ref{def:stat}, and they are continuous with respect to the distribution parameters. More importantly, all the statistical distances are differentiable, as each of the used operation is differentiable and the auto-differentiation techniques can be used to derive the overall gradient of the statistical distances with respect to the distribution parameters \citep{speelpenning1980compiling}. Theoretically, all the above distances can be used as the objective function in PDODE, while we will show in the numerical experiments that their performance can be drastically different.
\subsubsection{PDODE formulation}
To simulate the availability of multi-day traffic data, we assume that $I$ day's traffic counts data are collected, and $\tilde{\mathbf{X}}^{(i)}$ is the $i$th observed link flow data, $i= 1, 2, \cdots, I$, and $\tilde{\mathbf{X}}^{(i)}$ iid follows the distribution of $\tilde{\mathbf{X}}$.
Because the actual distributions of $\tilde{\mathbf{X}}$ and $\mathbf{X}$ are unknown, we use the Monte-Carlo approximation to approximate $\mathcal{L}_0$, as presented in Equation~\ref{eq:ereap}.
\begin{eqnarray}
\label{eq:ereap}
\mathcal{L} &=& \mathbb{E}_{\left({\boldsymbol \alpha}, {\boldsymbol \beta}\right) \sim
{\tilde{\mathbf{X}}}^{\bigotimes I} \bigotimes {\mathbf{X}}^{\bigotimes L}} \mathcal{M} \left(\boldsymbol \alpha, \boldsymbol \beta\right) \nonumber\\
&=&\frac{1}{IL}\sum_{i=1}^I \sum_{l=1}^{L} \mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right) \label{eq:L}
\end{eqnarray}
where $I, L$ are the number of samples from distributions of $\tilde{\mathbf{X}}$ and $\mathbf{X}$, respectively, and $\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}$ are the sampled distributions of $\tilde{\mathbf{X}}$ and $\mathbf{X}$, respectively.
By the law of large numbers (LLN), $\mathcal{L}$ converges to $\mathcal{L}_0$ when $I,L \to \infty$.
Combining the constraints in Equation~\ref{eq:vec} and the objective function in Equation~\ref{eq:L}, now we are ready to formulate the PDODE problem in Formulation~\ref{eq:pdode1}.
\begin{equation}
\label{eq:pdode1}
\begin{array}{rrcllll}
\vspace{5pt}
\displaystyle \min_{\vec{q}, {\boldsymbol \sigma}} & \multicolumn{4}{l}{\displaystyle \frac{1}{IL}\sum_{i=1}^I \sum_{l=1}^{L} \mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right)} &\\
\textrm{s.t.} & \left \{ \vec{C}^{(l)}, \vec{T}^{(l)}, {\boldsymbol \rho}^{(l)} \ \right\} &= & \Lambda(\vec{F}^{(l)}) & \forall l&\\
~ & \vec{p}^{(l)} &= & \Psi \left( \D(\vec{C}^{(l)}), \D(\vec{T}^{(l)}) \right)& \forall l &\\
~ & \mathbf{Q}^{(l)} &\sim& \mathcal{N}\left(\vec{q}, {\boldsymbol \sigma}^2\right)&\forall l&\\
~ & \vec{F}^{(l)} & = & \vec{p}^{(l)}\vec{Q}^{(l)} & \forall l&\\
~ & \mathbf{X}^{(l)} & = & {\boldsymbol \rho}^{(l)} \vec{F}^{(l)} &\forall l &
\end{array}
\end{equation}
where $\vec{C}^{(l)}, \vec{T}^{(l)}, \mathbf{X}^{(l)}, \vec{F}^{(l)}, \mathbf{Q}^{(l)}, {\boldsymbol \rho}^{(l)}, \vec{p}^{(l)}$ are the sample distributions of $\vec{C}, \vec{T}, \mathbf{X}, \vec{F}, \mathbf{Q}, {\boldsymbol \rho}, \vec{p}$, respectively. Formulation~\ref{eq:pdode1} searches for the optimal mean and standard deviation of the dynamic OD demand to minimize the statistical distance between the observed and estimated link flow distributions such that the DNL and travelers' behavior models are satisfied. We note that Formulation~\ref{eq:pdode1} can be extended to include the traffic speed, travel time, and historical OD demand data \citep{ma2019estimating}. It is straightforward to show that Formulation~\ref{eq:pdode1} is always feasible, as long as the sampled PDOD is feasible to the traffic simulator, as presented in Proposition~\ref{prop:fea}.
\begin{proposition}[Feasibility]
\label{prop:fea}
There exist a feasible solution $(\vec{q}, {\boldsymbol \sigma})$ to Formulation~\ref{eq:pdode1} if the non-negative support of the distribution $\mathcal{N}\left(\vec{q}, {\boldsymbol \sigma}^2\right)$ is feasible to the traffic simulator $\Lambda$.
\end{proposition}
To compute $\mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right)$, we first characterize the distribution of $\vec{X}^{(l)}$ by $\vec{X}^{(l)} = \Lambda(\vec{p}^{(l)} \vec{Q}^{(l)})$.
Hence the computation of the $\mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right)$ is based on the distribution of $\vec{Q}^{(l)}$, as the distribution of $\mathbf{X}^{(l)}$ is obtained from $\mathbf{Q}^{(l)}$. Additionally, the sample distribution $\mathbf{Q}^{(l)}$ is further generated from $\mathbf{Q}^{(l)} \sim \mathcal{N}\left(\vec{q}, {\boldsymbol \sigma}^2\right)$.
Formulation~\ref{eq:pdode1} is challenging to solve because the derivatives of the loss function with respect to $\vec{q}$ and ${\boldsymbol \sigma}$ are difficult to obtain. The reason is that $\mathbf{Q}^{(l)}$ is sampled from the Gaussian distribution, and it is difficult to compute $\frac{\partial \mathbf{Q}^{(l)}}{\partial \vec{q}}$ and $\frac{\partial \mathbf{Q}^{(l)}}{\partial {\boldsymbol \sigma}}$. Without the closed-form gradients, most existing studies adopt a two-step approach to estimate the PDOD. The first step estimates the OD demand mean and the second step estimates the standard deviation. The two steps are conducted iteratively until convergence \citep{ma2018statistical, yang2019estimating}.
In this paper, we propose a novel solution to estimate the mean and standard deviation simultaneously by casting the PDODE problem into computational graphs. Details will be discussed in the following section.
\section{Solution Algorithm}
\label{sec:solution}
In this section, a reparameterization trick is developed to enable the simultaneous estimation of mean and standard deviation of the dynamic OD demand. The PDODE formulation in Equation~\ref{eq:pdode1} is then cast into a computational graph. We then summarize the step-by-step solution framework for PDODE. Finally, the underdetermination issue of the PDODE problem is discussed .
\subsection{A key reparameterization trick}
To solve Formulation~\ref{eq:pdode1}, our objective is to directly evaluate the derivative of both mean and standard deviation of OD demand, {\em i.e.} $\frac{\partial \mathcal{L}}{\partial \mathbf{q}}$ and $\frac{\partial \mathcal{L}}{\partial {\boldsymbol \sigma}}$, then gradient descent methods can be used to search for the optimal solution.
We will leave the computation of $\frac{\partial \mathcal{L}}{\partial \mathbf{q}}$ and $\frac{\partial \mathcal{L}}{\partial {\boldsymbol\sigma}}$ in the next section, while this section addresses a key issue of evaluating $\frac{\partial \vec{Q}^{(l)}}{\partial \mathbf{q}}$ and $\frac{\partial \vec{Q}^{(l)}}{\partial {\boldsymbol\sigma}}$. The idea is actually simple and straightforward. Instead of directly sampling $\vec{Q}^{(l)}$ from $\mathcal{N}\left(\vec{q}, {\boldsymbol\sigma}^2\right)$, we conduct the following steps to generate $\vec{Q}^{(l)}$: 1) Sample ${\boldsymbol\nu}^{(l)} \in \mathbb{R}^{NK}$ from $\mathcal{N}\left(\vec{0}, \mathbf{1}\right)$; 2) Obtain $\vec{Q}^{(l)}$ by $\vec{Q}^{(l)} = \vec{q} + {\boldsymbol\sigma} \circ {\boldsymbol\nu}^{(l)}$, where $\circ $ represents the element-wise product.
Through the above reparameterization trick, we can compute the derivatives $\frac{\partial \vec{Q}^{(l)}}{\partial \mathbf{q}}$ and $\frac{\partial \vec{Q}^{(l)}}{\partial {\boldsymbol\sigma}}$ by Equation~\ref{eq:odd}.
\begin{equation}
\label{eq:odd}
\begin{array}{llllll}
\frac{\partial \vec{Q}^{(l)}}{\partial \mathbf{q}} &=& \vec{1}_{NK}\\
\frac{\partial \vec{Q}^{(l)}}{\partial {\boldsymbol\sigma}} &=& {\boldsymbol\nu}^{(l)}
\end{array}
\end{equation}
where $\vec{1}_{NK} \in \mathbb{R}^{NK}$ is a vector filled with $1$. This reparameterization trick is originally used to solve the variational autoencoder (VAE) \citep{kingma2013auto}, and we adapt it to solve the PDODE problem.
\subsection{Reformulating PDODE through computational graphs}
With the reparameterization trick discussed in the previous section, we can reformulate the PDODE problem in Equation~\ref{eq:pdode2}.
\begin{equation}
\label{eq:pdode2}
\begin{array}{rrcllll}
\vspace{5pt}
\displaystyle \min_{\vec{q}, {\boldsymbol \sigma}} & \multicolumn{4}{l}{\displaystyle \frac{1}{IL}\sum_{i=1}^I \sum_{l=1}^{L} \mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right)} &\\
\textrm{s.t.} & \left \{ \vec{C}^{(l)}, \vec{T}^{(l)}, {\boldsymbol \rho}^{(l)} \ \right\} &= & \Lambda(\vec{F}^{(l)}) & \forall l&\\
~ & \vec{p}^{(l)} &= & \Psi \left( \D(\vec{C}^{(l)}), \D(\vec{T}^{(l)}) \right)& \forall l &\\
~ & {\boldsymbol\nu}^{(l)} &\sim& \mathcal{N}\left(\vec{0}, \mathbf{1}\right)& \forall l&\\
~ & \mathbf{Q}^{(l)} & = & \vec{q} + {\boldsymbol\sigma}\circ{\boldsymbol\nu}^{(l)}& \forall l&\\
~ & \vec{F}^{(l)} & = & \vec{p}^{(l)}\vec{Q}^{(l)} & \forall l&\\
~ & \mathbf{X}^{(l)} & = & {\boldsymbol \rho}^{(l)} \vec{F}^{(l)} &\forall l &
\end{array}
\end{equation}
Extending the forward-backward algorithm proposed by \citet{ma2019estimating}, we can solve Formulation~\ref{eq:pdode2} through the forward-backward algorithm. The forward-backward algorithm consists of two major components: 1) the forward iteration computes the objective function of Formulation~\ref{eq:pdode2}; 2) the backward iteration evaluates the gradients of the objective function with respect to the mean and standard deviation of the dynamic OD demand ($\frac{\partial \mathcal{L}}{\partial \mathbf{q}}, \frac{\partial \mathcal{L}}{\partial {\boldsymbol\sigma}}$).
\begin{figure*}[h]
\centering
\includegraphics[width=0.95\linewidth]{diagram}
\caption{The computational graph for PDODE.}
\label{fig:fb}
\end{figure*}
{\bf Forward iteration.} In the forward iteration, we compute the objective function based on the sample distribution of observation $\tilde{\mathbf{X}}^{(i)}$ in a decomposed manner, as presented in Equation~\ref{eq:forward}.
\begin{equation}
\begin{array}{lllllll}
\label{eq:forward}
{\boldsymbol\nu}^{(l)} &\sim& \mathcal{N}\left(\vec{0}, \mathbf{1}\right)&\forall l\\
\mathbf{Q}^{(l)} & = & \vec{q} + {\boldsymbol\sigma} {\boldsymbol \nu}^{(l)}&\forall l\\
\vec{F}^{(l)} &=& \vec{p}^{(l)} \vec{Q}^{(l)}&\forall l\\
\mathbf{X}^{(l)} &=& {\boldsymbol \rho}^{(l)} \vec{F}^{(l)}&\forall l\\
\mathcal{L} &=& \frac{1}{L}\sum_{l=1}^{L} \mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right)\\
\end{array}
\end{equation}
{\bf Backward iteration.} The backward iteration evaluates the gradients of mean and standard deviation of the PDOD through the back-propagation (BP) method, as presented in Equation~\ref{eq:backward}.
\begin{equation}
\begin{array}{llllllll}
\label{eq:backward}
\frac{\partial \mathcal{L}}{\partial \mathbf{X}^{(l)}} &=& \frac{\partial \mathcal{M} \left(\tilde{\mathbf{X}}^{(i)}, \mathbf{X}^{(l)}\right)}{\partial \mathbf{X}^{(l)}} & \forall l\\
\vspace{5pt}
\frac{\partial \mathcal{L}}{\partial \vec{F}^{(l)}} &=& {{\boldsymbol \rho}^{(l)}}^T \frac{\partial \mathcal{L}}{\partial \mathbf{X}^{(l)}}& \forall l \\
\vspace{5pt}
\frac{\partial \mathcal{L}}{\partial \vec{Q}^{(l)}} &=& {\vec{p}^{(l)}}^T \frac{\partial \mathcal{L}}{\partial \vec{F}^{(l)}}& \forall l\\
\vspace{5pt}
\frac{\partial \mathcal{L}}{\partial \vec{q}} &= &\frac{\partial \mathcal{L}}{\partial \vec{Q}^{(l)}}& \forall l\\
\vspace{5pt}
\frac{\partial \mathcal{L}}{\partial {\boldsymbol\sigma}} &= & {\boldsymbol\nu}^{(l)}\circ \frac{\partial \mathcal{L}}{\partial \vec{Q}^{(l)}} & \forall l
\end{array}
\end{equation}
The forward-backward algorithm is presented in Figure~\ref{fig:fb}. Forward iteration is conducted through the solid line, during which the temporary matrices ${\boldsymbol\nu}, \mathbf{p}, {\boldsymbol\rho}$ are also prepared (through the dot dashed link). The objective $\mathcal{L}$ is computed at the end of $\mathcal{L}$, and the backward iteration is conducted through the dashed line, and both OD demand mean and standard deviation are updated simultaneously.
In one iteration of forward-backward algorithm, we first run the forward iteration to compute the objective function, then the backward iteration is performed to evaluate the gradient of the objective function with respect to $\mathbf{q}, {\boldsymbol\sigma}$.
With the forward-backward algorithm to compute the gradient of the objective function, we can solve the PDODE formulation in Equation~\ref{eq:pdode2} through gradient-based methods. For example, the projected gradient descent method can be used to iteratively update the OD demand. This paper adopts Adagrad, a gradient-based method using adaptive step sizes \cite{duchi2011adaptive}. As for the stopping criteria, Proposition~\ref{prop:stop} indicates that the following two conditions are equivalent: 1) in the forward iteration, the distribution of path cost, link cost, path flow, OD demand do not change; 2) in the backward iteration, $\frac{\partial \mathcal{L}}{\partial \vec{q}} = 0$ and $\frac{\partial \mathcal{L}}{\partial {\boldsymbol\sigma}} = 0$.
\begin{proposition}[Stopping criterion]
\label{prop:stop}
The PDODE formulation is solved when the forward and backward iterations converge, namely the the distributions of path cost, link cost, path flow, OD demand do not change, and $\frac{\partial \mathcal{L}}{\partial \vec{q}} = 0$ and $\frac{\partial \mathcal{L}}{\partial {\boldsymbol\sigma}} = 0$.
\end{proposition}
Since $\mathcal{L} \to \mathcal{L}_0$ when $I,L\to \infty$, we claim the PDODE problem is solved when $\frac{\partial \mathcal{L}}{\partial \vec{q}}$ and $\frac{\partial \mathcal{L}}{\partial {\boldsymbol\sigma}}$ are close to zero given a large $I$ and $L$.
\subsection{Solution Framework}
To summarize, the overall solution algorithm for PDODE is summarized in Table~\ref{tab:sol}.
\begin{table}[h]
\begin{tabular}{p{2.2cm}p{13.6cm}}
\textbf{Algorithm}& \textbf{[\textit{PDODE-FRAMEWORK}]} \\[3ex]\hline
\textit{Step 0} & \textit{Initialization.} Initialize the mean and standard deviation of dynamic OD demand $\vec{q}, {\boldsymbol\sigma}$. \\[3ex]\hline
\textit{Step 1} & \textit{Data preparation.} Randomly select a batch of observed data to form the sample distribution of $\tilde{\mathbf{X}}^{(i)}$.\\[3ex]\hline
\textit{Step 2} & \textit{Forward iteration.} Iterate over $l=1,\cdots,L$: for each $l$, sample ${\boldsymbol\nu}^{(l)}$, solve the DNL models and travelers' behavior model, and compute the objective function $\mathcal{L}$ based on Equation~\ref{eq:forward} with $\vec{q}, {\boldsymbol\sigma}$ \\[3ex]\hline
\textit{Step 3} & \textit{Backward iteration.} Compute the gradient of the mean and standard deviation of dynamic OD demand using the backward iteration presented in Equation~\ref{eq:backward}. \\[3ex]\hline
\textit{Step 4} & \textit{Update PDOD.} Update the mean and standard deviation of dynamic OD ($\vec{q}, {\boldsymbol\sigma}$) with the gradient-based projection method.\\[3ex]\hline
\textit{Step 5} & \textit{Batch Convergence check.} Continue when the changes of OD demand mean and standard deviation are within tolerance. Otherwise, go to Step 2.\\[3ex]\hline
\textit{Step 6} & \textit{Convergence check.} Iterate over $i=1,\cdots, I$. Stop when the changes of OD demand mean and standard deviation are within tolerance across different $i$. Otherwise, go to Step 1.\\[3ex]\hline
\end{tabular}
\caption{The PDODE solution framework.}
\label{tab:sol}
\end{table}
In practical applications, Step 3 and Step 4 can be conducted using the stochastic gradient projection methods to enhance the algorithm efficiency.
Additionally, Step 3 and Step 4 can be conducted using the auto-differentiation and deep learning packages, such as PyTorch, TensorFlow, JAX, etc, and both steps can be run on multi-core CPUs and GPUs efficiently.
\subsection{Underdetermination and evaluation criterion}
In this section, we discuss the underdetermination issue for the PDODE problem. It is well known that both static OD estimation and dynamic OD estimation problems are undetermined \citep{yang1995heuristic,ma2018statistical}. We claim that PDODE is also under-determined because the problem dimension is much higher than its deterministic version. In the case of PDODE, not only the OD demand mean but also the standard deviation need to be estimated.
Therefore, estimating exact PDOD accurately with limited observed data is challenging in most practical applications. Instead, since the objective of PDODE is to better reproduce the observed traffic conditions, we can evaluate the PDODE methods based on whether they can reproduce the network conditions accurately. We can evaluate the PDODE framework by measuring how well the traffic conditions can be evaluated for the observed and all links, respectively. Using this concept, we categorize the PDODE evaluation criterion into three levels as follows:
\begin{enumerate}[label=\roman*)]
\item Observed Links (\texttt{OL}): The traffic conditions simulated from the estimated PDOD on the observed links are accurate.
\item All Links (\texttt{AL}): The traffic conditions simulated from the estimated PDOD on all the links are accurate.
\item Dynamic OD demand (\texttt{OD}): The estimated PDOD is accurate.
\end{enumerate}
\begin{figure*}[h]
\centering
\includegraphics[width=0.75\linewidth]{ec}
\caption{An overview of the evaluation criterion in PDODE.}
\label{fig:ec}
\end{figure*}
The three evaluation criterion are summarized in Figure~\ref{fig:ec}. One can see that the objective of Formulation~\ref{eq:pdode2} is actually \texttt{OL}, and we include a series constraints in order to achieve \texttt{AL}. Specifically, the flow conservation and route choice model help to achieve \texttt{AL}. As for the \texttt{OD}, there is no guarantee for large-scale networks. Many recent studies also indicate the same observations \citep{osorio2019high, ma2019estimating, wollenstein2022joint}.
Overall, a PDODE framework that satisfies \texttt{OL} tends to overfit the observed data. We claim that a PDODE framework that satisfies \texttt{AL} is sufficient for most needs in traffic operation and management, as the ultimate goal for PDODE is to understand the dynamic traffic conditions on the networks. To achieve \texttt{OD}, a high-quality prior PDOD matrix is necessary to reduce the search space \citep{ma2018estimating}.
From the perspective of the underdetermination issue of PDODE, \texttt{OL} is always determined as it only focuses on the observed links. On general networks, \texttt{OD} is an under-determined problem as the cardinality of a network is much smaller than the dimension of OD demand. Whether \texttt{AL} is determined or not is based on the network topology and data availability, and hence it is promising to make full use of the proposed computational graphs to achieve \texttt{AL}, as the computational graphs have an advantage in multi-source data integration and fast computation. This further motivates the necessity to formulate the PDODE problem using computational graphs.
\section{Numerical experiments}
\label{sec:experiment}
In this section, we first examine the proposed PDODE framework on a small network. Different statistical distances are compared and the optimal one is selected. We further compare the PDODE with DDODE method, and the parameter sensitivity is discussed. In addition, the effectiveness and scalability of the PDODE framework are demonstrated on a real-world large-scale network: SR-41 corridor. All the experiments in this section are conducted on a desktop with Intel Core i7-6700K CPU 4.00GHz $\times$ 8, 2133 MHz 2 $\times$ 16GB RAM, 500GB SSD.
\subsection{A small network}
\subsubsection{Settings}
\label{sec:setting}
We first work on a small network with 13 nodes, 27 links, and 3 OD pairs, as presented in Figure~\ref{fig:31net}. There are in total 29 paths for the 3 OD pairs ($1 \to9$, $5 \to9$, and $10 \to9$). Links connecting node $1,5,9, 10$ are OD connectors, and the rest of links are standard roads with two lanes. The triangular fundamental diagrams (FD) are used for the standard links, in which the length of each road segment is $0.5$ mile, flow capacity is 2,000 vehicles/hour, and holding capacity is $200$ vehicles/mile. The free flow speed is uniform sampled from $20$ to $45$ miles/hour.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\linewidth]{network33}
\caption{\footnotesize{An overview of the small network.}}
\label{fig:31net}
\end{figure}
To evaluate the performance of the proposed method, we generate the mean and standard deviation of PDOD using a triangular pattern, as shown in Figure~\ref{fig:dod}. The PDOD is high enough to generate congestion. The observed flow is obtained by solving the statistical traffic equilibrium and then a Gaussian distributed noise is added to the observation. The performance of the PDODE estimation formulation is assessed by comparing the estimated flow
with the ``true'' flow (flow includes observed link flow, all link flow, and OD demand) \citep{antoniou2015towards}. We set the study period to 10 time intervals and each time interval lasts 100 seconds.
A Logit model with a dispersion factor $0.1$ is applied to the mean route travel time for modeling the travelers' behaviors.
Supposing 100 days' data are collected, the dynamic OD demand from the ``true'' distribution of dynamic OD demand is generated on each day, and the demand is loaded on the network with the route choice model and DNL model. We randomly select 12 links to be observed, and a random Gaussian noise $\mathcal{N}(0, 5)$ is further added to the observed link flow. Our task is to estimate the mean and standard deviation of PDOD using the observed 100 days' data.
We run the proposed PDODE presented in Table~\ref{tab:sol} with the projected stochastic gradient descent, and the solution algorithm is Adagrad \citep{duchi2011adaptive}. We use the loss function $\mathcal{L}$ to measure the efficiency of the proposed method, as presented in Equation~\ref{eq:ereap}.
Note that we loop all the 100 days' data once in each epoch.
\subsubsection{Comparing different statistical distances}
\begin{table*}[h]
\centering
\begin{tabular}{|l|cc|cc|cc|}
\hline
\multirow{2}{*}{\backslashbox{$\mathcal{M}$}{Accuracy}} & \multicolumn{2}{c|}{\texttt{OL}} & \multicolumn{2}{c|}{\texttt{AL}} & \multicolumn{2}{c|}{\texttt{OD}} \\
\cline {2-7}
~ & Mean & Std & Mean & Std & Mean & Std \\
\hline\hline
$\ell_1$-norm & {\bf 0.968}& 0.792& {\bf 0.997}& 0.806& {\bf 0.996}& 0.804\\
$\ell_2$-norm & 0.955 & {\bf 0.880}& 0.994& {\bf 0.897} & 0.985& {\bf 0.892} \\
2-Wasserstein distance & {\em 0.961} & {\em 0.843} & {\em 0.996} & {\em 0.861} & {\em 0.991}& {\em 0.860} \\
KL divergence & -0.575 & 0.027 & 0.508 & 0.062 & -0.592 & 0.027 \\
Bhattacharyya distance& -0.726 & -0.004 & 0.460 & 0.029 & -0.748 & -0.005 \\
\hline
\end{tabular}
\caption{Performance of different statistical distances in terms of R-squared score.}
\label{tab:compare}
\end{table*}
We first compare different statistical distances discussed in section~\ref{sec:obj}. Under the same settings in section~\ref{sec:setting}, different statistical distances are compared as the objective function in Formulation~\ref{eq:pdode2}, and the estimation results are presented in Table~\ref{tab:compare} in terms of R-squared score. We use the \texttt{r2\_score} function in sklearn to compute the R-squared score, and the score can be negative (because the model can be arbitrarily worse)\footnote{\url{https://scikit-learn.org/stable/modules/generated/sklearn.metrics.r2\_score.html}}.
One can see that neither KL divergence nor Bhattacharyya distance can yield proper estimation of PDOD, which may be due to the complicated formulations of its objective function, and gradient explosion and vanishing with respect to the objective function significantly affect the estimation accuracy. The other three statistical distances achieve satisfactory accuracy. Using $\ell_1$-norm and $\ell_2$-norm can achieve the best estimation of PDOD mean and standard deviation, respectively. Both objectives perform stably, which is probably attributed to the simple forms of their gradients. This finding is also consistent with many existing literature \citep{shao2014estimation,shao2015estimation}. The 2-Wasserstein distance achieves a balanced performance in terms of estimating both mean and standard deviation, which might be because the 2-Wasserstein distance compares the probability density functions, instead of directly comparing the parameters of the two distributions. For the rest of the experiments, we choose 2-Wasserstein distance as the objective function.
\subsubsection{Basic estimation results}
We present the detailed estimation results using 2-Wasserstein distance as the objective function. The estimated and ``true'' PDOD are compared in Figure~\ref{fig:dod}. One can see that the proposed PDODE framework accurately estimates the mean and standard deviation of the PDOD. Particularly, both surging and decreasing trends are quickly captured.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth]{odflow}
\caption{Comparison between the ``true'' and estimated OD demand (first row: mean; second row: standard deviation; first column: $1\to9$; second column: $5\to9$; third column: $10\to9$; unit: vehicle/100seconds).}
\label{fig:dod}
\end{figure}
\subsubsection{Comparing with the deterministic DODE}
To demonstrate the advantages of the PDODE framework, we also run the standard DDODE framework proposed by \citet{ma2019estimating} using the same setting and data presented in section~\ref{sec:setting}. Because the DDODE framework does not estimate the standard deviation, so we only evaluate the estimation accuracy of the mean. The comparison is conducted by plotting the estimated OD demand mean, observed link flow and all link flow against ``true'' flow for both PDODE and DDODE frameworks, as presented in Figure~\ref{fig:31comp}. The algorithm performs well when the scattered points are close to $y=x$ line.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth]{flow}
\caption{Comparison between the ``true'' and estimated flow in terms of \texttt{OL}, \texttt{AL}, and \texttt{OD} (first row: the proposed PDODE framework; second row: the standard DDODE framework; unit:vehicle/100seconds).}
\label{fig:31comp}
\end{figure}
It can be seen from Figure~\ref{fig:31comp}, the PDODE framework can better reproduce the ``true'' traffic flow. Firstly, DDODE can fit the observed link flow better as it directly optimize the gap between the observed and estimated link flow. However, the DDODE framework tends to overfit the noisy data because it does not model the variance of the flow explicitly. PDODE can provide a better estimation for those unobserved links and OD demand through a comprehensive modeling of the flow distribution. To summarize, DDODE achieves a higher accuracy on observed links (\texttt{OL}), while PDODE outperforms DDODE in terms of \texttt{AL} and \texttt{OD}.
To quantify the error, we compute the R-squared scores between the ``true'' and estimated flow for both PDODE and DODE, as presented in Table~\ref{tab:sr}.
\begin{table}[h]
\centering
\begin{tabular}{|c|ccc|}
\hline
\backslashbox{Formulation}{Accuracy} & \texttt{OL} & \texttt{AL} & \texttt{OD} \\
\hline\hline
PDODE & 0.961 & {\bf 0.996} & {\bf 0.991} \\
DDODE & {\bf 0.963} & 0.979 & 0.857 \\
\hline
\end{tabular}
\caption{R-squared scores between the ``true'' and estimated flow for PDODE and DODE.}
\label{tab:sr}
\end{table}
The R-squared score between the ``true'' and estimated OD demand and all links for PDODE is higher than that for DODE, while the differences between the R-squared scores for observed link flow are relatively small. This further explains the overfitting issue of the DDODE on the observed links, and the above experiments verify the illustrative example presented in section~\ref{sec:example}.
\subsubsection{Sensitivity analysis.}
We also conduct sensitivity analysis regarding the proposed PDODE framework.
{\bf Impact of travel time.}
If the travel time of each link on the network is also observed, the proposed PDODE framework can be extended to incorporate the data. To be specific, we use the travel time information to calibrate the DAR matrix using the approach presented in \citet{ma2018estimating}.
It is expected that the the estimation accuracy can further improved. The comparison of estimation accuracy is presented in Table~\ref{tab:compare2}. One can see that the inclusion of travel time data is beneficial to all the estimates (\texttt{OL}, \texttt{AL}, \texttt{OD}). Particularly, the estimation accuracy of standard deviation can be improved significantly by over 5\%.
\begin{table*}[h]
\centering
\begin{tabular}{|c|cc|cc|cc|}
\hline
\multirow{2}{*}{\backslashbox{$\mathcal{M}$}{Accuracy}} & \multicolumn{2}{c|}{\texttt{OL}} & \multicolumn{2}{c|}{\texttt{AL}} & \multicolumn{2}{c|}{\texttt{OD}} \\
\cline {2-7}
~ & Mean & Std & Mean & Std & Mean & Std \\
\hline\hline
PDODE & 0.961 & 0.843 & 0.996 & 0.861 & 0.991& 0.860 \\
PDODE + travel time & 0.997& 0.925& 0.997& 0.921& 0.998& 0.908 \\
\hline
Improvement & +3.746\% & +9.727\% & +0.100\% & +9.969\% & +0.706\% & +5.581\%\\
\hline
\end{tabular}
\caption{Effects of considering travel time data.}
\label{tab:compare2}
\end{table*}
{\bf Adaptive gradient descent methods.}
We compare different adaptive gradient descent methods, and the convergence curves are shown in Figure~\ref{fig:conv}. Note that the conventional gradient descent (GD) or stochastic gradient descent (SGD) cannot converge and the loss does not decrease, so their curves are not shown in the figure. One can see that the Adagrad converges quickly within 20 epochs, and the whole 200 epochs take less than 10 minutes. Both Adam and AdaDelta can also converge after 60 epochs, while the curves are not as stable as the Adagrad. This is the reason we choose Adagrad in the proposed framework.
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth]{converge}
\caption{Convergence curves of different adaptive gradient descent methods.}
\label{fig:conv}
\end{figure}
Sensitivity analysis regarding the learning rates, number of data samples, number of CPU cores, and noise level have also been conducted, and the results are similar to the previous study. For details, readers are referred to \citet{ma2019estimating}.
\subsection{A large-scale network: SR-41 corridor}
In this section, we perform the proposed PDODE framework on a large-scale network. The SR-41 corridor is located in the City of Fresno, California. It consists of one major freeway and two parallel arterial roads. These roads are connected with local streets, as presented in Figure~\ref{fig:srnet}. The network contains 1,441 nodes, 2,413 links and 7,110 OD pairs \citep{liu2006streamlined, zhang2008developing}. We consider 6 time intervals and each time interval last 15 minutes. The ``true'' dynamic OD mean is generated from $\texttt{Unif}(0,5)$ and the standard deviation is generated from $\texttt{Unif}(0,1)$. It is assumed $500$ links are observed. The statistical traffic equilibrium is solved with Logit mode, and we generate $10$ days' data under the equilibrium. We run the proposed PDODE framework, and $\mathcal{L}$ with 2-Wasserstein distance is used to measure the efficiency of the algorithm. No historical OD demand is used, and the initial PDOD is randomly generated for the proposed solution algorithm. The convergence curve is presented in Figure~\ref{fig:convsr}.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=\textwidth]{network_sr41}
\caption{\footnotesize{Overview of the SR41 corridor.}}
\label{fig:srnet}
\end{subfigure}
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=\textwidth]{sr41conv}
\caption{\footnotesize{Convergence curve of the proposed PDODE framework.}}
\label{fig:convsr}
\end{subfigure}
\caption{Network overview and the algorithm convergence curve for SR41 corridor.}
\label{fig:srover}
\end{figure}
One can see the proposed framework performs well and the objective function converges quickly. Each epoch takes 37 minutes and hence the algorithm takes 3,700 minutes ($\sim$ 62 hours) to finish 100 epochs.
As discussed in previous sections, we do not compare the OD demand as the estimation of OD demand is under-determined, and it is challenging to fully recover the exact dynamic OD demand on such large-scale network without historical OD demand, analogous to DDODE or static ODE in the literature. Instead, we focus on the \texttt{OL} and \texttt{AL} to assess the performance of the proposed PDODE framework.
We plot the ``true'' and estimated mean of link flow on observed links and all links in Figure~\ref{fig:srcomp}, respectively.
One can see that PDODE can reproduce the flow on observed links accurately, while the accuracy on all links is relatively lower. This observation is different from the small network, which implies extra difficulties and potential challenges of estimating dynamic network flow on large-scale networks, in extremely high dimensions. Quantitatively, the R-squared score is 0.949 on the observed links and 0.851 on all links. Both R-squared scores are satisfactory, and the R-squared score for \texttt{AL} is higher than that estimated by the DDODE framework ($0.823$). Hence we conclude that the proposed PDODE framework performs well on the large-scale network.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth]{sr41compare}
\caption{Comparison between the ``true'' and estimated mean of link flow in PDODE framework (unit:vehicle/15minutes).}
\label{fig:srcomp}
\end{figure}
\section{Conclusions}
\label{sec:con}
This paper presents a data-driven framework for the probabilistic dynamic origin-destination demand estimation (PDODE) problem. The PDODE problem is rigorously formulated on general networks.
Different statistical distances ({\em e.g.}, $\ell_p$-norm, Wasserstein distance, KL divergence, Bhattacharyya distance) are tested as the objective function. All the variables involved in the PDODE formulation are vectorized, and the proposed framework is cast into a computational graph.
Both mean and standard deviation of the PDOD can be simultaneously estimated through a novel reparameterization trick. The underdetermination issues of the PDODE problem are also discussed, and three different evaluation criterion (\texttt{OL}, \texttt{AL}, \texttt{OD}) are presented.
The proposed PDODE framework is examined on a small network as well as a real-world
large-scale network. The loss function reduces quickly on both networks and the time consumption is satisfactory. $\ell_1$ and $\ell_2$ norms have advantages in estimating the mean and standard deviation of dynamic OD demand, respectively, and the 2-Wasserstein distance achieves a balanced accuracy in estimating both mean and standard deviation. We also compared the DDODE framework with the proposed PDODE framework. The experiment results show that the DDODE framework tends to overfit on \texttt{OL}, and PDODE can achieve better estimation on \texttt{AL} and \texttt{OD}.
In the near future, we will extend the existing PDODE formulation to estimate the spatio-temporal covariance of the dynamic OD demand. The covariance (correlation) of dynamic OD demand can further help public agencies to better understand the intercorrelation of network dynamics and further improve the effectiveness of the operation/management strategies. Low-rank or sparsity regularization for the covariance matrix of the PDOD might be necessary. The choice of statistical distances can be better justified through theoretical derivations. The computational graph also has great potential in incorporating multi-source data \citep{ma2019estimating}, and it is interesting to explore the possibility of estimating the PDOD using emerging data sources, such as vehicle trajectory \citep{ma2019measuring} and automatic vehicle identification (AVI) data \citep{cao2021day}. The under-determination issue remains a critical challenge for the OD estimation problem (including both DDODE and PDODE), and this study demonstrates the possibility of mitigating the overfitting issue by considering the standard deviation. We believe this sheds light on overcoming the under-determination issues in general OD estimation problems.
\section*{Supplementary Materials}
The proposed PDODE framework is implemented with PyTorch and open-sourced on Github (\url{https://github.com/Lemma1/Probabilistic-OD-Estimation}).
\ACKNOWLEDGMENT{
The work described in this paper was supported by U.S. National Science Foundation CMMI-1751448. The first author was supported by the National Natural Science Foundation of China (No. 52102385) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU/25209221). The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein.
}
\clearpage
\bibliographystyle{informs2014trsc}
\bibliography{ref}
\end{document} | 184,408 |
TITLE: Please help me solve this probem on polynomials
QUESTION [0 upvotes]: Let
$$f(x)=a_0x^n+a_1x^{n-1}+a_2 x^{n-2}+\cdots+a_n $$
be a polynomial of degree $n$ with real coefficients and $a_1^2<a_2$.
Show that not all roots of $f(x)$ can be real.
Please help me solve this problem in a simple way as I am only a high school student.
(image of problem)
REPLY [1 votes]: Let's call the roots (real or complex) of the polynomial as $r_i$. Then the polynomial can be written as
$$f(x)=a_0(x-r_1)(x-r_2)(x-r_3)...(x-r_n)$$
Developing it a bit we get:
$$f(x) = a_0x^n -(r_1+r_2+r_3+...+r_n)x^{n-1} + 2(r_1 r_2 + r_1 r_3 +...+r_2r_3+r_2r_4+... )x^{n-2} +... $$
Where you can see that $a_1=-(r_1+r_2+r_3+...+r_n)$ and $a_2=2(pairs\;summatory)$
Now, calculate $a_1^2$
$$a_1^2= r_1^2+r_2^2+...+r_n^2 \;+\; 2(r_1 r_2 + r_1 r_3 +...+r_2r_3+r_2r_4+... )$$
If $a_1^2 < a_2$ then we have
$$r_1^2+r_2^2+...+r_n^2 \;+\; 2(r_1 r_2 + r_1 r_3 +...+r_2r_3+r_2r_4+... ) < 2(r_1 r_2 + r_1 r_3 +...+r_2r_3+r_2r_4+... )$$
Which results in $$r_1^2+r_2^2+...+r_n^2 < 0$$
Because squares are always $>=0$ for real numbers the result above ($<0$) is only possible with complex numbers.
That means at at least one of $r_i$ is complex, not all roots are real. | 7,021 |
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Hungary and Finland will play the second match of the ECC T10 International 2022. It has been scheduled to play at Cartama Oval Spain. Let us mention that both the teams were seen lacking form in their recent matches. It would definitely spice things up as they will leave no stone unturned to deliver a memorable performance that results in their victory. Everyone is so excited to watch the match as they want to see which team will clutch the victory and which team will have to feel content with getting defeated. Check HUN VS FIN match details here.
It is reported that the teams will face each other for the very first time this season. As it would be their first match, they would want it to be in their favour, and to attain that, the teams will try to deliver a performance that outdoes their rival team. However, the same efforts will be put in by both teams. They will try to build momentum from the start only. Due to this reason, the forthcoming match has become one of the most trending matches on the internet.
HUN VS FIN Match Details
Match- HUN VS FIN
League- European Cricket Championship
Date- September 19, 2022
Time- 5:00 PM IST
Venue- Cartama Oval, Cartama
Let us add that neither of the teams has been in form in their most recent matches, as both are heading into this match on a three-match losing streak. They both would badly want to snatch the victory from their rival team no matter what as they are looking forward to going ahead in the league having good points in their account and if everything went well, might emerge as the winner too.
Playing XI
Hungary: Khaibar Deldar, Safi Zahir, Zeechan Kukikhel, Asanka Weligamage, Abhishek Kheterpal, Abhishek Ahuja Stan(C), Maaz Bhaiji, Satyadeep Ashwathnarayan, Vinoth Ravindran, Harsh Mandhyan, Sandeep Mohandas
Finland: JOP Scamans, A Abdul Quadir, NG Collins, Ziaur Rehman, PF Gallagher(C), Amjad Sher, PK Garhwal, B Khan, MB Tambe, M Imran, R Muhammad
The excitement is quite evident among the fans as they are eager to watch both the teams perform in today’s match and to give their best performance. Everyone seems to be curious to know about the results so that it can put full stop to their curiosity. It would definitely be quite an interesting match which will see the teams going against one another. Stay connected with Social Telecast for more updates.
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The day that shook Kenya’s worldLeave a comment
September 23, 2013 by leighnewlands
[Some images may be sensitive to viewers] “The feeling of danger was everywhere” “They said, ‘What is the name of …
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April this year Cape Town, South Africa welcomed the likes of two girls from the UK. Their names? Em and … | 349,763 |
TITLE: Limit of normal hazard rate
QUESTION [4 upvotes]: I'm trying to work out the asymptotic behavior of the normal hazard rate as $x$ gets very large. To be clear, that's the behavior of
$$ h(x) = \frac{ \phi(x)}{1-\Phi(x)} \qquad \text{ as } \qquad x \rightarrow \infty$$
Where $\phi(\cdot)$ and $\Phi(\cdot)$ are the pdf and cdf respectively of the standard normal.
I don't think anyone on this site or elsewhere online has addressed this question specifically. The closest I found was this, but that is much more general than what I want.
In the limit, this looks like it is linear (picture below), but I can't quite show why or to what limit. Both numerator and denominator go to 0 as $x$ gets very large.
My best attempt at figuring this out was to apply L'Hopital's rule.
\begin{align*}
\lim_{x\rightarrow\infty} h(x) &= \frac{ \lim_{x\rightarrow\infty} \phi'(x) }{ \lim_{x\rightarrow\infty} -\phi(x)} \\
&= \frac{ \lim_{x\rightarrow\infty} -x \phi(x) }{ \lim_{x\rightarrow\infty} -\phi(x)}
\end{align*}
This still fails as both the top and bottom converge to zero. The limits don't exist, so I'm no good here.
Now I know that the next line is not ok, but I tried it anyway, because I had no better ideas. What if I "cancel" the $\phi(\cdot)$ functions in the numerator and denominator? That is, I tried the following:
$$ \lim_{x\rightarrow\infty} h(x) \overset{?}{=} x$$
It turns out that this is a pretty good approximation to the limiting behavior. See the picture below ($h(x)$ solid, candidate $h'(x)$ dashed). But I haven't proved anything, which is annoying. It also turns out that this limit doesn't work in some other applications (not discussed here!).
So, to summarize my questions:
What is the asymptotic behavior of the normal hazard rate? I couldn't find a reference.
Is $h'(x)=x$ in the limit?
If so, why? My abuse of L'Hopital's rule isn't the reason.
REPLY [6 votes]: A lower bound for the hazard is $h(s)>s,s>0$. I have established an upper bound, which might be new.
For $s>0$,
$$
h\left( s\right) <\frac{1}{%
s}+s.
$$
Proof:
Note that
$$
-\left[ \phi _{0}\left( s\right) \frac{s}{1+s^{2}}%
\right] ^{\prime }=\phi _{0}\left( s\right) \left[ 1-2\frac{1}{%
\left( s^{2}+1\right) ^{2}}\right]
$$
and that
$$
1-2\frac{1}{\left( t^{2}+1\right) ^{2}}<1.
$$
Then
$$
\phi _{0}\left( s\right) \frac{s}{1+s^{2}}%
=\int_{s}^{\infty }\phi _{0}\left( t\right) \left( 1-2\frac{1}{%
\left( t^{2}+1\right) ^{2}}\right) dt<1-\Phi_{0}\left(
s\right) .
$$
The conclusion follows since $s>0$ implies that $\frac{s}{1+s^{2}}>0$.
Remark This improves the bound $h(s)<\frac{s^3}{s^2 -1}$ provided by Feller (1957) Lemma 2, Chapter VII. | 129,551 |
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\begin{document}
\centerline{}
\centerline{}
\centerline {\Large{\bf Some Properties of Kenmotsu Manifolds Admitting }}
\centerline{\Large{\bf a Semi-symmetric Non-metric Connection}}
\centerline{}
\centerline{\bf {S. K. Chaubey}}
\centerline{Section of Mathematics, Department of IT, Shinas college of technology,}
\centerline{Shinas, P.O. Box 77, Postal Code 324, Sultanate of Oman.}
\centerline{Email: sk$22_{-}[email protected]}
\centerline{}
\centerline{\bf {A. C. Pandey}}
\centerline{Department of Mathematics, Bramanand P. G. College, Kanpur$-208004$, U. P., India. }
\centerline{Email: [email protected]}
\centerline{}
\centerline{\bf {N. V. C. Shukla}}
\centerline{Department of Mathematics and Astronomy, Lucknow University -226007, U.P., India. }
\centerline{Email: [email protected]}
\begin{abstract}
The aim of this paper is to study generalized recurrent, generalized Ricci-recurrent, weakly symmetric and weakly Ricci-symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection.
\end{abstract}
{\bf Subject Classification:} \textbf{$53C15$, $53B05$, $53C25$.} \\
{\bf Keywords:} Kenmotsu manifold, semi-symmetric non-metric connection, generalized recurrent manifold, generalized Ricci-recurrent manifold, weakly symmetric manifold, weakly Ricci-symmetric manifold.
\section{Introduction}
Let $(M_{n},g)$ be a Riemannian manifold of dimension $n$. A linear connection $\nabla$ in $(M_{n},g)$, whose torsion tensor $T$ of type $(1,2)$ is defined as
\begin{equation}
T(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y],
\end{equation}
for arbitrary vector fields $X$ and $Y$, is said to be torsion free or symmetric if $T$ vanishes, otherwise it is non-symmetric. If the connection $\nabla$ satisfy $\nabla{g}=0$ in $(M_{n},g)$, then it is called metric connection otherwise it is non-metric. Friedmann and Schouten \cite{friedmann} introduced the notion of semi-symmetric linear connection on a differentiable manifold. Hayden \cite{hayden} introduced the idea of semi-symmetric linear connection with non-zero torsion tensor on a Riemannian manifold. The idea of semi-symmetric metric connection on Riemannian manifold was introduced by Yano \cite{yano}. He proved that a Riemannian manifold with respect to the semi-symmetric metric connection has vanishing curvature tensor if and only if it is conformally flat. This result was generalize for vanishing Ricci tensor of the semi-symmetric metric connection by T. Imai (\cite{imai1}, \cite{imai2}). Various properties of such connection have studied in (\cite{add4}, \cite{add5}) and by many other geometers. Agashe and Chafle \cite{agashe1} defined and studied a semi-symmetric non-metric connection in a Riemannian manifold. This was further developed by Agashe and Chafle \cite{agashe2}, De and Kamilya \cite{de1}, Pandey and Ojha \cite{pandey}, Chaturvedi and Pandey \cite{chaturvedi} and others. Sengupta, De and Binh \cite{sengupta}, De and Sengupta \cite{de2} defined new type of semi-symmetric non-metric connections on a Riemannian manifold and studied some geometrical properties with respect to such connections. In this connection, the properties of non-metric connections have studied in (\cite{ozgur3}, \cite{kumar}, \cite{dubey}, \cite{add1}, \cite{add2}) and many others. In $2008$, Tripathi introduced the generalized form of a new connection in Riemannian manifold \cite{tripathi2}. Chaubey \cite{chaubey1, chaubeyadd1} defined semi-symmetric non-metric connections on an almost contact metric manifold and studied its different geometrical properties. Some properties of such connections have been noticed in (\cite{jaiswal}, \cite{chaubey3}, \cite{skchaubey_2011}, \cite{addsk4}, \cite{add6}) and others.
\newline
In 1972, K. Kenmotsu \cite{kenmotsu} introduced a class of contact Riemann manifold known as Kenmotsu Manifold. He studied that if a Kenmotsu manifold satisfies the condition R(X,Y)Z = 0, then the manifold is of negative curvature -1, where R is the Riemannian curvature tensor of type (1,3) and R(X,Y)Z is derivative of tensor algebra at each point of the tangent space. Several properties of Kenmotsu Manifold have been studied by Sinha and Srivastav \cite{srivastava}, De \cite{ucd}, De and Pathak \cite{pathak}, Chaubey et al. (\cite{chaubeyadd2}, \cite{addsk3}, \cite{skchaubey_2012}) and many others. Ozgur \cite{ozgur2} studied generalised recurrent Kenmotsu manifold and proved that if M be a generalised recurrent Kenmotsu manifold and generalised Ricci Recurrent Kenmotsu manifold then $\beta = \alpha$ holds on M. Sular studies the generalised recurrent and generalised Ricci recurrent Kenmotsu manifolds with respect to semi symmetric metric connection and proved that $\beta = 2\alpha$ where $\alpha$ and $\beta$ are smooth functions and M is generalised recurrent and generalised Ricci recurrent Kenmotsu manifold admitting a semi-symmetric connection \cite{sular}. In the present paper, we studied the properties of semi-symmetric non-metric connection in Kenmotsu manifolds.
\newline
The present paper is organized as follows. Section $2$ is preliminaries in which basic concepts of Kenmotsu manifolds are given. Section $3$ deals with the brief account of semi-symmetric non-metric connection. In section $4$, we define a generalized recurrent Kenmotsu manifolds with respect to the semi-symmetric non-metric connection and studied its some properties. Section $5$ is concerned with the weakly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection.
\section{Preliminaries}
An $n$-dimensional Riemannian manifold $(M_n,g)$ of class $C^{\infty}$ with a $1$-form $\eta$, the associated vector field $\xi$ and a $(1,1)$ tensor field $\phi$ satisfying
\begin{equation}
\label{2.1}
{\phi}^2{X}+X={\eta(X)}{\xi},
\end{equation}
\begin{equation}
\label{2.2}
{\phi}{\xi}=0,\hspace{.5cm}{\eta({\phi{X}})}=0,\hspace{.5cm}{\eta(\xi)}=1,
\end{equation}
for arbitrary vector field $X$, is called an almost contact manifold. This system $({\phi},{\xi},{\eta})$ is called an almost contact structure to $M_n$ \cite{blair}. If the associated Riemannian metric $g$ in $M_n$ satisfy
\begin{equation}
\label{2.3}
g({\phi}X,{\phi}Y)=g(X,Y)-{\eta}(X){\eta}(Y),
\end{equation}
for arbitrary vector fields $X$, $Y$ in $M_n$, then $(M_n,g)$ is said to be an almost contact metric manifold. Putting ${\xi}$ for $X$
in (\ref{2.3}) and using (\ref{2.2}), we obtain
\begin{equation}
\label{2.4}
g({\xi},Y)={\eta}(Y).
\end{equation}
Also,
\begin{equation}
\label{2.5}
\varphi(X,Y){\stackrel{\mathrm{def}}{=}}g({\phi}X,Y)
\end{equation}
gives
\begin{equation}
\label{2.6}
\varphi(X,Y)+\varphi(Y,X)=0.
\end{equation}
where $\varphi = d\eta$ is 2-form.
\newline
If moreover
\begin{equation}
\label{2.7}
(D_X{\phi})(Y)=g({\phi{X}},Y){\xi}-{\eta({Y})}{\phi{X}} ,
\end{equation}
\begin{equation}
\label{2.8}
D_X{\xi}=X-{\eta({X})}{\xi} ,
\end{equation}
hold in $(M_n,g)$, where $D$ being the Levi-Civita connection of the Riemannian metric $g$, then $(M_n,g)$ is called a Kenmotsu manifold \cite{kenmotsu}.
Also the following relations hold in a Kenmotsu manifold
\begin{equation}
\label{2.9}
K(X,Y){\xi}=\eta({X})Y-\eta{(Y)}X,
\end{equation}
\begin{equation}
\label{2.10}
K({\xi},X)Y=\eta(Y){X}-g(X,Y){\xi},
\end{equation}
\begin{equation}
\label{2.11}
S(X,{\xi})=-(n-1){\eta(X)},
\end{equation}
\begin{equation}
\label{2.12}
(D_X{\eta})(Y)=g(X,Y)-{\eta(X)}{\eta(Y)}
\end{equation}
for arbitrary vector fields $X$ and $Y$, where $K$ and $S$ denote the Riemannian curvature and Ricci tensors of the connection $D$ respectively.
\section{Semi-symmetric non-metric connection}
A linear connection $\nabla$ on $(M_n,g)$ is said to be a semi-symmetric non-metric connection if the torsion tensor $T$ of the connection $\nabla$ and the Riemannian metric $g$ satisfy the following conditions:
\begin{equation}
\label{3.1}
T(X,Y)=2\varphi(X,Y)\xi,
\end{equation}
\begin{equation}
\label{3.2}
(\nabla_{X}g)(Y,Z)=-\eta(Y)\varphi(X,Z)-\eta(Z)\varphi(X,Y),
\end{equation}
for arbitrary vector fields $X$, $Y$ and $Z$, where $\eta$ is $1-$form on $(M_{n},g)$ with $\xi$ as associated vector field.
If $D$ denotes the Levi-Civita connection, then the semi-symmetric non-metric connection \cite{chaubeyadd1,chaubey3} $\nabla$ on $(M_{n},g)$ is defined as
\begin{equation}
\label{3.3}
\nabla_{X}Y=D_{X}Y+g(\phi{X},Y)\xi,
\end{equation}
for arbitrary vector fields $X$ and $Y$.
The curvature tensor $R$ of the semi-symmetric non-metric connection \cite{chaubey3} $\nabla$ is defined as
\begin{eqnarray}
\label{3.4}
R(X,Y)Z&=&K(X,Y)Z+g(\phi{Y},Z)D_{X}\xi-g(\phi{X},Z)D_{Y}\xi
\nonumber\\&&+g\left( (D_{X}\phi)(Y)-(D_{Y}\phi)(X),Z\right)\xi.
\end{eqnarray}
From (\ref{2.2}), (\ref{2.4}), (\ref{2.7}) and (\ref{2.8}), it follows that
\begin{equation}
\label{3.5}
R(X,Y)Z=K(X,Y)Z+g(\phi{Y},Z)X-g(\phi{X},Z)Y+2\eta(Z)g(\phi{X},Y)\xi
\end{equation}
which give
\begin{equation}
\label{3.6}
\tilde{S}(Y,Z)=S(Y,Z)+(n-1)g(\phi{Y},Z)
\end{equation}
and
\begin{equation}
\label{3.7}
\tilde{r}=r.
\end{equation}
Here $\tilde{S}$ and $\tilde{r}$ denote the Ricci tensor and scalar curvature with respect to the semi-symmetric non-metric connection $\nabla$ and $r$ is the scalar curvature with respect to the Levi-Civita connection $D$. From (\ref{3.7}) we leads the following corollary:
\begin{cor}
Let $M_{n}$ be an $n-$dimensional Kenmotsu manifold equipped with a semi-symmetric non-metric connection $\nabla$, then the scalar curvature with respect to semi-symmetric non-metric connection is equal to scalar curvature with respect to Levi-Civita connection.
\end{cor}
Replacing $Z$ by $\xi$ in (\ref{3.5}) and (\ref{3.6}) and then using (\ref{2.2}) and (\ref{2.4}), we get
\begin{equation}
\label{3.8}
R(X,Y)\xi=K(X,Y)\xi+2g(\phi{X},Y)\xi
\end{equation}
and
\begin{equation}
\label{3.9}
\tilde{S}(Y,\xi)=S(Y,\xi).
\end{equation}
\section{ Generalized Recurrent Kenmotsu Manifolds }
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called generalized recurrent manifold \cite{de3} if its curvature tensor $K$ satisfies the condition
\begin{equation}
\label{4.1}
(D_{X}K)(Y,Z)W=A(X)K(Y,Z)W+B(X)\left[ g(Z,W)Y-g(Y,W)Z\right] ,
\end{equation}
where $A$ and $B\neq{0}$ are $1-$forms defined as
\begin{equation}
\label{4.2}
A(X)=g(X,\rho_{1}),{\hspace{10.5pt}} B(X)=g(X,\rho_{2}),
\end{equation}
for arbitrary vector fields $X$, $Y$, $Z$ and $W$. Here $\rho_{1}$ and $\rho_{2}$ are the vector fields associated with the $1-$forms $A$ and $B$ respectively.}
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called generalized Ricci-recurrent \cite{de3} if its Ricci tensor $S$ satisfies the condition
\begin{equation}
\label{4.3}
(D_{X}S)(Y,Z)=A(X)S(Y,Z)+(n-1)B(X)g(Y,Z),
\end{equation}
for arbitrary vector fields $X$, $Y$ and $Z$, where $A$ and $B$ are defined as in (\ref{4.2}).}
In the similar fashion, we defined the following definitions :
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called generalized recurrent with respect to the semi-symmetric non-metric connection $\nabla$ if its curvature tensor $R$ satisfies the condition
\begin{equation}
\label{4.4}
(\nabla_{X}R)(Y,Z)W=A(X)R(Y,Z)W+B(X)\left[ g(Z,W)Y-g(Y,W)Z\right] ,
\end{equation}
for arbitrary vector fields $X$, $Y$, $Z$ and $W$.
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called generalized Ricci-recurrent with respect to the semi-symmetric non-metric connection $\nabla$ if its Ricci tensor $\tilde{S}$ satisfies the condition
\begin{equation}
\label{4.5}
(\nabla_{X}\tilde{S})(Y,Z)=A(X)\tilde{S}(Y,Z)+(n-1)B(X)g(Y,Z),
\end{equation}
for arbitrary vector fields $X$, $Y$, $Z$, where $A$ and $B$ are defined as in (\ref{4.2}).}
Now we consider the generalized recurrent and generalized Ricci-recurrent Kenmotsu manifolds admitting the semi-symmetric non-metric connection $\nabla$ and prove the following theorems:
\begin{thm}
Let $M_{n}$ be an $n-$dimensional generalized recurrent Kenmotsu manifold equipped with a semi-symmetric non-metric connection $\nabla$. Then $B=A$ holds on $M_{n}$.
\end{thm}
\begin{proof}
Replacing $Y$ and $W$ by $\xi$ in (\ref{4.4}) and using (\ref{2.2}) and (\ref{2.4}), we obtain
\begin{equation}
\label{4.6}
(\nabla_{X}R)(\xi,Z)\xi=A(X)R(\xi,Z)\xi+B(X)\left[ \eta(Z)\xi-Z\right].
\end{equation}
In consequence of (\ref{2.9}) and (\ref{3.8}), (\ref{4.6}) becomes
\begin{equation}
\label{4.7}
(\nabla_{X}R)(\xi,Z)\xi=\left( B(X)-A(X)\right) \left[ \eta(Z)\xi-Z\right].
\end{equation}
It can be easily seen that
\begin{equation}
\label{4.8}
(\nabla_{X}R)(\xi,Z)\xi=\nabla_{X}R(\xi,Z)\xi-R(\nabla_{X}\xi,Z)\xi-R(\xi,\nabla_{X}Z)\xi-R(\xi,Z)\nabla_{X}\xi.
\end{equation}
From (\ref{2.2}), (\ref{2.9}), (\ref{3.8}) and (\ref{4.8}), it follows that
\begin{equation}
\label{4.9}
(\nabla_{X}R)(\xi,Z)\xi=0.
\end{equation}
In view of (\ref{4.7}) and (\ref{4.9}), we get
\begin{equation}
\label{4.10}
\left( B(X)-A(X)\right) \left[ \eta(Z)\xi-Z\right]=0.
\end{equation}
Since $Z\neq{\eta(Z)\xi}$ in general, therefore $B=A$.
\end{proof}
\begin{thm}
If an $n-$dimensional generalized Ricci-recurrent Kenmotsu manifold $M_{n}$ admitting a semi-symmetric non-metric connection $\nabla$, then $B=A$ holds on $M_{n}$.
\end{thm}
\begin{proof}
Replacing $Z$ by $\xi$ in (\ref{4.5}) and then using (\ref{2.4}), (\ref{2.11}) and (\ref{3.9}), we obtain
\begin{equation}
\label{4.11}
(\nabla_{X}\tilde{S})(Y,\xi)=(n-1)\eta(Y)\left[ B(X)-A(X)\right].
\end{equation}
It is obvious that
\begin{equation}
\label{4.12}
(\nabla_{X}\tilde{S})(Y,\xi)=\nabla_{X}\tilde{S}(Y,\xi)-\tilde{S}(\nabla_{X}Y,\xi)-\tilde{S}(Y,\nabla_{X}\xi).
\end{equation}
In consequence of (\ref{2.2}), (\ref{2.4}), (\ref{2.11}), (\ref{2.12}), (\ref{3.3}) and (\ref{3.9}), (\ref{4.12}) becomes
\begin{equation}
\label{4.13}
(\nabla_{X}\tilde{S})(Y,\xi)=-(n-1)g(X,Y)+2(n-1)g(\phi{X},Y)-S(X,Y).
\end{equation}
From (\ref{4.11}) and (\ref{4.13}), it follows that
\begin{equation}
\label{4.14}
(n-1)\eta(Y)\left[ B(X)-A(X)\right]=-(n-1)g(X,Y)+2(n-1)g(\phi{X},Y)-S(X,Y).
\end{equation}
Putting $Y=\xi$ in (\ref{4.14}) and using (\ref{2.2}), (\ref{2.4}) and (\ref{2.11}), we obtain $B=A$.
\end{proof}
\section{ Weakly symmetric Kenmotsu manifolds }
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called pseudo symmetric \cite{chaki} if there is a $1-$form $A$ on $M_{n}$ such that
\begin{eqnarray}
\label{5.1}
(D_{X}K)(Y,Z)W&=&2A(X)K(Y,Z)W+A(Y)K(X,Z)W+A(Z)K(Y,X)W\nonumber\\&&+A(W)K(Y,Z)X+g(K(Y,Z)W,X)\rho_{1},
\end{eqnarray}
where $D$ is the Levi-Civita connection and $X$, $Y$, $Z$ and $W$ are arbitrary vector fields on $M_{n}$. The vector field $\rho_{1}$ associated with the $1-$form $A$ is defined by $A(X)=g(X,\rho_{1})$.}
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called weakly symmetric \cite{tamassy1, tamassy2} if there are $1-$forms $A$, $B$, $C$ and $D$ on $M_{n}$ such that
\begin{eqnarray}
\label{5.2}
(D_{X}K)(Y,Z)W&=&A(X)K(Y,Z)W+B(Y)K(X,Z)W+C(Z)K(Y,X)W\nonumber\\&&+D(W)K(Y,Z)X+g(K(Y,Z)W,X)\sigma,
\end{eqnarray}
where $X$, $Y$, $Z$, $W$ are arbitrary vector fields on $M_{n}$. The vector field $\sigma$ associated with the $1-$form $p$ is defined as $p(X)=g(X,\sigma)$. A weakly symmetric manifold $M_{n}$ is said to be pseudo symmetric if $B=C=D=A$, $\sigma=\rho_{1}$ and $A$ is replaced by $2A$, locally symmetric if $A=B=C=D=0$ and $\sigma=0$. A weakly symmetric manifold is said to be proper if at least one of the $1-$forms $A$, $B$, $C$ and $D$ is non zero or $\sigma{\neq}0$.}
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called weakly Ricci-symmetric \cite{tamassy1, tamassy2} if there are $1-$forms $\alpha$, $\beta$ and $\gamma$ on $M_{n}$ such that
\begin{equation}
\label{5.3}
(D_{X}S)(Y,Z)=\alpha(X)S(Y,Z)+\beta(Y)S(X,Z)+\gamma(Z)S(X,Y),
\end{equation}
where $X$, $Y$ and $Z$ are arbitrary vector fields on $M_{n}$. A weakly Ricci-symmetric manifold $M_{n}$ is called pseudo Ricci-symmetric if $\alpha=\beta=\gamma$.}
Contracting (\ref{5.2}) with respect to $Y$, we get
\begin{eqnarray}
\label{5.4}
(D_{X}S)(Z,W)&=&A(X)S(Z,W)+B(K(X,Z)W)+C(Z)S(W,X)\nonumber\\&&+D(W)S(X,Z)+p(K(X,W)Z),
\end{eqnarray}
where $p$ is defined as $p(X)=g(X,\sigma)$ for arbitrary vector field $X$. The author \cite{add3} studied the properties of weakly and weakly Ricci symmetric manifolds with examples.
Similarly we define the following definitions:
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called weakly symmetric with respect to the semi-symmetric non-metric connection $\nabla$ if there are $1-$forms $A$, $B$, $C$ and $D$ on $M_{n}$ such that
\begin{eqnarray}
\label{5.5}
(\nabla_{X}R)(Y,Z)W&=&A(X)R(Y,Z)W+B(Y)R(X,Z)W+C(Z)R(Y,X)W\nonumber\\&&+D(W)R(Y,Z)X+g(R(Y,Z)W,X)\sigma,
\end{eqnarray}
where $X$, $Y$, $Z$, $W$ are arbitrary vector fields on $M_{n}$ and the $1-$forms $A$, $B$, $C$, $D$ and the vector field $\sigma$ are defined previously.
{\defi A non-flat $n-$dimensional differentiable manifold $M_{n}$, $(n>3)$, is called weakly Ricci-symmetric with respect to the semi-symmetric non-metric connection $\nabla$ if there are $1-$forms $\alpha$, $\beta$ and $\gamma$ on $M_{n}$ such that
\begin{equation}
\label{5.6}
(\nabla_{X}\tilde{S})(Y,Z)=\alpha(X)\tilde{S}(Y,Z)+\beta(Y)\tilde{S}(X,Z)+\gamma(Z)\tilde{S}(X,Y),
\end{equation}
where $X$, $Y$, $Z$ are arbitrary vector fields on $M_{n}$.
Contracting (\ref{5.5})with $Y$, we get
\begin{eqnarray}
\label{5.7}
(\nabla_{X}\tilde{S})(Z,W)&=&A(X)\tilde{S}(Z,W)+B(R(X,Z)W)+C(Z)\tilde{S}(W,X)\nonumber\\&&+D(W)\tilde{S}(X,Z)+p(R(X,W)Z),
\end{eqnarray}
where $p$ is defined as $p(X)=g(X,\sigma)$ for arbitrary vector field $X$.
$\ddot{O}$zg$\ddot{u}$r \cite{ozgur1} considered weakly symmetric and weakly Ricci-symmetric Kenmotsu manifolds and proved the following theorems:
\begin{thm} There is no weakly symmetric Kenmotsu manifold $M$, $(n>3)$, unless $A+C+D$ is everywhere zero.
\end{thm}
\begin{thm}
There is no weakly Ricci-symmetric Kenmotsu manifold $M$, $(n>3)$, unless $\alpha+\beta+\gamma$ is everywhere zero.
\end{thm}
Sular \cite{sular} considered weakly symmetric and weakly Ricci-symmetric Kenmotsu manifolds with respect to the semi-symmetric metric connection and proved the following results:
\begin{thm}
There is no weakly symmetric Kenmotsu manifold $M$ admitting a semi-symmetric metric connection, $(n>3)$, unless $A+C+D$ is everywhere zero.
\end{thm}
\begin{thm}
There is no weakly Ricci-symmetric Kenmotsu manifold $M$ admitting a semi-symmetric metric connection, $(n>3)$, unless $\alpha+\beta+\gamma$ is everywhere zero.
\end{thm}
Now we consider the weakly symmetric and weakly Ricci-symmetric Kenmotsu manifolds admitting the semi-symmetric non-metric connection $\nabla$ and prove the following theorems:
\begin{thm}
Let $M_{n}$, $(n>3)$ be an $n-$dimensional weakly symmetric Kenmotsu manifold admitting a semi-symmetric non-metric connection $\nabla$ then there is no $M_{n}$, unless $A+C+D$ is everywhere zero.
\end{thm}
\begin{proof}
Replacing $W$ by $\xi$ in (\ref{5.7}) and using (\ref{2.2}), (\ref{2.4}), (\ref{2.9}), (\ref{2.10}), (\ref{2.11}), (\ref{3.5}), (\ref{3.8}) and (\ref{3.9}) we obtain
\begin{eqnarray}
\label{5.8}
(\nabla_{X}\tilde{S})(Z,\xi)&=&-(n-1)A(X)\eta(Z)+\eta(X)B(Z)-\eta(Z)B(X)\nonumber\\&&
-(n-1)C(Z)\eta(X)+D(\xi)S(X,Z)-\eta(Z)p(X)\nonumber\\&&
+p(\xi)g(X,Z)+(n-1)D(\xi)g(\phi{X},Z)-p(\xi)g(\phi{X},Z).
\end{eqnarray}
From (\ref{4.13}) and (\ref{5.8}), it follows that
\begin{eqnarray}
\label{5.9}
&&-(n-1)g(X,Z)+2(n-1)g(\phi{X},Z)-S(X,Z)\nonumber\\&&
=-(n-1)A(X)\eta(Z)+\eta(X)B(Z)-\eta(Z)B(X)\nonumber\\&&
-(n-1)C(Z)\eta(X)+D(\xi)S(X,Z)-\eta(Z)p(X)\nonumber\\&&
+p(\xi)g(X,Z)+(n-1)D(\xi)g(\phi{X},Z)-p(\xi)g(\phi{X},Z).
\end{eqnarray}
Replacing $X$ and $Z$ by $\xi$ in (\ref{5.9}) and using (\ref{2.2}), (\ref{2.4}) and (\ref{2.11}), we get
\begin{equation}
\label{5.10}
A(\xi)+C(\xi)+D(\xi)=0.
\end{equation}
Putting $Z=\xi$ in (\ref{5.7}) and using (\ref{2.2}), (\ref{2.4}), (\ref{2.9}), (\ref{2.10}), (\ref{2.11}), (\ref{3.5}), (\ref{3.8}), we obtain
\begin{eqnarray}
\label{5.11}
(\nabla_{X}\tilde{S})(\xi,W)&=&-(n-1)A(X)\eta(W)+g(X,W)B(\xi)-\eta(W)B(X)+\eta(X)p(W)\nonumber\\&&
-g(\phi{X},W)B(\xi)+C(\xi)S(W,X)+(n-1)C(\xi)g(\phi{W},X)\nonumber\\&&
-(n-1)D(W)\eta(X)-\eta(W)p(X)+2g(\phi{X},W)p(\xi).
\end{eqnarray}
In consequence of (\ref{4.13}) and (\ref{5.11}), we have
\begin{eqnarray}
\label{5.12}
&&-(n-1)g(X,W)+2(n-1)g(\phi{X},W)-S(X,W)\nonumber\\&&
=-(n-1)A(X)\eta(W)+g(X,W)B(\xi)-\eta(W)B(X)+\eta(X)p(W)\nonumber\\&&
-g(\phi{X},W)B(\xi)+C(\xi)S(W,X)+(n-1)C(\xi)g(\phi{W},X)\nonumber\\&&
-(n-1)D(W)\eta(X)-\eta(W)p(X)+2g(\phi{X},W)p(\xi).
\end{eqnarray}
Putting $W=\xi$ in (\ref{5.12}) and using (\ref{2.2}), (\ref{2.4}) and (\ref{2.11}), we get
\begin{eqnarray}
\label{5.13}
&&\eta(X)B(\xi)-p(X)-B(X)-(n-1)C(\xi)\eta(X)\nonumber\\&&
-(n-1)D(\xi)\eta(X)+\eta(X)p(\xi)-(n-1)A(X)=0.
\end{eqnarray}
Replacing $X$ with $\xi$ in (\ref{5.12}) and then using (\ref{2.2}), (\ref{2.4}) and (\ref{2.11}), we find
\begin{equation}
\label{5.14}
-(n-1)A(\xi)\eta(W)-(n-1)C(\xi)\eta(W)-(n-1)D(W)+p(W)-\eta(W)p(\xi)=0.
\end{equation}
Replacing $W$ by $X$ in (\ref{5.14}), we get
\begin{equation}
\label{5.15}
-(n-1)A(\xi)\eta(X)-(n-1)C(\xi)\eta(X)-(n-1)D(X)+p(X)-\eta(X)p(\xi)=0.
\end{equation}
Adding (\ref{5.13}) and (\ref{5.15}) and using (\ref{5.10}), we obtain
\begin{equation}
\label{5.16}
\eta(X)B(\xi)-(n-1)A(X)-(n-1)D(X)-B(X)-(n-1)C(\xi)\eta(X)=0.
\end{equation}
Taking $X=\xi$ in (\ref{5.9}) and then using (\ref{2.2}), (\ref{2.4}) and (\ref{2.11}), we find
\begin{equation}
\label{5.17}
B(Z)-(n-1)A(\xi)\eta(Z)-\eta(Z)B(\xi)-(n-1)C(Z)-(n-1)D(\xi)\eta(Z)=0.
\end{equation}
Replacing $Z$ by $X$ in (\ref{5.17}), we get
\begin{equation}
\label{5.18}
B(X)-(n-1)A(\xi)\eta(X)-\eta(X)B(\xi)-(n-1)C(X)-(n-1)D(\xi)\eta(X)=0.
\end{equation}
Adding (\ref{5.16}) and (\ref{5.18}) and using (\ref{5.10}), we have
\begin{equation}
A(X)+C(X)+D(X)=0.
\end{equation}
Hence the statement of the theorem.
\end{proof}
\begin{thm}
Let $M_{n}$, $(n>3)$, be an $n-$dimensional weakly Ricci-symmetric Kenmotsu manifold admitting a semi-symmetric non-metric connection $\nabla$ then there is no $M_{n}$, unless $\alpha+\beta+\gamma$ is everywhere zero.
\end{thm}
\begin{proof}
Putting $Z=\xi$ in (\ref{5.6}) and then using (\ref{2.11}), (\ref{3.6}) and (\ref{3.9}), we find
\begin{eqnarray}
\label{5.19}
(\nabla_{X}\tilde{S})(Y,\xi)&=&-(n-1)\alpha(X)\eta(Y)-(n-1)\beta(Y)\eta(X)\nonumber\\&&
+\gamma(\xi)S(X,Y)+(n-1)\gamma(\xi)g(\phi{X},Y).
\end{eqnarray}
In consequence of (\ref{4.13}) and (\ref{5.19}), we have
\begin{eqnarray}
\label{5.20}
&&-(n-1)g(X,Y)+2(n-1)g(\phi{X},Y)-S(X,Y)\nonumber\\&&
=-(n-1)\alpha(X)\eta(Y)-(n-1)\beta(Y)\eta(X)\nonumber\\&&
+\gamma(\xi)S(X,Y)+(n-1)\gamma(\xi)g(\phi{X},Y).
\end{eqnarray}
Taking $X=Y=\xi$ in (\ref{5.20}) and using (\ref{2.2}), (\ref{2.4}) and (\ref{2.11}), we find
\begin{equation}
\label{5.21}
\alpha(\xi)+\beta(\xi)+\gamma(\xi)=0.
\end{equation}
Replacing $X$ by $\xi$ in (\ref{5.20}) and using (\ref{2.2}), (\ref{2.4}), (\ref{2.11}) and (\ref{5.21}), we get
\begin{equation}
\label{5.22}
\beta(Y)=\beta(\xi)\eta(Y).
\end{equation}
Again replacing $Y$ by $\xi$ in (\ref{5.20}) and using (\ref{2.2}), (\ref{2.4}), (\ref{2.11}) and (\ref{5.21}), we obtain
\begin{equation}
\label{5.23}
\alpha(X)=\alpha(\xi)\eta(X).
\end{equation}
From (\ref{2.2}), (\ref{2.8}), (\ref{2.11}), (\ref{2.12}), (\ref{3.3}) and (\ref{3.9}), it follows that
\begin{equation}
(\nabla_{\xi}\tilde{S})(\xi,X)=0.
\end{equation}
In view of (\ref{5.6}), above equation becomes
\begin{equation}
\label{5.24}
\alpha(\xi)\tilde{S}(\xi,X)+\beta(\xi)\tilde{S}(\xi,X)+\gamma(X)\tilde{S}(\xi,\xi)=0.
\end{equation}
With the help of (\ref{2.2}), (\ref{2.4}), (\ref{2.11}) and (\ref{3.9}), equation (\ref{5.24}) gives
\begin{equation}
\label{5.25}
\gamma(X)=\gamma(\xi)\eta(X).
\end{equation}
Adding (\ref{5.22}), (\ref{5.23}) and (\ref{5.25}) and using (\ref{5.21}), we get the statement of the theorem.
\end{proof} | 167,581 |
Because market in the You.Azines. slowly and gradually will continue to restore its ground, several brokers are considering this time around as being a chance to change his or her market place. With the amount of providers abandoning-or no less than significantly chopping back-their marketing and advertising programs to spend less, others are moving directly into benefit from the advertising and marketing avoid. In other words, these are getting a great questionable method as a way to put themselves throughout prime situation if the market place begins to rise.
In most elements of Europe, conversely, the market industry will continue to keep very hot as well as brokers are trying to find the easiest method to boost their enterprise. They may be trying to expand the actual achieve of the marketing and advertising zobacz stronę domową and take full advantage of cash flow possibilities. Whether within the You.Ersus. or Canada, numerous agents we're actually talking to believe that it is now time to produce your cross over in to the extremely high-end market.
Traditionally, luxurious real-estate is one of the toughest niches in an attempt to enter. The reason why? There are several frequent reasons. It may be the use of any prominent adviser currently ensconced in the neighborhood or undeniable fact that anyone by now has a peer in the property company. It might be as the providers themselves not have the patience to work in a very normally slower-paced market (less purchases to serve, more challenging opposition as well as sluggish revenue procedure). Perhaps they're not really ready for the unique challenges a new high-end market creates.
In my experience, it is often a mix of these kinds of reasons that inhibits most real estate agents through turning out to be successful within luxury real estate. There are lots of what exactly you need to understand prior to you making your massive bounce into the following budget range. We've assembled a list of several elements that will help determine if any go on to luxury real estate property meets your needs.
#1. Determine what You get Directly into
Agents often produce a window blind leap directly into high end property because they believe that is certainly "where the money will be.Inch Naturally, it is relatively simple numbers. When you get a similar separated, its smart chatting homes with greater prices. Theoretically, you can create more cash by performing a lesser number of transactions. Similarly, that is certainly accurate, however, if you are going directly into luxury property with this particular mentality, you are probably going to fail.
Indeed, your revenue every transaction increases drastically. That is certainly wonderful, there is however commonly a fresh set of difficulties presented any time working a high-end industry: your competitive buy-ins tend to be increased, sociable arenas are much much more shut, politics vary, where there are lots of additional circumstances i will certainly details during this information. Furthermore, marketing and also servicing costs are normally much more facing high end properties and also clients. Each clients anticipate countless require more as well as the attributes on their own require more attention (advertising, hosting, images, and so on.) for you to entice a more elaborate crowd.
Carol Barkin regarding Greater toronto area, New york is a huge profitable Sales rep for 25 decades, nonetheless it took the woman's a serious amounts of create the girl company in her high-end areas (both in the location plus the lakefront pastime marketplace around an hour outdoors Greater toronto area). "For us, the largest challenge has been producing which 1st link,In . the girl says. "They already have restricted sociable contacts as well as know how to find what they want, consequently developing interactions is often a matter of trust. It's important to connect with clientele being a good friend as well as a valuable fellow, not simply represent yourself being a supplier."
#2. Endurance, Patience, Persistence
It really is apparent in which high-end real estate property is a different dog when compared with classic home marketplaces. The idea has a tendency to transfer significantly sluggish. Usually, you'll find a lesser number of homes available on the market at any moment and there tend to be fewer buyers out there with all the way to acquire view website this sort of high-priced qualities. The stakes are higher for everyone concerned. So on average, it will take significantly extended to trade one of them homes. Additionally, there is a lot associated with opposition available for the small group involving attributes, so that it frequently calls for a lot more tolerance to destroy into the market and make a strong customers.
This can be a case the place that the finish usually court warrants the actual means if you have the correct knowing along with commitment moving in. Even though results are harder to come by and it takes more time for them to sell, the massive verify after the actual purchase is worth it. However, not all brokers possess the tummy to hold back more time between commission assessments. Often, this can be a hurdle in which halts these right where they are.
"In my own expertise in high-end real estate, 6 months out there are few things. Typically, it's much more 9 for the itemizing to offer,In . states Robin the boy wonder. "Also, should they be not necessarily actually determined to market, you will waste a lot of time and cash in advertising and marketing. In some cases, I'll alter my personal commission rate so the marketing prices are covered by the seller. It can help to be able to cancel out the time that it usually takes to offer. You additionally should never go into high end property without having take advantage the financial institution. It's really a long-term method to construct your business so if you're not necessarily prepared, it might bust you swiftly.In .
#3. Realize it. Stay This. Ensure that it stays Distinctive.
Another reason in which a number of agents fight to find their ground in a ultra high-end market is which they are not able to connect with the actual clients as well as converse properly. You might be handling a much savvier and in most cases more demanding masses who know very well what they want and so are used to receiving what they desire. Now, that you do not automatically ought to live in the luxurious neighborhood you are targeting, however you must look as if you perform. The way you dress, you skill to be able to community of their sectors, the way you contact these kind of advanced people, the caliber of the marketing and advertising materials-you need to be creating your own link along with develop a strong specialist picture. When they don't buy into a person being a luxury property expert that's tapped into their own community, they are not since planning to do business with anyone.
Port Jeffcoat III is surely an agent that's in the operation involving changing his or her market focus via high-end the game of golf areas inside Central Florida to be able to really high-end waterside attributes alongside Florida's Room Seacoast. Via his marketing and advertising existence to his individual display in order to his / her providing methods, every little thing he is doing is usually to help their tutaj znalazłem impression as being a high end real estate property consultant. He is often striking and also unarguable in their approach as they never ever would like to lose trustworthiness.
Think it is like every high-end product that is in demand for the lack and exclusivity. To be able a representative devoted to high-end components, you, your own marketing and advertising image, along with the support knowledge alone have to mirror the most good quality. In case you appear along with work like the very best adviser around, men and women dream to work with you.
"When I require a listing demonstration, I conduct an appointment with the vendor to ensure they are willing to follow our suggestions,In . Jack port claims. "At every possibility, I would like to remind them exactly why they're selecting myself. They are fully aware I am a luxurious real estate property skilled that is only for a unique gang of customers. Right from the start, they may be instilled with the notion when they wish to possess a profitable sale made, they must stick to our lead. That provides me with the top of hand and also maintains us positioned because market consultant.Inches
In addition, keep in mind that high-end real estate property isn't just the identical coming from different regions. The waterfront community within California may have some other list of problems compared to a mountain location neighborhood throughout Denver colorado or perhaps a the downtown area high-rise within Greater. Every now and then, "high-end" could possibly be $400,000 and up. In other business owners, price ranges could possibly be inside multi-millions. When you are looking at your own personal display and how anyone market place oneself, make sure to correctly existing the area of interest and appearance impressive.
"Always look bigger than you are,Inches says The boy wonder Milonakis. "You have to have extraordinary marketing materials. They must get people to feel good about employing you. The idea bottles their own vanity knowing they're dealing with the best.In .
#4. Image will be Everything, Particularly in Marketing
In terms of your own marketing materials, top quality is essential. You can't place on your own as being a high-end broker if your materials search unsophisticated. Any first-rate individual sales brochure along with energetic site tend to be absolutely essential. Your individual leaflet must take the place of your small business minute card when you meet any buyer. It requires to look sharp as well as sense extraordinary at the very first glimpse (outstanding images, great glossy cardstock, sophisticated creating, clean up design). It needs to echo your current character, and also relate with the luxury industry for your niche. In such a way, you're a representative of this kind of life-style and your advertising need to present in which. The idea displays your distinct expertise along with highlights the service/knowledge positive aspects that produce you a professional on this distinctive market.
It is rather essential that you will not sacrifice quality the following or it's going to display. You just are unable to bogus high-end top quality. You should end up being dedicated to trading the money to complete your marketing right or people will look out of this.
Basically, your brochure and all various other ads must be with the maximum good quality. This consists of your house promoting. You must no less than possess a tabloid-size lustrous flyer/brochure that you apply to promote each and every residence. Your holding should be fantastic. The actual images have to be really expert. Naturally, you ought to maintain your house advertising items top quality evidently with your own individual impression (logo design, shades, print styles, and so on.) which means you will not drop your own personal identification.
"My sales brochure will be quality the ones affiliate the actual piece using its sender,In . Hazel Barkin claims. "I send out it before finding someone to comfortable these up. This provides me with a lot more credibilty along with shows my expertise in the market industry they are concerned about.Inch
The same is specially accurate in relation to your internet site. It has to echo the quality of your brochure along with other printing supplies. It must look sharpened along with sense representative of your current luxurious market. 2 of the providers We mention with-Jack Jeffcoat along with The boy wonder Milonakis-are equally really in the process involving revamping his or her compaigns to higher goal his or her high-end patrons. Though both of them happen to be extremely lookity profitable with their latest activities, they understand it can be worth the investment to adopt their marketing and advertising to the next level to market a unique high end specialized niche.
One particular bold technique Jack port makes use of is usually to feature only attributes above a selected cost about their website. Can he acquire entries from lower prices? Sure, if the scenario requires that. But their graphic are extra real estate property skilled and his awesome web site is one much more way to show. "If certainly one of my high-end leads would go to our site as well as views a bunch of low-priced item listings, it's definitely not aiding our result in," Connector affirms. "Like a doctor, specialists earn more money and gain much more credibility, i really wish to be referred to as a high-end listing consultant in every aspect of our marketing and advertising.Inches
In terms of , in addition, you must ensure you happen to be very participating in your web website. You can't simply placed any site-no matter how wonderful that looks-and assume it to get company within the end. You will need to actively post information-links, content, websites, work schedule events, local community info and so on.: making it an origin that people need to come back to often. Your own productive wedding on the spot will allow you to raised talk with your audience. And of course, in addition, it boosts your current Search engine optimisation (search engine marketing) that may help you make more sales opportunities via all of the search engines like yahoo.
#5. Be ready to Back It Up
As well as ensuring that your current marketing strategy and private presentation tend to be representative of the industry, you should additionally be certain that you're totally in-tune with the marketplace itself. If you do not realize every thing that is certainly taking place around you, you won't be in a position to establish yourself as a high end professional. That is one area that you will not be able to be able to phony on your path through a financial transaction along with small knowledge as well as encounter. Consumers will expect many require more from you, so you have to be able to back the claims being an expert-in relation to its equally knowing about it along with your services encounter.
"Expectations from company is various and, generally, they may be more strenuous. They desire one to be accessible to supply answers and data,Inches Carol Barkin states any time referring to the actual customers your woman works together with. "In the finish, they have to make their particular selections. They may be collecting advice as well as expert recommendations from me so they can visit their own a conclusion."
In spite of this, never ever underestimate the particular customers' dependence on up-to-date info. Be positive throughout going for typical updates (one or more contact weekly) on marketplace task. Always continue to be current with exactly what is going on out there. Expression trips quickly inside high-class real estate property, therefore make sure you recognize what is going on on-what entries have got distributed, depending on how much, how much time these were available on the market, etc. If you aren't throughout the industry, customers will be around you. How as well as that which you connect could make all of them be ok with the experience
"No matter what, Personally, i contact every one of my personal customers on Wednesday using a detailed market place bring up to date,Inches Connector Jeffcoat claims. "I try to continually know what's happening on the market. If virtually any property markets, I want to keep in mind it and also talk to each and every consumer so they understand what's happening.Inch
Next, make sure your service knowledge reflects your own marketing and advertising graphic. You need to be able to supply on the promises by causing the customer feel special during the entire procedure. Consider it as the real difference between your Ritz-Carlton and also the Marriott. It is a completely different knowledge as soon as an individual walk-through the actual gates associated with sometimes hotel, and why you shell out significantly far more to remain at the Ritz. Picture your current property program being a luxurious knowledge. That will make that you simply beneficial item out there.
Will be the High end Industry Good for you?
Ultimately, that's so that you can decide. You should be equipped for the initial challenges and tough opposition located in the whole world of high-end real estate property. Make positive you happen to be patient jako przykład ample to manage a new slow-moving industry. You need to be willing to invest time and cash it takes not only to brand yourself as being a high-class specialist, nevertheless to be able to support it together with greater standards and services information and also experience. If you're prepared for the particular high-end industry holds, it's really a really rewarding spot to trade within the long-run. And also regardless if you are in the sluggish market place or even a hot market, right this moment could be the time for you to go ahead and take huge step! | 388,057 |
TITLE: Why is the class of isomorphisms perfect? Lurie, p.824
QUESTION [2 upvotes]: How does one prove that example in pg.824, A.2.6.11, satisfies axiom (4) ?
We want to show that for $C$ a presentable category, $W$ it sclass of isomorphism,
There exists a small set $W_0\subseteq W$ such that every morphism to $W$ can be obtained as a filtered colimit of morphisms belong to $W_0$.
It is not clear to me how to pick $W_0$ - with the axioms of presentable category, in A.1.1.1.
REPLY [1 votes]: I will use the more standardised terminology for this setting, see for example Locally Presentable and Accessible Categories by Adámek and Rosický. I will put the corresponding terminology from Lurie's document you linked behind it in brackets.
The definition from Lurie of a presentable category is also nonstandard. The usual definition is as follows:
A category $\mathcal{C}$ is called a locally presentable category (presentable category) if the following hold:
$\mathcal{C}$ is cocomplete,
there is a small set $\mathcal{A}$ of $\lambda$-presentable objects ($\lambda$-compact objects), such that every object in $\mathcal{C}$ is a $\lambda$-filtered colimit of a diagram with objects from $\mathcal{A}$.
Furthermore, I think that when Lurie says "filtered colimit" he means "$\lambda$-filtered for some $\lambda$", because in a lot of other contexts "filtered colimit" means "$\omega$-filtered colimit".
The reason this is relevant for your question is because this way it will pretty much follow from the definition. Let $\mathcal{A}$ be the set as in the definition above, then we can take $W_0$ to be the set of all identity arrows on objects in $\mathcal{A}$. Let $f: X \to Y$ be any isomorphism. We have that $X$ is the colimit of some $\lambda$-filtered diagram $(A_i)_{i \in I}$ in $\mathcal{A}$. Let $(p_i: A_i \to X)_{i \in I}$ denote the coprojections. Then $(fp_i: A_i \to Y)_{i \in I}$ form a cocone, which is also colimiting since $f$ is an isomorphism. So $f: X \to Y$ is the induced arrow between the colimits of $(A_i)_{i \in I}$ and, well, $(A_i)_{i \in I}$. In other words, $f$ is the $\lambda$-filtered colimit of $(Id_{A_i})_{i \in I}$. | 58,569 |
TITLE: $Q(a,i)$ is isomorphic to a quotient of $ \mathbb Q[X,Y]$
QUESTION [0 upvotes]: Let $a\in \mathbb C$ be a 3rd root of 2, i.e. $a$ has minimal polynomial $X^3-2$ over $ \mathbb Q$.
Claim: $ \mathbb Q[X,Y]/(X^3-2,Y^2+1) \cong \mathbb Q(a,i)$
How do I see this, do I need to consider the evaluation map $f \mapsto f(a,i)$ or is it easier to work with $Q[X,Y]\cong Q[X][Y]$ and evaluation in one variable?
REPLY [1 votes]: Note that we clearly have a surjective homomorphism
$$\mathbb Q[X,Y]/(X^3-2,Y^2+1) \to \mathbb Q(a,i),$$
given by evaluation. To show injectivity, it suffices to show that the left hand side is actually a field:
Note that, for any ring (commutative, with $1$), we have
$$R[X,Y]/(f(X),g(Y)) \cong (R[X]/(f))[Y]/(g(Y)) \cong (R[Y]/(g))[X]/(f(X)).$$
In particular
$$\mathbb Q[X,Y]/(X^3-2,Y^2+1) \cong (\mathbb Q[X]/(X^3-2))[Y]/(Y^2+1) \cong \mathbb Q(a)[Y]/(Y^2+1)$$
and
$$\mathbb Q[X,Y]/(X^3-2,Y^2+1) \cong (\mathbb Q[Y]/(Y^2+1))[X]/(X^3-2) \cong \mathbb Q(i)[X]/(X^3-2)$$
Hence, in order to conclude, you have to show one of the following equaivalent statements:
$Y^2+1$ is irreducible over $\mathbb Q(a)$,
$X^3-2$ is irreducible over $\mathbb Q(i)$.
This is quite easy, you can either do some concrete calculations or argue with the degree of the field extensions (This always works, when they are co-prime).
Let me elaborate on the isomorphism
$$R[X,Y]/(f(X),g(Y)) \cong (R[X]/(f))[Y]/(g(Y)).$$
Let $A$ be any ring and $I \subset J \subset A$ ideals. We have the well known isomorphism theorem $$A/J \cong (A/I)/(J/I).$$
Using this with $I=(a), J=(a,b)$, it says
$$A/(a,b) \cong (A/(a))/(\overline b)$$
where $\overline b$ is the residue class of $b$ in $A/(a)$. You should keep this in mind as "If we divide out two elements, we can divide them out one after another".
In our situation, we obtain
$$R[X,Y]/(f(X),g(Y)) = (R[X,Y]/(f(X)))/(\overline g(Y)).$$
Now let $A$ be again any ring and $I \subset A$ some ideal. Then $I[x] = IA[x]$ is an ideal of $A[x]$, and we have $A[x]/I[x]=(A/I)[x]$.
Using this with $A=R[X]$ and $I=(f(X))$, we obtain
$$R[X,Y]/(f(X)) = (R[X]/(f))[Y].$$
Summarizing, this yields
$$R[X,Y]/(f(X),g(Y)) = (R[X,Y]/(f(X)))/(\overline g(Y)) = (R[X]/(f))[Y] / (\overline g(Y)).$$
Note that the coefficient of $g$ are elements of $\mathbb Q$ in our situation, hence they are not touched by dividing out $f$. So we can forget about the \overline and just write $g(Y)$. | 189,982 |
TITLE: Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exist?
QUESTION [2 upvotes]: For$$\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$$
Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exist? If the answer is no, why?
Does $\bar{z}$ represents $a-bi$?
REPLY [7 votes]: If $z = a + bi$, then $\bar{z} = a - bi$. Equivalently, if $z = r e^{i\theta}$, $\bar{z} = r e^{-i \theta}$.
Take $z = re^{i\theta}$ and as $z \to 0$, we have $r \to 0$.
Plug in $z = r^{i\theta}$ in $\dfrac{\bar{z}^2}{z^2}$. Now, let $r \to 0$ and see what happens for different $\theta$'s. | 180,953 |
TITLE: How to prove that two parametrizations of a surface M give the same tangent plane
QUESTION [2 upvotes]: If you have 2 parametrizations of a surface $M$, $x$ and $y$, such that for some point $P \in M$, $x(u_0,v_0)=y(s_0,t_0)=P$. How do you show that the tangent plane of $M$ at $P$ are the same? That is how do I show that $\text{Span}(x_u,x_v)=\text{Span}(y_s,y_t)$.
REPLY [0 votes]: Ted Shifrin's answer can be generalized:
Let $A$ be an affine space with a real and normed translation space $V$ and let $x\colon U\subset\mathbf{R}^n\to A$ and $y\colon U'\subset\mathbf{R}^n\to A$ be injective maps defined on open subsets of $\mathbf{R}^n$ such that the transition maps $t:=y^{-1}\circ x\colon U\cap U'\to U\cap U'$ and $t^{-1}=x^{-1}\circ y\colon U\cap U'\to U\cap U'$ are differentiable in $a\in U\cap U'$ and $b:=y^{-1}(x(a))\in U\cap U'$. Finally, we require that $x$ and $y$ are differentiable in $a$ and $b$, respectively*.
In this case, $Dt(a)\colon\mathbf{R}^n\to\mathbf{R}^n$ and $Dt^{-1}(b)\colon\mathbf{R}^n\to\mathbf{R}^n$ are isomorphisms inverse to each other** and according to the chain rule,
\begin{equation}
Dy(b)=Dx(a)\circ Dt^{-1}(b)
\end{equation}
and
\begin{equation}
Dx(a)=Dy(b)\circ Dt(a).
\end{equation}
Thus, $Dx(a)(\mathbf{R}^n)=Dy(b)(\mathbf{R}^n)=:S\subset V$.
In your notation,
\begin{equation}
x_i=\lim\nolimits_{\delta\to 0}\frac{x(a_1,\ldots,a_{i-1},a_i+\delta,a_{i+1},\ldots,a_n)-x(a_1,\ldots,a_n)}{\delta}=Dx(a)(e_i)
\end{equation}
and $\text{span}\{x_1,\ldots,x_n\}=S$.*** Thus, we just proved the result you asked for.
Remark: $(x_1,\ldots,x_n)$ is a basis of $S$ if and only if $Dx(a)$ is injective and because of the two centered equations, $Dx(a)$ is injective if and only if $Dy(b)$ is injective.
*If $X$ and $Y$ are affine spaces with real and normed translation spaces $V$ and $W$, a map $f\colon D\subset X\to Y$ is called differentiable in $p\in D$, if $p$ is an inner point of $D$, $f$ is continuous in $p$ and a linear map $A\colon V\to W$ with the following property exsits: For each $0<\epsilon$, there is a $0<\delta$ such that
\begin{equation}
\|f(p+v)-f(p)-A(v)\|\leq\epsilon\|v\|
\end{equation}
for all $v\in V$ with $\|v\|<\delta$. If $f$ is differentiable in $p$, there is a unique linear map with this property, called the derivative of $Df(p)$ of $f$ in $p$.
**You can prove this by using the chain rule and the fact that a map $f\colon A\to B$ is injective if and only if there is a map $g\colon B\to A$ such that $g\circ f=\text{id}_A$ and surjective if and only if there is a map $g\colon B\to A$ such that $f\circ g=\text{id}_B$.
*** If $f\colon D\subset\mathbf{R}^n\to A$ is differentiable in $a=(a_1,\ldots,a_n)\in D$, the partial derivatives
\begin{equation}
\partial_i f(a)=\lim\nolimits_{\delta\to 0}\frac{x(a_1,\ldots,a_{i-1},a_i+\delta,a_{i+1},\ldots,a_n)-x(a_1,\ldots,a_n)}{\delta}\in V
\end{equation}
exist for all $i\in\{1,\ldots,n\}$ and
\begin{equation}
Df(a)(x_1,\ldots,x_n)=\sum_{i=1}^n x_i\cdot \partial_i f(a).
\end{equation} | 100,115 |
If you’ve never earned a college degree and you’re serious about learning skills for a new career, Paul Bushong of Aspen Glen wants you to apply for a scholarship he’s funding at Colorado Mountain College. Depending on the student’s campus and program, the scholarship provides up to $5,000 a year and is renewable for the second year of study, for a total of up to $10,000.
Fast Forward Scholarship at CMC donor Paul Bushong and CMC dean and vice president Dr. Heather Exby.
Maybe you didn’t attend or finish college and you’re in mid-career. Life in the valley is difficult on your current salary; perhaps you want to learn a new vocation but don’t know how you can afford the time and money.
If this describes you, the funds for you to succeed are available through the Colorado Mountain College Foundation.
Nontraditional students, who typically have fewer scholarship options, are encouraged to apply, regardless of their age. Grade point averages and participation in extracurricular activities aren’t considered critical for the new Fast Forward Scholarship at Colorado Mountain College, either.
“There are two important requirements,” said Bushong. “One, they have to be serious about their education. And two, they have to genuinely need the money. We’re trying to find those who’ve slipped through the cracks.”
The Fast Forward Scholarship began with CMC and the Aspen Community Foundation, in 2014. That initial scholarship is geared for high school seniors or those with GEDs who want to pursue workforce and job skills training. Instead of funding a typical bachelor’s degree, Fast Forward is for those who are looking for work-ready education. So far, 62 scholarships worth well over $300,000 have funded two-year degrees and certificates at accredited schools of the recipients’ choice. (Five additional scholarships were given this summer through CMC.) The Aspen Community Foundation still manages that initial scholarship program for traditional-aged students who want to go to college outside of the CMC district.
New scholarship at CMC for students of all ages
The new scholarship at CMC is for traditional college-aged students as well as nontraditional – sometimes referred to as “new traditional” – students, or those who have been out of high school for years.
Sophie Pittenger is in her first semester as a recipient of this new Fast Forward Scholarship at Colorado Mountain College. A 2018 Glenwood Springs High School graduate, she was born and raised in the Roaring Fork Valley. During her senior year she had missed the March application deadline for most CMC Foundation scholarships, but thanks to Fast Forward’s three annual application deadlines, she is now attending CMC Spring Valley pursuing an associate degree in outdoor education. Her goal is to work for Outward Bound, the National Outdoor Leadership School or the U.S. Forest Service.
“College is a lot different than high school,” Pittenger said. “I like it more than high school. It’s really nice that I don’t have to worry about trying to figure out finances because that can be overwhelming.”
As well as searching for applicants for Fast Forward at Colorado Mountain College, Bushong is also looking for donors to join him in his efforts. Currently, he is the major contributor to the scholarship.
“I believe CMC offers the biggest educational bargain in the U.S.,” said Bushong in a 2016 interview with the Glenwood Springs Post Independent.
Apply for the Fast Forward Scholarship at Colorado Mountain College
Criteria and other details:
- No age limit
- No annual deadline for applying
- Scholarship offered three times a year to coincide with the beginning of CMC’s three semesters in August, January and May
- Offered at any CMC campus although applicant must be a resident of the Glenwood Springs or Carbondale areas
- Applicant must be a first-time degree seeker
- Applicant must be enrolled full-time and demonstrate financial need
- Applicant must be working toward a particular certificate or career and technical associate degree, including accounting-bookkeeping, avalanche science, criminal justice, culinary arts, early childhood education, elementary teacher education, EMT paramedic, fire science technology, hospitality management, medical assistant, natural resource management, nursing, outdoor education, photography, ski and snowboard business, ski area operations, surgical technologist, veterinary technology, welding and many more.
For a complete list of applicable programs and more information, contact Jeanne Golay at the CMC Foundation at [email protected] or 970-947-8304. | 336,601 |
1
April
Queensland Department of Education - Adelaide, SA
Education, Childcare & Training
Source: uWorkin
Gabbinbar State School are seeking a dedicated LOTE (Chinese) teacher to join the school community for Term 2, 2021. As the School's LOTE teacher you will be required to design and deliver an engaging curriculum, based on the Australian Curriculum, to Year 4-6.
Our Staff And Students Show Our Commitment To The Vision Through Our Values
Gabbinbar is an indigenous word meaning beautiful place. Our vision for Gabbinbar is creating a beautiful place to belong, inspire and grow.
- Achieving your best
- Being respectful and responsible
- Caring and confident
This position is a temporary part-time (0.3FTE) position commencifor Term 2, 2021, with possibility of extension.
Please refer to our website Service s (including State Delivered Kindergarten programs) require an exemption card issued by Blue Card Services. For more information on blue cards and exemption cards, please CLICK HERE ..
Occupational group Education & Training
Queensland Department of Education
Adelaide, SA
Education, Childcare & Training
APPLY | 21,865 |
TITLE: Derivative of $f(x) := \min(H(a+x),H(b-x))$ in the sense of distributions (Schwarz)
QUESTION [0 upvotes]: Let $a$ and $b$ be real numbers and consider the function $f:\mathbb R \to \mathbb R$ defined by $$f(x) := \min(H(a+x),H(b-x)),$$ where $H:\mathbb R \to \mathbb R$ is the Heaviside function defined by
$$
H(u) = \begin{cases}0,&\mbox{ if }u \le 0,\\
1,&\mbox{ if }u>0.
\end{cases}
$$
Question. In the sense of distributions, what is the derivative of $f$ ?
I know that $H$ is locally-integrable and its derivative is the well-known Dirac distribution $\delta$. However, I'm very new to the theory of distributions, and I don't quite know how to go about the above problem.
REPLY [0 votes]: $$ f^{\prime}(x)~=~H(a+b) [\delta(x+a)-\delta(x-b)] .$$ | 120,372 |
- VariShapes
You can download the font "VariShapes" VariShapes maybe you will like other figure fonts.
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TITLE: Prove that number of non-cross partitions of a set is Catalan number
QUESTION [6 upvotes]: The title should be clear, how to prove that number of non-cross partitions of a set is Catalan number?
For example, if we have a set S = {1,2,3,4}, the number of non-cross partitions is 14, because of all the 15 possible partitions S, {{1,3},{2,4}} are excluded.
REPLY [3 votes]: The most straightforward way of proving that the number of noncrossing partitions is given by the Catalan number is introducing a bijection between the set of noncrossing partitions of $[n]$ and the set of balanced parentheses or any other well-known set whose cardinality is characterized by the Catalan number. This notes (http://www.maths.usyd.edu.au/u/kooc/catalan/cat46conv2.pdf) provides an algorithm of finding noncrossing partition of $[n]$ from a sequence of balanced parentheses; checking that the algorithm is a bijection is not hard.
Murasaki diagrams are closely related if you are interested: http://www.maths.usyd.edu.au/u/kooc/catalan/cat58mura.pdf | 173,282 |
TITLE: Does this inequality have a name : $2 \langle x , y \rangle \leqslant \langle x , x \rangle + \langle y , y \rangle $
QUESTION [3 upvotes]: Does the inequality $2 \langle x , y \rangle \leqslant \langle x , x \rangle + \langle y , y \rangle $, where $$ \langle \cdot, \cdot \rangle $$ denotes scalar product, have a name?
I've tried looking at several inequalities on wikipedia but I didn't find this one. And of course googling doesn't work for this purpose.
REPLY [6 votes]: It is the development of
$$ \langle x-y,x-y \rangle \geq 0$$
and it follows from the positive definitness of the scalar product.
Apart of the above proof of the inequality, and as a response to the comments to the question, here are a few reasons as to why this inequality should be true, at a first glance:
a scalar product has the properties of the multiplication on the real line, so the inequality $2xy\leq x^2+y^2$ should pop up while looking at the given inequality;
Cauchy Schwarz immediatley implies the inequality:
$$2\langle x,y \rangle \leq 2 \|x\|\|y\| \leq\|x\|^2+\|y\|^2 $$ | 2,196 |
I just started toying around with Hexographer -- I've decided I love it, so I went ahead and bought the pay-version (the free version gets me everything I need, frankly, but I try to support products I like and plan to use). It's got a short learning curve and it felt pretty intuitive to me. Just for fun I redid my West Kingdom campaign map, which looks like this using Hexographer (and a little bit of Microsoft's Picture It! photo-editing program).
And a close-up of the hexes immediately surrounding Fairbrook.
I'm also a huge fan and have used it for the Yendo and Stonehell maps. What is the Picture It! program? Haven't heard of that one before.
"Picture it!" is an old, photo-editing program that used to come free with certain older editions of windows (that's how I got it). It still is for me the the most intuitive and user-friendly photo-editing program I've ever come across. Unfortunately it does have some compatibility issues with Windows 8, but I can still manage to make it do (most of) the things I want it to do. I have read that certain versions of PI are fully compatible with Windows 8 (just not my particular version of PI), and I'm thinking of trying to pick up a used one to try it out.
That looks great!
Thanks Dan!
Thanks Dan!
So my character is going to toss back waaay too many pints at the Northstar Inn, then stumble over to the Blackwell ruins in the middle of the night. What's the worst that can happen? ;)
Ha! If your character starts throwing back pints at the Northstar, he'll be dragging himself on his belly to the ruins, not stumbling (that dwarven ale is some strong stuff!). A whole new meaning for the term "dungeon crawl."
That looks great Chris. I really like how clean it is. I want to roll some dice in that world.
Thanks, Tim. Be careful what you wish for though. :) | 269,014 |
\begin{document}
\begin{frontmatter}
\title{Estimation of Low-Rank Covariance Function}
\author[m1]{Koltchinskii, V.\footnote{[email protected]}\footnote{Supported in part by NSF Grants DMS-1207808 and CCF-1415498}}
\author[m1]{Lounici, K.\footnote{[email protected]}\footnote{Supported in part by NSF CAREER Grant DMS-1454515 and Simons Collaboration Grant 315477}}
\author[m2]{Tsybakov, A.B.\footnote{[email protected]}\footnote{Supported by GENES and by the French National Research Agency (ANR) under the grants
IPANEMA (ANR-13-BSH1-0004-02) and Labex ECODEC (ANR - 11-LABEX-0047)}}
\address[m1]{Georgia Institute of Technology, 686 Cherry St, Atlanta GA 30332, USA}
\address[m2]{Laboratoire de Statistique, CREST-ENSAE, 3, av. P.Larousse, 92240 Malakoff, France.}
\begin{abstract}
We consider the problem of estimating a low rank covariance function $K(t,u)$ of a Gaussian process $S(t), t\in [0,1]$ based on $n$ i.i.d. copies of $S$ observed in a white noise. We suggest a new estimation procedure adapting simultaneously to the low rank structure and the smoothness of the covariance function. The new procedure is based on nuclear norm penalization and exhibits superior performances as compared to the sample covariance function by a polynomial factor in the sample size $n$. Other results include a minimax lower bound for estimation of low-rank covariance functions showing that our procedure is optimal as well as a scheme to estimate the unknown noise variance of the Gaussian process.
\end{abstract}
\begin{keyword}
Gaussian process \sep Low rank Covariance Function \sep Nuclear norm \sep Empirical risk minimization \sep Minimax lower bounds \sep Adaptation
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{intro}
Let $X(t), t\in [0,1]$ be a Gaussian process satisfying the following stochastic differential equation:
\begin{align}\label{model}
dX(t) = S(t)dt + \sigma dW(t),\quad t\in [0,1],
\end{align}
where $W$ is the standard Brownian motion, $\sigma>0$ is the noise level, and
$$
S(t)=\sum_{k= 1}^r \sqrt{\lambda_k}\xi_k \varphi_k(t),\quad t\in [0,1].
$$
Here $\xi_k$ are i.i.d. standard Gaussian random variables independent of the Brownian motion $W,$ $\{\varphi_k\}_{k=1}^r$ are unknown orthonormal functions in $L_2[0,1],$ possibly, with $r=\infty$, and the coefficients $\lambda_k>0$ are unknown and such that $\sum_{k= 1}^r \lambda_k<\infty$. The value of $r$ is also unknown.
Assume that we observe $n$ i.i.d. copies $X_1(t),\dots, X_n(t)$ of the process $X(t)$. In this paper, we study the problem of estimation of the covariance function of the stochastic
process $S(\cdot),$
\begin{align}
\label{K}
K(t,u) =\E(S(t)S(u))=\sum_{k= 1}^{r} \lambda_k\varphi_k(t) \varphi_k(u), \quad t,u\in [0,1],
\end{align}
based on the observations $\{X_1(t),\dots, X_n(t), t\in [0,1]\}$. If $r=\infty,$ the sum in \eqref{K} is understood in the sense of $L_2([0,1]\times [0,1])$-convergence.
In short, (\ref{model}) is a model of a ``signal'' (Gaussian stochastic process $S$) observed in a Gaussian white noise and the
goal is to estimate the covariance of the signal based on a sample of such observations.
Statistical estimation of covariance functions has already received some attention in the literature. However, somewhat different setting was considered where the trajectories $X_i(\cdot)$ are observed at discrete time locations:
$$
Y_{i,j} = S(T_{i,j}) + \sigma \xi_{i,j},\quad 1\leq i \leq n,\; 1\leq j \leq m,
$$
where $\xi_{i,j}$ are i.i.d. $\cN(0,1)$ and, for each $i$, the points $T_{i,j}$, $1\leq j \leq m$, are equispaced in the interval $[0,1]$ or independent random variables with uniform distribution on $[0,1]$. In this setting, \cite{YaoMullerWang2005} proposed a local smoothing estimation procedure assuming that the trajectories $X_i(\cdot)$ are well approximated by the projection on the linear span of functions $\varphi_1,\dots,\varphi_k$ for some known fixed $k$ chosen by cross-validation.
This procedure is computationally intensive as it requires to compute the eigenvalues and the inverse for $n$ distinct $m\times m$ empirical covariance matrices of the trajectories $X_i$, $1\leq i \leq n$, at each of the cross-validation steps.
The results in \cite{YaoMullerWang2005} provide theoretical guarantees for estimation of the covariance function and its eigenfunctions under the condition that the previous approximation is sufficiently precise.
\cite{HallMullerWang2006} consider the same methodology and study the effect of the sampling rate on the estimation rate of the eigenfunctions. In a similar framework, \cite{BuneaXiao2013} propose a simpler procedure to estimate the eigenfunctions and obtain theoretical guarantees on the estimation error.
Their approach involves a dimension reduction step where the selection of the relevant eigenfunctions is performed by thresholding the eigenvalues of a correctly constructed empirical covariance matrix. In a similar setting, \cite{Bigot2010} consider the estimation of the covariance matrix of the process $S$ at sample points rather than that of the covariance function. This problem can be reduced to multivariate regression and \cite{Bigot2010} develop a model selection approach to it resulting in some oracle inequalities.
Noteworthy, strong regularity conditions are usually imposed on the eigenfunctions $\f_k$ in the existing literature. In \cite{HallMullerWang2006} the eigenfunctions are assumed to
admit bounded derivatives of order at least two. In addition, the optimal bandwidth choice in the local smoothing approach used in \cite{HallMullerWang2006,YaoMullerWang2005} requires the knowledge of smoothness degree of the eigenfunctions. In \cite{BuneaXiao2013}, the eigenfunctions are assumed to be continuously differentiable with bounded derivatives, the sequence of eigenvalues belongs to a Sobolev ball with regularity $\beta>0$ and the optimal choice of the threshold in the dimension reduction step depends on $\beta$.
An interesting question is what are the optimal rates of estimation of the covariance function in a minimax sense. To our knowledge, it was not addressed in the literature.
In this paper, we assume that the trajectories $X_i(\cdot)$ are fully observed in time. Our aim is to understand the influence of the structure of the covariance function $K$ on the estimation rate. The main contributions of this paper are as follows:
\begin{enumerate}
\item We propose a simple data-driven procedure to estimate the covariance function and prove oracle inequalities for it based on recent results on high-dimensional matrix estimation.
\item We show that the proposed method is minimax optimal for estimation of $K$ in the $L_2$-norm whereas the empirical covariance estimator is suboptimal.
\end{enumerate}
\section{Definitions and notations}\label{method}
Let $e_1(\cdot),e_2(\cdot),\ldots$ be an orthonormal basis of $L_2[0,1]$, which is assumed to be fixed throughout the paper.
Denote by $\|\cdot\|_2$ the norms either of $L_2[0,1]$ or of $L_2([0,1]\times [0,1])$ (according to the context) and by
$\langle \cdot,\cdot \rangle$ the corresponding inner products. For any integer $l\geq 1$,
consider the orthogonal projection $S^{(l)}= \sum_{k=1}^l \langle e_k,S\rangle e_k$ of $S$ onto the linear span of $\left\lbrace e_1,\dots,e_l \right\rbrace$. Set
\begin{align}\label{eq:proj}
\dot{X}^{(l)}= \sum_{k=1}^l \int_0^1 e_k(t)dX(t)\ e_k, \ \
\dot{W}^{(l)}=\sum_{k=1}^l \int_0^1 e_k(t)dW(t)\ e_k.
\end{align}
In view of (\ref{model}), we have
\begin{align*}
\dot{X}^{(l)} = S^{(l)} + \sigma \dot{W}^{(l)}.
\end{align*}
Similarly to \eqref{eq:proj}, we define the processes
\begin{align*}
\dot{X}_i^{(l)}= \sum_{k=1}^l \int_0^1 e_k(t)dX_i(t)\ e_k, \quad i=1,\dots, n,
\end{align*}
and consider
the empirical covariance function
\begin{align*}
R_n^{(l)}(t,u) = \frac{1}{n}\sum_{i=1}^n \dot{X}^{(l)}_i(t)\dot{X}_i^{(l)}(u),\quad t,u\in [0,1].
\end{align*}
Note that the expectation of $R_n^{(l)}(t,u)$ is
\begin{align*}
\E \left[R_n^{(l)}(t,u) \right] &= \E \left[S^{(l)}(t) S^{(l)}(u)\right] + \sigma^2 I^{(l)}(t,u)\\
&= K^{(l)}(t,u) + \sigma^2 I^{(l)}(t,u),
\end{align*}
with $ I^{(l)}(t,u) = \sum_{k=1}^l e_k(t)e_k(u)$ and
$$
K^{(l)}(t,u) = \E\left[S^{(l)}(t) S^{(l)}(u)\right] = \sum_{m=1}^r \l_m\f_m^{(l)}(t) \f_m^{(l)}(u)
$$
where $ \f_m^{(l)} = \sum_{k=1}^l \langle e_k,\f_m\rangle e_k$ is the orthogonal projection of $\f_m$ onto the linear span of
$\left\lbrace e_1,\dots,e_l \right\rbrace$. In what follows, we will consider the set of functions
\begin{align*}
{\mathcal S}_l = \biggl\{\sum_{j,k=1}^l s_{jk}(e_j\otimes e_k): \ s_{jk}=s_{kj}, \ j,k=1,\dots, l\biggr\}
\end{align*}
where $(e_j\otimes e_k)(t,s)=e_j(t)e_k(s).$
The set ${\mathcal S}_l$ consists of all symmetric kernels belonging to the linear
span of $\{e_j \otimes e_k: j,k=1,\dots, l\}.$
Note that $K$ is not necessarily in $\mathcal S_l$ while $R_n^{(l)},K^{(l)},I^{(l)}\in \mathcal S_{l}$. It is easy to see that $K^{(l)}$ is the orthogonal projection of $K$ onto $\mathcal S_l$.
If no ambiguity is caused, for any $A\in \mathcal S_l,$ we will use the same symbol $A$ to denote the corresponding symmetric $l\times l$ matrix.
For any function $A\in \mathcal S_l$ or any $l\times l$ matrix $A$ we denote by $\|A\|_1$ and $\|A\|_\infty$ its nuclear and spectral norms, respectively. The trace and the rank of matrix $A$ are denoted by ${\rm tr}(A)$ and ${\rm rank}(A)$, and its Frobenius norm by $\|A\|_F$. Writing $A\ge 0$ for a matrix $A$ means that $A$ is non-negative definite.
\section{Nuclear norm penalized estimator and its convergence rate}
In this section, we assume that the noise level $\sigma$ is known. For an integer $l\ge 1$, we define the estimator $\hat A^{(l)}$ of $K$ as a solution of the following penalized minimization problem
\begin{align}\label{Lasso}
\hat A^{(l)} \in \mathrm{argmin}_{A\in \mathcal S_l, A\geq 0} \left(\|R_n^{(l)}-A - \sigma^2 I^{(l)}\|_{2}^2 + \mu \|A\|_1 \right),
\end{align}
where $\mu>0$ is a regularization parameter to be tuned. Note that here we have $\|A\|_1 = \mathrm{tr}(A)$. The solution of \eqref{Lasso} is explicitly expressed via soft thesholding of the eigenvalues of the matrix $R_n^{(l)}- \sigma^2 I^{(l)}$ (cf. \cite{KoltLouTsy2011}).
The next theorem easily follows from the argument in the proof of Theorem 1 in \cite{KoltLouTsy2011} (see also \cite{lounici2014}).
\begin{theorem}\label{thm-main}
Let $n,l\geq 1$ be integers and let $X_1(\cdot),\dots,X_n(\cdot)$ be i.i.d. realizations of the process $X(\cdot)$ satisfying (\ref{model}). If $\mu \geq 2 \| R_n^{(l)}-K^{(l)} - \sigma^2 I^{(l)} \|_{\infty}$ then, for any $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$ we have
\begin{align*}
\| \hat A^{(l)} - K \|_2^2 &\leq \inf_{A\in \mathcal S_l, A\geq 0}
\left\lbrace \|A - K\|_2^2 + \min\left\lbrace 2\mu \|A\|_1, \frac{(1+\sqrt{2})^2}{8} \mu^2 \mathrm{rank}(A) \right\rbrace \right\rbrace.
\end{align*}
\end{theorem}
This theorem is a deterministic fact as soon as we have a proper bound on a single random variable, namely, the spectral norm $\| R_n^{(l)}-K^{(l)} - \sigma^2 I^{(l)} \|_{\infty}$. In other words, all stochastic effects in our problem are localized in the behaviour of this random variable and the choice of $\mu$ is driven by it as well. The next lemma provides a probabilistic bound on this random variable.
\begin{lemma}\label{lem-sparse}
Let $n,l\geq 1$ be integers and let $X_1(\cdot),\dots,X_n(\cdot)$ be i.i.d. realizations of the process $X(\cdot)$ satisfying (\ref{model}).
Set
$
\lambda_{\max}=\sup_{1 \leq j \leq r}\lambda_j.
$
For any $t>0$ and $l\geq 1$, define
\begin{align}\label{deltanlt}
\delta_n(l,t) = \max\left\lbrace \sqrt{\frac{l + t}{n}},\frac{l + t}{n} \right\rbrace.
\end{align}
Then, with probability at least $1-e^{-t}$, for any $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$ we have
$$
\| R_n^{(l)}-K^{(l)} - \sigma^2 I^{(l)} \|_{\infty} \leq C(\lambda_{\max}+\sigma^2) \delta_n(l,t),
$$
for some absolute constant $C>0$.
\end{lemma}
\begin{proof}
Set ${\rm x}_i (l)= (\int_0^1 e_1(t)dX_i(t),\ldots,\int_0^1 e_l(t)dX_i(t) )^\top$ for any $1\leq i \leq n$ and $\hat B_{n,l} = \frac{1}{n} \sum_{i=1}^n {\rm x}_i (l){\rm x}_i(l)^\top$. Note that ${\rm x}_i(l)$ are i.i.d. normal random vectors with mean 0 and covariance matrix $B_l= K^{(l)} + \sigma^2 I^{(l)}$. Also $\|R_n^{(l)}-K^{(l)} - \sigma^2 I^{(l)}\|_{\infty}= \|\hat B_{n,l} - B_l\|_{\infty}$. Here, $I^{(l)}$ is the $l\times l$ identity matrix.
Next,
$$
\|\hat B_{n,l} - B_l\|_\infty \leq \|B_l\|_\infty \left\|\frac{1}{n}\sum_{i=1}^n Z_i Z_i^\top - I^{(l)}\right\|_\infty \leq (\lambda_{\max}+\sigma^2)\left\|\frac{1}{n}\sum_{i=1}^n Z_i Z_i^\top - I^{(l)}\right\|_\infty
$$
where $Z_1,\ldots,Z_n$ are i.i.d. standard normal vectors in $\R^l$. Here we also used the fact that the following
representation holds for random vectors ${\rm x}_i (l):$ ${\rm x}_i (l)=B_l^{1/2}Z_i.$
Applying Theorem 5.39 in \cite{Vershynin} to the random variable $\left \|\frac{1}{n}\sum_{i=1}^n Z_i Z_i^\top - I^{(l)}\right\|_\infty$ we get the result.
\end{proof}
Theorem \ref{thm-main} with Lemma \ref{lem-sparse} immediately imply the following result.
\begin{theorem}\label{cor-main}
Let $n,l\geq 1$ be integers and let $X_1(\cdot),\dots,X_n(\cdot)$ be i.i.d. realizations of the process $X(\cdot)$ satisfying (\ref{model}).
Take
$$
\mu = c (\lambda_{\max}+\sigma^2)\delta_{n}(l,t),
$$
for some sufficiently large absolute constant $c>0$. Define
$$
v_{n}(A,l,t)=\min\left\lbrace (\lambda_{\max}+\sigma^2)\mathrm{tr}(A) \delta_{n}(l,t), (\lambda_{\max}+\sigma^2) ^2 \mathrm{rank}(A)\delta_{n}^2(l,t) \right\rbrace.
$$
Let $t>0$. Then, with probability at least $1-e^{-t}$, for any $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$ we have
\begin{align}\label{oracle}
\| \hat A^{(l)} - K \|_2^2 &\leq \inf_{A\in \mathcal S_l, A\geq 0}
\left\lbrace \|A - K\|_2^2 + C v_{n}(A,l,t) \right\rbrace
\end{align}
with some absolute constant $C>0$.
\end{theorem}
The bound \eqref{oracle} is the main oracle inequality that we will use now to obtain minimax bounds on the risk of the estimator $ \hat A^{(l)}$. It is easy to check that
$$
v_{n}(A,l,t)\leq (\lambda_{\max}+\sigma^2)^2 {\rm rank}(A)\frac{l+t}{n}.
$$
The above bound is trivial if $l+t\leq n.$ In the case $l+t>n,$ it follows from the bound
$$
(\lambda_{\max}+\sigma^2){\rm tr}(A)\frac{l+t}{n} \leq (\lambda_{\max}+\sigma^2)\lambda_{\max}{\rm rank}(A)\frac{l+t}{n}
\leq (\lambda_{\max}+\sigma^2)^2{\rm rank}(A)\frac{l+t}{n}.
$$
Combining Theorem \ref{cor-main} with the fact that, for a random variable $\eta$, $\E[|\eta|] = \int_{0}^\infty \mathbb P(|\eta|\geq t)dt$ and taking $A= K^{(l)}$,
\begin{align}\label{borne lasso r-fini}
\E[\| \hat A^{(l)} - K \|_2^2 ] \leq \|K^{(l)} - K\|_2^2 + C (\lambda_{\max}+\sigma^2)^2 \frac{(r\wedge l)l}{n}
\end{align}
for some absolute constant $C>0$, where we have used that $\mathrm{rank}(K^{(l)})\le r\wedge l$. This inequality is valid for all $K$ of the form \eqref{K}, with finite or infinite $r$.
As a corollary, we get the following bound on the minimax risk over the class of covariance functions that admit a finite expansion with respect to the basis~$\{e_k\}$. Denote by ${\mathcal K}_{r,l}(\l_{\max})$ the class of all covariance functions satisfying \eqref{K} such that
$K\in \mathcal S_l$ and $\|K\|_\infty \le \l_{\max}$ where $\l_{\max}$ is a finite positive constant. Note that the system of functions $\{\f_k\}$ in this definition is not fixed and varies among all orthonormal systems in $L_2[0,1]$.
\begin{corollary}\label{cor:finite}
Under the assumptions of Theorem~\ref{cor-main}, we have
\begin{align*}\label{}
\sup_{K\in{\mathcal K}_{r,l}(\l_{\max}) }\E[\| \hat A^{(l)} - K \|_2^2 ] \leq C (\lambda_{\max}+\sigma^2)^2 \frac{(r\wedge l)l}{n}
\end{align*}
for some absolute constant $C>0$.
\end{corollary}
It is interesting to compare the estimator $\hat A^{(l)}$ with the other natural estimator, which is the corrected empirical covariance function $$\bar A^{(l)} \triangleq R_n^{(l)} - \sigma^{2}I^{(l)}.$$
We have the following expression for the risk of $\bar A^{(l)}$.
\begin{proposition}\label{prop:empir_covar} For any $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$ we have
\begin{align*}
\E[\| \bar A^{(l)} - K \|_2^2 ] = \|K^{(l)} - K\|_2^2 +\frac{\|B_l\|_2^2 + [\tr(B_l)]^2}{n}
\end{align*}
where $B_l= K^{(l)} + \sigma^2 I^{(l)}$.
\end{proposition}
\begin{proof}
Set for brevity $B=B_l$, $ \hat B_n= \hat B_{n,l}$, ${\rm x}_i={\rm x}_i(l)$. Note that $\E(\bar A^{(l)}) = K^{(l)}$.
The bias-variance decomposition of the risk of $\bar A^{(l)}$ yields
\begin{align*}\label{}
\E[\| \bar A^{(l)} - K\|_2^2 ] & = \|K^{(l)} - K\|_2^2 + \E[\|R_n^{(l)} - \E(R_n^{(l)})\|_2^2].
\end{align*}
Here, $\E[\|R_n^{(l)} - \E(R_n^{(l)})\|_2^2] =\E[\| \hat B_n - B\|_F^2] = \E\big[\big\|\frac1n \sum_{i=1}^n W_i\big\|_F ^2\big]$ where $W_i ={\rm x}_i {\rm x}_i^\top - \E [{\rm x}_i {\rm x}_i^\top]$. Since the matrices $W_i$ are i.i.d. we find $ \E\big[\big\|\frac1n \sum_{i=1}^n W_i\big\|_F ^2\big]=
\E \,\tr \big(\frac1{n^2} \sum_{i,j=1}^n W_i^\top W_j \big) = \frac1n \tr\big(\E (W_1^\top W_1)\big)= \frac1n \big(\E (|{\rm x}_1|_2^4) - \tr(B^\top B)\big)$
where $|\cdot|_2$ denotes the Euclidean norm. Here, $\E (|{\rm x}_1|_2^4) - \tr(B^\top B) = \|B\|_2^2 + [\tr(B)]^2$ and the result follows.
\end{proof}
Since $\tr (B)\ge \s^2 l$, Proposition~\ref{prop:empir_covar} implies
\begin{align}\label{lower_empir0}
\E[\| \bar A^{(l)} - K \|_2^2 ] &\ge \|K^{(l)} - K\|_2^2 + \frac{ \sigma^4 l^2}{n},\\
\inf_{K }\E[\| \bar A^{(l)} - K \|_2^2 ] &\geq \frac{ \sigma^4 l^2}{n}\label{lower_empir}
\end{align}
where $\inf_{K }$ is the infimum over all $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$. Comparing \eqref{lower_empir}
with Corollary~\ref{cor:finite} we see that the risk of the empirical estimator $\bar A^{(l)}$ on the class ${\mathcal K}_{r,l}$ is of the order greater than the risk of our estimator $\hat A^{(l)}$ when $r$ is smaller than $l$.
Our estimator also outperforms the estimator $\bar A^{(l)}$ for kernels $K$ that do not admit a finite expansion with respect to the basis $\{e_k\},$ but satisfy some regularity conditions. To this end, we introduce a specific norm that can be naturally interpreted as a version of the Sobolev norm for covariance functions. Fix the smoothness parameter $s>0$. For any symmetric function $K \,:\, [0,1]^2 \rightarrow \R$, we define
\begin{align*}
\|K\|_{s,2} &:= \|\Delta^s K\|_2 =
\left( \sum_{k, k'\geq 1} k^{2s} \langle Ke_k, e_{k'} \rangle^2 \right)^{1/2},
\end{align*}
where $\Delta$ is an operator admitting the matrix representation $\mathrm{diag}(1,2,\cdots,k,\cdots)$ w.r.t the basis $(e_k)_{k\geq 1}$. Note that the norm $\|K\|_{s,2}$ depends on the basis $\{e_k\}$ but we do not indicate this dependence in the notation since $\{e_k\}$ is fixed.
Note also that if $K$ admits spectral representation (\ref{K}), then
$$
\|K\|_{s,2}=\left(\mathrm{tr}(\Delta^{2s} K^2)\right)^{1/2} = \left(\sum_{k=1}^r \lambda_k^2 \|\varphi_k\|_{s,2}^2\right)^{1/2},
$$
where we use the notation
$$
\|\varphi\|_{s,2}= \|\Delta^s \varphi\| = \left( \sum_{k\geq 1} k^{2 s} \langle \varphi,e_k \rangle^2 \right)^{1/2}
$$
for a Sobolev type norm of a function $\varphi\,:\, [0,1] \rightarrow \R.$
\begin{assum}
\label{smooth_kernel}
Suppose the covariance function $K$ has finite rank $r$ and there exist constants $\l_{\max}>0$, $s>0$ and $\rho\geq 1$ such that $\|K\|_{\infty}\leq \lambda_{\max}$ and $\|K\|_{s,2}\leq \rho$.
\end{assum}
Denote by $\overline{\mathcal K}_r(s,\rho;\lambda_{\max})$ the class of all
kernels $K$ satisfying Assumption \ref{smooth_kernel}.
\begin{theorem}
\label{th:smooth}
Given $r\geq 1, s>0, \rho>0$ and $\lambda_{\max}>0,$
set
$$
\ell := \max\left(\left\lceil \left(\frac{\rho^2}{(\lambda_{\max}+\sigma^2)^2}\frac{n}{r}\right)^{1/(2s+1)}\right\rceil,
\left\lceil \left(\frac{\rho^2 n}{(\lambda_{\max}+\sigma^2)^2}\right)^{1/(2s+2)}\right\rceil\right).
$$
Then, with some absolute constant $C>0,$
\begin{align}
\label{eq:th:smooth}
&\sup_{K\in \overline{\mathcal K}_r(s,\rho;\lambda_{\max})} \E[\| \hat A^{(\ell)} - K \|_2^2 ] \leq
\\
&
\nonumber
C \min
\left((\lambda_{\max}+\sigma^2)^{4s/(2s+1)}\rho^{2/(2s+1)} \left(\frac{r}{n}\right)^{2s/(2s+1)},
(\lambda_{\max}+\sigma^2)^{2s/(s+1)}\rho^{2/(s+1)} n^{-s/(s+1)}\right).
\end{align}
\end{theorem}
\begin{proof}
Since $K$ satisfies Assumption \ref{smooth_kernel}, we have for any $l\geq 1$ that
$$
\|K - K^{(l)}\|_2^2 =
\sum_{k\geq l+1} \sum_{k'=1}^{\infty}\langle Ke_k, e_{k'}\rangle^2 + \sum_{k'\geq l+1}\sum_{k=1}^l \langle Ke_k, e_{k'}\rangle^2
$$
$$
\leq (l+1)^{-2s}\sum_{k\geq l+1} \sum_{k'=1}^{\infty}k^{2s}\langle Ke_k, e_{k'}\rangle^2
+ (l+1)^{-2s}\sum_{k'\geq l+1} \sum_{k=1}^{\infty}(k')^{2s}\langle Ke_k, e_{k'}\rangle^2\leq 2\rho^2 l^{-2s}.
$$
Combining the previous display with (\ref{borne lasso r-fini}), we find that, for any $l\ge1$,
\begin{align*}\label{}
\E[\| \hat A^{(l)} - K \|_2^2 ] \leq \rho^2 l^{-2s} + C (\lambda_{\max}+\sigma^2)^2 \frac{(r\wedge l)l}{n}\,.
\end{align*}
The minimum of the right-hand side of this inequality is achieved for $l$ of the order of $\ell$. By setting $l=\ell,$ we obtain \eqref{eq:th:smooth}.
\end{proof}
Note that, if the rank $r$ is small, the problem of estimation of covariance function $K$ reduces to estimation
of a small number $r$ of eigenfunctions and eigenvalues of $K.$
The rate in (\ref{eq:th:smooth}) is, in this case, of the order $O(n^{-2s/(2s+1)}),$
which coincides with a standard minimax error rate of estimation of a function of one
variable of smoothness $s.$ On the other hand, when the rank $r$ is large (say, $r=+\infty$),
the estimation error rate becomes $O(n^{-s/(s+1)}),$ which is the minimax rate of estimation
of a function of two variables of smoothness $s.$ Similar error rates where studied earlier
in matrix completion problems for smooth kernels on graphs (see \cite{Kolt}).
We consider now a class of kernels determined by the following assumption, which can be interpreted as a Sobolev type condition on the individual eigenfunctions $\varphi_j$.
\begin{assum}\label{bias-cond}
The value $r$ is finite and there exist constants $s>0$, $c_*>0$ such that, for any $1\leq j \leq r$,
$
\|\varphi_j\|_{s,2} \leq c_*.
$
\end{assum}
Denote by $\cK_{r}(s,c_*;\l_{\max})$ the class of all kernels $K$ defined by \eqref{K} with eigenfunctions $\f_j$ satisfying Assumption~\ref{bias-cond} and such that $\|K\|_\infty < \l_{\max}$.
\begin{theorem}\label{th:nonpar}
Let $l_1 =\max \big( \lceil n^{\frac{1}{2s+1}}\rceil, \, \lceil (rn)^{\frac{1}{2(s+1)}}\rceil \big)$, $n\ge 1$, $1\le r<\infty$. For any $s>0$, $c_*>0$, $\l_{\max}>0$ we have
\begin{align}\label{eq:th:nonpar}
\sup_{K\in \cK_{r}(s,c_*;\l_{\max})} \E[\| \hat A^{(l_1)} - K \|_2^2 ] \leq C \min\big(rn^{-\frac{2s}{2s+1}}, \, r^{\frac{1}{s+1}}n^{-\frac{s}{s+1}}\big)
\end{align}
where $C>0$ is a constant depending only on $\lambda_{\max}, \sigma$ and $c_*$.
\end{theorem}
\begin{proof} It is enough to observe that, for all $K\in \cK_{r}(s,c_*;\l_{\max}),$
$$
\|K\|_{s,2}^2 = \sum_{k=1}^r \lambda_k^2 \|\varphi_k\|_{s,2}^2 \leq c_*^2 \lambda_{\max}^2 r,
$$
implying that
$\cK_{r}(s,c_*;\l_{\max})\subset \overline{\mathcal K}_r(s,\rho;\lambda_{\max})$
with $\rho =c_{\ast}\lambda_{\max}\sqrt{r}.$ Bound \eqref{eq:th:nonpar} now follows
from \eqref{eq:th:smooth}.
\end{proof}
When $r$ is a fixed constant and $n$ is large, the rate in \eqref{eq:th:nonpar} is $O(n^{-\frac{2s}{2s+1}})$. The next theorem shows that this rate cannot be achieved by the corrected empirical covariance estimator $ \bar A^{(l)}$ whatever is the choice of $l$.
\begin{theorem}\label{th:nonpar:empir_covar}
Let $n\ge 1$, $1\le r<\infty$. There exists $c_*>0$ such that for any $s>0$, $\l_{\max}>0$ we have
\begin{align}\label{eq:th:nonpar:empir_covar}
\inf_{l\ge1} \sup_{K\in \cK_{r}(s,c_*;\l_{\max})} \E[\| \bar A^{(l)} - K \|_2^2 ] \geq C n^{-\frac{s}{s+1}}
\end{align}
where $C>0$ is a constant that can depend only on $\lambda_{\max}, \sigma$, $s$ and $c_*$.
\end{theorem}
\begin{proof}
Fix $l\geq 1$ and consider the function
$$
\f_1(t) = C_1\left(\sum_{k=1}^l \frac{e_k(t)}{k^{s+1}} + \sum_{k=l+1}^{2l} \frac{e_k(t)}{k^{s+1/2}}\right), \quad t\in[0,1],
$$
where $C_1$ is a normalizing constant, depending only on $s,$ such that $\|\f_1\|_2=1$.
By an easy computation, $\|\f_1\|_{s,2}\leq c'$ for a constant $c'$ depending only on $s.$
Set $\bar K(t,u)=\l_{\max}\f_1(t)\f_1(u)$. Then $\bar K\in \cK_{r}(s,c_*;\l_{\max})$ with $c_*=c'$.
Due to \eqref{lower_empir0},
\begin{align}\nonumber
\sup_{K\in \cK_{r}(s,c_*;\l_{\max})} \E[\| \bar A^{(l)} - K \|_2^2 ] &\ge \sup_{K\in \cK_{r}(s,c_*;\l_{\max})} \|K^{(l)} - K\|_2^2 + \frac{ \sigma^4 l^2}{n}\\
&\ge \|\bar K^{(l)} - \bar K\|_2^2 + \frac{ \sigma^4 l^2}{n}.\label{eq:th:nonpar:empir_covar1}
\end{align}
Observe that
$$
\varphi_1\otimes \varphi_1=\varphi_1^{(l)}\otimes \varphi_1^{(l)}+ (\varphi_1-\varphi_1^{(l)})\otimes \varphi_1^{(l)}+
\varphi_1\otimes (\varphi_1-\varphi_j^{(l)}).
$$
Therefore,
\begin{align*}
& \| \f_1\otimes \f_1 - \varphi_1^{(l)}\otimes \varphi_1^{(l)} \|_2^2
= \|(\varphi_1-\varphi_1^{(l)})\otimes \varphi_1^{(l)}\|_2^2+ \|\varphi_1\otimes (\varphi_1-\varphi_1^{(l)})\|_2^2
\\
&\qquad \qquad \geq \|\varphi_1\|_2^2\|\varphi_1-\varphi_1^{(l)}\|_2^2
=\|\varphi_1 - \varphi_1^{(l)}\|_2^2.
\end{align*}
This implies that
$$
\|\bar K^{(l)}-\bar K \|_{2}^2\ge \l_{\max}^2\|\varphi_{1} - \varphi_{1}^{(l)}\|_2^2\ge c\l_{\max}^2 l^{-2s}
$$
for some constant $c>0$ depending only on $s.$
Using this inequality in \eqref{eq:th:nonpar:empir_covar1} and taking the minimum over $l\ge 1,$ we obtain the result.
\end{proof}
\section{Adaptive Estimation}\label{adap}
We observe that the optimal choice of the parameter $l$ in theorems \ref{th:smooth} and \ref{th:nonpar} depends on the unknown parameters
$\rho,$ $s$ and $r$ that quantify respectively the smoothness of the eigenfunctions of $K$ and their number. In this section, we propose an adaptive estimator, which does not depend on $s$ and $r$ that attains the same rate as in Theorem \ref{th:smooth} or in Theorem \ref{th:nonpar}.
First, we describe a general method of aggregating estimators.
Assume without loss of generality that the sample size $n$ is even. We split the sample of $n$ trajectories $\mathbb X =\{X_1,\ldots, X_n\}$ into two parts of equal size $n/2$, denoted $\mathbb X_1 = \{X_1,\ldots, X_{n/2}\}$ and $\mathbb X_2= \{X_{n/2+1},\ldots, X_n\}$. Fix an integer $L$.
Using the sample $\mathbb X_1$, we construct a family of estimators $A^{(1)},\ldots, A^{(L)}$ such that $A^{(l)}\in \mathcal S_l$, $1\leq l \leq L$. These can be, for example, the estimators $\hat A^{(1)},\ldots, \hat A^{(L)}$ defined in (\ref{Lasso}).
Consider the following adaptive selector of $l$:
\begin{align}\label{hatl}
\hat l = \argmin_{1\leq l \leq L}\{ \| A^{(l)} \|_2^2 - 2 \langle A^{(l)}, \tilde R_n^{(l)}-\s^2I^{(l)}\rangle \},
\end{align}
where $\tilde R_n^{(l)}(t,u)= \frac{2}{n}\sum_{i=n/2+1}^n \dot{X}_i^{(l)}(t) \dot{X}_i^{(l)}(u)$ is the projected empirical covariance function associated to the second subsample $\mathbb X_{2}$.
In the following theorem we assume that the first subsample is frozen, so we state the result for non-random functions $A^{(l)}\in \mathcal S_l$, $1\leq l \leq L$.
\begin{theorem} \label{adap-scheme}
Let $A^{(l)}$, $1\leq l \leq L$, be functions such that $A^{(l)}\in \mathcal S_l$.
For any $t>0$, with probability at least $1-e^{-t}$ with respect to the subsample $\mathbb X_{2}$ we have
\begin{align*}
\| A^{(\hat l)} - K \|_2^2 &\leq 2 \min_{1\leq l \leq L}\| A^{(l)} - K \|_2^2
+
C[\lambda_{\max}\vee \sigma^2]^2 \max\left\lbrace \frac{t+\log L}{n},\left(\frac{t+\log L}{n}\right)^2 \right\rbrace
\end{align*}
for all $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$.
Here, $C>0$ is an absolute constant.
\end{theorem}
\begin{proof} Fix an arbitrary $\bar l \in \{1,\dots,L\}$.
Note that, by definition, $\{\mathcal S_l\}_{l\geq 1}$ is a nested sequence satisfying
$$
\mathcal S_{l+1} = \mathcal S_l \oplus \mathrm{l.s.}\left\lbrace e_{j}\otimes e_{l+1} + e_{l+1} \otimes e_{j},\, 1\leq j \leq l \right\rbrace.
$$
Consequently, for any $1\leq l,l'\leq L$, we have
$\langle A^{(l)}, \tilde R_n^{(l)} \rangle = \langle A^{(l)}, \tilde R_n^{(l\vee l')} \rangle$. Similarly $\langle A^{(l)}, K \rangle = \langle A^{(l)}, K^{(l)} \rangle = \langle A^{(l)}, K^{(l\vee l')} \rangle$.
Combining this observation with (\ref{hatl}), we get
\begin{align*}
&\|A^{(\hat l)} - K\|_2^2 - \|A^{(\bar l)} - K\|_2^2 \\
&\hspace{1.5cm}= \| A^{(\hat l)}\|_2^2 - 2\langle A^{(\hat l)},K^{(\hat l)} \rangle - [\|A^{(\bar l)}\|_2^2 - 2\langle A^{(\bar l)}, K^{(\bar l)}\rangle]\\
&\hspace{1.5cm}\leq \|\hat A^{(\hat l)}\|_2^2 - 2\langle A^{(\hat l)},\tilde R_n^{(\hat l)} - \sigma^2 I^{(\hat l)} \rangle - [\|A^{(\bar l)}\|_2^2 - 2\langle A^{(\bar l)}, \tilde R_n^{(\bar l)} - \sigma^2 I^{(\bar l)}\rangle]\\
&\hspace{6cm} + 2\langle A^{(\hat l) } - A^{(\bar l)}, \tilde R_n^{(\hat l\vee \bar l )} - K^{(\hat l\vee \bar l )} -\sigma^2 I^{(\hat l \vee \bar l)}\rangle\\
&\hspace{1.5cm}\leq 2\langle A^{(\hat l )} - A^{(\bar l)}, \tilde R_n^{(\hat l\vee \bar l )} - K^{(\hat l\vee \bar l )} - \sigma^2 I^{(\hat l \vee \bar l)}\rangle.
\end{align*}
Here, $
K^{(\hat l\vee \bar l )} +\sigma^2 I^{(\hat l \vee \bar l)}= \E[ \tilde R_n^{(\hat l\vee \bar l )}]
$. Setting for brevity $m=\hat l\vee \bar l$ we deduce from the previous display that
\begin{align*}
\|A^{(\hat l)} - K\|_2^2 - \|A^{(\bar l)} - K\|_2^2 &\le 2 U \|A^{(\hat l )} - A^{(\bar l)}\|_2
\le \frac16 \|A^{(\hat l )} - A^{(\bar l)}\|_2^2 +6U^2
\end{align*}
where $U \triangleq \max_{l=1,\dots,L}
\langle U_l, \tilde R_n^{(m)} - \E[ \tilde R_n^{(m)}]\rangle$ with $U_l= (A^{(\hat l )} - A^{(\bar l)})/\|A^{(\hat l )} - A^{(\bar l)}\|_2$ if $A^{(\hat l )} \ne A^{(\bar l)}$ and $U_l=0$ otherwise. It follows from the last display and the bound
$$
\frac16 \|A^{(\hat l )} - A^{(\bar l)}\|_2^2 \leq \frac{1}{3}\|A^{(\hat l)}-K\|_2^2 + \frac{1}{3}\|A^{(\bar l)}-K\|_2^2
$$
that
\begin{align}\label{U2}
\|A^{(\hat l)} - K\|_2^2 \le 2 \|A^{(\bar l)} - K\|_2^2 +9 U^2.
\end{align}
Since $\bar l$ is arbitrary, to complete the proof it suffices to bound the random variable $U$ in probability. We first obtain a bound for each of the variables $\zeta_l= \langle U_l, \tilde R_n^{(m)} - \E[ \tilde R_n^{(m)}]\rangle$. Note that associating $U_l$ with the corresponding $m\times m$ matrices that we will also denote by $U_l$, we can write $\zeta_l =\langle U_l, \hat B - B \rangle$ where $\hat B = (2/n) \sum_{i=n/2+1}^n {\rm x}_i (m){\rm x}_i(m)^\top$, $B=K^{(m)} + \sigma^2 I^{(m)} = \E[{\rm x}_i(m){\rm x}_i(m)^\top]$, and
${\rm x}_i(m)$ are i.i.d. normal vectors with mean 0 and covariance matrix $B$ (cf. the proof of Lemma~\ref{lem-sparse}) and $\langle \cdot, \cdot \rangle$ is the inner product of matrices. It follows that
\begin{align*}
\zeta_l&= \langle B^{1/2} U_l B^{1/2}, \,\frac{2}{n}\sum_{i=n/2+1}^n Z_i Z_i^\top - I^{(m)}\rangle\\
&= \tr\Big( \frac{2}{n}\sum_{i=n/2+1}^n B^{1/2} U_l B^{1/2}Z_i Z_i^\top - B^{1/2} U_l B^{1/2}\Big)\\
&= \frac{2}{n}\sum_{i=n/2+1}^n Z_i^\top DZ_i - \tr(D)
\end{align*}
where $Z_1,\ldots,Z_n$ are i.i.d. standard normal vectors in $\R^m$ and $D=B^{1/2} U_l B^{1/2}$. By the Hanson-Wright inequality (see, e.g., \cite{RudelsonVershynin}) we have that for any $t>0$, with probability at least $1-e^{-t}$,
\begin{align}\label{HW}
\left|\frac{2}{n}\sum_{i=n/2+1}^n Z_i^\top DZ_i - \tr(D)\right| \le C\left(\frac{\|D\|_\infty t}{n} + \|D\|_F\sqrt{\frac{t}{n}}\right)
\end{align}
where $C>0$ is an absolute constant. Since $\|U_l\|_2\le1$ when considering $U_l$ as a function (which is equivalent to $\|U_l\|_F\le1$ when considering $U_l$ as a matrix) and $\|B\|_\infty\le \l_{\max} +\s^2$ we have $\|D\|_\infty\le \|D\|_F\le \l_{\max} +\s^2$. Thus, with probability at least $1-e^{-t}$
\begin{align*}
|\zeta_l| \le C (\lambda_{\max} \vee \sigma^2) \left( \sqrt{\frac{t}{n}} + \frac{t}{n}\right)
\end{align*}
where $C>0$ is an absolute constant.
The union bound argument gives that, with probability at least $1-e^{-t}$,
$$
U^2= \max_{l=1,\dots,L} \zeta_l^2 \le C (\lambda_{\max} \vee \sigma^2)^2 \left( \sqrt{\frac{t+ \log L}{n}} + \frac{t+ \log L}{n}\right)^2
$$
where $C>0$ is an absolute constant. Combining this with \eqref{U2} proves the theorem.
\end{proof}
We now apply Theorem \ref{adap-scheme} to $A^{(l)}=\hat A^{(l)}$ where the estimators $\hat A^{(1)},\ldots, \hat A^{(L)}$ are defined in (\ref{Lasso}).
Combining Theorems \ref{cor-main}, \ref{adap-scheme} and the fact that, for a random variable $\eta$, $\E[|\eta|] = \int_{0}^{\infty} \mathbb P \left(|\eta|\geq t \right)dt$ we get the following result.
\begin{theorem}\label{th:model_sel}
Let each of the estimators $\hat A^{(l)}$ satisfy the conditions of Theorem \ref{cor-main}. Then
$$
\E\left[\|\hat A^{(\hat l)} - K\|_2^2\right] \le C\min_{1\leq l \leq L} \inf_{A \in \mathcal S_l,\, A\geq 0} \left\lbrace \|A-K\|_2^2 + v_n(A,l,l) \right\rbrace +C[\lambda_{\max}\vee \sigma^2]^2 \frac{\log L}{n}
$$
for all $K$ satisfying \eqref{K} with $\sum_{k= 1}^r \lambda_k<\infty$.
Here,
$C>0$ is an absolute constant.
\end{theorem}
We now fix $L=n$. Using Theorem~\ref{th:model_sel}, Theorem~\ref{th:smooth}, Theorem~\ref{th:nonpar} and Corollary~\ref{cor:finite}, we obtain the following result.
\begin{theorem}\label{th:adapt}
Let each of the estimators $A^{(l)}=\hat A^{(l)}$ satisfy the conditions of Theorem \ref{cor-main}. Let $\hat A^{(\hat l)}$ be the aggregated estimator with $\hat l$ defined in (\ref{hatl}) with $L=n$.
(i) For any $r\geq 1$, $c_*>0$ and $s>0$ such that $1\le r\leq n^{1+2s}$, we have
\begin{align*}
\sup_{K\in \cK_{r}(s,c_*;\l_{\max})} \E\|\hat A^{(\hat l)} - K\|_2^2 \leq C \min\big(rn^{-\frac{2s}{2s+1}}, \, r^{\frac{1}{s+1}}n^{-\frac{s}{s+1}}\big),
\end{align*}
where $C>0$ is a constant that can depend only on $\lambda_{\max},\sigma^2, c_*$, and $s$.
(ii) For any $r\geq 1$, $\rho\geq 1$, $s>0$, $\l_{\max}>0$, $\sigma^2\geq 0$ such that $\rho^2 \leq (\l_{\max} + \sigma^2)^2 \min \big( r n^{2s} , n^{1+2s} \big)$,
we have
\begin{align*}
\sup_{K\in \overline{\cK}_{r}(s,\rho;\l_{\max})} \E\|\hat A^{(\hat l)} - K\|_2^2 \leq C \min\left( \left(\frac{r}{n}\right)^{2s/(2s+1)}, \, n^{-s/(s+1)}\right),
\end{align*}
where $C>0$ is a constant that can depend only on $\lambda_{\max},\sigma^2,\rho$, and $s$.
(iii) If $(r\wedge l)l\ge \log n$ and $l\le n$, then for any $\l_{\max}>0$,
\begin{align*}\label{}
\sup_{K\in{\mathcal K}_{r,l}(\l_{\max}) }\E[\| \hat A^{(\hat l)} - K \|_2^2 ] \leq C \frac{(r\wedge l)l}{n}
\end{align*}
where $C>0$ is a constant that can depend only on $\lambda_{\max}$ and $\sigma^2$.
\end{theorem}
The conditions $r\leq n^{1+2s}$ and $\rho^2 \leq (\l_{\max} + \sigma^2)^2 \min \big( r n^{2s} , n^{1+2s} \big)$ are rather mild. Indeed, if $r$ and $\rho$ are fixed quantities, then these conditions are satisfied for $n$ large enough. Theorem~\ref{th:adapt} shows that the estimator $\hat A^{(\hat l)}$ is adaptive to the unknown parameters $r$ and $s$ on the scale of classes $\overline{\cK}_{r}(s,\rho;\l_{\max})$ and $ \cK_{r}(s,c_*;\l_{\max})$ that no price is paid in the rate as compared to the non-adaptive estimators of Theorems~\ref{th:smooth} and \ref{th:nonpar}. The same estimator is adaptive on the scale of classes ${\mathcal K}_{r,l}(\l_{\max})$, again with no price to be paid, for a wide range of values of $l$ and $r$.
\section{Estimation of $\sigma^2$}\label{noise-est}
We now tackle the estimation of the unknown variance $\sigma^2$. We use the simple idea that $\langle e_l, S\rangle$ becomes negligible for large $l$ when Assumption \ref{bias-cond} is satisfied. Therefore, we propose the following (biased) estimator of $\sigma^2$ based on an independent copy $X$ of the process (\ref{model}):
\begin{align}
\hat\sigma^2 = \frac{1}{M} \left\|\dot X^{(L+M)}-\dot X^{(L)}\right\|_2^2,\quad L=e^{n}, M\geq 1.
\end{align}
\begin{theorem}
Let $n,l\geq 1$ be integers and let $X_1(\cdot),\dots,X_n(\cdot)$ be i.i.d. realizations of the process $X(\cdot)$ satisfying (\ref{model}). Let Assumption \ref{bias-cond} be satisfied. For any $t>0$, we have with probability at least $1-e^{-t}$
$$
|\hat\sigma^2 - \sigma^2| \lesssim \max\left\lbrace c_{*}^2r \lambda_{\max} L^{-2s} ( 1\vee \sqrt{t}\vee t) , \sigma^2 \sqrt{\frac{t}{M}} , \frac{t}{M}\right\rbrace.
$$
\end{theorem}
\begin{proof}
We have, in view of Plancherel inequality, that
\begin{align}\label{sigma-est}
&\hat\sigma^2 - \sigma^2= \frac{1}{M} \left\| S^{(L+M)} - S^{(L)} \right\|_2^2 + \frac{2}{M}\langle S^{(L+M)} - S^{(L)},\dot W^{(L+M)}-\dot W^{(L)}\rangle\notag\\
&\hspace{6cm}+ \frac{1}{M}\left\|\dot W^{(L+M)}-\dot W^{(L)}\right\|_2^2- \sigma^2\notag\\
&= \frac{1}{M} \sum_{l=L}^{L+M} \langle S,e_l\rangle^2 + \frac{2}{M}\sum_{l=L}^{L+M}\langle S,e_l\rangle z_l+ \frac{1}{M}\sum_{l=L}^{L+M} z_l^2- \sigma^2 = I+II+III,
\end{align}
where $z_L,\ldots,z_{L+M}$ are i.i.d. standard normal random variables also independent from $S$.
We now take the expectation
\begin{align*}
\E\left[\hat\sigma^2\right] - \sigma^2= \frac{1}{M} \sum_{l=L}^{L+M}\sum_{j=1}^r \lambda_j \langle \varphi_j, e_l\rangle^2.
\end{align*}
Note that $\langle \varphi_j, e_l\rangle^2 \leq \|\varphi_j - \varphi_j^{(l-1)}\|_2^2$. In view of Assumption \ref{bias-cond}, we get
\begin{align*}
\frac{1}{M} \sum_{l=L}^{L+M}\sum_{j=1}^r \lambda_j \langle \varphi_j, e_l\rangle^2 &\leq r \lambda_{\max} \frac{c_*^2}{M}\sum_{l=L-1}^{L+M-1} l^{-2s}\lesssim c_*^2 r \lambda_{\max} L^{-2s}.
\end{align*}
The bound in probability follows easily from the representation (\ref{sigma-est}). Indeed, the second term can be treated using standard deviations bounds for Gaussian combined with a conditioning argument. The third term can be treated with a standard deviation inequality for chi-square distributions. The first term can be treated using (\ref{HW}) again. More specifically, set $\xi = (\xi_1,\ldots,\xi_r)^\top$ and $A = (a_{j,j'})_{1\leq j,j'\leq r}$ with
$$
a_{j,j'} = \frac{\sqrt{\lambda_j \lambda_{j'}}}{M}\sum_{l=L}^{L+M} \langle \varphi_j, e_l \rangle \langle \varphi_{j'}, e_l\rangle.
$$
Then, we have
$$
\frac{1}{M} \sum_{l=L}^{L+M} \langle S,e_l\rangle^2 - \E\left[\frac{1}{M} \sum_{l=L}^{L+M} \langle S,e_l\rangle^2 \right]= \xi^\top A \xi - \E[\xi^\top A \xi],
$$
with $\|A\|_F\lesssim c_{*}^2 r\lambda_{\max} L^{-2s}$ and $\|A\|_{\infty} \lesssim c_*^2 \sqrt{r } \lambda_{\max} L^{-2s}$.
An union bound argument gives the result. Details of the proof are omitted here.
\end{proof}
\section{Minimax lower bound}\label{lower bound}
In this section, we show that the upper bounds of Corollary~\ref{cor:finite} and Theorem~\ref{th:nonpar} cannot be improved in a minimax sense.\begin{theorem}\label{th:finite:lower}
Let $1\le r< \infty$ and let $\l_{\max}>0$ be a given constant.
Then there exist absolute constants $c_0>0$ and $0<c_1<1$ such that, for any integers $n$ and $l$ satisfying $l\geq 2$, $n\geq l$, we have
$$
\inf_{\hat K_n}\sup_{K \in {\mathcal K}_{r,l}(\l_{\max}) } \mathbb P \left( \|\hat K_n - K\|_2^2 \geq c_0 [\lambda_{\max} \wedge \sigma^2]^2\frac{(r\wedge l)l}{n} \right) >c_1
$$
where $\inf_{\hat K_n}$ denotes the infimum over all estimators of $K$.
\end{theorem}
\begin{proof}
Let first $r\le l/2$.
Consider the vector-functions $e(t)=(e_1(t),\dots, e_l(t))$ and $\f (t)=(\f_1(t),\dots, \f_r(t))$ and a subset of ${\mathcal K}_{r,l}(\l_{\max})$ composed of kernels $K$ satisfying \eqref{K} with $\l_j\equiv \g$ and
$$
\f(t)= H e(t)
$$
for suitable $\g>0$ and suitable $r\times l$ matrices $H$. Orthonormality of functions $\f_j$ implies that $H$ must satisfy $HH^\top=I_r$ where $I_r$ is the $r\times r$ identity matrix, i.e., the rows of $H$ should be orthonormal. To each such matrix $H$ we associate a linear subspace $U_H$ of $\R^l$, which is the linear span of the $r$ rows of $H$. Clearly, ${\rm dim} (U_H)=r$ and $H^\top H$ is the orthogonal projector onto $U_H$ in $\R^l$.
Note that the set of all such spaces $U_H$ is the Grassmannian manifold $G_{r}(\mathbb R^l)$, i.e., the set of $r$-dimensional linear subspaces of $\mathbb R^l$. The Grassmannian manifold $G_{r}(\mathbb R^l)$ is a smooth manifold of dimension $d= r(l-r)$. A natural metric $d(\cdot,\cdot)$ on $G_{r}(\mathbb R^l)$ is defined as follows: for $U,\bar U \in G_{r}(\mathbb R^l)$,
$$
d(U,\bar U) \triangleq \|P_U-P_{\bar U}\|_F = \|H^\top H-\bar H^\top \bar H\|_F
$$
where $P_U$ is the orthogonal projector onto $U$ and $H$, $\bar H$ are the $r\times l$ matrices with orthonormal rows associated to $U$ and $\bar U$ respectively. We refer to \cite{Pertti} and \cite{MilnorStasheff} for more details on the Grassmannian manifold.
From now on, we will identify $U\in G_{r}(\mathbb R^l)$ with the associated orthogonal projector $P_U=H^\top H$.
The behavior of entropy numbers of the Grassmannian manifold is well studied (\cite{Szarek1982}, see also Proposition 8 in \cite{Pajor1998}). In particular, for any $\epsilon \in (0,1)$ there exists a family of orthogonal projectors $\mathcal U \subset G_{r}(\mathbb R^l)$ such that
\begin{align}\label{Grassman-entropy}
|\mathcal U | \geq \left\lfloor \frac{\bar c}{\epsilon} \right\rfloor^{d}\quad\text{and}\quad \bar c \epsilon \sqrt{r} \le \|P-Q\|_F \le \frac1{\bar c} \epsilon \sqrt{r}, \; \forall P,Q \in \mathcal{U}, \; P \neq Q,
\end{align}
for some small enough universal constant $\bar c>0$. Here $|\mathcal U |$ denotes the cardinality of $\mathcal U$. We take in what follows $\epsilon = 1/2$. Set $N = |\mathcal U |$ and $
\mathcal U = \left\lbrace P_1,\dots,P_N \right\rbrace$. The associated $H$-matrices will be denoted by $H_1,\dots,H_N$. Let $K_j$ be a kernel of the form \eqref{K} with eigenvalues $\l_i\equiv \g, i=1,\dots, r,$ and
$$
\f(t)= H_j e(t), \quad j=1,\dots, N,
$$
where $\gamma =a(\sigma^2\wedge \lambda_{\max})\sqrt{\frac{l}{n}}$ and $a \in (0,1)$ is an absolute constant to be chosen later. Consider the set $\cK'=\{K_1,\dots, K_N\}$. Clearly, we have $\cK' \subset {\mathcal K}_{r,l}(\l_{\max})$.
We now evaluate the Kullback-Leibler divergence between two probability measures induced by the observations $\{X_1(t),\dots, X_n(t), t\in [0,1]\}$ corresponding to the kernels $K_1$ and $K_j$ (with $j\ne 1$). Using the Girsanov formula and the fact that $K_j$ is bilinear in $\{e_k\}$ it is easy to check that this divergence is equal to
the Kullback-Leibler divergence between the $n$-product distributions of the associated Gaussian vectors $ \left( \int_0^1 e_1(t)d X (t),\dots, \int_0^1 e_l(t)d X (t)\right)$. If $K=K_j$ this vector is distributed as ${\mathcal N}\left(0,\Sigma_j\right)$ with $\Sigma_j = \sigma^2 I_{l} +
\gamma P_j = (\sigma^2+ \gamma)P_j + \sigma^2 P_j^{\perp}$ and $P_j^{\perp} = I_{ l} - P_j$. Denote the corresponding Gaussian measure by $\PP_j$ and by $\PP_j^{\otimes n}$ its $n$-product. Let ${\rm KL}(\PP,\QQ)$ be the Kullback-Leibler divergence between two probability measures $\PP$ and $\QQ$.
It is easy to see that all matrices $\Sigma_j$ have the same eigenvalues. Thus, for any $2\leq j \leq N$ we have
\begin{align*}
{\rm KL}(\PP_1^{\otimes n}, \PP_j^{\otimes n}) &= n \,{\rm KL}(\PP_1, \PP_j) \notag\\
&= \frac{n}{2} \left[ \mathrm{tr}(\Sigma_1^{-1} \Sigma_j) - l- \log \left( \mathrm{det}(\Sigma_1^{-1}\Sigma_j) \right) \right]\notag\\
&= \frac{n}{2} \left[ \mathrm{tr}(\Sigma_1^{-1}(\Sigma_j-\Sigma_1)\right].
\end{align*}
Now, $\Sigma_1^{-1} = \frac{1}{\sigma^2 + \gamma}P_1 + \frac{1}{\sigma^2}P_1^{\perp}$, which yields
\begin{align*}
\mathrm{tr}(\Sigma_1^{-1}(\Sigma_j-\Sigma_1)) &= \frac{\gamma}{\sigma^2 + \gamma}\mathrm{tr}(P_1 (P_j - P_1))+ \frac{\gamma}{\sigma^2}\mathrm{tr}(P_1^{\perp}(P_j-P_1))\\
&= \left(\frac{\gamma}{\sigma^2 + \gamma} - \frac{\gamma}{\sigma^2}\right) \left( \mathrm{tr}(P_1P_j) - r\right)\\
& = \frac{\gamma^2}{2(\sigma^2 + \gamma)\sigma^2}\|P_1- P_j\|_F^2\\
&\leq \frac{r\gamma^2}{8 \bar c^2 (\sigma^2 + \gamma)\sigma^2}
\end{align*}
where for the last inequality we have used \eqref{Grassman-entropy} with $\epsilon=1/2$, and the fact that $\mathrm{tr}(P_1P_j) = r - \|P_1 - P_j\|_F^2/2$.
Combining the last two displays, we find
\begin{align*}
{\rm KL}(\PP_1^{\otimes n}, \PP_j^{\otimes n}) &\leq a^2\frac{(\lambda_{\max} \wedge \sigma^2)^2}{8\bar c^2(\sigma^2 + \gamma)\sigma^2}rl ,\quad \forall \ 2\leq j \leq N.
\end{align*}
Recall that we assume $r\le l/2$, so that the dimension of the Grassmannian satisfies $d =r(l-r) \geq rl/2$. Consequently, in view of (\ref{Grassman-entropy}), we have that $\log |\mathcal U|\ge {\tilde c}rl $ for some absolute constant ${\tilde c}>0$. Thus, we get
\begin{align*}
{\rm KL}(\PP_1^{\otimes n}, \PP_j^{\otimes n}) &\leq \frac{1}{16}\log |\mathcal U|,\quad \forall \ 2\leq j \leq N,
\end{align*}
provided $a>0$ is taken sufficiently small independently of $r,l,n,\sigma,\lambda_{\max}$.
Next, for any $1\leq i,j\leq N$ with $i\neq j$,
$$
\| K_i - K_j \|_2^2 = \gamma^2 \|H_i^\top H_i-H_j^\top H_j\|_F^2=\gamma^2 \|P_i - P_j\|_F^2 \geq c a^2 [\sigma^4\wedge \lambda_{\max}^2]\frac{ r l }{n},
$$
where $c>0$ is a absolute constant and the last inequality is due to~(\ref{Grassman-entropy}).
The result now follows from the last two displays by application of Theorem 2.5 in \cite{Tsybakov-book}.
Finally, consider the case $r> l/2 $. Note that the classes ${\mathcal K}_{r,l}(\l_{\max})$ are nested in $r$. Assuming w.l.o.g. that $l$ is even, we get that the minimax risk over ${\mathcal K}_{r,l}(\l_{\max})$ is bounded from below by the minimax risk on ${\mathcal K}_{l/2,l}(\l_{\max})$.
But the minimax risk on ${\mathcal K}_{l/2,l}(\l_{\max})$ has been already treated above and we have proved that the lower rate is of the order $l^2/n$, which is the desired rate when $r>l/2$.
\end{proof}
\begin{remark}
It is possible to prove a minimax lower bound ensuring that the bound in Theorem \ref{th:smooth} is optimal at least regarding the $n$ dependence. Indeed, by a similar argument to that used in the proof of Theorem \ref{th:finite:lower}, we can prove the existence of an absolute constant $0<c_2<1$ and a constant $c_3>0$ possibly depending on $\sigma^2,\lambda_{\max},\rho,r$ such that, for any integer $n\geq 1$ we have
$$
\inf_{\hat K_n}\sup_{K \in \overline{\cK}_r(s,\rho;\lambda_{\max})} \mathbb P \left( \|\hat K_n - K\|_2^2 \geq c_3
\min\left(n^{-\frac{2s}{2s+1}}, n^{-s/(s+1)}\right)\right) >c_2
$$
where $\inf_{\hat K_n}$ denotes the infimum over all estimators of $K$. Specifying the dependence of the minimax rate on parameters $\sigma^2,\lambda_{\max},\rho,r$ remains an interesting open question.
\end{remark} | 53,163 |
Waylaid By Two Downturns Downtown, Will There Be A Third?August 14, 2013
First approved for development over a decade ago with plans for an 11-story hotel and restaurant to rise on the little downtown parking lot at 72 Ellis Street, the owners of the parcel were given until 2004 to start construction on the site:
).
Getting rid of that parking lot would be divine. It’s a horrible eyesore.
1 yr seems appropriate. **** or get off the pot (site)
crap :/ i had dreams of a 30+ story condo tower going up i even made a sketch. This tower would have killer views of union sq , hopefully they increase to 15 floors?
It’s ridiculous how long they’ve been sitting on this. Hopefully they are serious about building this time.
Considering there’s almost nothing in the hotel pipeline, you’d think investors would be eager to build this thing already. | 307,577 |
TITLE: Can we build the set of realizable types for a binary product of structures out of the set of realizable types for each structure individually?
QUESTION [0 upvotes]: I'm trying to understand types better in model theory and, in particular, how the set of all types interacts with various ways of building new models out of old ones (e.g. the Cartesian product $N \times M$).
What do we know about the set of all types and the set of all realizable types for a binary Cartesian product of models?
Suppose $M$ is a structure. Let $\mathrm{TY}(M)$ denote the set of all $n$-types of $M$ over any subset of $M$ and for any value of $n$.
Let $\mathrm{RT}(M)$ denote the subset of $\mathrm{TY}(M)$ consisting of exactly the realized types. For all types $p(x)$, $p(x)$ is in $\mathrm{RT}(M)$ if and only if $p(x)$ is in $\mathrm{TY}(M)$ and there exists some $b \in M^n$ such that $M \models p(b)$.
Fix a signature $\sigma$. Let $M$ and $N$ be structures over $\sigma$.
Let $M \times N$ be the Cartesian product of $M$ and $N$, with the interpretations of functions and relations defined componentwise as usual.
I'm interested in whether the following are true. Assuming they're false, I'm interested in what we do know about the set of all types and the set of realizable types in this setting.
$$ \mathrm{TY}(M \times N) = \mathrm{TY}(M) \cap \mathrm{TY}(N) \tag{1} $$
$$ \mathrm{RT}(M \times N) = \mathrm{RT}(M) \cap \mathrm{RT}(N) \tag{2} $$
I can prove the first claim.
For an ordinary parameter-free sentence $\varphi$, $M \times N \models \varphi$ holds if and only if $M \models \varphi$ holds and $N \models \varphi$ holds. This falls out of the way that the interpretation of function symbols and the interpretation of relation symbols are defined in $M \times N$.
For a sentence with paramters $\varphi(A)$, the same argument holds. If $\varphi(A)$ holds in $M \times N$, then it holds in $M$ and $N$ individually. Also, every possible pair $(m, n)$ is in $M \times N$, so if $\varphi(A)$ holds at some $m$ in $M$ and at some $n$ in $N$ for each parameter, then we can build a the values of the parameter $(m_1, n_1), (m_2, n_2) \cdots$ and each $(m_i, n_i)$ will be in the domain of $M \times N$.
If this holds for every sentence individually, then $p(x)$ is finitely satisfiable in $M \times N$ if and only if it is finitely satisfiable in $M$ and finitely satisfiable in $N$ because finite satisfiability can be assessed by only ever examining one sentence at a time.
For realizable types, I'm a lot less certain. Is there a way to build the set of realizable types over the Cartesian product of $M$ and $N$ out of the set of realizable types for each structure individually?
REPLY [3 votes]: Contra your claim, (1) isn't true at all. Already at the parameter-free level we have a problem: Cartesian products do not preserve arbitrary sentences. For a trivial example, take two finite structures each of size $>1$. For a nontrivial example, consider the standard exercise that $\mathbb{Z}\not\equiv\mathbb{Z}\times\mathbb{Z}$ as groups. And when we bring parameters into the picture, the statement doesn't even parse since $M\times N\not\subseteq M\cap N$ in general.
Unfortunately I don't see offhand any good sense in which the set of types of $M\times N$ can be built from the sets of types of $M$ and of $N$ separately, let alone in which realization plays well with this. | 215,751 |
Owl Love Card + Pin Combo
$ 11.99
Owl always love you! Give this to someone deer, so they know you're always watching them lovingly from a tree.
This 2-in-1 combo includes a card and gift (a pin!) together.
- 4.25" x 5.5" card
- Printed in the USA on thick matte paper
- Comes w/ matching enamel pin
- Packaged in a poly bag | 106,434 |
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