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Episode 8
recorded January 16, 2011
Welcome to This Week in Location Based Marketing where we rehash the news that matters in the location based marketing world.
Hosts: Rob Woodbridge & Asif Khan
NEW! Subscribe to TWiLBM: Audio feed | Video Feed
Show highlights:
1. Groupon Raises $950M And Moving To IPO
2. Living Social Buys Majority Stake in Let’s Bonus
3. O2 and Orange To Offer Brands Location-Based Services
4. Poynt Awarded Location Patent
5. Apple’s find a friend iOS feature
Resource of the Week
MomentFeed
Tool of the Week
eMarketer Report: Beyond the Check-in: Best Practices for Location Based Marketing
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Listen to the audio version now: [audio:]
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\begin{document}
\fontsize{12pt}{14pt} \textwidth=14cm \textheight=21 cm
\numberwithin{equation}{section}
\title{Sheaves on non-reduced curves in a projective surface.}
\author{Yao Yuan}
\subjclass[2010]{Primary 14D05}
\thanks{The author is supported by NSFC 21022107 and NSFC 11771229. }
\begin{abstract}Sheaves on non-reduced curves can appear in moduli space of 1-dimensional semistable sheaves over a surface, and moduli space of Higgs bundles as well. We estimate the dimension of the stack $\bm_{X}(nC,\chi)$ of pure sheaves supported at the non-reduced curve $nC~(n\geq 2)$ with $C$ an integral curve on $X$. We prove that the Hilbert-Chow morphism $h_{L,\chi}:\mm^H_X(L,\chi)\ra \ls$ sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for $X$ Fano or with trivial canonical line bundle and $\ls$ contains integral curves.
~~~
\textbf{Keywords:} 1-dimensional pure sheaves on projective surfaces, Hilbert-Chow morphism, Hitchin fibrations, stacks.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction.}
\subsection{Motiviations.} Sheaves on non-reduced curves can appear in two types of moduli stacks $\mm^H_X(L,\chi)$ and $\mm_{C,D}^{Higgs}(n,\chi)$ as follows.
\begin{enumerate}
\item[(A)] $\mm^H_X(L,\chi)$ parametrizes semistable sheaves $\mf$ with respect to the polarization $H$ on a projective surface $X$, supported on a curve in the linear system $\ls$ and of Euler characteristic $\chi$. We have a Hilbert-Chow morphism
\begin{equation}\label{HCmap}h_{L,\chi}:\mm^H_X(L,\chi)\ra \ls,~~\mf\mapsto \text{supp}(\mf).\end{equation}
In general $\ls$ contains singular curves and also curves with non-reduced components.
~~
\item[(B)] $\mm_{C,D}^{Higgs}(n,\chi)$ parametrizes semistable Higgs bundles $(\me,\Theta)$ with respect to the effective divisor $D$ on a smooth curves $C$ with $\me$ rank $n$ and Euler characteristic $\chi$. We have the Hitchin fibration
\begin{equation}\label{HF}h_{D,\chi}:\mm_{C,D}^{Higgs}(n,\chi)\ra \bigoplus_{i=1}^n H^0(C,\mo_C(iD)),~~(\me,\Theta)\mapsto \text{char}(\Theta).\end{equation}
Denote by $\Tot(\mo_C(D))$ the total space of the the line bundle $\mo_C(D)$. Let $p:\Tot(\mo_C(D))\ra C$ be the projection, then every Higgs bundle $(\me,\Theta)$ on $C$ gives a 1-dimensional pure sheaf $\mf_{\me}$ on $\Tot(\mo_C(D))$ via the following exact sequence
\[0\ra p^*\me\xrightarrow{\lambda\times p^*\text{id}_{\me}-p^*\Theta}p^*\me(D)\ra \mf_{\me}\ra 0. \]
We naturally have that $\text{supp}(\mf_{\me})$ is defined by the equation $\text{char}(\Theta)$ where $\lambda$ is the variable on the fiber of $p$. Hence the fiber of the Hitchin fibration over $\lambda^n$, which is also called the central fiber of the Hitchin fibration, consists of sheaves (with some semistablity) on the non-reduced curve in $\Tot(\mo_C(D))$ defined by $\lambda^n$.
Let $X:=\mathbb{P}(\mo_C(D)\oplus\mo_C)$ be the ruled surface over $C$. Then $X$ is projective and the central fiber of $h_{D,\chi}$ consists of sheaves (with some semistablity) on the non-reduced curve $nC_e$ with $C_e$ the section satisfying $C_e^2=\text{deg}(D)$ and $C_e.K_X=-\text{deg}(D)+2g_C-2$.
\end{enumerate}
People have to consider the dimension of each fibers of $h_{L,\chi}$ in $(A)$ or $h_{D,\chi}$ in $(B)$, if they want to study the flatness of the fibrations, or if they want to compute the cohomology of sheaves (constructible or coherent) over the moduli space $\mm^H_X(L,\chi)$ ($\mm_{C,D}^{Higgs}(n,\chi)$, resp.) via the fibration $h_{L,\chi}$ ($h_{D,\chi}$, resp.).
The fibers of $h_{L,\chi}$ ($h_{D,\chi}$, resp.) over integral supports are their compactified Jacobians and hence are of equal dimension denoted by $N_L$ ($N_{C,D,n}$, resp.) which only depends on $\ls$ ($(C,D,n)$, resp.). But the fibers over non-integral supports are much more complicated, among which the worst are fibers over non-reduced curves. People expect those fibers would also have dimension $N_L$ or $N_{D,C,n}$ respectively, but in principle they can be of larger dimension.
So we pose the following question.
\begin{que}\label{mque}Whether all fibers of $h_{L,\chi}$ in $(A)$ ($h_{D,\chi}$ in $(B)$, resp.) are of the expected dimension $N_{L}$ ($N_{C,D,n}$, resp.)?
\end{que}
Some results are already known for Question \ref{mque}. For instance, Ginzburg showed in \cite{Gin} that the central fiber of $h_{D,\chi}$ is of expected dimension $N_{C,D,n}$ for $D$ the canonical divisor of $C$; Chaudouard and Laumon showed
in \cite{CL} that the central fiber of $h_{D,\chi}$ is of dimension $\leq N_{C,D,n}$ for $\text{deg}(D)>2g-2$; and finally Maulik and Shen showed in \cite{MS} that the fibers of $h_{L,\chi}$ are of the expected dimension $N_L$ for $X$ a toric del Pezzo surface and $L$ ample effective on $X$. There seems to be no more general result. Especially the lack of the estimate of the dimension of all fibers of $h_{L,\chi}$ prevents the main result in \cite{MS} from generalizing to all del Pezzo surfaces not necessarily toric (see the paragraph after Remark 0.2 in \cite{MS}).
In this paper we give a complete answer to Question \ref{mque}.
\subsection{Notations \& Conventions.} All through the paper, let $X$ be a projective surface over an algebraically closed field $\Bbbk$. Let $C$ be an integral curve on $X$. Let $\delta_C\in H^0(X,\mo_X(C))$ be the section defining $C$. Denote by $nC$ the non-reduced curve with multiplicity $n$ over $C$, i.e. the 1-dimensional closed subscheme of $X$ defined by $\delta_C^n$.
Denote by $\overline{\bm_X}(nC,\chi)$ the stack of all 1-dimensional sheaves with schematic supported $nC$ and Euler characteristic $\chi$, and let $\bm_X(nC,\chi)\subset\overline{\bm_X}(nC,\chi)$ be the substack consisting of 1-dimensional pure sheaves. Then we have $h^{-1}_{\mo_X(nC),\chi}(nC)\subset \bm_X(nC,\chi)$ where $h_{\mo_X(nC),\chi}$ is the Hilbert-Chow morphism defined in (\ref{HCmap}).
Because the stack $\mathbf{Vect}_C(n,\chi)$ of rank $n$ Euler characteristic $\chi$ vector bundles over $C$ is a substack of $\bm_X(nC,\chi)$, we have $$\dim\overline{\bm_X}(nC,\chi)\geq\dim \bm_X(nC,\chi)\geq \dim\mathbf{Vect}_C(n,\chi)=n^2(g_C-1).$$
We use $K_X$ to denote both the canonical divisor of $X$ and the canonical line bundle as well. For any two (not necessarily integral) curves $C',C''$, we write $C'.K_X$ the intersection number of the divisor class of $C'$ with $K_X$, $C'.C''$ the intersection number of the divisor classes of these two curves and $C'^2:=C'.C'$.
Denote by $g_{C'}$ the arithmetic genus of $C'$. We have $g_{C'}-1=\frac{C'^2+C'.K_X}2.$
For a sheaf $\mf$ on $X$, let $\mf(\Sigma):=\mf\otimes\mo_X(\Sigma)$ for $\Sigma$ a curve or a divisor class.
Let $K(X),K(C)$ be the Grothendieck groups of coherent sheaves on $X$ and $C$ respectively. Let $\star$ stands for $X$ or $C$. Denote by $\chi_{\star}(-,-):K(\star)\times K(\star)\ra \mathbb{Z}$ the bilinear integral form on $K(\star)$ such that for every two coherent sheaves $\ms,\mt$ on $\star$,
$$\chi_{\star}([\ms],[\mt])=\sum_{j\geq 0}(-1)^j\dim\Ext^j_{\mo_{\star}}(\ms,\mt).$$
\subsection{Results \& Applications.}Our main result is the following theorem.
\begin{thm}[See Theorem \ref{main4}]\label{main1}For any integral curve $C\subset X$, we have
$$\dim \bm_X(nC,\chi)\begin{cases}\leq \frac{n^2C^2}2+\frac{nC.K_X}2=g_{nC}-1,\text{ if }C.K_X\leq 0\\ \\ =\frac{n^2C^2}2+\frac{n^2C.K_X}2=n^2(g_{C}-1),\text{ if }C.K_X>0\end{cases}.$$
\end{thm}
Theorem \ref{main1} suggests that in order to have a positive answer to Question 1.1, it is reasonable to ask the surface $X$ in $(A)$ to be Fano or with $K_X$ trivial and to ask the divisor $D$ in $(B)$ to satisfy $\text{deg}(D)\geq 2g-2$.
\begin{coro}\label{mcoro}Let $C'=\displaystyle{\prod_{j=1}^s}n_jC_j$ be a curve in the linear system $\ls$ on $X$ with $C_j$ pairwise distinct integral curves. If $C_j.K_X\leq 0$ for $j=1,\cdots,s$, then
\[\dim h_{L,\chi}^{-1}(C')\leq g_{C'}-1=N_{L}.\]
In particular, if $X$ is Fano or with $K_X$ trivial, and if either $\ls$ contains integral curves, or $H^1(\mo_X)=H^1(L)=0$ and $\mm^H_X(L,\chi)^s$, the substack of $\mm^H_X(L,\chi)$ consisting of stable sheaves, is not empty, then the Hilbert-Chow morphism $h_{L,\chi}$ in (\ref{HCmap}) has all fibers the expected dimension $g_{\ls}-1$ with $g_{\ls}$ the arithmetic genus of any curve in $\ls$.
\end{coro}
\begin{proof}Every sheaf in $h_{L,\chi}^{-1}(C')$ can be realized as an excessive extension of pure sheaves on $n_jC_j, ~j=1,\cdots,s$. Since $C_j$ are pairwise distinct integral curves, for sheaves $\mf_i\in \bm_X(n_jC_j,\chi_j)$ we have
\[\Hom_{\mo_X}(\mf_i,\mf_j)=\Ext^2_{\mo_X}(\mf_i,\mf_j)=0,~\dim\Ext^1_{\mo_X}(\mf_i,\mf_j)=-\chi_X(\mf_i,\mf_j),~\forall~i\neq j.\]
Hence we have
\begin{eqnarray}\dim h_{L,\chi}^{-1}(C')&\leq& \sum_{j=1}^s\dim\bm_{X}(n_jC_j,\chi_j)-\sum_{i<j}\chi_X(\mf_i,\mf_j)\nonumber\\
&=&\sum_{j=1}^s\left(\frac{n_j^2C_j^2}2+\frac{n_jC_j.K_X}2\right)+\sum_{i<j}n_in_jC_i.C_j\nonumber\\
&=&\frac{(C')^2}2+\frac{C'.K_X}2=g_{C'}-1=N_L.\end{eqnarray}
Let $X$ be Fano or with $K_X$ trivial. If $\ls$ contains integral curves, then by semicontinuity every fiber of $h_{L,\chi}$ is of dimension no less than $g_{\ls}-1$. If $H^1(\mo_X)=H^1(L)=0$, then $\mm^H_X(L,\chi)^s\neq \emptyset$ is smooth of dimension $g_{\ls}-1+\dim\ls$. Every fiber of $h_{L,\chi}$ is a closed subscheme of $\mm^H_X(L,\chi)$ defined by $\dim\ls$ equations, hence of dimension no less than $g_{\ls}-1$. We have proved the corollary.
\end{proof}
With Corollary \ref{mcoro}, we can generalize Theorem 0.1 in \cite{MS} to all del Pezzo surfaces. For the reader's convenience, we write the explicit statement as follows.
\begin{thm}[Generalization of Theorem 0.1 in \cite{MS} to all del Pezzo surfaces]\label{ms}
~Let $X$ be a del Pezzo surface \emph{not necessarily toric} with polarization $H$, and let $L$ be an ample curve class on $X$. Let $M^H_X(L,\chi)$ be the coarse moudli space of 1-dimensional semistable sheaves with schematic supports in $\ls$ and Euler characteristic $\chi$. Then we have for any $\chi,\chi'\in\mathbb{Z}$, there are isomorphisms of graded vector spaces
\[\text{IH}^*(M^H_X(L,\chi))\cong \text{IH}^*(M^H_X(L,\chi')),\]
where $\text{IH}^*(-)$ denotes the intersection cohomology. Moreover, those isomorphisms respect perverse and Hodge filtrations carried by these vector spaces.
\end{thm}
We refer to \cite{MS} for the details of the proof of Theorem \ref{ms} while Proposition 2.6 in \cite{MS} can be extended to $X$ any del Pezzo surface by Corollary \ref{mcoro}.
Another application of Corollary \ref{mcoro} is the following theorem which generalizes Theorem 1.1 in \cite{Yuan4}.
\begin{thm}\label{genY}Let $X=\p^2$ and $H$ the hyperplan class on $X$. Let $M_{\p^2}(d,\chi)$ be the coarse moudli space of 1-dimensional semistable sheaves with schematic supports in $|dH|$ and Euler characteristic $\chi$. Then for $0\leq i\leq 2d-3$ we have
\[\begin{cases}b_i^v(M_{\p^2}(d,\chi))=0,\text{ for }i\text{ odd }\\
b_i^v(M_{\p^2}(d,\chi))=b_{i}^v(X^{[\frac{d(d-3)}2-\chi_0]}),\text{ for }i\text{ even }\\
h^{p,i-p}_v(M_{\p^2}(d,\chi))=0,\text{ for }p\neq i-p\\
h^{p,i-p}_v(M_{\p^2}(d,\chi))=h^{p,i-p}_v(X^{[\frac{d(d-3)}2-\chi_0]}),\text{ for }i=2p
\end{cases}\]
where $b_i^v$ ($h^{p,q}_v$, resp.) denotes the $i$-th virtual Betti number ($(p,q)$-th virtual Hodge number, resp. ), $X^{[n]}$ is the Hilbert scheme of $n$-points on $X$ and finally $\chi_0\equiv \chi~(d)$ with $-2d-1\leq \chi_0\leq -d+1$.
\end{thm}
Theorem \ref{genY} follows immediately from Theorem 6.11 in \cite{Yuan4} and Corollary \ref{mcoro} helps us to improve the estimate of the codimension of the subscheme of $M_{\p^2}(d,\chi)$ consisting of sheaves supported at non-integral curves. Notice that by \cite{Bou}, $h^{p,q}_v(M_{\p^2}(d,\chi))=0$ for any $p\neq q$.
By Theorem \ref{genY} both the virtual Betti number $b_i^v(M_{\p^2}(d,\chi))$ and the virtual Hodge number $h^{p,q}_v(M_{\p^2}(d,\chi))$ stabilize as $d\ra\infty$. Write down the generating function
\[Z(t,q):=\sum_{d\geq 0} q^d\left(\sum_{i=0}^{2\dim M_{\p^2}(d,\chi)}b_i^v(M_{\p^2}(d,\chi))t^i\right).\]
Then the coefficient of $t^i$ in $(1-q)Z(t,q)$ is a polynomial in $q$. However, whether $Z(t,q)$ is a rational function is still a wildly open question.
\subsection{Acknowledgements.} I would like to thank L. G\"ottsche for leading me to study 1-dimensional sheaves over surfaces when I was his PhD student. I also would like to thank Junliang Shen for answering my questions on their paper \cite{MS}.
\section{Filtrations for sheaves on non-reduced curves.}
For any $\mf\in \overline{\bm_X}(nC,\chi)$, we can describe $\mf$ via two filtrations as in the following proposition.
\begin{prop}\label{2Fil}Let $\mf\in \overline{\bm_X}(nC,\chi)$, then there are two filtrations of $\mf$:
\begin{enumerate}
\item[(1)]The so-called \textbf{the lower filtration of $\mf$}:
\[0=\mf_0\subsetneq \mf_1\subsetneq\cdots\subsetneq \mf_l=\mf,\]
such that $Q_i:=\mf_{i}/\mf_{i-1}$ are coherent sheaves on $C$ with rank $t_i$. $\sum t_i=n$, and moreover there are injections $f^i_{\mf}:Q_{i}(-C)\hookrightarrow Q_{i-1}$ induced by $\mf$ for all $2\leq i\leq l$.
\item[(2)]The so-called \textbf{the upper filtration of $\mf$}:
\[0=\mf^0\subsetneq \mf^1\subsetneq\cdots\subsetneq \mf^m=\mf,\]
such that $R_i:=\mf^{i}/\mf^{i-1}$ are coherent sheaves on $C$ with rank $r_i$. $\sum r_i=n$, and moreover there are surjections $g^i_{\mf}:R_{i}(-C)\twoheadrightarrow R_{i-1}$ induced by $\mf$ for all $2\leq i\leq m$.
\end{enumerate}
Moreover we have
\begin{enumerate}
\item[(i)] $l=m$.
\item[(ii)] $\forall~1\leq i\leq m,$ $t_i=r_{m-i+1}$.
\end{enumerate}
\end{prop}
Proposition \ref{2Fil} is not difficult to prove. One sees easily that $\mf_i=\Ker(\mf\xrightarrow{.\delta_C^i}\mf(iC))$ and $\mf^i=\Ima(\mf((i-m)C)\xrightarrow{.\delta_C^{m-i}}\mf)$. In particular we have $Q_i\cong \mf_i/\Tor^1_{\mo_X}(\mf_i,\mo_{(i-1)C}((i-1)C))$ and $R_i\cong \mf^i\otimes\mo_C$. The reader can figure out the proof of Proposition \ref{2Fil} by him/her-self or look at Proposition 5.7, Proposition 5.10 and Lemma 5.11 in \cite{Yuan4} for more details.
\begin{rem}\label{sup}Use the same notations as in Proposition \ref{2Fil}, we can see that $m\leq n$ and $m=\max\{k|\mf\xrightarrow{.\delta_C^{k-1}}\mf((k-1)C)\text{ is not zero}\}$.
\end{rem}
Recall that $K(C)$ is the Grothendieck group of coherent sheaves on $C$. For any class $\beta\in K(C)$, denote by $r(\beta)$ ($\chi(\beta)$, resp.) the rank (Euler characteristic, resp.) of $\beta$.
Obviously both upper and lower filtrations are uniquely determined by $\mf$. Hence we can stratify $\overline{\bm_X}(nC,\chi)$ by the filtration type.
Let $\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}$ ($\overline{\bm_X}(nC,\chi)_{\beta_1,\cdots,\beta_m}$, resp.) with $\beta_1,\cdots,\beta_m\in K(C)$ be the substack of $\overline{\bm_X}(nC,\chi)$ consists of sheaves with upper (lower, resp.) filtrations satisfying $[R_i]=\beta_i$ ($[Q_i]=\beta_i$, resp.). Notice that $\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}$ ($\overline{\bm_X}(nC,\chi)_{\beta_1,\cdots,\beta_m}$, resp.) is not empty only if $r(\beta_i)\geq r(\beta_{i-1})$ ($r(\beta_i)\leq r(\beta_{i-1})$, resp.) for all $2\leq i\leq m$.
Define
\[\overline{\bm_X}(nC,\chi)^{r_1,\cdots,r_m}:=\coprod_{\substack{r(\beta_i)=r_i\\ 1\leq i\leq m}}\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m},\]
\[\overline{\bm_X}(nC,\chi)_{r'_1,\cdots,r'_m}:=\coprod_{\substack{r(\beta'_i)=r'_i\\ 1\leq i\leq m}}\overline{\bm_X}(nC,\chi)_{\beta'_1,\cdots,\beta'_m};\]
\[\bm_X(nC,\chi)^{r_1,\cdots,r_m}:=\bm_X(nC,\chi)\cap \overline{\bm_X}(nC,\chi)^{r_1,\cdots,r_m};\]
\[\bm_X(nC,\chi)_{r'_1,\cdots,r'_m}:=\bm_X(nC,\chi)\cap\overline{\bm_X}(nC,\chi)_{r'_1,\cdots,r'_m}.\]
By Proposition \ref{2Fil} we have
\[\overline{\bm_X}(nC,\chi)^{r_1,\cdots,r_m}=\overline{\bm_X}(nC,\chi)_{r_m,\cdots,r_1};\]
$$\bm_X(nC,\chi)^{r_1,\cdots,r_m}=\bm_X(nC,\chi)_{r_m,\cdots,r_1}.$$
The following lemma is straightforward.
\begin{lemma}\label{exfil}Let $\mf\in\overline{\bm_X}(nC,\chi)^{r_1,\cdots,r_m}$, and let $\mf',\mf''$ lie in the following two exact sequences over $X$
\[0\ra \mf\ra\mf'\ra T'\ra 0,~~0\ra T''\ra \mf''\ra \mf\ra0 ;\]
where $T',T''$ are zero-dimensional sheaves of $\mo_{nC}$-module. Then we have
\[\mf'~(\mf'',~\text{resp.})\in \overline{\bm_X}(nC,\chi'(\chi'',\text{ resp.}))^{r_1,\cdots,r_m},\]
where $\chi'=\chi+\text{length}(T')$ and $\chi''=\chi+\text{length}(T'')$.
\end{lemma}
\begin{rem}\label{tfuF}Use the same notations as in Proposition \ref{2Fil}. If $\mf\in\bm_X(nC,\chi)$, i.e. $\mf$ is pure, then the factors $Q_i$ of the lower filtration are torsion free over $C$ while the factors $R_i$ of the upper filtration may still contain torsion.
We can take another upper filtration
\[0=\widetilde{\mf}^0\subsetneq \widetilde{\mf}^1\subsetneq\cdots\subsetneq \widetilde{\mf}^m=\mf,\]
such that $\widetilde{R}_i:=\widetilde{\mf}^{i}/\widetilde{\mf}^{i-1}$ are torsion free sheaves on $C$ and every $\widetilde{\mf}^i$ is an extension of some zero dimensional sheaf by $\mf^i$ in the upper filtration. Actually $\widetilde{R}_i$ is the maximal torison-free quotient of $\widetilde{\mf}_i\otimes\mo_C$ and $r(\widetilde{R}_i)=r(R_i)$ for all $1\leq i\leq m$ by Lemma \ref{exfil}.
There are also morphisms $\widetilde{g}^i_{\mf}:\widetilde{R}_{i}(-C)\rightarrow \widetilde{R}_{i-1}$ induced by $\mf$ for all $2\leq i\leq m$. But
$\widetilde{g}^i_{\mf}$ is not necessary surjective. We call this filtration \textbf{the torsion-free upper filtration of $\mf$}.
\end{rem}
\section{The case with reduced curve smooth.}
In this section, we prove our main theorem for $C$ a smooth curve.
\begin{prop}\label{smc}Let $C$ be a smooth curve, then we have
\[\dim \overline{\bm_X}(nC,\chi)^{r_1,\cdots,r_m}\leq \frac{n^2C^2}2+\frac{C.K_X}2\left(\displaystyle{\sum_{j=1}^m}r_j^2\right).\]
In particular,
$$\dim \overline{\bm_X}(nC,\chi)\begin{cases}\leq \frac{n^2C^2}2+\frac{nC.K_X}2=g_{nC}-1,\text{ if }C.K_X\leq 0\\ \\ =\frac{n^2C^2}2+\frac{n^2C.K_X}2=n^2(g_{C}-1),\text{ if }C.K_X>0\end{cases}.$$
\end{prop}
\begin{proof}
Since the possible choices of $(\beta_1,\cdots,\beta_m)$ such that $r(\beta_j)=r_j,~j=1,\cdots,m$ form a discrete set. Hence Proposition \ref{smc} follows straightforward from the following lemma.\end{proof}
\begin{lemma}\label{smcb}For every $(\beta_1,\cdots,\beta_m)\in K(C)^m$, we have
\[\dim \overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}\leq \frac{n^2C^2}2+\frac{C.K_X}2\left(\displaystyle{\sum_{j=1}^m}r(\beta_j)^2\right).\]
\end{lemma}
Before proving Lemma \ref{smcb}, we need to define some stacks. For any $\eta\in K(C)$, denote by $\bcoh_{\eta}$ the stack of coherent sheaves on $C$ of class $\eta$. Let $\underline{\eta}=(\eta_1,\cdots,\eta_m)\in K(C)^{m}$, denote by $\widehat{\bcoh_{\underline{\eta}}}$ the stack of chains $\mc_{\bullet}$
\[\mc_{\bullet}:=[\xymatrix@C=1.2cm{\ms_m\ar@{->>}[r]^{d_m\quad}&\ms_{m-1}(C)\ar@{->>}[r]^{~d_{m-1}} &\cdots \ar@{->>}[r]^{d_2\qquad}& \ms_1((m-1)C) }],\]
where $\ms_i$ are coherent sheaves on $C$ of class $\eta_i$.
Two chains $\mc_{\bullet},\mc_{\bullet}'$ are isomorphic if we have the following commutative diagram with vertical arrows all isomorphisms
\[\xymatrix@C=1.2cm{\ms_m\ar@{->>}[r]^{d_m\quad}\ar[d]_{\cong} &\ms_{m-1}(C)\ar@{->>}[r]^{~d_{m-1}}\ar[d]_{\cong} &\cdots \ar@{->>}[r]^{d_2\qquad}& \ms_1((m-1)C)\ar[d]_{\cong}\\
\ms'_m\ar@{->>}[r]^{d'_m\quad}&\ms'_{m-1}(C)\ar@{->>}[r]^{~d'_{m-1}} &\cdots \ar@{->>}[r]^{d'_2\qquad}& \ms'_1((m-1)C) }.\]
Let $\widetilde{\bcoh}_{\underline{\eta}}$ be the stack of the pairs $(\mh,\mh_{\bullet})$, where $\mh$ is a coherent sheaf on $C$ of class $\sum_{j=1}^m\eta_j$ and where $\mh_{\bullet}$ is a filtration
\[\mh_1\subset \mh_2\subset\cdots\subset \mh_m=\mh\]
satisfying $[\mh_k]=\sum_{j=1}^k\eta_j$ for $k=1,\cdots,m$.
Define $\gamma_i:=\beta_m-\beta_{m-i}\otimes[\mo_X(C)]$, $\alpha_i:=\gamma_i-\gamma_{i-1}$ (with $\beta_0=\gamma_0=0$) and $\underline{\alpha}=(\alpha_1,\cdots,\alpha_m)$. We have several natural maps as follows
\begin{equation}\label{nmaps}
\xymatrix{
\bcoh_{\beta_m}\times\cdots\times\bcoh_{\beta_1}&\widehat{\bcoh}_{\underline{\beta}}\ar[r]^{\Phi}\ar[l]_{\qquad\qquad\pi_m}&\widetilde{\bcoh}_{\underline{\alpha}}\ar[d]^{\pi_q}\\
\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}\ar[u]^{\pi_s}&&\bcoh_{\alpha_1}\times\cdots \times\bcoh_{\alpha_m}},
\end{equation}
where $\Phi$ ($\pi_m$, resp.) is defined by sending $[\xymatrix@C=0.5cm{\ms_m\ar@{->>}[r]^{d_m\quad}&\ms_{m-1}(C)\ar@{->>}[r]^{~d_{m-1}} &\cdots \ar@{->>}[r]^{d_2\qquad}& \ms_1((m-1)C) }]$ to $(\mh=\ms_m,\mh_i=\Ker d_{m-i+1}\circ\cdots\circ d_m)$ ($(\ms_m,\cdots,\ms_1)$, resp.), $\pi_q$ is defined by sending $(\mh,\mh_{\bullet})$ to $(\mh_1,\mh_2/\mh_1,\cdots,\mh/\mh_{m-1})$, and finally $\pi_s$ is defined by sending $\mf$ to its factors $(R_m,\cdots,R_1)$ of the upper filtration.
\begin{lemma}\label{dimet}Let $\Phi,\pi_q,\pi_m,\pi_s,\underline{\beta},\underline{\alpha}$ be as in (\ref{nmaps}), then we have
\begin{enumerate}
\item[(i)] $\Phi$ is an isomorphism;
\item[(ii)] $\dim \pi_q^{-1}((\mh_1,\mh_2/\mh_1,\cdots,\mh/\mh_{m-1}))=-\displaystyle{\sum_{i<j}}\chi_C(\alpha_j,\alpha_i)$;
\item[(iii)] for every $(\ms_m,\cdots,\ms_1)\in \Ima(\pi_m)$
$$\dim \pi_m^{-1}((\ms_m,\cdots,\ms_1))=\displaystyle{\sum_{i=2}^m}\dim \Hom_{\mo_C}(\ms_i,\ms_{i-1}(C));$$
\item[(iv)] $\Ima(\pi_s)\subset\Ima(\pi_m)$;
\item[(v)] for every $(R_m,\cdots,R_1)\in \Ima(\pi_s)$
$$\dim \pi_s^{-1}((R_m,\cdots,R_1))\leq \displaystyle{\sum_{i<j}}r(\beta_i)r(\beta_j)C^2+\displaystyle{\sum_{i=1}^{m-1}}\dim\Hom_{\mo_C}(R_i,R_{i+1}(K_X)).$$
\end{enumerate}
\end{lemma}
\begin{proof}(i) is obvious.
(ii) is analogous to Proposition 3.1 (ii) in \cite{Sch}, hence we omit the proof here and refer to \cite{Sch}.
(iii) It is easy to see that $\dim \pi_m^{-1}((\ms_m,\cdots,\ms_1))=\displaystyle{\sum_{i=2}^m}\dim \Hom_{\mo_C}(\ms_i,\ms_{i-1}(C))^{sur}$ where $\Hom_{\mo_C}(\ms_i,\ms_{i-1}(C))^{sur}$ is the subset of $\Hom_{\mo_C}(\ms_i,\ms_{i-1}(C))$ consisting of surjective maps. But according to semicontinuity we have
\[\dim \Hom_{\mo_C}(\ms_i,\ms_{i-1}(C))^{sur}=\dim \Hom_{\mo_C}(\ms_i,\ms_{i-1}(C))\]
for any $(\ms_m,\cdots,\ms_1)\in \Ima(\pi_m)$.
(iv) is also obvous.
We prove (v) by induction on $m$. For $m=1$ there is nothing to prove.
For $m\geq 2$, we have the commutative diagram
\[\xymatrix{\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}\ar[r]^{\pi''_s\qquad\qquad}\ar[rd]_{\pi_s}& \bcoh_{\beta_m}\times \overline{\bm_X}((n-r(\beta_m))C,\chi)^{\beta_1,\cdots,\beta_{m-1}}\ar[d]^{\pi'_s\otimes Id_{\bcoh_{\beta_m}}}\\
&\bcoh_{\beta_m}\times\cdots\times\bcoh_{\beta_1}},\]
where $\pi''_s$ is defined by sending $\mf$ to $(R_m,\mf^{m-1})$ with $0=\mf^0\subsetneq \mf^1\subsetneq\cdots\subsetneq \mf^m=\mf$ the upper filtration.
We can get $\mf$ as an extension of $R_m$ by $\mf^{m-1}$
\[0\ra\mf^{m-1}\ra \mf\ra R_m\ra 0.\]
Notice that every element $\sigma\in \Aut_{\mo_X}(\mf)$ induces an element in $\Aut_{\mo_X}(R_m)\times \Aut_{\mo_X}(\mf^{m-1})$ because $R_m\cong \mf\otimes\mo_C$. Moreover $\Aut_{\mo_X}(R_m)\cong \Aut_{\mo_C}(R_m)$. Hence we have
\[\Aut_{\mo_X}(\mf)\xrightarrow{f_1}\Aut_{\mo_C}(R_m)\times\Aut_{\mo_X}(\mf^{m-1})\]
and $\Hom_{\mo_X}(R_m,\mf^{m-1})\subset \Ker(f_1)$. Denote by $\Ext^1_{\mo_X}(R_m,\mf^{m-1})_{\mf}$ the subset of $\Ext^1_{\mo_X}(R_m,\mf^{m-1})$ consists of extensions with middle term $\mf$, then $\Aut_{\mo_C}(R_m)\times\Aut_{\mo_X}(\mf^{m-1})$ acts on $\Ext^1_{\mo_X}(R_m,\mf^{m-1})_{\mf}$ transitively with stabilizer $\Ima(f_1)$. Therefore
\begin{eqnarray}\label{indest}
\dim (\pi''_s)^{-1}((R_m,\mf^{m-1}))&\leq& \dim \Ext^1_{\mo_X}(R_m,\mf^{m-1})-\dim \Hom_{\mo_X}(R_m,\mf^{m-1})\nonumber\\
&=&-\chi_{X}(R_m,\mf^{m-1})+\dim \Ext^2_{\mo_X}(R_m,\mf^{m-1})\nonumber
\end{eqnarray}
By Serre duality $$\dim \Ext^2_{\mo_X}(R_m,\mf^{m-1})=\dim \Hom_{\mo_X}(\mf^{m-1},R_m(K_X)).$$ Since $R_m$ is a sheaf of $\mo_C$-module, $\mf^{m-2}\subset \Ker(g),~\forall~g\in \Hom_{\mo_X}(\mf^{m-1},R_m(K_X))$. Hence
\begin{eqnarray}
\dim \Hom_{\mo_X}(\mf^{m-1},R_m(K_X))&=&\dim \Hom_{\mo_X}(R_{m-1},R_m(K_X))\nonumber\\
&=&\dim \Hom_{\mo_C}(R_{m-1},R_m(K_X))\nonumber
\end{eqnarray}
By Riemann-Roch $\chi_{X}(R_m,\mf^{m-1})=-\displaystyle{\sum_{j=1}^{m-1}}r(\beta_j)r(\beta_m)C^2$. Therefore
\[\dim (\pi''_s)^{-1}((R_m,\mf^{m-1}))\leq \sum_{j=1}^{m-1}r(\beta_j)r(\beta_m)C^2+\dim \Hom_{\mo_C}(R_{m-1},R_m(K_X))\]
On the other hand
\[\dim \pi_s^{-1}((R_m,\cdots,R_1))\leq \dim (\pi'_s)^{-1}((R_{m-1},\cdots,R_1))+\dim (\pi''_s)^{-1}((R_m,\mf^{m-1}))\]
We get (v) by applying induction assumption to $\mf^{m-1}$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{smcb}]As $C$ is smooth, $\dim\bcoh(\alpha)=-\chi_C(\alpha,\alpha)=(1-g_C)r(\alpha)^2$ for any $\alpha\in K(C)$ such that $\bcoh(\alpha)$ is not empty. Combine (i)-(v) in Lemma \ref{dimet} we have
\begin{eqnarray}\dim\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}&\leq& \sum_{i<j}r(\beta_i)r(\beta_j)C^2+\sum_{i=1}^{m-1}\dim\Hom_{\mo_C}(R_i,R_{i+1}(K_X))\nonumber\\
&&-\sum_{i\leq j}\chi_C(\alpha_j,\alpha_i)-\sum_{i=2}^m\dim \Hom_{\mo_C}(R_i,R_{i-1}(C))\nonumber
\end{eqnarray}
By Serre duality on $C$, we have $\dim\Hom_{\mo_C}(R_i,R_{i+1}(K_X))=\dim\Ext^1_{\mo_C}(R_{i+1},R_{i}(C))$ since $\mo_C(C+K_X)$ is the canonical line bundle on $C$. Therefore we have
\begin{equation}\label{es2}\dim\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}\leq \sum_{i<j}r(\beta_i)r(\beta_j)C^2-\sum_{i\leq j}\chi_C(\alpha_j,\alpha_i)-\sum_{i=2}^m\chi_{C}(R_i,R_{i-1}(C))
\end{equation}
It is easy to see that $\alpha_i=[R_{m-i+1}((i-1)C)]-[R_{m-i}(iC)]$. Therefore from (\ref{es2}) we have
\begin{eqnarray}\label{es3}\dim\overline{\bm_X}(nC,\chi)^{\beta_1,\cdots,\beta_m}&\leq& \sum_{i<j}r(\beta_i)r(\beta_j)C^2-\sum_{i=1}^m\chi_C(\alpha_i,\alpha_i)-\sum_{i=1}^{m-1}\chi_{C}(R_i,R_i)\nonumber\\
&&-\sum_{i=2}^m(\chi_{C}(R_i,R_{i-1}(C))+\chi_{C}(R_{i-1}(C),R_i))
\end{eqnarray}
Because we have
\[\chi_C(\alpha,\beta)+\chi_C(\beta,\alpha)=2(1-g_C)r(\alpha)r(\beta),~\forall~\alpha,\beta\in K(C),\]
we get Lemma \ref{smcb} from (\ref{es3}) by a direct computation.
\end{proof}
\begin{rem}\label{nofin}Although $\overline{\bm_X}(nC,\chi)$ has finite dimension, it is not of finite type and actually contains infinite many connected components. In general even $\bm_X(nC,\chi)$ is only locally of finite type. However, it will be an interesting question to ask whether the p-reduction of $\bm_X(nC,\chi)$ to a finite field $\mathbb{F}^q$ is of finite volume?
\end{rem}
\begin{rem}\label{locfree}If $\mf$ is a locally free sheaf of $\mo_{nC}$-module, then $R_m$ is torsion-free of rank 1 and hence $R_{m-i}\cong R_{m}(-iC)$ for all $i=1,\cdots, m-1$. Thus we have
\[\chi(\mf)=\sum_{j=1}^m\chi(R_j)=m\chi(R_m)-\frac{m(m-1)}2C^2.\]
Therefore if $m\not|~2\chi$, there is no locally free sheaf of $\mo_{nC}$-module in $\bm_X(nC,\chi)$.
\end{rem}
At the end of the section, we would like to state a result for extensions on $C\cong \p^1$ which generalizes Lemma 5.2 in \cite{Yuan4}. We won't need Lemma \ref{p1et} in the rest of the paper, the reader who only concerns the main theorem can also skip it.
\begin{lemma}\label{p1et}Let $C\cong \p^1$. Take an exact sequence on $X$
\begin{equation}\label{mide}0\ra\mo_{C}(s_1)\ra \me\ra\mo_C(s_2)\ra0.\end{equation}
If $s_1<s_2-C^2$, then $\me$ is a locally free sheaf of rank 2 on $C$ and hence splits into direct sum of two line bundles.
\end{lemma}
\begin{proof}We only need to show the following equality for all $s_1<s_2-C^2$
\begin{equation}\label{exd}\dim\Ext^1_{\mo_C}(\mo_C(s_2),\mo_C(s_1))=\dim\Ext_{\mo_X}^1(\mo_C(s_2),\mo_C(s_1)),.\end{equation}
By Serre duality we have LHS $=\dim H^0(\mo_{\p^1}(s_2-s_1-2))=\max\{0,s_2-s_1-1\}$ and
\footnotesize
\begin{eqnarray}\text{RHS }&=&-\chi(\mo_C(s_2),\mo_C(s_1))+\dim\Hom_{\mo_X}(\mo_C(s_2),\mo_C(s_1))+\dim\Ext_{\mo_X}^2(\mo_C(s_2),\mo_C(s_1))\nonumber\\
&=&C^2+\max\{0,s_1-s_2+1\}+\dim\Hom_{\mo_X}(\mo_C(s_1),\mo_C(s_2+K_X)).\nonumber\\
&=&C^2+\max\{0,s_1-s_2+1\}+\max\{0,s_2-s_1+1+C.K_X\}\nonumber\end{eqnarray}
\normalsize
If $C.K_X\geq -1$, then $C^2\leq-1$ and we have
\[\begin{cases}\text{LHS=RHS}=s_2-s_1-1,~\text{if }s_1\leq s_2-1\\
\text{LHS=RHS}=0,~\text{if }s_2-1<s_1\leq s_2-1-C^2
\end{cases}\]
If $C.K_X\leq -2$, then $C^2\geq0$ and we have $\text{LHS=RHS}=s_2-s_1-1,~\text{if }s_1\leq s_2-1-C^2$. Hence (\ref{exd}) holds if $s_1<s_2-C^2$.
\end{proof}
\section{Proof of the main theorem.}
In this section we let $C$ be any integral curve on $X$, not necessarily smooth. We want to prove the following theorem.
\begin{thm}\label{main4}For any integral curve $C\subset X$, we have
\[\dim \bm_X(nC,\chi)^{r_1,\cdots,r_m}\leq \frac{n^2C^2}2+\frac{C.K_X}2\left(\displaystyle{\sum_{j=1}^m}r_j^2\right).\]
In particular,
$$\dim \bm_X(nC,\chi)\begin{cases}\leq \frac{n^2C^2}2+\frac{nC.K_X}2=g_{nC}-1,\text{ if }C.K_X\leq 0\\ \\ =\frac{n^2C^2}2+\frac{n^2C.K_X}2=n^2(g_{C}-1),\text{ if }C.K_X>0\end{cases}.$$
\end{thm}
We proceed the proof of Theorem \ref{main4} by induction on the arithmetic genus $g_C$ of $C$. If $g_C=0$, then $C\cong \p^1$ and Theorem \ref{main4} follows immediately from Proposition \ref{smc}. Let's assume $g_C>0$ and $C$ is not smooth. Denote by $P$ a singular point of $C$ and $\theta\geq 2$ the multiplicity of $C$ at $P$.
Let $\widetilde{X}\xrightarrow{f}X$ be the blow-up at $P$ and $\widetilde{C}:=f^{-1}C$. Then $\widetilde{C}=C_0+\theta E$ with $C_0$ an integral curve with $g_{C_0}<g_{C}$ and $E\cong \p^1$ the exceptional divisors. We have analogous definitions for stacks $\overline{\bm_{\widetilde{X}}}(n\widetilde{C},\chi)$ and $\bm_{\widetilde{X}}(n\widetilde{C},\chi)$ although $\widetilde{C}$ is not integral.
\begin{prop}\label{inject}Let $C'\subset X$ be any curve not necessarily integral. Let $\widetilde{C'}=f^{-1}C'$. For every $\mf\in\bm_X(C',\chi)$, the pull-back $f^*\mf\in\bm_{\widetilde{X}}(\widetilde{C'},\chi)$. Moreover $f_*f^*\mf\cong \mf$ and $f^*:\bm_X(C',\chi)\ra\bm_{\widetilde{X}}(\widetilde{C'},\chi)$ is injective.
\end{prop}
\begin{proof}Since $\mf$ is pure, we can take a locally free resolution of it
\begin{equation}\label{reof}0\ra\ma\xrightarrow{A}\mb\ra\mf\ra0,\end{equation}
where $\ma,\mb$ are locally free.
Pull back (\ref{reof}) to $\widetilde{X}$ and we get
\[f^*\ma\xrightarrow{f^*A} f^*\mb\ra f^*\mf\ra 0,\]
where $f^*A$ has to be injective because it is generically injective and $f^*\ma$ is a locally free sheaf over $\widetilde{X}$. So we have a locally free resolution of length 1 for $f^*\mf$
\begin{equation}\label{reopf}0\ra f^*\ma\xrightarrow{f^*A} f^*\mb\ra f^*\mf\ra 0.\end{equation}
Hence $f^*\mf$ has to be pure of dimension 1 and obviously is supported at $f^*(C')=n\widetilde{C'}$.
Push forward (\ref{reopf}) to $X$ and we get
\[0\ra f_*f^*\ma\xrightarrow{f_*f^*A} f_*f^*\mb\ra f_*f^*\mf\ra R^1f_*f^*\ma,\]
where $R^1f_*f^*\ma\cong \ma\otimes R^1f_*\mo_{\widetilde{X}}=0$. As $R^if_*\mo_{\widetilde{X}}=0,\forall~i>0$ and $f_{*}\mo_{\widetilde{X}}\cong \mo_X$, it is easy to see $f_*f^*\ma\cong \ma, f_*f^*\mb\cong \mb, f_*f^*A=A$ and hence $f_*f^*\mf\cong \mf$. Also $\chi(f^*\mf)=\chi(R^{\bullet}f_*f^*\mf)=\chi(f_*f^*\mf)=\chi(\mf)$.
On the other hand for any $\mf_1,\mf_2\in \bm_X(C',\chi)$, we have
$$\Hom_{\mo_{\widetilde{X}}}(f^*\mf_1,f^*\mf_2)\cong \Hom_{\mo_{X}}(\mf_1,f_*f^*\mf_2)\cong \Hom_{\mo_{X}}(\mf_1,\mf_2).$$
The proposition is proved.
\end{proof}
We also have the following lemma.
\begin{lemma}\label{flat}For every pure 1-dimensional sheaf $\mf$ over $X$, $\Tor^1_{\mo_X}(\mf,\mo_{\widetilde{X}})=0$. In particular, any injective map $\mf_1\stackrel{\imath}{\hookrightarrow}\mf_2$ with $\mf_2/\mf_1$ purely of dimension one remains injective after pulled back to $\widetilde{X}$.
\end{lemma}
\begin{proof}Pull back (\ref{reof}) to $\widetilde{X}$ and we get
\[\Tor^1_{\mo_X}(\ma,\mo_{\widetilde{X}})\ra \Tor^1_{\mo_X}(\mf,\mo_{\widetilde{X}})\ra f^*\ma\xrightarrow{f^*A} f^*\mb\ra f^*\mf\ra 0.\]
As we have already seen in the proof of Proposition \ref{inject}, $f^*A$ is injective. $\Tor^1_{\mo_X}(\ma,\mo_{\widetilde{X}})=0$ by local freeness of $\ma$. Hence $\Tor^1_{\mo_X}(\mf,\mo_{\widetilde{X}})=0$.
\end{proof}
\begin{lemma}\label{filtype}Let $\mf\in\bm_X(nC,\chi)^{r_1,\cdots,r_m}$, then
$$f^*\mf\otimes \mo_{nC_0}\in\overline{\bm_{\widetilde{X}}}(nC_0,\chi_0)^{r_1,\cdots,r_m}$$
for some suitable $\chi_0$.
\end{lemma}
\begin{proof}For $\mf\in\bm_X(nC,\chi)^{r_1,\cdots,r_m}$ we can take its torsion-free upper filtration $\widetilde{\mf}^{\bullet}$ as in Remark \ref{tfuF}, hence by Lemma \ref{flat} the pull-back of the filtration $f^*\widetilde{\mf}^{\bullet}$ is still a filtration of $f^*\mf$ which generically coincides with the upper filtration of $f^*\mf\otimes\mo_{nC_0}$, hence
we have $f^*\mf\otimes \mo_{nC_0}\in\overline{\bm_{\widetilde{X}}}(nC_0,\chi_0)^{r_1,\cdots,r_m}.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{main4}]By Proposition \ref{inject}, it is enough to estimate the dimension of $f^*(\bm_{X}(nC,\chi))\subset \bm_{\widetilde{X}}(n\tilde{C},\chi)$. For any $\mf\in\bm_X(nC,\chi)^{r_1,\cdots,r_m}$, $f^*\mf$ lies in the following sequence
\begin{equation}\label{decom}0\ra \mf_{0}\ra f^*\mf\ra \mf_E\ra 0,\end{equation}
where $\mf_E$ is the torsion free quotient of $f^*\mf\otimes \mo_{n\theta E}$.
Since $\mf_{0}$ is the extension of the torsion free quotient of $f^*\mf\otimes\mo_{C^0}(-n\theta E)$ by a zero dimensional sheaf, by Lemma \ref{exfil} and Lemma \ref{filtype} we have $\mf_0\in \bm_{\widetilde{X}}(nC_0,\chi_0)^{r_1,\cdots,r_m}$ with some suitable $\chi_0$.
Take the upper filtration $0=\mf^0\subsetneq \mf^1\subsetneq \cdots \subsetneq\mf^m=\mf$. Then we have
\[f^*\mf\otimes \mo_{i\theta E}\cong f^*(\mf/\mf^{m-i})\otimes\mo_{i\theta E}, \forall~i=1,\cdots,m-1,\]
which is because $$f^*\mf\otimes \mo_{i\theta E}\cong f^*\mf\otimes \mo_{i\widetilde{C}}\otimes \mo_{i\theta E}\cong f^*(\mf\otimes\mo_{iC})\otimes \mo_{i\theta E}.$$
On the other hand the schematic support of $f^*(\mf/\mf^{m-i})$ is $\left(\displaystyle{\sum_{j=0}^{i+1}r_{m-j}}\right)\widetilde{C}+\eta_i E$ and $\eta_i=0$ iff $\mf/\mf^{m-i}$ contains no torsion supported at $P$. Hence we have
\[f^*\mf\otimes \mo_{n\theta E}\in \overline{\bm_{\widetilde{X}}}(n\theta E,\chi_E)^{r_1^1,\cdots,r_1^{\theta^1},\cdots,r_m^1,\cdots,r_m^{\theta^m}}\]
such that
\[f^*\mf\otimes\mo_{i\theta E}\in \overline{\bm_{\widetilde{X}}}(n\theta E,\chi_E)^{r_{m-i+1}^1,\cdots,r_{m-i+1}^{\theta^{m-i+1}},\cdots,r_m^1,\cdots,r_m^{\theta^m}},\forall~ i=1,\cdots,m.\]
Therefore \begin{equation}\label{ietheta}\theta^j\leq \theta,\forall ~j=1,\cdots,m.\end{equation}
Moreover we have
\begin{equation}\label{iesum}\sum_{j=k}^m(\sum_{t=1}^{\theta^j}r_j^t)\geq \theta(\sum_{j=k}^mr_j),k=2,\cdots,m;~\sum_{j=1}^m(\sum_{t=1}^{\theta}r_j^t)= \theta(\sum_{j=1}^mr_j).\end{equation}
Notice that we also have $\mf_E\in \bm_{\widetilde{X}}(n\theta E,\chi_E)^{r_1^1,\cdots,r_1^{\theta^1},\cdots,r_m^1,\cdots,r_m^{\theta^m}}$ by Lemma \ref{exfil}.
Denote by $\Delta$ the set of all $\underline{r}:=(r_1^1,\cdots,r_1^{\theta^1},\cdots,r_m^1,\cdots,r_m^{\theta^m})$ satisfying both (\ref{ietheta}) and (\ref{iesum}).
We have the map between stacks induced by (\ref{decom})
\[f^*(\bm_X(nC,\chi)^{r_1,\cdots,r_m})\xrightarrow{\pi_E}\coprod_{\chi_0} \bm_{\widetilde{X}}(nC_0,\chi_0)^{r_1,\cdots,r_m}\times \coprod_{\chi_E,\underline{r}\in \Delta} \bm_{\widetilde{X}}(n\theta E,\chi_E)^{\underline{r}}.\]
It is easy to see the fiber of $\pi_E$ is of dimension no bigger than $-\chi_X(\mf_{E},\mf_0)=nC_0.n(\theta E)=n^2C_0.\theta E$ since $\Ext^2(\mf_E,\mf_0)=0$. By applying the induction assumption to $C_0$ and $E$, we get
\begin{eqnarray}\dim \bm_X(nC,\chi)^{r_1,\cdots,r_m}&\leq& n^2C_0.(\theta E)+\frac{n^2C_0^2}2+\frac{C_0.K_{\widetilde{X}}}2\left(\sum_{j=1}^m r_j^2\right)\nonumber\\
&&+\max_{\underline{r}\in \Delta}\left\{\frac{n^2\theta^2 E^2}2+\frac{E.K_{\widetilde{X}}}2\left(\sum_{j=1}^m\left(\sum_{t=1}^{\theta^j}(r_j^t)^2\right)\right)\right\}.\end{eqnarray}
Since $E.K_{\widetilde{X}}=-1<0$, together with the following Lemma \ref{ineq} we get
\begin{eqnarray}\label{final}\dim \bm_X(n C,\chi)^{r_1,\cdots,r_m}&\leq& n^2C_0.(\theta E)+\frac{n^2C_0^2}2+\frac{C_0.K_{\widetilde{X}}}2\left(\sum_{j=1}^m r_j^2\right)\nonumber\\
&&+\frac{n^2\theta^2 E^2}2+\frac{\theta E.K_{\widetilde{X}}}2\left(\sum_{j=1}^m r_j^2\right)\nonumber\\
&=&\frac{n^2 (C_0+\theta E)^2}2+\frac{(C_0+\theta E).K_{\widetilde{X}}}2\left(\sum_{j=1}^m r_j^2\right)\nonumber\\
&=&\frac{n^2 (\widetilde{C})^2}2+\frac{(\widetilde{C}).K_{\widetilde{X}}}2\left(\sum_{j=1}^m r_j^2\right).\end{eqnarray}
By $\widetilde{C}^2=C^2$ and $\widetilde{C}.K_{\widetilde{X}}=\widetilde{C}.(f^*K_X+E)=C.K_X$, we get the theorem.
\end{proof}
\begin{lemma}\label{ineq}Let $r_m\geq r_{m-1}\geq\cdots\geq r_1$ and let $\theta\in \mathbb{Z}_{>0}$. If we have real numbers
$$r_m^1,r_m^2,\cdots,r_m^{\theta},r_{m-1}^1,\cdots, r_{m-1}^{\theta}, r_{m-2}^{1},\cdots, r_1^{\theta}$$
such that
\[\sum_{j=k}^m\left(\sum_{t=1}^{\theta}r_j^t\right)\geq \theta\left(\sum_{j=k}^mr_j\right),k=2,\cdots,m;~\sum_{j=1}^m\left(\sum_{t=1}^{\theta}r_j^t\right)= \theta\left(\sum_{j=1}^mr_j\right).\]
Then we have
\[\sum_{j=1}^m\left(\sum_{t=1}^{\theta}(r_j^t)^2\right)\geq \theta\left(\sum_{j=1}^m(r_j)^2\right)\]
and the equality holds iff $r_j^t=r_j$ for all $t=1,\cdots,\theta$ and $j=1,\cdots,m$.
In particular we can remove zeros in $\{r_j^t\}$ and ask rest of them to be positive, i.e.
\[r_m^1,r_m^2,\cdots,r_m^{\theta^1},r_{m-1}^1,\cdots, r_{m-1}^{\theta^2}, r_{m-2}^{1},\cdots, r_1^{\theta^m}>0\]
with $\theta^i\leq \theta$ for $i=1,\cdots,m$, then
\[\sum_{j=1}^m\left(\sum_{t=1}^{\theta^j}(r_j^t)^2\right)\geq \theta\left(\sum_{j=1}^m(r_j)^2\right)\]
and the equality holds iff $\theta^j=\theta$ and $r_j^t=r_j$ for all $t=1,\cdots,\theta$ and $j=1,\cdots,m$.
\end{lemma}
\begin{proof}Let $\epsilon_j^t:=r_j^t-r_j$, then we have
\[\sum_{j=k}^m\left(\sum_{t=1}^{\theta}\epsilon_j^t\right)\geq 0,k=2,\cdots,m;~\sum_{j=1}^m\left(\sum_{t=1}^{\theta}\epsilon_j^t\right)=0.\]
Hence
\begin{eqnarray}
&&\sum_{j=1}^m\left(\sum_{t=1}^{\theta}(r_j^t)^2\right)=\sum_{j=1}^m\left(\sum_{t=1}^{\theta}(r_j+\epsilon_j^t)^2\right)\nonumber\\
&=&\theta\left(\sum_{j=1}^m(r_j)^2\right)+\sum_{j=1}^m\left(\sum_{t=1}^{\theta}(\epsilon_j^t)^2\right)+2\sum_{j=1}^m\left(r_j\left(\sum_{t=1}^{\theta}\epsilon_j^t\right)\right)\nonumber\\
&=&\theta\left(\sum_{j=1}^m(r_j)^2\right)+\sum_{j=1}^m\left(\sum_{t=1}^{\theta}(\epsilon_j^t)^2\right)+2\sum_{k=2}^m\left((r_{k}-r_{k-1})\left(\sum_{j=k}^m\left(\sum_{t=1}^{\theta}\epsilon_j^t\right)\right)\right)\nonumber\\
&\geq &\theta\left(\sum_{j=1}^m(r_j)^2\right),\nonumber
\end{eqnarray}
where the last inequality is because $r_k\geq r_{k-1}$ and $\displaystyle{\sum_{j=k}^m\left(\sum_{t=1}^{\theta}\epsilon_j^t\right)}\geq 0$. It is easy to see the equality holds iff $\epsilon_j^t=0$ for all $t=1,\cdots,\theta$ and $j=1,\cdots,m.$
\end{proof}
\begin{rem}\label{algcl}As Theorem \ref{main4} only concerns the dimension, the assumption that the base field $\Bbbk$ is algebraically closed can be removed.
\end{rem}
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TITLE: Oriented cobordism classes represented by rational homology spheres
QUESTION [18 upvotes]: Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}({\Omega^{\text{SO}}_{5}}) \cong \mathbb Z/2\mathbb Z$. This motivates the following question.
Which classes in $\Omega^{\text{SO}}_{\ast}$ can be represented by rational homology spheres?
Of course, any such class is torsion, as all its composite Pontryagin numbers, as well as its signature, vanish.
REPLY [11 votes]: The necessary condition pointed out by Jens Reinhold is also sufficient: any torsion class $x = [M] \in \Omega^{SO}_d$ admits a representative where $M$ is a rational homology sphere.
EDIT: This is Theorem 8.3 in $\Lambda$-spheres by Barge, Lannes, Latour, and Vogel. They also calculate the group of rational homology spheres up to rational h-cobordism, and more. I'll leave my argument below:
To prove this, we first dispense with low-dimensional cases: in any dimension $d < 5$ the only torsion class is $0 = [S^d]$. The high dimensional case follows from Claims 1 and 2 below.
I'll write $MX$ for the Thom spectrum of a map $X \to BO$ and $\Omega^X_d \cong \pi_d(MX)$ for the bordism group of smooth $d$-manifolds equipped with $X$-structure. Representatives are smooth closed $d$-manifolds $M$ with some extra structure, which includes a continuous map $f: M \to X$.
Claim 1: if $d \geq 5$ and $X$ is simply connected and rationally $\lfloor d/2 \rfloor$-connected, then any class in $\Omega^X_d$ admits a representative where $M$ is a rational homology sphere.
Claim 2: There exists a simply connected space $X$ such that $\widetilde{H}_*(X;\mathbb{Z}[\frac12]) = 0$, and map $X \to BSO$ such that the image of the induced map $\Omega^X_d = \pi_d(MX) \to \pi_d(MSO) = \Omega_d^{SO}$ is precisely the torsion subgroup, for $d > 0$.
Proof of Claim 1: Starting from an arbitrary class in $\Omega^X_d$ we can use surgery to improve the representative. Since $X$ is simply connected and $d > 3$ we can use connected sum and then surgery on embeddings $S^1 \times D^{d-1} \hookrightarrow M$ to make $M$ simply connected. Slightly better, such surgeries can be used to make the map $M \to X$ be 2-connected, meaning that its homotopy fibers are simply connected. From now on we need not worry about basepoints and will write $\pi_{k+1}(X,M) = \pi_k(\mathrm{hofib}(M \to X))$. These are abelian groups for all $k$.
If there exists a $k < \lfloor d/2\rfloor$ with $\widetilde{H}_k(M;\mathbb{Q}) \neq 0$ we can choose $\lambda \in H_k(M;\mathbb{Q})$ and $\mu \in H_{d-k}(M;\mathbb{Q})$ with intersection number $\lambda \cdot \mu \neq 0$. If $d = 2k$ for even $k$ we can additionally assume $\lambda \cdot \lambda = 0$, since the signature of $M$ vanishes. The rational Hurewicz theorem implies that $\pi_k(M) \otimes \mathbb{Q} \to H_k(M;\mathbb{Q})$ is an isomorphism, and the long exact sequence implies that $\pi_{k+1}(X,M) \otimes \mathbb{Q} \to \pi_k(M)\otimes\mathbb{Q}$ is surjective. After replacing $\lambda$ by a non-zero multiple, we may therefore assume that it admits a lift to $\pi_{k+1}(X,M)$. Such an element can be represented by an embedding $j: S^k \times D^{d-k} \hookrightarrow M$, together with a null homotopy of the composition of $j$ with $M \to X$. In the case $k < d/2$ this follows from Smale-Hirsh theory, in the case $d = 2k$ we must also use $\lambda \cdot \lambda = 0$ to cancel any self-intersections. (Actually there could also be obstructions to this in the case $d=2k$ for odd $k$, but those obstructions vanish after multiplying $\lambda$ by 2.) The embedding and the nullhomotopy gives the necessary data to perform surgery on $M$ and to promote the surgered manifold to a representative for the same class in $\Omega^X_d$.
Performing the surgery gives a new manifold $M'$ where $H_k(M';\mathbb{Q})$ has strictly smaller dimension than $H_k(M;\mathbb{Q})$ and $\widetilde{H}_*(M';\mathbb{Q}) = 0$ for $* < k$. This is seen in the same way as in Kervaire-Milnor. The case $d > 2k+1$ is easy, similar to their Lemma 5.2. In the case $d = 2k+1$ the diagram on page 515 shows that we can kill the homology class $j[S^k]$ and at worst create some new torsion in $H_k(M')$. In the case $d = 2k$ the diagram on page 527 shows that we can kill the homology class $j[S^k]$ and at worst create some new torsion in $H_{k-1}(M')$.
In finitely many steps we arrive at a representative where $\widetilde{H}_k(M;\mathbb{Q}) = 0$ for all $k \leq \lfloor d/2\rfloor$. Poincaré duality then implies that $H_*(M;\mathbb{Q}) \cong H_*(S^d;\mathbb{Q})$. $\Box$.
Proof of Claim 2: Finiteness of the stable homotopy groups of spheres implies that $\pi_d(MX)$ is a torsion group for $d > 0$ for any such $X$. Therefore we can never hit more than the torsion in $\pi_d(MSO)$, all of which is exponent 2 by Wall's theorem. The difficult part is to construct an $X$ where all torsion is hit.
The non-trivial based map $S^1 \to BO$ factors through $\mathbb{R} P^\infty \to BO$, whose image in mod 2 homology generates the Pontryagin ring $H_*(BO;\mathbb{F}_2)$. We can freely extend to double loop maps
$$\Omega^2 S^3 \to \Omega^2 \Sigma^2 \mathbb{R}P^\infty \to BO$$
where the second map then induces a surjection on mod 2 homology. Both $\Omega^2 \Sigma^2 \mathbb{R}P^\infty$ and $BO$ split as $\mathbb{R} P^\infty$ times their 1-connected cover, so the induced map of 1-connected covers $\tau_{\geq 2}(\Omega^2 \Sigma^2 \mathbb{R}P^\infty) \to \tau_{\geq 2}(BO) = BSO$ also induces a surjection on mod 2 homology.
Now let $X = \tau_{\geq 2}(\Omega^2 \Sigma^2 \mathbb{R}P^\infty)$ with the map to $BSO$ constructed above. Take 1-connected covers of the double loop maps above, Thomify, 2-localize, and use the Hopkins-Mahowald theorem to get maps of $E_2$ ring spectra
$$H \mathbb{Z} _{(2)} \to MX_{(2)} \to MSO_{(2)}.$$
(See e.g. section 3 of this paper.)
We can view $MX_{(2)} \to MSO_{(2)}$ as a map of $H\mathbb{Z}_{(2)}$-module spectra, and hence $MX/2 \to MSO/2$ as a map of $H\mathbb{F}_2$-module spectra. The induced map $H_*(MX/2;\mathbb{F}_2) \to H_*(MSO/2;\mathbb{F}_2)$ is still surjective (it looks like two copies of $H_*(X;\mathbb{F}_2) \to H_*(BSO;\mathbb{F}_2))$, and inherits the structure of a module map over the mod 2 dual Steenrod algebra $\mathcal{A}^\vee = H_*(H\mathbb{F}_2;\mathbb{F}_2)$. Both modules are free, because any $H\mathbb{F}_2$-module spectrum splits as a wedge of suspensions of $H\mathbb{F}_2$. In fact the Hurewicz homomorphism $\pi_*(MX/2) \to H_*(MX/2;\mathbb{F}_2)$ induces an isomorphism
$$\mathcal{A}^\vee \otimes \pi_*(MX/2) \to H_*(MX/2;\mathbb{F}_2),$$
and similarly for $MSO$. Therefore the map $\pi_*(MX/2) \to \pi_*(MSO/2)$ may be identified with the map obtained by applying $\mathbb{F}_2 \otimes_{\mathcal{A}^\vee} (-)$ to the map on homology, showing that the induced map $\pi_*(MX/2) \to \pi_*(MSO/2)$ is also surjective. Now any 2-torsion class $x \in \pi_d(MSO)$ comes from $\pi_{d+1}(MSO/2)$, hence from $\pi_{d+1}(MX/2)$ and in particular from $\pi_d(MX)$. $\Box$
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\begin{document}
\title{Some new ordering results on stochastic comparisons of second largest order statistics from independent and interdependent heterogeneous distributions}
\author{{\large { Sangita {\bf Das}\thanks {Email address :
sangitadas118@gmail.com}~ and Suchandan {\bf
Kayal}\thanks {Email address (corresponding author):
kayals@nitrkl.ac.in,~suchandan.kayal@gmail.com}}} \\
{ \em \small {\it Department of Mathematics, National Institute of
Technology Rourkela, Rourkela-769008, India.}}}
\date{}
\maketitle
\begin{center}{\bf Abstract}
\end{center}
The second-largest order statistic is of special importance in reliability theory since it represents the time to failure of a $2$-out-of-$n$ system. Consider two $2$-out-of-$n$ systems with heterogeneous random lifetimes. The lifetimes are assumed to follow heterogeneous general exponentiated location-scale models. In this communication, the usual stochastic and reversed hazard rate orders between the systems' lifetimes are established under two cases. For the case of independent random lifetimes, the usual stochastic order and the reversed hazard rate order between the second-largest order statistics are obtained by using the concept of vector majorization and related orders. For the dependent case, the conditions under which the usual stochastic order between the second-largest order statistics holds are investigated. To illustrate the theoretical findings, some special cases of the exponentiated location-scale model are considered.
\\
\\
\noindent{\bf Keywords:} Second-largest order statistics; $2$-out-of-$n$ systems; stochastic order; reversed hazard rate order; majorization; Archimedean copula.
\\\\
{\bf Mathematics Subject Classification:} 60E15; 90B25.
\section{Introduction}
Stochastic orders are hugely popular mathematical tools, which have seen successful applications in various areas of research. Ideally, they are suited to model relationships of various characteristics across multiple heterogeneous samples. Specifically, in probability theory, the stochastic orderings are useful in comparing stochastic models and establishing probability inequalities. In economics, particularly in utility theory, they are useful to make decisions under risk. In reliability theory, the stochastic orders are utilized to derive reliability bounds. Various aging notions such as the new better than used and new worse than used are understood using the concept of stochastic orderings. They are also used in redundancy improvements and maintenance policies. Let us consider replacements upon failures. This is a very common in maintenance management. In this policy, the number of replacements, denoted by $N(t)$ in a time interval $(0,t)$ is of great importance in reliability theory, particularly its probability distribution. However, explicit formula for the cumulative distribution function of $N(t)$ is not available in general except some special cases. So, the stochastic bounds of the distribution function of $N(t)$ are useful from the practical point of view.
Order statistics play a vital role in various areas of probability and statistics. It has many useful interpretations in reliability theory, auction theory and in various other applied fields of research. Let $X_1,\cdots,X_n$ be $n$ random lifetimes. The $k$th order statistic ($k=1,\cdots,n$) is the $k$th smallest observation. It is denoted by $X_{k:n}.$ There are many important systems, we often face in reliability theory. One of these is the $k$-out-of-$n$ system, which is of huge importance. The mechanism of this system is that it works, if at least $k$ components out of $n$ operate. The order statistic $X_{n-k+1:n}$ characterizes the time to failure of the $k$-out-of-$n$ system. In particular, $k$-out-of-$n$ system reduces to the parallel and series systems if $k=1$ and $n$, respectively. Note that there have been substantial work on the stochastic comparison of the order statistics when the components' lifetimes have heterogeneous probability models. In the next paragraph, we present few recent developments.
\cite{khaledi2011stochastic} considered stochastic comparisons of the order statistics in the scale models. \cite{balakrishnan2013ordering} derived different ordering results between different order statistics according to the hazard rate, likelihood ratio, dispersive order, excess wealth order for the proportional hazard rate model. \cite{kochar2015stochastic} studied stochastic comparison of the largest order statistics, constructed from two sets of heterogeneous scaled-samples in terms of the likelihood ratio order. \cite{li2016stochastic} studied ordering properties of the extreme order statistics arising from scaled dependent samples. They obtained usual stochastic order, star order and dispersive order of the sample extremes.
\cite{bashkar2017stochastic} studied effect of heterogeneity on the order statistics arising from independent heterogeneous exponentiated scale samples. They used usual stochastic, reversed hazard rate and likelihood ratio orderings as the mathematical tools to compare order statistics. Further, in the presence of the Archimedean copula or survival copula for the random variables, they obtained the usual stochastic order of the sample extremes. \cite{torrado2017stochastic} addressed the problem of stochastic comparisons of the extreme order statistics from two sets of heterogeneous scale models. The author obtained various stochastic orderings when a set of parameters majorizes another set of parameters. For the location-scale distributed samples, \cite{hazra2017stochastic} obtained various stochastic ordering results between the maximum order statistics. \cite{hazra2018stochastic} considered independent heterogeneous location-scale models and obtained stochastic comparisons between the minimum order statistics in terms of various stochastic orders. \cite{fanggamma2019} developed comparison results between the lifetimes of series and parallel systems with heterogeneous
exponentiated gamma components. Very recently, exponentiated location-scale model, a generalized version of the location-scale model was considered by \cite{dasordering}. They obtained various ordering results between the extreme order statistics from independent heterogeneous exponentiated location-scale models. \cite{das2019ordering} studied comparison results between the extreme order statistics arising from heterogeneous dependent exponentiated location-scale models.
Note that $2$-out-of-$n$ system is a special case of the general $k$-out-of-$n$ system. For some comparison results between $k$-out-of-$n$ systems, one may refer to \cite{ding2013comparisons} and \cite{balakrishnan2018stochastic}. Recently, few researchers have studied stochastic comparison results between the second-largest order statistics. For instance, see \cite{fang2016stochastic} and \cite{balakrishnan2018necessary}. To the best of our knowledge, nobody has considered the stochastic comparison study between $k$-out-of-$n$ systems or $2$-out-of-$n$ systems, when the components' lifetimes follow exponentiated location-scale model. In this contribution, we consider $2$-out-of-$n$ systems. Here, the lifetimes of the components of the systems follow heterogeneous exponentiated location-scale models. Note that the $(n-1)$st order statistic represents the lifetime of a $2$-out-of-$n$ system. We will first compare the lifetimes of two $2$-out-of-$n$ systems in the sense of the usual stochastic order and the reversed hazard rate order when the components' lifetimes are heterogeneous and independently distributed. Then, we obtain comparison results for the heterogeneous dependent random lifetimes. It has been assumed that the dependence structure is coupled by the Archimedean copulas. Few supplementary results in addition to the main results are also presented. The $i$th random variable $X_{i}$, where $i=1,\cdots,n$ is said to follow exponentiated location-scale model with baseline distribution function $F_{b}$, if its cumulative distribution function is given by
\begin{eqnarray}\label{eq1.1}
F_i(x)\equiv
F_{X_i}(x;\lambda_i,\theta_i,\alpha_i,F_b)=\left[F_b\left(\frac{x-\lambda_i}{\theta_i}\right)\right]^{\alpha_i},~x>\lambda_i>0,~\alpha_i,~\theta_i>0.
\end{eqnarray}
In many practical applications such as in economy and medical study, the datasets are often skewed. In economics, the amount of gains is often small and the large losses are occasional. In this case, the dataset is (negatively) skewed. To capture skewness contained in the dataset, the skewness parameter $\alpha_i$ plays an important role.
Further, when considering lifetime models, the location parameter, here $\lambda_i$ represents a lower threshold of the lifetimes. Sometimes, it also represents guarantee time of an item. In hydrology and environmental science, the location parameter is used as a threshold parameter. It corresponds a minimum threshold value of the observed characteristic. Thus, the stochastic comparison results obtained in this paper not only have applications in reliability theory, but also in other applied fields of research.
The rest of the paper is laid out as follows. In Section $2$, we present some key definitions of the stochastic orders, majorization and related orders. The concept of copula and few well known lemmas are also provided. Section $3$ addresses main contribution of the paper. This section has two subsections. In Subsection $3.1$, it is shown that the usual stochastic and reversed hazard rate orders exist between the lifetimes of two $2$-out-of-$n$ systems under majorization-based sufficient conditions. Here, the components' lifetimes are taken to be heterogeneous and independent. The case of heterogeneous but dependent components' lifetimes is considered in Subsection $3.2$. Here, sufficient conditions, under which the usual stochastic order between the lifetimes of two $2$-out-of-$n$ systems holds are derived. Examples and counterexamples associated to the established results are presented throughout. Section $4$ concludes the paper.
Throughout the communication, the random variables are assumed to be absolutely continuous and nonnegative. The terms increasing and decreasing are used in wide sense. Prime denotes the derivative of a function.
\section{Background\setcounter{equation}{0}}
This section briefly reviews some of the basic concepts of stochastic orderings, majorization orderings, preliminary lemmas and copulas. These are essential to establish main results, which have been presented in the subsequent section. First, we consider the notion of stochastic orderings.
\subsection{Stochastic orderings}
Consider two nonnegative and absolutely continuous random variables $X$ and $Y$. Denote the probability density functions, cumulative distribution functions, survival functions and reversed hazard rate functions of $X$ and $Y$ by $f_{X}$ and $f_{Y}$, $F_{X}$ and $F_{Y}$, $\bar
F_{X}=1-F_{X}$ and $\bar F_{Y}=1-F_{Y}$ and $ \tilde r_{X}=f_{X}/
F_{X}$ and $\tilde r_{Y}=f_{Y}/
F_{Y}$, respectively.
\begin{definition}
$X$ is said to be smaller than $Y$ in the
\begin{itemize}
\item reversed hazard rate order (denoted by $X\leq_{rh}Y$)
if $\tilde{r}_{X}(x)\leq \tilde{r}_{Y}(x)$, for all $x>0$;
\item usual stochastic order (denoted by $X\leq_{st}Y$) if
$\bar F_{X}(x)\leq\bar F_{Y}(x)$, for all $x$.
\end{itemize}
\end{definition}
Note that the reversed hazard rate ordering implies the usual stochastic ordering. One may refer to \cite{shaked2007stochastic} for details on stochastic orderings and their applications in various contexts. Next, we consider the concept of the majorization and some associated orders.
\subsection{Majorization and related orders}
The notion of majorization plays a vital role in establishing various inequalities in the field of applied probability. Suppose we have two vectors of same dimension. Then, majorization is useful to compare these vectors in terms of the dispersion of their components. Let $\mathbb{A} \subset \mathbb{R}^{n}$, where $\mathbb{R}^{n}$ be
an $n$-dimensional Euclidean space. Denote $n$ dimensional vectors by $\boldsymbol{x} =
\left(x_{1},\cdots,x_{n}\right)$ and $\boldsymbol{y} =
\left(y_{1},\cdots,y_{n}\right)$ taken from $\mathbb{A}.$ Further, the order coordinates of the
vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ are denoted by $x_{1:n}\leq \cdots \leq x_{n:n}$ and
$y_{1:n}\leq\cdots \leq y_{n:n},$ respectively.
\begin{definition}\label{definition2.2}
A vector $\boldsymbol{x}$ is said to be
\begin{itemize}
\item majorized by another vector $\boldsymbol{y},$ (denoted
by $\boldsymbol{x}\preceq^{m} \boldsymbol{y}$), if for each $k=1,\cdots,n-1$, we have
$\sum\limits_{i=1}^{k}x_{i:n}\geq \sum\limits\limits_{i=1}^{k}y_{i:n}$ and
$\sum\limits\limits_{i=1}^{n}x_{i:n}=\sum\limits_{i=1}^{n}y_{i:n};$
\item weakly submajorized by another vector $\boldsymbol{y},$ denoted
by $\boldsymbol{x}\preceq_{w} \boldsymbol{y}$, if for each $k=1,\cdots,n$, we have
$\sum\limits_{i=k}^{n}x_{i:n}\leq \sum\limits_{i=k}^{n}y_{i:n};$
\item weakly supermajorized by another vector $\boldsymbol{y},$ denoted
by $\boldsymbol{x}\preceq^{w} \boldsymbol{y}$, if for each $k=1,\cdots,n$, we have
$\sum\limits_{i=1}^{k}x_{i:n}\geq \sum\limits_{i=1}^{k}y_{i:n};$
\item reciprocally majorized by another vector $\boldsymbol{y},$ denoted
by $\boldsymbol{x}\preceq^{rm}\boldsymbol{y}$, if $\sum\limits_{i=1}^{k}x_{i:n}^{-1}\le
\sum\limits_{i=1}^{k}y_{i:n}^{-1}$, for all $k=1,\ldots,n$.
\end{itemize}
\end{definition}
It is noted that $\bm{y}$ majorizes $\bm{x}$ means the components of $\bm{y}$ are more dispersed than that of $\bm{x}$ under the condition that the sum is fixed. One can easily prove that the majorization order implies both weakly supermajorization and weakly submajorization orders. Interested readers are referred to \cite{marshall2010} for an extensive and comprehensive details on the theory of majorization and its applications in the field of statistics. Next, we consider definition of the Schur-convex and Schur-concave functions.
\begin{definition}
A function $h:\mathbb{R}^n\rightarrow \mathbb{R}$ is said
to be Schur-convex (Schur-concave) on $\mathbb{R}^n$ if
$$\boldsymbol {x}\overset{m}{\succeq}\boldsymbol{ y}\Rightarrow
h(\boldsymbol { x})\geq( \leq)h(\boldsymbol { y}), \text{ for all } \boldsymbol { x},~ \boldsymbol
{ y} \in \mathbb{R}^n.$$
\end{definition}
The following notations will be used throughout the article.
$(i)~\mathcal{D}_{+}=\{(x_1,\cdots,x_n):x_{1}\geq
x_{2}\geq\cdots\geq x_{n}>0\}$, $(ii)~\mathcal{E}_{+}=\{(x_1,\cdots,x_n):0<x_{1}\leq
x_{2}\leq\cdots\leq x_{n}\}$ and $(iii)~\bm{1}_{n}=(1,\cdots,1).$
The following lemmas are helpful to establish the results in Section $3$. Denote by $h_{(k)}(\boldsymbol{ z})=\partial h(\boldsymbol{z})/\partial z_k$ the partial derivative of $h$ with respect to its $k$th argument
{\begin{lemma}(\cite{kundu2016some})\label{lem2.1a}
Let $h:\mathcal{D_+}\rightarrow \mathbb{ R}$ be a function, continuously differentiable on the interior of $\mathcal{D_+}.$ Then, for $\boldsymbol{x},~\boldsymbol{y}\in\mathcal{D_+},$
$$\boldsymbol{x}\succeq^{m}\boldsymbol{y}\text{ implies } h(\boldsymbol{x})\geq(\leq )~h(\boldsymbol{y}),$$
if and only if
$h_{(k)}(\boldsymbol{ z}) \text{ is decreasing (increasing) in } k=1,\cdots,n.$
\end{lemma}
\begin{lemma}(\cite{kundu2016some})\label{lem2.1b}
Let $h:\mathcal{E_+}\rightarrow \mathbb{ R}$ be a function, continuously differentiable on the interior of $\mathcal{E_+}.$ Then, for $\boldsymbol{x},~\boldsymbol{y}\in\mathcal{E_+},$
$$\boldsymbol{x}\succeq^{m}\boldsymbol{y}\text{ implies } h(\boldsymbol{x})\geq(\leq )~h(\boldsymbol{y}),$$
if and only if
$h_{(k)}(\boldsymbol{ z}) \text{ is increasing (decreasing) in } k=1,\cdots,n.$
\end{lemma}
\begin{lemma}(\cite{hazra2017stochastic})\label{lem2.1c}
Let $S\subseteq\mathbb{ R}^n_{+}.$
Further, let $h:S\rightarrow \mathbb{ R}$ be a function. Then, for $\boldsymbol{x},~\boldsymbol{y}\in S,$
$$\boldsymbol{x}\succeq^{rm}\boldsymbol{y}\text{ implies } h(\boldsymbol{x})\geq(\leq )~h(\boldsymbol{y}),$$
if and only if
\begin{itemize}
\item[(i)] $ h(\frac{1}{a_1},\cdots,\frac{1}{a_n})$ is Schur-convex (Schur-concave) in $(a_1,\cdots,a_n)\in S,$
\item[(ii)] $ h(\frac{1}{a_1},\cdots,\frac{1}{a_n})$ is increasing (decreasing) in $a_i$, for $i=1,\cdots,n,$
where $a_i=\frac{1}{x_i},$ for $i=1,\cdots,n.$
\end{itemize}
\end{lemma}
\subsection{Copula}
Let $\bm{X}=(X_1,\cdots,X_{n})$ be a nonnegative random vector. The univariate marginal distribution functions of $X_{1},\cdots,X_{n}$ are $F_1,\cdots,F_n$, respectively. The univariate survival functions of $X_{1},\cdots,X_{n}$ are respectively denoted by $\bar{F}_1,\cdots,\bar{F}_n$. Denote $\boldsymbol{v}=(v_1,\cdots,v_n)$. $C(\boldsymbol{v})$ and $\hat{C}(\boldsymbol{v})$ are called the copula and survival copula of $\boldsymbol{X}$, if there exist functions $C(\boldsymbol{v}):[0,1]^n\rightarrow [0,1]$ and
$\hat {C}(\boldsymbol{v}):[0,1]^n\rightarrow [0,1]$ such that for all $ x_i,~i\in \mathcal I_n, $ where $\mathcal I_n$ be the index set,
$$ F(x_1,\cdots,x_n)=C(F_1(x_1),\cdots,F_n(x_n))~\mbox{and}~
\bar{F}(x_1,\cdots,x_n)=\hat{C}(\bar{F_1}(x_1),\cdots,\bar{F_n}(x_n))$$ hold.
Consider $\psi:[0,\infty)\rightarrow[0,1]$ to be a nonincreasing and continuous function such that $\psi(0)=1$ and $\psi(\infty)=0.$ Further, let $\psi$ satisfy $(-1)^i{\psi}^{i}(x)\geq 0,~ i=0,1,\cdots,d-2$ and $(-1)^{d-2}{\psi}^{d-2}$ be nonincreasing and convex. Then, the generator $\psi$ is $d$-monotone. Furthermore, define, $\phi={\psi}^{-1}=\text{sup}\{x\in \mathcal R:\psi(x)>v\}$, the right continuous inverse of $\psi$. Then, a copula $C_{\psi}$ with generator $\psi$ is called Archimedean copula if $$C_{\psi}(v_1,\cdots,v_n)=\psi({\psi^{-1}(v_1)},\cdots,\psi^{-1}(v_n)),~\text{ for all } v_i\in[0,1],~i\in\mathcal{I}_n.$$ Interested readers may refer to \cite{nelsen2007} and \cite{mcneil2009multivariate} for more details on Archimedean copulas.
\section{Comparison results\setcounter{equation}{0}}
This section deals with the stochastic comparisons of the lifetimes of two $2$-out-of-$n$ systems with respect to the usual stochastic and reversed hazard rate orderings in the exponentiated location-scale models. It has been mentioned before that the second-largest order statistic represents the lifetime of a $2$-out-of-$n$ system. Thus, this problem is equivalent to comparing the second-largest order statistics arising from two sets of nonnegative random lifetimes. The random lifetimes can be independent or dependent. Firstly, consider the case of independent lifetimes.
\subsection{Independent lifetimes}
The random vector $\bm{X}=(X_{1},\cdots,X_{n})$ follows exponentiated location-scale model if the cumulative distribution function of the $i$th random variable $X_{i}$ is given by (\ref{eq1.1}), $i=1,\cdots,n.$ For convenience, we denote $\bm{X}\sim \mathbb{ELS}(
\bm{\lambda},\bm{\theta},\bm{\alpha};F_{b})$, where $F_{b}$ is the baseline distribution function, $\bm{\lambda}=(\lambda_1,\cdots,\lambda_n)$, $\bm{\theta}=
(\theta_1,\cdots,\theta_n)$ and $\bm{\alpha}=(\alpha_1,\cdots,\alpha_n)$. Denote by $\bm{Y}$ another random vector such that $\bm{Y}\sim \mathbb{ELS}(
\bm{\mu},\bm{\delta},\bm{\beta};F_{b})$, where $\bm{\mu}=(\mu_1,\cdots,\mu_n)$,
$\bm{\delta}=(\delta_1,\cdots,\delta_n)$ and $\bm{\beta}=(\beta_1,\cdots,\beta_n)$. Keeping some possible applications to the reliability theory in our mind, one can assume that the random vector $\bm{X}$ describes the random lifetimes of $n$ components of a $2$-out-of-$n$ system. Similarly, for the random vector $\bm{Y}$. Note that the cumulative distribution functions of $X_{n-1:n}$ and $Y_{n-1:n}$ are respectively given by (see, \cite{mesfioui2017stochastic})
\begin{eqnarray}\label{eq3.1}
F_{X_{n-1:n}}(x)=\sum\limits_{l=1}^{n}\left[\prod_{k\neq l}^{n}\left\{\left[F_{b}\left(\frac{x-\lambda_k}{\theta_k}\right)\right]^{\alpha_k}\right\}\right]-(n-1)\prod_{k=1}^{n}\left\{\left[F_{b}\left(\frac{x-\lambda_k}{\theta_k}\right)\right]^{\alpha_k}\right\},
\end{eqnarray}
where
$x>\max\{\lambda_k,~\forall ~k\}$ and
\begin{eqnarray}
G_{Y_{n-1:n}}(x)=\sum\limits_{l=1}^{n}\left[\prod_{k\neq l}^{n}\left\{\left[F_{b}\left(\frac{x-\mu_k}{\delta_k}\right)\right]^{\beta_k}\right\}\right]-(n-1)\prod_{k=1}^{n}\left\{\left[F_{b}\left(\frac{x-\mu_k}{\delta_k}\right)\right]^{\beta_k}\right\},
\end{eqnarray}
where $x>\max\{\mu_k,~\forall ~k\}$. Now, we are ready to present our main results. In the first theorem, we obtain conditions, under which the second-largest order statistics $X_{n-1:n}$ and $Y_{n-1:n}$ are comparable in the usual stochastic order. We refer to \cite{belzunce1998} and \cite{oliveira2015} for similar monotonicity conditions. The location parameters and the shape parameters are taken equal and fixed. Specifically, the following theorem states that the weak supermajorized scale parameter vector yields a $2$-out-of-$n$ system with larger reliability.
\begin{theorem}\label{th3.1}
For $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}=\lambda\boldsymbol{1}_{n}$ and $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha\boldsymbol{1}_{n}$, if $ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E_+}~( or~\mathcal{D_+})$ and $w^2 \tilde{r}_{b}(w)$ is increasing in $w>0$, then ${\boldsymbol\theta}\succeq^{w}{{\boldsymbol\delta}}\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$.
\end{theorem}
\begin{proof}
We only provide the proof of the case when $ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E_+}$. The other case can be finished in a similar manner. To prove the result, denote $\Psi_1\left({\boldsymbol \theta}\right)=F_{X_{n-1:n}}(x)$, where $F_{X_{n-1:n}}(x)$ is obtained from (\ref{eq3.1}). The partial derivative of $\Psi_1\left({\boldsymbol \theta}\right)$ with respect to $\theta_i$, for $i=1,\cdots,n$ is
\begin{align}\label{eq3.3}
\frac{\partial\Psi_1\left({\boldsymbol \theta}\right)}{\partial \theta_i}=-\left[\frac{\alpha[w^2{\tilde{r}_{b}}(w)]_{w=\left(\frac{x-\lambda}{\theta_i}\right)}}{x-\lambda}\right]\left[\sum\limits_{\overset{l=1}{l\neq i}}^{n}\prod_{k\neq l}^{n}d_{k}-(n-1)\prod_{k=1}^{n}d_{k}\right],
\end{align}
where $d_{k}=[F_{b}(\frac{x-\lambda}{\theta_k})]^{\alpha}$, for $k=1,\cdots,n.$ Now, using Lemma \ref{lem2.1b}, it is enough to show that $\Psi_1(\boldsymbol{\theta})$ is decreasing and Schur-convex with respect to $\boldsymbol{\theta}\in \mathcal{E_+}.$ Let
$1\leq i\leq j \leq n.$ Then, $\theta_i\leq\theta_j$. As a result, we get
$(\frac{x-\lambda}{\theta_i})\geq(\frac{x-\lambda}{\theta_j})$ and $[F_{b}(\frac{x-\lambda}{\theta_i})]^{\alpha}\geq[F_{b}(\frac{x-\lambda}{\theta_j})]^{\alpha}.$ Note that $\Psi_1(\boldsymbol{\theta})$ is decreasing and Schur-convex with respect to $\boldsymbol{\theta}\in \mathcal{E_+}$ is equivalent to show that $\frac{\partial\Psi_1\left({\boldsymbol \theta}\right)}{\partial \theta_i}$ given by (\ref{eq3.3}) is negative and increasing with respect to $\theta_i,$ for $i=1,\cdots,n.$
It is easy to check that $\frac{\partial\Psi_1\left({\boldsymbol \theta}\right)}{\partial \theta_i}\leq 0,$ since
\begin{eqnarray}\label{eq3.4}
\prod_{k\neq l}^{n} d_{k}\geq \prod_{k= 1}^{n}d_{k}\Rightarrow \sum\limits_{\overset{l=1}{l\neq i}}^{n}\prod_{k\neq l}^{n}d_{k}-(n-1)\prod_{k=1}^{n}d_{k}\geq 0.
\end{eqnarray}
Now, we will show that $\frac{\partial\Psi_1\left({\boldsymbol \theta}\right)}{\partial\theta_i}$ is increasing in $\theta_i.$
From the assumption that $w^2 \tilde{r}_{b}(w)$ is increasing, we obtain
\begin{equation}\label{eq3.5}
\Big[w^2 \tilde{r}_{b}(w)\Big]_{w=\left(\frac{x-\lambda}{\theta_i}\right)}\geq\Big[w^2 \tilde{r}_{b}(w)\Big]_{w=\left(\frac{x-\lambda}{\theta_j}\right)}.
\end{equation}
Further,
\begin{align}\label{eq3.6}
\sum\limits_{\overset{l=1}{l\neq i}}^{n}\prod_{k\neq l}^{n}d_{k}-\sum\limits_{\overset{l=1}{l\neq j}}^{n}\prod_{k\neq l}^{n}d_{k}
&=\prod_{k\neq \{i,j\}}^{n}d_{k}\left[\left[F_{b}\left(\frac{x-\lambda}{\theta_i}\right)\right]^{\alpha}-\left[F_{b}\left(\frac{x-\lambda}{\theta_j}\right)\right]^{\alpha}\right]\nonumber\\
&\geq 0.
\end{align}
Utilizing (\ref{eq3.5}) and (\ref{eq3.6}), it can be shown that $\frac{\partial\Psi_1\left({\boldsymbol \theta}\right)}{\partial\theta_i}-\frac{\partial\Psi_1\left({\boldsymbol \theta}\right)}{\partial\theta_j}$ is at most zero. Thus,
the required result follows by Theorem $A.8$ of \cite{marshall2010}. This completes the proof of the theorem.
\end{proof}
The example given below demonstrates Theorem \ref{th3.1}.
\begin{example}\label{exe3.1}
Consider two vectors $\boldsymbol{X}\sim \mathbb{ELS}(4,(5,9,10),4;(\frac{x}{100})^{0.2})$ and
$\boldsymbol{Y}\sim \mathbb{ELS}(4,$
$(7,10,12),4;(\frac{x}{100})^{0.2})$, where $0<x\le 100.$ For the assumed baseline distribution, $w^2 \tilde{r}_{b}(w)$ is increasing. Clearly, all the conditions of Theorem \ref{th3.1} are satisfied. Now,
we plot the graphs of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ in Figure $1a$. It is noticed that the plot of ${F}_{X_{2:3}}(x)$ is above the plot of ${F}_{Y_{2:3}}(x)$ for all $x$, which validates the result in Theorem \ref{th3.1}.
\end{example}
\begin{figure}[h]
\begin{center}
\subfigure[]{\label{c2}\includegraphics[height=2.41in]{secondl_example3_1.eps}}
\subfigure[]{\label{c1}\includegraphics[height=2.41in]{second_largest_exampleth3_4.eps}}
\caption{
(a) Plots of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ as in Example \ref{exe3.1}.
(b) Plot of $[\tilde{r}_{X_{2:3}}(x)-\tilde{r}_{Y_{2:3}}(x)]$ as in Example \ref{exe3.2}.
}
\end{center}
\end{figure}
Besides the baseline distribution as in Example \ref{exe3.1}, there is another distribution with cumulative distribution function $F_{b}(x)=\frac{x}{1+x},~x>0$, for which $w^2 \tilde{r}_{b}(w)$ is increasing. We have $(\theta_1,\cdots,\theta_n)\succeq^{w} (\theta,\cdots,\theta)$, for $n\theta\ge \sum_{i=1}^{n}\theta_i$. Using this fact, the following corollary immediately follows from Theorem \ref{th3.1}. This result is also useful to get bound of the time to failure of a $2$-out-of-$n$ system with heterogeneous components in terms of that with homogeneous components.
\begin{corollary}\label{cor3.1}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};{F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\lambda,\theta,\boldsymbol{\beta};F_{b}),$ with $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha \bm{1}_n$. Also, $\boldsymbol{\theta}\in\mathcal{D_+} ~(or~\mathcal{E_+})$. Then,
$n\theta\geq\sum_{i=1}^{n}\theta_i\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$, provided $w^2\tilde{r}_{b}(w)$ is increasing in $w>0$.
\end{corollary}
In the previous theorem, we have considered that the location parameters are the same and fixed. In the following theorem, we assume that the location parameters are the same but vector valued. The sufficient conditions here undergo little modification.
\begin{theorem}\label{th3.2}
Suppose $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}$, $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha \boldsymbol{1}_{n}$. Further, let $\boldsymbol{\lambda},~ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E_+}~(or~\mathcal{D_+})$ and $w \tilde{r}_{b}(w)$ be increasing in $w>0$. Then, ${\boldsymbol\delta}\succeq^{w}{{\boldsymbol\theta}}\Rightarrow Y_{n-1:n}\leq_{st}X_{n-1:n}$.
\end{theorem}
\begin{proof}
The proof of this theorem is similar to that of Theorem \ref{th3.1}. Thus, it is omitted for the sake of brevity.
\end{proof}
In the same vein as Corollary \ref{cor3.1}, the following corollary readily follows.
\begin{corollary}
For $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\delta,\alpha;F_{b})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\alpha;F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}$, we have $n\delta\ge\sum_{i=1}^{n}\delta_i\Rightarrow Y_{n-1:n}\leq_{st}X_{n-1:n}$, provided $\boldsymbol{\lambda},~\boldsymbol{\delta}\in\mathcal{E_+}~(or~\mathcal{D_+})$ and $w \tilde{r}_{b}(w)$ is increasing in $w>0$.
\end{corollary}
The following counterexample reveals that if $\boldsymbol{\lambda},~ \boldsymbol{\theta},~\boldsymbol{\delta}\notin\mathcal{E_+}~(or~\mathcal{D_+})$, then the result stated in Theorem \ref{th3.2} may not hold.
\begin{counterexample}\label{cex3.1}
Let us consider two vectors $\boldsymbol{X}\sim \mathbb{ELS}((3,4,5),(3,0.1,0.02),3;(\frac{x}{10})^{0.001})$ and
$\boldsymbol{Y}\sim \mathbb{ELS}((3,4,5),(2,0.03,0.01),3;(\frac{x}{10})^{0.001})$, where $0<x\le 10.$ Here, $w \tilde{r}_{b}(w)$ is increasing. The assumptions of Theorem \ref{th3.2} except the restrictions taken on the vectors of the parameters hold. Now, to check if the stated stochastic order holds,
we plot the graphs of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ in Figure $2a$. The graphs cross each other near the point $x=5.9$. This shows that the usual stochastic order in Theorem \ref{th3.2} can not be obtained, if one ignores the restrctions on the parameters vectors.
\end{counterexample}
\begin{figure}[h]
\begin{center}
\subfigure[]{\label{c2.0}\includegraphics[height=2.41in]{secondl_counterexample3_4.eps}}
\subfigure[]{\label{c1.0}\includegraphics[height=2.41in]{secondl_counterexample3_3.eps}}
\caption{
(a) Plots of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ as in Counterexample \ref{cex3.1}. (b) Plot of the difference $[{F}_{X_{2:3}}(x)-{F}_{Y_{2:3}}(x)]$ as in Counterexample \ref{cex3.3}.
}
\end{center}
\end{figure}
In the next result, we obtain sufficient conditions for the usual stochastic ordering between two second-largest order statistics, with the location and scale parameters being fixed. In particular, it proves that the weak supermajorized shape parameter vector produces a system with higher reliability.
\begin{theorem}\label{th3.3}
Suppose $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}=\lambda\boldsymbol{1}_{n}$ and $\boldsymbol{\theta}=\boldsymbol{\delta}=\theta\boldsymbol{1}_{n}$. Also, let $\boldsymbol{\beta}, ~\boldsymbol{\alpha}\in\mathcal{E_+}~(or~\mathcal{D_+})$. Then, $\boldsymbol\alpha\succeq^{w}\boldsymbol\beta\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$.
\end{theorem}
\begin{proof}
Denote $\Psi_2\left({\boldsymbol \alpha}\right)=F_{X_{n-1:n}}(x)$, where the distribution function of $X_{n-1:n}$ can be written from (\ref{eq3.1}) accordingly to the present set-up. The partial derivative of $\Psi_2\left({\boldsymbol \alpha}\right)$ with respect to $\alpha_i,$ for $i=1,\cdots,n$ is obtained as
\begin{equation}\label{eq3.7}
\frac{\partial\Psi_2\left({\boldsymbol \alpha}\right)}{\partial \alpha_i}=\left[\ln F\left(\frac{x-\lambda}{\theta}\right)\right]\left[\sum\limits_{\overset{l=1}{l\neq i}}^{n}\prod_{k\neq l}^{n}q_{k}-(n-1)\prod_{k=1}^{n}q_{k}\right],
\end{equation}
where $q_{k}=\left[F_{b}\left(\frac{x-\lambda}{\theta}\right)\right]^{\alpha_{k}},$ for $k=1,\cdots,n.$ The proof of this theorem will be completed if we show that the function $\Psi_2\left({\boldsymbol \alpha}\right)$ is decreasing and Schur-convex with respect to $\boldsymbol{\alpha}\in \mathcal{E_+}.$ This is equivalent to establish that the partial derivative $\frac{\partial\Psi_2\left({\boldsymbol \alpha}\right)}{\partial \alpha_i}$ given by (\ref{eq3.7}) is negative and increasing with respect to $\alpha_i,$ for $i=1,\cdots,n.$
Consider $1\leq i\leq j \leq n.$ Then, $\alpha_i\leq\alpha_j$ and
$\left[F_{b}\left(\frac{x-\lambda}{\theta}\right)\right]^{\alpha_{i}}\geq\left[F_{b}\left(\frac{x-\lambda}{\theta}\right)\right]^{\alpha_{j}}.$
Further, it is easy to check that $\frac{\partial\Psi_2\left({\boldsymbol \alpha}\right)}{\partial \alpha_i}$ is at most zero, since the first and second third-bracketed terms in (\ref{eq3.7}) are respectively negative and positive. Now, for $\alpha_i\leq\alpha_j$, consider
\begin{eqnarray}
\frac{\partial\Psi_2\left({\boldsymbol \alpha}\right)}{\partial \alpha_i}-\frac{\partial\Psi_2\left({\boldsymbol \alpha}\right)}{\partial \alpha_j}&=&
\left[\ln F\left(\frac{x-\lambda}{\theta}\right)\right]\left[\sum\limits_{\overset{l=1}{l\neq i}}^{n}\prod_{k\neq l}^{n}q_{k}-\sum\limits_{\overset{l=1}{l\neq j}}^{n}\prod_{k\neq l}^{n}q_{k}\right]\nonumber\\
&=& \left[\ln F\left(\frac{x-\lambda}{\theta}\right)\right]\prod_{k\neq \{i,j\}}^{n}q_{k}\left[\left[F_{b}\left(\frac{x-\lambda}{\theta}\right)\right]^{\alpha_{i}}-\left[F_{b}\left(\frac{x-\lambda}{\theta}\right)\right]^{\alpha_{j}}\right]\nonumber\\
&\le& 0.
\end{eqnarray}
This implies that $\frac{\partial\Psi_2\left({\boldsymbol \alpha}\right)}{\partial \alpha_i}$ is increasing with respect to $\alpha_i$, for $i=1,\cdots,n.$ Hence, the rest of the proof follows from
Theorem $A.8$ of \cite{marshall2010}. The proof for the case $\bm{\alpha}\in\mathcal{D_+}$ follows in a manner similar to that when $\bm{\alpha}\in\mathcal{E_+}$. So, it is omitted.
\end{proof}
Now, we obtain some comparison results between the second-largest order statistics in terms of the reversed hazard rate order. The reversed hazard rate function of $X_{n-1:n}$ is given by
\begin{align}\label{eq3.9}
\tilde{r}_{X_{n-1:n}}(x)=\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}\tilde{r}_{b}\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)+\left[\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}\left[\frac{\tilde{r}_b\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}{r_b\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}\right]'\right]\left[\sum\limits_{i=1}^{n}\left[\frac{\tilde{r}_b\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}{r_b\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}\right]+1\right]^{-1}.
\end{align}
The next consecutive four theorems provide conditions, under which the reversed hazard rate order between $X_{n-1:n}$ and $Y_{n-1:n}$ exists. For convenience of the presentation of the results, we first state the following conditions:
\begin{itemize}
\item[(C1)] $w^2[w\tilde{r}_{b}(w)]'$, $[\tilde{r}_{b}(w)/r_{b}(w)],$ $w^2[\tilde{r}_{b}(w)/r_{b}(w)]'$ and $w^2[w[\tilde{r}_{b}(w)/r_{b}(w)]']'$ are decreasing.
\item [(C2)] $\tilde{r}_{b}(w)$ is convex, $\tilde{r}_b(w)/r_b(w)$ is decreasing, convex and $[\tilde{r}_b(w)/r_b(w)]''$ is increasing.
\item[(C3)] $w\tilde{r}_{b}(w)$, $w^2[w\tilde{r}_{b}(w)]'$, $[\tilde{r}_{b}(w)/r_{b}(w)]$, $w[\tilde{r}_{b}(w)/r_{b}(w)]'$, $w^2[\tilde{r}_{b}(w)/r_{b}(w)]'$ and $w^2[w[\tilde{r}_{b}(w)/r_{b}(w)]']'$ are decreasing.
\item[(C4)] $w\tilde{r}_{b}(w)$, $[\tilde{r}_{b}(w)/r_{b}(w)]$, $w[\tilde{r}_{b}(w)/r_{b}(w)]'$ are decreasing, $w^2[\tilde{r}_{b}(w)/r_{b}(w)]'$, $w^2[w\tilde{r}_{b}(w)]'$ and $w^2[w[\tilde{r}_{b}(w)/r_{b}(w)]']'$ are increasing.
\end{itemize}
The result stated below reveals that a $2$-out-of-$n$ system with majorized scale parameter vector has larger reversed hazard rate.
\begin{theorem}\label{th3.4}
For $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};{F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}=\lambda \boldsymbol{1}_{n}$, $\boldsymbol{\alpha}=\boldsymbol{\beta}=\boldsymbol{1}_{n}$, if $ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{D_+}~(or~\mathcal{E_+})$, then
${\boldsymbol\theta}\succeq^{m}{\boldsymbol\delta}\Rightarrow X_{n-1:n}\leq_{rh}Y_{n-1:n}$, provided (C1) holds.
\end{theorem}
\begin{proof}
Under the assumptions made, the reversed hazard rate function of $X_{n-1:n}$ can be written as
\begin{equation}\label{eq3.10}
\tilde{r}_{X_{n-1:n}}(x)=\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}\tilde{r}_{b}\left(\frac{x-\lambda}{\theta_{i}}\right)+\left[\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda}{\theta_{i}}\right)\right]\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1\right]^{-1},
\end{equation}
where $h(\frac{x-\lambda}{\theta_{i}})=\tilde{r}_{b}(\frac{x-\lambda}{\theta_{i}})/r_{b}(\frac{x-\lambda}{\theta_{i}}).$ Denote $\Psi_{3}(\bm{\theta})=\tilde{r}_{X_{n-1:n}}(x)$, where $\tilde{r}_{X_{n-1:n}}(x)$ is given by (\ref{eq3.10}).
Differentiating $\Psi_{3}(\boldsymbol{\theta})$ partially with respect to $\theta_{i}$, for $i=1,\cdots,n$, we obtain
\begin{align}
\frac{\partial \Psi_{3}(\boldsymbol{\theta})}{\partial \theta_{i}}&=-\left[\frac{[w^2\tilde{r}_{b}(w)+w^3\tilde{r}^{'}_{b}(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{(x-{\lambda})^2}\right]-\left[\frac{\frac{1}{(x-{\lambda})^2}[w^2h'(w)+w^3h^{''}(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1}\right]\nonumber\\
&~~~~+\left[\frac{\frac{1}{(x-\lambda)}[w^2h^{'}(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda}{\theta_{i}}\right)}{\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1\right]^2}\right].
\end{align}
According to Lemma \ref{lem2.1a} (\ref{lem2.1b}), in proving the result, it is required to show that $\Psi_3(\boldsymbol{\theta})$ is Schur-concave with respect to $\boldsymbol{\theta}\in\mathcal{D_+}~(or~\mathcal{E_+})$.
Now, consider
\begin{eqnarray}
\frac{\partial \Psi_3(\boldsymbol{\theta})}{\partial \theta_i}-\frac{\partial \Psi_3(\boldsymbol{\theta})}{\partial \theta_j}\overset{sign}{=}T_1+T_2+T_3,
\end{eqnarray}
where
\begin{eqnarray}
T_1&=&\left[\frac{[w^2[w\tilde{r}_{b}(w)]']_{w=\left(\frac{x-\lambda}{\theta_{j}}\right)}}{(x-{\lambda})^2}-\frac{[w^2[w\tilde{r}_{b}(w)]']_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{(x-{\lambda})^2}\right],\\
T_2&=&\left[\frac{[w^2[wh'(w)]']_{w=\left(\frac{x-\lambda}{\theta_{j}}\right)}}{(x-{\lambda})^2}-\frac{[w^2[wh'(w)]']_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{(x-{\lambda})^2}\right]\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_i}\right)+1\right]^{-1}~\mbox{and}\\
T_3&=&\left[\frac{[w^2h'(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{(x-{\lambda})}-\frac{[w^2h'(w)]_{w=\left(\frac{x-\lambda}{\theta_{j}}\right)}}{(x-{\lambda})}\right]\left[\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda}{\theta_{i}}\right)\right]
\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1\right]^{-2}.\nonumber\\
\end{eqnarray}
Consider the case that $\boldsymbol{\theta}\in \mathcal{D_+}.$ The proof for the other case is similar. For $1\le i\le j \le n,$ we have $\theta_i\ge \theta_j$ implies $\frac{x-\lambda}{\theta_{i}}\le \frac{x-\lambda}{\theta_{j}}$. It is assumed that $w^2[w\tilde{r}_{b}(w)]'$ is decreasing. Therefore, $-w^2[w\tilde{r}_{b}(w)]'|_{w=\frac{x-\lambda}{\theta_{i}}}\le -w^2[w\tilde{r}_{b}(w)]'|_{w=\frac{x-\lambda}{\theta_{j}}}$. Further, $[\tilde{r}_{b}(w)/r_{b}(w)]$ and $w^2[\tilde{r}_{b}(w)/r_{b}(w)]'$ are decreasing. As a result, $\frac{1}{(x-\lambda)^2}[w^2h'(w)]|_{w=\frac{x-\lambda}{\theta_{j}}}\le \frac{1}{(x-\lambda)^2}[w^2h'(w)]|_{w=\frac{x-\lambda}{\theta_{i}}}\le0$. Again, $w^2[w[\tilde{r}_{b}(w)/r_{b}(w)]']'$ is decreasing. So, $-\frac{1}{(x-\lambda)^2}[w^2[wh'(w)]']|_{w=\frac{x-\lambda}{\theta_{i}}}\le- \frac{1}{(x-\lambda)^2}[w^2[wh'(w)]']|_{w=\frac{x-\lambda}{\theta{j}}}$. Combining these inequalities, we obtain that the values of the terms $T_1,~T_2$ and $T_3$ are at most zero. Thus,
$$ \frac{\partial \Psi_3(\boldsymbol{\theta})}{\partial \theta_i}-\frac{\partial \Psi_3(\boldsymbol{\theta})}{\partial \theta_j}\leq 0, \text{ for each } \bm{\theta}\in\mathcal{D_+}.$$ Hence, the rest of the proof readily follows.
\end{proof}
\begin{remark}\label{r1}
Let us consider the baseline distribution function as $F_{b}(x)=\frac{x}{x+1},~x>0.$ For this baseline distribution function, one can easily check that $w^2[w\tilde{r}_{b}(w)]'$, $[\tilde{r}_{b}(w)/r_{b}(w)]$, $w^2[\tilde{r}_{b}(w)/r_{b}(w)]'$ and $w^2[w[\tilde{r}_{b}(w)/r_{b}(w)]']'$ are decreasing. Thus, Theorem \ref{th3.4} can be applied for this baseline distribution.
\end{remark}
The following example provides an illustration of Theorem \ref{th3.4}.
\begin{example}\label{exe3.2}
Let us consider two vectors $\boldsymbol{X}\sim \mathbb{ELS}(4,(2,5,9),1;\frac{x}{x+1})$ and
$\boldsymbol{Y}\sim \mathbb{ELS}(4,(3,6,7)$
$,1;\frac{x}{x+1})$, where $x>0.$ One can easily check that all the conditions of Theorem \ref{th3.4} are satisfied. Thus, $X_{2:3}\leq_{rh}Y_{2:3}.$
We plot the graph of $[\tilde{r}_{X_{2:3}}(x)-\tilde{r}_{Y_{2:3}}(x)]$ as a function of $x$ in Figure $1b$. As expected, this function takes negative values for all $x>0$.
\end{example}
In the following theorem, we assume that the shape parameters are equal to $1.$ The scale parameters are equal but scaler valued. It is established that under some conditions, the majorized location parameter vector produces a $2$-out-of-$n$ system having smaller reversed hazard rate.
\begin{theorem}\label{th3.5}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\theta,\boldsymbol{\alpha};{F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\theta,\boldsymbol{\beta};F_{b}),$ with $\boldsymbol{\alpha}=\boldsymbol{\beta}=\boldsymbol{1}_{n}$ and $\boldsymbol{ \delta}=\boldsymbol{ \theta}=\theta \boldsymbol{1}_{n}$. Also, consider $\boldsymbol{\lambda},~\boldsymbol{\mu}\in\mathcal{D_+}~(or~\mathcal{E_+})$. Then,
${\boldsymbol\lambda}\succeq^{m}{\boldsymbol\mu}\Rightarrow Y_{n-1:n}\leq_{rh}X_{n-1:n}$, provided $(C2)$ holds.
\end{theorem}
\begin{proof}
Denote $\Psi_{4}(\bm{\lambda})=\tilde{r}_{X_{n-1:n}}(x)$, where the reversed hazard rate function of $X_{n-1:n}$ can be obtained from (\ref{eq3.9}). Differentiating $\Psi_{4}(\bm{\lambda})$ with respect to $\lambda_i$, $i=1,\cdots,n$ partially, we get
\begin{align}
\frac{\partial \Psi_{4}(\bm{\lambda})}{\partial \lambda_{i}}=&-\left[\frac{1}{{\theta}^2}\tilde{r}^{'}_{b}\left(\frac{x-\lambda_i}{\theta}\right)\right]-\left[ \frac{1}{{\theta}^2}h^{''}\left(\frac{x-\lambda_{i}}{\theta}\right)\right] \left[\sum\limits\limits_{i=1}^{n}h\left(\frac{x-\lambda_{i}}{\theta}\right)+1\right]^{-1}\nonumber\\
&+ \left[\frac{1}{\theta}h^{'}\left(\frac{x-\lambda_{i}}{\theta}\right)\sum\limits_{i=1}^{n}\frac{1}{\theta}h^{'}\left(\frac{x-\lambda_{i}}{\theta}\right)\right]\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda_{i}}{\theta}\right)+1\right]^{-2},
\end{align}
where $h\left(\frac{x-\lambda_{i}}{\theta}\right)=\tilde{r}_{b}\left(\frac{x-\lambda_{i}}{\theta}\right)/r_{b}\left(\frac{x-\lambda_{i}}{\theta}\right).$ To prove the stated result, it is sufficient to show that $\Psi_{4}(\bm{\lambda})$ is Schur-convex with respect to $\boldsymbol{\lambda}\in\mathcal{D_+}~(or~\mathcal{E_+}).$ This can be executed using a manner analogous to that of Theorem \ref{th3.4}. Thus, it is omitted for the sake of conciseness.
\end{proof}
Next theorem states sufficient conditions for the comparison of the second-largest order statistics, when the scale parameters are ordered according to the weakly supermajorization order.
\begin{theorem}\label{th3.6}
Assume that $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha}; {F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta}; F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}=\lambda \boldsymbol{1}_{n}$ and $\boldsymbol{\alpha}=\boldsymbol{\beta}=\boldsymbol{1}_{n}$. Further, assume $ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{D_+}~(or~\mathcal{E_+})$. Then,
${\boldsymbol\theta}\succeq^{w}{\boldsymbol\delta}\Rightarrow X_{n-1:n}\leq_{rh}Y_{n-1:n}$, provided $(C3)$ holds.
\end{theorem}
\begin{proof}
Under the assumed set-up, the reversed hazard rate function of $X_{n-1:n}$ can be written as
\begin{equation}\label{eq3.18}
\Psi_5(\boldsymbol{\theta})\overset{def}{=}\tilde{r}_{X_{n-1:n}}(x)=\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}\tilde{r}_{b}\left(\frac{x-\lambda}{\theta_{i}}\right)+\left[\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda}{\theta_{i}}\right)\right]\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1\right]^{-1},
\end{equation}
where $h(\frac{x-\lambda}{\theta_{i}})=\tilde{r}_{b}(\frac{x-\lambda}{\theta_{i}})/r_{b}(\frac{x-\lambda}{\theta_{i}}).$ The proof will be completed if we show that $\Psi_5(\boldsymbol{\theta})$ is increasing and Schur-concave with respect to $\boldsymbol{\theta}\in\mathcal{D_+}~(or~\mathcal{E_+}).$ Differentiating $\Psi_{5}(\boldsymbol{\theta})$ with respect to $\theta_{i}$, $i=1,\cdots,n$, we have
\begin{align}
\frac{\partial \Psi_5(\boldsymbol{\theta})}{\partial \theta_{i}}&=-\left[\frac{[w^2\tilde{r}_{b}(w)+w^3\tilde{r}^{'}(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{(x-{\lambda})^2}\right]-\left[\frac{\frac{1}{(x-{\lambda})^2}[w^2h'(w)+w^3h^{''}(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}}{\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1\right]}\right]\nonumber\\
&~~~~+\left[\frac{\frac{1}{(x-\lambda)}[w^2h^{'}(w)]_{w=\left(\frac{x-\lambda}{\theta_{i}}\right)}\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda}{\theta_{i}}\right)}{\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda}{\theta_{i}}\right)+1\right]^2}\right].
\end{align}
Based on the given assumptions, it can be shown that $\frac{\partial \Psi_{5}(\boldsymbol{\theta})}{\partial \theta_{i}}$ is at least zero. This implies that $\Psi_{5}(\boldsymbol{\theta})$ is increasing with respect to $\theta_{i}$, $i=1,\cdots,n$.
We omit the remaining details of the proof since it can be achieved using arguments similar to that of Theorem \ref{th3.4}.
\end{proof}
Similar to Corollary \ref{cor3.1}, we have the following corollary from the preceding theorem.
\begin{corollary}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
\lambda,\boldsymbol{\theta},\boldsymbol{1}_{n}; {F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\lambda,\delta,\boldsymbol{1}_{n}; F_{b}).$ Further, let $ \boldsymbol{\theta}\in\mathcal{D_+}~(or~\mathcal{E_+})$ and $(C3)$ hold. Then, $n\delta\ge \sum_{i=1}^{n}\theta_i\Rightarrow X_{n-1:n}\leq_{rh}Y_{n-1:n}$.
\end{corollary}
The following theorem provides the conditions, under which one can compare the reversed hazard rate functions of $X_{n-1:n}$ and $Y_{n-1:n}$, when the reciprocal of the scale parameters of two sets of heterogeneous random lifetimes are connected according to the reciprocally majorization order. In this theorem, we consider that the shape parameters are the same and equal to $1$. The location parameters are taken to be equal but vector-valued.
\begin{theorem}\label{th3.7}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha}; {F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta}; F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}$, $\boldsymbol{\alpha}=\boldsymbol{\beta}=\bm{1}_n$. Also, let $\boldsymbol{\lambda},~ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{D_+}~(or~\mathcal{E_+})$. Then,
$1/{\boldsymbol\theta}\succeq^{rm}1/{\boldsymbol\delta}\Rightarrow Y_{n-1:n}\leq_{rh}X_{n-1:n}$, provided $(C4)$ is satisfied.
\end{theorem}
\begin{proof}
Based on the given assumptions, the reversed hazard rate function of $X_{n-1:n}$ can be written as follows
\begin{equation}
\Psi_6(1/\boldsymbol{\theta})\overset{def}{=}\tilde{r}_{X_{n-1:n}}(x)=\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}\tilde{r}_{b}\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)+\left[\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)\right]\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)+1\right]^{-1},
\end{equation}
where $h(\frac{x-\lambda_{i}}{\theta_{i}})=\tilde{r}_{b}(\frac{x-\lambda_{i}}{\theta_{i}})/r_{b}(\frac{x-\lambda_{i}}{\theta_{i}}),$ for $i=1,\cdots,n.$ The partial derivative of $\Psi_6(1/\boldsymbol{\theta})$ with respect to $\theta_i$, for $i=1,\cdots,n,$ is given by
\begin{align}\label{eq3.21}
\frac{\partial \Psi_6(1/\boldsymbol{\theta})}{\partial \theta_{i}}&=-\left[\frac{[w^2\tilde{r}_{b}(w)+w^3\tilde{r}^{'}(w)]_{\left(\frac{x-\lambda_i}{\theta_{i}}\right)}}{(x-{\lambda_i})^2}\right]-\left[\frac{\frac{1}{(x-{\lambda_i})^2}[w^2h'(w)+w^3h^{''}(w)]_{w=\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}}{\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)+1\right]}\right]\nonumber\\
&~~~~+\left[\frac{\frac{1}{(x-\lambda_{i})}[w^2h^{'}(w)]_{w=\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}\sum\limits_{i=1}^{n}\frac{1}{\theta_{i}}h^{'}\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)}{\left[\sum\limits_{i=1}^{n}h\left(\frac{x-\lambda_{i}}{\theta_{i}}\right)+1\right]^2}\right].
\end{align}
Using Lemma \ref{lem2.1a} ( Lemma \ref{lem2.1b}) and Lemma \ref{lem2.1c}, we have to show that $\Psi_6(1/\boldsymbol{\theta})$ is increasing and Schur-convex with respect to $\boldsymbol{\theta}\in\mathcal{D_+}~(or~\mathcal{E_+}).$ Under the assumptions made, clearly, $\Psi_6(1/\boldsymbol{\theta})$ is increasing, since the derivative given by (\ref{eq3.21}) is nonnegative. Further, Schur-convexity of $\Psi_6(1/\boldsymbol{\theta})$ can be shown using the arguments similar to Theorem \ref{th3.4}. The details have been omitted. This completes the result.
\end{proof}
Note that $(\frac{1}{\theta_1},\cdots,\frac{1}{\theta_n})\succeq^{rm} (\frac{1}{\delta},\cdots,\frac{1}{\delta})$ holds for $n\delta\le \sum_{i=1}^{n}\theta_i$. Thus, we have the following corollary, which is an immediate consequence of Theorem \ref{th3.7}.
\begin{corollary}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha}; {F_{b}})$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\delta,\boldsymbol{\beta}; F_{b}),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}$, $\boldsymbol{\alpha}=\boldsymbol{\beta}=\bm{1}_n$. Also, let $\boldsymbol{\lambda},~ \boldsymbol{\theta}\in\mathcal{D_+}~(or~\mathcal{E_+})$. Then,
$n\delta \le \sum_{i=1}^{n}\theta_i\Rightarrow Y_{n-1:n}\leq_{rh}X_{n-1:n}$, provided $(C4)$ holds.
\end{corollary}
\subsection{Dependent lifetimes}
In the preceding subsection, we have considered $2$-out-of-$n$ systems having independent components' lifetimes. However, the components' lifetimes of a system may be dependent due to various factors. It can happen that the components have been produced by same company. So, naturally, there is a chance that the components are dependent. In this subsection, we consider two sets of exponentiated location-scale distributed random lifetimes associated with Archimedean copulas. Let $\bm{X}$ and $\bm{Y}$ be two vectors of $n$ dependent lifetimes such that $\bm{X}\sim \mathbb{ELS}(\bm{\lambda},\bm{\theta},\bm{\alpha};F_b,\psi_{1})$ and $\bm{Y}\sim \mathbb{ELS}(\bm{\mu},\bm{\delta},\bm{\beta};F_b,\psi_{2})$, where $X_{i}\sim\mathbb{ELS}(\lambda_i,\theta_i,\alpha_i;F_b,\psi_{1})$ and $Y_{i}\sim \mathbb{ELS}(\mu_i,\delta_i,\beta_i;F_b,\psi_{2})$, for $i=1,\cdots,n.$ We recall that $F_b$ is the baseline distribution function. The distribution functions of $X_{n-1:n}$ and $Y_{n-1:n}$ are respectively given by
\begin{equation}
{F}_{X_{n-1:n}}(x)=\sum\limits_{l=1}^{n}\psi_{1}\left[\sum\limits_{k\neq l}^{n}\phi_{1}\left\{\left[F_{b}\left(\frac{x-\lambda_k}{\theta_k}\right)\right]^{\alpha_k}\right\}\right]-(n-1)\psi_{1}\left[\sum\limits_{k=1}^{n}\phi_{1}\left\{\left[F_{b}\left(\frac{x-\lambda_k}{\theta_k}\right)\right]^{\alpha_k}\right\}\right],
\end{equation}
where $x>\max\{\lambda_k, \forall~ k\}$ and
\begin{equation}
{G}_{Y_{n-1:n}}(x)=\sum\limits_{l=1}^{n}\psi_{2}\left[\sum\limits_{k\neq l}^{n}\phi_{2}\left\{\left[F_{b}\left(\frac{x-\mu_k}{\delta_k}\right)\right]^{\beta_k}\right\}\right]-(n-1)\psi_{2}\left[\sum\limits_{k=1}^{n}\phi_{2}\left\{\left[F_{b}\left(\frac{x-\mu_k}{\delta_k}\right)\right]^{\beta_k}\right\}\right],
\end{equation}
where $x>\max\{\mu_k,\forall ~k\}.$ Now, we present sufficient conditions, for which the usual stochastic order holds between the second-largest order statistics arising from two sets of heterogeneous dependent samples under the assumption that the dependency structure is modeled by Archimedean copulas. In this regard, the following two lemmas are useful.
\begin{lemma}(\cite{li2015ordering})\label{lem3.1}
For two $n$-dimensional Archimedean copulas $C_{\psi_1}$ and $C_{\psi_2}$, if $\phi_2\circ\psi_1$ is super-additive, then $C_{\psi_1}(\boldsymbol{v})\leq C_{\psi_2}(\boldsymbol{v})$, for all $\boldsymbol{v}\in[0,1]^n.$ A function $f$ is said to be super-additive, if $ f(x)+f(y)\leq f(x+y),$ for all $x$ and $y$ in the domain of $f.$
\end{lemma}
\begin{lemma}\label{lem3.2}
For two $n$-dimensional Archimedean copulas $C_{\psi_1}$ and $C_{\psi_2}$, if $\phi_2\circ\psi_1$ is sub-additive, then $C_{\psi_2}(\boldsymbol{v})\leq C_{\psi_1}(\boldsymbol{v})$, for all $\boldsymbol{v}\in[0,1]^n.$ A function $f$ is said to be sub-additive, if $f(x+y)\le f(x)+f(y),$ for all $x$ and $y$ in the domain of $f.$
\end{lemma}
\begin{proof}
The proof is straightforward, and hence it is omitted.
\end{proof}
The following theorem states that if the scale parameters are connected with the weak supermajorization order, then under some conditions, there exists usual stochastic order between the second-largest order statistics. Here, we assume that the shape and scale parameters are the same and fixed.
\begin{theorem}\label{th3.8}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b},\psi_1)$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b},\psi_2),$ with $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha \boldsymbol{1}_n$ and $\boldsymbol{\mu}=\boldsymbol{\lambda}=\lambda \boldsymbol{ 1}_{n}$. Let $\phi_1=\psi^{-1}_1$, $\phi_2=\psi^{-1}_2$ and $\psi_1$ or $\psi_2$ be log-concave.
Further, assume $ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E}_+~(or~\mathcal{D_+})$ and $w^2 \tilde{r}_{b}(w)$ is increasing in $w>0$. Then,
\begin{itemize}
\item[(i)] ${{\boldsymbol\theta}}\succeq^{w}{{\boldsymbol\delta}}\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$, provided $\phi_2\circ\psi_1$ is sub-additive;
\item[(ii)] ${{\boldsymbol\delta}}\succeq^{w}{{\boldsymbol\theta}}\Rightarrow Y_{n-1:n}\leq_{st}X_{n-1:n}$, provided $\phi_2\circ\psi_1$ is super-additive.
\end{itemize}
\end{theorem}
\begin{proof} $(i)$ To prove the first part of the theorem, let us denote
\begin{equation}
J_1(x,\boldsymbol{ \theta}; F_{b},\psi_1)=\sum\limits_{l=1}^{n}\psi_1\left[\sum\limits_{k\neq l}^{n}\phi_1\left\{\left[F_{b}\left(\frac{x-\lambda}{\theta_k}\right)\right]^{\alpha}\right\}\right]-(n-1)\psi_1\left[\sum\limits_{k=1}^{n}\phi_1\left\{\left[F_{b}\left(\frac{x-\lambda}{\theta_k}\right)\right]^{\alpha}\right\}\right]
\end{equation}
and
\begin{equation}
J_2(x,\boldsymbol{ \delta}; F_{b},\psi_2)=\sum\limits_{l=1}^{n}\psi_2\left[\sum\limits_{k\neq l}^{n}\phi_2\left\{\left[F_{b}\left(\frac{x-\lambda}{\delta_k}\right)\right]^{\alpha}\right\}\right]-(n-1)\psi_2\left[\sum\limits_{k=1}^{n}\phi_2\left\{\left[F_{b}\left(\frac{x-\lambda}{\delta_k}\right)\right]^{\alpha}\right\}\right].
\end{equation}
Utilizing sub-additivity of $\phi_2\circ\psi_1$, and then from Lemma \ref{lem3.2}, we obtain
$
J_1(x,\boldsymbol{ \delta}; F_{b},\psi_1)\geq J_2(x,\boldsymbol{ \delta};F_{b},\psi_2).
$ Thus,
to prove the required result, we need to show that
$
J_1(x,\boldsymbol{ \theta}; F_{b},\psi_1)\geq J_1(x,\boldsymbol{ \delta};F_{b},\psi_1),
$
which is equivalent to showing that
$J_1(x,\boldsymbol{ \theta}; F_{b},\psi_1)$ is decreasing and Schur-convex with respect to $\boldsymbol{ \theta}\in\mathcal{E_+}~(or~\mathcal{D_+}).$
Further, denote
\begin{eqnarray}
\Phi_1\left({{\boldsymbol \theta}}\right)=J_1(x,\boldsymbol{ \theta}; F_{b},\psi_1).
\end{eqnarray}
After taking derivative of $\Phi_1\left({\boldsymbol \theta}\right)$ with respect to $\theta_i,$ for $i=1,\cdots,n$, we obtain
\begin{equation}\label{eqd3.25}
\frac{\partial\Phi_1\left({\boldsymbol \theta}\right)}{\partial \theta_i}=\varrho_{i1}({\boldsymbol \theta})\varrho_{1}(\theta_i),
\end{equation}
where
\begin{eqnarray*} \varrho_{i1}({\boldsymbol \theta})&=&(n-1)\psi_{1}'\left(\sum\limits_{k=1}^{n}t_k\right)-\sum\limits_{l\notin\{i,j\}}^{n}\psi_{1}'\left(\sum\limits_{k\neq l}^{n}t_k\right)
-\psi_{1}'\left(t_i+\sum\limits_{k\notin\{i,j\}}^{n}\left\{t_k\right\}\right),\\
\varrho_{1}(\theta_i)&=&\left[\frac{[w^2{\tilde{r}_b}(w)]_{w=\left({(x-\lambda)}{/\theta_i}\right)}}{x-\lambda}\right]\left[\frac{{\psi_{1}}({t_i})}{{\psi}'_{1}\left(t_i\right) }\right]
\end{eqnarray*}
and ${t_i=\phi_{1}\left[F_{b}\left(({x-\lambda}){/\theta_i}\right)\right]^{\alpha}}$, for $i=1,\cdots,n$. Let $\bm{\theta}\in\mathcal{E_+}$.
Then, for $1\leq i\leq j \leq n,$ $\theta_i\leq\theta_j.$ Therefore, $\left[F_{b}\left(({x-\lambda}){/\theta_i}\right)\right]^{\alpha}\geq\left[F_{b}\left(({x-\lambda}){/\theta_j}\right)\right]^{\alpha}$ and $({x-\lambda}){/\theta_i}\geq({x-\lambda}){/\theta_j}$. This gives $t_i\leq t_j$. Furthermore,
using the properties of the generator of an Archimedean copula, we have for $l=1,\cdots,n$,
\begin{eqnarray}
\psi_{1}'\left(\sum\limits_{k=1}^{n}t_k\right)&\geq& \psi_{1}'\left(\sum\limits_{k\neq l}^{n}t_k\right)\nonumber\\
\Rightarrow (n-1)\psi_{1}'\left(\sum\limits_{k=1}^{n}t_k\right)&\geq& \sum\limits_{l\neq i}^{n}\psi_{1}'\left(\sum\limits_{k\neq l}^{n}t_k\right),~i=1,\cdots,n.
\end{eqnarray}
Therefore,
\begin{align}
\varrho_{i1}(\boldsymbol{\theta})&=(n-1)\psi_{1}'\left(\sum\limits_{k=1}^{n}t_k\right)-\sum\limits_{l\notin\{i,j\}}^{n}\psi_{1}'\left(\sum\limits_{k\neq l}^{n}t_k\right)
-\psi_{1}'\left(\sum\limits_{k\neq j}^{n}\left\{t_k\right\}\right)\nonumber\\
&=(n-1)\psi_{1}'\left(\sum\limits_{k=1}^{n}t_k\right)-\sum\limits_{l\neq i}^{n}\psi_{1}'\left(\sum\limits_{k\neq l}^{n}t_k\right)\nonumber\\&\geq 0.
\end{align}
Hence, $\varrho_{i1}(\boldsymbol{\theta})$ is positive, as $\psi_{1}(x)$ is decreasing. Now, we have to show both $\varrho_{i1}(\boldsymbol{\theta})$ and $ \varrho_{1}(\theta_{i})$ are decreasing with respect to $\theta_i,$ for $i=1,\cdots,n.$ As we know, $\psi_{1}(x)$ is decreasing and convex. Thus, $\psi_{1}'(x)$ is negative and increasing. Also, $\phi_{1}[[F_{b}(\frac{x-\lambda}{\theta_i})]^{\alpha}]$ is increasing in $\theta_i,$ for $i=1,\cdots,n.$ Therefore, $\varrho_{i1}(\boldsymbol{\theta})$ is decreasing in $\theta_i,$ for all $i= 1,\cdots,n.$ Making use of the given assumptions, we have the following two inequalities:
\begin{align*}\label{eqd3.22}
\left[\frac{{\psi_{1}}({t_i})}{{\psi}_{1}'\left(t_i\right) }\right]\leq& \left[\frac{{\psi_{1}}({t_j})}{{\psi}_{1}'\left(t_j\right) }\right], \\
\left[w^2 \tilde{r}_{b}(w)\right]_{w=\left(\frac{x-\lambda}{\theta_i}\right)}\geq&\left[w^2 \tilde{r}_{b}(w)\right]_{w=\left(\frac{x-\lambda}{\theta_j}\right)}.
\end{align*}
Using these inequalities, one can easily check that $\varrho_1(\theta_i)$ is decreasing with respect to $\theta_i,$ for all $i=1,\cdots,n.$ Thus, $\Phi_1(\boldsymbol{\theta})$ is decreasing and Schur-convex with respect to $\boldsymbol{\theta}\in \mathcal{E}_+,$ by Lemma \ref{lem2.1b}. Rest of the proof can be proved by Theorem $A.8$ of \cite{marshall2010}. Note that the proof is similar when $\boldsymbol{\theta}\in \mathcal{D}_+$. Hence, it is omitted.\\
\\
$(ii)$
Utilizing super-additive property of $\phi_2\circ\psi_1$ and Lemma \ref{lem3.1}, we have
$
J_1(x,\boldsymbol{ \theta}; F_{b},\psi_1)\leq J_2(x,\boldsymbol{ \theta}; F_{b},\psi_2).
$
Now, to prove the stated result, it is enough to establish that
$
J_2(x,\boldsymbol{ \delta};F_{b},\psi_2)\geq J_2(x,\boldsymbol{ \theta};F_{b},\psi_2).
$
This is equivalent to establish that $J_2(x,\boldsymbol{ \delta};F_{b},\psi_2)$ is decreasing and Schur-convex with respect to $\boldsymbol{ \delta}\in\mathcal{E_+} ~(\text{or } \mathcal{D_+}).$ This follows in a similar vein to the proof of the first part, and hence it is not presented here.
\end{proof}
The following counterexample shows that the result in Theorem \ref{th3.8}$(i)$ does not hold if we do not consider all the assumptions. Here, the baseline distribution is taken as $F_{b}(x)=1-\exp(1-x^a),~x\geq1,~a>0$, for which $w^2\tilde{r}_b(w)$ is decreasing when $a=0.5.$
\begin{counterexample}\label{cex3.2}
Let us consider two $3$-dimensional vectors $\bm{X}$ and $\bm{Y}$ such that $\bm{X}\sim \mathbb{ELS}(
5,(2.5,6.5,3.1),0.1;1-\exp(1-x^{0.5}),e^{-x^\frac{1}{a_1}})$ and $\bm{Y}\sim \mathbb{ELS}(
5,(4.5,6.5,7.5),0.1;1-\exp(1-x^{0.5}),e^{-x^\frac{1}{a_2}}),~x>0,~a_1,~a_2\geq1.$
Note that $\psi_1$ and $\psi_2$ both are log-convex. Also, for $a_2\leq a_1,$ $\phi_2\circ\psi_1(x)=x^{\frac{a_2}{a_1}}$ is concave, implies $\phi_2\circ\psi_1$ is sub-additive. Consider $a_2=1.0001$ and $a_1=2.5$. Here, $\boldsymbol{\theta}$ does not belong to $\mathcal{E}_+ ~(\text{or }\mathcal{D}_+)$ and $\boldsymbol{\delta}\in\mathcal{E}_+.$ Clearly, all the conditions of Theorem \ref{th3.8}$(i)$ hold except the log-concavity of the generators, increasing property of $w^2\tilde{r}_{b}(w)$ and $\boldsymbol{\theta}\in\mathcal{E}_{+}~(or~\mathcal{D}_{+})$.
We plot the graphs of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ in Figure $3a$. It reveals that Theorem \ref{th3.8}$(i)$ does not hold.
\end{counterexample}
\begin{figure}[h]
\begin{center}
\subfigure[]{\label{c1.00}\includegraphics[height=5in]{secondl_counterexample3_2.eps}}
\caption{
(a) Plots of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ as in Counterexample \ref{cex3.2}.
}
\end{center}
\end{figure}
In the next result, we consider that location parameters are equal and vector-valued. The proof can be completed using arguments similar to that of Theorem \ref{th3.8}. Thus, it is omitted.
\begin{theorem}\label{th3.9}
Let $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b},\psi_1)$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b},\psi_2),$ with $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha \boldsymbol{1}_n$ and $\boldsymbol{\mu}=\boldsymbol{\lambda}$. Let $\phi_1=\psi^{-1}_1$, $\phi_2=\psi^{-1}_2$ and $\psi_1$ or $\psi_2$ be log-concave.
Also, assume $ \boldsymbol{\lambda},~ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E}_+~(or~\mathcal{D_+})$ and $ w\tilde{r}_{b}(w)$ is increasing in $w>0$. Then,
\begin{itemize}
\item[(i)] ${{\boldsymbol\theta}}\succeq^{w}{{\boldsymbol\delta}}\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$, provided $\phi_2\circ\psi_1$ is sub-additive;
\item[(ii)] ${{\boldsymbol\delta}}\succeq^{w}{{\boldsymbol\theta}}\Rightarrow Y_{n-1:n}\leq_{st}X_{n-1:n}$, provided $\phi_2\circ\psi_1$ is super-additive.
\end{itemize}
\end{theorem}
\begin{remark}
It is worth to mention that there are many Archimedean copulas, which satisfy super-additivity and sub-additivity of $\phi_2\circ\psi_1$ and log-concavity of $\psi_1$ and $\psi_2$.
\begin{itemize}
\item {\bf Independence copula:} Consider independence copula with generator $\psi_1(x)=\psi_2(x)=e^{-x},~x>0.$ For this copula, one can easily check that $\psi_{1}$ and $\psi_{2}$ both are log-concave. Further, $\phi_{2}\circ\psi_{1}(x)=x$, and clearly $\frac{d^2[\phi_{2}\circ\psi_{1}(x)]}{dx^2}$ is independent of $x$. Therefore, $\phi_{2}\circ\psi_{1}(x)$ satisfies both sub-additivity and super-additivity properties.
\item {\bf Gumbel copula:} Take Gumbel copula with generators $\psi_1(x)=e^{\frac{1}{a_1}(1-e^x)} $ and $\psi_2(x)=e^{\frac{1}{a_2}(1-e^x)},~x>0,~a_1,~a_2\in(0,1].$ Here, $\psi_{1}$ and $\psi_{2}$ both are log-concave. Again, $\phi_{2}\circ\psi_{1}(x)=\log(1-\frac{a_2}{a_1}(1-e^x)).$ Thus, $\frac{d^2[\phi_{2}\circ\psi_{1}(x)]}{dx^2}=\frac{a_2(a_1-a_2)e^{x}}{(a_1-{a_2}(1-e^x))^2}$. Therefore, for $a_1\geq a_2$ and $a_1\leq a_2$, $\phi_{2}\circ\psi_{1}(x)$ is super-additive and sub-additive, respectively.
\end{itemize}
\end{remark}
In the following, we consider an example, which illustrates Theorem \ref{th3.9}.
\begin{example}\label{exe3.3}
(i) Let us consider two vectors $\boldsymbol{X}\sim \mathbb{ELS}((4,6,8),(5,9,10),4;(\frac{x}{100})^{0.05},e^{\frac{1}{0.1}(1-e^x)})$ and
$\boldsymbol{Y}\sim \mathbb{ELS}((4,6,8),(7,10,12),4;(\frac{x}{100})^{0.05},e^{\frac{1}{0.5}(1-e^x)})$, where $0<x\leq 100.$ Here, $w \tilde{r}_{b}(w)$ is increasing. Further, it is not difficult to check that all the conditions of Theorem \ref{th3.9}(i) are satisfied. Now,
we plot the graphs of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ in Figure $4a$. It is seen that the graph of ${F}_{X_{2:3}}(x)$ is above the graph of ${F}_{Y_{2:3}}(x)$. That is, $X_{2:3}\leq_{st}Y_{2:3}$.\\\\
(ii) Let $\boldsymbol{X}\sim \mathbb{ELS}((2,4,6),(7,9,11),4;(\frac{x}{100})^{0.02},e^{\frac{1}{0.9}(1-e^x)})$ and
$\boldsymbol{Y}\sim \mathbb{ELS}((2,4,6),(2,3,5),4;$
$(\frac{x}{100})^{0.02},e^{\frac{1}{0.7}(1-e^x)})$, where $0<x\leq 100$. Clearly, $w \tilde{r}_{b}(w)$ is increasing. Note that, all the conditions of Theorem \ref{th3.9}(ii) are satisfied. The graphs of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ are depicted in Figure $4b$. This shows that $Y_{2:3}\le_{st}X_{2:3}$.
\end{example}
\begin{figure}[h]
\begin{center}
\subfigure[]{\label{c5}\includegraphics[height=2.46in]{example3_9i.eps}}
\subfigure[]{\label{c6}\includegraphics[height=2.46in]{example3_9ii.eps}}
\caption{
(a) Plots of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ as in Example \ref{exe3.3}(i). (b) Plots of ${F}_{X_{2:3}}(x)$ and ${F}_{Y_{2:3}}(x)$ as in Example \ref{exe3.3}(ii).
}
\end{center}
\end{figure}
In this part of the subsection, we concentrate on the sets of dependent samples sharing Archimedean copula with common generator.
\begin{theorem}\label{th3.10}
Suppose $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b},\psi)$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b},\psi),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}=\lambda\boldsymbol{1}_{n}$ and $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha \boldsymbol{1}_{n}$. Also, let $ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E_+}~(or ~\mathcal{D_+})$ and $w^2 \tilde{r}_{b}(w)$ be increasing in $w$. Then, ${{\boldsymbol\theta}}\succeq^{w}{{\boldsymbol\delta}}\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$, provided $\psi/\psi'$ is increasing.
\end{theorem}
\begin{proof}
Denote
\begin{eqnarray}\label{eq3.30}
\Phi_2\left({{\boldsymbol \theta}}\right)&=&F_{X_{n-1:n}}(x)\nonumber\\
&=&\sum\limits_{l=1}^{n}\psi\left[\sum\limits_{k\neq l}^{n}\phi\left\{\left[F_{b}\left(({x-\lambda}){/\theta_k}\right)\right]^{\alpha}\right\}\right]-(n-1)\psi\left[\sum\limits_{k=1}^{n}\phi\left\{\left[F_{b}\left({(x-\lambda)}{/\theta_k}\right)\right]^{\alpha}\right\}\right].\nonumber\\
\end{eqnarray}
Further, denote ${g_i=\phi\left[F_{b}\left(({x-\lambda}){/\theta_i}\right)\right]^{\alpha}},$ for $i=1,\cdots,n$.
Differentiating $\Phi_2\left({\boldsymbol \theta}\right)$ with respect to $\theta_i,$ for $i=1,\cdots,n$, we have
\begin{equation}\label{eqd3.31}
\frac{\partial\Phi_2\left({\boldsymbol \theta}\right)}{\partial \theta_i}=\eta_{i1}({\boldsymbol \theta})\eta_{1}(\theta_i),
\end{equation}
where
\begin{eqnarray*} \eta_{i1}({\boldsymbol \theta})&=&(n-1)\psi'\left(\sum\limits_{k=1}^{n}g_k\right)-\sum\limits_{l\notin\{i,j\}}^{n}\psi'\left(\sum\limits_{k\neq l}^{n}g_k\right)
-\psi'\left(g_i+\sum\limits_{k\notin\{i,j\}}^{n}\left\{g_k\right\}\right)\text{ and }\\
\eta_{1}(\theta_i)&=&\left[\frac{[w^2{\tilde{r}_b}(w)]_{w=\left({(x-\lambda)}{/\theta_i}\right)}}{x-\lambda}\right]\left[\frac{{\psi}({g_i})}{{\psi}'\left(g_i\right) }\right].
\end{eqnarray*}
To prove the required result, it is sufficient to show that $\Phi_2(\boldsymbol{\theta})$ given by (\ref{eq3.30}) is decreasing and Schur-convex with respect to $\boldsymbol{\theta}\in \mathcal{D_+}~(or~\mathcal{E_+}).$ This follows using similar arguments as in Theorem \ref{th3.8}, and hence it is omitted.
\end{proof}
In the previous theorem, we have considered the location parameters are same and fixed. The next result states that Theorem \ref{th3.10} also holds if the location parameters are taken same but vector-valued. However, the conditions will be modified a little. The proof can be completed using the arguments similar to that of Theorem \ref{th3.10}. Therefore, it is omitted for the sake of conciseness.
\begin{theorem}\label{th3.12}
Suppose $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha};F_{b},\psi)$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu},\boldsymbol{\delta},\boldsymbol{\beta};F_{b},\psi),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}$ and $\boldsymbol{\alpha}=\boldsymbol{\beta}=\alpha \boldsymbol{1}_{n}$. Also, let $\boldsymbol{\lambda},~ \boldsymbol{\theta},~\boldsymbol{\delta}\in\mathcal{E_+}~(or ~\mathcal{D_+})$ and $w \tilde{r}_{b}(w)$ be increasing in $w>0$. Then, ${{\boldsymbol\theta}}\succeq^{w}{{\boldsymbol\delta}}\Rightarrow X_{n-1:n}\leq_{st}Y_{n-1:n}$, provided $\psi/\psi'$ is increasing.
\end{theorem}
Next, we assume that the location parameters are equal and fixed. Similarly, for the scale parameters. It is shown that there exists usual stochastic order between the second-largest order statistics when the shape parameter vectors are connected with the weakly supermajorization order.
\begin{theorem}\label{th3.12.}
Assume that $\boldsymbol{X}\sim \mathbb{ELS}(
\boldsymbol{\lambda},\boldsymbol{\theta},\boldsymbol{\alpha},F_{b},\psi)$ and $\boldsymbol{Y}\sim \mathbb{ELS}(
\boldsymbol{\mu
},\boldsymbol{\delta},\boldsymbol{\beta},F_{b},\psi),$ with $\boldsymbol{\lambda}=\boldsymbol{\mu}=\lambda \boldsymbol{1}_n$ and $\boldsymbol{\theta}=\boldsymbol{\delta}=\theta\boldsymbol{1}_n$. Also, let $\boldsymbol{\alpha},~\boldsymbol{\beta}\in\mathcal{E_+}~(or~\mathcal{D_+})$. Then, ${\boldsymbol{\beta}}\succeq^{w}{\boldsymbol\alpha}\Rightarrow Y_{n-1:n}\leq_{st}X_{n-1:n}$, provided $\psi/\psi'$ is increasing.
\end{theorem}
\begin{remark}
If we take $\boldsymbol{\lambda}=\boldsymbol{\mu}=\boldsymbol{0}$, $\boldsymbol{\alpha}=\boldsymbol{\lambda},~\boldsymbol{ \beta}=\boldsymbol{\mu}$ and $\boldsymbol{\theta}=\boldsymbol{\delta}=\boldsymbol{1}_n$, then Theorem \ref{th3.12.} reduces to Theorem $6.2$ of \cite{fang2016stochastic}.
\end{remark}
The following counterexample reveals that if $\boldsymbol{\alpha},~ \boldsymbol{\beta}\notin\mathcal{E_+}~(or~\mathcal{D_+})$, then the result in Theorem \ref{th3.12.} may not hold.
\begin{counterexample}\label{cex3.3}
Let us consider two vectors $\boldsymbol{X}\sim \mathbb{ELS}(3,3,(2.5,10.5,3.1);1-exp(1-x^{0.5}),e^{-x})$ and
$\boldsymbol{Y}\sim \mathbb{ELS}(3,3,(0.5,6.5,7.5);1-exp(1-x^{0.5}),e^{-x})$, where $x\geq 1.$ Clearly, all the conditions except the restrictions taken on the vectors of the parameters are satisfied. Now,
we plot the graph of $[{F}_{X_{2:3}}(x)-{F}_{Y_{2:3}}(x)]$ in Figure $2b$. That shows that the usual stochastic order as in Theorem \ref{th3.12.} does not hold.
\end{counterexample}
\section{Conclusion}
This paper dealt with the ordering results between the second-largest order statistics, arising from two sets of exponentiated location-scale distributed data. The aim of the paper is two-fold. First, we considered the case of independent observations, and then dependent observations. The usual stochastic order and the reversed hazard rate order between the lifetimes of two $2$-out-of-$n$ systems have been obtained, when two sets of independent heterogeneous observations are available to us. We also considered two sets of heterogeneous dependent random observations. Usually, system components have dependent lifetimes due to the common environment. It has been assumed that the dependence structure is coupled by the Archimedean copulas. For the case of dependent observations, we obtained the usual stochastic order between the time to failures of two $2$-out-of-$n$ systems. Various examples and counterexamples have been considered to illustrate the established results. The results established in this paper will be helpful to the reliability theorists and practitioners to find out a better $2$-out-of-$n$ system.
\section*{Acknowledgements} Sangita Das thanks the
MHRD, Government of India for financial support. Suchandan Kayal acknowledges the partial financial support for this work under a grant MTR/2018/000350, SERB, India.
\section*{Disclosure statement}
{Both the authors states that there is no conflict of interest.}
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Special Education Teacher
Jean Adams
604 Harmony Lane
Pleasantville, CA 94588
(925) 555-1234
SUMMARY OF QUALIFICATIONS
• Current teaching certificate valid in Washington and Idaho for Elementary Education with an endorsement in Special Education.
• Quickly develops rapport with students, employees, and staff.
• Three years experience (Summers) working with handicapped individuals in a Developmental Disabilities Agency and teaching writing programs for handicapped individuals.
• Manages three employees and promotes self-directed work teams.
• Experienced with licensure surveys for Developmental Disabilities Agencies.
EDUCATION
Idaho State University, Pocatello, Idaho
1997
Bachelor of Science, Special Education
Developing Capable People Seminar, Temple Elementary
1997
Presented by Stacie Smith
Managing People with Handicaps Seminar, Temple Elementary
1996
Presented by Stacie Smith
RELEVANT EMPLOYMENT
Development at Home, Inc, Idaho Falls, Idaho
1994 - Present
Aide
• Traveled to clients' homes to teach cooking, cleaning, shopping, and budgeting.
• Assisted in writing, developing, and implementing program procedures.
• Taught life skills to clients; ensured the safety of the clients.
• Monitored facility maintenance and security
• Special project: worked with young boy, age 5, who would not speak. After nine months of intense therapy, patience, and special equipment, he began speaking broken words.
PRWRA Camile Carboneau - CC Computer Services - PO Box 50655 - Idaho Falls, Idaho 83405 - URL: - camille@ccComputer.com
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T.H.E. Journal, November 1998, p. 69
The Restructuring Educational Behaviors to Ultimately Improve Literacy in Dual-Languages (REBUILD) program has implemented videoconferencing as the primary tool to help limited English proficient students overcome obstacles to learning.
Through the use of videoconferencing, collaboration on lesson plans and teaching students at different campuses have improved test scores of students in the Gardena Complex of schools in the L.A. Unified School District.
Private technology companies have also been integral to the success of the program. Premio Computer, Lucent Technologies, and Zydacron all lent assistance, training, and support to the videoconferencing project.
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Quinlan Vos constructed his green bladed lightsaber while under the tutelage of Jedi Master Tholme. Vos' lightsaber inspired Obi-Wan Kenobi to make his own lightsaber bear a small resemblance to it beside Kenobi's own mentor, Qui-Gon Jinn. Vos wielded his lightsaber aggressively, and used it during the Yinchorri Uprising in 33 BBY, along with his master. After becoming a Jedi Knight, Vos continued to use the lightsaber, even during the Clone Wars.
Quinlan Vos wielding his lightsaber with the green blade ignited
When Vos became a double-agent for the Jedi, he brushed against the dark side of the Force, becoming one of Sith Lord Count Dooku's most trusted acolytes. After a mission to recover an ancient Sith holocron, Dooku removed the red Sith crystal from the holocron and presented it to Vos stating that it used to power Darth Andeddu's lightsaber.
After breaking free of the dark side, Vos returned to the Jedi Order, and replaced the red crystal with his original green crystal, using the weapon to kill his aunt and perform other deeds for himself and the Galactic Republic. After the bounty hunter Cad Bane freed Ziro the Hutt from prison, Vos teamed up with Obi-Wan Kenobi and headed for Nal Hutta in Hutt Space. After visiting Ziro's mother, Mama, the two headed for Teth, where they battled Bane and his droid.
During the siege of Saleucami in 19.5 BBY, Master Vos used the lightsaber to kill fallen Jedi Sora Bulq, after the latter killed Jedi Master Oppo Rancisis. After the siege, Vos was given orders to go to Kashyyyk, along with Masters Yoda and Luminara Unduli. When Supreme Chancellor Palpatine, as the Sith Master Darth Sidious, issued Order 66, Vos was still stationed on the Wookiee planet when Master Unduli was shot down and Yoda escaped the planet. Vos used his weapon to escape the Republic forces.
AppearancesEdit
- Star Wars: Republic: Twilight (First appearance)
- Star Wars: Republic: Infinity's End
- Heart of Fire (Appears in flashback(s))
- Star Wars: Republic: Darkness
- Star Wars: Republic: The Stark Hyperspace War (Appears in flashback(s))
- Star Wars 41: The Devaronian Version, Part 2 (Appears in flashback(s))
- Star Wars: Republic: Rite of)
- Star Wars: Republic: Siege of Saleucami
- Star Wars Episode III: Revenge of the Sith)}" > "Quinlan Vos: Jedi Without a Past" on Wizards.com (original article link, backup link)
- The New Essential Guide to Characters (Picture only)
- Star Wars: The Ultimate Visual Guide: Special Edition (Picture only)
- Jedi vs. Sith: The Essential Guide to the Force 137 (Picture only)
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Final Fantasy Sonic X5 - Funny Game
Final Fantasy Sonic X5 Description
Final
Game Tags
Users use these tags to search for games on our website:
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Video transcript
Use the TubeSift Bookmarker
Brady Snow: On this video, I’m going to show you how to find any YouTube ad. Why would you want to find a YouTube ad? You might want to reference it when you’re creating your own ad creatives on YouTube. Well, today we’re going to show you how you can do this. I’m also going to show you how you can use a YouTube ad spy tool to find any YouTube ad and browse a collection of YouTube ads in a specific niche category by specific advertiser from a domain, from a channel on YouTube, whatever it is, you can search the YouTube ad library using this ad spy tool. So let’s take a look at this.
So one way to find any YouTube ad is to bookmark it with the TubeSift Bookmarker tool. Let’s say I’m just browsing YouTube and an ad pops up. Let’s see if the ad comes up though. Yes, it’s an ad. So let’s say this is an ad I wanted to see. I could put bookmark ad here, and that’s using the TubeSift Bookmarker, which is a free Chrome extension. You can install it. No opt-in required just type in the TubeSift Bookmarker. And you’re going to be taken to this Chrome Web Store where you can add this. And you can go over here to where extension once you have it installed. So you can see that it’s being run by Shopify. So these are all the ads that I have bookmarked. If you don’t bookmark it, you can see the last 50 ads that you’ve been shown. So that’s one way to find it if you know you’ve just seen it and you have the TubeSift Bookmarker installed, you can just go ahead and see your ad history. Find the ad that you wanted to watch, like here’s one by MUD\WTR, it’s a health supplement beverage.
If I wanted to see that ad that’s that’s one, I can check out Motion Array. This was an ad for a motion graphic site, stock footage site that I saw. Here’s the one from LegalZoom, fiber art grid, all kinds of stuff. But when I hit bookmarks right here, I can just click it. You can also click this from the ad history where I just was. And there’s the ad. So that’s one way that you can find any YouTube ad, you just have your TubeSift Bookmarker, Chrome extension installed, and you just access that plugin right there. And access that interface, and you can see those ads easy, just like you see here. I bookmarked this one too for a video called [inaudible 00:02:19].
Use the best YouTube Ad library
So the other way to find any YouTube ad is to use a YouTube ad spy tool. This is just like an ad library. It’s similar to the Facebook ad library, but for YouTube, the one I really like because of the search features is Video Ad Vault. And I’m in my account here. Let’s say I just wanted to find those Shopify ads, because I knew that that ad I had seen was from Shopify. I can go through here and see all of the ads that they’re running. If there was a specific term in the Shopify ad, I could go through and search that too, sell online with Shopify. It might help me narrow down my ad results. Here you’re getting results from other companies as well. So if you want to see diversity of ads that are being run about Shopify offers, or maybe it’s Shopify education or software that helps you run on Shopify, you can see all those ads right there too.
Another thing I really like is if you do want is just search a specific advertiser’s domain. You can check domain right here and it does help to put in the https:/, but let’s just say I wanted to see Tubesift’s ad. That’s the company I work for here and we own Video Ad Vault. So here are ads that we’ve been running. You can see various angles, we’ve run. We run a bunch of different varieties. Another way to use this YouTube ad spy tool is to use the watch channels feature. So these are all advertisers that I’ve chosen to follow, and I can see all the ads that they roll out here. So anytime they roll out a new ad, I’ve chosen to turn on email alerts so I get notified every time in my email inbox when the new ad comes out. And I can just click and watch the ad itself.
Get unfair advantage over your competitors
So there’s lots of different ways you can use the search feature here. You can filter down, we have advanced exclusion filters where you can filter out videos that are within a certain time duration. If it’s under 15 seconds, only if you want to see that, or if you want to exclude ads that are under 15 seconds, that’s where you do that. There’s the bookmarker there, you can access that and get that installed as well, if you go to Video Ad Vault and check it out.
So those are the top ways to find any YouTube ad. Get that TubeSift Bookmarker. It’s free. We have a link in the description. You can check that out and get that installed. And if you’re watching this on the blog, we have links there as well. And then if you do want to use that YouTube ad spy tool, you can go to videoadvault.com and check that out. We have monthly or annual subscriptions. You’ll save a little bit of money with the annual and you also get access to weekly coaching sessions where you’re taught, not just how to use the YouTube ad spy tool, but gain major advantages when you’re advertising in your niche, but also about targeting, about making your ads, all kinds of different things. And it’s going to just help you dial in your YouTube ad campaigns and see massive ROI on your ad campaigns. So that’s all I’ve got, if you like this video, please subscribe. Thanks for watching. We’ll see you next time.
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Leonard Floyd has a pro future, but how soon could it begin? Photo: Scott Cunningham/Getty Images
To absolutely no one’s surprise, both Leonard Floyd and Todd Gurley have been tabbed as 2015 NFL Prospects. Despite being only a sophomore—Floyd is the NFL’s pre-requisite three-years removed from his graduating high school class—many view Floyd as a potential defensive superstar at the next level.
ESPN’s Todd McShay believes Floyd could be every bit the prospect Barkevious Mingo ($$$) was for the LSU Tigers—Mingo was eventually selected by the NFL’s Cleveland Browns (6th overall).
Said McShay:
…He explodes out of the blocks with elite get-off speed and acceleration, and he shows really good natural bend and body control. He has flashed the potential to effectively convert speed to power as a pass-rusher despite needing to improve his strength. There’s no doubt he has the speed and athleticism to handle coverage assignments as a 3-4 outside linebacker…
I don’t know that Floyd would consider making the jump after just one season under the guidance of his new position coach, but it’s clear that if he has a strong season, and opts to stay one more year, he could be a top pick come 2016—especially if he continues to put on weight and add muscle to his frame.
As for Gurley, everyone is well aware of his merits, and though McShay doesn’t feel Gurley rates as high on the board as Wisconsin’s Melvin Gordon, he does like what he’s seen from him thus far.
Gurley is a freight train. He’s a hard, downhill runner who is a load to bring down. He runs a little bit high and takes some big shots to his frame, but he’s the most powerful back we studied during the preseason. He’d be a perfect fit in a power-based rushing attack, and he compares favorably to Marshawn Lynch in that regard. But what separates Gurley from a lot of other big, punishing runners is his very good top-end speed. If he catches a crease, it’s a big problem for the defense. The only real negative with Gurley — outside of durability concerns — is his lack of ideal lateral agility. He takes a bit too long to gather before making cuts and he doesn’t show great initial acceleration off his plant foot. But in the right offense that weakness can be minimized to a certain degree.
You can bet Gurley looked to address the growing concerns about his “durability” this past off-season. And, according to position coach Bryan McClendon, Gurley is in the best shape of his career, so this could be the year everyone sees what Gurley can do when he’s not hampered by aches and pains.
No doubt Gurley’s physical prowess will be tested this season—as teams look to make senior quarterback Hutson Mason beat them by loading up the box and attacking the run game—but you can be sure he understands the payoff of doing well under such enormous pressure; it’s his year to truly show he can be a wire-to-wire player for the Bulldogs, and if he brings his ‘A’ game every week, the Bulldogs will definitely be in the thick of things come playoff time.
2 comments for “Georgia Football: Leonard Floyd and Todd Gurley Considered “Top NFL Prospects””
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: hill of tara […] […]
Sunday 30th May – Hill of Tara, County Meath. Ireland. This is the final post relating to my chakra work in Ireland in late May this year. It has been a long haul for anyone who’s been reading them all! I appreciate you taking the time to stick with it. If you want to read […]
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Head of Digital
1 day left
- Location
- Hertfordshire
- Salary
- £60k - £110k pa + Extensive package
- Posted
- 12 Mar 2019
- Closes
- 19 Mar 2019
- Ref
- 1631374
- Anna Jacobs
- Sector
- Financial Services
- Job Function
- Strategy and Planning
Demonstrable skillset and mind set required to succeed:
- Previous responsibility for driving digital initiatives within a programme of change as well as Business As Usual
- Ability to continuously monitor response and engagement rates, making recommendations as appropriate
- Accountability for leading and delivering digital marketing strategy and plans across all areas and channels (website, CRM, online content, social media, tech stack) of the business
- Able to provide digital thought leadership
- Analysis and monitoring of digital activity to ensure optimal usage of all channels
- Strong relationship management skills: ability to work with an array of internal stakeholders and external third parties. Acting as subject matter expert and partner is ensuring optimal delivery.
- You will have a forward thinking approach to the ongoing development of the digital marketing strategy. This is a newly created role, there is scope for the person to influence the ongoing evolution of this post and their contribution to the organisation.
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TITLE: Solve for X in this equation of sets.
QUESTION [0 upvotes]: Taking U to be the universe of discourse, let R be the set of subsets of U and define the operations +, *, for R by
$A + B \equiv A \cup B - A \cap B$
$A \circ B \equiv A \cap B$
for subsets A, B of U. Define $0 \equiv \emptyset$ and $e \equiv U$. Then $(R,0,e,+,\circ)$ is a ring.
Compute the solutions X to the equations below using only $A, B, +, \circ$
i. $A + X = 0$
ii. $A + X = B$
iii. $X = (A-B)(A+B)$
I understand that X in number i. should be A, and I found in ii. that X can be written as $B - A \cup (B' \cap A)$, but I can't write it in terms of $A,B,+,\circ$ like the question asks.
REPLY [2 votes]: How do we solve an equation of the form $a+x=b$ in the real numbers? We add $-a$ to both sides to get $x=b-a$. What does "$-a$" mean? $-a$ is the unique real number that, when added to $a$, will give you $0$.
So in essence, what we are doing is:
$$\begin{align*}
a+x &= b &\text{(assumption)}\\
-a+(a+x) &= -a+b&\\
(-a+a)+x &= -a+b &(+\text{ is associative)}\\
0+x &= -a+b&\text{(definition of }-a\text{)}\\
x & = -a+b &\text{(property of }0\text{)}
\end{align*}$$
We do the same thing in (i) and (ii). Only... what is "$-A$"? It's the unique subset of $U$ such that $A+A=\varnothing$.
So... what is the element which, when added to $A$, gives $\varnothing$? Once you find it, you can solve both $A+X=0$ and $A+X=B$, in terms of $A$, $B$, and $0$.
For (iii), you need to remember that $A-B$ means $A+(-B)$; so again, the first step is to figure out what is $-B$ in terms of $B$. Once you do, you can compute $A-B$, and compute $A+B$ (using the definition of $+$), and then compute their product (aka their intersection). This will give you $X$ in terms of $A$ and $B$.
Added. Henning raises a good point in comments: that in (iii), the $-$ on the right hand side might refer not to the $-$ operation in the ring, but rather to the set-theoretic difference of $A$ and $B$. This would be bad use of potentially confused and confusing notation; if that's what it means, then you would first compute $A\setminus B = \{a\in A\mid a\notin B\}$, then compute the "product" (intersection) with $A+B$ (aka their symmetric difference) and obtain an expression for $X$ from that.
REPLY [1 votes]: Hint: Draw Venn diagrams of $A+B$ and $A \circ B$. What happens when you add either of these to $A$?
And these operations are commutative, right? And we've already discovered that $A+A=0$, so what is $A+(A+B)$, I wonder....
Of course, you'll need to show that $+$ is associative--ie $A+(B+C)=(A+B)+C$ for all $A,B,C\in R$--which is doable (with a lot of element chasing).
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\begin{document}
\maketitle
\begin{abstract}
We determine the Jordan constants of groups $\mathrm{GL}_2(K)$, $\mathrm{SL}_2(K)$, $\mathrm{PGL}_2(K)$ and $\mathrm{PGL}_3(K)$ for any given field $K$ of characteristic 0.
\end{abstract}
\section{Introduction}
One of the commonly used approaches to study an infinite group is to study its finite subgroups. However, sometimes we want to avoid classifying all of its finite subgroups thoroughly because it may be too complicated. To do this, we will find the following definition useful, which originally comes from~\mbox{\cite[Definition 2.1]{popov}}. It measures to which extent a group is ``similar" to an abelian group.
\begin{definition}\label{def1.1}
For an arbitrary group $\Gamma$, its Jordan constant is defined as the minimal integer $J(\Gamma)$ such that any finite subgroup of $\Gamma$ has an abelian normal subgroup with index not exceeding $J(\Gamma)$.
\end{definition}
Throughout this paper, we work over a field $K$ of characteristic 0, unless stated otherwise. We use $\omega$ to denote a third primitive root of unity.
Our main result deals with the Jordan constant of $\mathrm{PGL}_3(K)$.
\begin{theorem}\label{thm1.4}
The Jordan constant of $\mathrm{PGL}_3(K)$ can attain only the following values: 360, 168, 60, 24, 12, 6. More explicitly:
(i) $J(\mathrm{PGL}_3(K))=360$ if and only if $\omega\in K$ and $\sqrt{5}\in K$;
(ii) $J(\mathrm{PGL}_3(K))=168$ if and only if at least one of $\omega$ and $\sqrt{5}$ is not in $K$, and $\sqrt{-7}\in K$;
(iii) $J(\mathrm{PGL}_3(K))=60$ if and only if $\sqrt{5}\in K$, but neither $\omega$ nor $\sqrt{-7}$ is in~$K$;
(iv) $J(\mathrm{PGL}_3(K))=24$ if and only if $\sqrt{5}\not\in K$, $\sqrt{-7}\not\in K$ and
\hspace{20pt} (a) either $\omega\in K$,
\hspace{20pt} (b) or $-1$ is the sum of two squares in $K$, and there exists $\zeta\in K$ such
\hspace{37pt} that $2\zeta^2$ is a root of unity;
(v) $J(\mathrm{PGL}_3(K))=12$ if and only if $-1$ is the sum of two squares in $K$, none of $\omega, \sqrt{5}$, and $\sqrt{-7}$ lies in $K$, and for any element $\zeta\in K$, $2\zeta^2$ is not a root of unity;
(vi) $J(\mathrm{PGL}_3(K))=6$ if and only if $-1$ is not the sum of two squares in $K$, and none of $\omega, \sqrt{5}$, and $\sqrt{-7}$ lies in $K$.
\end{theorem}
\begin{example}\label{eg1.5}
We can consider the most popular fields:
$$J(\mathrm{PGL}_3(\mathbb{C}))=360,\ J(\mathrm{PGL}_3(\mathbb{R}))=60,\ J(\mathrm{PGL}_3(\mathbb{Q}))=6.$$
\end{example}
To prove Theorem 1.2, we find the necessary and sufficient conditions for the group $\mathrm{PGL}_3(K)$ to contain a given finite primitive subgroup (see Definition \ref{def4.1}). This may be considered as a partial generalization of \cite[Proposition 1.1]{Beauville}, where the necessary and sufficient conditions for the existence of a given finite subgroup in $\mathrm{PGL}_2(K)$ are found. We expect that our results can be used to determine the possible Jordan constant of the group of birational automorphisms of $\mathbb{P}^2$ over any field $K$ of characteristic 0, which is currently known only over $\mathbb{C}, \mathbb{R}$ and $\mathbb{Q}$, see \cite{Yasinsky}.
As an auxiliary step of the proof of the main result, Jordan constant of~\mbox{$\mathrm{GL}_2(K)$} is determined in Section 3. We are also able to determine Jordan constants of $\mathrm{SL}_2(K)$ and $\mathrm{PGL}_2(K)$ as by-products of the proof of~\mbox{Theorem \ref{thm1.3}}.
\begin{theorem}\label{thm1.3}
The Jordan constant of $\mathrm{GL}_2(K)$ can attain only the following values: 60, 24, 12, 2. More explicitly:
(i) $J(\mathrm{GL}_2(K))=60$ if and only if $-1$ is the sum of two squares in $K$ and~\mbox{$\sqrt{5}\in K$};
(ii) $J(\mathrm{GL}_2(K))=24$ if and only if $-1$ is the sum of two squares in~$K$,~\mbox{$\sqrt{5}\not\in K$} and there exists $\zeta\in K$ such that $2\zeta^2$ is a root of unity;
(iii) $J(\mathrm{GL}_2(K))=12$ if and only if $-1$ is the sum of two squares in~$K$,~\mbox{$\sqrt{5}\not\in K$} and for any element $\zeta\in K$, $2\zeta^2$ is not a root of unity;
(iv) $J(\mathrm{GL}_2(K))=2$ if and only if $-1$ is not the sum of two squares in $K$.
\end{theorem}
\begin{corollary}\label{3.10}
The Jordan constant of $\mathrm{SL}_2(K)$ can attain only the following values: 60, 24, 12, 2. More explicitly:
(i) $J(\mathrm{GL}_2(K))=60$ if and only if $-1$ is the sum of two squares in $K$ and~\mbox{$\sqrt{5}\in K$};
(ii) $J(\mathrm{GL}_2(K))=24$ if and only if $-1$ is the sum of two squares in~$K$,~\mbox{$\sqrt{5}\not\in K$} and either $\sqrt{2}\in K$ or $\sqrt{-2}\in K$;
(iii) $J(\mathrm{GL}_2(K))=12$ if and only if $-1$ is the sum of two squares in $K$, and none of $\sqrt{5}, \sqrt{2}$ and $\sqrt{-2}$ is in $K$;
(iv) $J(\mathrm{GL}_2(K))=2$ if and only if $-1$ is not the sum of two squares in $K$.
\end{corollary}
\begin{proposition}\label{thm1.2}
The Jordan constant of $\mathrm{PGL}_2(K)$ can attain only the following values: 60, 6, 2. More explicitly:
(i) $J(\mathrm{PGL}_2(K))=60$ if and only if $-1$ is the sum of two squares in $K$ and~\mbox{$\sqrt{5}\in K$};
(ii) $J(\mathrm{PGL}_2(K))=6$ if and only if $-1$ is the sum of two squares in $K$ and~\mbox{$\sqrt{5}\not\in K$};
(iii) $J(\mathrm{PGL}_2(K))=2$ if and only if $-1$ is not the sum of two squares in $K$.
\end{proposition}
\begin{remark}
In some cases the criterion that $-1$ is the sum of two squares in $K$ holds automatically. For example, in claim (ii) of Theorem \ref{thm1.3}, if $2\zeta^2$ is a root of unity whose order $n$ is divisible by $4$, then $-1=x^2+x^2$, where~\mbox{$x=(2\zeta^2)^{\frac{n}{4}}$}. Moreover, in claim (ii) of Corollary \ref{3.10}, one has $-1=(\sqrt{-2})^2+1^2$ when~\mbox{$\sqrt{-2}\in K$}.
\end{remark}
The plan of the paper is as follows. We compute the Jordan constants of several finite groups in Section 2, based on which the Jordan constant of $\mathrm{GL}_2(K)$ is determined in Section 3. In Section 4, we classify all types of finite subgroups in $\mathrm{PGL}_3(K)$, and the criterion of the existence of each primitive subgroup is given in Section 5. Finally, the main result is proved in Section 6.
\noindent \textbf{Notation}: Throughout this essay, the cyclic group of order $r$ is denoted by~$C_r$; the dihedral group of order $2r$ is denoted by $D_r$; the finite field of order $q$ is denoted by $\mathbb{F}_q$; the algebraic closure of a field $K$ is denoted by $\bar{K}$; and the identity matrix is denoted by $I$.
\section{Preliminaries}
In this section, we compute the Jordan constants of several groups which will appear as subgroups of $\mathrm{PGL}_2(K)$ or $\mathrm{GL}_2(K)$.
\begin{lemma}\label{lem2.1}
(i) $J(S_n)=n!$, $J(A_n)=\frac{n!}{2}$, for $n\geq 5$.
(ii) $J(S_4)=6$, $J(A_4)=3$.
(iii) $J(D_n)=2$ for $n>2$.
\end{lemma}
\begin{proof}
For claim (i), it's well-known that $A_n$ is a simple group when $n\geq 5$, so~\mbox{$J(A_n)=|A_n|$}. Also, $A_n$ is a normal subgroup of $S_n$. If $S_n$ contains some other proper normal subgroup $N$, then $N\cap A_n$ is a normal subgroup of $A_n$, which implies that the intersection is either $\{1\}$ or $A_n$. The latter case implies~\mbox{$N=A_n$}. The previous case tells us $N\cap [S_n,S_n]=\{1\}$, hence $N$ is included in the center of $S_n$, which is trivial. That is, $S_n$ has no non-trivial normal subgroups other than $A_n$, while $A_n$ is non-abelian.
For claim (ii), the subgroup $$N=\{(1), (12)(34), (13)(24), (14)(23)\}$$ is abelian and normal in $A_4$. Note that every group of order 6 is either $C_6$ or symmetric group $S_3$. Clearly $A_4$ does not contain $C_6$. Also $A_4$ does not contain~$S_3$, as $S_3$ has exactly 3 order-$2$ elements, none of which commutes with the others. So $J(A_4)=3$.
Now let's consider normal subgroups of $S_4$. First of all, $N$ is also abelian and normal in $S_4$. Furthermore, a subgroup is normal only if it's a union of conjugacy classes and contains $\{1\}$. Conjugacy classes of $S_n$ are determined by cycle type, so it's easy to observe that the conjugacy classes in $S_4$ have cardinality 1, 3, 6 or 8. Therefore $S_4$ cannot have normal subgroup of order 6 or 8, and the only normal subgroup of order 12 is $A_4$, which is non-abelian.
For the last claim, note that $C_n\mathrel{\unlhd} D_n$, and $D_n$ is non-abelian when $n>2$.
\end{proof}
\begin{lemma}\label{lem2.2}
Let $G$ be a central extension of $D_n$ by a finite cyclic subgroup $C_r$, where $n\geq 3$, then $J(G)=2$.
\end{lemma}
\begin{proof}
We have a short exact sequence
$$1\rightarrow C_r\mathop{\rightarrow}^i G\mathop{\rightarrow}^{\pi} D_n\rightarrow 1.$$
Let $C_n\mathrel{\unlhd} D_n$ be the abelian normal subgroup of $D_n$, then $\pi^{-1}(C_n)$ is abelian and normal in $G$, with index 2. Note that $G$ is not abelian as $D_n$ is not abelian when $n\geq 3$, hence $J(G)=2$.
\end{proof}
\begin{lemma}\label{lem2.4}
Let $G$ be a nontrivial central extension of $A_5$ by a finite cyclic subgroup $C_r$, then $J(G)=60$.
\end{lemma}
\begin{proof}
Note that a maximal normal abelian subgroup of $G$ always contains~$C_r$, and it can be projected to a normal abelian subgroup of $A_5$, which is trivial. Therefore the maximal normal abelian subgroup of $G$ is exactly $ C_r$, and~\mbox{$J(G)=|A_5|=60$}.
\end{proof}
\section{Jordan constants of $\mathrm{PGL}_2(K)$ and $\mathrm{GL}_2(K)$}
We determine the Jordan constant of $\mathrm{GL}_2(K)$ in this section. The result is needed when we calculate the Jordan constants of finite intransitive subgroups of $\mathrm{PGL}_3(K)$. Proofs of Proposition \ref{thm1.2} and Corollary \ref{3.10} are also given in this section.
\begin{theorem}\label{3.1}
For an algebraically closed field $K$ of characteristic 0, a finite subgroup $G$ of $\mathrm{PGL}_2(K)$ must be of one of the following 5 types: $C_r$, $D_r$, alternating groups $A_4, A_5$ and symmetric group $S_4$.
\end{theorem}
\begin{proof}
We refer to \cite[Chapter X, Section 101-103]{Mi-Bl-Di} for determination of all finite subgroups of $\mathrm{SL_2}(K)$, and then consider the canonical projection to $\mathrm{PGL}_2(K)$.
\end{proof}
\begin{theorem}\label{3.2}
(i) $\mathrm{PGL}_2(K)$ contains $C_r$ and $D_r$ if and only if $K$ contains~\mbox{$\alpha+ \alpha^{-1}$} for some primitive $r$-th root of unity $\alpha$.
(ii) $\mathrm{PGL}_2(K)$ contains $A_4$ and $S_4$ if and only if $-1$ is the sum of two squares in $K$.
(iii) $\mathrm{PGL}_2(K)$ contains $A_5$ if and only if $-1$ is the sum of two squares and~\mbox{$\sqrt{5}\in K$}.
\end{theorem}
\begin{proof}
See \cite[Proposition 1.1]{Beauville}.
\end{proof}
Now we prove Proposition \ref{thm1.2}.
\begin{proof}
There are five types of subgroups $\mathrm{PGL}_2(K)$, listed in Theorem \ref{3.1}. Note that $A_4$ and $S_4$ appear in $\mathrm{PGL}_2(K)$ simultaneously, according to Theorem \ref{3.2}, and by Lemma \ref{lem2.1}, one has $J(S_4)>J(A_4)$. Hence we only consider $S_4$.
From Lemma \ref{lem2.1}, we know that the largest possible Jordan constant is~$J(A_5)=60$, and the criterion for the existence of $A_5$ in $\mathrm{PGL}_2(K)$ is given in Theorem \ref{3.2}, thus (i) is proved.
If $A_5$ does not exist in $\mathrm{PGL}_2(K)$, we will then consider $S_4$, thus (ii) can be proved again by Lemma \ref{lem2.1} and Theorem \ref{3.2}.
If $S_4$ does not exist in $\mathrm{PGL}_2(K)$ either, then finite subgroups of $\mathrm{PGL}_2(K)$ are all of the form $C_r$ or $D_r$, whose Jordan constants are at most $2$. It's always true that $\omega^{-1}+\omega=-1\in K$, hence $D_3$ is a subgroup of $\mathrm{PGL}_2(K)$ by~\mbox{Theorem \ref{3.2}}. Therefore $J(\mathrm{PGL}_2(K))=J(D_3)=2$, and (iii) is proved.
\end{proof}
\begin{proposition}\label{3.3}
Let $a,b\in K$ such that $-1=a^2+b^2$, consider the following matrices in $\mathrm{PGL}_2(K)$:
$$A=\begin{pmatrix}
-a & b\\
b &a
\end{pmatrix},\
B=\frac{1}{2}\begin{pmatrix}
-1+a+b & -1+a-b\\
1+a-b & -1-a-b
\end{pmatrix},\
C=\begin{pmatrix}
-a+1 & b\\
b &a+1
\end{pmatrix}.$$
Then $A$ and $B$ generate $A_4$, while $B$ and $C$ generate $S_4$. Moreover, any subgroup of $\mathrm{PGL}_2(K)$ which is isomorphic to $A_4$ is conjugate to $\left\langle A,B\right\rangle$, and any subgroup of $\mathrm{PGL}_2(K)$ which is isomorphic to $S_4$ is conjugate to $\left\langle B,C\right\rangle$.
\end{proposition}
\begin{proof}
It's known that $$A_4\cong\left\langle x,y\mid x^3=y^2=(xy)^3=1\right\rangle,$$ according to \cite[p138, Beispiel 19.8]{Huppert}. Here $x=(123), y=(12)(34)$.
We take $x=B, y=A$, then all relations are satisfied. Therefore $A$ and $B$ will generate a group which is a quotient of $A_4$, which is either $A_4$ or~\mbox{$A_4/C_2\times C_2\cong C_3$}. The latter case is impossible, because $y=A$ is an order-2 element, which does not exist in $C_3$. As a result, $\langle A, B\rangle\cong A_4$.
Also we have $$S_4\cong\left\langle x,y,z\mid x^2=y^2=z^2=(xy)^3=(yz)^3=(zx)^2=1\right\rangle,$$ according to \cite[p138, Beispiel 19.7]{Huppert}. Here $x=(14), y=(24), z=(23)$.
We take $x=CB, y=BC$, and $z=C^2B^2C$, then all relations are satisfied. Therefore they will generate a group which is a quotient of $S_4$, which is one of $S_4$,~\mbox{$S_4/A_4\cong C_2$} or $S_4/C_2\times C_2\cong S_3$. Note that $C=zyx, B=yC^{-1}$ and~\mbox{$A=C^2$}, so the group contains $A$ and $B$, hence the subgroup generated by $A$ and $B$, which is $A_4$, thus our group must be $S_4$. Additionally, $\langle x,y,z\rangle=\langle B, C\rangle$, so $B$ and $C$ generate $S_4$.
Finally, from \cite[Theorem 4.2]{Beauville}, we know that $\mathrm{PGL}_2(K)$ contains only one conjugacy class of subgroups isomorphic to $A_4$, and one conjugacy class of subgroups isomorphic to $S_4$.
\end{proof}
Now we study finite subgroups of $\mathrm{GL}_2(K)$. Let $\pi:\mathrm{GL}_2(K)\rightarrow \mathrm{PGL}_2(K)$ be the canonical projection.
\begin{corollary}\label{3.5}
Let $G$ be a finite subgroup of $\mathrm{GL}_2(K)$, then $G$ is a central extension of $C_r$, $D_r$, $A_4$, $S_4$ or $A_5$ by a finite cyclic subgroup.
\end{corollary}
\begin{proof}
Let $H\subset G$ be the subgroup of $G$ consisting of all scalar matrices, then it's finite and cyclic. After applying $\pi$ to $G$, the claim will be an obvious corollary of Theorem \ref{3.1}.
\end{proof}
\begin{lemma}\label{3.6}
Let $n\geq 2$ be an integer, and $r$ is an integer coprime to $n$. If there is a cyclic subgroup of order $r$ in $\mathrm{PGL}_n(K)$, then there is a cyclic subgroup of the same order in $\mathrm{SL}_n(K)$.
\end{lemma}
\begin{proof}
We can choose integers $u, v$ such that $un+vr=1$ since~\mbox{$\gcd (r,n)=1$}. Assume $A$ is a matrix in $\mathrm{GL}_n(K)$ such that $A^r=aI, a\in K^{\times}$, then we have~\mbox{$\det(A)^r=a^n$}. Consider $$B= \frac{1}{\det(A)^ua^v}A,$$ then $B^r=I$.
\end{proof}
\begin{lemma}\label{5.5}
Let $n\geq 2$ be an integer, and $r$ is an integer coprime to $n$. Then for any finite subgroup $G\subset\mathrm{PGL}_n(K)$ which is generated by its elements of order $r$, there exists a finite subgroup $\Tilde{G}\subset \mathrm{SL}_n(K)$ such that $\pi(\Tilde{G})=G$. If~\mbox{$|\Tilde{G}|=n|G|$}, then $K$ contains a primitive $n$-th root of unity.
\end{lemma}
\begin{proof}
Pick a set of generators of $G$, consisting of elements of order $r$. According to Lemma \ref{3.6}, each element in the set can be lifted to $\mathrm{SL}_n(K)$. Then let~\mbox{$\Tilde{G}\subset\mathrm{SL}_n(K)$} be the subgroup generated by those liftings.
\end{proof}
\begin{proposition}\label{3.4}
Let $G$ be a finite subgroup of $\mathrm{GL}_2(K)$.
(i) If $\pi(G)\cong A_4$, then $J(G)=12$.
(ii) If $\pi(G)\cong S_4$, then $J(G)=24$.
\end{proposition}
\begin{proof}
Let's first assume that $\pi(G)=\left\langle A,B\right\rangle$ or $\left\langle B, C\right\rangle$, where $A, B$ and $C$ are given in Proposition \ref{3.3}. Let $N\cong C_2\times C_2$ be the unique nontrivial abelian normal subgroup of both $A_4$ and $S_4$. Then $g_1=A$, and $$g_2=\begin{pmatrix}
0 & 1\\
-1 &0
\end{pmatrix}=BAB^{-1}$$
give a pair of generators of $N$.
For any lifting $h_1\in G$ of $g_1$, and $h'\in G$ of $B$, $h_2=h'h_1h'^{-1}$ will be a lifting of $g_2$, and it does not commute with $h_1$.
From Corollary \ref{3.5} we know that $G$ is a central extension of $A_4$ or $S_4$ by a finite cyclic subgroup $C_r$, which consists of all the scalar matrices in $G$. Note that a maximal abelian normal subgroup of $G$ must contain $C_r$. Let $H\subset G$ be such a subgroup. Then $\pi(H)$ is an abelian normal subgroup in $A_4$ or $S_4$, so $\pi(H)=\{1\}$ or $N$. If $\pi(H)=N$, then all liftings of $g_1$ and $g_2$ lie in $H$. In particular, we have $h_1\in H$ and $h_2\in H$. However, these two elements do not commute. The contradiction implies that $\pi(H)=\{1\}$. Hence $H=C_r$, and $$J(G)=\frac{|G|}{r}=|\pi(G)|.$$
Now we consider any $G$ such that $\pi(G)\cong A_4$ or $S_4$. By Proposition \ref{3.3} we know that $\pi(G)$ is conjugate to $\left\langle A,B\right\rangle$ or $\left\langle B, C\right\rangle$. Then our $G$ is conjugate to the subgroup we discussed in above paragraphs, by the same conjugation. Note that conjugation does not change Jordan constant.
\end{proof}
\begin{proposition}\label{3.7}
We have a finite subgroup $G\subset \mathrm{GL}_2(K)$ such that $\pi(G)\cong A_4$ if and only if $-1$ is the sum of two squares in $K$. If this is the case, then~\mbox{$J(G)= 12$}.
\end{proposition}
\begin{proof}
If $\pi(G)\cong A_4$, then $\mathrm{PGL}_2(K)$ contains $A_4$. Then $-1$ is the sum of two squares in $K$, by claim (ii) in Theorem \ref{3.2}.
Conversely, if $-1$ is the sum of two squares in $K$, then $\mathrm{PGL}_2(K)$ contains~$A_4$, again according to Theorem \ref{3.2}. Note that $A_4$ is generated by its order-3 elements, so we can lift all of them to $\mathrm{SL}_2(K)$ by Lemma \ref{5.5}. Let $G$ be the group generated by those lifted elements.
Finally, we obtain $J(G)=12$ according to Proposition \ref{3.4}.
\end{proof}
\begin{proposition}\label{3.8}
We have a finite subgroup $G\subset \mathrm{GL}_2(K)$ such that $\pi(G)\cong S_4$ if and only if $-1$ is the sum of two squares in $K$,and there exists $\zeta\in K$ such that $2\zeta^2\in K$ is a root of unity. If this is the case, then $J(G)=24$.
\end{proposition}
\begin{proof}
Suppose such $G$ exists, then $\mathrm{PGL}_2(K)$ contains $S_4$, therefore $-1$ is the sum of two squares in $K$, by claim (ii) in Theorem \ref{3.2}. Moreover, from Proposition \ref{3.3}, we know that $\mathrm{PGL}_2(K)$ contains only one conjugacy class of subgroups isomorphic to $S_4$. So we may assume it is generated by the matrices $B$ and~$C$ given in Proposition \ref{3.3}, without loss of generality. In particular, there is a matrix $g\in G$ such that $\pi(g)=C$. Assume $g=\zeta C$, then $\det(g)=2\zeta^2$ is a root of unity, as $|G|<\infty$.
Conversely, if $-1$ is the sum of two squares in $K$, we can firstly construct~$S_4$ in~$\mathrm{PGL}_3(K)$ via Proposition \ref{3.3}. Note that $B$ itself is an order-3 matrix in~$\mathrm{GL}_2(K)$. If $2\zeta^2$ is an $l$-th primitive root of unity in $K$, let $g=\zeta C$, and $G$ be the subgroup generated by $B$ and $g$. We observe that $\pi(G)\cong S_4$, and~\mbox{$g^{8l}=I$}. Let $$\mathrm{SL}^{24l}(K)= \{h\in\mathrm{GL}_2(K)\mid \ \det(h)^{24l}=1\}.$$
Let $\pi_{24l}$ be the restriction of $\pi$ to $\mathrm{SL}^{24l}(K)$, then $G\subset \pi_{24l}^{-1}(S_4)$, therefore $G$ is finite.
Finally, we obtain $J(G)=24$ according to Proposition \ref{3.4}.
\end{proof}
\begin{proposition}\label{3.9}
We have a finite subgroup $G\subset \mathrm{GL}_2(K)$ such that~\mbox{$\pi(G)\cong A_5$} if and only if $-1$ is the sum of two squares in $K$ and $\sqrt{5}\in K$. If this is the case, then $J(G)=60$.
\end{proposition}
\begin{proof}
If $\pi(G)\cong A_5$, then $\mathrm{PGL}_2(K)$ contains $A_5$. Then $-1$ is the sum of two squares in $K$, and $\sqrt{5}\in K$, by claim (iii) in Theorem \ref{3.2}.
Conversely, if $-1$ is the sum of two squares in $K$ and $\sqrt{5}\in K$, then $\mathrm{PGL}_2(K)$ contains $A_5$, again according to Theorem \ref{3.2}. Note that $A_5$ is generated by its order-3 elements, so we can lift all of them to $\mathrm{SL}_2(K)$ by Lemma \ref{5.5}. Let $G$ be the group generated by those lifted elements.
Finally, we obtain $J(G)=60$ according to Lemma \ref{lem2.4}.
\end{proof}
Now we are ready to prove Theorem \ref{thm1.3}.
\begin{proof}
If a finite subgroup $G\subset \mathrm{GL}_2(K)$ is a central extension of $C_r$ by a finite cyclic subgroup, then $G$ is still abelian, hence $J(G)=1$.
If $G$ is a central extension of $D_r$ by a finite cyclic subgroup, where $r>2$, then $J(G)=2$, by Lemma \ref{lem2.2}. Moreover, such subgroup always exists: $D_3\cong S_3$ has an obvious 2-dimensional irreducible faithful representation.
Let $G$ be a finite subgroup of $\mathrm{GL}_2(K)$, with $\pi(G)=A_4, S_4$ or $A_5$, then~$J(G)$ equals to $12, 24$ or $60$, respectively. We start from the case which give the largest possible Jordan constant. Then claim (i) is obtained by Proposition \ref{3.9}. If conditions in claim (i) are not satisfied, we may search for $\pi(G)=S_4$, then claim (ii) comes from Proposition \ref{3.8}. Furthermore, if conditions in both claim (i) and (ii) do not hold, we now search for $G$ such that $\pi(G)=A_4$, which is claim (iii), and it can be proved by referring to Proposition \ref{3.7}. Finally, if $\pi(G)$ cannot be any one of $A_4, S_4$ and $A_5$, then it should be $D_r$ or $C_r$. We already show that there always exists $G$ such that $\pi(G)=D_3$, with $J(G)=2$, so we come to claim (iv).
\end{proof}
Finally we provide a short proof of Corollary \ref{3.10}.
\begin{proof}
Note that in the proof of Theorem \ref{thm1.3} all the lifting matrices lie in~$\mathrm{SL}_2(K)$, except the lifting of the matrix $C$, which is one of the generators of~$S_4$, defined in Proposition \ref{3.3}. Assume that we have a lifting $\zeta C$ in $\mathrm{SL}_2(K)$, then $\zeta\in K$, and $1=\det (\zeta C)=2\zeta^2$. So $\zeta$ should be $\frac{1}{\sqrt{2}}$ or $\frac{1}{\sqrt{-2}}$, which is equivalent to say either $\sqrt{2}\in K$ or $\sqrt{-2}\in K$.
\end{proof}
\section{Finite subgroups in $\mathrm{PGL}_3(K)$}
In this section, we recall the concept of primitive subgroups, referring to \cite[Chapter 1, Definition 1.1]{Yau-Yu} and \cite[Chapter XI, Section 106]{Mi-Bl-Di}. We also list all types of finite subgroups in $\mathrm{PGL}_3(K)$.
\begin{definition}\label{def4.1}
We regard elements in $\mathrm{GL}_n(K)$ as linear automorphisms of a vector space $V=K^n$. Consider a subgroup $G$ of $\mathrm{GL}_n(K)$.
1. We say $G$ is intransitive if we can decompose the vector space $V$ into a direct sum of more than one subspaces $V=\bigoplus\limits_{i}V_i$, such that $g(V_i)=V_i$, for all $i$ and all $g\in G$. If such a decomposition does not exist, then we say $G$ is transitive.
2. We say $G$ is imprimitive, if it's intransitive, and we can decompose the the vector space $V$ into a direct sum of more than one subspaces $V=\bigoplus\limits_{i}V_i$, such that for each $i$ and each $g\in G$, $g(V_i)\subset V_j$ for some $j$.
3. We say $G$ is primitive, if it's neither transitive, nor imprimitive.
\end{definition}
\begin{theorem}\label{4.2}
For an algebraically closed field $K$ of characteristic 0, a finite subgroup $G$ of $\mathrm{SL}_3(K)$ is conjugate to a group of one of the following 12 types:
(A) diagonal group;
(B) group of the form $$\left\{\begin{pmatrix}
g & 0\\
0 & \det(g)^{-1}
\end{pmatrix}\mid g\in \Tilde{G}\right\},$$ where $\Tilde{G}\cong G$ is a finite subgroup of $\mathrm{GL}_2(K)$;
(C) group generated by group of type (A) and $T$;
(D) group generated by group of type (C) and $R_{a,b,c}$ for some $a,b$ and $c$ in~$K$;
(E) group of order 108 generated by S, T and V;
(F) group of order 216 generated by (E) and $UVU^{-1}$;
(G) group of order 648 generated by (E) and $U$;
(H) simple group of order 60 isomorphic to the alternating group $A_5$;
(I) simple group of order 168 isomorphic to $\mathrm{PSL}_2(\mathbb{F}_7)$;
(J) group of order 180 generated by (H) and F;
(K) group of order 504 generated by (I) and F;
(L) group of order 1080 with quotient $G/F$ isomorphic to the alternating group $A_6$.
Here
\begin{align*}
T=\begin{pmatrix}
0 & 1 & 0\\
0 & 0 & 1 \\
1 & 0 & 0
\end{pmatrix},\
S&=\begin{pmatrix}
1 & 0 & 0\\
0 & \omega & 0 \\
0 & 0 & \omega^2
\end{pmatrix}, \\
V=\frac{1}{\sqrt{-3}}\begin{pmatrix}
1 & 1 & 1\\
1 & \omega & \omega^2\\
1 & \omega^2 & \omega
\end{pmatrix},\
U&=\begin{pmatrix}
\epsilon & 0 & 0\\
0 & \epsilon & 0\\
0 & 0 & \epsilon\omega
\end{pmatrix},\
R_{a,b,c}=\begin{pmatrix}
a & 0 & 0\\
0 & 0 & b \\
0 & c & 0
\end{pmatrix}
\end{align*} with $abc=-1$.
Furthermore, $F=\{I,\omega I,\omega^2 I\}$ is the center of $\mathrm{SL}_3(K)$, and~\mbox{$\epsilon^3=\omega^2$}.
\end{theorem}
\begin{proof}
See \cite[Chapter XII]{Mi-Bl-Di} and \cite[Chapter 1]{Yau-Yu} for the case $K=\mathbb{C}$. For an arbitrary field $K$, note that every finite subgroup of $\mathrm{SL}_3(K)$ can be embedded into $\mathrm{SL}_3(\mathbb{C})$.
\end{proof}
\begin{remark}\label{4.3}
Here groups of type (A) and (B) are intransitive; groups of type~(C) and (D) are imprimitive; and all the remaining groups are primitive.
\end{remark}
Consider the natural projection $\pi:\mathrm{SL}_3(\bar{K})\rightarrow \mathrm{PGL}_3(\bar{K})$. In the sequel, we adopt the following notation: we say a group is of some type in $\mathrm{PGL}_3(K)$, for example, of type (E), when the group is isomorphic to $\pi(G)$ for some finite subgroup $G\subset \mathrm{SL}_3(\bar{K})$ of type (E).
\begin{corollary}\label{4.4}
A finite subgroup $G$ of $\mathrm{PGL}_3(K)$ is isomorphic to a group of one of the following 10 types:
(A) abelian group;
(B) finite subgroup of $\mathrm{GL}_2(K)$;
(C) group generated by group of type (A) and $T$;
(D) group generated by group of type (C) and $R'_{a,b,c}$ for some $a,b$ and $c$ in~$K$;
(E) group of order 36, which is isomorphic to $(C_3\times C_3)\rtimes C_4$, generated by~$S$, $T$ and $V$;
(F) group of order 72 generated by (E) and $UVU^{-1}$;
(G) Hessian group of order 216, which is isomorphic to $(C_3\times C_3)\rtimes \mathrm{SL}_2(\mathbb{F}_3)$, generated by (E) and $U$;
(H) the simple alternating group $A_5$ of order 60;
(I) the simple group $\mathrm{PSL}_2(\mathbb{F}_7)$ of order 168;
(L) the simple alternating group $A_6$ of order 360.
Here
$$
R'_{a,b,c}=\begin{pmatrix}
a & 0 & 0\\
0 & 0 & b \\
0 & c & 0
\end{pmatrix} $$
with $abc\neq 0$.
\end{corollary}
\begin{proof}
Let $G$ be a finite subgroup of $\mathrm{PGL}_3(K)$, then $G$ can be regarded as a finite subgroup of $\mathrm{PGL}_3(\bar{K})$ via the natural embedding $\mathrm{PGL}_3(K)\hookrightarrow \mathrm{PGL}_3(\bar{K})$. Let $\Tilde{G}\subset\mathrm{SL}_3(\bar{K})$ be the preimage of $G$ via $\pi$. Then $\Tilde{G}$ is conjugate to one of the 12 types of groups in Theorem \ref{4.2}, and $G$ is isomorphic to the projection of such a group.
Groups of type (A) may not be diagonal when the field $K$ is not algebraically closed, but they're still abelian.
Now let $G$ be a finite non-abelian (if it's abelian, then sort it into type (A)) subgroup of $\mathrm{PGL}_3(K)$ such that the corresponding $\Tilde{G}\subset \mathrm{SL}_3(\bar{K})$ is conjugate to a group of type (B). That is, we have a unique decomposition of the vector field $\bar{K}^{3}=V\oplus U$, where $\dim V=2, \dim U=1$, such that $\pi(V)$ and $\pi(U)$ are the only invariant line and invariant point of $G$ acting on $\mathbb{P}^2_{\bar{K}}$. Pick an arbitrary element $\sigma\in \mathrm{Gal}(\bar{K}/K)$, note that $G=\pi(\Tilde{G})=\pi(\sigma(\Tilde{G}))$, since $G$ is defined over~$K$. And $\pi(\sigma(V))$ is also an invariant line of $G$, hence it must be $\pi(V)$. That is, the subspace $V$ is preserved by the action of all elements in the Galois group, hence it is defined over $K$. Similarly, $\pi(U)$ is the unique invariant point of $G$, so it's preserved by all elements in the Galois group, hence it's a point over $K$. Therefore, we find an invariant line and an invariant point outside the line for $G$ when it's regarded as a subgroup of $\mathrm{Aut}(\mathbb{P}^2_{K})$. That is, $G$ is conjugate to a group in which every matrix is of the form
$$\begin{pmatrix}
g & 0\\
0 & a
\end{pmatrix},$$
where $g\in\mathrm{GL}_2(K), a\in K^{\times}$. Up to re-scaling, we may assume that $a=1$, and then $G$ is isomorphic to a finite subgroup in $\mathrm{GL}_2(K)$, consisting of all the $2\times 2$ matrices $g$ appearing in the upper left block.
Note that for a group $\tilde{G}\subset \mathrm{SL}_3(\bar{K})$ of type (A),
the projection $\pi(\Tilde{G})$ will always be abelian, i.e. of type (A). Therefore for a group $\Tilde{G}\subset \mathrm{SL}_3(\bar{K})$ of type~(C) or (D), its projection $\pi(\Tilde{G})$ is again generated by a group of type (A) and some other specific matrices.
A group $\Tilde{G}\subset \mathrm{SL}_3(\bar{K})$ of type (E)--(G) or (J)--(L) contains the kernel of the projection $F$, therefore $G\cong \pi(\Tilde{G})$ is isomorphic to $\Tilde{G}/F$, and $|G|=|\Tilde{G}|/3$. On the other hand, groups $\Tilde{G}\subset \mathrm{SL}_3(\bar{K})$ of type (H) and (I) are simple, so~\mbox{$\pi(\Tilde{G})\cong G$}. In particular, projections of groups of type (H) and (J) to $\mathrm{PGL}_3(\bar{K})$ are isomorphic to each other, and the same happens with groups of type (I) and (K).
\end{proof}
\begin{remark}\label{4.5}
We restrict ourselves to fields of characteristic 0 because Theorem \ref{4.2} does not hold for fields with positive characteristics. For instance, when~\mbox{$K= \bar{\mathbb{F}}_p$}, the group $\mathrm{PGL}_3(K)$ contains $\mathrm{PSL}_3(\mathbb{F}_q)$, where $q=p^{n}$. These subgroups are simple, hence the Jordan constant of $\mathrm{PGL}_3(K)$ reaches infinity.
\end{remark}
\section{Primitive finite subgroups in $\mathrm{PGL}_3(K)$}
In this section, we give the criterion of the existence of each primitive subgroup in $\mathrm{PGL}_3(K)$, for an arbitrary field $K$ of characteristic 0.
\begin{proposition}\label{5.1}
Let $n$ be an odd prime number, then $\mathrm{PGL}_3(K)$ has an cyclic subgroup of order $n$ if and only if there is an $n$-th primitive root of unity $\alpha$ and indices $1< i< j\leq n$, with $1+i+j\equiv 0 $ (mod n) such that $$\alpha+\alpha^i+\alpha^j\in K,\ \alpha^{-1}+\alpha^{-i}+\alpha^{-j}\in K.$$
\end{proposition}
\begin{proof}
The case $n=3$ is clear. For $n\neq 3$, we may lift a generator of the cyclic subgroup to a matrix $A\in \mathrm{SL}_3(K)$ of the same order, by Lemma \ref{3.6}. Let $f(x)$ be the characteristic polynomial of $A$.
Case 1: Suppose $f$ has a multiple root of order $3$, i.e. $f(x)=(x-\alpha)^3\in K[x]$, then $\alpha\in K$. Our conditions are clearly satisfied.
Case 2: Suppose $f$ has a multiple root of order $2$. Let $\zeta, \lambda$ be two distinct roots of $f$, and do the Euclidean Algorithm. Write $$\Phi_n(x)=f(x)q(x)+r(x),$$ with $\deg r(x)=1$ or $2$, where $\Phi_n(x)$ is the $n$-th cyclotomic polynomial. Note that both $\zeta$ and $\lambda$ are roots of $r(x)$, so $$r(x)=(x-\zeta)(x-\lambda)\in K[x],\ \zeta\lambda\in K.$$
If $\zeta\lambda=1$, then $\lambda=\zeta^{-1}$ and $\zeta+\lambda=\zeta+\zeta^{n-1}\in K$. We may pick $\alpha=\zeta$ and indices $(1,n-1,n)$ to fit the conditions. Otherwise, $K$ contains an $n$-th primitive root of unity $\zeta\lambda$, which will be chosen as $\alpha$.
Case 3: Suppose $f$ has three simple roots, denote one of them by $\alpha$, such that the other two can be presented as $\alpha^i$ and $\alpha^j$. The fact that $\alpha^{1+i+j}\in K$ implies that either $K$ contains an $n$-th primitive root of unity or $$1+i+j\equiv 0\ (\text{mod}\ n).$$ For the latter case, the coefficients of square and linear terms are $$-(\alpha+\alpha^i+\alpha^j), \ \alpha^{-1}+\alpha^{-i}+\alpha^{-j},$$ which are exactly what we want.
Conversely, if we have such $\lambda=\alpha+\alpha^i+\alpha^j$ and $\eta=\alpha^{-1}+\alpha^{-i}+\alpha^{-j}$, consider
$$A=\left(
\begin{array}{ccc}
\lambda+1 & 1 & -\lambda-\eta-2 \\
1 & 0 & 0 \\
1 & 0 & -1 \\
\end{array}
\right).$$
Its characteristic polynomial is $f(x)=x^3-\lambda x^2+\eta x-1$, so the three eigenvalues of $A$ are $\alpha,\alpha^i$ and $\alpha^j$, hence the order of $A$ is $n$.
\end{proof}
\begin{lemma}\label{5.2}
The group $\mathrm{PGL}_3(K)$ contains $C_3\times C_3$ if and only if $\omega\in K$.
\end{lemma}
\begin{proof}
If $\omega\in K$, then matrices $S$ and $T$ given in Theorem \ref{4.2} will generate~\mbox{$C_3\times C_3$}.
For the other direction, we will prove by contradiction. Assume $\omega$ is not in $K$, and $\mathrm{PGL}_3(K)$ contains $C_3\times C_3$. Let $A, B\in \mathrm{GL}_3(K)$ denote representatives of the generators of each $C_3$, respectively. Assume that~\mbox{$\det(A)=\lambda\in K$},~\mbox{$\det(B)=\mu\in K$}, then $A^3=\lambda I, B^3=\mu I$.
If $\lambda\in K$ does not have a cubic root in $K$, we may construct the field extension $L=K[x]/(x^3-\lambda)$ of degree $3$. If $\omega\in L$, then $K\subset K(\omega)\subset L$ is a subextension of $L$ of degree 2, which is impossible, as $3$ is not divisible by 2. Therefore $L$ contains exactly one cubic root of $\lambda$ and does not contain $\omega$. We may repeat this procedure once again if $\mu$ does not have a cubic root in the new field, and finally obtain a larger field $K\subset L$ such that $L$ contains cubic root of both $\lambda$ and $\mu$, but not $\omega$.
Now we may assume $A^3=B^3=I$, after replacing $A$ by $\frac{1}{\sqrt[\leftroot{-2}\uproot{2}3]{\lambda}}A$, $B$ by $\frac{1}{\sqrt[\leftroot{-2}\uproot{2}3]{\mu}}B$, and $K$ by $L$. These two matrices commute in $\mathrm{PGL}_3(K)$, thus there exists $\alpha\in K$ such that $AB=\alpha BA$. By computing determinants of matrices at both sides, we get $\alpha=1$ as $\omega\not\in K$. That is, they also commute in $\mathrm{SL}_3(K)$. Therefore $A$ and $B$ generate $C_3\times C_3$ in $\mathrm{SL}_3(K)$, thereby in $\mathrm{SL}_3(\bar{K})$, via the canonical embedding. Note that $C_3\times C_3$ has only one-dimensional irreducible representations since it's abelian, so we can diagonalize $A$ and $B$ simultaneously in $\mathrm{SL}_3(\bar{K})$. Eventually, we achieve a subgroup $C_3\times C_3\subset \mathrm{SL}_3(\bar{K})$ consisting of diagonal matrices, which does not contain any scalar matrices other than $I$. However, this is impossible, since $\mathrm{SL}_3(\bar{K})$ contains exactly 8 diagonal matrices of order 3, and two of them are scalar matrices.
\end{proof}
\begin{proposition}\label{5.3}
The group $\mathrm{PGL}_3(K)$ contains subgroups of type (E), (F) and~(G) if and only if $\omega\in K$.
\end{proposition}
\begin{proof}
If $\omega\in K$, then we refer to Theorem \ref{4.2} for the explicit construction.
Conversely, if $\mathrm{PGL}_3(K)$ contains a subgroup of one of these three types, then it contains a subgroup of the form $C_3\times C_3$. We are done after applying Lemma \ref{5.2}.
\end{proof}
\begin{proposition}\label{5.4}
The group $\mathrm{PGL}_3(K)$ contains a subgroup of type (H) if and only if $\sqrt{5}\in K$.
\end{proposition}
\begin{proof}
If we have $A_5$, then we have an element of order 5, then $\sqrt{5}\in K$, by Proposition \ref{5.1}.
Conversely, if $\sqrt{5}\in K$, it's known that $$A_5\cong\left\langle x,y\mid x^5=y^2=(xy)^3=1\right\rangle,$$ according to \cite[p140, Beispiel 19.9]{Huppert}. Here $y$ plays the role of (12)(45), and $x$ plays the role of (12345).
Take
\begin{equation}\label{1}
x=\begin{pmatrix}
-\sigma & -\tau & -1\\
-\tau & -1 &-\sigma\\
1 & \sigma& \tau
\end{pmatrix}, \ y=\begin{pmatrix} 1 &\sigma&\tau \\ \sigma& \tau & 1\\ \tau & 1 & \sigma \end{pmatrix},
\end{equation}
where $\sigma=\frac{1-\sqrt{5}}{2},\ \tau=\frac{1+\sqrt{5}}{2}$. Then $x$ and $y$ satisfy all the relations, hence they generate a group which is a quotient of $A_5$. It should be either $A_5$ or $\{1\}$ since~$A_5$ is a simple group. They cannot generate the trivial group because the order of $x$ is greater than 1.
\end{proof}
\begin{proposition}\label{5.6}
The group $\mathrm{PGL}_3(K)$ contains a subgroup of type (I) if and only if $\sqrt{-7}\in K$.
\end{proposition}
\begin{proof}
Note that $\mathrm{PSL}_2(\mathbb{F}_7)$ can be generated by its order-$2$ elements. Therefore, according to Lemma \ref{5.5} there is a finite subgroup $G\subset \mathrm{SL}_3(K)$ with $|G|$=168 or 504 which projects to $\mathrm{PSL}_2(\mathbb{F}_7)\subset \mathrm{PGL}_3(K)$. The only finite primitive subgroup of $\mathrm{SL}_3(K)$ (if there is any) with order 504 is $C_3\times \mathrm{PSL}_2(\mathbb{F}_7)$, so we always have a copy of $\mathrm{PSL}_2(\mathbb{F}_7)$ in $\mathrm{SL}_3(K)$. That is, we could give a three dimensional representation of $\mathrm{PSL}_2(\mathbb{F}_7)$ in $K$, which is clearly not a direct sum of three trivial one-dimensional representations. By checking the character table of $\mathrm{PSL}_2(\mathbb{F}_7)$ (see \cite[Section 1.1]{Elkies}), the square root of $-7$ will appear in $K$.
Conversely, formulas (1.6) and (1.7) in \cite{Elkies} give the explicit construction of~$\mathrm{PSL}_2(\mathbb{F}_7)$ when $\sqrt{-7}$ lies in $K$.
\end{proof}
\begin{proposition}\label{5.7}
The group $\mathrm{PGL}_3(K)$ contains a subgroup of type (L) if and only if $\sqrt{5}\in K$ and $\omega\in K$.
\end{proposition}
\begin{proof}
If $\sqrt{5}$ and $\omega$ lie in $K$, consider the group presentation given by ATLAS of Finite Group Representations \cite{ATLAS}:
$$A_6\cong <a,b\mid a^2=b^4=(ab)^5=1>.$$
Here in our case $a=(12)(45), b=(1243)(56)$.
We may pick $$z=\begin{pmatrix} 1&0&0 \\ 0&0&\omega \\ 0&\omega^2&0 \end{pmatrix},$$
and set $a=y, b=y(zx)^4$, where $x$ and $y$ are the matrices defined in equation~(\ref{1}). Then $a$ and $b$ satisfy all the relations, hence they generate a group which is a quotient of $A_6$. It should be either $A_6$ or $\{1\}$ since $A_6$ is a simple group. They cannot generate the trivial group because the order of $a$ is greater than 1.
Conversely, if we have $A_6$ in $\mathrm{PGL}_3(K)$, we will have $\sqrt{5}\in K$ as $A_6$ contains~$A_5$. Moreover, from Lemma \ref{5.5} we know there is a finite subgroup~\mbox{$G\subset \mathrm{SL}_3(K)$} with $|G|$=360 or 1080 whose projection to $\mathrm{PGL}_3(K)$ is isomorphic to $A_6$. However, $\mathrm{SL}_3(\bar{K})$ does not have a finite subgroup isomorphic to $A_6$, since the minimal dimension for a nontrivial representation of $A_6$ is~5, according to \cite{GroupNamesA6}. One can also derive this from Theorem \ref{4.2} alternatively. Therefore~$|G|=1080$ and~\mbox{$\omega\in K$}.
\end{proof}
\section{Jordan constant of $\mathrm{PGL}_3(K)$}
In this section, we prove our main result Theorem \ref{thm1.4}.
\begin{lemma}\label{6.1}
Let $G$ be a finite subgroup of $\mathrm{PGL}_3(K)$.
1. If G is a group of type (C), then $J(G)\leq 3$.
2. If G is a group of type (D), then $J(G)\leq 6$.
\end{lemma}
\begin{proof}
According to the construction given in Theorem \ref{4.2}, we have a surjective group homomorphism from a group of type (D) to $S_3$, with kernel of type (A), which is abelian, normal, and has index $|S_3|=6$. Similar arguments apply for groups of type (C).
\end{proof}
\begin{lemma}\label{6.2}
If $G$ is a group of type (G), then $J(G)=24$.
\end{lemma}
\begin{proof}
According to \cite{GroupNames}, the only non-trivial abelian normal subgroup of $G$ is~\mbox{$C_3\times C_3$}, whose index is 24.
\end{proof}
Now we are ready to prove Theorem \ref{thm1.4}:
\begin{proof}
Note that $\mathrm{PGL}_3(K)$ always contains $S_4$: the action of $S_4$ on $K^4$ via permutation of basis $\{e_i\}_{i=1,\ldots,4}$ gives rise to a four-dimensional representation of $S_4$. It has a three-dimensional invariant subspace spanned by $$\{e_1-e_2, e_2-e_3, e_3-e_4\}.$$ So $J(\mathrm{PGL}_3(K))$ is at least 6.
As a result, we don't need to care about groups of type (A), (C) and (D) as group of type (A) is abelian, and Jordan constant of group of type (C) or (D) cannot be more than 6, which is shown in Lemma \ref{6.1}. For group of type (B), we have a complete result in Theorem \ref{thm1.3}. Also, subgroups of type (E) and (F) always appear together with group of type (G), and the group of type (G) gives the largest Jordan constant among them, so we can omit groups of type (E) and~(F).
The Jordan constants of all possible finite subgroups of $\mathrm{PGL}_3(K)$, excepting those excluded in the previous paragraph, are 24, 60, 168 and 360, coming from a finite group $G$ of type (G), (H), (I) and (L), respectively. Combining with Theorem \ref{thm1.3}, all the possible Jordan constants of $\mathrm{PGL}_3(K)$ are:~\mbox{6, 12, 24, 60, 168} and 360.
We start from the largest possible Jordan constant 360, given by $A_6$, and the criterion for existence of $A_6$ is given in Proposition \ref{5.7}. If 360 cannot be achieved, then we search for the group contributing Jordan constant 168, which is of type (I). Its existence is examined by Proposition \ref{5.6}. If~\mbox{$J(\mathrm{PGL}_3(K))<168$}, we next search for finite subgroup $G$ such that $J(G)=60$, which is given by Theorem \ref{thm1.3} and Proposition \ref{5.4}. However, the condition for Theorem \ref{thm1.3} is stronger, so we only require Proposition \ref{5.4}. Next, if~\mbox{$J(\mathrm{PGL}_3(K))<60$}, we look for the conditions such that $J(\mathrm{PGL}_3(K))=24$, which is given by Theorem \ref{thm1.3} and Proposition \ref{5.3}. If $J(\mathrm{PGL}_3(K))<24$, we then look for the conditions such that $J(\mathrm{PGL}_3(K))=12$. This Jordan constant can only be achieved by groups of type (B), and the criterion is stated in~\mbox{Theorem}~\ref{thm1.3}.
If none of above conditions is satisfied, we see that~\mbox{$J(\mathrm{PGL}_3(K))=6$}, according to the discussion in the first paragraph of the proof.
\end{proof}
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\bibliography{main}
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| 5,092
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Nov 19, 2012
By Dale Goldhawk
11:30am ET | Bruce Matthews, the Deputy Registrar at RECO
12:15pm ET | Glen Murray, MPP Toronto Centre, Candidate for leader of the Ontario Liberal Party.
GUEST – Bruce Matthews, the Deputy Registrar at Real Estate Council of Ontario (RECO)
TOPIC – Home Inspectors. Who to look for and what they should be looking for.
INFO – Real Estate Council of Ontario (RECO). We are the not-for-profit organization that oversees real estate in the province on behalf of the Ontario government.
* An inspection condition is included to protect the buyer – it’s important to use it
* Examples where the failure to have an inspection carried out resulted in significant additional expense for the buyer (one recent example )
Additional information can be found at reco.on.ca. RECO’s core purpose is to foster confidence and uphold integrity in real estate transactions.
GUEST – Glen Murray, MPP Toronto Centre, Candidate for leader of the Ontario Liberal Party.
TOPIC – His announcement and bid for the Ontario Liberal Party Leadership.
INFO – Glen R. Murray is a Canadian politician and urban issues advocate. He served as the 41st Mayor of Winnipeg, Manitoba from 1998 to 2004, and was the first openly gay mayor of a large North American city.
| 309,439
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TITLE: Quantization of momentum in nanotubes
QUESTION [1 upvotes]: I'm reading about carbon nanotubes and how the momentum (lets call it $k_x$) is quantized along the circumferential direction and not along the cylindrical (call this $k_y$). I can follow the maths okay, but what I don't understand is the physical reason why $k_x$ CANNOT take any value and that it must be quantized?
EDIT
Here is a link to what I mean
It's much more simple than what you thought, is it to do with the modes the electron can take? So if it is not one of these values it would interfere with itself around the circumferance like the "particle in the box"?
REPLY [2 votes]: The reason is that you have periodic boundary conditions in the azimuthal direction while there are no special constraints along the cylinder axis (note that, as in the radial direction we have the $\pi$ bonds of the carbon lattice the electron's wavefunction must be strongly confined). Other way to see this, in the azimuthal direction you must have an integer number of electron wavelengths along the perimeter by continuity of the wavefunction $\psi$.
As a consequence, at room temperature your only degree of freedom comes along the cylinder axis and your nanotube is a quasi 1D system.
REPLY [0 votes]: it really makes sense if you take a slice of the bunch of nanofibers in any axial plane, and you'll notice that the fibers form roughly a regular lattice. A regular lattice implies that the longest wavelength not severely absorbed by the carbon walls are of the order of the lattice separation. Also the integer multiples of that wavelength will satisfy the same boundary conditions at the lattice walls.
Obviously, such rule cannot be anything else than a raw approximation. In real fibers, the nanotube separation will nearly never been completely regular
| 158,666
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TITLE: An inverse problem: Number fields attached to elliptic curves over Q
QUESTION [5 upvotes]: If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/Q and F/Q be elliptic curves and write Q(E[p]) (resp. Q(F[p])) for the number field obtained by adjoining the x and y coordinates of the p-torsion of E to Q. Then if Q(E[p]) = Q(F[p]) for infinitely many primes p, E/Q and F/Q are isogenous.
Learning of this result prompted me to wonder: suppose P is a finite set of primes. Then do there exist E/Q and F/Q such that Q(E[p]) = Q(F[p]) for each p in P with E/Q and F/Q not isogenous?
If not in general, what is known about the particular P for which the above question does (or doesn't) have an affirmative answer?
REPLY [8 votes]: Most people would say no. Indeed, there's a conjecture, most commonly ascribed to Frey, that for p a SINGLE large enough prime, E[p] isomorphic to F[p] implies that E and F are isogenous. I believe p=37 is thought to be large enough, but don't hold me to it.
| 13,072
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Nothing_0<<
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Some of the best salsa I have had
This salsa is so good. Love the tomatillos and the heat.
| 197,235
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Our Cybersurveillance team publishes this report, an in-depth analysis that serves as a framework for reflection for companies when defining and adapting their cybersecurity and communication plans to the digital environment.
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In the business world and in the public sphere there is a question that haunts all strategic plans, the Digital Transformation, but what is it really? We simplify the concept and the complex approach associated with it, discover the important points of it and identify the keys to be considered..
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TheNinny (1), ash422 (1), beanjeepin (1), baltimorekids (1), peachymomof2 (1)
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Sustainable Reforms for Innovation
In December 2015, the United Nations General Assembly declared 2017 as the International Year of Sustainable Tourism for Development.
Sep 14, 2017
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The Chengdu Hi-tech Industrial Development Zone (CDHT) established an offshore innovation & entrepreneurship base at the headquarters of Canada China Club in Toronto, Canada September 21 local time.
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\begin{document}
\newcommand{\aut}{\mathrm{Aut}}
\newcommand{\M}{\mathbb{M}}
\newcommand{\dih}[2]{DIH_{#1}(#2)}
\newcommand{\drt}[1]{\frac{1}{\sqrt{2}} #1}
\newcommand{\drtp}[1]{ \frac{1}{\sqrt{2}}\left( #1\right)}
\newcommand{\drtpp}[1]{ \left( \frac{1}{\sqrt{2}}\left( #1\right)\right)}
\newcommand{\mspan}{\mathrm{span}}
\newcommand{\Rspan}[1]{\mathrm{span}_{\mathbb{R}}(#1)}
\newcommand{\Dih}[1]{DIH_{#1}}
\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{coro}[thm]{Corollary}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{hyp}[thm]{Hypothesis}
\newtheorem{rem}[thm]{\bf Remark}
\newtheorem{de}[thm]{\bf Definition}
\newtheorem{nota}[thm]{\bf Notation}
\newtheorem{ex}[thm]{\bf Example}
\newtheorem{proc}[thm]{\bf Procedure}
\begin{center}
{\Large \bf Determinants for integral forms in lattice type vertex operator algebras}
\bigskip
3 April, 2018
\vskip 1cm
{\Large }
Chongying Dong
Department of Mathematics,
University of California,
Santa Cruz, CA 95064 USA
\&
School of Mathematics and Statistics,
Qingdao University,
Qingdao 266071 CHINA
{\tt dong@ucsc.edu}
\vskip 0.5cm
and
\vskip 0.5cm
Robert L. Griess Jr.
Department of Mathematics,
University of
Michigan,
Ann Arbor, MI 48109-1043 USA
{\tt rlg@umich.edu}
\smallskip
\end{center}
\begin{abstract}
We prove a determinant formula for the standard integral form of a lattice vertex operator algebra.
\end{abstract}
\section{Introduction}
We have studied group-invariant integral forms in vertex operator algebras \cite{DG,DG2}. In this article, we study standard integral forms in lattice vertex operator algebras and give the determinant of each homogeneous piece as a particular integral power of determinant of the input positive definite even integral lattice. When the lattice is unimodular, all these homogeneous pieces have determinant 1, already proved in \cite{DG}. For lattices of other determinants, there did not seem to be an obvious answer.
Borcherds stated without proof in \cite{B1986} that the determinant of a homogeneous piece was some (unspecified) integral power of the input lattice.
The standard integral form $V_{L.\ZZ}$ for a lattice vertex operator algebra $V_L$ is reviewed in Section 4, Definition \refpp{standardintform}. Lemma \refpp{l1} shows that our main theorem \refpp{maintheorem} is reduced to a study of determinants for integral forms within a certain symmetric algebra. The latter determinant is therefore our main object of study in this article.
\section{Background}
\begin{lem}\labtt{ballsinurns} If $x_1, \dots, x_k$ are variables, then the number of monomials $x_1^{a_1}\cdots x_k^{a_k}$, $a_i \in \ZZ_{\ge 0}$, of total degree $n$ is ${n+k-1}\choose {k-1}$.
\end{lem}
\pf This is essentially a counting result, called Balls in Urns. Monomials correspond to the set of $k-1$ marker balls to be chosen among a set of $n+k-1$ balls arranged in a straight line. One adds marker ball 0 at the very beginning and marker ball $k$ at the very end. The sequence $a_1, \dots, a_k$ gives the lengths of the gaps between successive marker balls. \eop
\begin{lem}\labtt{detindex}
If $J \le K$ are finite rank lattices and the index $|J:K|$ is finite, then $det(K)=|J:K|^2 det(J)$.
\end{lem}
\section{Symmetric algebas}
\begin{nota}\labtt{01} Let $H$ be a $k$-dimensional vector space over $\CC$ and let $t$ be a variable. For $r\ge 1$, let $H_r=H\otimes \CC t^{-r}$ be a copy of $H$, defined to have degree $r$, and set $h(-r)=h\otimes t^{-r}$ for
$h\in H.$ We shall work in the symmetric algebra $M(1)=\mathbb S [H[t^{-1}]t^{-1}]=\CC[h(-r)|h\in H]$ where $$H[t^{-1}]t^{-1}=\oplus_{r\geq 1}H\otimes \CC t^{-r}.$$
Then $$M(1)=\oplus_{n\geq 0}M(1)_n$$
is graded such that $M(1)_n$ is spanned by $h_1(-r) \cdots h_p(-r_p)$ for $h_1,...,h_p\in H$ and $r_1,...,r_p\in \NN$ with $\sum_ir_i=n.$
For a sequence $a=(a_1, \dots, a_n, \dots )$ of nonnegative integers which is almost all zero, define $wt(a):=\sum_{j\ge 1} \ ja_j$.
For $n\ge 0$, define $\mathcal A (n)$ to be the set of such $a$ of weight $n$. Note that $\mathcal A (0)=\emptyset$.
Suppose that $x_1, \dots, x_k$ is a basis of $H$. Then $L=\Z x_1+\cdots +\Z x_k$ is a free abelian group of rank $r$ in $H$ and $M(1)=\CC[x_1(-r), \cdots x_k(-r)| r \geq 1].$ Define $B(a):=B(x_1, \dots x_k; a)$ to be the $\ZZ$-span of all words $w_1\cdots w_n$ in $M(1)$ where $w_i$ is a product of length $a_i$ in the variables $x_j(-i)$, for $j\in \{1,2,\dots ,k\}$.
Finally, for an integer $n\ge 0$, define $B_L(n):=B(x_1, \dots x_k; n) :=\oplus _{a\in \mathcal A (n)} B(a)$ and $B_L:=\oplus _{n\ge 0} B_L(n)=\ZZ[x_1(-r), \cdots x_k(-r)|r \geq 1]$. Then $B_L$ is a subring of $R.$ Note that these objects are unchanged if $x_1, \dots, x_k$ is replaced by any basis of the $\ZZ$ span of $x_1, \dots, x_k$.
\end{nota}
\begin{nota}\labtt{01'}
Let $A_L$ be the $\ZZ$-submodule of $M(1)$ generated by $s_{\alpha,n}$ for $\alpha_i\in \{x_1,\dots x_k\}$ and $n\geq 0$ where
$$E^-(-\alpha,z):=\exp\left(\sum_{n>0}\frac{\alpha(-n)}{n}z^n\right)=\sum_{n\geq 0}s_{\alpha,n}z^n.$$
Although we do not use the vertex operator algebra structure on $M(1)$, we use the notations $E^-(-\alpha,z)$ and $s_{\alpha,n}$ from \cite{flm} and \cite{DG} here.
Then $A_L$ has a $\ZZ$-base $B_1B_2\cdots B_k$ where
$$B_i=\{s_{x_i,n_1}\cdots s_{x_i,n_q}|n_1\geq \cdots \geq n_q\geq 0\}$$
and $B_1\cdots B_k=\{u_1u_2\cdots u_k|u_i\in B_i\}.$
\end{nota}
We also set $A_L(n)=A_L\cap M(1)_n$ for all $n.$ The following result will be useful in computing the determinants for the lattice vertex operator algebras.
\begin{lem}\label{l1} $B_{L}$ is a subring of $A_L$ and the index $[A_L(n):B_L(n)]$ is independent of the base $\{x_1,...,x_k\}$ for any $n\geq 0.$
\end{lem}
\pf We first prove that $B_L$ is a subring of $A_L.$ It is good enough to show that $\alpha(-n)\in B_L$ for $\alpha\in\{x_1,...,x_k\}$ and $n\geq 0.$ Note that
$$E^-(-n\alpha,z)=E^-(-\alpha,z)^n=(\sum_{m\geq 0}s_{\alpha,m}z^m)^n.$$
So the coefficient $c_n$ of $z^n$ in $E^-(-n\alpha,z)$ lies in $A_L.$ Clearly, $c_n$ is also the coefficient of $z^n$ in $(\sum_{m=0}^ns_{\alpha,m}z^m)^n.$
A straightforward computation shows that
$a_n=\alpha(-n)+u$ where $u$ is a $\ZZ$-linear combination of elements of the form $s_{\alpha,m_1}s_{\alpha,m_2}\cdots $ with$m_i<n$ and $m_1+m_2+\cdots =n.$ As a result, $\alpha(-n)\in A_L.$
To show that the index $[A_L(n):B_L(n)]$ is independent of the base $\{x_1,...,x_k\}$ for any $n\geq 0$, we let $\{y_1,...,y_k\}$ be another basis of $H$
and $K=\ZZ y_1+\cdots +\ZZ y_k.$
Then a group isomorphism $f$ from $L$ to $K$ by sending $x_i$ to $y_i$ induces a ring isomorphism $\hat f$ from $R$ to itself such
that $\hat f(A_L)=A_K$ and $\hat f(B_L)=B_K.$ It is evident that $\hat f$ is a degree preserving map. As a result, $\hat f(A_L(n))=A_K(n)$ and $\hat f(B_L)=B_K.$ Thus, $[A_L(n):B_{L}(n)]=[A_K(n):B_K(n)].$
\eop
Note that both $A_L$ and $B_L$ are $\ZZ$-forms of $M(1).$
\begin{lem} \labtt{02}
(i) $rank(S^m(H))={{m+k-1}\choose {k-1}}$;
(ii) $rank(B(a))=\prod_{j=1}^n {{a_j+k-1}\choose {k-1}}$;
(iii) $wt(B(a))=\sum_{j\ge 1} \ ja_j$;
(iv) $rank(B(n))=\sum_{a: wt(a)=n} rank(B(a))$.
\end{lem}
\pf Straightforward, with Lemma \refpp{ballsinurns}.
\eop
\bigskip
So far, there is no bilinear form in this discussion. We shall introduce forms later, after Corollary \ref{06}.
\bigskip
We now compare what happens to the $B(a)$ when $x_1, \dots, x_n$ is replaced by another basis. We already noted in Notation \ref{01} that $B(x_1, \dots x_k; a) = B(y_1, \dots y_k; a)$ if $span_{\ZZ}(x_1, \dots x_k)=span_{\ZZ}(y_1, \dots y_k)$.
Using the proof of Lemma \ref{l1} we can easily have:
\begin{lem} \labtt {03} If $x_1, \dots x_k$ and $y_1, \dots y_k$ are bases and if
$span_{\ZZ}(x_1, \dots x_k)$ contains $span_{\ZZ}(y_1, \dots y_k)$, then for any invertible linear transformation $T$ on $H$,
$B(x_1, \dots x_k; a)/B(y_1, \dots y_k; a) \cong B(Tx_1, \dots Tx_k; a)/B(Ty_1, \dots Ty_k; a)$. In particular, we have equality of indices $|B(x_1, \dots x_k; a) : B(y_1, \dots y_k; a) |=|B(Tx_1, \dots Tx_k; a) : B(Ty_1, \dots Ty_k; a)|$.
\end{lem}
\begin{lem} \labtt {04} Suppose that $p>0$ is an integer. Then
$B(x_1, x_2, \dots , x_k; a)$ contains $B(px_1, px_2, \dots , px_k; a)$ and the index is
$p^{N(k,a)}$ where
$N(k,a):= \prod_{j=1}^n a_j{{a_j+k-1}\choose {k-1}}$.
\end{lem}
\pf
The free abelian group $B(x_1, x_2, \dots , x_k; a)$ has basis consisting of monomials in the $x_t(-j)$.
Such a monomial has a unique expression $w_1\cdots w_{wt(a)}$,
where $w_j$ is a monomial in the $x_t(-j)$.
There are ${{a_j+k-1}\choose {k-1}}$ such $w_j$.
The formula for $N(k,a)$ is now clear.
\eop
\begin{lem} \labtt {05} Suppose that $p>0$ is an integer. Then
$B(x_1, x_2, \dots , x_k; a)$ contains $B(px_1, x_2, \dots , x_k; a)$ and the index is
$p^{{\frac 1k} N(k,a)}$ where
$N(k,a):= \prod_{j=1}^n a_j{{a_j+k-1}\choose {k-1}}$.
\end{lem}
\pf Observe that we have a chain $$span_{\ZZ}(x_1, x_2, x_3, \dots x_k) > span_{\ZZ}(px_1, x_2, x_3, \dots x_k) > $$ $$span_{\ZZ}(px_1, px_2, x_3, \dots x_k) > \dots > span_{\ZZ}(px_1,p x_2, px_3, \dots px_k).$$ By Lemma \refpp{03},
the indices for each containment
$$B(x_1, x_2, x_3, \dots x_k; a) > B(px_1, x_2, x_3, \dots x_k; a) > $$
$$B(px_1, px_2, x_3, \dots x_k; a) > \dots > B(px_1,p x_2, px_3, \dots px_k; a)$$ are equal. We then deduce the result from Lemma \refpp{04}.
\eop
\begin{coro}\labtt{06}
In the notation of Lemma \refpp{05}, the index $$|B(x_1, x_2, \dots , x_k;a):B(px_1, x_2, \dots , x_k;a)|$$ is $p^{{\frac 1k} \sum_{a \in \mathcal A (n)} N(k,a)}$ =
$p^{{\frac 1k}\sum_{(a_j)=a \in \mathcal A (n)} \prod_{j=1}^n a_j{{a_j+k-1}\choose {k-1}}}$.
\end{coro}
Now assume that $H$ has a nondegenerate symmetric bilinear form $\<\cdot|\cdot\>.$ Then we can make $M(1)$ an irreducible module for the affine algebra
$$\hat H=H\otimes \CC[t,t^{-1}]\oplus \CC K$$
such that $H\otimes \CC[t]$ annihilates $\bf 1$ and the central element $K$ acts as $1.$ We abbrevuate $h\otimes t^m$ by writing $h(m)$ for $h\in H$ and $m\in \ZZ.$
\begin{nota}\label{ndsbf} There is a unique
nondegenerate symmetric bilinear form $\<\cdot|\cdot\>$ on $M(1)$ such that $\<\vac|\vac\>=1$ and $\<h(m)u|v\>=-\<u | h(-m)v\>$ for $u,v\in M(1)$ and $h\in H$ (see \cite{L}, \cite{DG}).
\end{nota}
Furthermore, $B_L(n)=B(x_1,...,x_k; n)$ is a $\ZZ$-form of $M(1)_n.$ Note that if $L=\ZZ x_1+\cdots +\ZZ x_k$ is rational lattice of $H$ in the sense that for any $\alpha,\beta\in L$
$\<\alpha|\beta\>\in\QQ$, then $B(x_1,...,x_k; n)$ is also a rational lattice, due to the form.
In the notation of Corollary \refpp{06}, we have
\begin{coro}\labtt{07} Assume existence of the form as in Notation \ref{ndsbf}.
For $k\ge 1$ and $n\ge 0$, define
$$S(k,n):= {\frac 1k} \sum_{a \in \mathcal A (n)} N(k,a) ={\frac 1k}\sum_{a=(a_j) \in \mathcal A (n)} \prod_{j=1}^n a_j{{a_j+k-1}\choose {k-1}}.$$
Then $S(k,0)=0$ and
$$\det(B(px_1, x_2, \dots , x_k; n)) = \det(B(x_1, x_2, \dots , x_k; n))p^{2S(k,n)}$$
for all $k\ge 1$ and $n\ge 0$.
\end{coro}
\begin{rem}\labtt {08} This presentation helps us understand the ``homogeneous part'' of the standard integral form in the symmetric algebra spanned over $\ZZ$ by all monomials made from a basis. The integral form involves expressions like Schur functions which have fractional coefficients so are not in the homogeneous part. We shall study the quotient of that integral form by its homogeneous part.
\end{rem}
\section{Integral forms of $M(1)$}
Let $L$ be a positive definite integral lattice with basis $x_1, \dots , x_k$ and we denote the form on $L$ by $\ip \cdot\cdot.$ We recall the standard integral form for the lattice vertex operator algebra based on $L$.
Note from \cite{flm} that $M(1): =\CC[x_i(-n)|i=1,...,k; n>0]$ is the Heisenberg vertex operator algebra and $V_L=M(1)\otimes \CC^{\epsilon}[L]$ is the corresponding lattice vertex operator algebra where $\epsilon$ is a bimultiplicative map from $L\times L\to \<\pm 1\>$
such that $\epsilon(\alpha,\beta)\epsilon(\beta,\alpha)=(-1)^{\ip \alpha\beta}$ and $\epsilon(\alpha,\alpha)=(-1)^{\ip \alpha\alpha/2}$
and where $\CC^{\epsilon}[L]=\oplus_{\alpha\in L}\CC e^{\alpha}$ is the twisted group algebra. There is a unique nondegenerate symmetric invariant bilinear form $\ip\cdot\cdot$ on $V_L$ such that
$$\ip {e^{\alpha}}{e^{\beta}}=\delta_{\alpha+\beta,0}$$
and
$$\ip{\alpha(n)u}{v}=-\ip u{\alpha(-n)v}$$
for all $u,v\in V_L$ $\alpha\in L$ and $n\in {\ZZ}$ (see \cite{B1986}, \cite{DG}).
Recall the subring $A_L$ from Section 3. Then $A_L$ is a $\ZZ$-form of $M(1)$ in the sense that $A_L$ is a vertex algebra over $\Z,$ $\ip{u}{v}\in\Z$ for $u,v\in A_L$ and
$M(1)=\CC\otimes_{\Z}A_L$ \cite{DG}.
We also set $(V_L)_\ZZ=\oplus_{\alpha\in L}A_L\otimes e^{\alpha}.$ Then $(V_L)_\ZZ$ is a vertex operator algebra over $\ZZ$ generated by $e^{\pm x_i}$ for $i=1,...,k$ and is a free $\ZZ$-module
such that $V_L=\CC\otimes_{\ZZ}(V_L)_{\ZZ}.$
\begin{de}\label{standardintform} $(V_L)_\ZZ=\oplus_{\alpha\in L}A_L\otimes e^{\alpha}.$ is the {\it standard integral form} in the lattice vertex operator algebra $V_L$.
\end{de}
Let $(V_L)_{\ZZ,n}$ consists of vectors of weight $n$ in $(V_L)_{\ZZ}.$ To get $det((V_L)_{\ZZ, n})$, we first understand $\det(A_L(n))$ in terms of $\det(L).$ Recall
$B_{L}=\Z[x_i(-n)\mid i=1,...,k; \, n>0]$ and $B_L(n)=B_L\cap M(1)_n$ for $n\geq 0.$ By Lemma \ref{l1}, $[A_L(n):B_L(n)]$ only depends on the rank of $L$ and integer $n.$
Using Lemma \ref{l1} we can give an explicit expression of $[A_L(n):B_L(n)]$. We define numbers $b_0:=1$ and for $n>0$,
$$b_n : =\prod_{a=(a_1,a_2,\cdots)\in \mathcal A (n)}\prod_{i\geq 1} i^{a_i}\cdot a_i! .$$
\begin{lem}\label{l2} The index $[A_L(n):B_L(n)]$ is the square root of
$$\prod_{n_1,...,n_k\geq 0, \, \sum n_i=n }b_{n_1}\cdots b_{n_k}$$
for $n\geq 0.$
\end{lem}
\pf By Lemma \ref{l1}, $[A_L(n):B_L(n)]$ is independent of lattice $L.$ So we can choose $L=\ZZ x_1+ \cdots +\ZZ x_k$ such that
$\{x_1,...,x_k\}$ is an orthonormal basis of $H$ for convenience of computation. Then
$A_L(n)$ is a unimodular lattice by Proposition 3.6 of \cite{DG}. It is easy to show that
\begin{eqnarray*}
& &\ \ \ \ip {x_i(-p)^s}{x_i(-p)^s}=(-1)^s\ip{{\bf 1}}{x_i(p)^sx_i(-p){\bf 1}}\\
& &=(-1)^ss!p^s\ip{{\bf 1}}{{\bf 1}}\\
& &=(-1)^ss!p^s
\end{eqnarray*}
for any $i,s$. This shows that $|\det (B_L(n))|$ equals $\prod_{n_1,...,n_k\geq 0, \, \sum n_i=n}b_{n_1}\cdots b_{n_k}.$ Since $|\det (B_L(n))|=[A_L(n):B_{L,n}]^2,$ the result follows immediately. \eop
\begin{lem}\label{l3} Let $A_1,A_2, C_1, C_2$ be lattices with the same rank such that $C_i \subset A_i$ for $i=1,2,$ $C_2\subset C_1.$ Then $|\det (A_2)|= \frac{[C_1:C_2]^2[A_1:C_1]^2|\det (A_1)}{[A_2:C_2]^2}.$
In particular, if $\det (A_1)=1$ and $[A_1:C_1]=[A_2:C_2]$ then $|\det (A_2)|=[C_1:C_2]^2.$
\end{lem}
\pf The result follows from the following relations
$$ |\det (C_i)|=|\det (A_i)|[A_i:C_i]^2,\det (C_2)=\det (C_1)[C_1:C_2]^2$$
for $i=1,2.$
\eop
\begin{thm}\label{t1} Let $L$ be an positive definite integral lattice with a base $\{x_1,...,x_k\}$ as before. Then for each $n\geq 0,$ $|\det(A_L(n))|$ is an integer power of $\det(L).$ In fact, $|\det (A_L(n))|=\det(L)^{2S(k,n)}$, where $S(k,n)$ is given by Lemma \refpp{07}.
\end{thm}
\pf We prove the theorem in several steps. If $K$ contains the sublattice $J$ with finite index, then one may deduce the results for $K$ from those for $J$, and conversely, from the results of Section 3.
The result $|\det (A_L(n))|=1$ when $L$ is unimodular was proved in
\cite{DG}. Let $p$ be a positive integer.
Case (a): Let $0\ne p \in \ZZ$ and $L=\Z pe_1\oplus \Z e_2\oplus\cdots \oplus\Z e_k$ be a sublattice of $\Z^k=\Z e_1\oplus \cdots \oplus\Z e_k$ where $\{e_1,...,e_k\}$ is the standard orthonormal basis of $\RR^k.$
Using Lemmas \ref{l1}, \ref{l2} with $A_1=A_{\Z^k}(n),$ $A_2=A_L(n),$ $C_1=B_{\Z^k}(n)$ and $C_2=B_L(n)$ gives $|\det (A_L(n))|=[B_{\Z^k}(n):B_L(n)]^2.$ Note that $\det(L)=p^2.$
By Corollary \ref{07}, $|\det (A_L(n))|=p^{2S(k,n)}=\det (L)^{2S(k,n)}.$
Case (b): Let $L=\Z p_1e_1\oplus \Z p_2e_2\oplus\cdots \oplus\Z p_ke_k$ for any positive integers $p_1,...,p_k.$ Then $\det ( L)=p_1^2\cdots p_k^2$ and $|\det (A_L(n))|=\det (L)^{2S(k,n)}$ by Case (a).
Case (c): Let $T$ be a positive integer such that $L$ is a rank $k$ sublattice of $K=\frac{1}{T}\Z^k$, i.e., $L \subset \QQ^k$. Then $\det (L)=[K:L]^2T^{-2k}.$ There exist
a base $\{u_1,...,u_k\}$ of $K$ and positive integers $p_1,...,p_k$ such that $\{p_1u_1,...,p_ku_k\}$ is a base of $L.$ This implies that $[K:L]=p_1\cdots p_k.$ From Lemma \ref{l3} and
discussion in Case (b),
we see that $|\det (A_L(n))|=[B_K(n):B_L(n)]^2|\det (A_K(n))|=(p_1\cdots p_k)^{2S(k,n)}|\det (A_K(n))|.$ On the other hand,
$$1=|\det(A_{\Z^k}(n))|=[B_K(n): B_{\Z^k}(n)]^2|\det (A_K(n))|=T^{{2k S(k,n)}}|\det (A_K(n))|.$$ Thus
$$|\det (A_L(n))|=(p_1\cdots p_k)^{2S(k,n)} T^{-{2kS(k,n)}}=\det (L)^{2S(k,n)}.$$
Case (d): Let $L$ be an arbitrary integral lattice in Euclidean space $\RR^k.$
The problem with applying (c) is that $L$ is not necessarily a sublattice of $\QQ^k.$ However, we can use a sequence of rational lattices $\{L_i|i\in\NN\}$ such that
``$\lim_{i\to \infty}L_i=L$''. We fix a base $\{v_1,...,v_k\}$ of $L$ and take linearly independent vectors $v^i_1,...,v^i_k\in \QQ^k$ such
that $|v^i_j-v_j|<\frac{1}{i}$ for all $i,j.$ It is clear that $\lim_{i\to \infty}\det (L_i)=\det (L)$ and $\lim_{i\to\infty}|\det (A_{L_i}(n))|=|\det (A_L(n))|$ for any $n\geq 0.$
It follows from Case (c) that $|\det (A_L(n))|=\det (L)^{2S(k,n)},$ as desired. \eop
\section{Integral forms of $V_L$}
We now assume that $L$ is a positive definite even lattice. Recall that $(V_L)_\ZZ=\oplus_{\alpha\in L}A_L\otimes e^{\alpha}.$ Also recall $(V_L)_{\ZZ,n}$ from Section 4. We determine $\det ((V_L)_{\ZZ,n})$ in this section.
For $m\geq 0$ we set $L_{2m}=\{\alpha\in L|\ip \alpha\alpha=2m\}.$ Define $Y_0:=L_o=\{0\}$.
For $m\ge 1$, let $Y_{2m}$ be a subset of $L_{2m}$ such that $2|Y_{2m}|=|L_{2m}|$ and $L_{2m}=Y_{2m}\cup (-Y_{2m}).$ For $\alpha\in L$ we set
$W^{\alpha}=M(1)_{L}\otimes e^{\alpha}+M(1)_L\otimes e^{-\alpha}\subset (V_L)_{\ZZ}.$ Let $W^{\alpha}_n=W^{\alpha}\cap (V)_{L,n}.$ Then $W^{\alpha}_n\ne 0$ if and only if $\alpha\in L_{2m}$ and $m\leq n.$
In this case, $W^{\alpha}_n=A_L(n-m)\otimes e^{\alpha}+A_L(n-m)\otimes e^{-\alpha}.$ Observe that
$$(V_L)_{\ZZ,n}=\oplus_{m=0}^n\oplus_{\alpha\in Y_{2m}}W^{\alpha}_n$$
and $\ip {W^{\alpha}}{W^{\beta}}=0$ if $\alpha\ne \beta.$ So
$$\det( (V_L)_{\ZZ,n})=\prod_{m=0}^n\prod_{\alpha\in Y_{2m}}\det (W^\alpha_m).$$
From the definition of the bilinear form, we know that
for $\alpha\in Y_{2m}$ with $m\ne 0$
$$|\det (W^{\alpha}_n)|=\det (A_L(n-m))^2.$$
Also, $det(W_0^0)=det(\ZZ \vac)=1$.
Here is our main theorem, an immediate consequence of Theorem \ref{t1}.
\begin{thm}\label{maintheorem} For all $n\ge 0$, we have
$$|\det ((V_L)_{\ZZ,n})|=\prod_{m=0}^n\det( L)^{|L_{2m}|{2S(k,n-m)}}.$$
\end{thm}
\section{Acknowledgements}
C. Dong was supported by China NSF grant 11871351.
R. Griess was supported by funds from his Collegiate Professorship and Distinguished University Professorship at the University of Michigan.
| 105,602
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Name: Yattaman film ita
File size: 389mb
Language: English
Rating: 3/10
Download
10 Apr - 86 min - Uploaded by Ward Entertainment More Funny Spiderman, Super Man and Frozen Elsa and Superhero in real life. 7 May - 2 min - Uploaded by majin goku speriamo presto anche in italia il film di yattaman ^^.
4 Jul - 43 min - Uploaded by maker bopa YATTAMAN! Il film live action (Sub Ita)!!! - Duration: kintamafansub , views. 31 Oct - 6 min - Uploaded by kintamafansub Il FILM COMPLETO lo potete vedere qui: dentgoogmimes.host 3 Jul - 2 min - Uploaded by CG Entertainment Yattaman - Trailer dal prolifico Tarantino d'oriente Takashi Miike, autore di film. 29 Dec - 11 min - Uploaded by ico Yattaman Film - La Vergine Di Ferro Film & Animation Yattaman - Yatterking.
This is "Yattaman - Il film () Completo in Italiano" by CPR Agency on Vimeo, the home for high quality videos and the people who love. Yatterman (Japanese: ヤッターマン, Hepburn: Yattāman) is a Japanese anime television series broadcast from January 1, to January 27, , comprising .
Chords for YATTAMAN! Il film live action (Sub Ita)!!!. Play along with guitar, ukulele, or piano with interactive chords and diagrams. Includes transpose, capo hints. YATTAMAN! Il film live action (Sub Ita)!!! Il FILM COMPLETO lo potete vedere qui: dentgoogmimes.host Commentate! Commentate!Commentate!.
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| 246,135
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"Hi! Uncle Eddie here! Labor Day's coming up and I thought I'd post again about the best way to make a burger. The best burger I know of is John K's "Manly Cartoonist Burger." I put up the recipe way back in August of 2007.
Rather than print that all over again, I thought I'd discuss it side by side with Kenny Shopsin's burger theories. Shopsin wrote a cookbook that I'm perusing called "Eat Me," which features recipes from his famous New york restaurant. I thought you might find the contrast interesting."
"Um...a word of warning: burger theorists are feisty people. If John hears what he considers burger heresy, there's half a chance that he'll trash the place while I'm shooting."
"Okay, let's start! Well, to begin with, both cooks agree that you want ground beef that's 20-25% fat. Don't worry about the high fat content, it cooks away. John puts islands of chopped sirloin around the paddy so you get different flavor sensations with every bite, but the restaurant guy uses one type of meat overall.
John says add an egg (1 egg for 4 people), but don't compress the meat much when you put it in and never squeeze it, because that makes the cooking more difficult. It's good to have some air inside. A little pepper, chopped green onions, garlic, and chilli pepper, but never salt. Salt dries out the meat. If you want salt, add it after the burger's cooked."
"It's important to let the burger cook for five minutes undisturbed, except to turn it over at the midway point. You don't want to poke it more than necessary because that lets the juices escape."
"Well, John passionately insists that burgers have to be cooked on hot charcoal, with a lid on half the time . The restaurant guy says no...it has to be on a really hot, pre-heated iron frying pan with a lid. Charcoal and frying pan: that difference defines the two types of people that exist in the universe. I hope these guys never meet because they'd probably kill each other."
"Anyway, after 5 minutes the restaurant guy relies sticks a meat thermometer right in the middle of the burger. 120 - 125 degrees for rare, 140 - 145 degrees for medium."
"Okay, that leaves one more subject...the bun!"
"John says you need a fresh pastry shop bun, something with sesame or poppy seeds. The book recommends Martin's Potato Buns, which you can probably get at the supermarket.
Mmmm, I gotta give it to John on this one. You can't beat a nice, fresh Kaiser roll. Besides, the restaurant guy owns a business and he's gotta be tempted to cut corners."
"You put a little butter on the roll to help it toast better, and something on top of it to press it down on the pan. You only toast it lightly so it's soft on the inside and crispy on the outside."
"Now the burger is assembled and spatulaed onto a plate. You take it over to stove where mushrooms have been cooking in bacon grease. Ladle some mushrooms and bacon on, then move to the condiment table, which contains Romaine lettuce, pickles, raw radishes, celery and fresh onions."
"Here we go with the controversy again. John prefers his onions raw and juicy. He says that's because you need to feel a little pain with your pleasure. He slices the onion only when it's ready for use.
The restaurant guy likes his thin sliced and fried in peanut oil til they're a gnarly brown/black that don't even look like onions anymore.
"And that's it. As I said, John's Manly Cartoonist burger is the best I've ever had. It even looks good! Even so, I'll try Shopsin's burger next time I'm in New york. All this reminds me that a good burger is a thing of beauty. It's not given to man to lay his eyes on a better Labor Day meal!'
P.S. At the supermarket where I usually score my Kaiser rolls, they tell me that Poppy seeds have been discontinued on rolls. Kali's Dad speculates that even though the seeds can't get you high, they can put something in your system that responds positively to drug tests.
P.P.S. Vincent Waller ate at Shopsin's and describes it in a comment.
P.P.P. S. I'M OFF FOR THE WEEKEND! 'BE BACK MONDAY NIGHT!
| 262,322
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TITLE: What is category theory useful for?
QUESTION [133 upvotes]: Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory?
In Wikipedia it says it is to formalize. As far as I can tell, it sort of generalizes a bunch of fields in mathematics, like in topology, graphs and groups we have isomorphisms and automorphisms and homomorphisms, it seems to me like it just finds a bunch of results that are true to these functions in a bunch of disciplines. But why would generalizing that be of use? I mean, wouldn't it be easier to prove them specifically in each of the important disciplines?
REPLY [8 votes]: Category theory is not something instead of other mathematical theories but an other and very interacting perspective. The search for theorems and their proofs will never end as long as there are still mathematicians left. Even the familiar system of natural numbers will always hide secrets and any theory that helps to prove or counter prove just one single conjecture is worth to be further developed.
The category language might or might not be fruitful for a certain study, but it is always a perspective worth considering. In number theory there are less and more complex functions. The less complex functions are the first to be studied but more complex functions, as the increasing prime number function $p: \mathbb Z_+\rightarrow\mathbb Z_+$, seems resist any attempt of finally survey.
There is a category with all functions $f: \mathbb Z_+\rightarrow\mathbb Z_+$ as objects and pair of commuting functions $\alpha,\beta: \mathbb Z_+\rightarrow\mathbb Z_+$,
$(\beta f=g\alpha)$, are the morphisms $\displaystyle f\overset{(\alpha,\beta)}{\longrightarrow}g$.
Investigating such a category (I don't know if it is done) could bring a new approach to some issues. After all, it is not only the prime number function that is unresolved. When using categories there is also a possibility to find functors to other categories, such that hard problems in the primary category is transformed to a less hard problem in the secondary category.
| 32,510
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\begin{document}
\title{Quantum Cuntz-Krieger algebras}
\author{Michael Brannan}
\address{Michael Brannan \\ Department of Mathematics\\
Texas A\&M University \\
College Station, TX 77840 \\
USA}
\email{mbrannan@tamu.edu}
\author{Kari Eifler}
\address{Kari Eifler \\ Department of Mathematics\\
Texas A\&M University \\
College Station, TX 77840\\
USA}
\email{keifler@tamu.edu}
\author{Christian Voigt}
\address{Christian Voigt \\ School of Mathematics and Statistics \\
University of Glasgow \\
University Place \\
Glasgow G12 8QQ \\
United Kingdom}
\email{christian.voigt@glasgow.ac.uk}
\author{Moritz Weber}
\address{Moritz Weber \\ Saarland University \\
Faculty of Mathematics \\
Postbox 151150 \\
D-66041 Saarbr\"ucken \\
Germany}
\email{weber@math.uni-sb.de}
\begin{abstract}
Motivated by the theory of Cuntz-Krieger algebras we define and study $ C^\ast $-algebras associated to directed quantum graphs. For classical graphs
the $ C^\ast $-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear.
We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single
matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix
block up to $ KK $-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding
quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular
that these $ C^\ast $-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of $ KK $-theory.
We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation.
We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
\end{abstract}
\subjclass[2020]{46L55, 46L67, 81P40, 19K35}
\maketitle
\section{Introduction}
Cuntz-Krieger algebras were introduced in \cite{CKalgebras}, generalizing the Cuntz algebras in \cite{Cuntzalgebras}.
These algebras have intimate connections with symbolic dynamics, and have been studied intensively in the framework of graph $ C^\ast $-algebras over the
past decades, thus providing a rich supply of interesting examples \cite{Raeburngraph}. The structure of graph $ C^\ast $-algebras is understood to an impressive
level of detail, and many properties can be interpreted geometrically in terms of the underlying graphs. Motivated by this success, the original constructions
and results have been generalized in several directions, including higher rank graphs \cite{KPhigherrankgraphcstar}, Exel-Laca algebras \cite{ExelLacaalgebras}
and ultragraph algebras \cite{Tomfordeunified}, among others.
The aim of the present paper is to study a generalization of Cuntz-Krieger algebras of a quite different flavor, based on the concept of a quantum graph. The
latter notion goes back to work of Erdos-Katavolos-Shulman \cite{EKSrankonesubspaces} and Weaver \cite{Weaverquantumrelations}, and was subsequently developed
further by Duan-Severini-Winter \cite{DSWnoncommutativegraphs} and Musto-Reutter-Verdon \cite{MRVmorita}. Quantum graphs play an intriguing role in the study
of the graph isomorphism game in quantum information via their connections with quantum symmetries of graphs \cite{BCEHPSWbigalois}. Moreover, based on the
use of quantum symmetries, fascinating results on the graph theoretic interpretation of quantum isomorphisms between finite graphs were recently obtained
by Man\v{c}inska-Roberson \cite{MRplanar}.
Our main idea is to replace the matrix $ A $ in the definition of the Cuntz-Krieger algebra $ \O_A $ by the quantum adjacency matrix of a directed quantum graph.
Roughly speaking, this means that the standard generators in a Cuntz-Krieger algebra are replaced by matrix-valued valued partial isometries, with matrix sizes
determined by the quantum graph, and the Cuntz-Krieger relations are expressed using the quantum adjacency matrix of the quantum graph,
in analogy to the scalar case.
An important difference to the classical situation is that the matrix partial isometries are not required to have mutually orthogonal ranges,
as this condition does not generalize to matrices in a natural way. Therefore, the quantum Cuntz-Krieger algebra of a classical graph is typically not
isomorphic to an ordinary Cuntz-Krieger algebra, and will often neither be unital nor nuclear. However, we show that free Cuntz-Krieger algebras, or equivalently,
quantum Cuntz-Krieger algebras associated with classical graphs, are always $ KK $-equivalent to Cuntz-Krieger algebras.
Our main results concern the quantum Cuntz-Krieger algebras associated with two basic examples of quantum graphs, namely the trivial and complete quantum graphs
associated to an arbitrary measured finite-dimensional $ C^\ast $-algebra $ (B,\psi) $. The first example we consider in detail, namely the quantum Cuntz-Krieger
algebra of the trivial quantum graph $ TM_N $ on the full matrix algebra $ M_N(\mathbb{C}) $, can be viewed as a non-unital version of Brown's universal algebra
generated by the entries of a unitary $ N \times N $-matrix \cite{Brownext}. For $ N > 1 $, the quantum Cuntz-Krieger algebra of $ TM_N $ is neither unital,
nuclear nor simple, but it is always $ KK $-equivalent to $ C(S^1) $. We exhibit a description of matrices over this algebra in terms of a free product. The
second example, namely the quantum Cuntz-Krieger algebra associated to the complete quantum graph $ K(B,\psi) $, is even more intriguing. We show
that this $ C^\ast $-algebra always admits a canonical quotient map onto the Cuntz algebra $ \O_n $, where $ n = \dim(B) $. Moreover, for certain
quantum complete graphs we are able to show that this map is an isomorphism. This fact, which seems far from obvious from the defining relations, is
proved using monoidal equivalence of the quantum automorphism groups of the underlying quantum graphs. In particular, our results show that for $ N > 1 $
the quantum Cuntz-Krieger algebras of the complete quantum graphs $ K(M_N(\mathbb{C}), \tr) $ are unital, nuclear, simple, and pairwise non-isomorphic,
even on the level of $ KK $-theory.
We also discuss how quantum symmetries of directed quantum graphs induce quantum symmetries of their associated quantum Cuntz-Krieger algebras in general.
This is particularly interesting when one tries to relate quantum Cuntz-Krieger algebras associated to graphs which are quantum isomorphic, as in our
analysis of the examples mentioned above. In particular, we indicate how the notion of a unitary error basis \cite{Wernerteleportation},
which is well-known in quantum information theory, can be used to find finite-dimensional quantum isomorphisms, which in turn induce crossed product relations
between quantum Cuntz-Krieger algebras. In a sense, the existence of quantum symmetries can be viewed as a substitute for the gauge action which features
prominently in the study of ordinary Cuntz-Krieger algebras. While there exists a gauge action in the quantum case as well, it seems to be of limited use
for understanding the structure of quantum Cuntz-Krieger algebras in general.
Let us briefly explain how the paper is organized. In section \ref{secck} we collect some background material on graphs and their
associated $ C^\ast $-algebras, and introduce free graph $ C^\ast $ algebras and free Cuntz-Krieger algebras. We show that these algebras are $ KK $-equivalent
to ordinary graph $ C^\ast $-algebras and Cuntz-Krieger algebras, respectively. After reviewing some facts about finite quantum spaces,
that is, measured finite-dimensional $ C^\ast $-algebras, we define directed quantum graphs in section \ref{secqck}. We then introduce our main object of
study, namely quantum Cuntz-Krieger algebras. In section \ref{secexamples} we discuss some examples of quantum graphs and their associated $ C^* $-algebras.
We show that the quantum Cuntz-Krieger algebras associated with classical graphs lead precisely to free Cuntz-Krieger algebras, and look at several
concrete examples of quantum graphs. We also discuss two natural operations on directed quantum graphs, obtained by taking direct sums and tensor products
of their underlying $ C^\ast $-algebras, respectively. Section \ref{secamplification} is concerned with a general procedure to assign quantum graphs to
classical graphs, essentially by replacing all vertices with matrix blocks of a fixed size. We analyze the structure of the resulting quantum Cuntz-Krieger
algebras, and show that they are always $ KK $-equivalent to their classical counterparts. This allows one in particular
to determine the $ K $-theory of the quantum Cuntz-Krieger algebra of the trivial quantum graph on a matrix algebra mentioned above.
In section \ref{secquantumsymmetry} we explain how quantum symmetries of quantum graphs induce actions on quantum Cuntz-Krieger algebras.
We also discuss the canonical gauge action, in analogy to the classical situation.
The construction of quantum symmetries works in fact at the level of linking algebras associated with arbitrary quantum isomorphisms of quantum graphs.
This is used together with the some unitary error basis constructions in section \ref{secunitaryerror} to study the structure of
the quantum Cuntz-Krieger algebras of the trivial and complete quantum graphs associated to a full matrix algebra equipped with its standard trace. In the final
section \ref{secquantumcomplete} we gather the required results from the preceding sections to furnish a proof of our main theorem for
quantum Cuntz-Krieger algebras of complete quantum graphs.
Let us conclude with some remarks on notation. The closed linear span of a subset $ X $ of a Banach space is denoted by $ [X] $. If $ F $ is a finite set
and $ A $ a $ C^\ast $-algebra we shall write $ M_F(A) $ for the $ C^\ast $-algebra of matrices indexed by elements from $ F $ with entries in $ A $.
We write $ \otimes $ both for algebraic tensor products and for the minimal tensor product of $ C^\ast $-algebras. For operators
on multiple tensor products we use the leg numbering notation.
\subsection*{Acknowledgements} The authors are indebted to Li Gao for fruitful discussions on unitary error bases and their connections with quantum isomorphisms.
MB and KE were partially supported by NSF Grant DMS-2000331. CV and MW were partially supported by SFB-TRR 195 ``Symbolic Tools in Mathematics and their
Application'' at Saarland University. Parts of this project were completed while the authors participated in the March 2019 Thematic Program
``New Developments in Free Probability and its Applications'' at CRM (Montreal) and the October 2019 Mini-Workshop ``Operator Algebraic Quantum Groups''
at Mathematisches Forschungsinstitut Oberwolfach. The authors gratefully acknowledge the support and productive research environments provided by these institutes.
\section{Cuntz-Krieger algebras} \label{secck}
In this section we review the definition of Cuntz-Krieger algebras and graph $ C^\ast $-algebras \cite{CKalgebras}, \cite{EnomotoWatatanigraph}, \cite{KPRck},
\cite{Raeburngraph}, and introduce a free variant of these algebras. Our conventions for graphs and graph $ C^\ast $-algebras will follow \cite{KPRck}.
\subsection{Graphs}
A directed graph $ E = (E^0, E^1) $ consists of a set $ E^0 $ of vertices and a set $ E^1 $ of edges,
together with source and range maps $ s, r: E^1 \rightarrow E^0 $. A vertex $ v \in E^0 $ is called a sink iff $ s^{-1}(v) $ is empty, and a source iff $ r^{-1}(v) $
is empty. That is, a sink is a vertex which emits no edges, and a source is a vertex which receives no edges. A self-loop is an edge with $ s(e) = r(e) $.
The graph $ E $ is called simple if the map $ E^1 \rightarrow E^0 \times E^0, e \mapsto (s(e), r(e)) $ is injective.
The line graph $ LE $ of $ E $ is the directed graph with vertex set $ EL^0 = E $, edge set
$$
EL^1 = \{(e,f) \mid r(e) = s(f) \} \subset E \times E,
$$
and the source and range maps $ s, r: LE^1 \rightarrow LE^0 $ given by projection to the first and second coordinates, respectively.
By construction, the line graph $ LE $ is simple.
The adjacency matrix of $ E = (E^0, E^1) $ is the $ E^0 \times E^0 $-matrix
$$
B_E(v,w) = |\{e \in E^1 \mid s(e) = v, r(e) = w \}|,
$$
and the edge matrix of $ E $ is the $ E^1 \times E^1 $-matrix with entries
$$
A_E(e,f) =
\begin{cases}
1 & r(e) = s(f) \\
0 & \text{else}.
\end{cases}
$$
Note that the edge matrix $ A_E $ of $ E $ equals the adjacency matrix $ B_{LE} $ of $ LE $.
We will only be interested in finite directed graphs in the sequel, that is, directed graphs $ E = (E^0,E^1) $ such that both $ E^0 $ and $ E^1 $ are
finite sets. This requirement can be substantially relaxed \cite{KPRck}.
\subsection{Graph $ C^\ast $-algebras and Cuntz-Krieger algebras}
We recall the definition of the graph $ C^\ast $-algebra of a finite directed graph $ E = (E^0, E^1) $.
A Cuntz-Krieger $ E $-family in a $ C^\ast $-algebra $ D $ consists of mutually orthogonal projections $ p_v \in D $ for all $ v \in E^0 $ together with partial
isometries $ s_e \in D $ for all $ e \in E^1 $ such that
\begin{bnum}
\item[a)] $ s_e^* s_e = p_{r(e)} $ for all edges $ e \in E^1 $
\item[b)] $ p_v = \sum_{s(e) = v} s_e s_e^* $ whenever $ v \in E^0 $ is not a sink.
\end{bnum}
The graph $ C^\ast $-algebra of $ E $ can then be defined as follows.
\begin{definition} \label{defgraphcstar}
Let $ E = (E^0, E^1) $ be a finite directed graph. The graph $ C^\ast $-algebra $ C^\ast(E) $ is the universal $ C^\ast $-algebra generated by a
Cuntz-Krieger $ E $-family. We write $ P_v $ and $ S_e $ for the corresponding projections and partial isometries in $ C^\ast(E) $, associated
with $ v \in E^0 $ and $ e \in E^1 $, respectively.
\end{definition}
That is, given any Cuntz-Krieger $ E $-family in a $ C^\ast $-algebra $ D $, with projections $ p_v $ for $ v \in E^0 $ and partial isometries $ s_e $
for $ e \in E^1 $, there exists a unique $ \ast $-homomorphism $ \phi: C^\ast(E) \rightarrow D $ such that $ \phi(P_v) = p_v $ and $ \phi(S_e) = s_e $.
Next we recall the definition of Cuntz-Krieger algebras \cite{CKalgebras}. If $ A \in M_N(\mathbb{Z}) $ is a matrix with entries $ A(i,j) \in \{0,1 \} $
then a Cuntz-Krieger $ A $-family in a $ C^\ast $-algebra $ D $ consists of partial isometries $ s_1, \dots, s_N \in D $
with mutually orthogonal ranges such that the Cuntz-Krieger relations
$$
s_i^* s_i = \sum_{j = 1}^N A(i,j) s_j s_j^*
$$
hold for all $ 1 \leq i \leq N $.
\begin{definition} \label{defck}
Let $ A \in M_N(\mathbb{Z}) $ be a matrix with entries $ A(i,j) \in \{0,1 \} $. The Cuntz-Krieger algebra $ \O_A $ is the $ C^\ast $-algebra
generated by a universal Cuntz-Krieger $ A $-family, that is, it is the universal $ C^\ast $-algebra generated by partial isometries $ S_1, \dots, S_N $
with mutually orthogonal ranges, satisfying
$$
S_i^* S_i = \sum_{j = 1}^N A(i,j) S_j S_j^*
$$
for all $ 1 \leq i \leq N $.
\end{definition}
In contrast to \cite{CKalgebras}, we do not make any further assumptions on the matrix $ A $ in Definition \ref{defck} in the sequel, except that it should
have entries in $ \{0,1\} $. Accordingly, the algebras $ \O_A $ may sometimes be rather degenerate or even trivial, as for instance if $ A = 0 $.
However, we have adopted this setting for the sake of consistency with our definitions in the quantum case further below.
If $ E $ is a graph with no sinks and no sources then the graph $ C^\ast $-algebra $ C^\ast(E) $ can be canonically identified with the
Cuntz-Krieger algebra associated with the edge matrix $ A_E $ of $ E $.
In particular, the projections in $ C^\ast(E) $ associated to vertices of $ E $ need not be mentioned explicitly in this case.
We note that the graph $ C^\ast $-algebra of a graph $ E $ with no sinks and no sources is completely determined by the line graph $ LE $ of $ E $,
keeping in mind that the edge matrix $ A_E $ equals the adjacency matrix $ B_{LE} $, see also \cite{MRSrepresentationsck}. Viewing $ C^\ast(E) $
as being associated with the line graph of $ E $ motivates our generalizations further below, where we will replace the matrix $ A $ in Definition \ref{defck}
with the quantum adjacency matrix of a directed quantum graph.
\begin{remark}
It is known that all graph $ C^\ast $-algebras of finite directed graphs without sinks are isomorphic to Cuntz-Krieger algebras \cite{ARcorners}.
\end{remark}
\subsection{Free graph $ C^\ast $-algebras and free Cuntz-Krieger algebras}
Borrowing terminology from \cite{BSliberation}, we shall now consider ``liberated'' analogues of graph $ C^\ast $-algebras and Cuntz-Krieger algebras.
In the case of graphs, the input for this construction is a finite directed graph $ E = (E^0, E^1) $ as above. By a free Cuntz-Krieger $ E $-family
in a $ C^\ast $-algebra $ D $ we shall mean a collection of projections $ p_v \in D $ for all $ v \in E^0 $ together with partial isometries $ s_e \in D $
for all $ e \in E^1 $ such that
\begin{bnum}
\item[a)] $ s_e^* s_e = p_{r(e)} $ for all edges $ e \in E^1 $
\item[b)] $ p_v = \sum_{s(e) = v} s_e s_e^* $ whenever $ v \in E^0 $ is not a sink.
\end{bnum}
That is, the only difference to an ordinary Cuntz-Krieger $ E $-family is that the projections $ p_v $ are no longer required to be mutually orthogonal.
Stipulating that the $ p_v $ are mutually orthogonal is equivalent to saying that the $ C^\ast $-algebra generated by the projections $ p_v $
is commutative. In the same way as in the liberation of matrix groups \cite{BSliberation}, removing commutation relations of this type leads to the following
free version of the notion of a graph $ C^\ast $-algebra.
\begin{definition} \label{defliberatedgraphcstar}
Let $ E = (E^0, E^1) $ be a finite directed graph. The free graph $ C^\ast $-algebra $ \FC^\ast(E) $ is the universal $ C^\ast $-algebra generated by a
free Cuntz-Krieger $ E $-family. We write $ P_v $ and $ S_e $ for the corresponding projections and partial isometries in $ \FC^\ast(E) $, associated
with $ v \in E^0 $ and $ e \in E^1 $, respectively.
\end{definition}
Of course, a similar definition can be made in the Cuntz-Krieger case as well. For the sake of definiteness, let us say that
a free Cuntz-Krieger $ A $-family in a $ C^\ast $-algebra $ D $, associated with a matrix $ A \in M_N(\mathbb{Z}) $ with entries $ A(i,j) \in \{0,1 \} $,
consists of partial isometries $ s_1, \dots, s_N \in D $ such that the Cuntz-Krieger relations
$$
s_i^* s_i = \sum_{j = 1}^N A(i,j) s_j s_j^*
$$
hold for all $ 1 \leq i \leq N $.
\begin{definition} \label{defliberatedck}
Let $ A \in M_N(\mathbb{Z}) $ be a matrix with entries $ A(i,j) \in \{0,1 \} $. The free Cuntz-Krieger algebra $ \FO_A $ is the universal $ C^* $-algebra
generated by partial isometries $ S_1, \dots, S_N $, satisfying the relations
$$
S_i^* S_i = \sum_{j = 1}^N A(i,j) S_j S_j^*
$$
for all $ i $.
\end{definition}
We note that free graph $ C^\ast $-algebras and free Cuntz-Krieger algebras always exist, keeping in mind that the norms of all generators are uniformly
bounded in any representation of the universal $ \ast $-algebra generated by a free Cuntz-Krieger family.
Let us also remark that the free graph $ C^* $-algebra of a finite directed graph $ E $ with no sinks and no sources agrees with the free Cuntz-Krieger algebra
associated with the edge matrix $ A_E $.
For any finite directed graph $ E $ and any matrix $ A $ as above there are canonical surjective $ \ast $-homomorphisms $ \pi: \FC^\ast(E) \rightarrow C^\ast(E) $
and $ \pi: \FO_A \rightarrow \O_A $, respectively, obtained directly from the universal property. These maps are not isomorphisms in general.
For instance, if $ E $ is the graph with two vertices and no edges then $ C^\ast(E) = \mathbb{C} \oplus \mathbb{C} $,
whereas $ \FC^\ast(E) = \mathbb{C} * \mathbb{C} $ is the non-unital free product of two copies of $ \mathbb{C} $.
However, we note that if $ E $ is the graph with a single vertex and $ N $ self-loops then the canonical projection induces an
isomorphism $ \FC^\ast(E) \cong C^\ast(E) $, identifying the free graph $ C^\ast $-algebra with the Cuntz algebra $ \O_N $.
Let us now elaborate on the relation between $ \FC^\ast(E) $ and $ C^\ast(E) $ for an arbitrary finite directed graph $ E $,
and similarly on the relation between $ \FO_A $ and $ \O_A $.
\begin{theorem} \label{graphversusfreegraph}
Let $ E $ be a finite directed graph. Then the canonical projection map $ \FC^\ast(E) \rightarrow C^\ast(E) $ is a $ KK $-equivalence.
Similarly, if $ A \in M_N(\mathbb{Z}) $ is a matrix with entries $ A(i,j) \in \{0,1 \} $ then the canonical projection $ \FO_A \rightarrow \O_A $
is a $ KK $-equivalence.
\end{theorem}
\begin{proof}
The proof is analogous for graph algebras and Cuntz-Krieger algebras, therefore we shall restrict attention to the case of graph algebras.
Adapting a well-known argument from \cite{Cuntzfreeproduct},
we will show more generally that $ C^\ast(E) $ and $ \FC^\ast(E) $ cannot be distinguished by any homotopy invariant functor on the category
of $ C^\ast $-algebras which is stable under tensoring with finite matrix algebras.
Firstly, we claim that there exists a $ \ast $-homomorphism $ \phi: C^\ast(E) \rightarrow M_{E^0}(\FC^\ast(E)) $ satisfying
\begin{align*}
\phi(P_v)_{xy} &= \delta_{x,v} \delta_{y,v} P_v, \\
\phi(S_e)_{xy} &= \delta_{x, s(e)} \delta_{y, r(e)} S_e
\end{align*}
for $ v \in E^0 $ and $ e \in E^1 $. For this it suffices to show that the elements $ \phi(P_v), \phi(S_e) $ in $ M_{E^0}(\FC^\ast(E)) $ given by the above
formulas define a Cuntz-Krieger $ E $-family. Clearly, the elements $ P_v $ are mutually orthogonal projections,
and the elements $ \phi(S_e) $ are partial isometries such that
$$
(\phi(S_e)^* \phi(S_e))_{xy} = \delta_{x, r(e)} \delta_{y, r(e)} S_e^* S_e = \delta_{x, r(e)} \delta_{y, r(e)} P_{r(e)} = \phi(P_{r(e)})_{xy}
$$
and
\begin{align*}
\phi(P_v)_{xy} = \delta_{x, v} \delta_{y, v} P_v
&= \delta_{x, v} \delta_{y, v} \sum_{s(f) = v} S_f S_f^* \\
&= \sum_{s(f) = v} \delta_{x, s(f)} \delta_{y, s(f)}S_f S_f^* \\
&= \sum_{s(f) = v} \sum_{z \in E^0} \phi(S_f)_{xz} (\phi(S_f)_{yz})^* \\
&= \sum_{s(f) = v} (\phi(S_f) \phi(S_f)^*)_{xy}
\end{align*}
if $ v \in E^0 $ is not a sink, as required.
Recall that we write $ \pi: \FC^\ast(E) \rightarrow C^*(E) $ for the canonical projection.
Fixing a vertex $ w \in E^0 $, we claim that $ M_{E^0}(\pi) \circ \phi $ is homotopic to the embedding $ \iota $ of $ C^\ast(E) $
into the corner of $ M_{E^0}(C^\ast(E)) $ corresponding to $ w $.
For this we consider the $ \ast $-homomorphisms $ \mu_t: C^\ast(E) \rightarrow M_{E^0}(C^\ast(E)) $ for $ t \in [0,1] $ given by
$$
\mu_t(P_v) = u^v_t \iota(P_v) (u^v_t)^*, \qquad \mu_t(S_e) = u^{s(e)}_t \iota(S_e) (u^{r(e)}_t)^*,
$$
where $ u^x_t $ for $ x \in E^0 $ with $ x \neq w $ is the rotation matrix
$$
u_t =
\begin{pmatrix}
\cos(2 \pi t) & \sin(2 \pi t) \\
-\sin(2 \pi t) & \cos(2 \pi t)
\end{pmatrix}
$$
placed in the block corresponding to the indices $ w $ and $ x $, and $ u^x_t = \id $ for $ x = w $. In a similar way as above one checks that $ \mu_t $
preserves the relations for $ C^\ast(E) $. Indeed, the elements $ \mu_t(P_v) $ are mutually orthogonal projections
since $ P_v, P_w $ for $ v \neq w $ are orthogonal in $ C^\ast(E) $ and the unitaries $ u^x_t $ have scalar entries.
Moreover, for $ t \in [0,1] $ the elements $ \mu_t(S_e) $ are partial isometries such that
$$
\mu_t(S_e)^* \mu_t(S_e) = u^{r(e)}_t \iota(S_e^* S_e) (u^{r(e)}_t)^* = u^{r(e)}_t \iota(P_{r(e)}) (u^{r(e)}_t)^* = \mu_t(P_{r(e)}),
$$
and
\begin{align*}
\mu_t(P_v) &= u^v_t \iota(P_v) (u^v_t)^*
= \sum_{s(f) = v} u^{s(f)}_t \iota(S_f S_f^*) (u^{s(f)}_t)^*
= \sum_{s(f) = v} \mu_t(S_f) \mu_t(S_f)^*
\end{align*}
if $ v $ is not a sink. By construction we have $ \mu_0 = \iota $ and $ \mu_1 = M_{E^0}(\pi) \circ \phi $.
The composition $ \phi \circ \pi $ looks the same as $ M_{E^0}(\pi) \circ \phi $ on generators, and a similar homotopy shows that $ \phi \circ \pi $
is homotopic to the embedding $ \FC^\ast(E) \rightarrow M_{E^0}(\FC^\ast(E)) $ associated with a fixed vertex $ w $. This finishes the proof.
\end{proof}
\section{Quantum Cuntz-Krieger algebras} \label{secqck}
In this section we define our quantum analogue of Cuntz-Krieger algebras. Since the input for this construction is the quantum adjacency
matrix of a directed quantum graph, we shall first review the concept of a quantum graph.
\subsection{Quantum graphs} \label{parqgraph}
The notion of a quantum graph has been considered with some variations by a number of authors, see \cite{EKSrankonesubspaces}, \cite{Weaverquantumrelations},
\cite{DSWnoncommutativegraphs}, \cite{MRVmorita}, \cite{BCEHPSWbigalois}. We will follow the approach in \cite{MRVmorita}, \cite{BCEHPSWbigalois}, and adapt
it to the setting of directed graphs.
Assume that $ B $ is a finite dimensional $ C^\ast $-algebra $ B $ and let $ \psi: B \rightarrow \mathbb{C} $ be a faithful state. We
denote by $ L^2(B) = L^2(B, \psi) $ the Hilbert space obtained by equipping $ B $ with the inner product $ \bra x, y \ket = \psi(x^* y) $.
Moreover let us write $ m: B \otimes B \rightarrow B $ for the multiplication map of $ B $ and $ m^* $ for its adjoint, noting that $ m $ can be
viewed as a linear operator $ L^2(B) \otimes L^2(B) \rightarrow L^2(B) $.
If $ B = C(X) $ is the algebra of functions on a finite set $ X $ then states on $ B $ correspond to probability measures on $ X $.
The most natural choice is to take for $ \psi $ the state corresponding to the uniform measure in this case.
For an arbitrary finite dimensional $ C^\ast $-algebra $ B $ we have the following well-known condition, singling out certain natural choices among all possible
states on $ B $ in a similar way \cite{Banicafusscatalan}.
\begin{definition} \label{deffqs}
Let $ B $ be a finite dimensional $ C^\ast $-algebra and $ \delta > 0 $. A faithful state $ \psi: B \rightarrow \mathbb{C} $ is called a $ \delta $-form
if $ m m^* = \delta^2 \id $.
By a finite quantum space $ (B, \psi) $ we shall mean a finite dimensional $ C^\ast $-algebra $ B $ together with
a $ \delta $-form $ \psi: B \rightarrow \mathbb{C} $.
\end{definition}
If $ B $ is a finite dimensional $ C^\ast $-algebra then we have $ B \cong \bigoplus_{a = 1}^d M_{N_a}(\mathbb{C}) $ for some $ N_1, \dots, N_d $.
A state $ \psi $ on $ B $ can be written uniquely in the form
$$
\psi(x) = \sum_{a = 1}^d \Tr(Q_{(a)} x_i)
$$
for $ x = (x_1, \dots, x_d) $, where the $ Q_{(a)} \in M_{N_a}(\mathbb{C}) $ are positive matrices satisfying $ \sum_{a = 1}^d \Tr(Q_{(a)}) = 1 $.
Then $ \psi $ is a $ \delta $-form iff $ Q_{(a)} $ is invertible and $ \Tr(Q_{(a)}^{-1}) = \delta^2 $ for all $ a $. Here $ \Tr $ denotes the natural
trace, given by summing all diagonal terms of a matrix.
Note that we may assume without loss of generality that all matrices $ Q_{(a)} $ in the definition of $ \psi $ are diagonal. We shall say that $ (B, \psi) $
as above is in standard form in this case.
Any finite dimensional $ C^\ast $-algebra $ B $ admits a unique tracial $ \delta $-form for a uniquely determined value of $ \delta $. Explicitly,
this is the tracial state given by
$$
\tr(x) = \frac{1}{\dim(B)} \sum_{a = 1}^d N_a \Tr(x_i),
$$
and we have $ \delta^2 = \dim(B) $. Note that if $ B = C(X) $ is commutative then this corresponds to the uniform measure on $ X $, and $ \delta^2 $ is the
cardinality of $ X $.
For later purposes it will be useful to record an explicit formula for the adjoint of the multiplication map in a finite quantum space.
\begin{lemma} \label{mstarcomputation}
Let $ (B, \psi) $ be a finite quantum space in standard form as described above, and consider the linear basis of $ B $ given by the adapted matrix units
$$
f^{(a)}_{ij} = (Q_{(a)}^{-1/2})_{ii} e^{(a)}_{ij} (Q_{(a)}^{-1/2})_{jj},
$$
where $ e^{(a)}_{ij} $ in $ M_{N_a}(\mathbb{C}) $ are the standard matrix units.
Then we have $ (f^{(a)}_{ij})^* = f^{(a)}_{ji} $ and
$$
m^*(f^{(a)}_{ij}) = \sum_k f^{(a)}_{ik} \otimes f^{(a)}_{kj}
$$
for all $ a,i,j $.
\end{lemma}
\begin{proof}
Since the matrices $ Q_{(a)} $ are positive we clearly have $ (f^{(a)}_{ij})^* = f^{(a)}_{ji} $. Moreover, observing
$$
f^{(b)}_{rs} f^{(c)}_{pq} = \delta_{bc} (Q_{(b)}^{-1})_{sp} f^{(b)}_{rq}
$$
and $ \psi(f^{(a)}_{kl}) = \delta_{kl} $, we compute
\begin{align*}
\bra f^{(a)}_{ij}, m(f^{(b)}_{rs} \otimes f^{(c)}_{pq}) \ket
&= \delta_{bc} \psi(f^{(a)}_{ji} (Q_{(b)}^{-1})_{sp} f^{(b)}_{rq})
= \delta_{abc} (Q_{(a)}^{-1})_{sp} (Q_{(a)}^{-1})_{ir} \delta_{jq}
\end{align*}
and
\begin{align*}
\sum_k \bra f^{(a)}_{ik} \otimes f^{(a)}_{kj}, f^{(b)}_{rs} \otimes f^{(c)}_{pq} \ket
&= \sum_k \psi(f^{(a)}_{ki} f^{(b)}_{rs}) \psi(f^{(a)}_{jk} f^{(c)}_{pq}) \\
&= \delta_{ab} \delta_{ac} \sum_k (Q_{(a)}^{-1})_{ir} (Q_{(a)}^{-1})_{kp} \psi(f^{(a)}_{ks}) \psi(f^{(a)}_{jq}) \\
&= \delta_{abc} (Q_{(a)}^{-1})_{ir} (Q_{(a)}^{-1})_{sp} \delta_{jq}.
\end{align*}
This yields the claim.
\end{proof}
Let us now discuss the concept of a quantum graph. We shall be interested in directed quantum graphs in the following sense.
\begin{definition} \label{defdirectedqgraph}
Let $ B $ be a finite dimensional $ C^\ast $-algebra and $ \psi: B \rightarrow \mathbb{C} $ a $ \delta $-form. A linear operator $ A: L^2(B) \rightarrow L^2(B) $
is called a quantum adjacency matrix if
$$
m(A \otimes A) m^* = \delta^2 A.
$$
A directed quantum graph $ \G = (B, \psi, A) $ is a finite quantum space $ (B, \psi) $ together with a quantum adjacency matrix.
\end{definition}
In order to explain Definition \ref{defdirectedqgraph} let us consider the case that $ B = C(X) $ is the quantum space associated with a finite set $ X $,
with $ \psi $ being given by the uniform measure. A linear operator $ A: L^2(B) \rightarrow L^2(B) $ can be identified canonically with a matrix
in $ M_X(\mathbb{C}) $. Moreover, a straightforward calculation shows that
$$
\frac{1}{|X|} m(A \otimes B) m^*
$$
is the Schur product of $ A, B \in M_X(\mathbb{C}) $, given by entrywise multiplication.
Hence $ A $ is a quantum adjacency matrix iff it is an idempotent with respect to the Schur product in this case, which is equivalent to saying that $ A $ has
entries in $ \{0,1\} $.
According to the above discussion, every simple finite directed classical graph $ E = (E^0, E^1) $ gives rise to a directed quantum graph
in a natural way. More precisely, if $ A_E $ denotes the adjacency matrix of $ E $ then we obtain a
directed quantum graph structure on $ B = C(E^0) $ by taking the state $ \psi $ which corresponds to counting measure,
and the operator $ A: L^2(B) \rightarrow L^2(B) $ given by $ A(e_i) = \sum_j A(i,j) e_j $.
Conversely, every directed quantum graph structure on a finite dimensional commutative $ C^\ast $-algebra $ B = C(X) $ arises from
a simple finite directed graph on the vertex set $ X $.
For a general finite quantum space $ (B, \psi) $ it will be convenient for our considerations further below to write down the quantum adjacency
matrix condition in terms of bases.
\begin{lemma} \label{adjacencycomputation}
Let $ (B, \psi) $ be a finite quantum space in standard form. Then a linear operator $ A: L^2(B) \rightarrow L^2(B) $, given by
$$
A(f^{(a)}_{ij}) = \sum_{brs} A_{ija}^{rsb} f^{(b)}_{rs}
$$
in terms of the adapted matrix units, is a directed quantum adjacency matrix iff
$$
\sum_{ks} (Q_{(b)}^{-1})_{ss} A_{ika}^{rsb} A_{kja}^{snb} = \delta^2 A_{ija}^{rnb}
$$
for all $ a,b,i,j,r,n $.
\end{lemma}
\begin{proof}
Using Lemma \ref{mstarcomputation} we calculate
\begin{align*}
m(A \otimes A) m^*(f^{(a)}_{ij}) &= \sum_k A(f^{(a)}_{ik}) A(f^{(a)}_{kj}) \\
&= \sum_k \sum_{brs} \sum_{cmn} A_{ika}^{rsb} f^{(b)}_{rs} A_{kja}^{mnc} f^{(c)}_{mn} \\
&= \sum_k \sum_{brsn} (Q_{(b)}^{-1})_{ss} A_{ika}^{rsb} A_{kja}^{snb} f^{(b)}_{rn},
\end{align*}
so that comparing coefficients yields the claim.
\end{proof}
We point out that there is a rich supply of directed quantum adjacency matrices and quantum graphs. Let $ B $ be a finite dimensional $ C^\ast $-algebra and
let $ \tr $ be the unique tracial $ \delta $-form on $ B $. Every element $ P \in B \otimes B^{op} $ has a Choi-Jamio\l kowski form, that is, there exists a
unique linear map $ A: B \rightarrow B $ such that
$$
P = P_A = \frac{1}{\dim(B)} (1 \otimes A)m^*(1),
$$
where $ m^*: B \rightarrow B \otimes B $ is the adjoint of multiplication with respect to $ \tr $.
Then $ A $ is a quantum adjacency matrix with respect to $ (B, \tr) $ iff $ P $ is idempotent, that is, iff $ P^2 = P $.
Moreover, idempotents in $ B \otimes B^{op} $ can be naturally obtained as follows. Assume that $ B \hookrightarrow B(\H) $ is unitally embedded into
the algebra of bounded operators on some finite dimensional Hilbert space $ \H $, and let $ B' \subset B(\H) $ be the commutant of $ B $.
Then $ B \otimes B^{op} $ identifies with the space of all completely bounded $ B' $-$ B' $-bimodule maps from $ B(\H) $ to itself.
In particular, idempotents in $ B \otimes B^{op} $ are the same thing as direct sum decompositions $ B(\H) = S \oplus R $ of $ B' $-$ B' $-bimodules.
Taking $ B = M_N(\mathbb{C}) $ and the identity embedding into $ B(\mathbb{C}^N) = M_N(\mathbb{C}) $ we see that there is a bijective
correspondence between quantum graph structures on $ (M_N(\mathbb{C}), \tr) $ and vector space direct sum decompositions $ M_N(\mathbb{C}) = S \oplus R $.
\begin{remark}
One could work more generally with arbitrary faithful positive linear functionals $ \psi $ instead of $ \delta $-forms, by modifying the
defining relation of a quantum adjacency matrix in Definition \ref{defdirectedqgraph} to
$$
m(A \otimes A) m^* = A mm^*.
$$
We will however restrict ourselves to $ \delta $-forms in the sequel, as this will allow us to remain closer to the classical theory in the
commutative case.
\end{remark}
\begin{remark}
The definition of a quantum graph in \cite{MRVmorita}, \cite{BCEHPSWbigalois} contains further conditions on the quantum adjacency matrix. If $ B = C(X) $
is commutative then these conditions correspond to requiring that the matrix $ A \in M_X(\mathbb{C}) $ is symmetric and has entries $ 1 $
on the diagonal, respectively. That is, the quantum graphs considered in these papers are undirected and have all self-loops. Neither of these conditions
is appropriate in connection with Cuntz-Krieger algebras.
\end{remark}
\subsection{Quantum Cuntz-Krieger algebras}
Let us now define the quantum Cuntz-Krieger algebra associated to a directed quantum graph. Comparing with the definition of graph $ C^\ast $-algebras,
we note that the quantum graph used as an input in our definition may be thought of as an analogue of the line graph of a classical graph.
If $ \G = (B, \psi, A) $ is a directed quantum graph then we shall say that a quantum Cuntz-Krieger $ \G $-family in a $ C^\ast $-algebra $ D $ is a linear
map $ s: B \rightarrow D $ such that
\begin{bnum}
\item[a)] $ \mu_D(\id \otimes \mu_D)(s \otimes s^* \otimes s)(\id \otimes m^*)m^* = s $
\item[b)] $ \mu_D(s^* \otimes s)m^* = \mu_D(s \otimes s^*) m^* A $,
\end{bnum}
where $ \mu_D: D \otimes D \rightarrow D $ is the multiplication map for $ D $ and $ s^*(b) = s(b^*)^* $ for $ b \in B $.
We also recall that $ m^* $ denotes the adjoint of the multiplication map for $ B $ with respect to the inner product given by $ \psi $.
\begin{definition} \label{defqck}
Let $ \G = (B, \psi, A) $ be a directed quantum graph. The quantum Cuntz-Krieger algebra $ \FO(\G) $ is the universal $ C^\ast $-algebra
generated by a quantum Cuntz-Krieger $ \G $-family $ S: B \rightarrow \FO(\G) $.
\end{definition}
In other words, the quantum Cuntz-Krieger algebra $ \FO(\G) $ satisfies the following universal property. If $ D $ is a $ C^\ast $-algebra
and $ s: B \rightarrow D $ a quantum Cuntz-Krieger $ \G $-family, then there exists a unique $ \ast $-homomorphism $ \varphi: \FO(\G) \rightarrow D $
such that $ \varphi(S(b)) = s(b) $ for all $ b \in B $.
\begin{remark}
We note that Definition \ref{defqck} makes sense for a finite dimensional $ C^\ast $-algebra $ B $ together with a faithful positive linear
functional $ \psi $ and an arbitrary linear map $ A: L^2(B) \rightarrow L^2(B) $. At this level of generality one can shift information from $ \psi $ into
the matrix $ A $ and vice versa, without changing the resulting $ C^\ast $-algebra.
Our definition will allow us to remain closer to the standard setup for Cuntz-Krieger algebras.
\end{remark}
It is not difficult to check that
the quantum Cuntz-Krieger algebra $ \FO(\G) $ always exists. This is done most easily by rewriting Definition \ref{defqck} in terms of a linear basis for
the algebra $ B $. In the sequel we shall say that a directed quantum graph $ \G = (B, \psi, A) $ is in standard form if its underlying finite quantum space
is, compare paragraph \ref{parqgraph}.
\begin{prop} \label{qckconcrete}
Let $ \G = (B, \psi, A) $ be a directed quantum graph in standard form, and let
\begin{align*}
A(f^{(a)}_{ij}) = \sum_{brs} A_{ija}^{rsb} f^{(b)}_{rs}
\end{align*}
be the quantum adjacency matrix written in terms of the adapted matrix units as discussed further above.
Then the quantum Cuntz-Krieger algebra $ \FO(\G) $ identifies with the universal $ C^\ast $-algebra $ \FO_A $ with generators $ S^{(a)}_{ij} $
for $ 1 \leq a \leq d $ and $ 1 \leq i,j \leq N_a $, satisfying the relations
\begin{align*}
\sum_{rs} S^{(a)}_{ir} (S^{(a)}_{sr})^* S^{(a)}_{sj} &= S^{(a)}_{ij} \\
\sum_l (S^{(a)}_{li})^* S^{(a)}_{lj} &= \sum_{brs} A^{rsb}_{ija} \sum_l S^{(b)}_{rl} (S^{(b)}_{sl})^*
\end{align*}
for all $ a,i,j $.
\end{prop}
\begin{proof}
Let us first consider the elements $ S^{(a)}_{ij} = S(f^{(a)}_{ij}) $ in $ \FO(\G) $. If $ \mu $ denotes the multiplication map for $ \FO(\G) $, then
according to Lemma \ref{mstarcomputation} we get
\begin{align*}
\sum_{rs} S^{(a)}_{ir} (S^{(a)}_{sr})^* S^{(a)}_{sj} &= \sum_{rs} S(f^{(a)}_{ir}) S^*(f^{(a)}_{rs}) S(f^{(a)}_{sj}) \\
&= \sum_{rs} \mu(\id \otimes \mu)(S \otimes S^* \otimes S)(f^{(a)}_{ir} \otimes f^{(a)}_{rs} \otimes f^{(a)}_{sj}) \\
&= \mu(\id \otimes \mu)(S \otimes S^* \otimes S)(\id \otimes m^*)m^*(f^{(a)}_{ij}) \\
&= S(f^{(a)}_{ij}) = S^{(a)}_{ij},
\end{align*}
and similarly
\begin{align*}
\sum_r (S^{(a)}_{ri})^* S^{(a)}_{rj} &= \sum_r \mu(S^* \otimes S)(f^{(a)}_{ir} \otimes f^{(a)}_{rj}) \\
&= \mu(S^* \otimes S)m^*(f^{(a)}_{ij}) \\
&= \mu(S \otimes S^*)m^* A(f^{(a)}_{ij}) \\
&= \sum_{brs} A_{ija}^{rsb} \mu(S \otimes S^*)m^* (f^{(b)}_{rs}) \\
&= \sum_{brsl} A_{ija}^{rsb} \mu(S \otimes S^*)(f^{(b)}_{rl} \otimes f^{(b)}_{ls}) \\
&= \sum_{brsl} A_{ija}^{rsb} S^{(b)}_{rl} (S^{(b)}_{sl})^*.
\end{align*}
Hence, by the definition of $ \FO_A $, there exists a unique $ \ast $-homomorphism $ \phi: \FO_A \rightarrow \FO(\G) $ such
that $ \phi(S^{(a)}_{ij}) = S(f^{(a)}_{ij}) $ for all $ a,i,j $.
Conversely, let us define a linear map $ s: B \rightarrow \FO_A $ by $ s(f^{(a)}_{ij}) = S^{(a)}_{ij} $.
Essentially the same computation as above shows that $ s $ defines a quantum Cuntz-Krieger $ \G $-family in $ \FO_A $, so that there
exists a unique $ \ast $-homomorphism $ \psi: \FO(\G) \rightarrow \FO_A $ satisfying $ \psi(S(b)) = s(b) $ for all $ b \in B $.
It is straightforward to check that the maps $ \phi $ and $ \psi $ are mutually inverse isomorphisms.
\end{proof}
Using matrix notation we can rephrase the relations from Proposition \ref{qckconcrete} in a very concise way.
More precisely, writing $ S^{(a)} \in M_{N_a}(\FO(\G)) $ for the matrix with entries $ S^{(a)}_{ij} = S(f^{(a)}_{ij}) $
and $ \hat{A} $ for the $ d \times d $-matrix with coefficients $ \hat{A}^b_a = A_{ija}^{rsb} $ we obtain
\begin{align*}
S^{(a)} (S^{(a)})^* S^{(a)} &= S^{(a)} \\
(S^{(a)})^* S^{(a)} &= \sum_b \hat{A}^b_a S^{(b)} (S^{(b)})^*
\end{align*}
for all $ 1 \leq a \leq d $. The first formula says that the elements $ S^{(a)} \in M_{N_a}(\FO(\G)) $ are partial isometries. This means in particular that
their entries are bounded in norm by $ 1 $, which implies in turn that the universal $ C^\ast $-algebras $ \FO_A $ and $ \FO(\G) $
always exist. The second formula can be viewed as a matrix-valued version of the classical Cuntz-Krieger relation.
\begin{remark}
From Proposition \ref{qckconcrete} and the above remarks it may appear at first sight that $ \FO(\G) \cong \FO_A $ does not depend on
the $ \delta $-form $ \psi $ in $ \G = (B, \psi, A) $. However, recall from Lemma \ref{adjacencycomputation} that the choice of $ \psi $ is
reflected in the defining relations for the coefficients $ A_{ija}^{rsb} $ of the quantum adjacency matrix.
\end{remark}
\begin{remark}
As will be discussed in more detail at the start of the next section, the notation $ \FO_A $ used in Proposition \ref{qckconcrete} is compatible
with our notation for free Cuntz-Krieger algebras introduced in Definition \ref{defliberatedck}.
\end{remark}
\section{Examples} \label{secexamples}
In this section we take a look at some examples of quantum graphs and their associated quantum Cuntz-Krieger algebras in the sense of
Definition \ref{defqck}.
\subsection{Classical graphs} \label{parclassical}
Assume that $ E = (E^0, E^1) $ is a finite simple directed graph with $ N $ vertices.
The directed quantum graph $ \G $ associated with $ E $ has $ B = C(E^0) = \mathbb{C}^N $ as underlying $ C^\ast $-algebra. We work with the canonical
basis $ e_1, \dots, e_N $ of minimal projections in $ B $ and the normalized standard trace $ \tr: B \rightarrow \mathbb{C} $. That is, $ \tr(e_i) = 1/N $
for all $ i $, and we have $ m(e_i \otimes e_j) = \delta_{ij} e_i $ and $ m^*(e_i) = N e_i \otimes e_i $.
If $ B_E $ denotes the adjacency matrix of $ E $ then
$$
A(e_i) = \sum_{j = 1}^N B_E(i,j) e_j
$$
determines a quantum adjacency matrix $ A: L^2(B) \rightarrow L^2(B) $.
\begin{prop} \label{qgraphalgebraclassical}
Let $ E $ be a finite simple directed graph and let $ \G = (B, \psi, A) $ be the quantum graph corresponding to $ E $ as above. Then the
free Cuntz-Krieger algebra associated with the adjacency matrix $ B_E $ of $ E $ is canonically isomorphic to the quantum Cuntz-Krieger algebra $ \FO(\G) $.
\end{prop}
\begin{proof}
This can be viewed as a special case of Proposition \ref{qckconcrete}, but let us write down the key formulas explicitly.
Note that $ \tr $ is a $ \delta $-form with $ \delta^2 = N $ and consider $ S_i = N S(e_i) \in \FO(\G) $.
Then the defining relations for a free Cuntz-Krieger $ B_E $-family are obtained from
\begin{align*}
S_i S_i^* S_i &= N^3 \mu(\id \otimes \mu)(S(e_i) \otimes S^*(e_i) \otimes S(e_i)) \\
&= N \mu(\id \otimes \mu)(S \otimes S^* \otimes S)(\id \otimes m^*)m^*(e_i) \\
&= N S(e_i) = S_i
\end{align*}
and
\begin{align*}
S_i^* S_i &= N^2 \mu(S^* \otimes S)(e_i \otimes e_i) \\
&= N\mu(S^* \otimes S)m^*(e_i) \\
&= N \mu(S \otimes S^*)m^*(A(e_i)) \\
&= N^2 \sum_{j = 1}^N B_E(i,j) \mu(S \otimes S^*)(e_j \otimes e_j) \\
&= \sum_{j = 1}^N B_E(i,j) S_j S_j^*
\end{align*}
for all $ i $. This yields a $ \ast $-homomorphism $ \FO_{B_E} \rightarrow \FO(\G) $.
Similarly, one checks that the linear map $ s: B \rightarrow \FO_{B_E} $ given by $ s(e_i) = \frac{1}{N} S_i $ is a quantum Cuntz-Krieger $ \G $-family,
which induces a $ \ast $-homomorphism $ \FO(\G) \rightarrow \FO_{B_E} $. These maps are mutually inverse isomorphisms.
\end{proof}
It follows from the remarks after Definition \ref{defdirectedqgraph} that every quantum Cuntz-Krieger algebra $ \FO(\G) $ over a directed quantum
graph $ \G = (B, \psi, A) $ with $ B $ abelian
is a free Cuntz-Krieger algebra associated to some $ 0,1 $-matrix, and conversely, all free Cuntz-Krieger algebras arise in this way.
Let us also point out that already quantum Cuntz-Krieger algebras associated with classical graphs as in Proposition \ref{qgraphalgebraclassical}
may fail to be unital. This is of course in contrast to the situation for ordinary Cuntz-Krieger algebras.
\subsection{Complete quantum graphs and quantum Cuntz algebras} \label{parQKn}
Let us next consider an arbitrary finite quantum space $ (B, \psi) $ in standard form, using the same notation as after Definition \ref{deffqs}.
Following \cite{BCEHPSWbigalois}, we can form the {\it complete quantum graph} on $ (B,\psi) $, which is the directed quantum
graph $ K(B,\psi) = (B,\psi,A) $ with quantum adjacency matrix $ A: L^2(B) \to L^2(B) $ given by $ A(b) = \delta^2 \psi(b)1 $.
In terms of the adapted matrix units $ f_{ij}^{(a)} \in B $ defined in Lemma \ref{mstarcomputation} we get
$$
A(f_{ij}^{(a)}) = \delta_{ij} \delta^2 1 = \sum_{b} \sum_k \delta_{ij}\delta^2 (Q_{(a)})_{kk}f^{(b)}_{kk}.
$$
Therefore, relative to this basis, we have the matrix representation $ A = (A^{klb}_{ija}) $, where
\begin{align*}
A^{klb}_{ija} = \delta_{ij}\delta_{kl} \delta^2(Q_{(b)})_{kk}.
\end{align*}
It follows from Proposition \ref{qckconcrete} and the preceding discussion that the quantum Cuntz-Krieger algebra $ \FO(K(B,\psi)) $ is the
universal $ C^\ast $-algebra with generators $ S_{ij}^{(a)} $ for $ 1 \leq a \leq d, 1 \leq i,j \leq N_a $ and relations
\begin{align*}
\sum_{rs} S^{(a)}_{ir} (S^{(a)}_{sr})^* S^{(a)}_{sj} &= S^{(a)}_{ij}, \\
\sum_r (S^{(a)}_{ri})^* S^{(a)}_{rj} &= \delta_{ij} \delta^2 \sum_{b} \sum_{kl} (Q_{(b)})_{kk}S^{(b)}_{kl} (S^{(b)}_{kl})^*
\end{align*}
for all $ a,i,j $.
\begin{example} \label{completematrixqgraph}
Let us consider explicitly the special case of the complete quantum graph $ K(M_N(\mathbb{C}), \tr) $ on a full matrix algebra $ B = M_N(\mathbb{C}) $.
The $ C^\ast $-algebra $ \FO(K(M_N(\mathbb{C}), \tr)) $ has generators $ S_{ij} $ for $ 1 \leq i,j \leq N $ satisfying the relations
\begin{align*}
\sum_{kl} S_{ik} S^*_{lk} S_{lj} &= S_{ij} \\
\sum_r S^*_{ri} S_{rj} &= \delta_{ij} N \sum_{rs} S_{rs} S^*_{rs}
\end{align*}
for all $ i,j $.
\end{example}
Note that when $ B = \mathbb{C}^d $ is abelian, Proposition \ref{qgraphalgebraclassical} implies that $ \FO(K(\mathbb{C}^d, \tr)) $ is nothing other than
the free Cuntz-Krieger algebra associated to the complete graph $ K_d $,
or equivalently, the free graph $ C^\ast $-algebra associated to the graph with a single vertex and $ d $ self-loops.
Thus $ \FO(K(\mathbb{C}^d, \tr)) $ identifies with the Cuntz algebra $ \O_d $, compare the remarks after Definition \ref{defliberatedck}.
With this in mind, we may call any quantum Cuntz-Krieger algebra of the form $ \FO(K(B,\psi)) $ a {\it quantum Cuntz algebra}.
The algebras obtained in this way are in fact rather closely related to Cuntz algebras, as we discuss next.
\begin{lemma} \label{QCChomo}
Let $ \FO(K(B,\psi)) $ be as above and write $ n = \dim(B) $. Then there exists a surjective $ \ast $-homomorphism $ \phi: \FO(K(B,\psi)) \rightarrow \O_n $
such that
$$
\phi(S_{ij}^{(a)}) = \frac{1}{(Q_{(a)})_{ii}^{1/2} \delta} s_{ij}^{(a)}
$$
for all $ a,i,j $, where $ s_{ij}^{(a)} $ are standard generators of the Cuntz algebra $ \O_n $.
\end{lemma}
\begin{proof}
We just have to check that the elements $ \phi(S^{(a)}_{ij}) $ satisfies the defining relations of $ \FO(K(B,\psi)) $ from above. Indeed, we obtain
\begin{align*}
\sum_{rs} \phi(S^{(a)}_{ir}) \phi(S^{(a)}_{sr})^* \phi(S^{(a)}_{sj})
&= \sum_{rs} \frac{1}{(Q_{(a)})_{ii}^{1/2}(Q_{(a)})_{ss} \delta^3}s^{(a)}_{ir} (s^{(a)}_{sr})^* s^{(a)}_{sj} \\
&=\sum_{s} \frac{1}{(Q_{(a)})_{ii}^{1/2}(Q_{(a)})_{ss} \delta^3}s^{(a)}_{ij} \\
& = \frac{1}{(Q_{(a)})_{ii}^{1/2} \delta} s^{(a)}_{ij} \\
&= \phi(S_{ij}^{(a)}),
\end{align*}
and similarly
\begin{align*}
\sum_r \phi(S^{(a)}_{ri})^* \phi(S^{(a)}_{rj}) &= \sum_r \frac{1}{(Q_{(a)})_{rr} \delta^2}(s^{(a)}_{ri})^* s^{(a)}_{rj} \\
&= \delta_{ij}\\
&= \delta_{ij} \sum_{bkl} s^{(b)}_{kl} (s^{(b)}_{kl})^* \\
&= \delta_{ij} \delta^2 \sum_{bkl} (Q_{(b)})_{kk} \phi(S^{(b)}_{kl}) \phi(S^{(b)}_{kl})^*
\end{align*}
as required.
\end{proof}
\begin{remark}
Lemma \ref{QCChomo} implies in particular that the canonical linear map $ S: B \rightarrow \FO(K(B,\psi)) $ is injective.
This is not always the case for general quantum Cuntz-Krieger algebras. An explicit example will be given in Example \ref{noninj} further below.
\end{remark}
Our main structure result regarding the quantum Cuntz algebras $ \FO(K(B,\psi)) $ can be stated as follows.
\begin{theorem} \label{quantumcompletemain}
Let $ B $ be an $ n $-dimensional $ C^\ast $-algebra and let $ \psi: B \rightarrow \mathbb{C} $ be a $ \delta $-form satisfying $ \delta^2 \in \mathbb{N} $.
Then $ \FO(K(B,\psi)) \cong \O_n $.
\end{theorem}
We will prove Theorem \ref{quantumcompletemain} using methods from the theory of quantum groups in section \ref{secquantumcomplete}.
Under the hypothesis $ \delta^2 \in \mathbb{N} $, Theorem \ref{quantumcompletemain} implies that
the $ \ast $-homomorphism $ \phi: \FO(K(B,\psi)) \rightarrow \O_n $ constructed in Lemma \ref{QCChomo} is an isomorphism.
It seems remarkable that the relations defining $ \FO(K(B,\psi)) $ do indeed characterize the Cuntz algebra $ \O_n $, at least when we restrict
to $ \delta $-forms satisfying the above integrality condition. Already in the special case $ (B, \psi) = (M_N(\mathbb{C}), \tr) $
from Example \ref{completematrixqgraph} it seems not even obvious that $ \FO(K(B, \psi)) $ is {\it unital}. In fact, an easy argument shows that the
element $ e = N^2 \sum_{kl} S_{kl} (S_{kl})^* \in \FO(K(M_N(\mathbb{C}),\tr)) $ satisfies $ S_{ij} e = S_{ij} $ for all $ 1 \leq i,j \leq N $.
In section \ref{secquantumcomplete} we will verify in particular the less evident relation $ e S_{ij} = S_{ij} $ for all $ i,j $.
We note at the same time that $ \FO(K(M_N(\mathbb{C}),\tr)) $ is very different from the universal $ C^\ast $-algebra generated
by the coefficients of a $ N \times N $-matrix $ S = (S_{ij}) $ satisfying $ S^* S = \id $, as the latter algebra admits many characters.
\subsection{Trivial quantum graphs} \label{parQTn}
If $ (B,\psi) $ is a finite quantum space as above, then the {\it trivial quantum graph} $ T(B,\psi) $ on $ (B,\psi) $ is given by the quantum
adjacency matrix $ A = \id $, so that we have the matrix representation $ A^{klb}_{ija} = \delta_{ab} \delta_{ik} \delta_{jl} $.
Using Proposition \ref{qckconcrete} we see that the quantum Cuntz-Krieger algebra $ \FO(T(B, \psi)) $ is the universal $ C^\ast $-algebra with
generators $ S_{ij}^{(a)} $ for $ 1 \leq a \leq d, 1 \leq i,j \leq N_a $, and relations
\begin{align*}
\sum_{kl} S_{ik}^{(a)}(S_{lk}^{(a)})^* S_{lj}^{(a)} &= S_{ij}^{(a)} \\
\sum_k (S_{ki}^{(a)})^* S_{kj}^{(a)} &= \sum_k S_{ik}^{(a)} (S_{jk}^{(a)})^*
\end{align*}
for all $ a,i,j $. We note that $ \FO(T(B, \psi)) $ is independent of the $ \delta $-form $ \psi $ on $ B $, and we will therefore also
write $ \FO(TB) $ instead of $ \FO(T(B,\psi)) $ in the sequel.
\begin{example}
Let us consider explicitly the special case of the trivial quantum graph $ TM_N = TM_N(\mathbb{C}) $ on a full matrix algebra $ B = M_N(\mathbb{C}) $.
The $ C^\ast $-algebra $ \FO(TM_N) $ has generators $ S_{ij} $ for $ 1 \leq i,j \leq N $ satisfying the relations
\begin{align*}
\sum_{kl} S_{ik} S^*_{lk} S_{lj} &= S_{ij} \\
\sum_k S^*_{ki} S_{kj} &= \sum_k S_{ik} S^*_{jk}
\end{align*}
for all $ i,j $.
It is easy to check that $ \FO(TM_N) $ maps onto Brown's algebra \cite{Brownext}, that is, the universal $ C^\ast $-algebra $ U_N^{nc} $
generated by the entries of a unitary $ N \times N $-matrix $ u = (u_{ij}) $, by sending $ S_{ij} $ to $ u_{ij} $.
This shows in particular that $ \FO(TM_N) $ for $ N > 1 $ is not nuclear.
We may also map $ \FO(TM_N) $ onto the non-unital free product $ \mathbb{C} * \cdots * \mathbb{C} $ of $ N $ copies of $ \mathbb{C} $,
by sending $ S_{ij} $ to $ \delta_{ij} 1_i $, where $ 1_i $ denotes the unit element in the $ i $-th copy of $ \mathbb{C} $.
It follows that $ \FO(TM_N) $ is not unital for $ N > 1 $.
In our study of amplifications in section \ref{secamplification} we will obtain the following result on the structure of $ \FO(TM_N) $ as a special case
of Theorem \ref{amplificationmain}.
\begin{theorem} \label{Ktheoryquantumtrivial}
Let $ TM_N $ be the trivial quantum graph as above. Then there exists a $ \ast $-isomorphism
$$
M_N(\FO(TM_N)^+) \cong M_N(\mathbb{C}) *_1 (C(S^1) \oplus \mathbb{C}),
$$
and the quantum Cuntz-Krieger algebra $ \FO(TM_N) $ is $ KK $-equivalent to $ C(S^1) $ for all $ N \in \mathbb{N} $. In particular
\begin{align*}
K_0(\FO(TM_N)) &= \mathbb{Z}, \\
K_1(\FO(TM_N)) &= \mathbb{Z}.
\end{align*}
\end{theorem}
Here $ *_1 $ denotes the unital free product and $ \FO(TM_N)^+ $ is the minimal unitarization of $ \FO(TM_N) $.
With little extra effort one can also determine generators for the $ K $-groups in Theorem \ref{Ktheoryquantumtrivial}. More precisely, if we
write $ S = (S_{ij}) $ for the matrix of generators of $ \FO(TM_N) $, then these are represented by the projection $ S^* S \in M_N(\FO(TM_N)) $
and the unitary $ S - (1 - S^* S) \in M_N(\FO(TM_N)^+) $, respectively.
\end{example}
\begin{remark}
Combining Theorem \ref{Ktheoryquantumtrivial} and Proposition \ref{sumfreeproduct} below one can determine
the $ K $-theory of $ \FO(TB) $ for general $ B $. More precisely, if $ B \cong \bigoplus_{a = 1}^d M_{N_a}(\mathbb{C}) $ then we obtain
\begin{align*}
K_0(\FO(TB)) &= \mathbb{Z}^d, \\
K_1(\FO(TB)) &= \mathbb{Z}^d,
\end{align*}
taking into account \cite{Cuntzfreeproduct}.
\end{remark}
\subsection{Diagonal quantum graphs} \label{pardiagonal}
A natural generalization of the trivial quantum graphs described in the previous paragraph are the {\it diagonal quantum graphs}. Here, we take $ (B,\psi) $
again to be an arbitrary finite quantum space in standard form, but replace the trivial quantum adjacency matrix $ A = \id $ with a map of the form
$$
A(f_{ij}^{(a)}) = x_{ij}^{(a)} f_{ij}^{(a)}
$$
for some suitable complex numbers $ x^{(a)}_{ij} \in \mathbb{C} $ for $ 1 \leq a \leq d, 1 \leq i,j \leq N_a $. Note that if $ B $ is abelian then the
associated adjacency matrix is a diagonal matrix with entries in $ \{0,1 \} $. That is, the only edges possible are self-loops, and we recover precisely
the classical notion of a diagonal graph.
In the non-commutative setting the notion of a diagonal graph is somewhat richer. Namely, Lemma \ref{adjacencycomputation} shows that the only requirements
on the coefficients $ x_{ij}^{(a)} $ are
$$
\sum_s (Q_{(b)}^{-1})_{ss} x_{ks}^{(b)} x_{sl}^{(b)} = \delta^2 x_{kl}^{(b)}
$$
for all $ 1 \leq b \leq d, 1 \leq k,l \leq N_b $.
\begin{example} \label{noninj}
Let $ B = M_N(\mathbb{C}) $ be equipped with the $ \delta $-form $ \psi $ corresponding to the
diagonal matrix $ Q $ with entries $ q_1, \dots, q_N $ satisfying $ q_1 + \cdots + q_N = 1 $. Moreover let $ A $ be the diagonal quantum adjacency matrix with
coefficients $ A^{ij}_{kl} = x_{ij} \delta_{ik} \delta_{jl} $ for some scalars $ x_{ij} $ satisfying $ \sum_s q_s^{-1} x_{ks} x_{sl} = \delta^2 x_{kl} $.
The quantum Cuntz-Krieger algebra $ \FO(\G) $ associated with the diagonal quantum graph $ \G = (B, \psi, A) $ has generators $ S_{ij} $ for $ 1 \leq i,j \leq N $
satisfying the relations
\begin{align*}
\sum_{kl} S_{ik} S_{lk}^* S_{lj} &= S_{ij} \\
\sum_k S^*_{ki} S_{kj} &= \sum_k x_{ij} S_{ik} S^*_{jk}
\end{align*}
for all $ i,j $.
Consider the special case $ x_{11} = q_1 \delta^2 $ and $ x_{ij} = 0 $ else. From the second relation above we
get $ \sum_i S^*_{ij} S_{ij} = 0 $ for $ j > 1 $, and hence $ S_{ij} = 0 $ for all $ 1 \leq i \leq N $ and $ j > 1 $.
This shows that the canonical linear map $ S: B \rightarrow \FO(\G) $ in the definition of a quantum Cuntz-Krieger algebra need not be injective.
One may interpret this as a reflection of the fact that we work with rather general quantum adjacency matrices. It would be
interesting to identify a suitable condition on directed quantum graphs $ \G $ which ensures that the map $ S: B \rightarrow \FO(\G) $ is injective.
Note also that we have $ \sum_l S_{i1} S_{l1}^* S_{l1} = S_{i1} $ and $ \sum_k S^*_{k1} S_{k1} = x_{11} S_{11} S^*_{11} $ in the above special case.
Hence for all complex numbers $ \epsilon_1, \dots, \epsilon_N $ satisfying $ |\epsilon_1|^2 + \cdots + |\epsilon_N|^2 = 1 $
and $ x_{11} |\epsilon_1|^2 = 1 $ there exists a $ \ast $-homomorphism $ \epsilon: \FO(\G) \rightarrow \mathbb{C} $ satisfying
$$
\epsilon(S_{ij}) =
\begin{cases}
\epsilon_i & j = 1 \\
0 & j > 1.
\end{cases}
$$
It follows in particular that the $ C^\ast $-algebra $ \FO(\G) $ admits a trace.
\end{example}
\subsection{Direct sums and tensor products of quantum graphs} \label{parsumtensor}
Assume that $ \G_1 = (B_1, \psi_1, A_1) $ and $ \G_2 = (B_2, \psi_2, A_2) $ are directed quantum graphs. We obtain a finite quantum space structure
on the direct sum $ B_1 \oplus B_2 $ by considering the state
$$
\psi = \frac{\delta_1^2}{\delta^2} \psi_1 \oplus \frac{\delta_2^2}{\delta^2} \psi_2,
$$
with $ \delta^2 = \delta_1^2 + \delta_2^2 $.
It is easy to check that $ A = A_1 \oplus A_2 $ defines a quantum adjacency matrix on $ (B_1 \oplus B_2, \psi) $,
so that $ \G_1 \oplus \G_2 = (B_1 \oplus B_2, \psi, A) $ is a directed quantum graph. Classically, this construction corresponds to taking the
disjoint union of graphs.
\begin{prop} \label{sumfreeproduct}
Let $ \G_1 = (B_1, \psi_1, A_1) $ and $ \G_2 = (B_2, \psi_2, A_2) $ be directed quantum graphs. Then
$$
\FO(\G_1 \oplus \G_2) \cong \FO(\G_1) * \FO(\G_2)
$$
is the non-unital free product of $ \FO(\G_1) $ and $ \FO(\G_2) $.
\end{prop}
\begin{proof}
This follows directly from the universal properties of the algebras involved, noting that the quantum adjacency matrix $ A_1 \oplus A_2 $
does not mix generators from $ B_1 $ and $ B_2 $.
\end{proof}
We can also form tensor products in a natural way. If $ \G_1 = (B_1, \psi_1, A_1) $ and $ \G_2 = (B_2, \psi_2, A_2) $ are directed quantum
graphs then $ \psi = \psi_1 \otimes \psi_2 $ is a $ \delta $-form on the tensor product $ B_1 \otimes B_2 $ with $ \delta = \delta_1 \delta_2 $.
Moreover $ A = A_1 \otimes A_2 $ defines a quantum adjacency matrix on $ (B_1 \otimes B_2, \psi) $.
We let $ \G_1 \otimes \G_2 $ be the corresponding directed quantum graph.
Compared to the case of direct sums, it seems less obvious how to describe the structure of $ \FO(\G_1 \otimes \G_2) $ in terms of $ \FO(\G_1) $
and $ \FO(\G_2) $ in general. We shall discuss a special case in the next section.
\section{Amplification} \label{secamplification}
In this section we study examples of quantum Cuntz-Krieger algebras obtained from classical graphs by replacing
the vertices with matrix blocks. This \emph{amplification} procedure is a special case of the
tensor product construction for quantum graphs described in paragraph \ref{parsumtensor}.
Given a directed quantum graph $ \G = (B, \psi, A) $ and $ N \in \mathbb{N} $ we define the amplification $ M_N(\G) $ of $ \G $ to be the tensor
product $ M_N(\G) = \G \otimes TM_N $, where $ TM_N $ is the trivial quantum graph on $ M_N(\mathbb{C}) $
as defined in paragraph \ref{parQTn}. Explicitly, $ M_N(\G) $ is the directed quantum graph with underlying $ C^\ast $-algebra $ B \otimes M_N(\mathbb{C}) $,
state $ \phi = \psi \otimes \tr $, and quantum adjacency matrix $ A^{(N)} = A \otimes \id $.
In the sequel we shall restrict ourselves to the case that $ \G $ is associated with a classical graph. Recall from paragraph \ref{parclassical}
that if $ E = (E^0, E^1) $ is a simple finite directed classical graph then the adjacency matrix $ B_E $ of $ E $ induces canonically a directed
quantum graph structure on $ C(E^0) $ with its unique $ \delta $-form.
\begin{lemma} \label{matrixampqgraphrelations}
Let $ E = (E^0, E^1) $ be a simple finite directed classical graph and denote by $ \G = (C(E^0), \tr, B_E) $ the directed quantum graph
corresponding to $ E $. Then the quantum Cuntz-Krieger algebra $ \FO(M_N(\G)) $ associated with the amplification $ M_N(\G) $ is
the universal $ C^\ast $-algebra with generators $ S_{eij} $ for $ e \in E^0 $ and $ 1 \leq i,j \leq N $, satisfying the relations
\begin{align*}
\sum_{rs} S_{eir} S^*_{esr} S_{esj} &= S_{eij} \\
\sum_k S^*_{eki} S_{ekj} &= \sum_k \sum_{f \in E^0} B_E(e,f) S_{fik} S_{fjk}^*.
\end{align*}
\end{lemma}
\begin{proof}
Consider the generators $ S_{eij} = S(f^{(e)}_{ij}) $ in $ \FO(M_N(\G)) $ associated with the
adapted matrix units $ f^{(e)}_{ij} = n N \delta_e \otimes e_{ij} $, where $ e \in E^0 $ and $ n $ is the number of vertices of $ E $.
Noting that the quantum adjacency matrix of $ M_N(\G) $ is given by
$$
A^{(N)}(f^{(e)}_{ij}) = \sum_{f \in E^0} B_E(e,f) f^{(f)}_{ij},
$$
the assertion is a direct consequence of Proposition \ref{qckconcrete}.
\end{proof}
We will follow arguments of McClanahan \cite{McClanahanunitarymatrix} to study the structure of the quantum Cuntz-Krieger algebras
in Lemma \ref{matrixampqgraphrelations}. As a first step we discuss a slight strengthening of Theorem 2.3 in \cite{McClanahanunitarymatrix}.
If $ A $ is a $ C^\ast $-algebra we write $ A^+ $ for the unital $ C^\ast $-algebra obtained by adjoining an identity element to $ A $,
and if $ A,B $ are unital $ C^\ast $-algebras we denote by $ A *_1 B $ their unital free product.
\begin{prop} \label{unitalfreeproductkk}
Let $ A $ be a separable $ C^\ast $-algebra. Then $ M_N(\mathbb{C}) *_1 A^+ $ is $ KK $-equivalent to $ A^+ $.
\end{prop}
\begin{proof}
This fact is certainly known to experts, but we shall give the details for the convenience of the reader.
Note first that $ A^+ $ is $ KK $-equivalent to the direct sum $ A \oplus \mathbb{C} $.
This equivalence is implemented by taking the direct sum of the canonical $ * $-homomorphisms $ A \rightarrow A^+ $
and $ \mathbb{C} \rightarrow A^+ $ at the level of $ KK $-theory.
We consider the unital $ * $-homomorphism $ \phi: M_N(\mathbb{C}) *_1 A^+ \rightarrow M_N(\mathbb{C}) \otimes A^+ $ given by
$$
\phi(e_{ij}) = e_{ij} \otimes 1, \qquad \phi(a) = e_{11} \otimes a,
$$
for $ 1 \leq i,j \leq N $ and $ a \in A $, and view this as a class $ [\phi] \in KK(M_N(\mathbb{C}) *_1 A^+, A^+) $.
In the opposite direction we define a map $ \psi_A: A \rightarrow M_N(\mathbb{C}) \otimes (M_N(\mathbb{C}) *_1 A^+) $ by
$$
\psi_A(a) = \sum_{kl} e_{kl} \otimes e_{1k} a e_{l1}.
$$
Then
\begin{align*}
\psi_A(a) \psi_A(b) &= \sum_{klrs} e_{kl} e_{rs} \otimes e_{1k} a e_{l1} e_{1r} b e_{s1} \\
&= \sum_{kls} e_{ks} \otimes e_{1k} a e_{l1} e_{1l} b e_{s1} \\
&= \sum_{ks} e_{ks} \otimes e_{1k} a b e_{s1} = \psi_A(ab)
\end{align*}
and $ \psi_A(a^*) = \psi_A(a)^* $, so that the map $ \psi_A $ is a $ * $-homomorphism.
Consider also the $ * $-homomorphism $ \psi_\mathbb{C}: \mathbb{C} \rightarrow M_N(\mathbb{C}) \otimes (M_N(\mathbb{C}) *_1 A^+) $
given by $ \psi_\mathbb{C}(1) = e_{11} \otimes e_{11} $. Combining the maps $ \psi_A $ and $ \psi_\mathbb{C} $, and using that $ A^+ $ is $ KK $-equivalent
to $ A \oplus \mathbb{C} $, we obtain a class in $ KK(A^+, M_N(\mathbb{C}) *_1 A^+) $, which we shall denote by $ [\psi] $.
We claim that the classes $ [\phi] $ and $ [\psi] $ are mutually inverse.
In order to determine the composition $ [\phi] \circ [\psi] \in KK(A^+, A^+) $ it suffices to
compute $ M_N(\phi) \circ \psi_A $ and $ M_N(\phi) \circ \psi_\mathbb{C} $, respectively.
We calculate
$$
(M_N(\phi) \circ \psi_A)(a) = \sum_{kl} e_{kl} \otimes \phi(e_{1k} a e_{l1}) = \sum_{kl} e_{kl} \otimes e_{1k} e_{11} e_{l1} \otimes a
= e_{11} \otimes e_{11} \otimes a
$$
for $ a \in A $ and $ (M_N(\phi) \circ \psi_\mathbb{C})(1) = M_N(\phi)(e_{11} \otimes e_{11}) = e_{11} \otimes e_{11} \otimes 1 $.
This immediately yields $ [\phi] \circ [\psi] = \id $.
Next consider $ [\psi] \circ [\phi] \in KK(M_N(\mathbb{C}) *_1 A^+, M_N(\mathbb{C}) *_1 A^+) $.
Let us write $ j_{A^+}: A^+ \rightarrow M_N(\mathbb{C}) *_1 A^+ $ and $ j_{M_N(\mathbb{C})}: M_N(\mathbb{C}) \rightarrow M_N(\mathbb{C}) *_1 A^+ $
for the canonical inclusion homomorphisms. Moreover write $ u: \mathbb{C} \rightarrow M_N(\mathbb{C}) \oplus A^+ $ for the
unit map. According to \cite{Germainfreeproduct}, \cite{FimaGermainamalgamated},
the suspension of $ M_N(\mathbb{C}) *_1 A^+ $ is $ KK $-equivalent to the cone of $ u $. In order to show $ [\psi] \circ [\phi] = \id $ it therefore suffices
to verify $ [\psi] \circ [\phi] \circ [j_{A^+}] = [j_{A^+}] $ and $ [\psi] \circ [\phi] \circ [j_{M_N(\mathbb{C})}] = [j_{M_N(\mathbb{C})}] $.
We calculate
$$
(M_N(\psi_A) \circ \phi)(a) = M_N(\psi)(e_{11} \otimes a) = \sum_{kl} e_{11} \otimes e_{kl} \otimes e_{1k} a e_{l1}
$$
for $ a \in A $.
Pick a continuous path of unitaries $ U_t $ in $ M_N(\mathbb{C}) \otimes M_N(\mathbb{C}) $ such that $ U_0 = \id $ and
$$
U_1(e_k \otimes e_1) = e_1 \otimes e_k
$$
for all $ k $, and push this into the last two tensor factors of $ M_N(\mathbb{C}) \otimes M_N(\mathbb{C}) \otimes (M_N(\mathbb{C}) *_1 A^+) $ via the
obvious map. Then conjugating $ (M_N(\psi_A) \circ \phi)(a) $ by $ U_1 $ gives $ e_{11} \otimes e_{11} \otimes a $ for all $ a \in A $.
It follows that $ [\psi] \circ [\phi] \circ [j_A] = [j_A] $, where we write $ j_A $ for the restriction of $ j_{A^+} $ to $ A \subset A^+ $.
Next we calculate
$$
(M_N(\psi_\mathbb{C}) \circ \phi)(1) = M_N(\psi_\mathbb{C})(1 \otimes 1) = 1 \otimes e_{11} \otimes e_{11}.
$$
Conjugating this with the unitary $ U_1 $ from above, pushed into the first and third tensor factors, gives $ e_{11} \otimes e_{11} \otimes 1 $.
Hence $ [\psi] \circ [\phi] \circ [j_\mathbb{C}] = [j_\mathbb{C}] $, where $ j_\mathbb{C} $ denotes the restriction of $ j_{A^+} $
to $ \mathbb{C} \subset A^+ $. Combining these two observations gives $ [\psi] \circ [\phi] \circ [j_{A^+}] = [j_{A^+}] $.
Finally, we have
$$
(M_N(\psi_\mathbb{C}) \circ \phi)(e_{ij}) = M_N(\psi_\mathbb{C})(e_{ij} \otimes 1) = e_{ij} \otimes e_{11} \otimes e_{11},
$$
so that conjugating $ (M_N(\psi_\mathbb{C}) \circ \phi)(e_{ij}) $ by $ U_1 $ in the first and third tensor factors
gives $ e_{11} \otimes e_{11} \otimes e_{ij} $ for all $ i,j $.
We conclude $ [\psi] \circ [\phi] \circ [j_{M_N(\mathbb{C})}] = [j_{M_N(\mathbb{C})}] $, and this finishes the proof.
\end{proof}
With these preparations in place let us now present our main result on amplified quantum Cuntz-Krieger algebras.
\begin{theorem} \label{amplificationmain}
Assume that $ E = (E^0, E^1) $ is a finite directed simple graph and let $ \G = (C(E^0), \tr, B_E) $ be the corresponding
directed quantum graph. Then the following holds.
\begin{bnum}
\item[a)] We have an isomorphism $ M_N(\FO(M_N(\G))^+) \cong M_N(\mathbb{C}) *_1 (\FO(\G)^+) $.
\item[b)] $ \FO(M_N(\G)) $ is $ KK $-equivalent to the classical Cuntz-Krieger algebra $ \O_{B_E} $.
\end{bnum}
\end{theorem}
\begin{proof}
$ a) $ In the sequel we shall write $ C = M_N(\mathbb{C}) *_1 (\FO(\G)^+) $ and $ D = \FO(M_N(\G)) $.
We define a $ \ast $-homomorphism $ g: D \rightarrow C $ by
$$
g(S_{eij}) = \sum_k e_{ki} S_e e_{jk}
$$
on generators. To check that this is well-defined we use Lemma \ref{matrixampqgraphrelations} to calculate
\begin{align*}
\sum_{kl} g(S_{eik}) g(S_{elk})^* g(S_{elj}) &= \sum_{rstkl} e_{ri} S_e e_{kr} e_{sk} S_e^* e_{ls} e_{tl} S_e e_{jt} \\
&= \sum_{rkl} e_{ri} S_e e_{kk} S_e^* e_{ll} S_e e_{jr} \\
&= \sum_r e_{ri} S_e S_e^* S_e e_{jr} \\
&= \sum_r e_{ri} S_e e_{jr} \\
&= g(S_{eij})
\end{align*}
and
\begin{align*}
\sum_k g(S_{eki})^* g(S_{ekj}) &= \sum_{rsk} e_{ri} S_e^* e_{kr} e_{sk} S_e e_{js} \\
&= \sum_r e_{ri} S_e^* S_e e_{jr} \\
&= \sum_r \sum_{f \in E^0} B_E(e,f) e_{ri} S_f S_f^* e_{jr} \\
&= \sum_k \sum_{f \in E^0} B_E(e,f) g(S_{fik}) g(S_{fjk})^*
\end{align*}
for $ e \in E^0 $ and $ 1 \leq i,j \leq N $.
Let $ g^+: D^+ \rightarrow C $ be the unital extension of $ g $. It is easy to see that the image of $ g^+ $ is contained in the relative
commutant $ M_N(\mathbb{C})' $ of $ M_N(\mathbb{C}) $ inside the free product. In fact, we have
$$
g(S_{eij}) e_{kl} = \sum_r e_{ri} S_e e_{jr} e_{kl} = e_{ki} S_e e_{jl} = \sum_r e_{kl} e_{ri} S_e e_{jr} = e_{kl} g(S_{eij})
$$
for all $ i,j,k,l $.
We can thus extend $ g^+ $ to a unital $ \ast $-homomorphism $ G: M_N(D^+) \rightarrow C $ by setting $ G(e_{ij}) = e_{ij} $
and $ G(x) = g(x) $ for $ x \in D^+ $.
Let us also define a unital $ \ast $-homomorphism $ F: C \rightarrow M_N(D^+) = D^+ \otimes M_N(\mathbb{C}) $ by
\begin{align*}
F(e_{ij}) &= 1 \otimes e_{ij} \\
F(S_e) &= \sum_{ij} S_{eij} \otimes e_{ij}.
\end{align*}
To see that this is well-defined we only need to check that these formulas define unital $ \ast $-homomorphisms from $ M_N(\mathbb{C}) $
and $ \FO(\G)^+ $ into $ M_N(D^+) $, respectively. For $ M_N(\mathbb{C}) $ this is obvious. For $ \FO(\G)^+ $ we need to check the free Cuntz-Krieger
relations for the elements $ F(S_e) $. In fact, each $ F(S_e) $ is a partial isometry by construction, and using Lemma \ref{matrixampqgraphrelations} we calculate
\begin{align*}
F(S_e)^* F(S_e) &= \sum_{ijkl} S_{eij}^* S_{ekl} \otimes e_{ji} e_{kl} \\
&= \sum_{ijl} S_{eij}^* S_{eil} \otimes e_{jl} \\
&= \sum_{ijl} \sum_{f \in E^0} B_E(e,f) S_{fji} S_{fli}^* \otimes e_{jl} \\
&= \sum_{f \in E^0} B_E(e,f) \sum_{ijkl} S_{fji} S_{flk}^* \otimes e_{ji} e_{kl} \\
&= \sum_{f \in E^0} B_E(e,f) F(S_f) F(S_f)^*
\end{align*}
as required.
Next observe that $ F \circ G: M_N(D^+) \rightarrow M_N(D^+) $ satisfies
\begin{align*}
(F \circ G)(S_{eij} \otimes 1) &= \sum_k F(e_{ki}) F(S_e) F(e_{jk}) \\
&= \sum_{krs} (1 \otimes e_{ki}) (S_{ers} \otimes e_{rs}) (1 \otimes e_{jk}) = S_{eij} \otimes 1
\end{align*}
for all $ e \in E^0 $ and $ (F \circ G)(e_{ij}) = e_{ij} $ for all $ i,j $. This implies $ F \circ G = \id $.
Similarly, we have
\begin{align*}
(G \circ F)(S_e) &= \sum_{ij} G(S_{eij} \otimes e_{ij}) = \sum_{ij} e_{ii} S_e e_{jj} = S_e
\end{align*}
for all $ e \in E^0 $, and $ (G \circ F)(e_{ij}) = e_{ij} $ for all $ i,j $. We conclude that $ G \circ F = \id $.
$ b) $ Clearly $ M_N(\FO(M_N(\G))^+) $ is $ KK $-equivalent to $ \FO(M_N(\G))^+ $.
According to Proposition \ref{unitalfreeproductkk}, we also know that $ M_N(\mathbb{C}) *_1 (\FO(\G)^+) $ is $ KK $-equivalent to $ \FO(\G)^+ $.
It is easy to check that these equivalences are both compatible with the canonical augmentation morphisms to $ \mathbb{C} $.
Hence $ \FO(M_N(\G)) $ is $ KK $-equivalent to $ \FO(\G) $. Finally, recall from Theorem \ref{graphversusfreegraph} that the free
Cuntz-Krieger algebra $ \FO(\G) = \FO_{B_E} $ is $ KK $-equivalent to $ \O_{B_E} $.
\end{proof}
Under some mild extra assumptions, Theorem \ref{amplificationmain} allows one to compute the $ K $-theory of $ \FO(M_N(\G)) $ in terms of the graph $ E $,
see \cite{Cuntzmarkov2} and chapter 7 in \cite{Raeburngraph}.
Finally, remark that if $ E $ is the graph with one vertex and one self-loop then we have $ \FO(\G) = \FO_{B_E} = C(S^1) $,
and $ \FO(M_N(\G)) = \FO(TM_N) $ is the quantum Cuntz-Krieger algebra of the trivial quantum graph on $ M_N(\mathbb{C}) $.
Therefore Theorem \ref{amplificationmain} implies Theorem \ref{Ktheoryquantumtrivial}.
\section{Quantum symmetries of quantum Cuntz-Krieger algebras} \label{secquantumsymmetry}
In this section we study how quantum symmetries and quantum isomorphisms of directed quantum graphs induce symmetries of their associated
quantum Cuntz-Krieger algebras. This will be useful in particular to exhibit relations between the $ C^\ast $-algebras corresponding to quantum
isomorphic quantum graphs.
\subsection{Gauge actions} \label{pargauge}
Before discussing quantum symmetries, let us first show that there is a canonical gauge action on quantum Cuntz-Krieger algebras, thus
providing very natural classical symmetries. This is analogous to the well-known gauge action on Cuntz-Krieger algebras and graph $ C^\ast $-algebras, which plays
a crucial role in the analysis of the structure of these $ C^\ast $-algebras, compare \cite{Raeburngraph}.
Let $ \G = (B, \psi, A) $ be a directed quantum graph, and let $ \FO(\G) $ be the corresponding quantum Cuntz-Krieger algebra.
For $ \lambda \in U(1) $ consider the linear map $ S_\lambda: B \rightarrow \FO(\G) $ given by
$$
S_\lambda(b) = \lambda S(b),
$$
where $ S: B \rightarrow \FO(\G) $ is the canonical linear map. Then we
have $ S_\lambda^*(b) = (\lambda S(b^*))^* = \overline{\lambda} S^*(b) $ for all $ b \in B $,
and using this relation it is easy to check that $ S_\lambda: B \rightarrow \FO(\G) $ is a quantum Cuntz-Krieger $ \G $-family.
By the universal property of $ \FO(\G) $ we obtain a corresponding automorphism $ \alpha_\lambda \in \Aut(\FO(\G)) $,
and these automorphisms combine to a strongly continuous action of $ U(1) $ on $ \FO(\G) $.
In terms of the generators of $ \FO(\G) $ as in Proposition \ref{qckconcrete} the gauge action is given by
$$
\alpha_\lambda(S^{(a)}_{ij}) = \lambda S^{(a)}_{ij},
$$
from which it is easy to determine the action on arbitrary noncommutative polynomials in the generators, and the
decomposition into spectral subspaces.
In some cases one may define more general gauge type actions. For instance, for the complete quantum graph $ K(M_N(\mathbb{C}), \tr) $ from paragraph \ref{parQKn}
and the trivial quantum graph $ TM_N $ from paragraph \ref{parQTn} we have an action of $ U(1) \times U(1)^N $, given by
$$
\alpha_{\lambda \mu}(S_{ij}) = \lambda \frac{\mu_i}{\mu_j} \,S_{ij}
$$
on generators. In fact, one may even extend this to an action of $ U(1) \times U(N) $ by setting
$$
\alpha_{\lambda U}(S) = \lambda U S U^*,
$$
where $ S = (S_{ij}) $ is the generating matrix partial isometry.
However, none of the above actions seems to suffice to obtain structural information about quantum Cuntz-Krieger algebras in the same way
as for classical graph algebras. In particular, the corresponding fixed point algebras tend to have a more complicated structure than in
the classical setting.
It turns out that this deficiency can be compensated to some extent by considering actions of compact quantum groups instead, and in
particular symmetries arising from suitable monoidal equivalences between quantum automorphism groups of directed quantum graphs. We will explain these
constructions in the following paragraphs.
\subsection{Compact quantum groups}
Let us first give a quick review of the definition of compact quantum groups and their action on $ C^\ast $-algebras. For more background and further information
we refer to \cite{Woronowiczleshouches}, \cite{NTlecturenotes}.
A compact quantum group $ G $ is given by a unital $ C^\ast $-algebra $ C(G) $ together with a
unital $ \ast $-homomorphism $ \Delta: C(G) \rightarrow C(G) \otimes C(G) $
such that $ (\Delta \otimes \id) \Delta = (\id \otimes \Delta) \Delta $ and the density conditions
$$
[\Delta(C(G)) (C(G) \otimes 1)] = C(G) \otimes C(G) = [\Delta(C(G)) (1 \otimes C(G))]
$$
hold.
We will mainly work with the canonical dense Hopf $ \ast $-algebra $ \Poly(G) \subset C(G) $, consisting of the matrix coefficients
of all finite dimensional unitary representations of $ G $. For the definition of unitary representations and their intertwiners see \cite{NTlecturenotes}.
The collection of all finite dimensional unitary representations of $ G $ forms naturally a $ C^\ast $-tensor category $ \Rep(G) $.
On the $ C^\ast $-level we will only consider the universal completions of $ \O(G) $ in the sequel, and always denote them by $ C(G) $.
With this in mind, a morphism $ H \rightarrow G $ of compact quantum groups is nothing but a $ \ast $-homomorphism $ C(G) \rightarrow C(H) $
compatible with the comultiplications. Equivalently, such a morphism is given by a homomorphism $ \Poly(G) \rightarrow \Poly(H) $ of
Hopf $ \ast $-algebras. One says that $ H $ is a quantum subgroup of $ G $ if there exists a morphism $ H \rightarrow G $
such that the corresponding homomorphism of Hopf $ \ast $-algebras $ \Poly(G) \rightarrow \Poly(H) $ is surjective.
By definition, an action of a compact quantum group $ G $ on a $ C^\ast $-algebra $ A $ is a $ \ast $-homomorphism $ \alpha: A \rightarrow A \otimes C(G) $
satisfying $ (\alpha \otimes \id)\alpha = (\id \otimes \Delta)\alpha $ and the density condition $ [(1 \otimes C(G)) \alpha(A)] = A \otimes C(G) $.
A $ C^\ast $-algebra $ A $ equipped with an action of $ G $ will also be called a $ G $-$ C^\ast $-algebra.
Every $ G $-$ C^\ast $-algebra $ A $ contains a canonical dense $ \ast $-subalgebra $ \A \subset A $, given by
the algebraic direct sum of the spectral subspaces of the action. Moreover, the map $ \alpha $ restricts to
a $ \ast $-homomorphism $ \alpha: \A \rightarrow \A \otimes \Poly(G) $, and this defines an algebra coaction in the sense of Hopf algebras.
In particular, one has $ (\id \otimes \epsilon)\alpha(a) = a $ for all $ a \in \A $, where $ \epsilon: \Poly(G) \rightarrow \mathbb{C} $
is the counit.
If $ A $ is a $ G $-$ C^\ast $-algebra then the fixed point subalgebra of $ A $ is defined by
$$
A^G = \{a \in A \mid \alpha(a) = a \otimes 1 \},
$$
and a unital $ G $-$ C^\ast $-algebra $ A $ is called ergodic if $ A^G = \mathbb{C} 1 $.
The same terminology is also used for $ \ast $-algebras equipped with algebra coactions of $ \Poly(G) $.
Let us now review the definition of quantum automorphism groups of finite quantum spaces in the sense of Definition \ref{deffqs}.
These quantum groups were introduced by Wang \cite{Wangqsymmetry} and studied further by Banica \cite{Banicageneric} and others.
If $ G $ is a compact quantum group and $ \omega: A \rightarrow \mathbb{C} $ a state on a $ G $-$ C^\ast $-algebra $ A $ with
action $ \alpha: A \rightarrow A \otimes C(G) $, then we say that the action preserves $ \omega $ if
$$
(\omega \otimes \id) \alpha(a) = \omega(a) 1
$$
for all $ a \in A $.
\begin{definition} \label{defqut}
Let $ (B, \psi) $ be a finite quantum space. The quantum automorphism group of $ (B, \psi) $ is the universal compact quantum group $ G^+(B,\psi) $
equipped with an action $ \beta: B \rightarrow B \otimes C(G^+(B, \psi)) $ which preserves $ \psi $.
\end{definition}
In other words, if $ G $ is a compact quantum group and $ \gamma: B \rightarrow B \otimes C(G) $ an action of $ G $ preserving $ \psi $, then
there exists a unique $ * $-homomorphism $ \pi: C(G^+(B, \psi)) \rightarrow C(G) $, compatible with the comultiplications, such that the diagram
$$
\xymatrix{
B \ar@{->}[r]^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \beta} \ar@{->}[rd]_{\gamma} & B \otimes C(G^+(B, \psi)) \ar@{->}[d]^{\id \otimes \pi} \\
& B \otimes C(G)
}
$$
is commutative.
The most prominent example of a quantum automorphism group is the quantum permutation group $ S_N^+ $. This is the quantum automorphism
group of $ B = \mathbb{C}^N $ with its unique $ \delta $-form. The corresponding $ C^\ast $-algebra $ C(S_N^+) = C(G^+(\mathbb{C}^N, \tr)) $
is the universal $ C^\ast $-algebra generated by projections $ u_{ij} $ for $ 1 \leq i,j \leq N $ such that
$$
\sum_k u_{ik} = 1 = \sum_k u_{kj}
$$
for all $ i,j $. These relations can be phrased by saying that the matrix $ u = (u_{ij}) $ is a magic
unitary. The comultiplication $ \Delta: C(S_N^+) \rightarrow C(S_N^+) \otimes C(S_N^+) $ is defined by
$$
\Delta(u_{ij}) = \sum_{k = 1}^n u_{ik} \otimes u_{kj}
$$
on generators.
\begin{remark}
Quantum automorphism groups can always be described explicitly in terms of generators and relations, see Proposition 2.10 in \cite{Mrozinskiso3deformations}.
More precisely, let us assume that $ (B, \psi) $ is a finite quantum space in standard form as in section \ref{parqgraph}, so that
$$
B = \bigoplus_{a = 1}^d M_{N_a}(\mathbb{C}), \qquad \psi(x) = \sum_{a = 1}^d \Tr(Q_{(a)} x_a)
$$
for $ x = (x_1, \dots, x_d) \in B $. Then the Hopf $ \ast $-algebra $ \Poly(G^+(B, \psi)) $ is generated
by elements $ v_{ija}^{rsb} $ for $ 1 \leq a, b \leq d $ and $ 1 \leq i,j \leq N_a, 1 \leq r,s \leq N_b $, satisfying the relations
\begin{itemize}
\item[(A1a)] $ \sum_w v_{kla}^{xwc} v_{rsb}^{wyc} = \delta_{ab} \delta_{lr} v_{ksa}^{xyc} $
\item[(A1b)] $ \sum_w (Q_{(c)})^{-1}_{ww} v^{srb}_{ywc} v^{lka}_{wxc} = \delta_{lr} \delta_{ab} (Q_{(a)})^{-1}_{ll} v^{ska}_{yxc} $
\item[(A2)] $ (v_{kla}^{xyc})^* = v_{lka}^{yxc} $
\item[(A3a)] $ \sum_{xb} (Q_{(b)})_{xx} v^{xxb}_{kla} = \delta_{kl} (Q_{(a)})_{kk} $
\item[(A3b)] $ \sum_{ka} v^{xyb}_{kka} = \delta_{xy} $.
\end{itemize}
In terms of the standard matrix units $ e^{(a)}_{ij} $ for $ B $ and the generators $ v_{ija}^{rsb} $, the defining
action $ \beta: B \rightarrow B \otimes \Poly(G^+(B,\psi)) $ is given by
$$
\beta(e^{(a)}_{ij}) = \sum_{bkl} e^{(b)}_{kl} \otimes v_{ija}^{klb},
$$
and the matrix $ v = (v_{ija}^{rsb}) $ is also called the fundamental matrix of $ G^+(B, \psi) $.
We will reobtain the above description of the $ \ast $-algebra $ \Poly(G^+(B, \psi)) $ as a special case of Proposition \ref{qisorelations} below.
\end{remark}
\subsection{Quantum symmetries of quantum graphs} \label{parqsym}
In this paragraph we discuss the quantum automorphism group of a directed quantum graph, and also quantum isomorphisms relating a pair
of directed quantum graphs.
Recall first that if $ E = (E^0, E^1) $ is a simple finite
graph then the automorphism group $ \Aut(E) $ consists of all bijections of $ E^0 $ which preserves the adjacency relation in $ E $.
If $ |E^0| = N $ and $ A \in M_N(\mathbb{Z}) $ is the adjacency matrix of $ E $, then this can be expressed as
$$
\Aut(E) = \{\sigma \in S_N \mid \sigma A = A \sigma\} \subset S_N,
$$
where one views elements of the symmetric group as permutation matrices. In \cite{Banicaqutgraph}, Banica defined the quantum automorphism group $ G^+(E) $
of the graph $ E $ via the $ C^\ast $-algebra
$$
C(G^+(E)) = C(S_N^+)/\langle u A = A u \rangle,
$$
where $ u = (u_{ij}) \in M_N(C(S_N^+)) $ denotes the defining magic unitary matrix. This yields a quantum subgroup of $ S_N^+ $, which contains the
classical automorphism group $ \Aut(E) $ as a quantum subgroup.
If $ \G = (B, \psi, A) $ is a directed quantum graph we shall say that an action $ \beta: B \rightarrow B \otimes C(G) $
of a compact quantum group $ G $ is compatible with $ A: B \rightarrow B $ if $ \beta \circ A = (A \otimes \id) \circ \beta $.
Motivated by the considerations in \cite{Banicaqutgraph}, we define the quantum automorphism group of a directed quantum graph as follows,
compare \cite{BCEHPSWbigalois}.
\begin{definition}
Let $ \G = (B, \psi, A) $ be a directed quantum graph. The quantum automorphism group $ G^+(\G) $ of $ \G $ is the universal compact quantum
group equipped with a $ \psi $-preserving action $ \beta: B \rightarrow B \otimes C(G^+(\G)) $ which is compatible with the quantum adjacency matrix $ A $.
\end{definition}
That is, if $ G $ is a compact quantum group and $ \gamma: B \rightarrow B \otimes C(G) $ an action of $ G $ which preserves $ \psi $
and is compatible with $ A $, then there exists a unique $ * $-homomorphism $ \pi: C(G^+(\G)) \rightarrow C(G) $, compatible with the comultiplications,
such that the diagram
$$
\xymatrix{
B \ar@{->}[r]^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \beta} \ar@{->}[rd]_{\gamma} & B \otimes C(G^+(\G)) \ar@{->}[d]^{\id \otimes \pi} \\
& B \otimes C(G)
}
$$
is commutative.
Comparing this with Definition \ref{defqut}, it is straightforward to check that $ C(G^+(\G)) $ can be identified with the quotient of $ C(G^+(B,\psi)) $
obtained by imposing the relation $ (1 \otimes A) v = v (1 \otimes A) $ on the fundamental matrix $ v $ of $ G^+(B,\psi) $.
\begin{remark}
If $ \G = K(M_N(\mathbb C), \tr) $ or $ \G = TM_N $ is the complete or the trivial quantum graph on $ M_N(\mathbb{C}) $, then it is easy to see that
compatibility with the quantum adjacency matrix is in fact automatic. That is, we have $ G^+(\G) = G^+(M_N(\mathbb{C}), \tr) $ in either case.
\end{remark}
Let us recall that two compact quantum groups $ G_1, G_2 $ are called monoidally equivalent if their representation categories $ \Rep(G_1) $ and $ \Rep(G_2) $
are unitarily equivalent as $ C^* $-tensor categories \cite{BdRV}, \cite{NTlecturenotes}. A monoidal equivalence is a unitary tensor
functor $ F: \Rep(G^+(\G_1)) \rightarrow \Rep(G^+(\G_2)) $ whose underlying functor is an equivalence.
Assume that $ \G_i = (B_i, \psi_i, A_i) $ are directed quantum graphs for $ i = 1,2 $. Then the quantum automorphism group $ G^+(\G_i) $ is a
quantum subgroup of $ G^+(B_i, \psi_i) $ such that the quantum adjacency matrix $ A_i $ is an intertwiner
for the defining representation $ B_i = L^2(B_i) $ of $ G^+(\G_i) $. Note also that the multiplication map $ m_i: B_i \otimes B_i \rightarrow B_i $
and the unit map $ u_i: \mathbb{C} \rightarrow B_i $ are intertwiners for the action of $ G^+(\G_i) $, so that $ B_i $
becomes a monoid object in the tensor category $ \Rep(G^+(\G_i)) $.
In analogy to \cite{BCEHPSWbigalois} we give the following definition.
\begin{definition} \label{defqiso}
Two directed quantum graphs $ \G_i = (B_i, \psi_i, A_i) $ for $ i = 1,2 $ are quantum isomorphic if there exists a monoidal
equivalence $ F: \Rep(G^+(\G_1)) \rightarrow \Rep(G^+(\G_2)) $ such that
\begin{bnum}
\item[a)] $ F $ maps the monoid object $ B_1 $ to the monoid object $ B_2 $.
\item[b)] $ F(A_1) = A_2 $.
\end{bnum}
We will write $ \G_1 \cong_q \G_2 $ in this case.
\end{definition}
From Definition \ref{defqiso} it is easy to see that the notion of quantum isomorphism is an equivalence relation on isomorphism classes of directed
quantum graphs. For concrete computations it is however more convenient to describe quantum isomorphisms in terms of bi-Galois
objects \cite{BCEHPSWbigalois}, sometimes also called linking algebras.
Concretely, if $ \G_i = (B_i, \psi_i, A_i) $ for $ i = 1,2 $ are directed quantum graphs then $ \O(G^+(\G_2, \G_1)) $ is the bi-Galois object
generated by the coefficients of a unital $ \ast $-homomorphism
$$
\beta_{ji}: B_i \to B_j \otimes \Poly(G^+(\G_j, \G_i))
$$
satisfying the conditions
$$
(\psi_j \otimes \id) \beta_{ji}(x) = \psi_i(x)1
$$
for all $ x \in B_i $ and
$$
(A_j \otimes \id) \beta_{ji} = \beta_{ji} A_i.
$$
Note that these conditions generalize the requirements on the action of the quantum automorphism group of a quantum graph to be state-preserving and
compatible with the quantum adjacency matrix, respectively.
We write $ C(G^+(\G_j, \G_i)) $ for the universal enveloping $ C^\ast $-algebra of $ \Poly(G^+(\G_j, \G_i)) $.
In exactly the same way as in \cite{BCEHPSWbigalois} one then arrives at the following characterization of quantum isomorphisms.
\begin{theorem} \label{starbigaloischar}
Let $ \G_1, \G_2 $ be directed quantum graphs. Then the following conditions are equivalent.
\begin{bnum}
\item[a)] $ \G_1 $ and $ \G_2 $ are quantum isomorphic.
\item[b)] $ \Poly(G^+(\G_2, \G_1)) $ is non-zero.
\item[c)] $ \Poly(G^+(\G_2, \G_1)) $ admits a nonzero faithful state.
\item[d)] $ C(G^+(\G_2, \G_1)) $ is non-zero.
\end{bnum}
\end{theorem}
If the equivalent conditions in Theorem \ref{starbigaloischar} are satisfied then $ \Poly(G^+(\G_2, \G_1)) $ is a
$ \Poly(G^+(\G_2)) $-$ \Poly(G^+(\G_1)) $ bi-Galois object in a natural way \cite{Shopfgalois}.
In particular, there exist ergodic left and right actions of $ G^+(\G_2) $ and $ G^+(\G_1) $ on $ \Poly(G^+(\G_2, \G_1)) $, respectively.
Moreover, $ \Poly(G^+(\G_2, \G_1)) $ is equipped with a unique faithful state which is invariant with respect to both actions.
For $ \G_1 = \G_2 $ and the identity monoidal equivalence, the $ \ast $-algebra $ \Poly(G^+(\G_2, \G_1)) $ equals $ \Poly(G^+(\G_1)) = \Poly(G^+(\G_2)) $,
both actions are implemented by the comultiplication, and the invariant faithful state is nothing but the Haar state.
\begin{remark}
The abelianization of $ \Poly(G^+(\G_2, \G_1)) $ is the algebra of coordinate functions on the space of ``classical isomorphisms''
between the quantum graphs $ \G_1 $ and $ \G_2 $, that is, the space of unital $ \ast $-isomorphisms $ \varphi: B_1 \to B_2 $ satisfying
$$
\psi_2 \circ \varphi = \psi_1, \qquad A_2 \circ \varphi = \varphi \circ A_1.
$$
If moreover each $ \G_i $ is associated with a classical directed graph $ E_i = (E_i^0, E_i^1) $ as in paragraph \ref{parclassical}, then by Gelfand duality
such a map $ \varphi $ corresponds precisely to a graph isomorphism $ \varphi_*: E_2 \to E_1 $ via $ \varphi(f) = f \circ \varphi_* $ for $ f \in C(E_1^0) $.
This is the reason for the ordering of the quantum graphs in our notation $ \Poly(G^+(\G_2, \G_1)) $.
\end{remark}
\begin{remark}
There is a canonical algebra isomorphism $ S: \Poly(G^+(\G_2, \G_1)) \to \Poly(G^+(\G_1, \G_2))^{op} $, which can be viewed as a generalization of the
antipode of the Hopf $ \ast $-algebra associated to a compact quantum group.
More precisely, if $ (e_m) $ and $ (f_n) $ are orthonormal bases for $ B_1 $ and $ B_2 $, respectively, and we
write $ \beta_{21}(e_m) = \sum_n f_n \otimes u_{nm} $, then $ u = (u_{ij}) \in \End(B_1, B_2) \otimes \Poly(G^+(\G_2, \G_1)) $ is a unitary matrix,
and there is an algebra isomorphism $ S: \Poly(G^+(\G_2, \G_1)) \to \Poly(G^+(\G_1, \G_2))^{op} $ given by
$$
(\id \otimes S)(u) = u^* = u^{-1}, \qquad (\id \otimes S)(u^*) = (J_2^t \bar{u} (J_1^{-1})^t),
$$
where $ (\bar{u})_{kl} = (u_{kl}^*), J_i: B_i \to B_i $ is the anti-linear involution map given by $ J_i(b) = b^* $ and $ t $ denotes
the transpose map. We refer to \cite{BCEHPSWbigalois} for more details.
\end{remark}
\begin{remark} \label{wealth}
We have a wealth of examples quantum isomorphisms between the complete quantum graphs $ K(B,\psi) $ introduced in paragraph \ref{parQKn}, and also between
the trivial quantum graphs $ T(B,\psi) $ introduced in paragraph \ref{parQTn}. Recall that $ K(B, \psi) $ (resp. $ T(B,\psi) $) is defined by equipping the finite
quantum space $ (B, \psi) $ with the quantum adjacency matrix $ A: L^2(B) \rightarrow L^2(B) $ given by $ A(b) = \delta^2 \psi(b) 1 $ (resp. $ A = \id $).
More precisely, if $ (B_i, \psi_i) $ are finite quantum spaces for $ i = 1,2 $, with $ \delta_i $-forms $ \psi_i $, then
$$
K(B_1, \psi_1) \cong_q K(B_2,\psi_2) \iff T(B_1, \psi_1) \cong_q T(B_2, \psi_2) \iff \delta_1 = \delta_2 .
$$
These equivalences follow from work of DeRijdt and Vander Vennet in \cite{dRV}, where unitary fiber functors on quantum automophism groups of finite quantum
spaces equipped with $ \delta $-forms were classified.
\end{remark}
Let $ \G_i = (B_i, \psi_i, A_i) $ be directed quantum graphs in standard form, in the sense explained in paragraph \ref{parqgraph}. Explicitly, we fix
multimatrix decompositions
$$
B_i = \bigoplus_{a = 1}^{d_i} M_{N^i_a}(\mathbb{C})
$$
and diagonal positive invertible matrices $ Q^i_{(a)} $ implementing $ \psi_i $. Let us express the quantum adjacency matrices relative to
the standard matrix units $ e_{kl}^{(a)} \in M_{N^i_a}(\mathbb{C}) $, so that
$$
A_i(e^{(a)}_{kl}) = \sum_{brs} (A_i)_{kla}^{rsb} e^{(b)}_{rs}.
$$
We then obtain the following result, compare \cite{Mrozinskiso3deformations} for the case $ \G_1 = \G_2 $.
\begin{prop} \label{qisorelations}
Let $ \G_1 $ and $ \G_2 $ be directed quantum graphs given as above. Then $ \Poly(G^+(\G_2, \G_1)) $ is the universal unital $ \ast $-algebra with
generators $ v_{ija}^{klb} $ for $ 1 \leq i,j \leq N^1_a $, $ 1 \leq k,l \leq N^2_b $, $ 1 \leq a \leq d_1 $, $ 1 \leq b \leq d_2 $, satisfying the
relations
\begin{itemize}
\item[(A1a)] $ \sum_w v_{kla}^{xwc} v_{rsb}^{wyc} = \delta_{ab} \delta_{lr} v_{ksa}^{xyc} $
\item[(A1b)] $ \sum_l (Q^1_{(a)})^{-1}_{ll} v_{mla}^{xwb} v_{lka}^{zyc} = \delta_{bc} \delta_{wz} (Q^2_{(c)})^{-1}_{zz} v_{mka}^{xyc} $
\item[(A2)] $ (v_{kla}^{xyb})^* = v_{lka}^{yxb} $
\item[(A3a)] $ \sum_{xb} (Q^2_{(b)})_{xx} v^{xxb}_{kla} = \delta_{kl} (Q^1_{(a)})_{kk} $
\item[(A3b)] $ \sum_{ka} v^{xyb}_{kka} = \delta_{xy} $
\item[(A4)] $ \sum_{rsb} (A_2)^{xyc}_{rsb} v^{rsb}_{kla} = \sum_{rsb} (A_1)_{kla}^{rsb} v_{rsb}^{xyc} $
\end{itemize}
for all admissible indices.
\end{prop}
\begin{proof}
The following argument is analogous to the one for Proposition 2.10 in \cite{Mrozinskiso3deformations}, compare \cite{Wangqsymmetry}.
Expressing the universal morphism $ \beta_{21}: B_1 \to B_2 \otimes \Poly(G^+(\G_2, \G_1))$ relative to the bases chosen as above, we can write
$$
\beta_{21}(e^{(a)}_{kl}) = \sum_{xyb} e_{xy}^{(b)} \otimes v^{xyb}_{kla}.
$$
Then $ \Poly(G^+(\G_2, \G_1)) $ is generated as a $ \ast $-algebra by the elements $ v^{xyb}_{kla} $. Now the conditions on this implementing
a bi-Galois object are equivalent to the equations listed above. More precisely, we have
\begin{bnum}
\item[$\bullet$] \emph{$ (A1a) \iff \beta_{21} $ is an algebra homomorphism}.
This follows from
$$
\beta_{21}(e^{(a)}_{kl}) \beta_{21}(e^{(b)}_{rs}) = \sum_{xwc myd} e_{xw}^{(c)} e_{my}^{(d)} \otimes v^{xwc}_{kla} v^{myd}_{rsb}
= \sum_{xwc n} e_{xy}^{(c)} \otimes v^{xwc}_{kla} v^{wyc}_{rsb}
$$
and
$$
\beta_{21}(e^{(a)}_{kl} e^{(b)}_{rs}) = \delta_{ab} \delta_{lr} \beta_{ji}(e^{(a)}_{ks}) = \sum_{xyc} \delta_{ab} \delta_{lr} e_{xy}^{(c)} \otimes v^{xyc}_{ksa}.
$$
\item[$\bullet$] \emph{$(A1b) \iff S: \Poly(G^+(\G_1, \G_2)) \to \Poly(G^+(\G_2, \G_1)) $ given by
$$
S(v^{kla}_{rsb}) = (Q^2_{(b)})_{ss} (Q^1_{(a)})^{-1}_{ll} v_{lka}^{srb}
$$
defines an algebra anti-isomorphism}. Indeed, we have
$$
\sum_l S(v^{lma}_{wxb}) S(v^{kla}_{yzc}) = \sum_l (Q^2_{(b)})_{xx} (Q^1_{(a)})^{-1}_{mm} v_{mla}^{xwb} (Q^2_{(c)})_{zz} (Q^1_{(a)})^{-1}_{ll} v_{lka}^{zyc}
$$
and
$$
\delta_{bc} \delta_{wz} S(v^{kma}_{yxc}) = \delta_{bc} \delta_{wz} (Q^2_{(c)})_{xx} (Q^1_{(a)})^{-1}_{mm} v_{mka}^{xyc},
$$
so this statement follows in combination with $ (A1a) $.
\item[$\bullet$] \emph{$(A2) \iff \beta_{21} $ is involutive}. This follows immediately from $ (e^{(a)}_{kl})^* = e^{(a)}_{lk} $.
\item[$\bullet$] \emph{$(A3a) \iff (\psi_2 \otimes \id) \circ \beta_{21}(b) = \psi_1(b)1 $ for all $ b \in B_1 $}.
This follows from
$$
(\psi_2 \otimes \id) \circ \beta_{21}(e^{(a)}_{kl}) = \sum_{xyb} \psi_2(e_{xy}^{(b)}) v^{xyb}_{kla} = \sum_{xb} (Q_{(b)}^2)_{xx} v^{xxb}_{kla}
$$
and
$$
\psi_1(e^{(a)}_{kl})1 = (Q_{(a)}^1)_{kk} \delta_{kl}.
$$
\item[$\bullet$] \emph{$(A3b) \iff \beta_{21} $ is unital}. This follows from
$$
\beta_{21}(1) = \sum_{ak} \beta_{21}(e^{(a)}_{kk}) = \sum_{xybak} e_{xy}^{(b)} \otimes v^{xyb}_{kka}.
$$
\item[$\bullet$] \emph{$(A4) \iff \beta_{21} \circ A_1 = (A_2 \otimes \id) \circ \beta_{21} $}. This follows from
$$
(\beta_{21} \circ A_1)(e^{(a)}_{kl}) = \sum_{rsb} (A_1)_{kla}^{rsb} \beta_{21}(e^{(b)}_{rs}) = \sum_{rsbxyc} (A_1)_{kla}^{rsb} e^{(c)}_{xy} \otimes v_{rsb}^{xyc}
$$
and
$$
(A_2 \otimes \id) \circ \beta_{21}(e^{(a)}_{kl}) = \sum_{rsb} A_2(e_{rs}^{(b)}) \otimes v^{rsb}_{kla}
= \sum_{rsbxyc} (A_2)_{rsb}^{xyc} e^{(c)}_{xy} \otimes v^{rsb}_{kla}.
$$
\end{bnum}
Combining these observations yields the claim.
\end{proof}
\subsection{Quantum symmetries of quantum Cuntz-Krieger algebras}
We shall now show that quantum automorphisms and quantum isomorphisms of directed quantum graphs lift naturally to the level of
their associated $ C^\ast $-algebras.
Firstly, we have the following lifting result for quantum symmetries, compare the work in \cite{SchmidtWeberqsym} on classical graph $ C^\ast $-algebras.
\begin{theorem} \label{qutprop}
Let $ \G = (B, \psi, A) $ be a directed quantum graph. Then the canonical action $ \beta: B \rightarrow B \otimes C(G^+(\G)) $
of the quantum automorphism group of $ \G $ induces an action $ \hat{\beta}: \FO(\G) \rightarrow \FO(\G) \otimes C(G^+(\G)) $ such that
$$
\hat{\beta}(S(b)) = (S \otimes \id)\beta(b)
$$
for all $ b \in B $.
\end{theorem}
The proof of Theorem \ref{qutprop} will be obtained as a special case of the more general Theorem \ref{qisolift} on quantum isomorphisms below.
Nonetheless, for the sake of clarity we have decided to state this important special case separately.
\begin{remark}
There are typically plenty of quantum automorphisms of $ \FO(\G) $, and in fact, even $ * $-automorphisms, which do not arise from quantum automorphisms
as in Theorem \ref{qutprop}. For instance, the gauge action on the free Cuntz-Krieger algebra associated with a classical directed graph cannot be described
this way, compare paragraph \ref{pargauge}.
\end{remark}
Now assume that $ \G_1, \G_2 $ are quantum isomorphic directed quantum graphs in standard form, with corresponding linking algebras $ \Poly(G^+(\G_j, \G_i)) $.
The associated $ \ast $-homomorphisms $ \beta_{ji}: B_i \to B_j \otimes \Poly(G^+(\G_j,\G_i)) $ for $ 1 \leq i, j \leq 2 $ are given by
$$
\beta_{ji}(e^{(a)}_{kl}) = \sum_{xyb} e_{xy}^{(b)} \otimes v^{xyb}_{kla}
$$
in terms of the standard matrix units. Here $ v^{xyb}_{kla} $ are the generators of $ \Poly(G^+(\G_j,\G_i)) $ as in Proposition \ref{qisorelations}.
\begin{theorem} \label{qisolift}
Let $ \G_i = (B_i, \psi_i, A_i) $ for $ i = 1,2 $ be directed quantum graphs and assume that $ \G_1 \cong_q \G_2 $. Then there exists $ \ast $-homomorphisms
$$
\hat{\beta}_{ji}: \FO(\G_i) \to \FO(\G_j) \otimes C(G^+(\G_j, \G_i))
$$
for $ 1 \leq i,j \leq 2 $ such that
$$
\hat{\beta}_{ji}(S_i(b)) = (S_j \otimes \id) \beta_{ji}(b)
$$
for all $ b \in B_i $.
\end{theorem}
\begin{proof}
Observe first that for $ i = j $ we are precisely in the situation of Theorem \ref{qutprop}, so that Theorem \ref{qutprop} is indeed
a special case of the claim at hand.
Let us write $ m_\O: \O \otimes \O \rightarrow \O $ for the multiplication in $ \O = \Poly(G^+(\G_j, \G_i)) $. We claim that
$$
(m_j^* \otimes \id)\beta_{ji} = (\id \otimes \id \otimes m_\O)(\id \otimes \sigma \otimes \id)(\beta_{ji} \otimes \beta_{ji}) m_i^*,
$$
where $ m_i: B_i \rightarrow B_i \rightarrow B_i $ denotes multiplication in $ B_i $ and $ \sigma $ is the flip map.
Indeed, rewriting Lemma \ref{mstarcomputation} in terms of the standard matrix units yields
$$
m_i^*(e^{(a)}_{kl}) = \sum_r (Q^i_{(a)})^{-1}_{rr}\, e^{(a)}_{kr} \otimes e^{(a)}_{rl},
$$
and using relation $ (A1b) $ from Proposition \ref{qisorelations} we get
\begin{align*}
(m_j^* \otimes \id)\beta_{ji}(e^{(a)}_{kl}) &= \sum_{xnb} m_j^*(e^{(b)}_{xn}) \otimes v^{xnb}_{kla} \\
&= \sum_{xybn} (Q_{(b)}^j)^{-1}_{yy} e^{(b)}_{xy} \otimes e^{(b)}_{yn} \otimes v^{xnb}_{kla} \\
&= \sum_{xybmnc} (Q_{(b)}^j)^{-1}_{yy} \delta_{bc} \delta_{ym} e^{(b)}_{xy} \otimes e^{(c)}_{mn} \otimes v^{xnb}_{kla} \\
&= \sum_{wxybmnc} (Q_{(a)}^i)^{-1}_{ww} e^{(b)}_{xy} \otimes e^{(c)}_{mn} \otimes v^{xyb}_{kwa} v^{mnc}_{wla} \\
&= \sum_w (Q_{(a)}^i)^{-1}_{ww} (\id \otimes \id \otimes m_\O)(\id \otimes \sigma \otimes \id)(\beta_{ji} \otimes \beta_{ji}) (e^{(a)}_{kw} \otimes e^{(a)}_{wl}) \\
&= (\id \otimes \id \otimes m_\O)(\id \otimes \sigma \otimes \id)(\beta_{ji} \otimes \beta_{ji}) m_i^*(e^{(a)}_{kl})
\end{align*}
as required.
Now consider the linear map $ s: B_i \rightarrow \FO(\G_j) \otimes C(G^+(\G_j, \G_i)) = D $ given by $ s = (S_j \otimes \id) \beta_{ji} $. Then
$$
s^*(b) = s(b^*)^* = (S_j \otimes \id)\beta_{ji}(b^*)^* = (S_j^* \otimes \id)\beta_{ji}(b),
$$
and we claim that $ s $ is a quantum Cuntz-Krieger $ \G_i $-family in $ D $. Writing $ \mu $ for the multiplication in $ \FO(\G_j) $
and $ \mu_D $ for the one in $ D $, our above considerations yield
\begin{align*}
&\mu_D (\id \otimes \mu_D)(s \otimes s^* \otimes s)(\id \otimes m_i^*)m_i^* \\
&= \mu_D (\id \otimes \mu_D)(S_j \otimes \id \otimes S_j^* \otimes \id \otimes S_j \otimes \id)(\beta_{ji} \otimes \beta_{ji} \otimes \beta_{ji})
(\id \otimes m_i^*)m_i^* \\
&= \mu_D (\id \otimes \id \otimes \mu \otimes \id)(S_j \otimes \id \otimes S_j^* \otimes S_j \otimes m_\O) \sigma_{45}
(\beta_{ji} \otimes \beta_{ji} \otimes \beta_{ji}) (\id \otimes m_i^*)m_i^* \\
&= \mu_D(\id \otimes \id \otimes \mu \otimes \id)(S_j \otimes \id \otimes S_j^* \otimes S_j \otimes \id)
(\id \otimes \id \otimes m_j^* \otimes \id)(\beta_{ji} \otimes \beta_{ji})m_i^* \\
&= (\mu \otimes \id)(\id \otimes \mu \otimes \id)(S_j \otimes S_j^* \otimes S_j \otimes m_\O)
(\id \otimes m_j^* \otimes \id)(\id \otimes \sigma \otimes \id)(\beta_{ji} \otimes \beta_{ji})m_i^* \\
&= (\mu \otimes \id)(\id \otimes \mu \otimes \id)(S_j \otimes S_j^* \otimes S_j \otimes \id)(\id \otimes m_j^* \otimes \id) (m_j^* \otimes \id)\beta_{ji} \\
&= (S_j \otimes \id)\beta_{ji} = s,
\end{align*}
and similarly
\begin{align*}
\mu_D (s^* \otimes s) m_i^*
&= (\mu \otimes m_\O) \sigma_{23} (S_j^* \otimes \id \otimes S_j \otimes \id)(\beta_{ji} \otimes \beta_{ji})m_i^* \\
&= (\mu \otimes \id)(S_j^* \otimes S_j \otimes m_\O) \sigma_{23} (\beta_{ji} \otimes \beta_{ji})m_i^* \\
&= (\mu \otimes \id)(S_j^* \otimes S_j \otimes \id)(m_j^* \otimes \id)\beta_{ji} \\
&= (\mu \otimes \id)(S_j \otimes S_j^* \otimes \id)(m_j^* \otimes \id)(A_j \otimes \id)\beta_{ji} \\
&= (\mu \otimes \id)(S_j \otimes S_j^* \otimes \id)(m_j^* \otimes \id)\beta_{ji} A_i \\
&= \mu_D (s \otimes s^*) m_i^* A_i,
\end{align*}
using the quantum Cuntz-Krieger relation for $ S_j $. Hence the universal property of $ \FO(\G_i) $ yields the claim.
\end{proof}
\begin{remark} \label{remhatinjective}
If we denote by $ \C_i \subset \FO(\G_i) $ the dense $ \ast $-subalgebra generated by $ S_i(B_i) $, then the restriction of the map $ \hat{\beta}_{ji} $
in Theorem \ref{qisolift} to $ \C_i $ is injective. Indeed, there exists a canonical unital $ \ast $-isomorphism
\begin{align*}
\theta_i^j: \Poly(G^+(\G_i)) &\to \Poly(G^+(\G_i, \G_j)) \Box_{\Poly(\G_j)} \Poly(G^+(\G_j, \G_i)),
\end{align*}
where
\begin{align*}
\Poly(G^+(\G_i, \G_j)) \Box_{\Poly(\G_j)} \Poly(G^+(\G_j, \G_i)) &= \{x \mid (\rho_j \otimes \id)(x) = (\id \otimes \lambda_j)(x) \} \\
&\subset \Poly(G^+(\G_i, \G_j)) \otimes \Poly(G^+(\G_j, \G_i)),
\end{align*}
and
\begin{align*}
\rho_j&: \Poly(G^+(\G_i, \G_j)) \rightarrow \Poly(G^+(\G_i, \G_j)) \otimes \Poly(G^+(\G_j)) \\
\lambda_j&: \Poly(G^+(\G_j, \G_i)) \rightarrow \Poly(G^+(\G_j)) \otimes \Poly(G^+(\G_j, \G_i))
\end{align*}
are the canonical ergodic actions of $ G^+(\G_j) $ on the linking algebras.
The map $ \theta_i^j $ satisfies
$$
(\hat{\beta}_{ij} \otimes \id) \hat{\beta}_{ji}(x) = (\id \otimes \theta_i^j) \hat{\beta}_{ii}(x)
$$
for all $ x \in \C_i $.
If $ \epsilon_i: \Poly(G^+(\G_i, \G_j)) \Box_{\Poly(\G_j)} \Poly(G^+(\G_j, \G_i)) \cong \Poly(G^+(\G_i)) \to \mathbb{C} $ is the character given by
the counit of $ \Poly(G^+(\G_i)) $, this implies
$$
(\id \otimes \epsilon_i)(\hat{\beta}_{ij} \otimes \id) \hat{\beta}_{ji}(x) = x
$$
for $ x \in \C_i $. Hence the restriction of $ \hat{\beta}_{ji} $ to $ \C_i $ is indeed injective.
However, it is not clear whether the map $ \hat{\beta}_{ji}: \FO(\G_i) \to \FO(\G_j) \otimes C(G^+(\G_j, \G_i)) $ itself is injective. In
the following section we show that this is at least sometimes the case.
\end{remark}
\section{Unitary error bases and finite dimensional quantum symmetries} \label{secunitaryerror}
In this section we apply the general results of the previous section to certain pairs of complete quantum graphs and
trivial quantum graphs, respectively. More precisely, we fix $ N \in \mathbb{N} $ and consider
\begin{align*}
\G_1^K(N) &= K_{N^2} = K(\mathbb{C}^{N^2}, \tr) \\
\G_2^K(N) &= K(M_N(\mathbb{C}), \tr)
\end{align*}
and
\begin{align*}
\G_1^T(N) &= T_{N^2} = T(\mathbb{C}^{N^2}, \tr) \\
\G_2^T(N) &= T(M_N(\mathbb{C}), \tr) = TM_N,
\end{align*}
compare section \ref{secexamples}.
The similarity between these pairs stems from the fact that we have canonical identifications
\begin{align*}
G^+(\G_1^K(N)) &= G^+(\G_1^T(N)) = S_{N^2}^+, \\
G^+(\G_2^K(N)) &= G^+(\G_2^T(N)) = G^+(M_N(\mathbb{C}), \tr),
\end{align*}
respectively. We will therefore also use the short hand notation $ G^+(\G_1(N)) $ and $ G^+(\G_2(N)) $ for these quantum automorphism groups.
We recall that $ G^+(\G_1(N)) $ and $ G^+(\G_2(N)) $ are monoidally equivalent, and that we have quantum isomorphisms $ \G_1^K(N) \cong_q \G_2^K(N) $
and $ \G_1^T(N) \cong_q \G_2^T(N) $, see the remarks in paragraph \ref{parqsym}. This means in particular that there exists a
bi-Galois object $ \Poly(G^+(\G_2(N), \G_1(N))) $ linking $ G^+(\G_1(N)) $ and $ G^+(\G_2(N)) $. If $ X $ is a set of cardinality $ N^2 $, then
this $ \ast $-algebra can be described in terms of generators $ v^{rs}_x $ with $ 1 \leq r,s \leq N $ and $ x \in X $, satisfying the relations as
in Proposition \ref{qisorelations}.
\subsection{Representations from unitary error bases}
With some inspiration from quantum information theory, we shall now construct unital $ * $-homomorphisms from the linking algebra $ \Poly(G^+(\G_2(N), \G_1(N))) $
to $ M_N(\mathbb{C}) $. The key tool in this construction is the notion of a unitary error basis \cite{Wernerteleportation}.
\begin{definition}
Let $ N \in \mathbb{N} $ and let $ X $ be a finite set of cardinality $ N^2 $. A {\it unitary error basis} for $ M_N(\mathbb{C}) $ is a
basis $ \mathcal{W} = \{w_x\}_{x \in X} $ for $ M_N(\mathbb{C}) $ consisting of unitary matrices that are orthonormal with respect to the normalized
trace inner product, so that
$$
\tr(w_x^*w_y) = \delta_{xy}
$$
for all $ x, y \in X $.
\end{definition}
Unitary error bases play a fundamental role in quantum information theory. In particular, they form a one-to-one correspondence with ``tight'' quantum
teleportation and superdense coding schemes \cite{Wernerteleportation}.
\begin{prop} \label{constructpiW}
Let $ N \in \mathbb{N} $ and assume that $ \mathcal{W} = \{w_x\}_{x \in X} $ is a unitary error basis for $ M_N(\mathbb{C}) $. With the notation as above,
there exists a unital $ \ast $-representation $ \pi_{\mathcal{W}}: \Poly(G^+(\G_2(N),\G_1(N))) \to M_N(\mathbb{C}) $ such that
$$
\pi_{\mathcal{W}}(v^{rs}_x) = \frac{1}{N} w_x^* e_{rs} w_x
$$
for all $ r,s,x $.
\end{prop}
\begin{proof}
Recalling that we write $ e_{rs} \in M_N(\mathbb{C}) $ for the standard matrix units, let us define
$$
V^{rs}_x = \frac{1}{N} w_x^* e_{rs} w_x
$$
for all $ 1 \leq r,s \leq N $ and $ x \in X $. It suffices to check that the elements $ V^{rs}_x \in M_N(\mathbb{C}) $ satisfy
the relations in Proposition \ref{qisorelations}.
In order to do this, we recall from Theorem 1 in \cite{Wernerteleportation} that a unitary error basis
can be equivalently characterized by the following properties for a family of unitaries $ \mathcal{W} = \{w_x\}_{x \in X} \subset M_N(\mathbb{C}) $:
\begin{bnum}
\item[a)] ({\it Depolarizing property}): $ \sum_{x \in X} w_x^*a w_x = N \Tr(a) 1 $ for $ a \in M_N(\mathbb{C}) $.
\item[b)] ({\it Maximally entangled basis property}): If $ \Omega = \frac{1}{\sqrt{N}}\sum_{i = 1}^N e_{i} \otimes e_{i} \in \mathbb{C}^N \otimes \mathbb{C}^N $
is a maximally entangled state and $ \Omega_x = (w_x \otimes 1)\Omega $, then $ \{\Omega_x\}_{x \in X} $ is an orthonormal basis
for $ \mathbb{C}^N \otimes \mathbb{C}^N $.
\end{bnum}
Observing that $ Q^1 = N^{-2} \id $ and $ Q_2 = N^{-1} \id $ we therefore we have to verify the following relations:
\begin{bnum}
\item[$\bullet$] \emph{$(A1a) \iff \sum_w V_{x}^{rw} V_{y}^{ws} = \delta_{xy} V_{x}^{rs} $}. This follows from
\begin{align*}
\sum_t V_{x}^{rt} V_{y}^{ts} &= N^{-2} \sum_t w_x^*e_{rt} w_x w_y^*e_{ts}w_y \\
&= N^{-2} \Tr(w_xw_y^*) w_x^*(e_{rs})w_y = \delta_{xy} N^{-1} w_x^*(e_{rs})w_x = \delta_{xy} V_{x}^{rs}.
\end{align*}
\item[$\bullet$] \emph{$ (A1b) \iff V^{ji}_{x} V^{sr}_{x} = \delta_{is} N^{-1} V^{jr}_{x} $}.
This follows directly from
$$
(w_x^*e_{ji}w_x)(w_x^*e_{sr}w_x) = \delta_{is} w_x^*e_{jr}w_x.
$$
\item[$\bullet$] \emph{$ (A2) \iff (V^{ij}_{x})^* = V^{ji}_{x} $}. This is immediate.
\item[$\bullet$] \emph{$ (A3a) \iff \sum_i N V^{ii}_{x} = 1 $}. This follows from
$$
\sum_i N V^{ii}_{x} = \sum_i w_x^* e_{ii} w_x = w_x^* w_x = 1.
$$
\item[$\bullet$] \emph{$ (A3b) \iff \sum_{z} V^{ij}_{z} = \delta_{ij}1 $}. This is the depolarizing property of $ \mathcal{W} $.
\item[$\bullet$] \emph{$ (A4) \iff \sum_{rs} (A_2)_{rs}^{ij} V^{rs}_{x} = \sum_y (A_1)^{y}_{x} V_{y}^{ij} $}.
For the trivial quantum graphs this is obvious. In the case of complete quantum graphs we have $ (A_1)^x_y = 1, (A_2)^{ij}_{kl} = N \delta_{ij} \delta_{kl} $
for all $ x,y,i,j,k,l $. Combining this with relations (A3a) and (A3b) yields the claim. More precisely, using (A3a) we obtain
\begin{align*}
\text{(A4)} &\iff \delta_{ij}\sum_s N V^{ss}_x = \sum_{y} V_{y}^{ij} \\
&\iff \delta_{ij}1 = \sum_y V^{ij}_y \\
&\iff \text{(A3b)}
\end{align*}
as required.
\end{bnum}
This completes the proof.
\end{proof}
\begin{remark} \label{pauli}
It is easy to construct examples of unitary error bases.
Let $ X, Z \in M_N(\mathbb{C}) $ be the generalized Pauli matrices given by their action on the standard basis $ |0 \ket, \dots, |N - 1 \ket $
of $ \mathbb{C}^N $ according to the formulas
$$
X|j \ket = \omega^j |j \ket, \qquad Z|j \ket = |j + 1 \ket,
$$
where we write $ \omega = e^{\frac{2\pi i}{N}} $ and calculate modulo $ N $. Then $ \mathcal{W} = \{X^jZ^k\}_{0 \leq j,k \leq N-1} $ is a unitary error
basis for $ M_N(\mathbb{C}) $.
\end{remark}
\subsection{Applications}
We shall use Proposition \ref{constructpiW} to study the structure of the quantum Cuntz-Krieger algebras $ \FO(\G_2^K(N)) = \FO(K(M_N,\tr)) $
and $ \FO(\G_2^T(N)) = \FO(TM_N) $, by comparing them with $ \FO(\G_1^K(N)) $ and $ \FO(\G_1^T(N)) $, respectively.
Recall from paragraph \ref{parQKn} that $ \FO(\G_1^K(N)) $ identifies canonically with the Cuntz algebra $ \O_{N^2} $.
Note also that $ \FO(\G_1^T(N)) = \ast_{N^2} C(S^1) $ is the non-unital free product of $ N^2 $ copies of $ C(S^1) $, compare Proposition \ref{sumfreeproduct}.
\begin{prop} \label{Cuntzembedding}
There are injective $ \ast $-homomorphisms
\begin{align*}
\pi_N^K&: \O_{N^2} \hookrightarrow M_N(\FO(K(M_N, \tr))) \\
\sigma_N^K&: \FO(K(M_N, \tr)) \hookrightarrow M_N(\O_{N^2})
\end{align*}
and
\begin{align*}
\pi_N^T&: \ast_{N^2} C(S^1) \hookrightarrow M_N(\FO(TM_N)) \\
\sigma_N^T&: \FO(TM_N) \hookrightarrow M_N(\ast_{N^2}C(S^1))
\end{align*}
for all $ N \in \mathbb{N} $.
\end{prop}
\begin{proof}
The construction of these maps for trivial quantum graphs is virtually identical to the one for complete quantum graphs. In order to treat both cases
simultaneously we will therefore write $ \G_1(N) $ and $ \G_2(N) $ to denote either $ \G_1^K(N) $ and $ \G_2^K(N) $, or $ \G_1^T(N) $
and $ \G_2^T(N) $, respectively. Our task is then to define injective $ \ast $-homomorphisms
\begin{align*}
\pi_N&: \FO(\G_1(N)) \hookrightarrow M_N(\FO(\G_2(N))) \\
\sigma_N&: \FO(\G_2(N)) \hookrightarrow M_N(\FO(\G_1(N)))
\end{align*}
for $ N \in \mathbb{N} $.
Since the quantum graphs $ \G_1(N) $ and $ \G_2(N) $ are quantum isomorphic, Theorem \ref{qisolift} yields natural $ \ast $-homomorphisms
\begin{align*}
&\hat{\beta}: \FO(\G_1(N)) \to \FO(\G_2(N)) \otimes C(G^+(\G_2(N), \G_1(N))) \\
&\hat{\gamma}: \FO(\G_2(N)) \to \FO(\G_1(N)) \otimes C(G^+(\G_2(N), \G_1(N)))^{op},
\end{align*}
taking into account that $ C(G^+(\G_1(N), \G_2(N))) \cong C(G^+(\G_2(N), \G_1(N)))^{op} $.
In order to give explicit formulas for these maps let $ X $ be a set of cardinality $ N^2 $ and denote the standard generators
of $ \FO(\G_1(N)) $ by $ S_x = S_1(N^2 e_x) $ for $ x \in X $. Similarly, write $ S_{rs} = S_2(N e_{rs}) $ for the standard generators
of $ \FO(\G_2(N)) = \FO(\G_2) $. Here $ S_1: \mathbb{C}^{N^2} \rightarrow \FO(\G_1(N)) $ and $ S_2: M_N(\mathbb{C}) \rightarrow \FO(\G_2(N)) $ are the
canonical linear maps.
Then we calculate
\begin{align*}
\hat{\beta}(S_x) &= N \sum_{rs} S_{rs} \otimes v^{rs}_x, \\
\hat{\gamma}(S_{rs}) &= \sum_x S_x \otimes v^{sr}_x,
\end{align*}
where $ v^{rs}_x $ for $ 1 \leq r,s \leq N, x \in X $ are the standard generators of the linking algebra $ \O(G^+(\G_2(N), \G_1(N))) $,
see Proposition \ref{qisorelations}.
Consider now the $ \ast $-homomorphism $ \pi_\mathcal{W}: \Poly(G^+(\G_2(N),\G_1(N))) \to M_N(\mathbb{C}) $ obtained in Proposition \ref{constructpiW}.
Slicing the maps $ \hat{\beta}, \hat{\gamma} $ with $ \pi_\mathcal{W} $, we define the desired $ \ast $-homomorphims $ \pi_N, \sigma_N $ by
\begin{align*}
\pi_N &= (\id \otimes \pi_{\mathcal W}) \hat{\beta}, \\
\sigma_N &= (\id \otimes t) (\id \otimes \pi_{\mathcal W}^{op}) \hat{\gamma},
\end{align*}
using the isomorphism $ t: M_N(\mathbb{C})^{op} \cong M_N(\mathbb{C}) $ given by sending a matrix $ Y $ to its transpose $ Y^t $.
Concretely, if we let $ V^{rs}_x $ be constructed out of a unitary error basis as in Proposition \ref{constructpiW} then we have
\begin{align*}
\pi_N(S_x) &= N \sum_{rs} S_{rs} \otimes V^{rs}_x, \\
\sigma_N(S_{rs}) &= \sum_x S_x \otimes (V^{sr}_x)^t.
\end{align*}
Let us denote by $ m: M_N(\mathbb{C})^{op} \otimes M_N(\mathbb{C}) \to M_N(\mathbb{C}), m(a^{op} \otimes b) = ab $ the multiplication map.
Using the relations in Proposition \ref{constructpiW} we readily see that
$$
(\id \otimes m)(\id \otimes t \otimes \id)(\sigma_N \otimes \id)\pi_N(a) = a \otimes 1
$$
for all $ a $ contained in the $ \ast $-algebra generated by the elements $ S_x $ for $ x \in X $.
Similarly, we have
$$
(\id \otimes m)(\id \otimes t \otimes \id)(\pi_N \otimes \id)\sigma_N(b) = b \otimes 1
$$
for all $ b $ contained in the $ \ast $-algebra generated by the elements $ S_{rs} $.
Since $ m $ is completely bounded, it follows by continuity that $ \pi_N $ and $ \sigma_N $ are injective.
\end{proof}
Let us continue to use the notation from above and denote the embeddings obtained in Proposition \ref{Cuntzembedding} by $ \pi_N $ and $ \sigma_N $,
referring to either the trivial or complete quantum graphs $ \G_1(N), \G_2(N) $. Following ideas of Gao, Harris and Junge \cite{GHJteleportation}, we shall
refine these embeddings and realize each of $ \FO(\G_1(N)) $ and $ \FO(\G_2(N)) $ as an iterated crossed product of the other algebra with respect to
certain $ \mathbb{Z}_N $-actions, up to tensoring with matrices. This is indeed very much related to the work in \cite{GHJteleportation}, which exhibited a
similar connection between free group $ C^\ast $-algebras and Brown's universal non-commutative unitary algebras.
In the sequel we write again $ \omega = e^{\frac{2 \pi i}{N}} $ and calculate modulo $ N $. We relabel the generators of $ \FO(\G_i(N)) $
in the proof of Proposition \ref{Cuntzembedding} by $ S^{(i)}_{kl} $ for $ i = 1,2 $, with indices $ 0 \leq k,l \leq N-1 $, and let $ X, Z $ be the
generalized Pauli matrices from Remark \ref{pauli}.
Let us consider the following order $ N $ automorphisms $ \alpha_j \in \Aut(\FO(\G_2(N))) $ for $ j = 1,2 $ given on generators by
$$
\alpha_1(S^{(2)}_{kl}) = \omega^{k-l} S^{(2)}_{kl} \qquad \alpha_2(S^{(2)}_{kl}) = S^{(2)}_{k-1, l-1}.
$$
Note that $ \alpha_1 $ and $ \alpha_2 $ can be viewed as examples of gauge automorphisms as in paragraph \ref{pargauge}.
More precisely, they are the gauge automorphisms associated with the unitaries $ X, Z $ in the sense that
$$
(\alpha_1 \otimes \id)(S) = (1 \otimes X) S (1 \otimes X^*) \qquad (\alpha_2 \otimes \id)(S) = (1 \otimes Z^*) S (1 \otimes Z)
$$
for $ S = (S^{(2)}_{ij}) \in \FO(\G_2(N)) \otimes M_N(\mathbb{C}) $.
Similarly, we define order $ N $ automorphisms $ \beta_j \in \Aut(\FO(\G_1(N))) $ for $ j = 1,2 $ by
$$
\beta_1(S^{(1)}_{kl}) = S^{(1)}_{k-1,l} \qquad \beta_2(S^{(1)}_{kl}) = S^{(1)}_{k, l-1}.
$$
Clearly, all these automorphisms define actions of $ \mathbb{Z}_N $ on $ \FO(\G_2(N)) $ and $ \FO(\G_1(N)) $, respectively.
From the relation $ XZ = \omega ZX $ it follows that both pairs of actions $ \alpha_1, \alpha_2 $ and $ \beta_1, \beta_2 $ mutually commute.
Let us now consider the iterated crossed products
$$
\FO(\G_2(N)) \rtimes_{\alpha_1} \mathbb{Z}_N \rtimes_{\alpha_2} \mathbb{Z}_N, \qquad \FO(\G_1(N)) \rtimes_{\beta_1} \mathbb{Z}_N \rtimes_{\beta_2} \mathbb{Z}_N,
$$
where $ \alpha_2, \beta_2 $ are naturally extended to the crossed products by letting $ \mathbb{Z}_N $ act on itself through appropriate dual actions.
More precisely, given $ a \in \FO(\G_2(N)), b \in \FO(\G_1(N)) $ and $ g \in \mathbb{Z}_N \cong \{0, \ldots, N-1\} $, we let
$$
\alpha_2(a u_g) = \alpha_2(a) \omega^g u_g, \qquad \beta_2(b u_g) = \beta_2(g) \omega^{g} u_g.
$$
Abstractly, the algebra $ \FO(\G_2(N)) \rtimes_{\alpha_1} \mathbb{Z}_N \rtimes_{\alpha_2} \mathbb{Z}_N $ is the universal $ C^\ast $-algebra spanned by
elements of the form
$$
x = \sum_{j,k = 0}^{N - 1} a_{jk} v^j w^k,
$$
where $ a_{jk} \in \FO(\G_2(N)) $ and $ v, w $ are unitaries, such that the relations
$$
v^N = w^N = 1, \qquad v a_{jk} = \alpha_1(a_{jk}) v, \qquad w a_{jk} = \alpha_2(a_{jk}) w = a_{jk}, \qquad wv = \omega vw
$$
are satisfied. A similar description holds for $ \FO(\G_1(N)) \rtimes_{\beta_1} \mathbb{Z}_N \rtimes_{\beta_2} \mathbb{Z}_N $.
Our aim is to establish the following description of the iterated crossed products obtained in this way.
\begin{theorem} \label{crossedproducts}
For the double crossed products with respect to the actions of $ \mathbb{Z}_N $ introduced above one obtains $ \ast $-isomorphisms
\begin{align*}
M_N(\FO(K(M_N(\mathbb{C}), \tr))) &\cong \O_{N^2} \rtimes_{\beta_1} \mathbb{Z}_N \rtimes_{\beta_2} \mathbb{Z}_N \\
M_N(\O_{N^2}) &\cong \FO(K(M_N(\mathbb{C}), \tr)) \rtimes_{\alpha_1} \mathbb{Z}_N \rtimes_{\alpha_2} \mathbb{Z}_N
\end{align*}
and
\begin{align*}
M_N(\FO(TM_N)) &\cong (\ast_{N^2} C(S^1) )\rtimes_{\beta_1} \mathbb{Z}_N \rtimes_{\beta_2} \mathbb{Z}_N \\
M_N(\ast_{N^2}C(S^1)) &\cong \FO(TM_N)) \rtimes_{\alpha_1} \mathbb{Z}_N \rtimes_{\alpha_2} \mathbb{Z}_N
\end{align*}
for all $ N \in \mathbb{N} $.
\end{theorem}
In order to prove Theorem \ref{crossedproducts} we will construct the required isomorphisms explicitly, using again uniform notation
to treat the cases of trivial and complete quantum graphs simultaneously.
Consider the unitary error basis $ \W = \{X^j Z^k\}_{0 \leq j,k \leq N - 1} $ for $ M_N(\mathbb{C}) $ described in Remark \ref{pauli}.
Moreover let $ \pi_N: \FO(\G_1(N)) \to \FO(\G_2(N)) \otimes M_N(\mathbb{C}) $ and $ \sigma_N: \FO(\G_2(N)) \to \FO(\G_1(N)) \otimes M_N(\mathbb{C}) $ be the
corresponding embeddings constructed in the proof of Proposition \ref{Cuntzembedding}. That is, if we set
$$
V^{rs}_{lm} = \frac{1}{N}(X^lZ^m)^*e_{rs}(X^lZ^m) = \frac{1}{N}\omega^{-(r-s)l}e_{r-m,s-m},
$$
and use our previous notation for the generators of $ \FO(\G_j(N)) $, then we obtain
$$
\pi_N(S^{(1)}_{lm}) = N \sum_{0 \le r,s \le N-1} S^{(2)}_{rs} \otimes V^{rs}_{lm}
= \sum_{0 \le r,s \le N-1} S^{(2)}_{rs} \otimes \omega^{-(r-s)l} e_{r-m,s-m}
$$
and
$$
\sigma_N(S^{(2)}_{jk}) = \sum_{0 \le l,m \le N-1} S^{(1)}_{lm} \otimes (V^{kj}_{lm})^{t}
= \frac{1}{N} \sum_{0 \le l,m \le N-1} S^{(1)}_{lm} \otimes \omega^{(j-k)l} e_{j-m,k-m}.
$$
From the above formulas we can easily see that
\begin{align*}
(1 \otimes X)\sigma_N(S^{(2)}_{jk})(1 \otimes X^*) &= \sigma_N(\alpha_1(S^{(2)}_{jk})) \\
(1 \otimes X)\pi_N(S^{(1)}_{jk})(1 \otimes X^*) &= \pi_N(\beta_1(S^{(1)}_{jk})).
\end{align*}
Hence $ (\sigma_N, (1 \otimes X)) $ defines a covariant representation of the dynamical system $ (\FO(\G_2(N)), \mathbb{Z}_N, \alpha_1) $,
and similarly $ (\pi_N, 1 \otimes X) $ defines a covariant representation of $ (\FO(\G_1(N)), \mathbb{Z}_N, \beta_1) $.
As a consequence, we obtain $ \ast $-homomorphisms
\begin{align*}
\sigma_N'&: \FO(\G_2(N)) \rtimes_{\alpha_1} \mathbb{Z}_N \to \FO(\G_1(N)) \otimes M_N(\mathbb{C}) \\
\pi_N'&: \FO(\G_1(N)) \rtimes_{\beta_1} \mathbb{Z}_N \to \FO(\G_2(N)) \otimes M_N(\mathbb{C})
\end{align*}
satisfying
\begin{align*}
\sigma_N'(a) &= \sigma_N(a), \qquad \sigma_N'(v) = 1 \otimes X, \\
\pi_N'(b) &= \pi_N(b), \qquad \pi_N'(v) = 1 \otimes X,
\end{align*}
where $ a \in \FO(\G_2(N)) $ and $ b \in \FO(\G_1(N)) $, respectively.
Similarly, we compute
\begin{align*}
(1 \otimes Z^*) \sigma_N(S^{(2)}_{jk})(1 \otimes Z) &= \sigma_N(\alpha_2(S^{(2)}_{jk})), \\
(1 \otimes Z^*)(1 \otimes X) &= \omega (1 \otimes X)(1 \otimes Z^*), \\
(1 \otimes Z^*) \pi_N(S^{(1)}_{jk})(1 \otimes Z) &= \pi_N(\beta_2(S^{(1)}_{jk})).
\end{align*}
Hence $ (\sigma_N', (1 \otimes Z^*)) $ defines a covariant representation of the dynamical
system $ (\FO(\G_2(N)) \rtimes_{\alpha_1} \mathbb{Z}_N, \mathbb{Z}_N, \alpha_2) $,
and $ (\pi_N', (1 \otimes Z^*)) $ defines a covariant representation of $ (\FO(\G_1(N)) \rtimes_{\beta_1} \mathbb{Z}_N, \mathbb{Z}_N, \beta_2) $.
In the same way as before we obtain associated $ \ast $-homomorphisms
\begin{align*}
\sigma_N''&: \FO(\G_2(N)) \rtimes_{\alpha_1} \mathbb{Z}_N \rtimes_{\alpha_2} \mathbb{Z}_N \to \FO(\G_1(N)) \otimes M_N(\mathbb{C}) \\
\pi_N''&: \FO(\G_1(N)) \rtimes_{\beta_1} \mathbb{Z}_N \rtimes_{\beta_2} \mathbb{Z}_N \to \FO(\G_2(N)) \otimes M_N(\mathbb{C}),
\end{align*}
satisfying
\begin{align*}
\sigma_N''(a) &= \sigma_N(a), \qquad \sigma_N''(v) = \sigma_N'(v) = 1 \otimes X, \qquad \sigma_N''(w) = 1 \otimes Z^*, \\
\pi_N''(b) &= \pi_N(b), \qquad \pi_N''(v) = \pi_N'(v) = 1 \otimes X, \qquad \pi_N''(w) = 1 \otimes Z^*,
\end{align*}
respectively, where $ a \in \FO(\G_2(N)), b \in \FO(\G_1(N)) $.
With these constructions in place, Theorem \ref{crossedproducts} is a consequence of the following assertion.
\begin{theorem}
The maps $ \sigma_N'' $ and $ \pi_N'' $ are isomorphisms.
\end{theorem}
\begin{proof}
Using $ C^\ast(X, Z^*) = M_N(\mathbb{C}) $ and the description of $ \sigma_N'' $ given above it is easy to see that $ \sigma_N'' $ is surjective.
Explicitly, the range of $ \sigma_N'' $ contains $ \sigma_N(\FO(\G_2(N))) $ and $ 1 \otimes M_N(\mathbb{C}) $,
and these two algebras generate $ \FO(\G_1(N)) \otimes M_N(\mathbb{C}) $.
To show that $ \sigma_N'' $ is injective consider the $ \ast $-homomorphism
$$
(\pi_N \otimes \id) \sigma_N: \FO(\G_2(N)) \to \FO(\G_2(N)) \otimes M_N(\mathbb{C}) \otimes M_N(\mathbb{C}).
$$
If we denote by $ \{|\xi_{jk} \ket \}_{0 \le j,k \le N-1} \subset \mathbb{C}^N \otimes \mathbb{C}^N $ the orthonormal basis of maximally entangled
vectors given by
$$
\xi_{jk} = \frac{1}{\sqrt{N}} \sum_{r = 0}^{N-1} X^j Z^k |r \ket \otimes | r\ket,
$$
then one obtains
\begin{align*}
(\pi_N \otimes \id) \sigma_N''(S^{(2)}_{jk}) &= \frac{1}{N} \sum_{lmrs} S^{(2)}_{rs} \otimes \omega^{-(r-s)l} e_{r-m, s-m} \otimes \omega^{(j-k)l}e_{j-m,k-m} \\
&= \sum_{sm} S^{(2)}_{s+j-k,s} \otimes e_{s+j-k-m, s-m} \otimes e_{j-m, k-m} \\
&= \sum_{nm} S^{(2)}_{n+j, n+k} \otimes e_{n+j-m,n+k-m} \otimes e_{j-m,k-m} \\
&= \sum_{ln} \alpha_1^{-l}\alpha_2^{-n}(S^{(2)}_{jk}) \otimes |\xi_{l,n}\ket \bra \xi_{l,n}|.
\end{align*}
Next, we define a unitary $ V $ on $ \mathbb{C}^N \otimes \mathbb{C}^N $ by setting $ V(|j\ket \otimes |k\ket) = \omega^{-jk}|\xi_{jk} \ket $.
Then we have
$$
V^*(1 \otimes X)V = Z \otimes 1,
\qquad V^*(1 \otimes Z^*) V = X \otimes Z.
$$
Thus, if we consider the $ \ast $-homomorphism
$$
\Phi: \FO(\G_2(N)) \rtimes_{\alpha_1} \mathbb{Z}_N \rtimes_{\alpha_2} \mathbb{Z}_N \to \FO(\G_2(N)) \otimes M_N(\mathbb{C}) \otimes M_N(\mathbb{C})
$$
given by $ \Phi = \ad(1 \otimes V^*)(\pi_N \otimes \id)\sigma_N'' $, then we get
$$
\Phi(a) = \sum_{l,n} \alpha_1^{-l}\alpha_2^{-n}(a) \otimes e_{ll} \otimes e_{nn}
$$
for all $ a \in \FO(\G_2(N)) $, and also
$$
\Phi(v) = (1 \otimes V^*)(\pi_N \otimes \id) \sigma_N''(v)(1 \otimes V) = 1 \otimes V^*(1 \otimes X)V = 1 \otimes Z \otimes 1
$$
and
$$
\Phi(w) = (1 \otimes V^*)(\pi_N \otimes \id) \sigma_N''(w)(1 \otimes V) = 1 \otimes V^*(1 \otimes Z^*)V = 1 \otimes X \otimes Z.
$$
From these formulas it follows that the image of $ \Phi $ is exactly the reduced crossed
product $ \FO(\G_2(N)) \rtimes_{\alpha_1,r} \mathbb{Z}_N \rtimes_{\alpha_2,r} \mathbb{Z}_N $,
and $ \Phi $ is none other than the canonical quotient map from the full crossed product to the reduced crossed product.
Since $ \mathbb{Z}_N $ is finite, and hence amenable, the map $ \Phi $ is an isomorphism, forcing $ \sigma_N''$ to be injective. This proves the claim
for $ \sigma_N'' $.
For $ \pi_N'' $ one proceeds in a similar way, essentially by swapping the roles of the maps $ \pi_N $ and $ \sigma_N $ and repeating the above arguments.
\end{proof}
\begin{remark}
The first pair of isomorphisms in Theorem \ref{crossedproducts}
should not come as a great surprise, given that Theorem \ref{quantumcompletemain} in section \ref{secexamples} already asserts an
isomorphism $ \FO(K(M_N(\mathbb{C}), \tr)) \cong \O_{N^2} $.
In fact, the latter isomorphism can be verified by considering the injective $ \ast $-homomorphism $ \sigma_N^K: \FO(K(M_N(\mathbb{C}), \tr)) \to M_N(\O_{N^2}) $
obtained in Proposition \ref{Cuntzembedding} and inspecting the relations in the proof of Proposition \ref{constructpiW}. In the next section we will
prove Theorem \ref{quantumcompletemain} in full generality.
\end{remark}
\begin{remark}
Taking into account the identification $ \FO(K(M_N(\mathbb{C}), \tr)) \cong \O_{N^2} $, the statement for complete quantum graphs in Theorem \ref{crossedproducts}
is reminiscent of Takesaki-Takai duality. However, the isomorphisms are slightly different.
Note also that the $ C^\ast $-algebras $ *_{N^2} C(S^1) $ and $ \FO(TM_N) $ are not even Morita equivalent, compare Theorem \ref{Ktheoryquantumtrivial}.
\end{remark}
\begin{remark}
Using the isomorphism from Theorem \ref{Ktheoryquantumtrivial} we see that $ \pi_N^T $ induces an
embedding $ \ast_{N^2} C(S^1) \rightarrow M_N(\mathbb{C}) *_1 (C(S^1) \oplus \mathbb{C}) $. In the notation used above this maps the
generators $ S^{(1)}_{kl} $ to $ \sum_{rs} \omega^{k(s - r)} e_{r - l,r} S e_{s, s - l} $, where $ S $ denotes the standard generator
of $ C(S^1) \subset C(S^1) \oplus \mathbb{C} $.
\end{remark}
\begin{remark}
It seems natural to look at pairs of quantum Cuntz-Krieger algebras associated to quantum isomorphic quantum graphs
beyond the cases considered in Theorem \ref{crossedproducts}. Finding ``small'' representations of linking algebras could potentially allow one
to transfer properties like unitality, nuclearity, or existence of traces from one algebra to the other, without a priori knowing
whether the algebras are isomorphic or not.
\end{remark}
\section{The structure of complete quantum Cuntz-Krieger algebras} \label{secquantumcomplete}
In this final section we discuss our main result on the structure of complete quantum Cuntz-Krieger algebras, that is, we
provide the proof of Theorem \ref{quantumcompletemain} stated in section \ref{secexamples}.
Let us begin with a simple lemma.
\begin{lemma} \label{cuntz-sym}
Let $ A $ be a non-zero unital $ C^\ast $-algebra and let $ n_1, n \in \mathbb{N} $. Moreover assume that $ u = (u_{xy}) \in M_{n_1, n}(A) $ is a
rectangular unitary matrix with coefficients in $ A $. Let $ s_x $ for $ 1 \leq x \leq n_1 $ be the standard generators of $ \O_{n_1} $ and
define elements $ \hat{s}_y \in \O_{n_1} \otimes A $ for $ 1 \leq y \leq n $ by
$$
\hat{s}_y = \sum_{x = 1}^{n_1} s_x \otimes u_{xy}.
$$
Then the elements $ \hat{s}_y $ satisfy the defining relations of $ \O_n $ and
$$
C^\ast(\hat{s}_1, \ldots, \hat{s}_n) \cong \O_n.
$$
\end{lemma}
\begin{proof}
In order to verify the Cuntz relations we calculate
\begin{align*}
\hat s_z^*\hat s_y &= \sum_{x_1,x_2} s_{x_1}^*s_{x_2} \otimes u_{x_1z}^*u_{x_2y} \\
&= \sum_{x_1} 1 \otimes u_{x_1z}^*u_{x_1y} \\
&= \delta_{y,z}(1 \otimes 1)
\end{align*}
and
\begin{align*}
\sum_y \hat{s}_y \hat{s}_y^* &= \sum_{y, x_1, x_2} s_{x_1} s_{x_2}^* \otimes u_{x_1y} u_{x_2y}^* \\
&= \sum_{x_1, x_2} s_{x_1} s_{x_2}^* \otimes \delta_{x_1,x_2}1 \\
&= 1 \otimes 1.
\end{align*}
Since $ \O_n $ is simple this yields the claim.
\end{proof}
Now let us fix a complete quantum graph $ K(B,\psi) $ satisfying the hypotheses of Theorem \ref{quantumcompletemain},
that is, $ (B, \psi) $ is a finite quantum space in standard form such that $ \psi: B \rightarrow \mathbb{C} $ is a $ \delta $-form
with $ \delta^2 \in \mathbb{N} $. We shall use the same notation that as after Definition \ref{deffqs}, so
that $ B = \bigoplus_{a = 1}^d M_{N_a}(\mathbb{C}) $ and $ \psi(x) = \sum_{a = 1}^d \Tr(Q_{(a)} x_i) $ for $ x = (x_1, \dots, x_d) $.
By Remark \ref{wealth}, we have a quantum isomorphism $ K(B,\psi) \cong_q K_{\delta^2} $.
Denote by $ v^x_{ija} $ for $ 1 \leq a \leq d, 1 \leq i,j \leq N_a, 1 \leq x \leq \delta^2 $ the standard generators of
the $ C^\ast $-algebra $ A = C(G^+(K_{\delta^2}, K(B,\psi))) $ given in Proposition \ref{qisorelations}.
Moreover let $ n = \dim(B) $ and consider the rectangular matrix $ u = (u_{ija}^x) \in M_{\delta^2,n}(A) $ given by
$$
u_{ija}^x = (Q_{(a)})^{-1/2}_{jj} \delta^{-1} v_{ija}^x.
$$
Using the relations in Proposition \ref{qisorelations} one obtains
\begin{align*}
(u^*u)_{ija, klb} &= \sum_x (Q_{(a)})^{-1/2}_{jj} \delta^{-1} (v_{ija}^x)^* (Q_{(b)})^{-1/2}_{ll} \delta^{-1} v_{klb}^x \\
&= \sum_x (Q_{(a)})^{-1/2}_{jj} (Q_{(b)})^{-1/2}_{ll} \delta^{-2} v_{jia}^x v_{klb}^x \\
&= \sum_x (Q_{(a)})^{-1/2}_{jj} (Q_{(a)})^{-1/2}_{ll} \delta^{-2} \delta_{ab} \delta_{ik} v_{jla}^x \\
&= \sum_x (Q_{(a)})^{-1/2}_{jj} (Q_{(a)})^{-1/2}_{ll} (Q_{(a)})_{jj} \delta_{ab} \delta_{ik} \delta_{jl} \\
&= \delta_{ab} \delta_{ik} \delta_{jl}
\end{align*}
and
\begin{align*}
(uu^*)_{xy} &= \sum_{ija} (Q_{(a)})^{-1/2}_{jj}\delta^{-1} v_{ija}^x (Q_{(a)})^{-1/2}_{jj}\delta^{-1} (v_{ija}^y)^* \\
&= \sum_{ija} (Q_{(a)})^{-1}_{jj} \delta^{-2} v_{ija}^x v_{jia}^y \\
&= \sum_{ia} \delta_{xy} v_{iia}^x \\
&= \delta_{xy}.
\end{align*}
We conclude that $ u^*u = 1_{M_{n}(A)} $ and $ uu^* = 1_{M_{\delta^2}(A)} $, or equivalently, that $ u $ is unitary.
Next, we consider the $ \ast $-homomorphism $ \hat{\beta}: \FO(K(B,\psi)) \rightarrow \O_{\delta^2} \otimes A $ from Theorem \ref{qisolift},
which satisfies
$$
\hat{\beta}(S(e_{ij}^{(a)})) = \sum_{x} S(e_x) \otimes v_{jia}^x
$$
in terms of the standard matrix units. Equivalently, if we write $ S_{ij}^{(a)} = S(f_{ij}^{(a)}) $,
where $ f_{ij}^{(a)} = (Q_{(a)})^{-1/2}_{ii} e^{(a)}_{ij} (Q_{(a)})_{jj}^{-1/2}$ are the adapted matrix units for $ (B,\psi) $, and $ s_x = S(\delta^2e_x) $
for the canonical Cuntz isometries generating $ \O_{\delta^2} $, then
\begin{align*}
\hat{\beta}(S_{ij}^{(a)}) &=(Q_{(a)})^{-1/2}_{ii}(Q_{(a)})^{-1/2}_{jj}\delta^{-2}\sum_{x} s_x \otimes v_{ija}^x\\
&= (Q_{(a)})_{ii}^{-1/2} \delta^{-1} \sum_{x} s_x \otimes u^x_{ija}.
\end{align*}
Hence the unitarity of the matrix $ u = (u_{ija}^x) $ combined with Lemma \ref{cuntz-sym} implies that the
elements $ (Q_{(a)})^{1/2}_{ii} \delta \hat{\beta}(S_{ij}^{(a)}) $ form an $ n $-tuple of Cuntz isometries in $ \O_{\delta^2} \otimes A $.
According to Remark \ref{remhatinjective}, the restriction of $ \hat{\beta} $ to the $ \ast $-algebra generated by the $ S_{ij}^{(a)} $ is injective.
This shows that $ \FO(K(B,\psi)) $ is unital with unit
$$
e = \sum_{ija} (Q_{(a)})_{ii} \delta^2 S_{ij}^{(a)}(S_{ij}^{(a)})^*,
$$
and that the elements $ (Q_{(a)})^{1/2}_{ii} \delta S_{ij}^{(a)} $ form an $ n $-tuple of Cuntz isometries generating $ \FO(K(B,\psi)) $.
This completes the proof of Theorem \ref{quantumcompletemain}.
\begin{remark}
It seems reasonable to expect that $ \FO(K(B,\psi)) \cong \O_n $ for all choices of $ \delta $-forms $ \psi $, but we are unable to supply a proof.
Note that when $ \delta^2 \notin \mathbb{N} $, we no longer have a quantum isomorphism between $ K(B,\psi) $ and a classical complete graph, and
therefore a different approach would be needed.
\end{remark}
\bibliographystyle{hacm}
\bibliography{cvoigt}
\end{document}
| 110,770
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TITLE: Why do we put this term to the power of n/2 instead of n in this MLE?
QUESTION [0 upvotes]: Let say we want the estimator of something within the normal distribution. Let's take $\sigma^2$ as an example.
We would therefore write $$L(\sigma^2) = \prod_{i=1}^n \dfrac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{- (x - \mu)^2}{2\sigma^2}}$$
When trying to extract terms outside the product symbol we get this: $$\left(\frac{1}{\sqrt{2\pi}}\right)^n \left(\frac{1}{\sigma^2} \right)^{n/2} \prod_{i=1}^n e^{ \frac{-1}{2} \cdot \frac{(x - \mu)^2}{\sigma^2}}$$
My question is why when this $\dfrac{1}{\sigma^2}$ is put out of the product symbol do we make it to the power of $n/2$ instead of $n$ in contrast to what we did to $\dfrac{1}{\sqrt{2\pi}}$?
REPLY [1 votes]: You could have written $\left(\frac{1}{\sqrt{2\pi \sigma^2}}\right)^n \prod_{i=1}^n e^{ \frac{-1}{2} \cdot \frac{(x_i - \mu)^2}{\sigma^2}}$
or $\left(\frac{1}{\sqrt{2\pi}}\right)^n \left(\frac{1}{\sqrt{\sigma^2}} \right)^{n}\prod_{i=1}^n e^{ \frac{-1}{2} \cdot \frac{(x_i - \mu)^2}{\sigma^2}}$
or $\left(\frac{1}{\sqrt{2\pi}}\right)^n \left(\frac{1}{\sigma} \right)^{n}\prod_{i=1}^n e^{ \frac{-1}{2} \cdot \frac{(x_i - \mu)^2}{\sigma^2}}$
or $\left(\frac{1}{\sqrt{2\pi }}\right)^n\left(\frac{1}{\sigma^2} \right)^{n/2}\prod_{i=1}^n e^{ \frac{-1}{2} \cdot \frac{(x_i - \mu)^2}{\sigma^2}}$
as they are all equivalent,
and in any case multiplying the likelihood by a positive constant does not affect the maximum likelihood estimate
| 72,398
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What Happened on July 15
Significant Events
- 1099 City of Jerusalem is captured and plundered by Christian forces during the First Crusade
- 1381 John Ball, a leader in the Peasants' Revolt, is hung, drawn and quartered in the presence of Richard II of England.
- 1410 Battle of Grunwald (Second
- 1799 The Rosetta Stone is found in the Egyptian village of Rosetta by French Captain Pierre-François Bouchard during Napoleon's Egyptian Campaign
- 1955 18 Nobel laureates sign the Mainau Declaration against nuclear weapons, later co-signed by 34 more laureates
Famous Birthdays
Rembrandt van Rijn
(1606 - 1669)
Emmeline Pankhurst
(1858 - 1928)
Arianna Huffington
67 Years Old
Kim Alexis
57 Years Old
Forest Whitaker
56 Years Old
Famous Weddings
- 1500 "Blood Wedding" of Astorre Baglione & Lavinia Colonna in Perugia family Baglione massacre
- 1952 Gerald D. Lascelles (under English princess Mary) weds Angela Dowding
- 1965 Actress Joan Rivers (32) weds TV producer Edgar Rosenberg
- 1999 "Grateful Dead" guitarist Bob Weir (51) weds Natasha Muenter (31) in Mill Valley, California
- 2005 Heavyweight boxing champion Lennox Lewis (39) weds former Miss Jamaica runner-up Violet Chang at William Knibb Memorial Baptist Church in Falmouth, Jamaica
Famous Divorces
- 1996 Prince Charles and Princess Diana sign divorce papers
- 2016 Actress Gwyneth Paltrow and singer Chris Martin divorce after 12
| 418,083
|
Posts Tagged<<
Big Data Facilitates Cancer Research
Project Data Sphere (PDS), an online platform sharing clinical trial data for use in cancer research, announced its official opening earlier this week. The goal of the project is to accelerate drug discovery and research to improve the lives of cancer patients around the world. 9 data sets are being
| 175,582
|
A logo is a symbol that represents your business and the values that your business stands for. It is an important part of branding and creating an image of the business in the eyes of your customers. A well designed and unique logo goes a long way in building brand identity for a business. Logos of companies like Nike, Apple and McDonald’s are some well known examples of this.
A good logo captures the essence of the business and represents it before the viewer. Its basic function is to attract the attention and captivate the mind of the viewer and thus help the business in building its image and brand. In that respect, an animated logo is much more effective that a static logo. How would you like the letters of the logo of your business coming dancing out with different light effects, in changing colors and with various rotational or spinning movements of the sign? Imagine a rotating globe, a flying bird or a belle dancer in motion as compared to the static images. Which would be more attractive or captivating in the eyes of the viewer? The answer is quite obvious. Animated logo is a new emerging trend that is fast becoming popular, especially among web-based businesses.
An animated logo can be beneficial for your business in more than one ways. It goes a long way in building the brand identity of your business. Everybody is attracted by animated images and text. It creates a unique image in the mind of the customers and the potential customers. It distinguishes your business from the others with static logos and leaves a lasting impact on the mind of the viewer. That will help in building the brand and ultimately growth in the business.
A well designed animated logo can send a more powerful message than a static logo. Through an animated image, you can convey something even in a few seconds that you wouldn’t be able to convey through a static image. It could be uncorking of a bottle of a drink, ripple and rise of a wave or a sportsman in action. A static image can never convey the thought an animated image can. It opens new horizons to your creative ideas to create effective and attention grabbing logos.
An animated logo can be a potent marketing weapon for your business. An attractive and well designed animated logo will certainly leave impression on the mind of the potential customers and that would later translate into increased business volume. Being a part of your advertisement campaign, it would make it more effective, as everyone likes and remembers a catchy animation. It is even more important if your business is online, because animated logo will always be visible on your website and your online advertisements and will give you a competitive edge over your rivals. Yahoo has recognized it and it has started using animated logos on its various pages.
To sum up, a picture is worth a thousand words. Similarly an animated picture is worth a million words!
| 255,983
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Beats Music service debuts for iPhone, Android
The new streaming music app is slick, but you'll have to pay $9.99 a month to listen.
Can't get your fill of streaming music with the likes of Pandora, Spotify, and Rdio? There's a new music service in town.
Debuting for the iPhone and Android devices on Tuesday, the Beats Music app works more or less like its rivals. It acts as an online radio station to serve you tunes from your favorite artists and genres.
The service claims access to more than 20 million songs, which you can listen to online or download to your device to listen offline.
But unlike Pandora and similar services, Beats Music doesn't offer a free, ad-supported version. Your only option is to shell out $9.99 a month for the privilege of subscribing. Fortunately, you can try the service through a free 7-day trial.
Related stories
After you download the app and sign up for your account, Beats Music poses a couple of simple questions. You're asked to choose your favorite genres and three of your favorite artists. The service then cooks up a lineup of songs based on your preferences.
You can tag the songs you like or don't like so the service can learn what to play. You also can add a song to your library and playlists. If an individual song is featured on an album, you can access that album to hear the rest of the tracks.
The iOS version of the app is optimized for the iPhone, though it's also compatible with the iPad. A version for Windows Phone is due out this Friday, according to a tweet from Beats Music CEO Ian Rogers picked up by Windows Phone Central. A Windows Store edition also is on the way, as posted on the Beats Music Web site.
Beats Music did a nice job with the app and the service. But will music lovers pony up $10 a month for yet another streaming service when free options abound?
| 143,121
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ESPN lets us know that 2013 really was a special year for Georgia.
Georgia was one of six teams last season to pass for at least 4,000 yards and rush for at least 2,000. Those six teams averaged 10.5 wins. UGA won eight.
My first thought was to blame that on the injury situation, but those stats were compiled despite the injuries. That leaves the usual suspects – defense and special teams.
How good could last year’s team have been if those two areas had merely been competent?
43 responses to “Today’s random factoid”
Other than our field goal kickers it’s been a while since we had competent special teams. It still strikes me as odd given that early in Richt’s tenure at Georgia we were much more aggressive on special teams.
As Interesting as it is confounding the Dawgs two best ST seasons in recent vintage were the two with the worst overall results: 2009 & 2010, I think Richt & Co. treat STs something like a training ground for the incoming recruits. If said recruits are intrinsically flawed or tend towards the soft the field position game can tilt against UGa.
Rule changes about how players can launch to block kicks and punts have a lot to do with that. Many of Boss’s blocks would be flagged now. As far as punt returns I have no idea.
yet other teams seem to block ours…..
It was a crazy year, and many of the unnoticed factors were to blame. Certainly, Grantham was incompetent. I don’t know another way to frame it. He knows defense, but he was not good at translating that knowledge to 20 year olds to play at a high level. That’s obvious.
While Bobo and Murray overcame the injuries to produce on offense, they killed us in other ways. Tray couldn’t stay healthy at safety and it killed us. JHC got a little banged up. We played 4 games with both Tray and JHC at the safeties: South Carolina, North Texas, LSU and Auburn. The defense looked better in those games than many others. Safety was one position we really couldn’t afford suspensions or injuries and we had both.
Also, the injuries killed our special teams. Some of those guys were playing special teams. Would Malcom have been our PR? JSW was playing gunner on the punt team. Does JSW get a crack at KOR? Or Malcom? At one point, we had JJ Green or Dawson returning KO because no one was left. Keith was out, we couldn’t risk Todd, and JSW and Malcom were sidelined. The injuries mounted the fear of putting our starters on special teams because we had so few starters left. It was a bad snowball.
The defense I saw against LSU and Auburn was atrocious.
But they were enough. They slowed LSU down enough. And, they slowed Auburn down enough to win if not for a miracle play. With those players on the field, we would have stoned Vandy and UT, and slowed down Mizzou much more.
It wasn’t just a miracle play, though. It was a miracle play enabled by a defensive mistake.
As good as 2008 should have been. Once again thanks to a turd at DC we piss away an overloaded Offense.
Such a shame to see this happen again. Lets hope with Pruitt that Richt gets back to a 50% success in hiring d coordinators. Lets also hope that the stars align with both Offense and D. That would be amazing to see.
Bobo lost his crayon.
Grantham found it, and kept it. Hopefully he packed it when he left for UL.
And don’t forget the offensive line.
Well, 2 losses were attributable to special teams gaffes, so there’s your answer I guess. I’d add Mizzou and Auburn to the “if these units were merely competent” equation too.
What most people don’t remember is that, the way everything ended up last year, if we had just beaten Mizzou, we get a rematch with Auburn in Atlanta. Wonder how that might have turned out?? Thanks, Todd, again.
Last year’s team would have been in the conversation the first weekend in December if those two had been merely competent for most of the year. Those who claim our offensive philosophy is the problem just flat don’t understand football. Grantham underperformed with a defense full of NFL prospects. We including me should have known what was going to happen with a bunch of young talent that needed to be coached.
Grantham played favorites, failed to develop his talent, stubbornly played wrong schemes, passed the buck and blame to his players, and other stuff.
But perhaps most of all, he was a lousy teacher. Plus, he failed to teach the little things, the things that make players fundamentally sound, and failed to see that his coaches did the same. It was as if those things were too elementary for him to fool with.
It’s amazing, in one sense, that our defense did as well as it did, because it was in spite of him. What he knows doesn’t matter. The only thing that matters is what his players know.
I’m sorry he ever stepped foot in Athens.
~~~
Well stated, Ivey. I was suckered into the belief that Grantham was going to fix the mess Martinez created and, honestly, Coach Richt enabled for a couple of years. He took complex schemes and added that on to poor fundamentals and no teaching. What we got last year on defense was the result – guys playing on talent alone and hoping for the best.
I’m going to throw this out there, and will be prepared to get shouted down: Georgia wasn’t THAT bad on special teams last year.
Our field goal kicking was second in the conference –
Our punt team allowed the third fewest yards, though we did allow two to be returned for scores.
Our kickoff team allowed the 7th fewest yards per return. Right behind Bama.
Where we DID suck was in returning the ball, perhaps because of our head coach’s strategy. We were dead last averaging 2.92 yards per game, which MUST include fair catches, though the stats aren’t clear. Obviously our offense could move the ball though.
And we were also last in kickoff returns per game.
Our punting was middle of the pack.
So, we allowed two punt returns and a kickoff return, and we continue to bleed field position based on strategy. That strategy, at least last year, worked as our offense scored a LOT of points and gained a LOT of yards.
I don’t think the data is there to call our special teams a dumpster fire, though the botched punt stands out pretty hard. Mistakes at critical times often do.
If you want a culprit for last year’s record, maybe look to being 10th in scoring defense and 8th in total defense. Oh, let’s not forget 12th in opponent red zone conversion rates. It’s not rocket surgery, people scored a lot of points on us last year.
This^^. The problem on STs has been coaching and coaches decision-making.
I tend to vehemently disagree with CMR’s return strategy. HOWEVER, last year we had #11 and #3 back there… so there’s at least a justification to not trying to return it.
That said, I’d love to see some data on fumbles during fair catches vs fumbles trying to return the ball. It seems to me that there can’t be that much of a difference, and we should be returning punts instead of giving up hundreds of yards a year and valuable field position. In other words, if you’re going to fumble it 5% of the time no matter what, at least put someone back there to move it forward.
We have a Thomas Brown sighting.
Hate these stupid trolls.
Quit bashing our team idiots.
Hey dude, I was defending our special teams. Just cause I use statistics doesn’t make me:
a) a Thomas Brown/troll
b) a team basher.
Were you suggesting our defense was good last year? Cause that it sucked was my point.
Work on your reading comprehension before you start breathing fire.
Our defense gave up @36 pts per game vs ranked teams, same as Auburn, and they got to the BCS championship.
You saying Auburn didn’t deserve to get there after beating us and Alabama?
Todd, is that you?
Seriously, if this is deep 9-layer sarcasm, then hats off to you sir. Well played.
Senator, please get Thomas Brown troll off this site.
Done.
Although I suspect you and I have a different idea about who the troll is.
Thank you.
Ha!
I see that Thomas Brown changed his name ag…oh, you fixed it. Nice. :-)
We had enough special teams screw-ups to call them a dumpster fire:
1) Clemson – botched field goal snap (difference maker)
2) South Carolina – dropped punt snap (ensuing TD tied the game just before half)
3) North Texas – kickoff return for TD/punt blocked for TD – wasted a good effort by the D
4) Tennessee – blocked punt for TD (changed momentum) and penalty on punt block gave UT the opportunity to get a 1st down and turned into a TD
5) Vandy – muffed punt (turned into a TD) and muffed punt snap (difference making TD)
6) Auburn – attempt to jump over the wedge on a punt gave Auburn a 1st down (I believe turned into a TD)
7) Nebraska – muffed punt (turned into a TD)
Yes, Morgan had a great year. Barber was having a good year until the UT game (snap issues not withstanding). Our return teams gave us zero field position. I think they were pretty terrible taken as a whole.
Great points. I’m going to remove N Texas as a difference maker from that list, but can’t argue with the rest. And we beat SC and UT.
My only retort is to ask: If we had a defense that gave up 5 points less a game, would those plays be difference makers? And my other only retort (sorry) is to ask: How do you coach away botched snaps and catches? I honestly don’t know if there’s an answer or not to the last question.
Excellent points, Spence. I understand your perspective on UNT, but the special teams snafus did make the Sanford crowd much more uncomfortable than it should have been. The South Carolina screw-up did change the complexion of the game. Remember they were driving with the first possession of the 2nd half and Shaw fumbled in our territory. It had the potential to change the game. The special teams screw-ups in the UT game made that game much more competitive than it should have been.
I agree with you about the defense. If they give up five fewer points per game, they don’t matter, but the special teams’ issues took away some possessions and put the defense in some very bad field position. For a team with as much youth as we had on defense, the offense and special teams couldn’t afford to give up poor field position as well. Unfortunately the special teams did that a lot.
If We had managed to stay healthy last year we would’ve won ten regular season games. The offense was something else but you can’t overcome that level of ineptitude in 2 of the 3 phases of the game week in and week out.
Spencer, a 3rd name really? You trollhead.
Look, our team will win the BCS this year, now go back to the Varsity and eat your lunch ok?
What?
I know the feeling.
a second name for Spencer troll?
One of the few times I can to a Gator, “I feel your pain …” Seriously, Timmy, you’re a good Gator fan that is welcome anytime.
With a serviceable D, we probably win MO, Vandy, AU and a good chance we beat Clempson. Then Alabama in the SECCG and then FSU in the MNC, We would have then fired Richt to promote Grantham who in turn hires Pruitt, they both run off Bobo, we hire Lane Kiffin as our OCoordinator, who sabotages Grantham and we start the season as Kiffin as our Head Coach and Grantham goes back to coaching the DLine in Cleveland.
Wow. I’ve never seen mainc depression so perfectly elaborated in a paragraph before. Well done!
Jack, when you get a coke from McDonalds, do you giggle like a school girl then immediately yell at them that it’s empty, no matter the level of the soda?
Get a life you stupid trollface
Of course it’s defense and special teams. And a little bit is also offense (mostly OL). But the overriding factor in it all is sloppy football – beating yourselves. That’s been our trademark for years now, and why our reputation is what it is, both around the conference and nationally.
It’s a reputation that’s well-deserved, I’m sorry to say. It’s why we’ve underachieved. Certainly most of it, the greater part of it, has come from the defensive side of the ball, and it has, from time to time, bled over into ST’s.
So, it may sound like a broken record, but that’s what’s so exciting about everything that’s going on in Athens now – we have a legitimate chance to turn all that around, and become a well-coached, efficient team, that is opportunistic and doesn’t give games away.
Because that’s what our record of the last 8 years reflects.
~~~
| 160,118
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Kaleidoscope live!
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Time: 17:00 - 17:45
Title: Driving down a country road at night with no lights while looking out the back window: Prediction, gender, and dementia
Papers:
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Brushaber D, et al. Proposed research criteria for prodromal behavioural
variant frontotemporal dementia. Brain 2022; 145(3): 1079–97.
- Krstacic JE, Carr BM, Yaligar AR, Kuruvilla AS, Helali JS, Saragossi J, et al.
Academic medicine’s glass ceiling: author’s gender in top three medical
research journals impacts probability of future publication success. PLoS One2022; 17(4): e0261209.
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progress and challenges. Mol Psychiatry [Epub ahead of print] 1 Apr 2022.
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Meet the hosts:
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Prof Sukhi S. Shergill
A multilingual MD/PhD, Sukhi is a Professor of Psychiatry and Systems Neuroscience at the Institute of Psychiatry, Psychology, and Neuroscience at King’s College London and a PsychiatristView profile
| 404,950
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TITLE: Reference or proof for an integral inequality
QUESTION [2 upvotes]: The following seems believable and quasi-intuitive to me, but it also doesn't quite seem trivial, and I'm not sure whether I've seen it stated before.
Let $f$ be a complex-valued integrable function on $[a,b]$. We then have
$$
\left|\int_a^b f(x) dx\right| \le \int_a^b |f(x)| dx
$$
and furthermore (this is the part I care more about) the inequality is an equality if and only if
$$
f(x) = e^{ic}|f(x)| \qquad \mbox{almost everywhere, where $c$ is a constant.}
$$
Speaking very informally, if the integral of $f(x)$ "adds up" to have the same modulus as the integral of $|f(x)|$, it seems like this could only happen if $f(x)$ always "points the same way", i.e. if $f(x)$ maintains the same argument (almost everywhere).
My questions are: 1. Does anyone know a reference for this fact? (And does it have a name)? 2. Does it have a simple self-contained proof? (Or would a proof from more fundamental principles be somewhat involved, approximating an integral with a sum and so forth?)
REPLY [3 votes]: The fact that this is complex valued is not actually useful; this is more generally true in inner product spaces, so let's prove it there. In particular, let
$$v=\int_{a}^bf(x)\,dx.$$
and let $\hat{v}$ be the unit vector in the same direction as $v$. Notice that the left side of your inequality is then $\hat{v}\cdot v=\|v\|$. Thus, we have that your inequality is the same as (bringing the dot product inside the integral by linearity)
$$\int_{a}^b\hat v \cdot f(x)\,dx\leq \int_{a}^b|f(x)|\,dx$$
But this is obvious because $\hat v \cdot f(x) \leq |f(x)|$ for all $x$. Then, in order to obtain equality, you obviously need that $\hat v \cdot f(x) = |f(x)|$ for almost all $x$, as if there were a set of positive measure for which it were less, the former integral would be less too. This is essentially what your condition stated.
| 49,029
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| 413,600
|
您要查找的是不是：
- The coach saved his star player for a trump card. 教练保留他的明星选手，作为他的王牌。
- Have it charged to my credit card. 把它记在我的信用卡的帐号里。
- The police asked him to show his identity card. 警察让他出示身份证。
- punch card transcriber 穿孔卡抄录器,穿孔卡转录器
- She sent me a Christmas card last year. 去年寄过一张圣诞卡。
- He fanned out the card in his hand. 他把牌在手上展成扇形。
- Place your card face up on the table, please. 请把你的牌面朝上摊在桌子上。
- card transcriber 卡片读数器,卡片抄录器
- A post card and three eight- cent stamp, please. 请给我买一张明信片和三张8分邮票。
- It looks as if Frank's holding all the high card. 看起来弗兰克好像掌握全部大牌。
- Play the highest card in your longest suit. 打你手中最长套的大牌。
- Can I pay by check or credit card? 我能用支票或信用卡吗？
- Now it's time that you played your trump card. 现在是你摊王牌的时候了。
- No one understood how I did the card trick. 谁也没有看出来我是怎样玩纸牌戏法的。
- I just want to talk to him. Here is my card. 我只是想跟他谈谈。这是我的名片。
- Can I charge it on my visa card? 我可以有VISA卡付款。
- Please fill in the arrival card. 请填写这张入境卡。
- A card, die, or domino with three pips. 三点的牌、骰子或骨牌
- He pencilled the number on a card and handed it to me. 他用铅笔把号码写在一张卡片上交给了我。
- 今日热词
| 354,497
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[index] [32863] [27519] [16340] [13747] [23052] [25177] [21671] [2227] [6280] [25926]
| 32,639
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TITLE: Initial and Final Objects in a Category
QUESTION [7 upvotes]: I understand the definition of initial and final objects in a category: that an object $\frak{I}$ is initial in a category $\frak{C}$ if for every other object in $\frak{C}$ (we'll just call it $\frak{O}$), there is a unique morphism $\frak{I} \rightarrow \frak{O}$. Similarly for final objects with the arrow reversed - there should be a unique morphism from that object to the final object.
My question is a little bit vague. In some sense, I want to understand how knowledge of the initial and final objects in a set help you unravel the content of the category in some way. For example, In $\frak{Set}$, the only initial object is $\emptyset$, and the final objects are all the singletons. On the other hand, in $\frak{Grp}$, the trivial group $\{e\}$ is both initial $and$ final.
The proof of these claims is pretty trivial. However, I expect that this should help me to understand exactly what makes these categories (and their objects) different. In this case, the difference arises from the requirement that the morphisms in $\frak{Grp}$ must preserve identities. How can I go about understanding these differences in other categories? What is the general principle at work when parsing language involving initial and final objects in general? That is to say, what mathematical content do these objects contain?
REPLY [5 votes]: Zhen Lin is correct that you almost always care about initial/final objects in related categories rather than (or rather, in addition to) the main category of interest. This isn't too surprising since a category can have at most one initial object up to unique isomorphism. These related categories are almost always comma categories.
The significance of initial/final objects is that they are one of three ways of formulating universal properties, a (or even the) core idea in category theory. Arguably, (1-)category theory is the study of universal properties.
I recently wrote a blog post providing an overview of these three ways of formulating universal properties and how each of these perspectives relate to different styles of doing category theory. Being familiar with all three perspectives and how they relate is, of course, important.
Briefly summarizing, as mentioned above, with initial objects you tend to make companion categories to study properties of some given category. Universal arrows can be formulated as initial objects in a particular comma category. Using universal arrows you usually use diagram chasing to prove theorems. This style doesn't involve defining other categories as much but more finding the universal arrows in your category. Finally, the third perspective is representability where you work directly with hom functors and can often utilize basic set theoretical tools.
| 192,047
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In my application, I am connecting multiple RS485 to Pi compute module through the multiplexer(just for selection of RS485) and finally connecting data lines to the MAX485 and then to pi on UART port (GPIO14, GPIO15) and RE/DE(GPIO13).
But there is no such library or driver support for MAX485-TO-PI communication. Am not able to get data on UART port.
Herewith I am attaching the block diagram of the connection: ... juMIrDbi-l
| 86,407
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DOUGLAS COUNTY, Colo. — Douglas County Commissioners now know how they’ll spend $10 million pledged toward school safety following the shooting at STEM School Highlands Ranch.
They’re going to let schools decide how to spend the money.
The Board of Douglas County Commissioners announced in May the re-appropriation of $10 million from their general fund toward public school safety and mental health services
RELATED: 'A guitar case with guns made it into the school': DougCo commissioners allocate $10M toward school safety
Later in the month, the board presented a proposal to the Douglas County School District that included adding about $3 million to bring on more School Resource Officers (SROs) but included a stipulation that the district and any participating private schools would have to match that $3 million.
The remaining $10 million was for one-time funding for “physical entryway security technology and mental health services for children.” The commissioner's plan in May was to create two committees that would develop a strategy for using these funds.
Instead, they decided the $10 million would be best spent giving schools what they think they need the most.
School leaders will be able to submit applications starting Wednesday and they can make a variety of safety and mental health-related requests such as an anti-bullying program, additions to building security, safety training and culture assessments.
Commissioners will be the ones making the final decision on which requests are approved.
Funding requests are due by Sept. 17.
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| 392,122
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| 185,980
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\begin{document}
\preprint{}
\title{Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities}
\author{Paul B. Slater}
\email{slater@kitp.ucsb.edu}
\affiliation{
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030\\
}
\date{\today}
\begin{abstract}
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio,
yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates
repeatedly close to nine decimal places to
$\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion
iterations, we do not presently perceive an exact value (rational or otherwise) for an estimate of 0.15709623 for the Bures two-rebit separability probability.
The Bures qubit-qutrit case--for which Khvedelidze and Rogojin gave an estimate of 0.0014--is analyzed too. The value of $\frac{1}{715}=\frac{1}{5 \cdot 11 \cdot 13} \approx 0.00139860$ is a well-fitting value to an estimate of 0.00139884. Interesting values ($\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11}$ and $\frac{625}{109531136}=\frac{5^4}{2^{12} \cdot 11^2 \cdot 13 \cdot 17}$) are conjectured for the Hilbert-Schmidt (HS) and Bures qubit-qudit ($2 \times 4$) positive-partial-transpose (PPT)-probabilities. We re-examine, strongly supporting, conjectures that the HS qubit-{\it qutrit} and rebit-{\it retrit} separability probabilities are
$\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}$, respectively. Prior studies have demonstrated that the HS two-rebit separability probability is $\frac{29}{64}$ and strongly pointed to the HS two-qubit counterpart being $\frac{8}{33}$, and a certain operator monotone one (other than the Bures) being $1 -\frac{256}{27 \pi^2}$.
\end{abstract}
\pacs{Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w}
\keywords{separability probabilities, two-qubits, two-rebits, Hilbert-Schmidt measure, random matrix theory, quaternions, PPT-probabilities, operator monotone functions, Bures measure, Lovas-Andai functions, quasirandom sequences, golden ratio, qubit-qutrit, rebit-retrit, inverse normal cumulative distribution}
\maketitle
\section{Introduction}
It has now been formally proven by Lovas and Andai \cite[Thm. 2]{lovas2017invariance} that the separability probability with respect to Hilbert-Schmidt (flat/Euclidean/Frobenius) measure \cite{zyczkowski2003hilbert} \cite[sec. 13.3]{bengtsson2017geometry} of the 9-dimensional convex set of
two-rebit states \cite{Caves2001} is $\frac{29}{64}=\frac{29}{2^6}$.
(``For quantum mechanics defined over real vector spaces the simplest composite systems are two-rebits systems" \cite{batle2003understanding}.) Additionally, the multifaceted evidence \cite{slater2017master,khvedelidze2018generation,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson}--including a recent ``master'' extension \cite{slater2017master,slater2018extensions} of the Lovas-Andai framework to {\it generalized} two-qubit states--is strongly compelling that the corresponding value for the 15-dimensional convex set of two-qubit states is $\frac{8}{33}=\frac{2^3}{3 \cdot 11}$ (with that of the 27-dimensional convex set of two-quater[nionic]bits being $\frac{26}{323}=\frac{2 \cdot 13}{17 \cdot 19}$ [cf. \cite{adler1995quaternionic}], among other still higher-dimensional companion random-matrix related results). A still further extension to the use of induced measures--reducing to the Hilbert-Schmidt case for the case $k=0$--has been found \cite[sec. XII]{slater2018extensions}--yielding, for example, $\frac{61}{143}$ for $k=1$.
(The parameter $k$ is the difference [$k= K-N$] between the dimensions [$K,N$,with $K\geq N$] of the subsystems of the pure state bipartite system in which the density matrix is regarded as being embedded \cite{zyczkowski2001induced}.)
Further, appealing hypotheses parallel to these rational-valued results have been advanced--based on extensive sampling--that the Hilbert-Schmidt separability probabilities for the 35-dimensional qubit-{\it qutrit} and 20-dimensional rebit-{\it retrit} states are
$\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}$, respectively \cite[eqs. (15),(20)]{slater2018extensions} \cite[eq. (33)]{milz2014volumes}. (These will be further examined in sec.~\ref{HSsection} below.)
Certainly, one can, however, still aspire to a yet greater ``intuitive'' understanding of these assertions, particularly in some ``geometric/visual'' sense
[cf. \cite{szarek2006structure,samuel2018lorentzian,avron2009entanglement,braga2010geometrical,gamel2016entangled,jevtic2014quantum}], as well as further formalized proofs. It would be of interest, as well, to compare/contrast these finite-dimensional studies with those other quantum-information-theoretic ones, presented in the recent comprehensive volume of Aubrun and Szarek
\cite{aubrun2017alice}, in which the quite different concepts of {\it asymptotic geometric analysis} are employed.
By a separability probability, in the above discussion, we mean the ratio of the volume of the separable states to the volume of all (separable and entangled) states, as proposed, apparently first, by {\.Z}yczkowski, Horodecki, Sanpera and Lewenstein \cite{zyczkowski1998volume} (cf. \cite{petz1996geometries,e20020146,singh2014relative,batle2014geometric}). The present author was, then, led--pursuing an interest in ``Bayesian quantum mechanics" \cite{slater1994bayesian,slater1995quantum} and the concept of a ``quantum Jeffreys prior" \cite{kwek1999quantum}--to investigate how such separability probabilities might depend upon the choice of various possible measures on the quantum states \cite{petz1996geometries}.
\subsection{Partitioning of separability/PPT-probabilities}
\subsubsection{Hilbert-Schmidt and Bures cases}
A phenomenon apparently restricted to the Hilbert-Schmidt ($k=0$) case of induced measure, is that the positive-partial-transpose (PPT) states are equally divided probability-wise between those for which the determinant $|\rho^{PT}|$ of the partial transpose of the density matrix ($\rho$) exceeds the determinant $|\rho|$ of the density matrix itself and {\it vice versa}. (Also, along somewhat similar lines, the Hilbert-Schmidt PPT-probability for minimally degenerate [having a single zero eigenvalue] states is {\it half} that for nondegenerate states \cite{szarek2006structure}. The PPT-property is, of course, equivalent--by the Peres-Horodecki criterion--to separability for $4 \times 4$ and $6 \times 6$ density matrices \cite[sec. 16.6.C]{bengtsson2017geometry}.) Quite contrastingly, based on some 122,000,000 two-qubit density matrices randomly generated with respect
to Bures measure, of the 8,945,951 separable ones found, 5,894,648 of them (that is, 65.89$\%$) had $|\rho^{PT}| >|\rho|$, clearly distinct from simply 50$\%$ (cf. \cite[Tabs. 1, 2]{khvedelidze2018generation} \cite{KhvRog}).
\subsubsection{Induced measures, in general}
A formula
for that part, $Q(k,\alpha)$, of the {\it total} separability probability, $P_{sep}(k,\alpha)$,
for generalized (real [$\alpha=1$], complex [$\alpha=2$], quaternionic [$\alpha=4$],\ldots) two-qubit states endowed with random induced measure, for which the determinantal inequality $|\rho^{PT}| >|\rho|$ holds was given in \cite[p. 26]{slater2016formulas}. It took the form $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,\ldots 9$.
(The factors $G^k_2(\alpha)$ are sums of polynomial-weighted generalized hypergeometric functions $_pF_{p-1}, p \geq 7$, all with argument $z = \frac{27}{64}$.)
Here $\rho$ denotes a $4 \times 4$ density matrix, obtained by tracing over the pure states in $4 \times (4 +k)$-dimensions, and $\rho^{PT}$, its partial transpose. Further, $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ for the standard (15-dimensional) convex set of (complex) two-qubit states.
Further, in the indicated ($k=0$) Hilbert-Schmidt case, we can apparently employ
the formula \cite[p. 26]{slater2016formulas}
\begin{equation} \label{InducedMeasureCase}
\mathcal{P}_{sep/PPT}(0,\alpha)= 2 Q(0,\alpha)= 1-
\frac{\sqrt{\pi } 2^{-\frac{9 \alpha}{2}-\frac{5}{2}} \Gamma \left(\frac{3 (\alpha+1)}{2}\right)
\Gamma \left(\frac{5 \alpha}{4}+\frac{19}{8}\right) \Gamma (2 \alpha+2) \Gamma \left(\frac{5
\alpha}{2}+2\right)}{\Gamma (\alpha)} \times
\end{equation}
\begin{displaymath}
\, _6\tilde{F}_5\left(1,\alpha+\frac{3}{2},\frac{5 \alpha}{4}+1,\frac{1}{4} (5 \alpha+6),\frac{5
\alpha}{4}+\frac{19}{8},\frac{3 (\alpha+1)}{2};\frac{\alpha+4}{2},\frac{5
\alpha}{4}+\frac{11}{8},\frac{1}{4} (5 \alpha+7),\frac{1}{4} (5 \alpha+9),2 (\alpha+1);1\right).
\end{displaymath}
That is, for $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with
(the real case) $Q(0,1) = \frac{29}{128}$, (the standard complex case) $Q(0,2)=\frac{4}{33}$, and
(the quaternionic case) $Q(0,4)= \frac{13}{323}$---the three simply
equalling--by the equipartitioning result noted above--$ P_{sep}(0,\alpha)/2$.
More generally, $Q(k,\alpha)$ gives that portion, for induced measure, parameterized by $k$, of the total separability/PPT-probability for which the determinantal inequality
$|\rho^{PT}| >|\rho|$ holds \cite[eq. (84)]{slater2017master}.
\section{Use of Bures Measure}
Of particular initial interest was the the Bures/statistical distinguishability (minimal monotone) measure \cite{slater2000exact,sarkar2019bures,vsafranek2017discontinuities,forrester2016relating, braunstein1994statistical}. (``The Bures metric
plays a distinguished role since it is the only metric which is also monotone,
Fisher-adjusted, Fubini-Study-adjusted, and Riemannian" \cite{forrester2016relating}. Bej and Deb have recently ``shown that if a qubit gets entangled with another ancillary qubit then negativity, up to a constant factor, is equal to square root of a specific Riemannian metric defined on the metric space corresponding to the state space of the qubit" \cite{bej2018geometry}.)
In \cite[sec. VII.C]{slater2017master}, we recently reported, building upon analyses of Lovas and Andai \cite[sec. 4]{lovas2017invariance}, a two-qubit separability probability equal to $1 -\frac{256}{27 \pi^2} =1- \frac{2^8}{3^3 \pi^2} \approx 0.0393251$. This was based on another
(of the infinite family of) operator monotone functions, namely
$\sqrt{x}$. (The Bures measure itself is associated with the operator monotone function $\frac{1+x}{2}$.) (Let us note that the complementary ``entanglement probability'' is simply $\frac{256}{27 \pi^2} \approx 0.960675$. There appears to be no intrinsic reason
to prefer/privilege one of these two forms (separability, entanglement) of probability to the other [cf. \cite{dunkl2015separability}]. We observe that the variable denoted $K_s = \frac{(s+1)^{s+1}}{s^s}$, equalling $\frac{256}{27} = \frac{4^4}{3^3}$, for $s=3$, is frequently employed as an upper limit of integration in the Penson-{\.Z}yczkowski paper, ``Product of Ginibre matrices: Fuss-Catalan and Raney
distributions'' \cite[eqs. (2), (3)]{penson2011product}.)
Interestingly, Lovas and Andai ``argue that from the separability probability point of view, the main difference between the Hilbert-Schmidt measure and the volume form
generated by the operator monotone function $x \rightarrow \sqrt{x}$ is a special distribution on the unit ball in operator norm of
$2 \times 2$ matrices, more precisely in the Hilbert-Schmidt case one faces a uniform distribution on the whole unit ball and for
monotone volume forms one obtains uniform distribution on the surface of the unit ball'' \cite[p. 2]{lovas2017invariance}.
\subsection{Osipov-Sommers-{\.Z}yzckowski Interpolation Formula}
Of central importance in our analyses below will be the construction of Osipov, Sommers and {\.Z}yzckowski of an interpolation between the generation of random
density matrices with respect to Hilbert-Schmidt and those with respect to Bures measures \cite[eq. (24)]{al2010random} (cf. \cite[eq. (33)]{borot2012purity}). This formula takes the form
\begin{equation} \label{JointBuresHSformula}
\rho_x= \frac{(y \mathbb{I} +x U) AA^{\dagger}(y \mathbb{I}+x U^{\dagger})}{\mbox{Tr} (y \mathbb{I}+x U) A A^{\dagger} (y \mathbb{I} +x U^{\dagger})},
\end{equation}
where $y=1-x$, with $x=0$ yielding a Hilbert-Schmidt density matrix $\rho_0$, and $x=\frac{1}{2}$, the Bures counterpart $\rho_{1/2}$.
Here, $A$ is an $N \times N$ complex-valued random matrix pertaining to the Ginibre ensemble (with real and imaginary parts of each of the $N^2$ entries being independent standard normal random variates). Further, $U$ is a random
unitary matrix distributed according to the Haar measure on $U(N)$. (Of course, $N=4$ in the basic two-qubit case of first interest here.)
It is an intriguing hypothesis that the Bures two-qubit separability probability also assumes a strikingly elegant form (such as the indicated $\frac{8}{33}$, $1-\frac{256}{27 \pi^2}$). (``Observe that the Bures volume of the set of mixed states is equal to the volume of an $(N^2-1)$-dimensional hemisphere of radius $R_B=\frac{1}{2}$'' \cite[p. 415]{bengtsson2017geometry}. It is also noted there that $R_B$ times the area-volume ratio asymptotically increases with the dimensionality $D=N^2-1$, which is typical for hemispheres.)
\subsection{Prior Estimations of Bures separability probabilities}
In the relatively early (2002) work
\cite{slater2002priori}, we had conjectured a Bures two-qubit separability probability equal to $\frac{8}{11 \pi^2} \approx .0736881$. But it was later proposed
in 2005 \cite{slater2005silver}, in part motivated by the lower-dimensional {\it exact} Bures results reported in \cite{slater2000exact}, that the value might be $\frac{1680 \sigma_{Ag}}{\pi^8} \approx 0.07334$, where $\sigma_{Ag}= \sqrt{2}-1 \approx 0.414214$ is the ``silver mean''. Both of these studies \cite{slater2002priori,slater2005silver} were conducted using quasi-Monte Carlo procedures, before the developmentof the indicated Osipov-Sommers-{\.Z}yczkowski methodology (\ref{JointBuresHSformula}) for generating density matrices, random with respect to Bures measure \cite{al2010random}. More recently, in \cite[sec. X.B.1]{slater2018extensions}, we reported, using this Ginibre ensemble-based formula (\ref{JointBuresHSformula}) an estimate of 0.0733181043 based on 4,372,000,000 realizations, using simply standard
(independent) random normal variate generation. (Khvedelidze and Rogojin
gave a value of 0.0733 \cite[Table 1]{khvedelidze2018generation} \cite{KhvRog}.)
Performing a parallel (but much smaller) computation in the two-rebit case, based on forty million random density matrices (6,286,209 of them being separable), we obtained a corresponding (slightly corrected) Bures separability probability estimate of 0.1571469. (In doing so, we took, as required, the now real-entried Ginibre matrix A to be $4 \times 5$ \cite[eqs. (24), (28)]{al2010random}, and not $4 \times 4$ as in the two-qubit calculation.)
\subsection{Application of Quasirandom Methodology to Bures Two-Rebit and Two-Qubit Cases}
We now importantly examine the question of whether Bures two-qubit and two-rebit separability probability estimation can be accelerated--with superior convergence properties--by, rather than using, as typically done,
{\it independently}-generated normal variates for the Ginibre ensembles at each iteration, making use of normal variates {\it jointly}-generated by employing low-discrepancy (quasi-Monte Carlo) sequences \cite{leobacher2014introduction}. In particular, we have employed an ``open-ended'' sequence (based on extensions of the golden ratio \cite{livio2008golden}) recently introduced by Martin Roberts in the detailed presentation ``The Unreasonable Effectiveness
of Quasirandom Sequences'' \cite{Roberts}.
Roberts notes: ``The solution to the
$d$-dimensional problem, depends on a special constant $\phi_d$, where $\phi_d$ is the value of the smallest, positive real-value of x such that''
\begin{equation}
x^{d+1}=x+1,
\end{equation}
($d=1$, yielding the golden ratio, and $d=2$, the ``plastic constant'' \cite{Roberts32D}).
The $n$-th terms in the quasirandom (Korobov) sequence take the form
\begin{equation} \label{QR}
(\alpha _0+n \vec{\alpha}) \bmod 1, n = 1, 2, 3, \ldots
\end{equation}
where we have the $d$-dimensional vector,
\begin{equation} \label{quasirandompoints}
\vec{\alpha} =(\frac{1}{\phi_d},\frac{1}{\phi_d^2},\frac{1}{\phi_d^3},\ldots,\frac{1}{\phi_d^d}). "
\end{equation}
The additive constant $\alpha_0$ is typically taken to be 0. ``However, there are some arguments, relating to symmetry, that suggest that $\alpha_0=\frac{1}{2}$
is a better choice,'' Roberts observes.
These points (\ref{QR}), {\it uniformly} distributed in the $d$-dimensional hypercube $[0,1]^d$, can be converted to (quasirandomly distributed) normal variates, required for implementation of the Osipov-Sommers-{\.Z}yczkowski formula (\ref{JointBuresHSformula}), using the inverse of the cumulative distribution function \cite[Chap. 2]{devroye1986}.
Impressively, in this regard, Henrik Schumacher developed for us a specialized algorithm that accelerated the default Mathematica command InverseCDF for the normal distribution approximately {\it ten-fold}, as reported in the highly-discussed post \cite{Schumacher}--allowing us to vastly increase the realization rate.
We take $d=36$ and 64 in the Roberts methodology, using the Osipov-Sommers-{\.Z}yczkowski (real and complex) interpolation formulas to estimate the Bures two-rebit and two-qubit separability probabilities, respectively. In the two-qubit case, 32 of the 64 variates are used in generating the Ginibre matrix $A$, and the other 32, for the unitary matrix $U$.
(A subsidiary question--which appeared in the discussion with Roberts \cite{Roberts32D}--is the relative effectiveness of employing--to avoid possible ``correlation'' effects--the {\it same} 32-dimensional sequence but at different $n$'s for $A$ and $U$, rather than a single 64-dimensional one, as pursued here. A small analysis of ours in this regard did not indicate this to be a meritorious approach.) In the two-rebit case, 20 variates are used to generate
the $4 \times 5$ matrix A, and the other 16 for an orthogonal $4 \times 4$ matrix $O$.
In Figs.~\ref{fig:twoqubitplot} and \ref{fig:tworebitplot} we show the development of the Bures separability probability estimation procedure in the two cases at hand. (Much earlier versions of these [$\alpha_0=\frac{1}{2}$] plots--together with [less intensive] estimates using $\alpha_0=0$--were displayed as Figs. 5 and 6 in \cite{slater2018extensions}.)
\begin{figure}
\includegraphics[]{BuresQubit2.pdf}
\caption{Two-qubit Bures separability probability estimates--divided by $\frac{25}{341}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. This ratio is equal to 1 to nearly eight decimal places at: 1,445,000,000; 10,850,000,000; 11,500,000,000; and 16,075,000,000 iterations. Estimates are recorded at intervals of five million iterations.}
\label{fig:twoqubitplot}
\end{figure}
\begin{figure}
\includegraphics[]{BuresRebit2.pdf}
\caption{Two-rebit Bures separability probability estimates as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\label{fig:tworebitplot}
\end{figure}
\subsubsection{Two-qubit Bures analysis}
Using the indicated, possibly superior parameter value
$\alpha_0= \frac{1}{2}$ in (\ref{QR}), this quasirandom/normal-variate-generation procedure
yielded a two-qubit estimate, based on 16,895,000,000 iterations, of 0.073313759. This is closely fitted by the two (themselves very near) values
$\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.07331378299$ and (as suggested by the WolframAlpha.com site) $\frac{\sqrt{51}}{\pi ^4} \approx 0.07331377752$. (Informally, Charles Dunkl wrote: "I would hate to think that the answer is $\frac{\sqrt{51}}{\pi^4}$- that is just ugly. One hopes for a rational number.") At 10,850,000,000 iterations, interestingly, the estimate of 0.0733137814 matched $\frac{25}{341}$ to nearly eight decimal places. The estimate of 0.0733137847 obtained at the considerably smaller number of iterations of 1,445,000,000, was essentially as close too.
The Hilbert-Schmidt measure counterpart is (still subject to formal proof) essentially known to be $\frac{8}{33} = \frac{2^3}{3 \cdot 11}$ \cite{slater2017master,khvedelidze2018generation,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson}.
\subsubsection{Two-rebit Bures analysis} \label{BuresTwoRebit}
In the two-rebit instance, we obtained a Bures estimate, based on 23,460,000,000
iterations, of 0.157096234. This is presumably, at least as accurate as the considerably, just noted, smaller sample based two-qubit estimate--apparently corresponding to $\frac{25}{341}$. Nevertheless, we do not presently perceive any possible exact--rational or otherwise--fits to this estimate.
While, the Hilbert-Schmidt two-rebit separability probability
has been proven by Lovas and Andai to be $\frac{29}{64} = \frac{29}{2^6}$ \cite[Thm. 2]{lovas2017invariance}, somewhat similarly to this Bures result, the two-rebit separability probability, 0.2622301318, based on the other monotone ($\sqrt{x}$) measure, did not seem to have an obvious exact underlying formula.
\section{Examination of Hilbert-Schmidt Qubit-Qutrit and Rebit-Retrit Separability Conjectures} \label{HSsection}
\subsection{Prior studies}
Based on extensive (standard) random sampling of independent normal variates, in \cite[eqs. (15),(20)]{slater2018extensions}, we have conjectured that the Hilbert-Schmidt separability probabilities for the 35-dimensional qubit-{\it qutrit} and 20-dimensional rebit-{\it retrit} states are (also interestingly rational-valued)
$\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3} =0.027$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775796$, respectively .
In particular, on the basis of 2,900,000,000 randomly-generated
qubit-qutrit density matrices, an estimate (with 78,293,301
separable density matrices found) was obtained, yielding an associated separability probability of 0.026997690. (Milz and Strunz had given a confidence interval of $0.02700 \pm 0.00016$ for this probability \cite[eq. (33)]{milz2014volumes}, while Khvedelidze and Rogojin reported an estimate of 0.0270 \cite[Tab. 1]{khvedelidze2018generation}--but also only 0.0014 for the Bures counterpart [sec.~\ref{BuresQubitQutrit}].)
Further, on the basis of 3,530,000,000 randomly-generated
rebit-retrit density matrices, with respect to Hilbert-Schmidt measure, an estimate (with 462,704,503
separable density matrices found) was obtained for an associated separability probability of 0.1310777629. The associated
$95\%$ confidence interval is $[0.131067, 0.131089]$.
\subsection{New studies}
Applying the quasirandom methodology here to further appraise this pair of conjectures, we obtain Figs.~\ref{fig:QuasiRandomQubitQutrit} and \ref{fig:QuasiRandomRebitRetrit}. (We take the dimensions $d$ of the sequences of normal variates generated to be 72 and 42, respectively.)
\begin{figure}
\centering
\includegraphics{QuasiRandomQubitQutrit.pdf}
\caption{Qubit-qutrit Hilbert-Schmidt separability probability estimates--divided by $\frac{27}{1000}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\label{fig:QuasiRandomQubitQutrit}
\end{figure}
\begin{figure}
\centering
\includegraphics{QuasiRandomRebitRetrit.pdf}
\caption{Rebit-retrit Hilbert-Schmidt separability probability estimates--divided by $\frac{860}{6561} = \frac{2^2 \cdot 5 \cdot 43}{3^8}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\label{fig:QuasiRandomRebitRetrit}
\end{figure}
Interestingly, as in Fig.~\ref{fig:twoqubitplot}, we observe some
drift away--with increasing iterations--from early particularly close fits
to the two conjectures. But, as in Fig.~\ref{fig:twoqubitplot}--assuming the validity
of the conjectures--we might anticipate the estimates re-approaching more closely
the conjectured values. It would seem that any presumed eventual convergence is not simply a
straightforward monotonic process--perhaps somewhat comprehensible in view of the very high dimensionalities (72, 42) of the sequences involved. (The last recorded separability probabilities--in these ongoing analyses--were 0.0269923 and 0.1310848, based on 1,850,000,000 and
2,415,000,000 iterations, respectively.)
In \cite[App. B]{slater2017master}, we reported an effort to extend the innovative framework of Lovas and Andai \cite{lovas2017invariance} to such qubit-qutrit and rebit-retrit settings. (One aspect of interest pertaining to the original $4 \times 4$ density matrix study of Lovas and Andai \cite{lovas2017invariance}, was that it (surprisingly) appeared possible in \cite{slater2017master} to extend the original Lovas-Andai framework by restricting our analyses to $4 \times 4$ density matrices in which the two $2 \times 2$ diagonal blocks were themselves diagonal.)
\section{An 8-dimensional ($X$-states) rebit-retrit scenario} \label{8D}
Along similar lines, let us consider an 8-dimensional ($X$-states) rebit-retrit scenario, in which now the only non-zero entries of $\rho$ are those on the diagonal and anti-diagonal--so that the two $3 \times 3$ diagonal blocks are themselves diagonal. Also, let us employ the "separability function" framework developed in
\cite[eq. (5)]{slater2008extended}, where the variable
$\eta= \frac{\rho_{11} \rho_{66}}{\rho_{33} \rho_{44}}$
was employed.
Then, with the use of the Mathematica command GenericCylindricalDecomposition--employed to enforce the positivity of leading minors of the density matrix and its partial transpose--we are able to formally establish that the associated rebit-retrit Hilbert-Schmidt separability probability is $\frac{16}{3 \pi^2} \approx 0.54038$ \cite{dunkl2015separability}.
(This value also holds for the two-rebit and two-retrit $X$-states, while $\frac{2}{5}$ is the two-qubit $X$-states probability
\cite[sec. VIII]{slater2018extensions}.) The value $\frac{16}{3 \pi^2}$ is obtained--through integration using the output of this Mathematica command--by taking the ratio of
\begin{equation} \label{8dNumerator}
\int_{\eta=0}^1 \frac{\pi \eta \left(-3 \eta ^2+(\eta +4) \eta \log (\eta )+\log (\eta
)+3\right)}{40320 (\eta -1)^5} d \eta =\frac{\pi}{967680}= \frac{\pi}{2^{10} \cdot 3^3 \cdot 5 \cdot 7}
\end{equation}
to
\begin{equation} \label{8dDenominator}
\int_{\eta=0}^1 \frac{\pi \sqrt{\eta } \left(-3 \eta ^2+(\eta +4) \eta \log (\eta )+\log (\eta
)+3\right)}{40320 (\eta -1)^5} d \eta =\frac{\pi^3}{5160960} =\frac{\pi^3}{2^{14} \cdot 3^2 \cdot 5 \cdot 7},
\end{equation}
where $\sqrt{\eta}$ plays the role of separability function, and is the added factor--that is, $\eta= (\sqrt{\eta})^2$--in the first of the two integrands immediately above.
\section{Application in higher dimensions of master Lovas-Andai generalized two-qubit formulas}
We investigated extending this 8-dimensional rebit-retrit analysis to a 10-dimensional one, by replacing two previously zero entries, so that the two off-diagonal $3 \times 3$ blocks now themselves form X-patterns. The counterpart of the denominator function (\ref{8dDenominator}) is, then,
\begin{equation}
\int_{\eta=0}^1 \frac{\pi \eta (3 (\eta +1) (\eta (\eta +8)+1) \log (\eta )-(\eta -1) (\eta (11 \eta
+38)+11))}{1209600 (\eta -1)^7}d \eta = \frac{\pi}{29030400} =
\end{equation}
\begin{displaymath}
\frac{\pi}{2^{11} \cdot 3^4 \cdot 5^2 \cdot 7}.
\end{displaymath}
We, then, need to find the appropriate separability function--corresponding to $\sqrt{\eta}$ in (\ref{8dNumerator})--to insert into this integrand--for the numerator--to complete the calculation of the separability probability ratio. In this regard, we were able to, preliminarily, utilize a sub-optimal separability function (based on the enforcement of the positivity of the determinant of the $5 \times 5$ leading submatrix of the partial transpose--but not yet the full determinant),
\begin{equation}
\frac{2 \left(\sqrt{(1-\eta) \eta}+\sin ^{-1}\left(\sqrt{\eta}\right)\right)}{\pi },
\end{equation}
which yields an upper bound on the separability probability of
$\frac{919}{5} -264 \log (2) \approx 0.809144$.
Then--using the full determinant--we were able to construct the actual separability
function \cite{Heidecki,Student},
\begin{equation} \label{sepfunct}
\frac{2 \left(\varepsilon ^2 \left(4 \text{Li}_2(\varepsilon )-\text{Li}_2\left(\varepsilon
^2\right)\right)+\varepsilon ^4 \left(-\tanh ^{-1}(\varepsilon )\right)+\varepsilon ^3-\varepsilon
+\tanh ^{-1}(\varepsilon )\right)}{\pi ^2 \varepsilon ^2},
\end{equation}
where the dilogarithm is indicated, and $\epsilon^2=\eta$.
The corresponding separability probability was, then, shown to be \cite{Student}
\begin{equation}
\frac{272}{45 \pi^2} \approx 0.612430.
\end{equation}
(We have also found very strongly convincing numerical evidence that the same separability probability holds, if instead of considering ten-dimensional rebit-retrit scenarios in which the two off-diagonal $3 \times 3$ blocks have $X$-patterns, one considers that the two diagonal $3 \times 3$ blocks do.)
Further, it appears remarkable that the 10-dimensional rebit-retrit separability function (\ref{sepfunct}) turned out to be completely identical with the (polylogarithmic) Lovas-Andai two-rebit function $\tilde{\chi}_1(\varepsilon)$ \cite[eq.(2)]{slater2017master} \cite[eq. (9)]{lovas2017invariance}.
Then, in light of this finding, it appears reasonable to entertain an hypothesis that the Lovas-Andai two-qubit function $\tilde{\chi}_2(\varepsilon)=\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$ and two-quaterbit function $\tilde{\chi}_{4}(\varepsilon)=\frac{1}{35} \varepsilon ^4 \left(15 \varepsilon ^4-64 \varepsilon ^2+84\right)$ play parallel roles when the associated sets of density matrices share the same zero-nonzero pattern as the two ten-dimensional sets of rebit-retrit density matrices just considered (those with either the two off-diagonal or the two diagonal $3 \times 3$ blocks having $X$-patterns).
Pursuing such an hypothesis, and employing polar and ``hyper-polar'' coordinates in the very same manner as was done in \cite{slater2017master}, we can readily perform computations, in these higher-dimensional settings, leading to a presumptive qubit-qutrit separability probability of $\frac{5}{3} \left(112 \pi ^2-1105\right) = \frac{560 \pi ^2}{3}-\frac{5525}{3} \approx 0.659488$ and a quaterbit-quatertrit separability probablility of $\frac{8962661573}{4725}-192192 \pi ^2 \approx 0.583115$. (Let us interestingly note that $1105 = 5 \cdot 13 \cdot 17$, $112=2^4 \cdot 7$,$192192-2^6 \cdot 3 \cdot 7 \cdot 11 \cdot 13$ and $4725 = 3^3 \cdot 5^2 \cdot 7$. Also, we will note that $8962661573 = 193 \cdot 46438661$.)
We have tried directly computing/approximating--an apparently rather challenging task--this hypothesized qubit-qutrit separability probability value of $\frac{5}{3} \left(112 \pi ^2-1105\right) \approx 0.659488$--that is, without simply assuming the applicability of $\tilde{\chi}_2(\varepsilon)$, but have only obtained a value of 0.67696 \cite{qubitqutrit}.
\section{Enlarged Two-Retrit $X$-states}
It has been established--as previously noted (sec.~\ref{8D})--that the Hilbert-Schmidt separability-PPT probabilities are all equal to $\frac{16}{3 \pi^2}$ for the two-rebit, rebit-retrit and two-retrit $X$-states.
Then, continuing along the lines we have just been investigating, we considered a scenario in which the two-retrit $X$-states gained a non-zero (1,2)-entry.
Then, we, in fact, were able to determine that the Hilbert-Schmidt PPT-probability for this scenario was $\frac{65}{36 \pi}$, making use of a separability function $\frac{8 \left(\sqrt{1-u^2} u^2-\sqrt{1-u^2}+1\right)}{3 \pi u}$, where $u=\sqrt{\frac{\rho_{33} \rho_{77}}{\rho_{11} \rho_{99}}}$. (We found identical results when the entry chosen to be non-zero was the (1,4)--and not the (1,2)--one.)
\section{Bures Qubit-Qutrit Analysis} \label{BuresQubitQutrit}
In Table 1 of their recent study, "On the generation of random ensembles of qubits and qutrits: Computing separability probabilities for fixed rank states'' \cite{khvedelidze2018generation}, Khvedelidze and Rogojin report an estimate (no sample size being given) of 0.0014 for the separability probability of the 35-dimensional convex set of qubit-qutrit states. We undertook a study of this issue, once again employing the quasirandom methodology advanced by Roberts (with the sequence dimension parameter $d$ now equal to $144=2 \cdot 72$), in implementing the Osipov-Sommers-{\.Z}yczkowski formula (\ref{JointBuresHSformula}) given above with $x=\frac{1}{2}$. (For the companion Bures rebit-retrit estimation, we would have a smaller $d$, that is, 78--but given our Bures two-rebit analysis above (sec.~\ref{BuresTwoRebit}), we were not optimistic in being able to advance a possible exact value.) In Fig.~\ref{fig:BuresQubitQutrit} we show a (scaled) plot of our corresponding computations. The estimates--recorded at intervals of one million--are in general agreement with the reported value of Khvedelidze and Rogojin. The last value (after 3,174 million iterations) was $\frac{1479997}{1058000000}= \approx 0.001398863$. This can be well-fitted by $\frac{1}{715} =\frac{1}{5 \cdot 11 \cdot 13} \approx 0.00139860$.
\begin{figure}
\centering
\includegraphics{BuresQubitQutrit.pdf}
\caption{Qubit-qutrit Bures separability probability estimates--divided by $\frac{1}{715} = \frac{1}{5 \cdot 11 \cdot 13}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of one million iterations.}
\label{fig:BuresQubitQutrit}
\end{figure}
\subsection{Higher-Dimensional Bures Analyses}
To estimate the Bures qubit-qudit ("ququart") bipartite ($2 \times 4$) PPT-probability, we employed a 256-dimensional quasirandom sequence, obtaining 4,760 PPT density matrices in 830 million realizations, yielding an estimated probability of $5.7349398 \cdot 10^{-6}$. An interesting candidate for a possible corresponding exact value is
$\frac{625}{109531136}=\frac{25^2}{2^{12} \cdot 11^2 \cdot 13 \cdot 17} \approx 5.70614003 \cdot 10^{-6}$ (Fig.~\ref{fig:BuresQubitQudit}).
\begin{figure}
\centering
\includegraphics{BuresQubitQudit.pdf}
\caption{Qubit-qudit ($2 \times 4$) Bures PPT-probability estimates--divided by $\frac{625}{109531136}=\frac{25^2}{2^{12} \cdot 11^2 \cdot 13 \cdot 17}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of one million iterations.}
\label{fig:BuresQubitQudit}
\end{figure}
For the Bures two-qutrit scenario, employing a 324-dimensional sequence, only 43 PPT density matrices were generated in 678 million realizations, yielding an estimate of $6.3421829 \cdot 10^{-8}$ (Fig.~\ref{fig:BuresQubitQudit2x5}). (It would be of interest to relate this last very small PPT-probability estimation to the asymptotic analyses of Aubrun and Szarek \cite{aubrun2017alice}, as well as Ye \cite{ye2009bures}.)
\begin{figure}
\centering
\includegraphics{BuresQubitQudit2x5.pdf}
\caption{Two-qutrit Bures PPT-probability estimates as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of one million iterations.}
\label{fig:BuresQubitQudit2x5}
\end{figure}
\section{Higher-Dimensional Hilbert-Schmidt Analyses}
Further, in
\cite[sec. 3,5]{slater2018extensions}, we had suggested Hilbert-Schmidt PPT-probability hypotheses for the $2 \times 4$ and $2 \times 5$ qubit-qudit systems of $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$ and $\frac{125}{4790016} = \frac{5^3}{2^8 \cdot 3^5 \cdot 5 \cdot 7 \cdot 11} \approx 0.0000260959$, and $\frac{201}{8192} = \frac{3 \cdot 67}{2^{13}} \approx 0.0245361$
and $\frac{29058}{9765625}= \frac{2 \cdot 3 \cdot 29 \cdot 167}{5^{10}} \approx 0.00297554$ for their respective rebit-redit analogues.
For the Hilbert-Schmidt $2 \times 4$ qubit-qudit and two-qutrit scenarios, using the quasirandom procedure introduced by Martin Roberts \cite{Roberts}, we have obtained PPT-probability estimates of 0.0012928963 and 0.00010275452 based on 2,104 and 1,768 million iterations, respectively (Figs.~\ref{fig:HSQubitQudit} and \ref{fig:HSTwoQutrit}).
In further regard to Hilbert-Schmidt two-{\it qutrit} probabilities, an estimate of 0.00010218 based on 100 million random realizations was reported in sec. III.A of ``Invariance of Bipartite Separability and
PPT-Probabilities over Casimir Invariants of Reduced States'' \cite{slater2016invariance}. (An intriguing possible corresponding exact value is $\frac{323}{3161088}=\frac{17 \cdot 19}{2^{10} \cdot 3^2 \cdot 7^3} \approx 0.000102180009$--or $\frac{11}{107653}=\frac{11}{7^2 \cdot 13^3} \approx 0.000102180153$.)
\subsection{Use of realignment criterion for (bound-)entanglment estimations}
Also, in an auxiliary $2 \times 4$ qubit-qudit analysis, based on 795 million iterations,
use of the realignment criterion \cite{chen2002matrix} yielded an
estimate of 0.000234478 for the bound-entangled probability and 0.942343 (conjecturally, $\frac{589}{625}=\frac{17 \cdot 31}{5^4} \approx 0.9424$) for the entanglement probability, in general. (The PPT-probability was, once again, well fitted--to almost five decimal places--by $\frac{16}{12375}$.) In that analysis, we were not able to detect any finite probability at all of genuinely tripartite entanglement using the Greenberger-Horne-Zeilinger test set out in Example 3 in \cite{bae2018entanglement}.
\begin{figure}
\centering
\includegraphics{HSQubitQudit.pdf}
\caption{Qubit-qudit ($2 \times 4$) Hilbert-Schmidt PPT-probability estimates--divided by the conjectured value $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of one million iterations.}
\label{fig:HSQubitQudit}
\end{figure}
\begin{figure}
\centering
\includegraphics{HSTwoQutrit.pdf}
\caption{Two-qutrit Hilbert-Schmidt PPT-probability estimates as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of one million iterations.}
\label{fig:HSTwoQutrit}
\end{figure}
However, in a parallel two-qutrit study, the realignment test for entanglement was not passed by any randomly generated states (cf. \cite[sec. IV]{gabdulin2019investigating}).
\subsection{The question of optimality of 64D low-discrepancy sequences}
It may be of interest to the reader to here include a response of Martin Roberts to a query as to whether
to calculate a 64D integral, it is optimal or not to use a 64D low-discrepancy sequence, as employed above in the two-qubit case.
Roberts interestingly replied: ``It depends. In theory, the convergence rate of simple random sampling is O(1/n), whereas for low discrepancy sequences it is O($\frac{\log(N)^d}{N}$). The $\log(N)^d$ term implies that in theory for some large D, and very large N, the convergence rate of quasirandom sequences is inferior to simple random sampling. However, the classic Big O notation ignores two things, which in practice are crucial.
(i) Big O notation is for $N \rightarrow \infty$. For finite N, the constants of proportionality play a big role in determining which one is more efficient.
(ii) it has been found that for many high dimensional integrals (especially the finance, computer vision, and natural language processing) although they may outwardly look like high dimensional functions they are in fact really relatively low dimensional problems embedded in a high dimensional manifold. Therefore the pragmatic D in the above expression, is really the 'intrinsic' D. This is why finance options-pricing which is based on integrations over a few hundred dimensions are still more efficient with quasirandom sampling."
\section{Concluding Remarks}
We should stress that the problem of {\it formally deriving} the Bures two-rebit and two-qubit separability probabilities, and, thus, testing the candidate value ($\frac{25}{341}$) advanced here (Fig.~\ref{fig:twoqubitplot}), certainly currently seems intractable--even, it would seem, in the pioneering framework of Lovas and Andai \cite{lovas2017invariance}. (Perhaps some formal advances can be made, in such regards, with respect to $X$-states [cf. \cite{xiong2017geometric}].)
Let us note that the ``master Lovas-Andai" formula for {\it generalized} two-qubit Hilbert-Schmidt ($k=0$) separability probabilities reported in \cite[sec. VIII.A]{slater2017master}
\begin{equation} \label{MasterFormula}
\tilde{\chi}_{d,0}(\varepsilon) \equiv \tilde{\chi_d}(\varepsilon)= \frac{\varepsilon ^d \Gamma (d+1)^3 \,
_3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3
d}{2}+1;\varepsilon ^2\right)}{\Gamma \left(\frac{d}{2}+1\right)^2},
\end{equation}
($\varepsilon$ being a singular-value ratio, and $d$--{\it not} the quasirandom dimension parameter--the random-matrix Dyson index)
has been recently extended to apply to the still more general class of ``induced measures" \cite{zyczkowski2001induced}, giving expressions for $\chi_{d,k}(\varepsilon)$ \cite{slater2018extensions}. (Also, we have sought to develop an alternative framework to that of Lovas and Andai, in the context of
``Slater separability functions'', but not yet fully successfully \cite{LovasAndaiAlternative1,LovasAndaiAlternative2}.)
As specific illustrations here of (\ref{MasterFormula})--with the assistance of C. Dunkl--are the formulas \cite[sec. B.3.c]{slater2018extensions}--with $z=\varepsilon^2$--for $\chi_{2,k},\chi_{4.k}$ and
$\chi_{6,k}$:
\begin{align*}
\chi_{2,k}\left( z\right) & =1+\left( 1-z\right) ^{k+1}\left(
-1+\frac{1}{k+3}z\right) ,\\
\chi_{4.k}\left( z\right) & =1+\left( 1-z\right) ^{k+1}\left(
-1-\left( k+1\right) z+\frac{2\left( 2k^{2}+14k+21\right) }{\left(
k+5\right) \left( k+6\right) }z^{2}-\frac{6\left( k+3\right) }{\left(
k+6\right) \left( k+7\right) }z^{3}\right) ,\\
\chi_{6,k}\left( z\right) & =1+\left( 1-z\right) ^{k+1}\{-1-\left(
k+1\right) z-\frac{\left( k+1\right) \left( k+2\right) }{2}z^{2}\\
& +\frac{3\left( 3k^{4}+60k^{3}+432k^{2}+1230k+1264\right) }{2\left(
k+7\right) \left( k+8\right) \left( k+9\right) }z^{3}-\frac{6\left(
k+4)(3k^{2}+33k+80\right) }{\left( k+8\right) \left( k+9\right) \left(
k+10\right) }z^{4}\\
& +\frac{30\left( k+4\right) \left( k+5\right) }{\left( k+9\right)
\left( k+10\right) \left( k+11\right) }z^{5}\}.
\end{align*}
In section 4 of their recent study \cite{lovas2017invariance}, Lovas and Andai extended their analyses from one involving the (non-monotone \cite{ozawa2000entanglement}) Hilbert-Schmidt measure to
one based on the operator monotone function $\sqrt{x}$. They were able to conclude (for the case $d=1$ [a Dyson-type random-matrix index]) that the applicable ``separability function" in this case,
$\tilde{\eta}_d(\varepsilon)$,
is precisely the same as the Hilbert-Schmidt counterpart $\tilde{\chi}_d(\varepsilon)$.
Now, quite strikingly, we obtained \cite{slater2017master}, using this function, for the two-qubit ($d=2$) analysis, the ratio of
$\frac{\pi ^2}{2}-\frac{128}{27}$ to $\frac{\pi^2}{2}$, that is,
\begin{equation}
\mathcal{P}_{sep.\sqrt{x}}(\mathbb{C}) = 1-\frac{256}{27 \pi ^2} =1 -\frac{4^4}{3^3 \pi^2} \approx 0.0393251.
\end{equation}
(We observe that such results--as with the Hilbert-Schmidt value of
$\frac{8}{33}$--interestingly appear to reach their most simple/elegant in the [standard, 15-dimensional] two-{\it qubit} setting, where the off-diagonal entries of the density matrix are, in general, complex-valued.)
Lovas and Andai have shown that the two-rebit separability probability based on the operator monotone function $\sqrt{x}$ is approximately 0.26223001318, asserting that the integrand can be evaluate[d] only numerically". Nevertheless, we
investigated--so far, rather not too productively, as with the Bures two-rebit estimate 0.157096234 above (sec.~\ref{BuresTwoRebit})--the possibility of finding an exact, underlying value for this statistic. (Our investigation, in this regard, is reported in \cite{StackExchange}. It involved first performing a series expansion of the elliptic and hypergeometric functions in their integrand. We were able to then integrate this series expansion, but only over a restricted range--rather than $[0,\infty]$--of the two indices. Numerical summation over this restricted set yielded a value of only 0.0042727 [reported in \cite{StackExchange}] {\it vs.} 0.26223001318.)
It would be of substantial interest to compare/contrast the relative merits of our quasirandom estimations above of the two-rebit and two-qubit Bures separability probabilities in the 36- and 64-dimensional settings employed with earlier studies
(largely involving Euler-angle parameterizations of $4 \times 4$ density matrices \cite{tilma2002parametrization}), in which 9- and 15-dimensional integration problems were addressed \cite{slater2005silver,slater2009eigenvalues} (cf. \cite{maziero2015random}). In the higher-dimensional frameworks used here, the integrands are effectively unity, with each randomly generated matrix being effectively assigned equal weight, while not so in the other cases indicated. In \cite{ExperimentalData}, we asked the question ``Can `experimental data from a quantum computer' be used to test separability probability conjectures?'', following the analyses of Smart, Schuster and Mazziotti in their article \cite{ssm}, ``Experimental data from a quantum computer verifies the generalized Pauli exclusion principle'', in which
``quantum many-fermion states are randomly prepared on the quantum computer and tested for constraint violations''.
So, in brief summary, let us state that at this stage of our continuing investigations, it appears that we have a set of three exact-valued measure-dependent two-qubit separability probabilities ($\frac{8}{33}$ [Hilbert-Schmidt], $1-\frac{256}{27 \pi^2}$ [operator monotone $\sqrt{x}$], $\frac{25}{341}$ [Bures--minimal monotone $\frac{1+x}{2}$]), but only one two-rebit one ($\frac{29}{64}$ [Hilbert-Schmidt]).
The [apparent lesser than $\frac{25}{341}$] separability probabilities for other members--Kubo-Mori, Wigner-Yanase,\ldots--of the monotone family have been estimated in \cite{slater2005silver}--cf. \cite{singh2014relative,batle2014geometric}. But since there is, at present, no apparent mechanism available for generating density matrices random with respect to such measures [cf. \cite[sec. V.B]{puchala2011probability} in regard to superfidelity], the quasirandom procedure seems unavailable for them. (Also the use of measures that are non-monotone in nature--in addition to the well-studied Hilbert-Schmidt one--would be of interest, for example, the Monge \cite{zyczkowski2001monge} and Husimi \cite{slater2006quantum,rexiti2018volume} measures.) However, separability/PPT-probabilities can be so analyzed for the class of induced measures \cite{zyczkowski2001induced}.
Let us pose the following problem: construct a function $f$ that yields the separability probabilities associated with the monotone metrics. That is, we would have (the Bures case) $f(\frac{1+t}{2})=\frac{25}{341} =0.0733138$, $f(\sqrt{t}) =1 -\frac{256}{27 \pi^2}=0.0393251$ and $f(\frac{2 t}{1+t}) =0$. Additionally, $f(\frac{t^{(t-1)}}{e}) \approx 0.0609965$, $f(\frac{1}{4} \left(\sqrt{t}+1\right)^2) \approx 0.0503391$ and $f(\frac{(t-1)}{\log{t}}) \approx .0346801$ \cite[Tab. II]{slater2005silver} and also $f(\frac{1+6 t +t^2}{4 +4 t}) \approx 0.0475438$ \cite[Tab. I]{slater2005silver}.
\begin{acknowledgements}
This research was supported by the National Science Foundation under Grant No. NSF PHY-1748958.
\end{acknowledgements}
\bibliography{main}
\end{document}
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Preprocessing and postprocessing for vector quantization
5596659
Preprocessing and postprocessing for vector quantization
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(16 images)
Inventor:
Normile, et al.
Date Issued:
January 21, 1997
Application:
07/938,942
Filed:
September 1, 1992
Inventors:
Normile; James O.
(Sunnyvale, CA)
Wang; Katherine S.
(San Jose, CA)
Wu; Hsi-Jung
(Cupertino, CA)
Assignee:
Apple Computer, Inc.
(Cupertino, CA)
Primary Examiner:
Boudreau; Leo
Assistant Examiner:
Tran; Phuoc
Attorney Or Agent:
Blakely, Sokoloff, Taylor & Zafman
U.S. Class:
348/422.1
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;
382/299
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Other References:
Buzo, A. and Gray, R., "Speech Coding Baded Upon Vector Quantization" in: IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.ASSP-18, No. 5 (Oct. 1980) pp. 562-574..
Goldberg, M. and Sun, H., "Image Sequence Coding Using Vector Quantization" in: IEEE Transactions on Communications, vol. COM-34, No. 7, (Jul. 1986) pp. 380-397..
N. M. Nasrabadi and R. A. King, "Image Coding Using Vector Quantization: A Review," IEEE Trans. Commun., vol. COM-36, pp. 957-971 (Aug. 1988)..
R. M. Gray, "Vector Quantization," IEEE ASSP Mag., vol. 1, pp. 4-29 (Apr. 1984)..
Nasrabadi et al., "Interframe Hierarchical Address-Vector Quantization," IEEE Journal on Selected Areas in Communications, vol. 10, No. 5, pp. 960-967, Jun. 1992..
Abstract:
Improved proprecessing and postprocessing for vector quantization, for example, for encoding an image. In one embodiment, the luminosity of the vectors in an image are used to determine the codes for vector quantization. In another embodiment, a median filter is performed to eliminate motion artifacts. In another embodiment, temporal filtering is applied once the difference between an earlier frame and a current frame exceeds a threshold. Embodiments are also provided for adaptive temporal filtering based on temporal "no change" blocks and their errors. Embodiments are provided for different regions of images which reference different codebooks, and regions of variable size. Embodiments are also provided for shared and variable size codebooks for different images or other data.
Claim:
What is claimed is:
1. A method of encoding an image comprising the following steps:
a. adaptively preprocessing a first bitstream representative of said image in order to eliminate spatial redundancy by selectively subsampling individual vectors in said image based upon luminance of said individual vectors, reduce noise andcontrol data rate, and generate a second bitstream on which to perform vector quantization, wherein said individual vectors include a block of adjacent pixels in said image;.
2. The method of claim 1 wherein said selective subsampling is performed according to a different weight assigned to different positions of the subsampled individual vectors in said image.
3. The method of claim 2 wherein said weight assigned to central regions of said image is less than said weight assigned to border regions of said image such that said border regions of said image are subsampled prior to said central regions ofsaid image given similar characteristics of said central border regions of said image.
4. The method of claim 1 wherein said selective subsampling is performed according to a different weight assigned to different magnitudes of luminance of the subsampled individual vectors in said image.
5. The method of claim 4 wherein said weight increases as the magnitude of luminance of the individual subsampled vectors in said image increases.
6. The method of claim 5 wherein said different weight is generated by a rate control mechanism.
7. The method of claim 4 wherein said different weight is generated by a rate control mechanism.
8. The method of claim 1 wherein said selective subsampling is performed on individual vectors having edge transitions less than a first luminance threshold to preserve edge detail in said image.
9. The method of claim 8 wherein the first threshold is generated by a rate control mechanism.
10. The method of claim 1 further comprising the step of limiting the number of vectors chosen for subsampling in said image to a first number of vectors.
11. The method of claim 10 wherein the first number of vectors may be a value ranging from zero to the number of vectors in said image.
12. A method of encoding an image comprising the following steps:
a. adaptively preprocessing a first bitstream representative of said image in order to eliminate temporal redundancy by adaptively adjusting a difference threshold to determine whether blocks in said bitstream should be encoded according to ratecontrol demands, and generate a second bitstream on which to perform vector quantization;.
13. The method of claim 12 wherein said preprocessing step comprises determining whether a difference between a first vector in said image and a second vector in a previous image at a same spatial position as said first vector is less than afirst threshold, then tagging said first vector as a no-change block, said tagging indicating that said vector quantization should not be performed on said first vector, said first threshold being adaptively adjusted based upon rate control demandsdetermined by examining said fourth bitstream and errors in encoding previous vectors in said first bitstream.
14. The method of claim 13 wherein said preprocessing step further includes the step of determining whether said first vector in said image has been tagged as a no-change block for a first interval of time from previous images, and if so, thendetermining whether said difference is less than a second threshold, said second threshold being less than said first threshold, and if so, then tagging said fast vector as a no-change block.
15. The method of claim 14 wherein the first interval of time comprises a number of frames that said first vector has been tagged as a no-change block from previous images.
16. The method of claim 14 wherein the fast interval of time is determined based upon a fast age associated with the fast vector, said first age having an initial value which is pseudo-randomly generated and associated with said first vectorupon an initial determination in a previous image that said vector should be tagged as a no-change block.
17. A method of encoding an image comprising the following steps:
a. based upon local image characteristics of said image, adaptively determining which filtering method in a first set of filtering methods best preserves detail in said image given rate control demands and which eliminates spatial redundancy insaid image to generate a first bitstream on which to perform vector quantization;
b. preprocessing said first bitstream using said filtering method in order to generate a second bitstream which is free of said spatial redundancy;
c. applying vector quantization to said second bitstream in order to create a third bitstream comprising a codebook and indices referencing said codebook; and.
18. The method of claim 17 wherein said first set of filtering methods comprises a horizontal filtering method.
19. The method of claim 17 wherein said first set of filtering methods comprises a vertical filtering method.
20. The method of claim 17 wherein said first set of filtering methods comprises a upward diagonal filtering method.
21. The method of claim 17 wherein said first set of filtering methods comprises a downward diagonal filtering method.
22. An apparatus for encoding an image comprising:
a. means for adaptively preprocessing a first bitstream representative of said image in order to eliminate spatial.
23. An apparatus for encoding an image comprising:
a. means for adaptively preprocessing a first bitstream representative of said image in order to eliminate temporal packed indices in said thirdbitstream and an associated offset value each for each packed index from said single base address to reference entries in said codebook.
24. An apparatus for encoding an image comprising:
a. means for adaptively determining based upon local image characteristics of said image a filter in a first set of filters which yields a minimum absolute error compared to vectors in a first bitstream representative of said image in order toeliminate spatial redundancy to generate a first bitstream on which to perform vector quantization;
b. means for preprocessing said first bitstream using said filter in order to generate a second bitstream which is free of said spatial redundancy;
c. means for applying vector quantization to said second bitstream in order to create a third bitstream comprising a codebook and indices referencing said codebook; and.
25. An apparatus for decoding an image comprising:
a. means for creating a current codebook from a;
b. means for receiving a coded bitstream including block header identifiers in said coded bitstream, each said block header blocks in said coded bitstream have been vector quantized;;
f. means for upsampling said retrieved codebook entries for said vectors determined to be subsampled by said subsampling determining means; and
g. means for displaying each of the vectors contained in said codebook entries retrieved by said retrieving means on a display means.
26. The apparatus of claim 25 wherein the means for detecting the reference to the first codebook entry in the previous codebook includes means for detecting a bitmap referencing said first codebook entry in said previous codebook.
27. The apparatus of claim 25 wherein the means for upsampling codebook entries comprises means for replicating pixels from a codebook entry.
28. The apparatus of claim 25 wherein the means for upsampling codebook entries comprises means for interpolating pixels from a codebook entry.
29. The apparatus of claim 25 wherein the means for retrieving entries from said current codebook comprises means for using one index of said indices to reference said current codebook.
30. The apparatus of claim 29 wherein the means for using the index to reference said current codebook comprises means for adding said one index to a base address to calculate an actual index and using said actual index to reference said entryin said current codebook.
31. An apparatus for decoding an image comprising:
a. means for creating a current codebook from codebook entries transmitted by an encoding means;
b. means for receiving a ceded bitstream including block header identifiers in said coded bitstream each said block head block in said coded bitstream have been only vector quantized with no subsampling:; and
f. means for upsampling said retrieved codebook entries for said vectors determined to be subsampled by said subsampling determining means.
32. The apparatus of claim 31 further comprising means for displaying vectors comprising an image based upon entries retrieved from said current codebook.
33. The apparatus of claim 31 wherein said upsampling means comprises means for interpolating pixel values from said entries received from said current codebook.
34. The apparatus of claim 31 wherein said upsampling means comprises means for replicating pixel values from said entries received from said current codebook.
35. The apparatus of claim 31 wherein said vector is represented using luminance and chrominance information.
36. The apparatus of claim 31 wherein said vector is represented using a block of luminance and a single chrominance value.
37. An apparatus for decoding an image comprising:
a. means for receiving a reference to a previous codebook, wherein said reference to a previous codebook refers to one of a plurality of codebooks in said apparatus and each of said plurality of codebooks is used for decoding different spatialregions of said image;
b. means for creating a current codebook from said;
c. means for receiving codebook indices, said codebook indices referencing entries in said current codebook; and
d. means for retrieving entries from said current codebook based upon said codebook indices received by said codebook receiving means.
38. A method of encoding an image comprising the following steps:
a. adaptively preprocessing a first bitstream representative of said image in order to eliminate redundancy by selectively subsampling individual vectors in said image based upon data rate demands and the luminance of each of said vectors, andgenerate a second bitstream on which to perform vector quantization, wherein said individual vectors include a block of adjacent pixels in said image;
b. applying vector quantization to said second bitstream in order to create a third bitstream comprising indices referencing a.
39. The method of claim 38 wherein said selective subsampling is performed according to a different weight assigned to different positions of the subsampled individual vectors in said image.
40. The method of claim 39 wherein said weight assigned to central regions of said image is less than said weight assigned to border regions of said image such that said border regions of said image are subsampled prior to said central regionsof said image given similar characteristics of said central and border regions of said image.
41. The method of claim 38 wherein said selective subsampling is performed according to a different weight assigned to different magnitudes of luminanee of the subsampled individual vectors in said image.
42. The method of claim 41 wherein said weight increases as the magnitude of luminanee of the individual subsampled vectors in said image increases.
43. The method of claim 38 wherein said selective subsampling is performed on individual vectors having edge transitions less than a first threshold to preserve edge detail in said image.
44. The method of claim 43 wherein the first threshold is generated by a rate control mechanism.
45. The method of claim 38 further comprising the step of limiting the numbers of vectors chosen for subsampling in said image is limited to a first number of vectors.
46. The method of claim 45 wherein the first number of vectors may be a value ranging from zero to the number of vectors in said image.
47. An apparatus for encoding an image comprising:
a. preprocessing circuitry for adaptively preprocessing a first bitstream representative of said image in order to eliminate redundancy based upon data rate demands, and generate a second bitstream on which to perform vector quantization;
b. vector quantization circuitry for encoding said second bitstream in order to create a third bitstream comprising indices referencing a codebook; and
c. index packing circuitry for processing said indices in said third bitstream in order to generate a fourth bitstream, said fourth bitstream including a single base address into said codebook for a plurality said indices in said third bitstreamand a plurality of associated offset values from said single base address to reference entries in said codebook .
Description:
BACKGROUND OF THE INVENTIONinformation. This is due to the large amount of data to transmit or store for representing high resolution full image video information. Generally, apparatus such as shown in FIGS. 1a, 1b, and 1c are employed in order to compress and decompress aninput image for vector quantization based techniques. For instance, as shown in FIG. 1a, an image 100 may be input to an encoder 101 which applies spatial or temporal proprocessing to an input image or sequence of images in order to reduce theredundancy or otherwise reduce the amount of information contained in the input image 100. Encoder 101 generates a compressed image 102 which is substantially smaller than the original image 100. In certain prior art systems, the encoder 101 uses acodebook 105 which is used for matching given pixel patterns in the input images 100, so that the pixel patterns are mapped to alternative pixel patterns in the compressed images 102. In this manner, each area in the image may be addressed byreferencing an element in the codebook by an index, instead of transmitting the particular color or other graphics information. Although in some prior art applications, quality is lost in compressed images 102, substantial savings are incurred by thereductioncontained within compressed images 102. Decoder 131 requires the use of the same codebook 105 which was used to encode the image. Generally, in prior art systems, the codebook is unique as associated with a given image or set of images which arecompressed and/or decorepressedcodebook is used for decoding the images it was generated from. The codebook can also be generated once by optimizing it for a long training sequence which is meant to be a reasonable representation of the statistics of sequences of images to be codedin the future. This training codebook is meant to be representative of a large range of image characteristics. The training codebook is often fixed at the encoder and decoder, but pieces of the codebook may also be improved adaptively. In some priorart.times.2 pixel blocks such as 201 and 202whichinclude generating a codebook; and (b) decode an image. Each of the vectors such as 201,202, etc. in image 200 may be used to represent image 200. Thus, an image may be represented by references to elements in a codebook which each are approximationsofimage.
Such prior art apparatus, such as discussed with reference to FIGS. 1a through 1c, are implemented in a device known as a codec (coder/decoder) which generates a compressed bitstream for a sequence of images from the corresponding codebook, andusesother types of data. Such input data may be applied to a preprocessor 320 wherein certain parameters are adjusted to preprocess the data in order to make encoding/decoding an easier task. Preprocessor 320 then feeds into a vector quantizer 330 whichuses vector quantization to encode the image in some manner, which equivalently reduces redundancies. Then, vector quantizer 330 outputs to a packing/coding process 340 to further compress the bitstream. A rate control mechanism 345 receivesinformationimage.imageadi").sequences is quality, which may not be acceptable for many sequences which may not be similar to image(s) in the training sequence. Overall performance is also a concern. Some prior art techniques require an inordinate amount of processing and still donot achieve acceptable compression while not being able to perform the compression in realtime. Demands for fast decoding capability are often even more stringent or real time playback is not possible. Most prior art systems also have a computationallyexpensive decoder.
SUMMARY AND OBJECTS OF THE INVENTION
One of the objects of the present invention is to provide an apparatus and method for efficiently generating codebooks by vector quantization, reducing spatial and temporal redundancy in images, and associated processing of images in order toconserve a method and apparatus for encoding data which sends preprocessed blocks to a vector quantizer. The vector quantizer represents the image blocks by indices into a table ofrepresentative vectors, referred to as a codebook, which minimize some fidelity criterion. In one embodiment, the luminance and chrominance (YUV) of the image vectors are used to determine the codebook and the indices to the codebook associated with animage or image set. In another embodiment, a temporal filter which adapts to how much change occurs is performed to reduce motion artifacts. In another embodiment, image vectors are not sent if the difference between the previous decoded frame and acurrent frame is less than an adaptive threshold. Embodiments also provide for different regions of images which reference different codebooks, and regions of variable size. Embodiments also provide for different blocktypes which reference differentcodebooks. Embodiments are also provided for shared and variable size codebooks for vector quantization. Embodiments are also provided for prefiltering which avoid edges in the image. Embodiments are also provided for a rate control scheme. Embodiments are also provided for reducing spatial redundancy by adaptive spatial subsampling, and reducing temporal redundancy by temporal subsampling.
BRIEF DESCRIPTION OF THE DRAWINGS an scheme for dividing an image into vectors comprising 2.times.2 pixel blocks.
FIG. 3 shows a functional block diagram of a prior an.
DETAILED DESCRIPTION
The present invention is related to improved methods of vector quantization. In the following description, for the purposes of explanation, specific types of data, applications, data structures, pointers, indices, and formats are set forth inorder to provide a thorough understanding of the present invention. It will be apparent, however, to one skilled in the art, that the present invention may be practiced without these specific details. In other instances, well-known structures and dataareand various static and dynamic storage devices. This also may include a special purpose video coder or decoder which is designed to provide for special purpose applications. Of course, it can be appreciated by one skilled in the art that the methodsand apparatus of the preferred embodiment may be implemented in discrete logic devices, firmware, an application specific integrated circuit (AS IC)intoquantization, such as in the audio field, and the specific discussion of video information in this application should not be viewed as limiting the present invention.
PREPROCESSING
The data rate at the output from the codec is used to control the amount of information which is allowed to reach the vector quantization process via the preprocessor 320. This is done at two levels--global and local. Global changes to thespatial resolution are made by applying a lowpass input filter to the input image, which changes the bandwidth of the image. The passband width of this filter varies with the error in the required data rate. As the error decreases, the bandwidth of theinput filter increases allowing more information to reach the codec. Conversely as the error in desired data rate increases, the input filters bandwidth decreases, limiting the information which reaches the codec. Global changes to the temporalresolution are made by determining the difference between current and previous frames. If the change is below a threshold, then the current frame is skipped. The threshold is determined from the data rate error. Another global mechanism by which thetemporalthe preferred embodiment implements an improved vector quantizer as shown as 330 in FIG. 3, which is very efficient at producing a small set of representative image vectors, referred to as the codebook, from a very large set of vectors, such as an imageto be encoded. The image(s) reconstructed by decoder 351 from the codebook generated by such a vector quantizer will be close to the original in terms of some criterion. The performance of the overall compression/decompression scheme is furtherimproved in the preferred embodiment by controlling the content of the bitstream prior to vector quantizer 330 by a preprocessor 320. This preprocessing can be transparent to vector quantizer 330. Preprocessor 320 substantially reduces the amount ofinformation used to code the image with a minimum loss of quality. Tags are used in the preferred embodiment to designate vectors that don't change in time instead of coding them. These are known as "no-change" blocks because they don't changeaccording to some threshold. Blocks are also processed using spatial subsampling in the preferred embodiment to achieve better compression. Further, preprocessor 320 can also change the characteristics of the image space in order to increase speed orto improve quality, such as by performing a transformation from an encoding represented in red, green and blue (RGB) to an encoding represented using luminance and chrominance (YUV).
NO-CHANGE BLOCKS
In a preferred embodiment, a series of decisions are made in order to determine whether to encode an image vector or to send a "no-change" block tag. In the case of a "no-change" block, compression is almost always improved because an index doesnot have to be sent for that image block. Encoding speed is improved because there are less image vectors to create a codebook from and find an index for. Decoding time is also improved because the new block does not have to be placed on the screenover the decoded block from the previous frame. Thus, instead of transmitting an index referring to an element in a codebook, a no-change tag is sent by preprocessor 320 and passed by vector quantizer 330 specifying that the block has not changedsubstantially from a previous frame's block at the same position. This is shown and discussed with reference to FIG. 4. Process 400 starts at step 401 and retrieves the next block in frame N at step 402. This image block of frame N is then compared byprethreshold p, .mu. as a no-change block is put through a more rigorous test before being tagged as a no-change block. The number of frames over which the block has beena no-change block, referred to as the "age" of the block, is checked to make sure it has not exceeded a maximum allowable age. If it has not exceeded the maximum allowable age, the block remains a "no-change" block. If it has exceeded the maximumallowable age, the error between that block and the block in the same location of the previous decoded frame is compared to a tighter threshold, for example, .mu./2. This is done in order to prevent no-change blocks from remaining in a given locationfor a long period of time, which can be noticeable to the viewer. A side effect of using block aging occurs when a large number of blocks age and reach the maximum age together. This results in a sudden datarate increase, which can trigger subsequentlarge fluctuations in datarate unrelated to image content. To prevent this from occurring, each block is initialized in the preferred embodiment with varying starting ages, which are reset periodically. This can be done randomly, but if it is done incontiguous sections of the image, aging will break up the bitstream with block headers less often. The main disadvantage of aging "no-change" blocks is a higher datarate, so it is most appropriate for use when the desired datarate does not demand veryhighlost by the blockheader overhead required to tell the decoder that subsequent block(s) are "no-change," then the "no-change" block is changed back to the blocktype preceding or following it. An example of when this occurs in the current embodiment iswhen there is a single 4.times.4NC (4-2.times.2 no-change) block in the middle of stream of subsampled blocks. The single 4.times.4NC block requires one header preceding it and one header following it to separate it from the stream of subsampled blocks,yielding 16 bits assuming one byte per blockheader. If the single 4.times.4NC block were changed to a subsampled block, it would only require one 8-bit index (for a 256 entry codebook), which is less costly than keeping it as a 4.times.4NC in terms ofthein regions of greater intensity (Weber's Law). The threshold .mu. is initially determined in the preferred embodiment from the user's quality settings, but is allowed to vary from its initial value by adapting to rate control demands and to a previousseries of frames' mean squared error (frame.sub.-- mse). The approach used in the preferred embodiment is to calculate the no-change threshold and .mu. as follows:
long.sub.-- term.sub.-- error, which will be discussed in more detail below in the discussion of the improved rate control mechanism 345, provides a benchmark for achieving the required datarate over a period of time. No-change blocks will beflagged more frequently if the long.sub.-- term.sub.-- error indicates that the datarate is too high. Conversely, no-change blocks will be flagged less frequently if the long.sub.-- term.sub.-- error indicates that the datarate produced is even lowerthan desired. Instead of reacting instantaneously, .mu. is buffered by .beta., which effectively controls the time constant (or "delay") of the reaction time to changing the datarate. This prevents oscillatory datarates and also allows a tolerance formore complex images with a lot of variation to generate more bits, and less complex images with less variation to generate less bits, instead of being driven entirely by a datarate. Because of the range of quality achievable in a given sequence, theno-change threshold .mu. maintains the quality of the most recently encoded part of the sequence by taking into account frame.sub.-- mse. Frame.sub.-- mse is also used by rate control 345 and will be discussed in more detail in the rate controlsection.
SPATIAL SUBSAMPLING
Another technique performed by preprocessor 320 in the preferred embodiment is that of spatial subsampling. Spatial subsampling is used to reduce the amount of information that is coded by vector quantizer 330. This results in faster encodingand more compression at the cost of some spatial quality. The primary challenge is to maintain high quality and compression. There are two approaches which can be taken by the preferred embodiment, each with different benefits. In the first approach,the image is separated into "smooth" and "detailed" regions by some measure, where blocks that are "smooth" are subsampled according to datarate demands. For example, "smooth" regions may be determined by comparing the mean squared error between theoriginal block and the corresponding subsampled and upsampled block. This is advantageous because "smooth" regions that are subsampled usually produce the least noticeable artifacts or error. An additional benefit to this approach occurs when twoseparate codebooks are generated for subsampled and 2.times.2C ("change") blocks, and each codebookalso affects the subsampling decision. The advantages of the second approach include the ability to efficiently (in terms of bits) communicate to the decoder which areas of the image to postfilter, and more efficient run length blockheader coding bycongregating subsample blocks together.
The subsampling process is discussed with reference to FIG. 5a. For subsampling, the image is divided into 4.times.4 blocks such as shown in FIG. 5a. Each 4.times.4 block is reduced to a 2.times.2 block such as 510 if it is selected to besubsampled. A filtering subsampling operation performed in the preferred embodiment actually uses a weighted average of each of the four 4.times.4 pixel blocks (e.g. block 518, comprising pixels 1-3, 5-7, 9-11 and 17-23) for representing the subsampledblock imagewere subsampled using either of these techniques, the number of vectors going into improved vector quantizer 330 would be reduced by a factor of 4, and therefore, the number of codebook indices in the final bitstream would also be reduced by a factor of4. In alternative embodiments, subsampling can also be done only in the horizontal direction, or only in vertical direction, or by more than just a factor of 2 in each direction by sampling blocks larger than 4.times.4 pixels into 2.times.2 pixelblocks. During decoding, improved decoder 351 detects, in a header preceding the indices, that the indices contained in a block such as 510 refer to subsampled blocks, and replicates each pixel by one in both the horizontal and the vertical directionsin order to recreate a full 4.times.4 block such as 520 (e.g. see, block 521 comprising 4 pixels, which each are equal to pixel 1 in the simple subsampling case). Note that block 521 can also be represented by four .gamma.s instead of four 1's, where.gamma. is a weighted average of block 518. In another alternative embodiment, the pixels between existing pixels can be interpolated from neighboring pixels in order to obtain better results. This, however, can have a detrimental effect on the speedof the decoder.
The method by which "smoothness" is determined is based on how much squared error would result if a block were to be subsampled. The subsampling operation may include filtering as well, as illustrated in the following error calculation. Thesquared error .epsilon. is calculated between each pixel of a 2.times.2 block such as 560 shown in FIG. 5b (comprising pixels a.sub.0 -a.sub.3) and the average .gamma. of its surrounding 4.times.4 block 555 (comprising pixels a.sub.0 -a.sub.3 andb.sub.0 -b.sub.11): ##EQU1## .gamma. calculated from block 518 is used in place of the value of pixel 1 in 2.times.2 block 521. If a 2.times.2 block such as 560 were to be subsampled, then the average of its surrounding 4.times.4.gamma. (block 555),would be transmitted instead of the four individual pixel values a.sub.0 -a.sub.3. The average .gamma. is useful in reducing blockiness. Thus, as shown with reference to FIG. 5, the value .gamma. is transmitted instead of the four original pixelvalues a.sub.0 -a.sub.3 of block 530. The squared error .epsilon. is then scaled by a weighting coefficient k to approximate the human visual system's luminanee sensitivity (or the SNR can be used as a rough approximation instead of MSE). Thus regionsof high luminanee are more easily subsampled assuming the subsampling errors are the same. The four scaled errors are then added to generate the error associated with each 2.times.2 block such as 560: ##EQU2## In order to rank a 4.times.4 block 500 as acandidate for subsampling, each of the subsampling errors .epsilon. from the four 2.times.2 blocks of pixels aligned at the corners within the 4.times.4 500 are added. Blocks are chosen for subsampling from smallest error to largest error blocks untilthe rate control determines that enough blocks have been subsampled to meet the desired frame size. In an alternative embodiment, edges in the image may be extracted by edge detection methods known to those skilled in the art in order to prevent edgesfrom being subsampled. Basing the decision to subsample on subsampling error has a tendency to preserve most edges, because subsampling and then upsampling across edges tend to produce the largest errors. But, it is also useful in some circumstances toexplicitlyavertical), each of which has a weighting associated with them. Obviously, the number of zones and their sizes can be fairly diverse. In one embodiment, weighting the border zones of the image may be performed so that it is more difficult to subsamplethenotice subsampling artifacts unless the motion is fast.
In the second approach of the preferred embodiment, zones are sorted according to their zonal errors, which is the average squared error .epsilon.: ##EQU3## and each zone is weighted according to its location to produce zone error ZE:
Blocks tagged for subsampling are subsampled in order of best to worst zones, in terms of zone error, until the number of subsampled blocks requested by rate control 345 is reached. Improved decoder 351 is able to determine from the inputbitstream 350 which zones have been subsampled and, depending on certain criteria (such as quality settings, etc.), may decide whether or not to postfilter (process 375) those zones during decoding in order to soften blockiness. Because subsampling isz.sub.-- mse are subsampled within the zone. The edge.sub.-- mse value is controlled by the rate control, so more blocks are preserved fromsubsampling if the compressed frame size desired is large.
In an alternative embodiment, the edge.sub.-- mse can be weighted so that edges in the image, extracted by edge detection methods known to those skilled in the art, are preserved from subsampling.
Directional Filtering
Spatial redundancy may also be reduced with minimal smearing of edges and detail by perforating "directional" filtering in an alternative embodiment. This processing performs a horizontal, vertical, upward diagonal and downward diagonal filterover orderto generate a filtered value for pixel 6. For example, in order to perform the "horizontal filter," the value may be represented as .function..sub.h wherein .function..sub.h is computed in the following manner:
wherein .alpha..sub.1, .alpha..sub.2, and .alpha..sub.3 are weighting coefficients. .alpha..sub.1, .alpha..sub.2, and .alpha..sub.3 may be equal to 0.25, 0.5, and 0.25, respectively, so that more weight is given to center pixel 6 of the3.times.3 block and the result .function..sub.h may be computed using computationally inexpensive shift operations. Note that these filters can be applied in three dimensional space as well, where the additional dimension is time in yet anotheralternative embodiment.
Comparing the results of these directional filters also gives information about the orientation of the edges in the image. The orientation of the edge may be extracted by comparing the ratio of the errors associated with orthogonal directionpairs. The first step is to select the direction which produced the minimum error, min.sub.-- directional.sub.-- error, and compare this error with the errors associated with the filter in the other three directions. Characteristics which wouldindicatefilter is very adaptive since it may vary its characteristics for every pixel according to the characteristics of the area around the pixel.
YUV TRANSFORMATION4## Performing code bookgeneration using YUV in vector quantizer 330 can improve clustering because of the tighter dynamic range and the relative decorrelation among components. Consequently, improvement in quality is noticeable. For situations where encoding speed isimportant,transformation may be done first and preprocessing such as subsampling can be done after the YUV transformation. At any rate, the resulting preprocessed data passed to improved VQ 330 is in YUV format.
IMPROVED VECTOR QUANTIZER
Vector Quantization (VQ) is an efficient way for representing blocks or vectors of data. A sequence of data, pixels, audio samples or sensor data is often quantized by treating each datum independently. This is referred to as scalarquantization. VQ, on the other hand, quantizes blocks or vectors of data. A primary issue with VQ is the need to find a set of representative vectors, termed a codebook, which is an acceptable approximation of the data set. Acceptability is usuallymeasured using the mean squared error between the original and reconstructed data set. A common technique for codebook generation is described in Linde, Y., Buzo, A., and Gray, R., "An Algorithm for Vector Quantizer Design," COM-28 IEEE Transactions onCommunications 1 (January 1980) (known as the "LBG" algorithm). A technique which employs the LBG algorithm to generate a codebook starts by sampling input vectors from an image in order to generate an initial estimate of the codebook. Then, each ofthe input vectors is compared with the codebook entries and associated with the closest matching codebook entry. Codebook entries are iteratively updated by calculating the mean vector associated. with each codebook entry and replacing the existingentry with the mean vector. Then, a determination is made whether the codebook then has improved significantly from a last iteration, and if not, the process repeats by comparing input vectors with codebook entries and re-associating, etc. This codebookgeneration may be done on a large sequence of images, the training set, or the codebook may be regenerated on each frame. In addition, this technique may be applied to binary trees used in certain prior art vector quantization systems for encodingefficiency.
The improved vector quantizer 330 is organized in a tree structure. Instead of a binary tree as used in certain prior art schemes, at the root of the tree, N child nodes 610, as shown in FIG. 6, are generated initially. This may be performedusing a variety of techniques. For example, in one embodiment, a segmenter may be used to extract representative centroids from the image to generate the N initial nodes which contain the centroid values. In another embodiment, the initial centroidsmay be determined from an image by extracting N vectors from the image itself. Prior art binary trees have relied simply upon the establishment of two initial nodes. Binary trees suffer from the disadvantage that the errors in the two initial nodespropagate down to the rest of the nodes in the tree. In the preferred embodiment, N nodes are used wherein the value N varies depending on image characteristics. This advantage is related to the fact that more initial nodes reduce the chances ofincorrectimproved from the N initial nodes by iteratively adjusting the values of the initial nodes and associating vectors with them at step 703. This iterative process is described below with reference to FIG. 8, which shows an iterative nodebinning/recalculation process. Then, at step 704, the node with the worst distortion is determined, where its distortion is calculated from a comparison between the node's centroid value and its associated vectors. In the preferred embodiment, meansquaredinto two children nodes at step 705. Of course, even though two children nodes are described and used in the preferred embodiment, more than two children nodes may be created in an alternative embodiment. Then, an iterative process upon the childrennodesnodes, such as 670 shown in FIG. 6, from the group of vectors associated with its parent node. In the case of a root node, all of the vectors of the image are used to cream representative centroids. Then, each of the vectors is associated (or "binned")with the node having the closest centroid. Then, at step 804, the error between the vectors associated with each of the centroids anal the centroid itself is determined. The error calculation may be performed using a variety of techniques, however, inthe preferred embodiment, a mean squared calculation is used. Once the error calculation has been determined at step 805, it is determined whether the change in the error has become less than a certain threshold value. In step 806, new centroids arecalculatedthe error values calculated. However, on subsequent iterations of the loop comprising steps 803 through 806, the change in error will become smaller, eventually becoming less than the threshold values. If the change in total error associated with thenodechange600. Thus, in the preferred embodiment, the total number of terminal nodes desired determines how many times nodes in VQ tree 600 will be split. Process 700 continues at step 704 through 707 until the desired number of terminal nodes in the tree havebeen unsigned long *vect.sub.-- index.sub.-- list;//pointer to list of vector indicesassociated with this node unsigned long num.sub.-- vect; //number of vectors associated with this node unsigned long distortion; //total distortion associated with this node unsigned long avg.sub.-- dist; //Average distortion associated with thisnode unsigned long peak.sub.-- dist; //Peak distortion associated with this node unsigned long percent.sub.-- dist; //percentage distortion associated with this node unsigned long num.sub.-- children; //number of children unsigned long ic.sub.--method; //method for initializing this node struct tnode **children; //pointer to a list of structures for the child nodes of this node struct tnode *parent; //pointer to the parent of this node unsigned char terminal; //flag to indicate if thisis a terminal node unsignedinformationthea given vector, for example, at step 803 in FIG. 8. In the early stages of tree generation the preferred embodiment modifies this calculation to weight large errors more heavily than is the case with squared error. In this manner, large errors areweighederror is used as the distortion measure, then the total distortion is the sum of the mean squared errors. The use of the other distortion measures, or combinations thereof, may be used in yet other alternative embodiments, each having certain advantagesaccording to image content, or desired quality.
Fourth, multiple retries are attempted in order to split nodes. Occasionally, an attempt to split a specific node fails. In this case, a number of other initial conditions are generated which will assist in leading to a successful split. Forexample, one way in which this may be performed is by adding noise to an initial split. For certain images characterized by flat or very smooth varying color or luminance areas, node splitting is difficult. A small amount of random noise is added tothe image vectors prior to splitting. The noise is pseudorandom and has a range between zero and two least significant bits of the input image data. One manner in which the noise is generated is to use a pseudorandom noise generator. This value isadded to each of the RGB components of each pixel of each vector to be encoded. The random noise added to each of the RGB components of each pixel will differentiate them enough in order to achieve a successful split. More generally, assuming that adecision610 comprising N initial nodes captures these features. Enhanced performance in terms of computational speed and improved image quality can be obtained by reusing the top layer of the tree from one frame to the next. This reuse may be overridden from ahigher level in the codec. For example in the casepreferred embodiment utilizes a technique which gives many of the advantages of mean residual VQ without the decoder complexity. The technique works as follows. The mean value is calculated for a large image or "zone," and then this mean is subtractedfrom.
VARIABLE SIZE, SHARED, AND MULTIPLE CODEBOOKS FOR IMAGES
In the preferred embodiment, each image is associated with a codebook which has been adapted to the characteristics of that image, rather than a universal codebook which has been trained, though a combination of fixed codebook and adaptivecodebook is also possible in alternative embodiments. In alternative embodiments, each image need not be limited to having exactly one codebook or a codebook of some fixed size. Alternative embodiments include using codebooks of variable size, sharingcodebooks among frames or sequences of frames, and multiple codebooks for the encoding of an image. In all of these alternative embodiments, the advantage is increased compression with minimal loss in quality. Quality may be improved as well.
Variable Size Codebooks
For a variable size codebook, the nodes in the tree are split until some criterion is met, which may occur before there are a specified number of terminal nodes. In one embodiment, the number of codebook vectors increases with the number ofblocks that change from the previous frame. In other words, the greater the number of no-change blocks, the smaller the codebook. In this embodiment, codebook size is obviously related to the picture size. A more robust criterion, which is used in thepreferred embodiment, depends on maintaining a frame mean squared error (not including no-change blocks). If 128 2.times.2 codebook vectors are used instead of 256, the net savings is 768 bytes in the frame. This savings is achieved because each2.times.2 block comprises a byte per pixel for luminance information and 1 byte each per 2.times.2 block for U and V chrominance information (in the YUV 4:1:1 case). Reducing the number of codebook vectors from 256 to 128 yields 128.cndot.6=768 bytestotal savings. For images where 128 codebook vectors give adequate quality in temps of MSE, the 768 bytes saved may be better used to reduce the number of subsampled blocks, and therefore improve perceived quality to a viewer.
Shared Codebooks
Another feature provided by the preferred embodiment is the use of shared codebooks. Having one or more frames share a codebook can take advantage of frames with similar content in order to reduce codebook overhead. Using a shared codebook cantake advantage of some temporal con, elation which cannot be efficiently encoded using no-change blocks. An example of such a case is a panned sequence. If two frames were to share a 256 element codebook, the savings would be equivalent to having eachframe use separate 128 element codebooks, but quality would be improved if the frames were not completely dissimilar. Obviously, the separate 128 element codebook case could use 7 bit indices instead of 8 bit indices, but the lack of byte alignmentmakes packing and unpacking the bitstream unwieldy. Reduced codebook overhead is not the only advantage to using a shared codebook. For example, temporal flickering can also be reduced by increasing the correlation in time among images by using thesameis encoded using the shared codebook, and the frame.sub.-- mse (the mean squared error between the original and decoded frame) is calculated. The shared codebook is replaced with a new codebook if the frame.sub.-- mse is greater than the frame.sub.--mse from the previous frame or the average frame.sub.-- mse from the previous frames by some percentage. If the frame.sub.-- mse passes this test, the shared codebook can still be entirely replaced if the number of blocks with an MSE over somepercentage of the average MSE (i.e. the worst blocks) for the entire frame is over some number. In this case, the encoder assumes that it is too difficult to fix the worst error blocks with only an update to the codebook, and will regenerate the entirecodethe terminal nodes of the tree (i.e. with a codebook vector). This is achieved by starting at the root of the tree, choosing which of the children is closer in terms of squared error, and choosing which of that child's children is a best match, and soforth. An image vector traverses down the tree from the root node toward a terminal node in this fashion. Using the structure of the tree instead of an exhaustive search to match image vectors with codebook vectors improves encode time, though anexhaustive asshown in FIG. 9b to tree 930. Terminal nodes that were discarded because they were either zero cells, such as 901, or became parents by splitting are tagged to be overwritten with new updated codebook vectors. Finally, new children from the node splitsoverwrite these codebook vectors which are tagged to be overwritten. The actual overwrite occurs in the decoder, which is given the overwrite information via the bitstream (see, discussion below). If there are no zero cells, each node split wouldrequireshared.
Multiple Codebooks
In yet another embodiment, multiple codebooks can be associated with an image by generating a separate codebook for each blocktype, or by generating separate codebooks for different regions of the image. The former case is very effective inincreasingbe "smooth" for subsampled regions and more "detailed" for blocks which are not subsampled. The block types are separated by the error calculation described in the section on spatial subsampling. The separation between "smooth" and "detailed" regionsoccurs even when the compression desired requires no subsampling, because the separate codebooks work very well when the "smooth" and "detailed" blocks are separately encoded. Note that each index is associated with a codebook via its blocktype, so thenumber of codebook vectors can be doubled without changing the bits per index, or increasing the VQ clustering time. This results in a noticeable improvement in quality. Also, the subsampled blocks codebook and 2.times.2C blocks codebook can be sharedwith the previous frame's codebook of the same type. In such a case, it is even more important to keep "smooth" regions and "detailed" regions separate so there is consistency within each codebook across several frames. Note that this separation intodetailed and smooth areas is a special case of the more general idea of defining separate trees for image categories. The categories can be determined with a classifier which identifies areas in an image with similar attributes. Each of these similarareas are then associated with its own tree. In the simple case described above, only two categories, smooth and detailed, are used. Other possible categorizations include edge areas, texture, and areas of similar statistics such as mean value orvariance.
As mentioned briefly, multiple trees may be associated with different regions in the image. This is effective in reducing the encode time and increasing the compression ratio. For example, a coarse grid (8 rectangles of equal size) is encodedwith. Thistechnique is particularly well suited for lower quality, higher compression ratios, faster encode modes. A compromise between using many small codebooks for small pieces of the image and one 256 entry codebook for the entire image can be most effectivein maintaining quality while gaining some additional compression where the quality won't suffer as much. In such a compromise, much smaller codebooks are used only for portions of the image that are very homogeneous and only require a few codebookvectors, and the regular 256 entry codebook is used for the rest of the image. If the portion of the image associated with a much smaller codebook is constrained to be rectangular, it will require almost no overhead in bits to tell the decoder when toswitch to the much smaller codebook, and hence the smaller indices (4-bits for 16 entry codebooks or 6 bits for 64 entry codebooks). If the region associated with each codebook is not constrained to be rectangular, the quality can be improved withsegmentation techniques known to those skilled in the art, which group similar pixels into a region.
RATE CONTROL
Rate control 345 is an important element of the improved video compression system when the compressed material is meant to be decoded over a limited bandwidth channel. To maintain N frames/second in a synchronous architecture, or over a networkor phone line, decoder 351 must be able to read one frame of data over the limited bandwidth channel, decode the information, and display the image on the screen in 1/Nth of second. Rate control 345 attempts to keep the maximum frame size below somenumber,.sub.-- frame.sub.-- length is calculated as: ##EQU5## The desired frame length for the current frame N is equal to thetarget.sub.-- frame.sub.-- length, dampened by an error term frame.sub.-- error which may be averaged over some number of frames, such as a second's worth of video data:
Note that frame.sub.-- error, which is the overshoot or undershoot that will be allowed, is averaged as an IIR (infinite impulse response) filter in a recursive fashion. This may also be implemented as an FIR (finite impulse response) filter inan alternative embodiment. The value of .alpha. affects how quickly the current frame error (target.sub.-- frame.sub.-- length-avg.sub.-- frame.sub.-- length) forces the long term frame error (frame.sub.-- error) to respond to it. Also, the currenterror is defined as the difference between the target.sub.-- frame.sub.-- length and the average of the frame lengths of some number of frames (avg.sub.-- frame.sub.-- length), such as a seconds worth of data. This rate control scheme maintains anaverage datarate over the past second that does not exceed the desired datarate. Fluctuations in frame size occur at the per frame level, but these fluctuations are dampened by averaging effects. These relationships are determined as follows: ##EQU6##
After the desired.sub.-- frame.sub.-- length is determined for frame N, it is used to influence the encoder parameters (ncthreshfactor and edge.sub.-- mse) which control how much temporal processing and spatial subsampling to apply in thoseembodiments where temporal filtering and spatial subsampling are used. These encoder parameters are set by the spatial and temporal quality preferences determined by the user, but they are allowed to fluctuate about their quality setting according tohow well the system is keeping up with its datarate demands. Rather than allowing these parameters to fluctuate considerably over a short period of time, they track a long term error calculated as follows:
Thus, the only distinction between the calculations for the long.sub.-- term.sub.-- error and the frame error is the difference between .alpha. and .beta.. Values which have been determined to be effective are .alpha.=0.20 and .beta.=0.02 whichare used in the preferred embodiment, although it can be appreciated by one skilled in the art that other weighting values of .alpha. and .beta. may be used.
If long.sub.-- term.sub.-- error is not used to control the values of encoder parameters for spatial subsampling and no-change blocks, the desired frame length can still be used to keep track of how well the datarate is being maintained, giventhat no-change and subsampling thresholds are determined only by the user's quality settings. However, this doesn't guarantee that subsampling and no-change blocks can reduce the frame size to the desired.sub.-- frame.sub.-- size. In such a case, thevalue long.sub.-- term.sub.-- error is used to reduce the quality by changing subsampling and no-change block parameters, ncthreshfactor and edge.sub.-- mse, and therefore reduce the datarate.
TRANSMISSION OF CODEBOOK INDICES
After an image has been associated with indices to a codebook via vector quantization by improved process 330, the bitstream can be packed more efficiently than prior art techniques to allow for the flexibility of future compatible changes to thebitstream and to communicate the information necessary to decode the image without creating excessive decoding overhead. The indices may each be transmitted as an index to the codebook or as offsets from a base index in the codebook. In the formercase, 8 bits are required per image vector to indicate which of the vectors of a 256 entry codebook is the best match. In the latter case, less bits may be required if there is a lot of correlation between indices, because the differences betweenindicesheaders which indicate what blocktype the following indices refer to. 2.times.2 change (2.times.2C), 2.times.2 no-change (2.times.2NC), 4.times.4 no-change (4.times.4NC), 4.times.4 change (4.times.4C), subsampled (4.times.4SS), different combinations ofmixed, forexample. A single sequence header 1001 precedes a sequence of images and specifies information about the sequence. Sequence header 1001 can be almost any length, and carries its length in one of its fields. Several fields currently defined for thesequence headers are shown in FIG. 11. Sequence header 100 1integer value which is stored as an unsigned long word in the preferred embodiment allowing sequence lengths of up to 2.sup.32 frames. The next field 1104 in the sequence header is currently reserved, and the following two fields 1105 and 1106 definethe width and height of the images in the sequence. The last field in sequence header 1001 is the version field 1107 which is an integer field defining the current version of the encoding/decoding apparatus being used. This is to distinguish newersequences from older sequences which may have additional features or lack certain features. This will allow backward and upward compatibility of sequences and encoding/decoding schemes. The sequence header may also contain an ASCII or character stringthat can identify the sequence of images (not shown).
Returning to FIG. 10, Chunk header 101 1 carries a chunk type which conveys information about the next chunk of frames, such as whether or not they use a shared codebook. The chunk header can also specify how many codebooks are used for thatchunk of frames. Chunk header 1011 precedes a "chunk" of frames in the sequence. A chunk is one or more frames which is distinguishable from another "chunk" in the preferred embodiment by such apparatus as a scene change detector algorithm. In anotherembodiment,code definestwhich need to be updated. This field is followed by vector updates 1371-1373 for each of the vectors which is being updated. In this manner, instead of the entire codebook being regenerated, only selected portions are updated, resulting in a furtherreductionreferencepattern indicates whether it is a null frame for skipped frames, an entirely subsampled frame, a keyframe, or a frame sharing a codebook with another frame. Other types of frames are contemplated within the spirit of the invention. The subsampled zonefieldwhether the following set of indices are 2.times.2C blocks (change blocks), 4.times.4NC blocks (no-change blocks), 4.times.4SS blocks (subsampled blocks), mixed blocks, or raw pixel values. If the first three bits specify that the blocktype is notmixed, the last 5 bits of header 1401 is an integer indicating how many indices 1402 follow the block header 1401. This is called a "runlength" block header. The blockheader may also specify mixed blocks, such as a mix of 2.times.2C and 2.times.2NCblocks. In such a case, the 5 bits in the header reserved for length specifies how many 4.times.4s of mixed 2.times.2C and 2.times.2NC blocks are encoded. Alternatively, one of these 5 bits may instead be used to allow for more mix possibilities. Abitmap follows, padded to the nearest byte. In the 2.times.2C-2.times.2NC mix example, the bitmap specifies with a "1" that the blocktype is 2.times.2C, and with a "0" that the blocktype is 2.times.2NC. The blocks can be mixed on a 4.times.4granheader type (e.g. "111111111000000000") would be better coded with the runlength header type. The blockheader that codes the blocks more efficiently is chosen. The bitmap header allows the efficient coding of short run blocks which can occurfrequently.
Because of the overhead of two bytes of a block type header 1401 before and after a block which is tagged as a "no-change" block in the middle of a stream of "change" blocks, the runlength blockheaders in the preferred embodiment only disturbsthe structure of the indices with headers if there at least 4 2.times.2 no-change blocks in a row. The runlength headers in the preferred embodiment requires that 4-2.times.2NC (no-change) blocks must occur together to make a 4.times.4NC (no-change)block, in order to distinguish them in the bitstream with headers such as 1410. A block header such as 1410 which indicates that the following N blocks are of the 4.times.4NC (no-change) type need not waste any bytes with indices since the previousframe's blocks in the same location are going to be used instead. Decoder 351 only needs to know how many blocks to skip over for the new image. 2.times.2C blocks indices such as 1402 do not need to occur in sets of 4 because actual pixel values may beused or even singular 2.times.2 blocks. If actual pixel values or singular 2.times.2C and 2.times.2NC blocks are not supported in some implementations, assuming 2.times.2C blocks occur in fours can increase the number of blocks associated with the2.times.2C blockheader such as 1401, and consequently decrease the effective overhead due to the blockheader. For example, a block may identify eight 2.times.2C (change) blocks and interpret that as meaning eight groups of 4 2.times.2C blocks, ifsingular 2.times.2 blocks are not supported. (See an example of this in FIG. 15, 16 where 2-2.times.2C blocks are interpreted as two sets of 4-2.times.2C blocks).
Additionally, the indices 1402 in FIG. 14 referring to the 2.times.2C blocks do not have to be from the same codebook as the indices 1421 referring to the 4.times.4SS blocks. This bitstream flexibility allows the support of higher quality atvery little reduction in compression by having more than 256 codebook vectors without having to jump to a non-byte aligned index size (such as an unwieldy 9 bits for 512 codebook vectors).
INDEX PACKING
If image blocks are in close proximity in the codebook and are also similar in RGB color space, it is advantageous to use a base address when coding the indices, instead of just listing them in the bitstream. Because the codebook vectors aregenerated by splitting "worst error" nodes, similar image vectors tend to be close together in the codebook. Because like image blocks tend to occur together in space in the image (i.e. there is spatial correlation among the blocks), index values thatare close together tend to occur together. Assignment of codebook indices can also be performed in such a way that differences in indices over space can be minimized. An example of how this may be used to reduce the number of bits losslessly is shownaddress.1601 is required to be transmitted defining the base address, and that differential coding is being used. Regions which have a large, variable set of codebook indices (from one end of the codebook to the other), are more efficiently coded using thetransmission of complete indices such as shown in FIG. 15, and regions which are similar on a block level are more efficiently coded using a bitstream such as 1600 shown in FIG. 16. Using offsets from a base address, as is shown in FIG. 16, is equallyloss in FIG. 1 through 16. It will, however,be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the present invention as set forth in the appended claims. The specification and drawings are, accordingly, to be regarded in anillustrative rather than a restrictive sense.
* * * * *
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Description
Keurig Descaling Solution Brewer Cleaner, Includes 28 oz. Descaling Solution, Compatible with Keurig Classic/1.0 & 2.0 K-Cup Pod Coffee Makers (28 Oz)
- IMPROVE COFFEE TASTE: by eradicating mineral buildup that may have an effect on the style of your drinks
- SAFE AND GENTLE: citric acid safely cleans inside your espresso maker with out harming your brewer
- FAST ACTING AND ODORLESS: cleanses and rinses away mineral construct up with out abandoning any odor or residual style
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| 120,600
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TITLE: Extension of linear independent set of vectors, with a span, that doesn't contain the standard basis
QUESTION [1 upvotes]: I need the following statement as part of a longer proof. It has been some time since I learned linear algebra and I can't figure out why this holds.
Let $d,N \in \mathbb{N}$ with $d < N$. Let $\mathcal{A} := \{a_1,...,a_d\}$ be a linear independent set of $d$ vectors from $\mathbb{R}^{N+1}$ ($a_1 \in \mathbb{R}^{N+1}$, $a_2 \in \mathbb{R}^{N+1}$,...), furthermore none of the vectors in the standard basis of $\mathbb{R}^{N+1}$ should be contained in $span(\mathcal{A})$ ($(1,0,...,0)^T \notin span(\mathcal{A})$ and $(0,1,0,...,0)^T \notin span(\mathcal{A})$, ...) . I'm looking for vectors $b_1,...,b_{N+1-d}$ such that $\tilde{\mathcal{A}} := \{a_1,...,a_d,b_1,...,b_{N+1-d}\}$ is still an linear independent set and none of the vectors in the standard basis of $\mathbb{R}^{N+1}$ is in $span(\tilde{\mathcal{A}})$.
My idea was to proof this through the basis extension theorem, but I'm struggleing finding a vectorspace $V$, such that $span(\mathcal{A}) \subset V$ and none of the vectors in the standard basis of $\mathbb{R}^{N+1}$ is in $V$. In other words I'm struggleing to find a vectorspace, that does not contain $(1,0,...,0)^T, (0,1,0,...,0)^T, ...$, but does contain every linear combination of vectors from $\mathcal{A}$.
Sorry for my spotty english and sorry if I oversee an easy solution. Every help is appreciated. Thank you in advance!
REPLY [1 votes]: If you have such an $\tilde{\mathcal{A}}$, then it would span all of $\mathbb{R}^{N+1}$ and thus contain every standard basis vector.
It is impossible to construct a linearly independent set of $N+1$ vectors that don’t contain the standard basis vectors (or any other vector, for that matter).
| 135,025
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~ - ~
Recently, I was at Ambulance HQ for some further training. I had my uniform on, whilst getting out the car a fellow got out of his car next to me. "You guys do an excellent job!" he stated, adding a few more compliments to the monologue. All I could stammer was "Ah, its an easygoing job, cheers, good fun, nice to meet you, thanks, good bye". Talk about an inadequate answer….
I felt silly after that. There he was, Joe Blow, thanking me, Yours Truly. Saying thanks on behalf of the public to my profession. And i wipe it away, play it down as if it were some kind of game, not worthy of mentioning.
~
Not long ago, a colleague unfortunately had the 'pleasure' of seeing the other side of the Emergency Department - as a patient. During treatment and assessment, the doctor walked up to my colleague and stated "I couldn't do your job".
Sure, our workspace may be a little volatile at times, strewn with uncertainty. Sometimes outnumbered, sometimes running out of hands. There we stand, in front of whoever called us for whatever reason. We stand in their world, but have to tell them that they now have to play by our rules. Need a hand? Call another crew, wait 5-50 minutes. Patient lying awkwardly in the bathroom? It might be awkward getting him or her out. Hasn't flushed the toilet? Might be a little smelly. Didn't quite hit the toilet bowl? Makes extrication easier, as the floor is already slippery.
But it beats having to go to work five out of seven days to an office, seeing the same people again and again, and sitting on yer bum staring at a screen.
Chances are, I couldn't do your job either.
~
"You must see some horrible things"
"Yes, and I'm looking at one right now"
In all honesty, most situations are abstracted for treatment (and survival). Patients become dehumanised, are a bundle of symptoms just waiting to be treated; and if they can talk, you try out a bad joke to test their sense of humour (and you can then reenter the real world again). Most situations are horrible for the people involved - mostly because people panic and common sense (if they ever had any) flies out of the window; or in a minor amount of cases, because someone is quite sick or even dead.
But it is the patients emergency. At the end of the day I don't know them. Their pain is visible to me, I can usually imagine that If it were me in the same situation I would be equally sad/distraught/griefstricken. But I'm not. I walk out of that job and that's that job emotionally done. The only memories are professional ones.
~
"Thank You"
"Your Welcome". No other reply necessary. Sweet, Sincere and Simple.
~
"You boys are heroes"
No we're not.
"Arm das Land das Helden noetig hat - Unlucky the nation that needs heroes"
(Galilei, in B. Brechts "Das Leben des Galileo).
| 129,080
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News.
Bob Mould Unveils New 'Tym Sky Patch' Distortion Pedal
Ex-Husker Du and Sugar guitarist has collaborated with Australia's Tym Guitars on a new signature distortion pedal.
'How...
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This ebook by means of Michael Banner explores and makes an attempt to appreciate the importance of Christian trust for a number of modern and arguable moral concerns together with euthanasia, the surroundings, biotechnology, abortion, the kinfolk, sexual ethics, and the distribution of scarce assets for healthiness care. Its value lies in its try to exhibit the an important distinction that Christian trust makes to an knowing of those matters, whereas whilst demonstrating many of the weaknesses and confusions of sure well known ways to them.
Read or Download Christian Ethics and Contemporary Moral Problems PDF
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The Oxford History of the Crusades
During this selection of essays, the tale of the Crusades is informed as by no means ahead of in an engrossing and entire background that levels from the preaching of the 1st campaign in 1095 to the legacy of crusading beliefs and imagery that maintains this present day. listed here are the tips of apologists, propagandists, and poets concerning the Crusades, in addition to the perceptions and causes of the crusaders themselves and the capability through which prior scholarship has claimed, but additionally by means of the main neglected and underestimated ladies in eire: the nuns. as soon as regarded as simply passive servants of the male clerical hierarchy, women's spiritual orders have been in could input and within which you can see God. even though it has frequently been prompt that Augustine not directly a person else in America's founding iteration, for he had come to precise conclusions approximately how top to keep up a conventional figuring out of Christianity in a global ever altering by way of the forces of the Enlightenment.
Additional info for Christian Ethics and Contemporary Moral Problems
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Top 10 Things a Woman Must Know About Male Life Partners - Stan Onodu
Are you always angered by most of the moves made by your life partner in your relationship? Women misunderstand men in more ways than one because they find it difficult to decode men's emotional make-ups. For joy and happiness to thrive in any given relationship, there must be understanding. In this piece, I want to let every woman in on 10 most important things that her choice life partner thinks and does.
Protection. Men naturally have the disposition to protect their life partners even during ordinary dating. They'll stand against anything that'll adversely affect their spouses' physical, spiritual and emotional well-being. But as a woman, if you discovered that your man is careless about all these, then you need to think again.
Leadership. Men are born to lead. They like to act that mentorship and father figure. This cuts across into marriage and relationships. The opposite sex must recognize this. A woman should never do anything to relegate her man.
Integrity. The hallmark for real and honourable men is integrity. They like to live by their words. Their word is their bond. And so, if your life partner begins to promise you one thing and keeps failing to live up to it, that is a warning sign that he no longer values you, loves you or cares no more about you.
Respect. Honourable men work hard to earn their respect. Man has an ego to protect and maintain. And fundamentally, he expects respect from his life partner. He wants a woman with a gentle and peaceable spirit by his side. Any woman, no matter how beautiful, captivating, attractive or sensual, that lacks respect and submissiveness for her man would be depriving herself of a vibrant and healthy love.
Power & Success. These - power & success - drive the whole being of a man. Men always desire to be in-charge, in control, in command even in the home. Men also love success. Most men work hard and smart to succeed. And when they fall short of this success, they don't feel too happy with themselves or even with their life partners. You know what? Those - power & success - are one of the strong reasons women swarm around men. After all, who wants to be by the side of a failure. You know, they say failure is an orphan.
Focus. There is time for everything, so says the preacher. Business before pleasure?! Most men strive to separate pleasure from business. For most men, business comes first. At work, they come focused and concentrate on getting a good job done. When it comes to love and romance, they also give their 100%. But women hardly take time to differentiate between business and pleasure insofar as a man after their hearts is involved.
Talking Less. The women need to understand that men are not given to talking a lot. Men are after facts and so, it seems they usually keep to themselves and unwilling to explain anything unless there is need. But more often than not, this is misconstrued by the ladies to mean.... he doesn't love me. This is not absolutely correct.
Living-In. A couple of women think that the easiest way to make a man pop-up the big question - Will you marry me? - is to move in with him. This is a big mistake because any decent man will lose faith and trust in the woman in question.
Character Change. Is there a behavioural change you want to see in your life partner? And you think you can effect that change because of your feminity, beauty, sexual prowess, cooking skills or vibrant personality? I think you're wrong! You can't change a man unless it has to come from him; unless he takes that decision. Don't forget: nagging will make matters worse.
Sex. Anytime, you have sex with a man that is not your husband, you've lost some degree of value, honour and respect from him. The truth is: when a man loves a woman, he'll honour and respect her, and sex will not be uppermost in his mind. By way of testing, he may demand it from you but if you give in, know for sure that the relationship is heading for the crash.
Ladies, it's important you get to understand your life partners. Get to establish your individuality and dignity in your relationships. Love and respect yourself and I can assure you, your man will love and respect you in return.
Are you of marriageable age and you're still struggling to choose a life partner? There is a leeway for you. You can learn all the secrets here: Article Source: Article Source:
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Start Slideshow
- Alex Randall's Taxidermied Rat LampWe've seen some pretty creepy lamps here at Inhabitat -- from <a href="">blood-powered bulbs</a> to living, growing <a href="">lamps made from bioluminescent hamster cells</a>... but none of them really compare to "The Most Nightmarish Lamp Ever Created." Made by bespoke lighting designer <a href="">Alex Randall</a>, this blood-curdling creation featuring a swarm of taxidermied rats has caused quite a stir in the design world. Don't worry though - the rats were deceased long before Randall got her hands on them, meaning this light fixture makes use of <a href="">reclaimed materials</a> too. Click through our slideshow to see more of Ms. Randall's freaky preserved animal pieces.1
- Squirrel Wall LightWeird and wonderful doesn't get any weirder or more wonderful than these <a href="">squirrel wall lights</a>.2
- Squirrel Wall LightImagine what a start these would give to your friends as they walk into your home and see them for the first time!3
- Duck Light FixtureThis tongue-in-cheek <a href="">duck desk lamp</a> highlights the jewel tones naturally found on the water fowl's plumage.4
- Duck Light FixturePlus it looks as if the animal is holding the wire of the lamp in its mouth -- as if it were some magical duck helper.5
- Pigeon PendantsThese impactful <a href="">pigeon pendants</a> were first created for a store in East London. Randall points out <a href="">on her site</a> that while pigeons are considered vermin, they are actually quite beautiful with shimmering green and purple feathers adorning their necks.6
- Taxidermied Pigeon Desk LampIf you love the pigeon pendants but don't have the space to hang thirty birds, here is a <a href="">desk lamp</a> featuring a single specimen carrying a light in its beak.7
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help
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\section{Basic triangulation simplification}
\label{app: cert}
In this appendix, we expand on Section~\ref{sec: cert} and detail how
the initial triangulation $\cT$ and Pachner moves transforming it into
$\cT_0$ are constructed from the input pair $(\Tdot, \alpha)$. Our
overall goal is to minimize the number of Pachner moves and,
especially, minimize the number that involve any arcs.
The SnapPy kernel \cite{SnapPy} provides two routines for trying to
simplify a triangulation. The first is \texttt{simplify}, which
greedily does various moves that immediately reduce the number of
tetrahedra, as well as random $4 \to 4$ moves in hopes of setting up
such a reduction; it is very similar to Algorithm 2.5 of
\cite{Burton2013} which is \texttt{intelligentSimplify} in Regina
\cite{Regina}. The second is \texttt{randomize}, which first does
$4 t$ random $2 \to 3$ moves, where $t$ is the number of tetrahedra,
and then calls \texttt{simplify}; it is key for escaping local minima in
the set of triangulations. In practice, one sometimes needs
\texttt{randomize} in order to reduce a layered filling triangulation
$\cT$ to $T_0$. Because it increases the number of tetrahedra
drastically, however temporarily, we work hard to avoid applying it
when there are arcs present. We modified \texttt{simplify} and
\texttt{randomize} so that one can specify a subcomplex of the
triangulation that is to remain unchanged. Our basic strategy is:
\begin{enumerate}
\item
\label{item: go go go}
Construct the layered filling triangulation $\cT$ from $(\Tdot,
\alpha)$.
\item
\label{item: side step}
Apply \texttt{simplify} and \texttt{randomize} extensively to $\cT$
with the provisio that each tetrahedron that is the core of a
filling layered solid torus is not modified. Call the new
triangulation $\cT'$. It contains a barycentric link $L'$
consisting of the cores of the layered solid tori, which is isotopic
to the original $L$ in $\cT$.
\item If \texttt{simplify} reduces $\cT'$ to $\cT_0$, record the
sequence of Pacher moves and consider $\big(\cT', L, (P_i)\big)$ a
candidate input for the core algorithm. Otherwise, throw it away.
\item Go back to \ref{item: go go go} until we have several candidates
for $\big(\cT', L, (P_i)\big)$ or we get tired. If no candidate is
found, raise an error; otherwise output the
one where $(P_i)$ is shortest.
\end{enumerate}
Despite needing \texttt{randomize} to simplify some triangulations of
$S^3$ to $\cT_0$, so far the above has always succeeded.
Finally, it turns out the last few Pachner moves are the most
expensive, since the link is usually quite complicated at that
point. Therefore, we built a look-up table of all triangulations of
$S^3$ with fewer than five tetrahedra, along with geodesic Pachner
move sequences reducing these triangulations to $\cT_0$. If, when
searching for a sequence of Pachner moves, we reduce the initial
triangulation to one with fewer than five tetrahedra, we can look up
whether we indeed have $S^3$, and we then append to the certificate
the geodesic Pachner move sequence reducing to $\cT_0$. While this
only shortens the sequence by a few moves, it gave us a major speedup.
For further details, see Appendix~\ref{app: trust but verify}.
| 164,506
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Mattel Uno Attack® Relaunch
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Mattel Uno Attack® Relaunch
Product Description. Contents Include 112 Uno Attack Cards Including Special Command Cards 1 Card Launcher 1 Instructions Sheet For 2 10 Players Ages 7 And Up Made In China
- Base Material: 5% Other 5% Or Less, 95% Cardboard
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| 34,746
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San Luis Obispo girls volleyball defeated No. 2 Beaumont Tuesday evening, claiming its first CIF State Southern Regional title.
The Tigers recently defeated No. 4 Nordhoff in the regional semifinals for an appearance in the finals. With being the No. 1 seed in Division 4, the team received home court advantage once again.
In front of a home crowd, the Tigers punched its first ticket to the CIF-State Championship after sweeping the Cougars, 25-18, 25-22, 25-23 . They will face the northern No. 1 team, Hilmar. The game is on Saturday, November 17 at Santiago Canyon College.
| 156,260
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2015 Farmers Market Salesperson
The Humble Pie Company is looking for sales people to sell our pies at the farmers markets around Rhode Island this summer. There are multiple slots open ranging from 6-7 hours each throughout the week to accommodate this economy’s busy, gig based worker.
Who are we? In brief, we are a bakery that sells delicious, honest pies made by hand using the freshest ingredients that Rhode Island farmers and producers have to offer. Visits us online at humblepiepvd.com for more details.
Now, before you get carried away imagining swimming in a sea of free pie, let us tell it to you straight. Here goes:
Hustlin’ ain’t easy. This is a physical job. Lugging, lifting, packing, reaching, squatting and high fiving. You should be able to lift 50# or so with ease.
In spite of the grunt work, you are nevertheless juggling delicate pastries. They require care and patience in their handling.
Since you are selling quality wares, you must look the part. Neat and clean in habits, appearance and personality is necessary...at least while working.
A registered vehicle, valid driver’s license, and the ability to stow things such as: pie containers, a folding table, tent, etc are required. Vehicles need not be quite as clean as your person.
You must be super reliable. Hey, we’re a new business operating on slim margins and trying to grow. No flakes. Yep, we sell on weekends and weekdays.
We’ll ask you how you’d sell our pies. Not in a grand overall plan kind of way, but you’d better wax poetic a little bit about the power of suggestion, capitalism, local food systems, etc. Impress the dickens out of us with crazy philosophical nonsense.
Now, what is in it for you? Well being the skilled sales maven you are, we would hire you as a contractor at a per market rate plus tips. Each market shift is about 6-7 hours. At the close of each market, you reap the benefits of pie as currency and may exchange remaining pies for whatever other vendors will give you. (And honestly, you can get a lot with pie.)
Time slots available, beginning in late May:
Wednesdays afternoon
Thursdays afternoon
Saturdays morning
Interested applicants can submit their resume with a cover letter. Please include a description of your availability and vehicle photo to daniel@humblepiepvd.com
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The Nineteenth Century Problem August 15, 2011Posted by Christopher Donohue in History of Economic Thought, History of the Human Sciences.
Tags: Adam Smith, Alexis de Tocqueville, Alfred Marshall, Archibald Alison, Arnold Guyot, Arthur de Gobineau, E.B. Tylor, Emile Durkheim, Franz Boas, Fustel de Coulanges, Hans Kohn, Henry Buckle, Hippolyte Taine, Jerry Muller, John Maynard Keynes, John Ruskin, Joseph Denniker, Karl Marx, Ludwig von Mises, Martin Heidegger, Matthew Arnold, Max Weber, Mosei Ostrogorski, Philip Mirowski, R.R. Marett, W.E.H. Lecky, Walter Bagehot
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The universal historian Henry T. Buckle (1821-1862) was last subject of a serious scholarly monograph in 1958. This is the fate of any number of nineteenth-century intellectuals. The first reason for the disappearance of these writers has been the inability to connect them to the catastrophic events of the twentieth century: the World Wars, National Socialism, the deradicalization of the European right after Nuremberg, the flight of the Marxist intellectuals, and so on. Second, the nineteenth century has been the province of sociologists and literary scholars. Such attention continues to be selective, judging from the ceaseless publications on the canonical sociologists: springtime for Weber, and winter for Gobineau and Bagehot.
Third, ignoring the nineteenth century allows anthropologists to get on with their own work. Fourth, and finally, while some nineteenth century economists get attention — Alfred Marshall (1842-1924) has been accumulating more slim volumes as the months go by — the impression I get from some not so cursory reading of the literature is that the with the exception of the proponents of “evolutionary” and “heterodox” economics, philosophers of economics, and Philip Mirowski, it’s Smith, Marx, Keynes, Hayek, Mises, or monograph wilderness.
The unifying thread behind all of these reasons is that of “relevance.” It’s quite easy to pitch another book on Heidegger, since academics are still haunted by the fact that brilliant people are also terrible, immoral human beings. How philosophy can be “political” is the key to Heidegger’s “relevance”. And of course Hayek or Keynes appear daily to save us all. What about poor Buckle? How is he relevant? The answer is not an easy one, but it has everything to do with how one really should do intellectual history.
As I mentioned in my post on Joseph Denniker, the central problem confronting the historian of nineteenth and early twentieth century ideas is how nineteenth century writers related to one another. It’s not merely a question of who said what first or who formulated the problem best. Tocqueville was not the only sociologist of American exceptionalism: Arnold Guyot and Mosei Ostrogorski produced variants of their own which are each interesting in their own way.
Thus, a way to tackle the nineteenth century problem is to identify a series of authors who addressed the same issue. This has been done admirably by Jerry Muller in his Mind and the Market and his more recent, more controversial Jews and Capitalism. Muller’s work is a concept-history or history of mentalities. Hans Kohn does the same in his history of the idea of nationalism. So does the nineteenth century Irish historian W.E.H. Lecky in his two volume History of European Morals from Augustus to Charlemagne.
As needed are accounts of the development of intellectual sub-disciplines in Europe and America, particularly geography, comparative linguistics, archaeology, primitive technology and material culture, agriculture and nutrition studies, and the discovery of ancient writing systems. Who has heard of the disciplines of “economic,” “political,” or “industrial” geography, or “industrial medicine”? What precisely were the connections between these geographies and economics, industrial medicine and studies of the labor problem, particularly the conditions of the working class, and geography and the sciences of ethnology and anthropology? How do historians of manners, the arts, or of national literature, such as Hippolyte Taine, John Ruskin, and Matthew Arnold fit into this picture?
The nineteenth century problem, which extends into the first decade of the twentieth, is that we have no sense of the nineteenth century “canon.” As historians, we have no idea what people in the nineteenth century read, who or where from they really got their ideas, and how ideas adapted and changed over time.
The problem of the nineteenth century is then of the “second order” intellectuals, those thinkers who while popular or novel in the nineteenth century, or who exerted a massive influence on better known social thinkers, have disappeared from historical consciousness. These include Fustel de Coulanges and Alfred Espinas, who had a decisive influence on Marx, Engels, Kautsky, and Durkheim, among others. Even more important was Archibald Alison, the progenitor of “economic determinism” and whose multivolume history of Europe was derided (and cited) by just about everyone. Perhaps most apparent is the lack of scholarship on E. B. Tylor, R. R. Marett, Gobineau, George Gliddon, and Robert Knox, more or less any anthropologist in the nineteenth century in Europe or America other than Lewis H. Morgan. Without a firm understanding of these thinkers we cannot truly assess Franz Boas’ “revolution.”
And here may be the salvation of nineteenth century intellectuals, connecting them to twentieth century revolutions and innovations. This requires, however, that we understand what they read, why they wrote, and what they argued.
[…] intellectual movements- evolutionary positivism and historicism. However, we can now recover these 19th century contexts on this […]
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TITLE: trace and involution permutations: Part II
QUESTION [3 upvotes]: This is a follow up on my earlier MO question.
Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $I_n:=\#\operatorname{Inv}(\mathfrak{S}_n)$. Let $\operatorname{tr}(\pi)$ be the number of fixed points of a permutation $\pi$. Call the functions $B_n(z)=\sum_{j=0}^nS(n,j)z^j$ as Bell polynomials; where $S(n,j)$ are Stirling numbers of the 2nd kind.
Some experimental evidence convinces me that it is possible to express of the following.
Question. Fix an integer $k\geq1$. Does this exponential generating function hold true?
$$\sum_{n=1}^{\infty}\frac{z^n}{n!}\sum_{\pi\in \operatorname{Inv}(\mathfrak{S}_n)}\operatorname{tr}(\pi)^k=B_k(z)\cdot e^{z+\frac12z^2}.$$
Note. The case $k=0$ is well-known. Moreover, $\sum_{\pi\in \operatorname{Inv}(\mathfrak{S}_n)}1=\sum_{\lambda\vdash n}f_{\lambda}$ where $f_{\lambda}$ is the number of SYT of shape $\lambda$.
REPLY [5 votes]: A generating function proof can be given as follows. First we can take a sum over a second variable, so we need to prove
$$\sum_{k,n=1}^{\infty}\frac{z^nt^k}{n!k!}\sum_{\pi\in \operatorname{Inv}(\mathfrak{S}_n)}\operatorname{tr}(\pi)^k=\sum_{k=1}^{\infty}\frac{t^k}{k!}B_k(z)\cdot e^{z+\frac12z^2}.$$
The left hand side is
$$\sum_{n=1}^{\infty}\frac{z^n}{n!}\sum_{\pi\in \operatorname{Inv}(\mathfrak{S}_n)}e^{t\operatorname{tr}(\pi)}=e^{e^tz+\frac{z^2}{2}}$$
by the exponential formula. (We are counting cycles of length 1 with weight e^t, length 2 with weight 1, and every other cycle length with weight 0.)
On the other hand the generating function for Bell polynomials is $e^{(e^t-1)z}$ so the right hand side evaluates to
$$e^{(e^t-1)z}\cdot e^{z+\frac{z^2}{2}}=e^{e^tz+\frac{z^2}{2}},$$
as desired.
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TITLE: How is $a_n=(1+1/n)^n$ monotonically increasing and bounded by $3$?
QUESTION [1 upvotes]: I was reading about how completeness is required for limits. And I came across this:
the sequence $a_n=(1+1/n)^n$ is monotonically increasing and bounded by 3 and so we expect it to converge, but that it does not converge within $\mathbb{Q}$. More generally it stands to reason that any sequence of real numbers which is increasing and bounded must converge to some real number. This is a consequence of completeness of $IR$
My question is: How is the mentioned sequence monotonically increasing and bounded by $3$ ?
REPLY [0 votes]: From Rudin's PMA Theorem $3.31$, by the Binomial theorem,
$$ t_n=\left( 1 + \frac{1}{n} \right)^n $$
$$= 1 + 1 + \underbrace{\frac{1}{2!}\left(1-\frac{1}{n}\right)}_{\text{term } 2} + \underbrace{\frac{1}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)}_{\text{term } 3} + \ldots + \frac{1}{n!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\cdots \left(1-\frac{n-1}{n}\right). $$
Since each bracket increases as $n$ increases, each term increases as $n$ increases also, because products and sums of increasing positive functions is also increasing.
Therefore, for all $n\geq 2,$ and all $2\leq k \leq n,$ the $k$-th term of $t_{n+1}$ is greater than the $k$-th term of $t_n.$
So we see that $\left( 1 + \frac{1}{n} \right)^n$ is increasing.
Furthermore, the Binomial expansion above is
$$ \leq 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \ldots + \frac{1}{n!} = 1 + 1 + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \frac{1}{1\cdot 2\cdots n} $$
$$ < 1+1+ \frac{1}{2} + \frac{1}{2^2} + \ldots + \frac{1}{2^{n-1}} <3. $$
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TITLE: A novelty integral for $\pi$
QUESTION [34 upvotes]: My lab friends always play a mentally challenging brain game every month to keep our mind running on all four cylinders and the last month challenge was to find a novelty expression for $\pi$. In order to stick to the rule, of course we must avoid the good old Ramanujan and online available expressions, for instance: the coolest ways of expressing $\pi$ on Quora. The winner of the last month challenge is this integral
$${\large\int_0^\infty}\frac{(1+x)\log(1+x)(2+\log x)\log\left(\!\frac{1+x}{2}\!\right)-2x\log(1+x)\log x}{x^{3/2}(1+x)\log^2x}\ dx={\Large\pi}$$
The equality is precise to at least thousand decimal places. Unfortunately, my friend who proposes this integral keeping the mystery to himself. I tried to crack this integral while waiting for a solution to be offered by one of my friends, but failed to get any.
I have tried to break this integral into two part:
$${\large\int_0^\infty}\frac{\log(1+x)(2+\log x)\log\left(\!\frac{1+x}{2}\!\right)}{x^{3/2}\log^2x}\ dx-2{\large\int_0^\infty}\frac{\log(1+x)}{\sqrt{x}(1+x)\log x}\ dx$$
but each integrals diverges. I have tried many substitutions like $x=y-1$, $x=\frac{1}{y}$, or $x=\tan^2y$ hoping for familiar functions, but couldn't get one. I also tried the method of differentiation under integral sign by introducing
$$I(s)={\large\int_0^\infty}x^{s}\cdot\frac{(1+x)\log(1+x)(2+\log x)\log\left(\!\frac{1+x}{2}\!\right)-2x\log(1+x)\log x}{(1+x)\log^2x}\ dx$$
and differentiating twice with respect to $s$ to get rid of $\log^2x$ couldn't work either. I have a strong feeling that I miss something completely obvious in my calculation. I'm not having much success in evaluating this integral since two weeks ago, so I thought it's about time to ask you for help. Can you help me out to prove it, please?
REPLY [29 votes]: The integrand can be broken up as
$$I=\int_0^{\infty} \left(\frac{2 \ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{3/2} \ln^2 x} +\frac{\ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{3/2} \ln x}-\frac{2\ln(1+x)}{x^{1/2} (1+x) \ln x}\right)dx.$$
But, by integration by parts,
$$\int \frac{2\ln(1+x)}{x^{1/2} (1+x) \ln x} dx= \int \frac{2\ln(1+x)}{x^{1/2} \ln x} d\left(\ln\left(\frac{1+x}{2}\right)\right)
\\=\small\frac{2 \ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{1/2} \ln x}-2\int \left( \frac{\ln\left(\frac{1+x}{2}\right)}{x^{1/2} (1+x) \ln x}-\frac{ \ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{3/2} \ln^2 x}-\frac{\ln(1+x)\ln\left(\frac{1+x}{2}\right)}{2 x^{3/2} \ln x}\right)dx$$
That is,
$$\int \left(\frac{2 \ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{3/2} \ln^2 x} +\frac{\ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{3/2} \ln x}-\frac{2\ln(1+x)}{x^{1/2} (1+x) \ln x}\right)dx \\= -\frac{2 \ln(1+x)\ln\left(\frac{1+x}{2}\right)}{x^{1/2} \ln x}+2 \int \frac{\ln\left(\frac{1+x}{2}\right)}{x^{1/2} (1+x) \ln x} dx$$
And the claim follows from Jack D'Aurizio's preliminary result.
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Welcome To
Safety Harbor Elementary
Safety Harbor Elementary, a true community school with a 88-year tradition of academic excellence, remains one of the most charming schools in Pinellas County.
Dedicated teachers, committed parents and willing members of an active community work together as a team to give Safety Harbor students every opportunity for academic success and a feeling of positive self worth.
Thank you for visiting our web site and we hope you enjoy your visit!
Welcome our New Principal
Cecilia Palmer
Classroom Partners for 2014-2015 school year
Kiwanis Club of Safety Harbor
Mathnasium
Barbara & John Duffy
The Scott Family
Salon West/Jill Somers
Frank & Katherine Mann
Matthew & Valerie Brown
The Beiter Family
Keith & Shelly Hurley/ASH Services
Bill & Lorraine Mazza
Clearwater Toyota
The Cabello Family
Lucile Casey/Berkshire Hathaway Home Services
Thank you!
Classroom Supply Lists
2014/2015
Campus Access for Parents & Visitors
Safety Harbor Elementary
2014-15
Effective immediately, all parents and visitors wishing to gain access to our
campus must adhere to the following procedures:
All visitors must sign in at the Front Office (No Exceptions)
All visitors are required to wear the appropriate issued Identification Badges
Visitors will only be granted campus access to specified areas for intended business
Any visitor will be stopped at the Front Office and denied campus access without specific business and arrangements
Thank you for your support in helping to keep our
Campus a Safe Learning Environment and a closed campus.
School Public Accountability Reports (SPAR Reports)
The annual school report summarizing our school’s results for last school year 2012-13.
Classroom Partners
Thank you to all our Classroom Partners in 2013/2014.
Classroom Partner Brochure
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TITLE: Prove that the sequence $(-1)^n (-n)^2/(9n^2)$ does not converge to a specific number
QUESTION [1 upvotes]: Prove that $\{(-1)^n (-n)^2/(9n^2)\}$ does not converge to $\frac{1}{9}$ or $\frac{-1}{9}$, I've already chosen an epsilon whats the next step. DO NOT SOLVE ALL THE WAY! Do I chose an $N$?
REPLY [1 votes]: No, you don't get to choose an $N$. The negation of "$x_n\to a$" is: there is $\epsilon>0$ such that for every $N$ there is $n>N$ for which $|x_n-a|\ge \epsilon$. This means:
you get to choose $\epsilon$
you have no control on $N$
you have to come up with $n$ such that $n>N$ and $|x_n-a|\ge \epsilon$.
It seems that one of $n=N+1$ and $n=N+2$ ought to work, if your choice of $\epsilon$ was good.
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Invisible Motion in Video
Magnifying hidden movement
About this video
Finding the Visible in the Invisible.
A team from the Massachusetts Institute of Technology (MIT) has developed a computer program that reveals colours and movement invisible to the naked eye.
Known as Eulerian Video Magnification the amplification process reveals imperceptible changes including small motions and colour shifts in a baby's face in response to their heart beat.
The resaerch team have suggested a range of applications for the video technology, from rescuers revealing if someone is breathing to identifying tiny movements in machinery indicating possible breakdown.
Themes
Details
Licence: Standard YouTube License
Related Links and Media
Related Videos
Video, 02:22
Chromosome 23 (X and Y): Colour blindness
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An analysis of the validity and reliability of risk assessment instruments for making predictions of violence appropriate for the employee’s assessment
For essays Guru need tonight by midnight central time$100
RA 2: Written Report
You 10- to 12-page report addressing the following:
-
In addition, in your report, identify at least one potential victim of the employee. Make a recommendation for action by the company to protect that victim. Prepare your findings and recommendations in such a way that the employee cannot effectively sue the company for slander. An axiom of law is that negative statements do not constitute slander if they can be supported. Your job is to write the report in such a way (with supportable conclusions) that the company can establish that its actions were prudent.
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2020 Virtual Educational Family Camp & Adult Retreat
“Keeping It Coastal”
September 25th through October 3rd
We are excited to welcome you to come spend time together virtually with friends and community members throughout South Carolina for our Annual Educational Family Camp Weekend and Adult Retreat. We are working hard to provide the same weekend experience we have all come to love in a virtual setting, spread out over a week, allowing many opportunities for connecting. We will be having event sessions for families and adults, caregivers & couples, advocates, individuals affected by VWD, and kids! Enjoy a virtual experience in a box and through your computer, Ipads, tablets, or phone. While it’s not the same as being together in person, we invite you to enjoy a wonderful time connecting with community members until we are together again. Please register to join us. “See you soon” !
Pre-Register here!-Closed
Pre-Registration Deadline Date: September 11, 2020 (to receive the family camp in a box enhancements and goodies)
Once you pre-register, watch for your “Beach in a Box” to arrive at your home. Next, you will be asked to scroll down this page and register again for each session you are planning to attend. The zoom link will be sent to your email. Then on the session date, you will log into the conference and enjoy!
Family Camp Invite and Registration Information
- Online Pre- Registration Link- closed
- Event Flyer
- Family Camp Virtual Agenda Program
- Photo release Information
- How to use Zoom with your Phone
- How to use Zoom with your Tablet or IPAD
- How to Use Zoom with a Laptop event, you agree to our photo and events waiver and Zoom policies and disclaimer release.
Schedule of Events:
Friday, September 25th – 6:30pm – Opening Night Welcome Session
Bombardier Blood
Register here!
Join the Chapter and your community for pizza, popcorn, camp kahoot trivia, sponsors welcomes, and a showing of Bombardier Blood – 7:30pm Viewing. The viewing will be done on our Zoom platform.
Order pizza to hang out with your family and friends while viewing this wonderful documentary with our community and receive reimbursement for your pizza upon receipts being sent to the chapter. Up to $40.00 per household will be reimbursed. Then, join us throughout the week for our many sessions for everyone!
About BOMBARDIER BLOOD
Chris Bombardier has never let severe hemophilia stop him from climbing some of the world’s tallest mountains. In 2017, Chris partnered with hemophiliac filmmaker Patrick James Lynch and his award-winning production team at Believe Limited to film his journey through Nepal to summit the world’s tallest peak, Mount Everest.
In his home state of Colorado, filming captured Chris’s training, interviews with Chris, his wife and family, and his hemophilia clinicians. In Nepal, production followed Chris to meetings with Nepalese hemophilia advocates and clinicians as well as to the homes of patients and families affected by the disease, where Chris heard the emotional stories of pain, suffering, and loss related to hemophilia. Production remained with Chris during his two months on Everest, capturing the physical, psychological, and emotional struggle of acclimating to the mountain and preparing for the greatest challenge of his life, all with the eyes of the global hemophilia community watching.
Leveraging powerful archival footage, the film also chronicles Chris’s incredible six-year journey of climbing the Seven Summits, the highest mountain on each continent, during which he receives his “wake-up call” about hemophilia in developing countries and begins his activist mission.
In Nepal, as in nearly all developing nations, there is no routine access to healthcare and lifesaving medicine for people suffering from hemophilia. Bombardier Blood is an inspiring and heart-warming adventure film that cinematically highlights both what is and is not possible when living with this rare disease, depending on a patient’s access to medicine and care.
Saturday, September 26th – 10am – 2pm – Advocacy Action Workshop
Join HSC for its Keeping it Coastal advocacy session. We’ll be having an informative training session, hear from SC Legislator Jason Elliott (SCHR-22) and others during our Town Hall meeting while discussing our concerns of the co-pay accumulator adjuster programs implemented in insurance plans and what the up coming legislative session will look like. Learn the ins and outs of insurance lingo and acronyms (co-pay, deductible, new words for 2021), understand our state resources, and hear some national updates. Most importantly, we’ll be preparing for our 2021 (possible virtual) legislative session and how WE can represent the Bleeding Disorders Community of South Carolina assuring access to care and treatment. This session is available to all members of HSC, our Coalition members and our Ambassadors!
Get involved in advocacy and YOU can make a difference for us all!
Register Here!
Tuesday, September 29th – 7:00 pm – Adult Retreat Session : “Planning Your Future”, Sponsored by Pfizer
Join us for an overview of the potential ways adults with bleeding disorders can plan for their financial futures. Emphasis is on defining retirement goals and identifying future needs, aspects of long-term care, and the benefits of a support network. Then, join the conversations with our special guest, Kyla Capers from the South Carolina Department of Consumer Affairs who will further discuss financial literacy covering saving and investing while planning your future.
Register Here!
Financial Planning for the Future Slides
Thursday, October 1st – 6:30 pm – VWD Education and Support: OBGYN Plus, Sponsored by Octapharma
Learning more about VWD is important, so we have invited our guests in VWD education to join us for conversations about this complicated disease. Help HSC and the VWD community as we participate in our global call to action to raise awareness and education about VWD. Learn about OBGYN education with VWD and share your knowledge with your participation! AND… enjoy a little fall holiday fun art project in the company of your VWD friends! (Hint: Burlap and Ribbons, You must pre-register for the conference (above) before September 11th to receive supplies)
Our Guest Speaker:
Speaker: Dr. Amber Federizo, APRN, FNP-BC
Nurse Practitioner Hemostasis and Family Board Certified
Hemostasis and Thrombosis Center of Nevada
Register Here!
Friday, October 2nd – 7:00pm – Couples & Caregivers Night Out: “Indulge” Sponsored by Genentech
COVID fatigue, homeschooling fatigue, media fatigue, caregiver fatigue, lists about fatigue, we hear you. Join us for ” A Night out for Couples and Caregivers. Enjoy a special viewing of the first episode of the YouTube series CHALLENGED ACCEPTED! CHALLENGE ACCEPTED is a series, created by Genentech, inspired by and made for the hemophilia community. Two boys with hemophilia A challenge host Justin Willman to take their parents out for the best date night ever. Guest starring Chef Graham Elliot. This episode talks about keeping relationships strong when raising children with bleeding disorders and taking care of oneself. Following the show we’ll be having a discussion together on self-care and ways to infuse some relaxation back into your life. This session is for caregivers too! (Pre-register for the conference (above) before September 11th to receive some indulging enhancements supplies)
Register Here!
Saturday, October 3rd – Kids Day, Family Sessions, Mental Health, Blood Brothers & Closing Retreat Sessions
Register Here!
(you will only need one login meeting ID and password for the entire day. Just login as ofter as you like throughout the day to attend different sessions)
10:00am – Chapter Coffee Hour!
After a long week of fun, support, and education, bring your coffee or favorite morning beverage and let’s hang out for some casual conversations. Share your support and knowledge with community members just like we did at camp last year in person! Learn about the “Turkey Trot Walk / Run” and join the fun. Help us reach out goal.
10:30am – Parents and Kids: A visit with the South Carolina Aquarium – “Critter Calls Ocean Tour” ~ Sharks, Stingrays, and more!
Before we have a tour with the SC Aquarium, kids will share their favorite beach toys, pets, stuffed animals, and beach stories with their friends from our community, building bonds for their futures. (Critter Calls Ocean Tour begins at 11:00am sharp) Following this session, parents will enjoy sharing conversations with discussions on going back to school, what you are discovering, and what you should know about children in school with bleeding disorders.
1:30pm – Building Connections: Seeking Support in a Time of Social Distance, Sponsored by Takeda
The worldwide pandemic has created unforeseen challenges for our health and well-being. Learn how to manage a bleeding disorder amidst shelter-in-place and social distancing mandates while building and maintaining supportive relationships.
4:00pm – Let’s Talk Mental Health with Patrick James Lynch, Sponsored By Sanofi Genzyme
Join Patrick James Lynch and Believe Limited, and the team behind Bombardier Blood, as we premier their new documentary about mental health in the bleeding disorders community. Let’s Talk is an immersive journey through the lives of five members of the U.S. bleeding disorders community, as they open their hearts and lives to show how we can gain strength through struggle, and that perhaps we aren’t so different after all. Produced in partnership with Mental Health Matters Too, the film is intended to spark conversation, increase awareness, and decrease stigma. Information to their website is intended to provide easy-to-use links, resources, and tools for connection, screening, and receiving support. You don’t want to miss this session!
View Informational Flyer Here! Content and Speakers
View Support Materials Here! Mental Health Matters Too – Resource Guide for the Bleeding Disorders Community
7:00pm – Closing Event– HSC Family Bingo Night, Turning the Stumbling Blocks into Stepping Stones with Perry Parker, Sponsored by CSL Behring
After a wonderful week of educational and supportive information, join us for the final night at camp and experience the joy of gathering together, talking with other community members while breaking out into groups and sharing some helpful hits and knowledge with each other. Thank our sponsors and hear from Perry Parker, CSL Behring’s Community Advocate and professional PGA Golfer who lives with hemophilia and shares his life stories promoting sports, exercise, and healthy lifestyles. Finish the night with us in a family fun game of “BINGO” ZOOM Style! Win prizes just for playing and attending. We are going to have a blast!
Download Extra Bingo Cards and Print Here!
Learn more about our speaker, Perry Parker!
————————————————————————————————————————————————————————-
Save the Date! We will return to the Marina Inn on September 24-26, 2021
Marina Inn at Grande Dunes
8121 Amalfi Place, Myrtle Beach, SC.
This annual recreational and educational event is available to individuals and families who either have a bleeding disorder, are carriers of a bleeding disorder, or have an immediate family member living in the same household diagnosed with a bleeding disorder. Our Family Camp and Adult Retreat is held in the Lowcountry over a three-day weekend and provides education and support to our community members who live in this region, while also providing all members a coastal venue experience.
What is the Family Camp – Adult Retreat?
The Family Camp and Adult Retreat provides community members with an extended opportunity to connect with others who are experiencing the same journeys while living with a bleeding disorder. Participants learn that they experience similar situations living with bleeding disorders, yet in different ways. For children and teens, they connect in a way that is unique to them; knowing they are not the only one with a bleeding disorder, and for a brief time, they can connect and forget what brings them to camp. They are among friends who just “get it”. Oftentimes, they make lifetime friendships. For the older adults, it is a special time to reconnect and find support in sharing their experiences and providing comfort to one another in friendship. Families leave with a renewed feeling of community and empowerment knowing they are not alone.
What do you get out of it?
We provided educational opportunities, empowerment, group activities, and supportive family fun connections with members of the South Carolina bleeding disorders community. Family Camp is attended by approximately 225 members and supportive affiliates; such as, guest speakers, national organizations, and healthcare / Industry representatives who are affiliated with the bleeding disorders community. Educational exhibit hall opportunities are offered for participants throughout the weekend to provide informational materials and one-on-one conversation opportunities on the most updated treatment therapies and support services.
Come Join Us!
The weekend is free of charge with a small reservation deposit. All meals, lodging, and activities are included. Space is limited and registration with HSC is required. Notification will be sent once an application is accepted. Applications are accepted on a first come / first serve basis, and a need-based weighted selection process will be implemented if we cannot accommodate all eligible applications. Register to join us and “Keep it Coastal”.
The Moby Dick Sponsors
The Great White Shark Sponsors
Educational Sponsors
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Please join us Thursday, March 7th We are looking forward to seeing you all on Thursday for our Great Start Collaborative Quarterly Meeting. Please R
We are looking forward to seeing you all on Thursday for our Great Start Collaborative Quarterly Meeting. Please RSVP by Wednesday at 8am for lunch and childcare needs.
In addition to working in our smaller Strategic Teams, we are excited to welcome Beth Washington, System Director of Community Health, Equity and Inclusion at Bronson Healthcare. Beth will guide us through "Together, We Create Health Equity." Please come prepared to participate fully as we have a lot of work to do to create success for families!
We will have a light lunch and networking from 12:30-1:00pm. The meeting will take place 1:00-3:00pm.
Our meeting will take place at Kalamazoo RESA West Campus, 4606 Croyden Ave., Kalamazoo. Please enter through the PET doors, entrance #4 on the east side of the building.
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\begin{document}
\begin{frontmatter}
\title{Open-source tools for dynamical analysis of Liley's mean-field cortex model}
\author{Kevin R. Green}
\author{Lennaert van Veen}
\address{Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, L1H 7K4 Ontario, Canada}
\begin{abstract}
Mean-field models of the mammalian cortex treat this part of the brain as a two-dimensional excitable medium.
The electrical potentials, generated by the excitatory and inhibitory neuron populations, are described by nonlinear, coupled,
partial differential equations, that are known to generate complicated spatio-temporal behaviour. We focus on the
model by Liley {\sl et al.} (Network: Comput. Neural Syst. (2002) 13, 67-113). Several reductions of this model have
been studied in detail, but a direct analysis of its spatio-temporal dynamics has, to the best of our knowledge, never
been attempted before. Here, we describe the implementation of implicit time-stepping of the model and the tangent linear model,
and solving for equilibria
and time-periodic solutions, using the open-source library PETSc. By using domain decomposition for
parallelization, and
iterative solving of linear problems, the code is capable of parsing some dynamics of a macroscopic slice of
cortical tissue with a sub-millimetre resolution.
\end{abstract}
\begin{keyword}
Mean-field modelling \sep hyperbolic partial differential equations \sep
numerical partial differential equations \sep
35Q92 \sep
65Y05
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:intro}
Models of cortical dynamics come in two main families: network models and mean-field models. The former describe many interacting
neurons, each with their own dynamical rules, while the latter describe electrical potentials, generated collectively
by many neurons, as continuous in space and time. These potentials can be thought of
as averages over a number of macrocolumns, groups of hundreds of thousands of neurons in columnar structures at the
surface of the cortex. The reason to abandon the description of individual
neurons and pass to the mean-field limit, in analogy to the thermodynamic limit in statistical physics, is twofold.
Firstly, the description and analysis of a substantial piece of the cortex with a network model is not tractable since it would contain
billions of neurons, and many times more connections between them. As demonstrated by recent publications,
such as by Izhikevich and Edelman \cite{izhi} or by the Blue Brain team \cite{bluebrain}, progress in super computing allows
for the simulation of ever larger neuronal networks, that reflect actual brain dynamics. However, it is hard
to see how the output of such models can be analysed, other than by purely statistical techniques.
In contrast, mean-field models can be analysed as infinite-dimensional dynamical systems.
The second advantage of the mean-field approach is that the electrical potentials, which appears as dependent
variables, are directly related to the electroencephalograph (EEG) \cite{Nunez2006}. The EEG
is usually measured with electrodes on the scalp or, in exceptional circumstances, directly on the surface of
the brain. In either case, the measured signal is not that of individual neurons, but that of many neurons, spread out
over a few square centimetres or millimetres. Thus, the way the signals of
individual neurons are smeared out by the spatial averaging of mean-field modelling is similar to the way they are mixed up in EEG
measurements.
Because of the direct link between
the local mean potential and the EEG, mean-field models are sometimes called EEG models.
The origin of mean-field modelling lies in the nineteen seventies, when pioneers like Walter Freeman \cite{Freeman1975},
Wilson \& Cowan \cite{Wilson1972} and Lopes da Silva \cite{LopesdaSilva1974} started to model components
of the human cortex with continuous fields. Over the past four decades, mean-field models have been used
to study a range of open questions in neuroscience, such as the generation of the alpha rhythm,
8-$13\,\mathrm{Hz}$
oscillations in the EEG (see, e.g., \cite{LopesdaSilva1974,Nunez1974}), epilepsy (see, e.g., \cite{breakspear,kramer,blenkinsop})
and anaesthesia \cite{Bojak2005}.
Also, they are used in models for sensorimotor control, pattern discrimination and target tracking \cite{quinton}.
Although mean-field models have been used in all these contexts, little analysis has been done on
their behaviour as spatially extended dynamical systems. In part, this is due to their staggering
complexity. The Liley model \cite{Liley2002} considered here, for instance, consists of fourteen
coupled Partial Differential Equations (PDEs) with strong nonlinearities, imposed by coupling between
the mean membrane potentials and the mean synaptic inputs. The model can be reduced to a system of
Ordinary Differential Equations (ODEs) by considering only spatially homogeneous solutions, and the
resulting system has been examined in detail using numerical bifurcation analysis
(see \cite{Frascoli2011} and references therein). In order to compute equilibria, periodic orbits and such
objects for the PDE model, we need a flexible, stable simulation code for the model and it linearization
that can run in parallel to scale up to a domain size of about $2500\,\mathrm{cm}^2$, the size of
a full-grown human cortex. We also need efficient, iterative solvers for linear problems with
large, sparse matrices. In this paper, we will show that all this can be accomplished in the
open-source software package PETSc \cite{petsc-user-ref}. Our implementation consists of
a number of functions in C that will be available publicly \cite{public}.
The goal of this computational work is similar to that of Coombes {\sl et al.}, who analysed
``spots'': rotationally symmetric, localized solutions in a model of a single neuron population
in two dimensions \cite{coombes}. The study of such special solutions will help us parse the
spatio-temporal dynamics of mean-field models. We will attempt to compute periodic orbits
and other special solutions in a full-fledged, two-population mean field model without imposing
any spatial symmetries.
\subsection{Liley's model}\label{model}
The model we use was first proposed by Liley {\sl et al.}~in 2002 \cite{Liley2002}.
The dependent variables are the mean inhibitory and excitatory membrane potential,
$h_i$ and $h_e$, the four mean synaptic inputs, originating from either population
and connecting to either, $I_{ee}$, $I_{ei}$, $I_{ie}$ and $I_{ii}$, and the excitatory
axonal activity in long-range fibers, connecting to either population, $\phi_{ee}$ and $\phi_{ei}$.
The model equations are
\begin{equation} \label{eq:Liley1}
\tau_k \frac{\partial h_k(\vec{x},t)}{\partial t} = h^r_k - h_k(\vec{x},t)
+ \frac{h^{eq}_{ek}-h_k(\vec{x},t)}{\left|h^{eq}_{ek}-h^r_e\right|}I_{ek}(\vec{x},t)
+ \frac{h^{eq}_{ik}-h_k(\vec{x},t)}{\left|h^{eq}_{ik}-h^r_e\right|}I_{ik}(\vec{x},t)
\end{equation}
\begin{equation} \label{eq:Liley2}
\left(\frac{\partial}{\partial t} + \gamma_{ek}\right)^2 I_{ek}(\vec{x},t) =
e \Gamma_{ek}\gamma_{ek}\left\{N^\beta_{ek}S_e\left[h_e(\vec{x},t)\right]
+p_{ek}+\phi_{ek}(\vec{x},t)\right\}
\end{equation}
\begin{equation} \label{eq:Liley3}
\left(\frac{d}{dt} + \gamma_{ik}\right)^2 I_{ik}(\vec{x},t) =
e \Gamma_{ek}\gamma_{ek}\left\{N^\beta_{ik}S_i\left[h_i(\vec{x},t)\right]
+p_{ik}\right\}
\end{equation}
\begin{equation} \label{eq:Liley4}
\left[\left(\frac{\partial}{\partial t}+v\Lambda\right)^2
-\frac{3}{2}v^2\nabla^2\right]\phi_{ek}(\vec{x},t) =
N^\alpha_{ek}v^2\Lambda^2 S_e\left[h_e(\vec{x},t)\right]
\end{equation}
\begin{equation} \label{eq:Liley5}
S_k\left[h_k\right] = S^{max}_k\left(
1 + \exp\left[-\sqrt{2}\frac{h_k-\mu_k}{\sigma_k}\right]
\right)^{-1}
\end{equation}
where index $k=\{e,i\}$ denotes excitatory or inhibitory. The meaning of the parameters, along with
some physiological bounds and the values used in our tests, are given in Table \ref{table:pars}. A detailed description
of these equations can be found in references \cite{Liley2002,Frascoli2011}. Here, we will focus
on the aspects of the model most relevant for the numerical implementation.
There are two sources of nonlinearity, related to the coupling of the synaptic inputs to the membrane
potential and vice versa. The former connection is quadratically nonlinear, while the latter is given
by the sigmoidal function $S$, which describes the onset of firing as the potential exceeds the threshold
value $\mu_{i,e}$. These nonlinearities tend to form sharp transitions of the potentials across the domain.
That is one reason why we opted for a finite-difference discretization over a pseudo-spectral approach.
Spectral accuracy would be of limited value in the presence of steep gradients and the finite-difference
scheme can be parallelized much more efficiently. The second reason is that we would like to be
able to change the geometry of the domain and the boundary conditions in future work. The finite-difference
scheme is more flexible in that respect.
The only spatial derivatives in the model are those in the equations for the long-range connections.
These are damped wave equations. We will discretize the Laplacian using a five-point stencil on a rectangular
grid. In previous work, Bojak \& Liley chose a second-order centered difference scheme for the time
derivatives \cite{Bojak2005}. A disadvantage of this approach is that the stability condition of this scheme dictates that we
set the time step inversely proportional to the grid spacing. In practice, they used a time step of $0.05\,\mathrm{ms}$.
To avoid this obstacle, we implemented the unconditionally stable implicit Euler method, as described in Sec.~\ref{timestepping}.
Other authors have used this model with an additional diffusive term in the equations for the membrane
potentials to model gap junctions \cite{steynross}. Inclusion of these terms can drastically change
the bifurcation behaviour, as they can cause Turing transitions to space-dependent equilibria.
Without the additional terms, a Hopf bifurcation from a spatially homogeneous
equilibrium to a space dependent periodic orbit or a saddle-node bifurcation of this equilibrium
can be the primary instability. The gap junction terms can readily be included in our implementation, and in Sec.~\ref{equilibria}
we will describe how to solve for equilibrium states that may depend on space.
\begin{sidewaystable}[htbf]
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Parameter & Definition & Minimum & Maximum & Value & Units \\
\hline
$h^r_e$ & resting excitatory membrane potential & $-80$ & $-60$ & -72.293 & mV\\
$h^r_i$ & resting inhibitory membrane potential & $-80$ & $-60$ & -67.261 & mV\\
$\tau_e$ & passive excitatory membrane decay time & $5$ & $150$ & 32.209 & ms\\
$\tau_i$ & passive inhibitory membrane decay time & $5$ & $150$ & 92.260 & ms\\
$h^{\mathrm{eq}}_{ee}$ & excitatory reversal potential & $-20$ & $10$ & 7.2583 & mV\\
$h^{\mathrm{eq}}_{ei}$ & excitatory reversal potential & $-20$ & $10$ & 9.8357 & mV\\
$h^{\mathrm{eq}}_{ie}$ & inhibitory reversal potential & $-90$ & $ h^r_k-5$ & -80.697 & mV\\
$h^{\mathrm{eq}}_{ii}$ & inhibitory reversal potential & $-90$ & $ h^r_k-5$ & -76.674 & mV\\
$\Gamma_{ee}$ & EPSP peak amplitude & $0.1$ & $2.0$ & 0.29835 & mV\\
$\Gamma_{ei}$ & EPSP peak amplitude & $0.1$ & $2.0$ & 1.1465 & mV\\
$\Gamma_{ie}$ & IPSP peak amplitude & $0.1$ & $2.0$ & 1.2615 & mV\\
$\Gamma_{ii}$ & IPSP peak amplitude & $0.1$ & $2.0$ & 0.20143 &mV\\
$\gamma_{ee}$& EPSP characteristic rate constant$^\ddagger$ & $100$ & $1,000$ & 122.68 & $\mathrm{s}^{-1}$\\
$\gamma_{ei}$& EPSP characteristic rate constant$^\ddagger$ & $100$ & $1,000$ & 982.51 & $\mathrm{s}^{-1}$\\
$\gamma_{ie}$& IPSP characteristic rate constant$^\ddagger$ & $10$ & $500$ & 293.10 & $\mathrm{s}^{-1}$\\
$\gamma_{ii}$& IPSP characteristic rate constant$^\ddagger$ & $10$ & $500$ & 111.40 & $\mathrm{s}^{-1}$\\
$N^\alpha_{ee}$ & no.\ of cortico-cortical synapses, target excitatory & $2000$ & $5000$ & 3228.0 & --\\
$N^\alpha_{ei}$ & no.\ of cortico-cortical synapses, target inhibitory & $1000$ & $3000$ & 2956.9 & --\\
$N^\beta_{ee}$ & no.\ of excitatory intracortical synapses & $2000$ & $5000$ & 4202.4 & --\\
$N^\beta_{ei}$ & no.\ of excitatory intracortical synapses & $2000$ & $5000$ & 3602.9 & --\\
$N^\beta_{ie}$ & no.\ of inhibitory intracortical synapses & $100$ & $1000$ & 443.71 & --\\
$N^\beta_{ii}$ & no.\ of inhibitory intracortical synapses & $100$ & $1000$ & 386.43 & --\\
$v$ & axonal conduction velocity & $100$ & $1,000$ & 116.12 & $\mathrm{cm}\,\mathrm{s}^{-1} $\\
$1/\Lambda$ & decay scale of cortico-cortical connectivity & $1$ & $10$ & 1.6423 & cm \\
$S^{\mathrm{max}}_e$ & maximum excitatory firing rate & $50$ & $500$ & 66.433 & $\mathrm{s}^{-1}$ \\
$S^{\mathrm{max}}_i$ & maximum inhibitory firing rate & $50$ & $500$ & 393.29 & $\mathrm{s}^{-1}$ \\
$\mu_e$ & excitatory firing threshold & $-55$ & $-40$ & -44.522 & mV \\
$\mu_i$ & inhibitory firing threshold & $-55$ & $-40$ & -43.086 & mV \\
$\sigma_e$ & standard deviation of excitatory firing threshold & $2$ & $7$ & 4.7068 & mV \\
$\sigma_i$ & standard deviation of inhibitory firing threshold & $2$ & $7$ & 2.9644 & mV \\
$p_{ee}$ & extracortical synaptic input rate & $0$ & $10,000$ & 2250.6 & $\mathrm{s}^{-1}$ \\
$p_{ei}$ & extracortical synaptic input rate & $0$ & $10,000$ & 4363.4 & $\mathrm{s}^{-1}$ \\
\hline
\end{tabular}
\caption{{\bf Meaning, ranges and values for the model parameters.}
The values used for the tests presented in Sec. \ref{examples} are taken from reference \cite{Bojak2007}.}
\label{table:pars}
\end{sidewaystable}
We will test our implementation by comparing to, and extending, the computations of oscillations
with a $40\,\mathrm{Hz}$
component by Bojak \& Liley \cite{Bojak2007}. The corresponding parameter values
are listed in Table~\ref{table:pars}. The $40\,\mathrm{Hz}$
oscillations arise spontaneously if the number of
local inhibitory-to-inhibitory connections is changed slightly. We introduce a scaling parameter
$r$ by replacing $N^{\beta}_{ii}\rightarrow r N^{\beta}_{ii}$. This is the only parameter that will
be varied in our tests.
\subsection{PETSc overview}
Rather than creating our code from scratch, we opted to work with the
Portable, Extensible Toolkit for Scientific Computation (PETSc): an open-source, object
oriented library that is designed for the scalable
solution and analysis of PDEs \cite{petsc-web-page,petsc-user-ref}.
PETSc is written in the C language, and is usable from C/C++ as
well as Fortran and Python. We use PETSc in conjunction with the Scalable Library for
Eigenvalue Problem Computations (SLEPc) \cite{Hernandez2005}, for the computation
of eigenspectra of equilibrium and periodic solutions. Since our implementation
uses some features of PETSc and SLEPc that are recent additions and are still being tested,
we use the development version of both projects.
PETSc is split up into multiple components to address the various problems associated with
solving PDEs numerically. For our purposes, we treat the {\tt DM} component, which handles
the topology of the discretization, as the most fundamental, from which we can easily derive
memory allocation and communication for distributed vectors ({\tt Vec}) and matrices ({\tt Mat}).
With vectors and matrices, we can now solve linear systems, such as those that
arise in Newton iteration
for implicit time-stepping and the computation of equilibria and periodic orbits. PETSc's component for this is
called {\tt KSP}, and it has numerous iterative solvers implemented, as well
as preconditioners, ({\tt PC}), to increase convergence rates. For implicit time-stepping, for example,
we use GMRES , preconditioned with incomplete LU (ILU) factorization, combined
with the block Jacobi method \cite{Saad1986,Saad2003}.
On top of the linear solvers come the nonlinear solvers, PETSc's {\tt SNES} component, which
implements a few different methods, such as globally convergent
Newton iteration with line search \cite{schnabel}.
Finally, PETSc provides a timestepping component, {\tt TS}, to obtain time dependant
solutions. Implemented here are numerous explicit and implicit schemes such as adaptive
stepsize Runge-Kutta and implicit Euler. The implicit schemes use the {\tt KSP} component.
A schematic of the hierarchy discussed here can be found in Fig.~\ref{fig:petsc-schematic}.
\begin{figure}[hbt]
\begin{center}
\epsfig{file=petsc-schematic.eps,width=0.9\textwidth}
\caption{\label{fig:petsc-schematic} Schematic representation of the components of PETSc
and SLEPc used in our code, and their relative hierarchy.}
\end{center}
\end{figure}
For our dynamical systems calculations we will frequently need to compute specific
eigenvalues and eigenvectors for system-sized matrices.
For this end, we use SLEPc, which implements
iterative eigenvalue solvers
using PETSc {\tt Vec} and {\tt Mat} distributed data structures. The component of
SLEPc that we use is {\tt EPS}, which has a few algorithms for iteratively
solving eigenproblems. Its default algorithm is Krylov-Schur
iteration.
\section{Model Implementation}
\subsection{Geometry}
Following earlier work by Bojak \& Liley (e.g. \cite{Bojak2005,Bojak2007}) we consider
the PDEs on a rectangular domain with periodic boundary conditions. On this domain,
we use a rectangular grid of $N_x$ by $N_y$ points.
In the tests presented
in Sec.~\ref{examples}, the domain and the grid are square. PETSc allows for more complicated
domain shapes and grids, that can be encoded in the {\tt DM} component, independent of the
higher-level components.
Within {\tt DM}, PETSc provides a simpler subcomponent, {\tt DMDA},
for working with finite differences on structured grids such as our rectangle.
If we specify a stencil to use for the spatial derivatives in the {\tt DMDA},
PETSc will automatically handle numerous things for parallel execution, such as
memory allocation and the communication setup for distributed vectors and for
the distributed Jacobian matrix.
\subsection{Fields}
To make use of PETSc's solvers, the model must be written as a system of equations that is
first order in time. This we achieve by introducing new states $J_{jk}$ and $\psi_{ek}$
according to
\begin{equation}\label{eq:1storder_1}
\frac{\partial I_{jk}}{\partial t} = J_{jk}-\gamma_{jk}I_{jk}
\end{equation}
\begin{equation}\label{eq:1storder_2}
\frac{\partial J_{jk}}{\partial t} =
e \Gamma_{jk}\gamma_{jk}\left\{N^\beta_{jk}S_j\left[h_j\right] + \phi_{jk}
+p_{jk}\right\} - \gamma_{jk}J_{jk}
\end{equation}
\begin{equation}
\frac{\partial \phi_{ek}}{\partial t} = \psi_{ek}-v^2\Lambda^2\phi_{ek}
\end{equation}
\begin{equation}\label{eq:1storder_3}
\frac{\partial \psi_{ek}}{\partial t} = v^2\Lambda^2N^\alpha_{ek}S_e\left[h_e\right] +
\frac{3}{2}v^2\nabla^2\phi_{ek}- v^2\Lambda^2\psi_{ek},
\end{equation}
with indices $j,k=\{e,i\}$.
We opted to use a struct, seen in Code.~\ref{fig:fieldstruct}, to store the fields,
rather than a triply indexed array.
\begin{code}[b]
\small
\begin{verbatim}
typedef struct _Field{
PetscReal h_e, h_i,
I_ee, J_ee,
I_ie, J_ie,
I_ei, J_ei,
I_ii, J_ii,
phi_ee, psi_ee,
phi_ei, psi_ei;
} Field;
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:fieldstruct}Struct for the fields.}
\end{code}
This allows the code to be more readable in the function and Jacobian evaluation routines.
For example, one accesses the $\phi_{ee}$ component at grid point $(x_i,y_j)$ simply as
{\tt u[j][i].phi\_ee}, provided that the elements of the array ({\tt Field **u;}) are stored
on the processor in which the call is made.
\subsection{Parameters}
All of the model parameters are stored in a struct designated as the application context.
The application context is how PETSc gets problem related parameters into the user-defined
functions needed by its solvers.
\begin{code}
\small
\begin{verbatim}
typedef struct _AppCtx{
PassiveReal hr_e, hr_i,
tau_e, tau_i,
heq_ee, heq_ie,
heq_ei, heq_ii,
Gamma_ee, Gamma_ie,
Gamma_ei, Gamma_ii,
gamma_ee, gamma_ie,
gamma_ei, gamma_ii,
Nalpha_ee, Nalpha_ei,
Nbeta_ee, Nbeta_ie,
Nbeta_ei, Nbeta_ii,
v, Lambda,
Smax_e, Smax_i,
mu_e, mu_i,
sigma_e, sigma_i,
p_ee, p_ei,
p_ie, p_ii;
...
} AppCtx;
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:appctxstruct}Application context struct with the model parameters.}
\end{code}
Similar to the fields, this allows readable code for the parameters. For example,
one accesses the $\Gamma_{ie}$ parameter as {\tt user->Gamma\_ie}, if {\tt user} is defined
as the pointer {\tt AppCtx *user;}.
How the parameters show up in our struct for the application
context is shown in Code~\ref{fig:appctxstruct}.
\subsection{User supplied functions}
In addition to the structs listed above, we need to provide PETSc with (at least)
a C function that computes the vector field for a given state. We call this function
{\tt FormFunction}, and from this PETSc is capable of approximating the Jacobian
with various finite difference methods. However,
we also supply a C function to explicitly compute the Jacobian, named {\tt FormJacobian},
because this allows for more efficient calculations, especially when looking at stepping
the variational equations in Sec.~\ref{sec:variational}.
\section{Timestepping}\label{timestepping}
We use the implicit Euler method to time-step the discretized equations. As mentioned
in Sec.~\ref{model}, this allows us to take larger time steps than feasible with
explicit methods. Since we are aiming to compute
periodic orbits, rather than to generate long time series, the first order accuracy of
the method is not an issue. Once a periodic orbit is computed, the time-step size can
be reduced to increase accuracy. Another option is to use Richardson extrapolation to
increase the order of accuracy, using the same nonlinear solving as described below.
\subsection{Mathematical basis}
We symbolically write the dynamical system as
\begin{equation}\label{eq:ds}
\dot{u}=f(u),\quad f:\mathbb{R}^{N}\rightarrow\mathbb{R}^N.
\end{equation}
where $N$ is the total number of unknowns after discretization, in our case $14\times N_x\times N_y$.
The implicit Euler scheme for time integration is given by
\begin{equation}\label{eq:beuler}
u_{n+1} = u_{n} + \mathrm{d}t\,f(u_{n+1})
\end{equation}
where the subscript represents the step number, $\mathrm{d}t$ the step size, and $u_{0}$
the initial conditions. This nonlinear equation is solved by Newton iteration:
\begin{equation}\label{eq:newton_beuler}
u^{k+1}_{n+1}= u^k_{n+1}+ \mathrm{d}u^k,
\end{equation}
where the superscript denotes the Newton iterate, and
$\mathrm{d}u^k$ is the solution to the linear system
\begin{equation}\label{eq:newton_beuler_lin}
\left(\mathbb{I}-\mathrm{d}t\left.\frac{\partial f}{\partial u}\right|_{u^k_{n+1}}\right)\mathrm{d}u^k
= \mathrm{d}t\,f(u^k_{n+1})-u^k_{n+1}+u^k_n,
\end{equation}
where $\partial f/\partial u$ denotes the $N\times N$ Jacobian matrix.
Provided that the initial approximation, $u^0_{n+1}$, is close enough
to the actual solution of equation \eqref{eq:beuler}, this iteration should converge
quadratically. This is achieved by making the initial approximation the
result of an explicit Euler step
\begin{equation}\label{eq:beuler_init}
u^0_{n+1}=u_n + \mathrm{d}t\,f(u_n).
\end{equation}
As we scale up the size of our problems, it becomes the linear solve in equation
\eqref{eq:newton_beuler_lin} that takes most time. This problem is handled by using
GMRES to solve the linear system.
For large time steps, the spectrum of the matrix in Eq.~\ref{eq:newton_beuler_lin} is spread out,
and we need to precondition it for iterative solving. We make use ILU, which has shown
to be reliable \cite{Sanchez2002,Saad1994} for this type of problem. If we use more than one processor, PETSc
uses distributed storage for the matrix, and combines ILU with block Jacobi preconditioning.
\subsection{Implementation}
PETSc provides a simple interface for timestepping in its {\tt TS} component. The basic
code required to set up a {\tt TS} is given in Code~\ref{fig:ts_setup}. With a {\tt TS}
set up like this, the timestepping parameters are set from command line arguments at run time.
For example, to do implicit Euler timestepping for $40.67\,\mathrm{ms}$
with a time step of $0.1\,\mathrm{ms}$, one needs to provide the arguments
\newline
{\tt -ts\_type beuler -ts\_dt 0.1 -ts\_final\_time 40.67}.
\newline
In this specific case, since the final time is not an integer number of timesteps, PETSc will
step past it, and interpolate at the desired time.
\begin{code}[t]
\begin{verbatim}
TS ts;
TSCreate(PETSC_COMM_WORLD,&ts);
TSSetProblemType(ts,TS_NONLINEAR);
TSSetExactFinalTime(ts);
TSSetRHSFunction(ts,PETSC_NULL,FormFunction,&user);
TSSetRHSJacobian(ts,J,J,FormJacobian,&user);
TSSetFromOptions(ts);
TSSolve(ts,u,PETSC_NULL)
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:ts_setup}PETSc code for setting up and running the timestepping.
{\tt FormFunction}
and {\tt FormJacobian} are user provided functions that compute the rhs of equation \eqref{eq:ds},
and its Jacobian respectively. {\tt J} is an appropriately allocated matrix to hold the Jacobian,
and {\tt u} a vector to hold the solutions.}
\end{code}
\section{Stepping of the variational equations}\label{sec:variational}
\subsection{Mathematical basis}
The variational equations for the dynamical system are written as
\begin{equation}\label{eq:vareq}
\dot{v} = \left.\frac{\partial f}{\partial u}\right|_uv,\quad v\in \mathbb{R}^N
\end{equation}
and must be integrated simultaneously with the dynamical system \eqref{eq:ds}.
Solving the variational equations allow us to compute the stability of solutions, and is
also an essential ingredient for the treatment of boundary value problems such as
those that arise in the computation of periodic orbits.
Performing implicit Euler timestepping on the variational equations \eqref{eq:vareq}
requires solutions of the linear problems
\begin{equation}\label{eq:beuler_vareq}
\left(\mathbb{I}-\mathrm{d}t\left.\frac{\partial f}{\partial u}\right|_{u_{n+1}}\right)v_{n+1}
= v_n.
\end{equation}
Since we already have the Jacobian of the dynamical system at timestep $n+1$, stepping the
variational equations requires only one additional $N\times N$ linear solve per time step.
\subsection{Implementation}
In PETSc, we implement the timestepping of the variational equations as a {\tt MATSHELL}.
A {\tt MATSHELL} allows users to define their own matrix type.
Within a {\tt MATSHELL}, one needs to give a context for storing the relevant data and write
functions for the desired matrix operation.
For example, we point the operation {\tt MATOP\_MULT} to
a function that takes the initial state of the variational system $v(0)$ as input,
and outputs the result $v(T)$ at the end of the timestepping.
The context we use for the time stepping of the variational equations is shown in
Code \ref{fig:PeriodIntegrationCtx}. The function we provide for {\tt MATOP\_MULT}
works by first taking a step of the {\tt TS}, then loading the Jacobian computed
from that step and solving equation \eqref{eq:beuler_vareq}. This is repeated until
the {\tt TS} reaches its end.
\begin{code}
\small
\begin{verbatim}
typedef struct _PeriodIntegrationCtx{
// timestepping of the original eqn
TS ts;
Mat tsJac;
Vec initState,endState,fullSol;
// additional requirements for variational eqn
Mat J,eye;
KSP ksp;
} PeriodIntegrationCtx;
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:PeriodIntegrationCtx}The {\tt MATSHELL} context for timestepping of the
variational equations. The {\tt TS} holds the relevant info for stepping the dynamical
system}
\end{code}
The {\tt MATSHELL} thus defined can be used by SLEPc for the iterative computation of eigen
pairs. In particular, we will use this approach to compute the Floquet multipliers of periodic
orbits.
\section{Equilibria}\label{equilibria}
Having set up the function {\tt FormFunction} for the right hand side of the dynamical
system, and its Jacobian computation {\tt FormJacobian}, also used for time integration,
we can set up equilibrium calculations using PETSc's {\tt SNES} component with very little
effort.
\subsection{Mathematical Basis}
Equilibrium solutions to the dynamical system \eqref{eq:ds} are solutions that satisfy
\begin{equation}\label{eq:equilibriumsolution}
f(u)=0.
\end{equation}
To solve this, we can set up a Newton iteration scheme
\begin{equation}\label{eq:newton_equilibrium}
u^{k+1} = u^{k}+\mathrm{d}u^k
\end{equation}
with $\mathrm{d}u$ coming from the solution of the linear system
\begin{equation}\label{eq:newton_equilibrium_lin}
\left.\frac{\partial f}{\partial u}\right|_{u^k}\mathrm{d}u^k = -f(u^k).
\end{equation}
As with the timestepping, if the initial guess is good enough this will converge
quadratically provided that $\left.\frac{\partial f}{\partial u}\right|_{u^k}$ is
nonsingular. Unlike the case of time stepping, though, we do not always have a
way to produce an initial approximation that is good enough.
For stable equilibrium solutions, we can
use timestepping to get close to an equilibrium, but this will
not work for unstable equilibria. One possible solution is using
globally convergent Newton methods. Using such methods we can find equilibria from
very coarse initial data, at the cost of computing many iterations. The line
search algorithm and the trust region approach (see, e.g. \cite{schnabel}) are implemented in the {\tt SNES} component.
Stability of equilibrium solutions follows from the spectrum of the Jacobian. Because of the spatial symmetries of
the model, these will mostly appear in groups. On a square domain, for instance, a single eigenvalue will be
associated with up to eight eigenvectors, with wavenumbers $(\pm k_x,\pm k_y)$ and $(\pm k_y,\pm k_x)$.
\subsection{Implementation}
Setting up and using a nonlinear solver within PETSc is straightforward, as shown in Code~\ref{fig:snes_equilibrium}.
The default algorithm used by {\tt SNES} is Newton's
method with line search.
\begin{code}
\small
\begin{verbatim}
SNES snes;
SNESCreate(PETSC_COMM_WORLD,&snes);
SNESSetFunction(snes,r,FormFunctionSNES,&user);
SNESSetJacobian(snes,J,J,FormJacobianSNES,&user);
SNESSetFromOptions(snes);
SNESSolve(snes,PETSC_NULL,u);
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:snes_equilibrium}Code snippet for solving for equilibria. Vectors
{\tt r} and {\tt u} are preallocated, with {\tt u} being the initial approximation, and
{\tt J} a preallocated matrix for the Jacobian.}
\end{code}
\section{Periodic solutions}\label{sec:periodic}
The primary instability in the Liley model is often a Hopf bifurcation, and periodic orbits
have been shown to play an important role in the dynamics of ODE reductions of the model
(e.g. \cite{Frascoli2011,VanVeen2006}). However, space dependent periodic orbits have not
previously been computed and studied. Using PETSc data structures for bordered matrices,
in conjunction with a {\tt MATSHELL} structure, we can solve for periodic orbits based on the
time stepping described in Secs.~\ref{timestepping} and \ref{sec:variational}.
\subsection{Mathematical basis}
Periodic orbits solve the boundary value problem
\begin{equation}\label{eq:periodic}
F(u,T) = \phi(u,T)-u = 0,
\end{equation}
where $\phi$ is the flow of the dynamical
system \eqref{eq:ds}, and $T$ is the period.
Our strategy for solving this equation is essentially that of Sanchez et al. \cite{Sanchez2004},
namely Newton iterations combined with unconditioned GMRES iteration.
Linearising Eq.~\ref{eq:periodic} gives
\begin{equation}\label{eq:periodic_diff}
\left(D_u\phi(u,T)-\mathbb{I}\right)\mathrm{d}u + f(\phi(u,T))\mathrm{d}T = -F(u,T),
\end{equation}
where $D_u\phi$ is a matrix of derivatives of the flow with respect to its initial condition.
Upon convergence, this is the monodromy matrix of the periodic orbit.
This results in $N$ equations in $N+1$ unknowns, which must be closed by a phase condition.
We opted for the use of a Poincar\'{e} plane involving one of the state variables, $u_k$:
\begin{equation}\label{eq:Poin_plane}
\phi_k(u,T)-C = 0,
\end{equation}
where $C$ is set appropriately, for instance to the time-mean value of $u_k$. This choice gives
the following bordered system
\begin{equation}\label{eq:periodic_bordered}
\left[
\begin{array}{cc}
\left(D_u\phi(u,T)-\mathbb{I}\right) & f(\phi(u,T)) \\
\left[D_u\phi(u,T)\right]_{k,.} & f_k(\phi(u,T))
\end{array}\right]
\left[\begin{array}{c}
\mathrm{d}u \\
\mathrm{d}T
\end{array}\right]
= \left[\begin{array}{c} -F(u,T) \\ C-\phi_k(u,T) \end{array}\right],
\end{equation}
where $[D_u\phi(u,T)]_{k,.}$ denotes the $k^{\rm th}$ row of the matrix $D_u\phi$.
An update can then be made to the approximate solution as
\begin{equation}\label{eq:Newton_update}
\left[\begin{array}{c}
u^{n+1} \\
T^{n+1}
\end{array}\right]
=\left[\begin{array}{c}
u^{n} \\
T^{n}
\end{array}\right] +
\left[\begin{array}{c}
\mathrm{d}u \\
\mathrm{d}T
\end{array}\right].
\end{equation}
The matrix $D_u\phi$ is dense, so we should avoid calculating and storing it explicitly.
Iterative solving of the linear problem, \eqref{eq:periodic_bordered}, requires the
computation of matrix-vector products, which are constructed from the integration of the
variational equation \eqref{eq:vareq} with $v=\mathrm{d}u$ and the vector field $f(\phi(u,T))$
at the end point of the approximately periodic orbit.
Since the governing PDE is dissipative, most of the eigenvalues of the monodromy matrix
are clustered around zero. This aids the convergence of GMRES, without any preconditioning.
Sanchez et al.\ \cite{Sanchez2004} provide bounds for the number of GMRES iterations
for the Navier-Stokes equation, and the convergence we observe for the Liley model is
qualitatively similar.
\subsection{Implementation}\label{implement:per}
The problem of creating a bordered matrix system in a distributed environment is
not a trivial one. The specific case that we have is one vector, $u$,
that is sparsely connected and distributed among processors, and one parameter,
$T$, that must exist and be synchronized across all processors.
PETSc's {\tt DM}
module has some recently introduced functionality that allows us to handle this in a
straightforward way, letting us make use of the {\tt DMDA} already used in the
other types of calculations.
{\tt DMRedundant} can be used for the $T$ component of our extended system, as it has
the precise behaviour that we require. Next, we use a {\tt DMComposite} to join together
the {\tt DMDA} of the grid with the {\tt DMRedundant} of the period. We can then derive
vectors from this {\tt DMComposite}, and use these vectors for PETSc's iterative linear
solvers. PETSc code that illustrates this idea is shown in Code~\ref{fig:dmcomposite}.
\begin{code}
\begin{verbatim}
DM packer, redT;
DMCompositeCreate(PETSC_COMM_WORLD,&packer);
DMRedundantCreate(PETSC_COMM_WORLD,0,1,&redT);
DMCompositeAddDM(packer,da);
DMCompositeAddDM(packer,redT);
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:dmcomposite}Additional {\tt DM} pieces for extended vectors as in equation
\eqref{eq:Newton_update}, assuming that {\tt da} is the {\tt DM} associated with the
grid structure. The numerical arguments in {\tt DMRedundantCreate} represent the
processor where the redundant entries live (in global vectors), and the number of
redundant entries respectively.}
\end{code}
The matrix multiplication is done through a {\tt MATSHELL}, and the struct that holds the
relevant data is found in Code~\ref{fig:periodrefine_struct}.
\begin{code}
\begin{verbatim}
typedef struct _PeriodFindCtx{
Mat *linTimeIntegration;
DM packer,redT;
Vec endState,f_at_endState;
} PeriodFindCtx;
\end{verbatim}\vspace{-10pt}
\caption{\label{fig:periodrefine_struct}For finding periodic solution, we need a method
for integrating the variational equations (the {\tt MATSHELL} discussed in section
\ref{sec:variational}), additional {\tt DM}s,
and space for holding $f$ evaluated at the state at the end of the integration}
\end{code}
\section{Example calculations}
\label{examples}
In this section, we present some computations that serve to validate our implementation
and to investigate its efficiency. All tests are based on the parameter set in Table~\ref{table:pars},
and the scaling of the number of local inhibitory-to-inhibitory connections, $r$, is varied
around the first bifurcation from an equilibrium to more complicated, spatio-temporal behaviour.
Fig.~\ref{fig:nsc} shows the neutral stability curve for the spatially homogeneous equilibrium,
which is the unique attractor of the model at small values of $r$. The primary transition is
a Hopf bifurcation with spatial wave numbers that depend on the system size. For systems smaller
than $2\times 2\,\mathrm{cm}^2$, the emerging periodic orbit is spatially homogeneous. For larger
systems, space dependent orbits emerge, and their typical length scale converges to about
$9.3\,\mathrm{cm}$
for large system sizes. These stability curves were computed by solving small eigenvalue problems
for each combination of wavenumbers, independent from the PETSc implementation. The eigenvalues
computed by Krylov-Schur iteration in SLEPc, presented in Sec.~\ref{ex:eq}, are in good agreement.
A partial bifurcation diagram, for spatially homogeneous solutions only, is shown in Fig.~\ref{fig:bifurcation_diagram}.
In this diagram, the Hopf bifurcation is subcritical, and time series analysis indicates that
the Hopf bifurcations associated with nonzero wave numbers are, too. The time series presented
in Sec.~\ref{ex:ts} was generated by starting from the equilibrium at $r=1$ and adding a finite-size
perturbation in the least stable direction, with wave number $|k_x|=|k_y|=1$.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=0.9\textwidth]{NSC.eps}
\caption{\label{fig:nsc}Neutral stability curve for the spatially homogeneous equilibrium of the Liley model with
parameters set according to Table~\ref{table:pars}. Shown is the scaling parameter, $r$, versus the linear
domain size, $L$, and wave numbers $\bm{k}=(k_x,k_y)$ are shown
in parenthesis. When varying $r$, only for very small domains
the primary instability is spatially homogeneous. For domain sizes over $12.5\times 12.5\text{cm}^2$
the location of the primary instability approaches $r=1.04$ and the length scale of the leading instability
approaches $L/\|\bm{k}\|=9.3\,\mathrm{cm}$.}
\end{center}
\end{figure}
\begin{figure}[hbt]
\epsfig{file=cont_40Hz.eps,width=0.9\textwidth}
\caption{\label{fig:bifurcation_diagram}Partial bifurcation diagram showing the primary transition from
a spatially homogeneous equilibrium to a space and time dependent periodic orbit. On the vertical axis
the scaling parameter $r$ is plotted, and on the vertical axis the (maximum of) the excitatory membrane
potential. The branch of periodic solutions shown with a dashed line is a spatially homogeneous branch
that is unstable to space-dependent perturbations.}
\end{figure}
\subsection{Timestepping}\label{ex:ts}
For the timestepping demonstration, we used a system size of $12.8\times 12.8\,\mathrm{cm}^2$
with $0.5\,\mathrm{mm}$
resolution, resulting in a $256\times 256$ grid, and $N=917,504$ unknowns in total. Setting the
parameter $r=1.0$, we initialize with the stable equilibrium solution perturbed by its least
stable eigenmode, shown in Fig.~\ref{fig:eigenmode}. Since the
equilibrium solution is stable, small perturbations just decay, but sufficiently large perturbations
grow. The snapshots
of Fig.~\ref{fig:time_snapshots} were taken after a transient time of 600ms. The membrane potentials
show behaviour that is nearly periodic, with a dominant period of 40Hz, as demonstrated by the
power spectrum shown in the last panel.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=.4\textwidth]{00580he_boxes.eps}\qquad\quad\includegraphics[width=.4\textwidth]{00586he_boxes.eps}\vspace{10pt}\\
\includegraphics[width=.4\textwidth]{00592he_boxes.eps}\qquad\quad\includegraphics[width=.4\textwidth]{40Hz.eps}
\end{center}
\caption{\label{fig:time_snapshots}Three snapshots of the excitatory membrane potential,
$6\,\mathrm{ms}$ apart, of a solution at $r=1$, near the primary Hopf
bifurcation. The domain size is $12.8 \times 12.8\,\mathrm{cm}^2$,
the resolution is $0.5\,\mathrm{mm}$ and the time-step
size $1\,\mathrm{ms}$.
The fourth panel shows the power spectrum of $h_e$, averaged over the region inside the black
square.}
\end{figure}
Since the time-stepping code lies at the core of the periodic orbit solver, we carefully investigated its
scaling with an increasing number of processors. Doubling the domain size, while keeping the grid spacing
fixed, gives a dynamical system with $N=3,670,016$ degrees of freedom. We time-stepped this system on
a small subcluster of $2.4\,\mathrm{GHz}$ AMD Opteron nodes with gigabit interconnects.
Apart from some minor load-balancing effects, the scaling is linear up to 16
processors, despite the relatively slow interconnects.
\begin{figure}[hbt]
\epsfig{file=wtime.eps,width=0.9\textwidth}
\caption{\label{fig:wtime}Wall time for the computation of 50 time steps of $1\,\mathrm{ms}$ each on a $25.6 \times 25.6\,\mathrm{cm}^2$
domain with $0.5\,\mathrm{mm}$ resolution. The fully implicit Euler steps are computed with Newton iterations, each of which
is solved for by GMRES, preconditioned with a combination of block Jacobi and ILU. The initial guess is given by
an explicit Euler step. Two or three Newton iterations are sufficient to reduce the residual by a factor of $10^8$.
About 80 Krylov vectors are computed by GMRES to bring the relative residual down to $10^{-5}$. The number of unknowns
is $N=3,670,016$.}
\end{figure}
\subsection{Equilibrium}\label{ex:eq}
We computed the whole equilibrium curve of Fig.~\ref{fig:bifurcation_diagram}
through parameter continuation, which is a trivial extension of the algorithm
for computing equilibria, presented in Sec.~\ref{equilibria}. For each computed equilibrium
solution, we took the Jacobian and used SLEPc to compute the
eigenvalues with the largest real parts. The result is shown in Fig.~\ref{fig:eigenmode}. As predicted
by the neutral stability curve computation, the $(1,1)$ mode turns unstable first, immediately followed by
the $(1,0)$ mode. Around $r=1.08$, the $(0,2)$ mode crosses the $(0,1)$ mode and proceeds to become the
most unstable mode for larger values of $r$.
The least stable
eigenmode for $r=1.046$, just after its eigenvalue has crossed zero,
is shown in Fig.~\ref{fig:eigenmode}.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=0.8\textwidth]{eq_cont_ev.eps}
\caption{\label{continuation}The real parts of the leading two eigenvalue pairs of the spatially homogeneous
equilibrium tracked in the scaling parameter $r$, for system size $L=12.8\,\mathrm{cm}$. The primary transition is tied
to wave numbers $|k_x|=|k_y|=1$. The other curves shown are for wave numbers $k_x=0$, $k_y=\pm 1$ and
$k_x=\pm 1$, $k_y=0$ and for $k_x=0$, $k_y=\pm 2$ and
$k_x=\pm 2$, $k_y=0$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{eigenmode_he.eps}\qquad\includegraphics[width=.45\textwidth]{eigenmode_hi.eps}
\end{center}
\caption{\label{fig:eigenmode}The real part of the least stable eigenmode of the stable
equilibrium located at $r=1.046$. Displayed are the excitatory (left) and inhibitory (right) membrane potentials.
The eigenvector, with wave numbers $(1,1)$, was computed by Arnoldi iteration and is scaled to have unit $L_2$ norm.}
\end{figure}
\subsection{Periodic solutions}\label{ex:per}
We tested the computation of periodic orbits on a smaller grid, namely $16\times 16$ points,
still with $0.5\,\mathrm{mm}$
resolution, and with $r=1.2$. The primary Hopf bifurcation is
sub critical, so there is no easy way to compute the branch of space-dependent periodic solutions.
Instead, we computed one of the spatially homogeneous orbits, for which an approximate solution
can readily be obtained from analysis of the ODE reduction of the model. In fact, the upper part
of the branch of periodic orbits shown in Fig.~\ref{fig:bifurcation_diagram} is stable to all
spatially homogeneous perturbations.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{NR_res.eps}\qquad\includegraphics[width=.45\textwidth]{GMRES_res.eps}
\end{center}
\caption{\label{fig:NR_GMRES_res}Residuals of the Newton iteration (left) and the corresponding
GMRES iterations (right). The latter is normalised by the norm of the right hand side of Eq.~\ref{eq:periodic_bordered},
i.e. the Newton residual. The tolerance was set at $10^{-8}$ for the Newton iteration and to $10^{-5}$
for the GMRES iteration. Note the super linear convergence of the former. }
\end{figure}
Starting from a coarse initial approximation, the Newton iterations converged faster than linear,
and each Newton step took between 8 and 11 GMRES iterations, out of a maximum of $N+1=3585$.
Subsequently, we computed the most unstable multipliers, using SLEPc with the {\tt MATSHELL}
that computes products with the bordered matrix, as described in Sec.~\ref{implement:per}.
The most unstable multiplier is $\mu_1=1.111$ and corresponds to a wave number $(1,1)$ perturbation.
\section{Conclusion and future improvements}
In the current paper, we have presented the basic implementation of the model and example computations
to validate it and test its performance. The code will be available publicly \cite{public}.
As it is built on top of PETSc, the user has access to a range of nonlinear and linear solvers and
preconditioners, which can be used to solve the boundary value problems that typically arise in
dynamical systems analysis. The periodic orbit computation, presented in Sec.~\ref{ex:per}, is a simple
example of such a boundary value problem, that has all the ingredients: a module for time-stepping
the system and perturbations and a representation of user-specified, bordered matrices.
The next step in the development of the code is the implementation of pseudo-arclength continuation
of equilibria and periodic orbits. This will enable us, for instance, to complement the bifurcation
diagram of the current test case, Fig.~\ref{fig:bifurcation_diagram}, with the branches of space-dependent
periodic solutions that actually regulate the observed dynamics, in contrast to the highly unstable
spatially homogeneous periodic orbits computed from an ODE reduction of the model.
We expect that our implementation will be useful to researchers
studying the dynamics of the Liley model, or similar models, such as the model with gap junctions
proposed by Steyn-Ross {\sl et al.} \cite{steynross}. Also, it could be useful for those who
incorporate a similar mean-field model in, for instance, the control of robotic motion
or network models of brain activity.
\section*{Acknowledgements}
LvV was supported by NSERC Grant nr. 355849-2008. Some of the computations were made
possible by the facilities of the Shared Hierarchical
Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada.
\bibliographystyle{elsarticle-num}
\bibliography{MFM_small}
\end{document}
| 202,186
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TITLE: The relationship between normal extension and the splitting field of polynomial
QUESTION [3 upvotes]: An algebraic extension $K/F$ is called a normal extension if any irreducible polynomial of $F[x]$, which has a root in $K$, can be completely decomposed in $K$.
I have know that a finite extension $K/F$ is a normal extension if and only if $K$ is a splitting field of a polynomial $f(x)\in F[x]$.
Is is right that if $K$ is the splitting field of a collection of polynomials in $F[x]$, this is, let $T$ denote a collection of polynomials and $S$ the set of roots of polynomials in $T$ ($S$ maybe be uncountable) then $K=F(S)$, $K/F$ is a normal extension? How to prove?
REPLY [2 votes]: Yes, this is true. To prove it, suppose $f\in F[x]$ is irreducible and has a root $\alpha$ in $K$. Then there is some finite subset $S_0\subseteq S$ such that $\alpha\in F(S_0)$. Let $S_1$ be the set of elements of $S$ which have the same minimal polynomial over $F$ as an element of $S_0$. Then $S_1$ is still finite, and $F(S_1)$ is now the splitting field of a finite set of polynomials and thus normal over $F$. But $\alpha\in F(S_0)\subseteq F(S_1)$, so all the roots of $f$ are in $F(S_1)$. Thus $f$ splits over $K$, and since $f$ was arbitrary, $K$ is normal over $F$.
| 147,531
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Food Rejection in Infants
Why bub says ‘Thanks, but no thanks’? Is this the beginning of picky eating?
It is important not to confuse food rejection with the dislike of a food. There are many reasons why babies may turn away from food; often dislike has nothing to do with it. The information below is designed to answer some of the questions about why babies may refuse foods. Picky eating in the toddler and preschool years where a child refuses to eat certain foods such as vegetables is a common stage that most children grow out of in their own time.
Beginning and progressing through the introduction of complimentary foods can be fun (hilarious at times), frustrating, confusing, exciting, rewarding and sometimes terrifying. But it is also an important aspect of developing your child’s life-long skills in healthy eating and socialisation. Your role is immense, but don’t overlook the power of a baby!
Why does a baby reject a food?
There may be many reasons why a baby is rejecting a food. The appearance, smell, texture of the food in their mouths, and of course the taste sensation – any one of these can be off-putting. Other reasons could include feelings of unwellness, teething, and mechanical issues related to chewing, moving food around the mouth and swallowing. The reasons can be varied.
Inherently, babies appear to have a preference for sweet and salty foods. It is wise to avoid offering sweeter foods such as fruit over more bitter foods such as vegetables. We suggest progressing through vegetables and offering them as often as is practical each day. Start on fruit only once baby has a good repertoire of vegetables under his or her belt.
Many parents report fussiness with food texture, which is often as a result of baby being offered smooth-textured food for too long as an infant. It seems that progressing from soft foods to coarse, textured and lumpy foods is really important when it comes to later eating habits.
There also appears to be a relationship between the duration of breastfeeding and picky eating. Breastfed bubs generally are less likely to be picky eaters because they are exposed to a wider array of tastes via breastmilk (Galloway et al., 2003). Another study found that parents of picky eaters tended to offer new foods less often (3-5 times) than parents of children who weren’t that fussy. Apparently, new foods should be offered between 10-15 times to improve your child’s acceptance of new tastes and textures. (Carruth et al., 2004).
Why doesn’t my baby like soilds?
Numbers mean a lot to us, how much does your baby weight, what percentile is he or she one, how old are they, when did you start and so on. While we know now that about six months is the ideal time to try baby on solids, some bubs just can’t be tempted. That’s okay. Moving from a liquid diet to a textured diet of food isn’t always an easy one. Aside from the strange tastes and textures mum is adamant you will enjoy, you also have to learn to move the ‘stuff’ about your mouth and try swallowing it. Babies need to develop the mechanisms in their mouths in order to adapt to solids too.
So if your bub seems to be gagging excessively or just down right uninterested in your lovingly made solids that is okay. You could try baby with a baby feeding mesh (pop some avocado or banana in) which are also fabulous for babies who are teething, or let baby suck the food of the spoon as apposed to spooning it into babies mouth. Try some different foods such as sweet potato, but be a little mindful of babies preference for sweet and salty when it comes to fruit early on. Also there is no harm in take a rain check and trying baby again in a few days.
If things just don’t get better and baby is 7 months’ish then see a health professional for more advice.
Getting off to a good start
Consistency is one of the most important aspects of parenting. When introducing a new food, try to ensure it is given in a familiar place, saving old favourites for new or novel situations. So when you are eating out, take a favourite food rather than attempt to introduce a new one.
Don’t confuse rejection with dislike, and being persistent is also important. If a food is rejected, try it again (up to 10-15 times in some cases). There can be many reasons why bub hasn’t accepted it first time around and allowing them to have another go (without any fuss) is a sure way of improving the outcome. You can try freezing and then taking out small amounts to offer a number of times over the day or week, to save on time and cost.
Also keep in mind that babies can vary the amount of food they eat from one meal to the next. Sometimes this is a consequence of the energy density of a particular meal which has filled them up. Sometimes too, a baby will wolf down a meal then reject it the next day. Though confounding, this is all quite normal.
How babies learn to eat
Our understanding of what a baby can eat and when is not only based on the development of their digestive and immune system but their oral-motor development. The World Health Organisation defines four phases in the introduction of ‘complimentary foods’ defined allowing finger foods.
- Stage four is self-feeding and nearing family meals.
A 6 month old begins to consciously suck from the breast or bottle. Around 7 months, bub will be making chomping motions with their mouths and even showing when they are ready for another mouthful right down to when they show you they have had enough. By 8-9 months, baby will shortly start to chew food (even those bubs without teeth are adept at chewing). Around this time babies can open their mouths and begin to use their upper lips to take food from the spoon. At 10-12 months, baby will be gaining a great deal more fluid from sipper cups and while their tongue may still protrude on the bottom of the cup, this just helps their stability. By one year they are able to maintain a good biting action (depending on their teeth). By about a year and a half they are quite adept at keeping food and drink well and truly in their mouths.
The importance of introducing ‘lumpy’ foods
It has been found that babies who are not exposed to lumpy foods may be more likely to develop fussy eating habits later on. Therefore, it is important to give babies of between 6 to 9 months a variety of foods. This helps prevent picky eating in later years. Introduce mashed over pureed foods around 8 to 9 months, and then at 9 months pieces of cooked (soft) vegetables and finger foods (see also our fact sheet on starting solids).
Suggestions for coping with food rejection
Don’t force or coerce a baby to eat a food they are rejecting. Try to set up a good meal-time routine and also avoid snack-eating too close to main meals, as main meals tend to be more nutritious.
Remember that food rejection is a normal behaviour for almost all toddlers and preschoolers. Meal times should always be family-orientated and enjoyable. Do not overestimate the influence of togetherness during meals for a child’s overall development. It is amazing how much a bub will be influenced by what is going on around them. This includes distractions, and don’t forget that they are always learning from you so be sure to set a good example. Take the plate away when they have finished without a fuss; they have plenty of time to try it again.
Ways to reduce food rejection
Don’t coax, beg or trick – it may backfire on you and cause more strife. Take it gently, don’t rush your child through their meal, teach them to eat slowly. It takes up to 20 minutes for the brain to tell the body that it has had enough; this will also avoid overeating.
Whenever possible, ensure your child does not eat alone. Introduce new foods in a positive family environment; we know that a relaxed, communicative and happy atmosphere helps foster positive eating habits. And eating in front of the TV is not advisable. We strongly suggest you make the meal and people the centre of attention.
Just because a food has been rejected, it doesn’t mean you shouldn’t try offering it again. As mentioned above, sometimes it takes numerous attempts before your child becomes accustomed to – and likes – new tastes and textures.
What do you class as an eating problem?
Occasionally a bub may have problems with eating. If any of the following occur, or if you are concerned, seek professional assistance.
- If bub vomits frequently, particularly if it is associated with pain or discomfort. This does not include normal ‘spitting up’.
- Problems moving from soft textures to coarser, lumpier foods and difficulty chewing. Doesn’t include habits from the over-reliance on commercially prepared foods which can be overly soft.
- Where baby has ongoing problems swallowing food often with choking and or gagging, particularly where this is have a negative effect on baby’s health and development.
- Excessive mouth-stuffing or storing of food in the mouth for long periods (this does not include the overindulgent baby who just likes to get in as much food at once). Instead look more for the tendency to pass the food from the mouth to the throat.
- Recurring or upsetting reactions such as skin reactions which may be a sign of an allergy or intolerance.
- Persistent diarrhoea as this is a major factor in the failure to thrive in many infants and should be resolved as quickly as possible.
- Persistent constipation which can cause a great deal of stress and discomfort and should be investigated.
- Where you feel the issue is behavioural.
- Where any of the above are causing you anxiety or you feel unable to cope.
Where do I go for help?
The first port of call is generally your local doctor who would in many cases refer you to either a paediatrician or a testing unit in a hospital (as in the case of a suspected allergy or intolerance).
Much of today’s guidelines on the introduction of foods are based on oral-motor ability. In other words, is your baby able to move certain types of food – such as coarse and finger foods – around their mouth and prepare it for swallowing? This is why speech pathologists are commonly called upon to help with feeding problems. A speech pathologist can assess your baby and help you and bub to progress through the stages of complimentary feeding. Speech pathologists may also be able to help with things such as drooling, problems with drinking from cups and straws, and where bubs are determined to jam food into their mouths.
The reality
Sometimes you can do everything right, textbook fashion, and still end up with a fussy-eating toddler. Don’t overlook a little one’s own personality and freedom of choice. Life is a mix of both our inherent patterns, including personality and motivation, and our environment. One we can control but the other we must work with.
Remember it’s up to parents and carers to offer nutritious food for children to choose from!
This information has been provided by Leanne Cooper from Sneakys baby and child nutrition. Leanne is a qualified nutritionist and mother of two very active boys.
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/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import algebra.big_operators.ring
import combinatorics.double_counting
import combinatorics.set_family.shadow
import data.rat.order
import tactic.linarith
/-!
# Lubell-Yamamoto-Meshalkin inequality and Sperner's theorem
This file proves the local LYM and LYM inequalities as well as Sperner's theorem.
## Main declarations
* `finset.card_div_choose_le_card_shadow_div_choose`: Local Lubell-Yamamoto-Meshalkin inequality.
The shadow of a set `𝒜` in a layer takes a greater proportion of its layer than `𝒜` does.
* `finset.sum_card_slice_div_choose_le_one`: Lubell-Yamamoto-Meshalkin inequality. The sum of
densities of `𝒜` in each layer is at most `1` for any antichain `𝒜`.
* `is_antichain.sperner`: Sperner's theorem. The size of any antichain in `finset α` is at most the
size of the maximal layer of `finset α`. It is a corollary of `sum_card_slice_div_choose_le_one`.
## TODO
Prove upward local LYM.
Provide equality cases. Local LYM gives that the equality case of LYM and Sperner is precisely when
`𝒜` is a middle layer.
`falling` could be useful more generally in grade orders.
## References
* http://b-mehta.github.io/maths-notes/iii/mich/combinatorics.pdf
* http://discretemath.imp.fu-berlin.de/DMII-2015-16/kruskal.pdf
## Tags
shadow, lym, slice, sperner, antichain
-/
open finset nat
open_locale big_operators finset_family
variables {𝕜 α : Type*} [linear_ordered_field 𝕜]
namespace finset
/-! ### Local LYM inequality -/
section local_lym
variables [decidable_eq α] [fintype α] {𝒜 : finset (finset α)} {r : ℕ}
/-- The downward **local LYM inequality**, with cancelled denominators. `𝒜` takes up less of `α^(r)`
(the finsets of card `r`) than `∂𝒜` takes up of `α^(r - 1)`. -/
lemma card_mul_le_card_shadow_mul (h𝒜 : (𝒜 : set (finset α)).sized r) :
𝒜.card * r ≤ (∂𝒜).card * (fintype.card α - r + 1) :=
begin
refine card_mul_le_card_mul' (⊆) (λ s hs, _) (λ s hs, _),
{ rw [←h𝒜 hs, ←card_image_of_inj_on s.erase_inj_on],
refine card_le_of_subset _,
simp_rw [image_subset_iff, mem_bipartite_below],
exact λ a ha, ⟨erase_mem_shadow hs ha, erase_subset _ _⟩ },
refine le_trans _ tsub_tsub_le_tsub_add,
rw [←h𝒜.shadow hs, ←card_compl, ←card_image_of_inj_on (insert_inj_on' _)],
refine card_le_of_subset (λ t ht, _),
apply_instance,
rw mem_bipartite_above at ht,
have : ∅ ∉ 𝒜,
{ rw [←mem_coe, h𝒜.empty_mem_iff, coe_eq_singleton],
rintro rfl,
rwa shadow_singleton_empty at hs },
obtain ⟨a, ha, rfl⟩ :=
exists_eq_insert_iff.2 ⟨ht.2, by rw [(sized_shadow_iff this).1 h𝒜.shadow ht.1, h𝒜.shadow hs]⟩,
exact mem_image_of_mem _ (mem_compl.2 ha),
end
/-- The downward **local LYM inequality**. `𝒜` takes up less of `α^(r)` (the finsets of card `r`)
than `∂𝒜` takes up of `α^(r - 1)`. -/
lemma card_div_choose_le_card_shadow_div_choose (hr : r ≠ 0) (h𝒜 : (𝒜 : set (finset α)).sized r) :
(𝒜.card : 𝕜) / (fintype.card α).choose r ≤ (∂𝒜).card / (fintype.card α).choose (r - 1) :=
begin
obtain hr' | hr' := lt_or_le (fintype.card α) r,
{ rw [choose_eq_zero_of_lt hr', cast_zero, div_zero],
exact div_nonneg (cast_nonneg _) (cast_nonneg _) },
replace h𝒜 := card_mul_le_card_shadow_mul h𝒜,
rw div_le_div_iff; norm_cast,
{ cases r,
{ exact (hr rfl).elim },
rw nat.succ_eq_add_one at *,
rw [tsub_add_eq_add_tsub hr', add_tsub_add_eq_tsub_right] at h𝒜,
apply le_of_mul_le_mul_right _ (pos_iff_ne_zero.2 hr),
convert nat.mul_le_mul_right ((fintype.card α).choose r) h𝒜 using 1,
{ simp [mul_assoc, nat.choose_succ_right_eq],
exact or.inl (mul_comm _ _) },
{ simp only [mul_assoc, choose_succ_right_eq, mul_eq_mul_left_iff],
exact or.inl (mul_comm _ _) } },
{ exact nat.choose_pos hr' },
{ exact nat.choose_pos (r.pred_le.trans hr') }
end
end local_lym
/-! ### LYM inequality -/
section lym
section falling
variables [decidable_eq α] (k : ℕ) (𝒜 : finset (finset α))
/-- `falling k 𝒜` is all the finsets of cardinality `k` which are a subset of something in `𝒜`. -/
def falling : finset (finset α) := 𝒜.sup $ powerset_len k
variables {𝒜 k} {s : finset α}
lemma mem_falling : s ∈ falling k 𝒜 ↔ (∃ t ∈ 𝒜, s ⊆ t) ∧ s.card = k :=
by simp_rw [falling, mem_sup, mem_powerset_len, exists_and_distrib_right]
variables (𝒜 k)
lemma sized_falling : (falling k 𝒜 : set (finset α)).sized k := λ s hs, (mem_falling.1 hs).2
lemma slice_subset_falling : 𝒜 # k ⊆ falling k 𝒜 :=
λ s hs, mem_falling.2 $ (mem_slice.1 hs).imp_left $ λ h, ⟨s, h, subset.refl _⟩
lemma falling_zero_subset : falling 0 𝒜 ⊆ {∅} :=
subset_singleton_iff'.2 $ λ t ht, card_eq_zero.1 $ sized_falling _ _ ht
lemma slice_union_shadow_falling_succ : 𝒜 # k ∪ ∂ (falling (k + 1) 𝒜) = falling k 𝒜 :=
begin
ext s,
simp_rw [mem_union, mem_slice, mem_shadow_iff, exists_prop, mem_falling],
split,
{ rintro (h | ⟨s, ⟨⟨t, ht, hst⟩, hs⟩, a, ha, rfl⟩),
{ exact ⟨⟨s, h.1, subset.refl _⟩, h.2⟩ },
refine ⟨⟨t, ht, (erase_subset _ _).trans hst⟩, _⟩,
rw [card_erase_of_mem ha, hs],
refl },
{ rintro ⟨⟨t, ht, hst⟩, hs⟩,
by_cases s ∈ 𝒜,
{ exact or.inl ⟨h, hs⟩ },
obtain ⟨a, ha, hst⟩ := ssubset_iff.1
(ssubset_of_subset_of_ne hst (ht.ne_of_not_mem h).symm),
refine or.inr ⟨insert a s, ⟨⟨t, ht, hst⟩, _⟩, a, mem_insert_self _ _, erase_insert ha⟩,
rw [card_insert_of_not_mem ha, hs] }
end
variables {𝒜 k}
/-- The shadow of `falling m 𝒜` is disjoint from the `n`-sized elements of `𝒜`, thanks to the
antichain property. -/
lemma _root_.is_antichain.disjoint_slice_shadow_falling {m n : ℕ}
(h𝒜 : is_antichain (⊆) (𝒜 : set (finset α))) :
disjoint (𝒜 # m) (∂ (falling n 𝒜)) :=
disjoint_right.2 $ λ s h₁ h₂,
begin
simp_rw [mem_shadow_iff, exists_prop, mem_falling] at h₁,
obtain ⟨s, ⟨⟨t, ht, hst⟩, hs⟩, a, ha, rfl⟩ := h₁,
refine h𝒜 (slice_subset h₂) ht _ ((erase_subset _ _).trans hst),
rintro rfl,
exact not_mem_erase _ _ (hst ha),
end
/-- A bound on any top part of the sum in LYM in terms of the size of `falling k 𝒜`. -/
lemma le_card_falling_div_choose [fintype α] (hk : k ≤ fintype.card α)
(h𝒜 : is_antichain (⊆) (𝒜 : set (finset α))) :
∑ r in range (k + 1),
((𝒜 # (fintype.card α - r)).card : 𝕜) / (fintype.card α).choose (fintype.card α - r)
≤ (falling (fintype.card α - k) 𝒜).card / (fintype.card α).choose (fintype.card α - k) :=
begin
induction k with k ih,
{ simp only [tsub_zero, cast_one, cast_le, sum_singleton, div_one, choose_self, range_one],
exact card_le_of_subset (slice_subset_falling _ _) },
rw succ_eq_add_one at *,
rw [sum_range_succ, ←slice_union_shadow_falling_succ,
card_disjoint_union h𝒜.disjoint_slice_shadow_falling, cast_add, _root_.add_div, add_comm],
rw [←tsub_tsub, tsub_add_cancel_of_le (le_tsub_of_add_le_left hk)],
exact add_le_add_left ((ih $ le_of_succ_le hk).trans $ card_div_choose_le_card_shadow_div_choose
(tsub_pos_iff_lt.2 $ nat.succ_le_iff.1 hk).ne' $ sized_falling _ _) _,
end
end falling
variables {𝒜 : finset (finset α)} {s : finset α} {k : ℕ}
/-- The **Lubell-Yamamoto-Meshalkin inequality**. If `𝒜` is an antichain, then the sum of the
proportion of elements it takes from each layer is less than `1`. -/
lemma sum_card_slice_div_choose_le_one [fintype α] (h𝒜 : is_antichain (⊆) (𝒜 : set (finset α))) :
∑ r in range (fintype.card α + 1), ((𝒜 # r).card : 𝕜) / (fintype.card α).choose r ≤ 1 :=
begin
classical,
rw ←sum_flip,
refine (le_card_falling_div_choose le_rfl h𝒜).trans _,
rw div_le_iff; norm_cast,
{ simpa only [nat.sub_self, one_mul, nat.choose_zero_right, falling]
using (sized_falling 0 𝒜).card_le },
{ rw [tsub_self, choose_zero_right],
exact zero_lt_one }
end
end lym
/-! ### Sperner's theorem -/
/-- **Sperner's theorem**. The size of an antichain in `finset α` is bounded by the size of the
maximal layer in `finset α`. This precisely means that `finset α` is a Sperner order. -/
lemma _root_.is_antichain.sperner [fintype α] {𝒜 : finset (finset α)}
(h𝒜 : is_antichain (⊆) (𝒜 : set (finset α))) :
𝒜.card ≤ (fintype.card α).choose (fintype.card α / 2) :=
begin
classical,
suffices : ∑ r in Iic (fintype.card α),
((𝒜 # r).card : ℚ) / (fintype.card α).choose (fintype.card α / 2) ≤ 1,
{ rwa [←sum_div, ←nat.cast_sum, div_le_one, cast_le, sum_card_slice] at this,
norm_cast,
exact choose_pos (nat.div_le_self _ _) },
rw [Iic_eq_Icc, ←Ico_succ_right, bot_eq_zero, Ico_zero_eq_range],
refine (sum_le_sum $ λ r hr, _).trans (sum_card_slice_div_choose_le_one h𝒜),
rw mem_range at hr,
refine div_le_div_of_le_left _ _ _; norm_cast,
{ exact nat.zero_le _ },
{ exact choose_pos (lt_succ_iff.1 hr) },
{ exact choose_le_middle _ _ }
end
end finset
| 194,228
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| 43,171
|
\begin{document}
\author{Ihyeok Seo}
\title[Unique continuation for Schr\"odinger operators]
{On unique continuation for Schr\"odinger operators of fractional and higher orders}
\address{School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea}
\email{ihseo@kias.re.kr}
\subjclass[2000]{Primary 35B60, 35J10.}
\thanks{\textit{Key words and phrases.} Unique continuation, Schr\"odinger operators.}
\begin{abstract}
In this note we study the property of unique continuation for solutions
of $|(-\Delta)^{\alpha/2}u|\leq|Vu|$, where
$V$ is in a function class of potentials including $\bigcup_{p>n/\alpha}L^p(\mathbb{R}^n)$ for $n-1\leq\alpha<n$.
In particular, when $n=2$, our result gives a unique continuation theorem
for the fractional ($1<\alpha<2$) Schr\"odinger operator $(-\Delta)^{\alpha/2}+V(x)$ in the full range of $\alpha$ values.
\end{abstract}
\maketitle
\section{Introduction}
As it is well-known, analytic functions that are representable by power series
have the property of unique continuation.
This means that they cannot vanish in any non-empty open set without being identically zero.
Note that a solution $u$ of the Cauchy-Riemann operator $\overline{\partial}$ in $\mathbb{R}^2$
(i.e., $\overline{\partial}u=0$) has the property since it is complex analytic.
The same result holds for the Laplace operator $\Delta$ in $\mathbb{R}^n$
since its solutions are harmonic functions that are still real analytic.
So it would be desirable to obtain such property for partial differential operators
whose solutions are not necessarily analytic, or even smooth.
Historically, the first such result is due to Carleman ~\cite{C}, who showed the property
for the Schr\"odinger operator $-\Delta+V(x)$ in $\mathbb{R}^2$
if $V\in L_{\textrm{loc}}^\infty(\mathbb{R}^2)$.
This was extended to higher dimensions $n\geq3$ by M\"{u}ller~\cite{M}.
Since then, a great deal of work was devoted to the case $V\in L_{\textrm{loc}}^p(\mathbb{R}^n)$, $p<\infty$.
This is because the potentials $V$ that arise in quantum physics need not be locally bounded
and more importantly it can be applied to the problem of absence of positive eigenvalues
of the Schr\"odinger operator.
Among others, Jerison and Kenig~\cite{JK} proved
the unique continuation for more general differential inequalities of the form $|\Delta u|\leq|Vu|$
when $V\in L_{\textrm{loc}}^{n/2}(\mathbb{R}^n)$ if $n>2$,
and $V\in \bigcup_{p>1}L_{\textrm{loc}}^p(\mathbb{R}^2)$ if $n=2$.
This result later turns out to be optimal in the context of $L^p$ potentials $V$ (\cite{KN,KT}),
and was extended to the higher orders
$|\Delta^m u|\leq|Vu|$ ($m\in\mathbb{N}$) when $V\in L_{\textrm{loc}}^{n/2m}(\mathbb{R}^n)$ if $n>2m$ (\cite{L}),
and $V\in \bigcup_{p>1}L_{\textrm{loc}}^p(\mathbb{R}^n)$ if $n=2m$ (\cite{Se}).
In this note we are concerned with more general cases
of fractional ($0<\alpha<2$) and higher ($\alpha>2$) orders:
\begin{equation}\label{FH}
|(-\Delta)^{\alpha/2}u|\leq|Vu|,
\end{equation}
where $(-\Delta)^{\alpha/2}$ is defined by means of the Fourier transform
$\mathcal{F}f$ (=\,$\widehat{f}$\,):
$$\mathcal{F}[(-\Delta)^{\alpha/2}f](\xi)=|\xi|^\alpha\widehat{f}(\xi).$$
To the best of our knowledge, all the known results on unique continuation for~\eqref{FH}
deal only with the case of even integers $\alpha=2m$, $m\in\mathbb{N}$.
When it comes to the other cases of $\alpha$, the difficulty comes from the fact that
$(-\Delta)^{\alpha/2}$ is a nonlocal operator
which means that $(-\Delta)^{\alpha/2}f(x)$ depends not just on $f(y)$ for $y$ near $x$ but on $f(y)$ for all $y$.
Moreover, it does not satisfy Leibnitz's rule of differentiation, in general.
In order to get around these difficulties,
we consider a function class $\mathcal{K}_\alpha$, $0<\alpha<n$, of potentials $V$ defined by
\begin{equation}\label{poten}
V\in\mathcal{K}_\alpha\quad\Leftrightarrow\quad
\lim_{r\rightarrow0}\sup_{x\in\mathbb{R}^n}\int_{|x-y|<r}\frac{|V(y)|}{|x-y|^{n-\alpha}}dy=0.
\end{equation}
The case $\alpha=2$ is just the usual Kato class introduced by Kato~\cite{K}
to study the self-adjointness of the Schr\"odinger operator.
Since then, it has played an important role in the study of many other properties of the operator (cf. \cite{Si}).
In order to extend these studies to $(-\Delta)^{\alpha/2}+V(x)$,
the class ~\eqref{poten} was introduced and used by several authors.
(See ~\cite{ZY} and the references therein.)
By making use of ~\eqref{poten}, we obtain here unique continuation results
for ~\eqref{FH} with $n-1\leq\alpha<n$.
Before stating our result precisely,
it should be emphasized that there are physical interests in the case $1<\alpha<2$
as well as the case $\alpha=2$.
Recently, following the path integral approach (\cite{FH}) to quantum mechanics,
Laskin~\cite{La,La2,La3} generalized the Feynman path integral to the L\'{e}vy one.
This generalization leads to fractional quantum mechanics governed by the fractional Schr\"odinger equation
$i\partial_t\psi=((-\Delta)^{\alpha/2}+V(x))\psi$, where $1<\alpha<2$.
The usual quantum mechanics corresponds to the case $\alpha=2$.
In particular, when $n=2$, Theorem~\ref{thm} below gives a unique continuation result for
the fractional Schr\"odinger operator $(-\Delta)^{\alpha/2}+V(x)$
in the full range of $\alpha$ values.
\begin{thm}\label{thm}
Let $n\geq2$ and $n-1\leq\alpha<n$.
Assume that $u$ is a nonzero solution of~\eqref{FH} such that
\begin{equation}\label{sol}
u\in L^1(\mathbb{R}^n)\quad\text{and}\quad(-\Delta)^{\alpha/2}u\in L^1(\mathbb{R}^n).
\end{equation}
Then it cannot vanish in any non-empty open set of $\mathbb{R}^n$ if $V$ is in the class $\mathcal{K}_\alpha$.
\end{thm}
\begin{rem}
It is currently conjectured\footnote{The conjecture was first formulated by B. Simon in his survey paper on Schr\"odinger semigroups~\cite{Si}.}
that ~\eqref{FH} for $\alpha=2$ has the unique continuation whenever $V\in\mathcal{K}_2$.
When $n=3$, it is known to be true (\cite{S}). In higher dimensions this conjecture is verified
for $\Delta u=Vu$ with radial potentials (\cite{FGL}).
More generally, it is natural to ask whether the theorem can hold for $0<\alpha<n-1$.
It should be also noted that the theorem can be applied to the stationary equation
$$((-\Delta)^{\alpha/2}+V(x))u=Eu,$$
and the same result holds for $E\in\mathbb{C}$ by noting that $(-\Delta)^{\alpha/2}u=(E-V(x))u$ and
the condition~\eqref{poten} is trivially satisfied for the constant $E$.
\end{rem}
Note that the class $\mathcal{K}_\alpha$ has the property that
$\bigcup_{p>n/\alpha}L^p(\mathbb{R}^n)\subset\mathcal{K}_\alpha\subset L_{\textrm{loc}}^1(\mathbb{R}^n)$.
In fact, if $V\in L^p(\mathbb{R}^n)$, we see that
\begin{equation}\label{pr1}
\sup_{x\in\mathbb{R}^n}\int_{|x-y|<r}|x-y|^{-(n-\alpha)}|V(y)|dy
\leq C\bigg(\int_{|y|<r}|y|^{-(n-\alpha)p^\prime}dy\bigg)^{1/p^\prime}
\end{equation}
by H\"{o}lder's inequality.
Using polar coordinates, if $p>n/\alpha$,
one can see that the right-hand side of ~\eqref{pr1} tends to $0$ as $r\rightarrow0$.
So it follows that $V\in\mathcal{K}_\alpha$.
On the other hand, if $V\in\mathcal{K}_\alpha$, there is $0<r_0<1$ so that
the left-hand side of ~\eqref{pr1} is less than $1$.
Hence we get
$$\sup_{x\in\mathbb{R}^n}\int_{|x-y|<r_0}|V(y)|dy\leq 1$$
since $|x|^{-(n-\alpha)}\geq1$ for $|x|<r_0$.
This implies that $V\in L_{\textrm{loc}}^1(\mathbb{R}^n)$.
\medskip
Throughout this paper, the letter $C$ stands for positive constants possibly different at each occurrence.
\section{Preliminary lemmas}
In this section we present some preliminary lemmas which will be used
for the proof of Theorem~\ref{thm}.
\begin{lem}\label{lemma}
Let $\phi_\alpha(y)=|y|^{-(n-\alpha)}$ for $0<\alpha<n$.
Then we have for $x\in\mathbb{R}^n$ and $N\geq1$,
\begin{equation}\label{7}
u(x)=C\int_{\mathbb{R}^n}
\big[\phi_\alpha(x-y)-\sum_{k=0}^{N-1}\frac{(x\cdot\nabla)^k}{k!}\phi_\alpha(-y)\big](-\Delta)^{\alpha/2}u(y)dy
\end{equation}
if $u$ satisfies~\eqref{sol} and has a compact support in $\mathbb{R}^n\setminus\{0\}$.
\end{lem}
\begin{proof}
It is enough to show that ~\eqref{7} holds for $u\in C_0^\infty(\mathbb{R}^n\setminus\{0\})$.
The remaining follows from this and a standard limiting argument involving a $C_0^\infty$ approximate identity.
First we claim that
\begin{equation}\label{1}
u(x)=C\int_{\mathbb{R}^n}\phi_\alpha(x-y)(-\Delta)^{\alpha/2}u(y)dy.
\end{equation}
Indeed, using the well-known fact (cf.~\cite{W}, p.23) that
$\widehat{|x|^{-\alpha}}(\xi)=C|\xi|^{-(n-\alpha)}$ in the sense of distributional Fourier transforms,
we see that
\begin{align*}
u(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\widehat{u}(\xi)d\xi
&=\int_{\mathbb{R}^n}|\xi|^{-\alpha}\mathcal{F}\big[(-\Delta)^{\alpha/2}u(\cdot+x)\big](\xi)d\xi\\
&=C\int_{\mathbb{R}^n}|y|^{-(n-\alpha)}(-\Delta)^{\alpha/2}u(y+x)dy,
\end{align*}
which gives ~\eqref{1}.
Now, we note that the $(N-1)^{th}$ degree Taylor polynomial of $u$ at $0$ must be zero,
since $u$ vanishes near $x=0$.
That is to say,
$$\sum_{|\beta|\leq N-1}\frac{D^\beta u(0)}{\beta!}x^\beta\equiv0,$$
where $\beta$ is the usual multiindex notation.
By ~\eqref{1}, this can be rewritten as
\begin{equation}\label{sdfs2}
C\int_{\mathbb{R}^n}\sum_{|\beta|\leq N-1}\frac{D^\beta \phi_\alpha(-y)}{\beta!}x^\beta(-\Delta)^{\alpha/2}u(y)dy
\equiv0.
\end{equation}
Then, we subtract ~\eqref{sdfs2} from both sides of ~\eqref{1} to conclude
\begin{equation*}
u(x)=C\int_{\mathbb{R}^n}\big[\phi_\alpha(x-y)-\sum_{|\beta|\leq N-1}\frac{D^\beta \phi_\alpha(-y)}{\beta!}x^\beta\big](-\Delta)^{\alpha/2}u(y)dy
\end{equation*}
which is same as ~\eqref{7}.
\end{proof}
Nextly, we recall from ~\cite{S} the following estimate on Taylor polynomial approximations to
$|x|^{-\beta}$, $\beta>0$.
\begin{lem}\label{lem}
Let $\psi_\beta(x)=|x|^{-\beta}$ for $0<\beta\leq1$.
Then one has
\begin{equation}\label{Taylor}
\Big|\psi_\beta(x-y)-\sum_{k=0}^{N-1}\frac{(x\cdot\nabla)^k}{k!}\psi_\beta(-y)\Big|\leq
C\Big(\frac{|x|}{|y|}\Big)^N\psi_\beta(x-y)
\end{equation}
for $x,y\in\mathbb{R}^n$ and $N\geq1$.
Moreover, this estimate is not valid if $\beta>1$.
\end{lem}
\section{Proof of Theorem~\ref{thm}}\label{Proof}
Without loss of generality, we need to show that $u$ must be identically zero
if $u$ vanishes in a sufficiently small neighborhood of $0\in\mathbb{R}^n$.
By using ~\eqref{FH}, ~\eqref{7} and ~\eqref{Taylor} with $\beta=n-\alpha$, if $n-1\leq\alpha<n$,
we see that
\begin{equation}\label{121}
|u(x)|\leq C\int_{\mathbb{R}^n}\big(\frac{|x|}{|y|}\big)^N\phi_\alpha(x-y)|V(y)u(y)|dy.
\end{equation}
Let $f(y)=|y|^{-N}|V(y)u(y)|$.
Since $u$ vanishes near the origin, it follows that
\begin{equation}\label{15789}
\|f\|_{L^1}=\int_{\mathbb{R}^n}|y|^{-N}|V(y)u(y)|dy<\infty.
\end{equation}
Here, we also used the fact (see Theorem 2.2 in~\cite{ZY}) that the condition ~\eqref{sol} implies
$Vu\in L^1$ if $V\in \mathcal{K}_\alpha$.
Hence, from~\eqref{121} we get
\begin{align}\label{lme}
\int_{|x|<r}|V(x)||x|^{-N}|u(x)|dx
\nonumber&\leq C\int_{\mathbb{R}^n}\bigg(\int_{|x|<r}\phi_\alpha(x-y)|V(x)|dx\bigg)f(y)dy\\
&\leq C\bigg(\sup_{y\in\mathbb{R}^n}\int_{|x|<r}\phi_\alpha(x-y)|V(x)|dx\bigg)\|f\|_{L^1}.
\end{align}
Now, we set
$$\eta(r)=\sup_{y\in\mathbb{R}^n}\int_{|x|<r}\phi_\alpha(x-y)|V(x)|dx.$$
Then the condition~\eqref{poten} implies $\lim_{r\rightarrow0}\eta(r)=0$.
In fact, we note that
$$\sup_{|y|<2r}\int_{|x|<r}\phi_\alpha(x-y)|V(x)|dx
\leq\sup_{y\in\mathbb{R}^n}\int_{|x-y|<4r}\phi_\alpha(x-y)|V(x)|dx$$
and
\begin{align*}
\sup_{|y|\geq2r}\int_{|x|<r}\phi_\alpha(x-y)|V(x)|dx
&\leq Cr^{\alpha-n}\int_{|x|<r}|V(x)|dx\\
&\leq Cr^{\alpha-n}r^{n-\alpha}\int_{|x|<r}\frac{|V(x)|}{|x|^{n-\alpha}}dx.
\end{align*}
Then it follows from ~\eqref{poten} that
$$\lim_{r\rightarrow0}\eta(r)\leq C\lim_{r\rightarrow0}\sup_{y\in\mathbb{R}^n}\int_{|x-y|<4r}\frac{|V(x)|}{|x-y|^{n-\alpha}}dx=0.$$
Hence, if we choose $r_0>0$ small enough, we see from ~\eqref{lme} that
\begin{equation*}
\int_{|x|<r_0}|V(x)||x|^{-N}|u(x)|dx\leq\frac12\|f\|_{L^1}.
\end{equation*}
Combining this with ~\eqref{15789}, we get
\begin{equation*}
\int_{|x|<r_0}|V(x)||x|^{-N}|u(x)|dx\leq\int_{|y|\geq r_0}|y|^{-N}|V(y)u(y)|dy,
\end{equation*}
so that
\begin{equation}\label{dks2}
\int_{|x|<r_0}|V(x)|\big(\frac{r_0}{|x|}\big)^N|u(x)|dx\leq\|Vu\|_{L^1}<\infty.
\end{equation}
Here we may assume that $|V|\geq1$,
since $|V|+1$ also satisfies ~\eqref{FH} and $\lim_{r\rightarrow0}\eta(r)=0$.
Indeed, to show the second one for $|V|+1$, we only need to show
\begin{equation}\label{357}
\lim_{r\rightarrow0}\sup_{y\in\mathbb{R}^n}\int_{|x|<r}\phi_\alpha(x-y)dx=0.
\end{equation}
Since $\phi_\alpha(x)=|x|^{-(n-\alpha)}$, it follows that
\begin{align*}
\sup_{y\in\mathbb{R}^n}\int_{|x|<r}\phi_\alpha(x-y)dx
&\leq\sup_{|y|<2r}\int_{|x|<r}|x-y|^{-(n-\alpha)}dx
+\sup_{|y|\geq2r}\int_{|x|<r}r^{-(n-\alpha)}dx\\
&\leq\sup_{y\in\mathbb{R}^n}\int_{|x-y|<4r}|x-y|^{-(n-\alpha)}dx+Cr^\alpha\\
&\leq Cr^\alpha.
\end{align*}
This gives ~\eqref{357}.
Therefore, from ~\eqref{dks2} we see that
$$\int_{|x|<\frac{r_0}2}2^N|u(x)|dx\leq
\int_{|x|<r_0}|V(x)|\big(\frac{r_0}{|x|}\big)^N|u(x)|dx<\infty.$$
By letting $N\rightarrow\infty$,
we get that $u$ vanishes in $\{|x|<\frac{r_0}2\}$.
Then, using a standard connectedness argument, we can conclude that $u$ must be identically zero in $\mathbb{R}^n$.
This completes the proof.
\bibliographystyle{plain}
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\begin{document}
\title[\textbf{On some products of finite groups}]{\textbf{On some products of finite groups}}
\author[A. Ballester-Bolinches]{A. Ballester-Bolinches}
\address{Department of Mathematics, Guangdong University of Education, 510310, Guangzhou,
People’s Republic of China}
\address{Departament de Matem\`atiques, Universitat de Val\`encia, Dr. Moliner 50, 46100 Burjassot, Val\`encia, Spain}
\email{adolfo.ballester@uv.es}
\author[S. Y. Madanha]{S. Y. Madanha}
\address{Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa}
\email{sesuai.madanha@up.ac.za}
\author[M. C. Pedraza-Aguilera]{M. C. Pedraza-Aguilera}
\address{Instituto Universitario de Matem\'atica Pura y Aplicada, Universitat Polit\`ecnica de Val\`encia, 46022 Camino de Vera, Val\`encia, Spain}
\email{mpedraza@mat.upv.es}
\author[X. Wu]{X . Wu}
\address{School of Mathematics, Suzhou University, Suzhou 215006 Jiang, People’s Republic of China}
\email{wxwjs1991@126.com}
\thanks{}
\subjclass[2010]{Primary 20D10, 20D20}
\date{\today}
\keywords{Finite groups, semidirect products, supersoluble groups, residuals}
\begin{abstract}
A classical result of Baer states that a finite group $ G $ which is the product of two normal supersoluble subgroups is supersoluble if and only if $ G' $ is nilpotent. In this article we show that if $ G=AB $ is the product of supersoluble (respectively, $ w $-supersoluble) subgroups $ A $ and $ B $, $ A $ is normal in $ G $, $ B $ permutes with every maximal subgroup of each Sylow subgroup of $ A $, then $ G $ is supersoluble (respectively, $ w $-supersoluble) provided that $ G' $ is nilpotent. We also investigate products of subgroups defined above when $ A\cap B=1 $ and obtain more general results.
\end{abstract}
\maketitle
\section{Introduction}
All groups considered here will be finite.
A significant number of articles investigating the properties of groups expressible as a product of two supersoluble subgroups were published since the $1957$ paper by Baer \cite{Bae57} in which he proved that a normal product $G = AB$ of two supersoluble subgroups $A$ and $B$ is supersoluble provided that the derived subgroup $G'$ is nilpotent. There has been many generalisations of this theorem. Instead of having normal subgroups, certain permutability conditions were imposed on the factors. The case in which $A$ permutes with every subgroup of $B$ and $B$ permutes with every subgroup of $A$, that is, when $G$ is a \emph{mutually permutable product} of $A$ and $B$ is in fact one of the most interesting cases and has been investigated in detail (see \cite{BBERA10}, for a thorough review of results in this context and also \cite{AFG92} for general results on products).
In this article we study a weak form of a normal product arising quite frequently in the structural study of mutually permutable products and appears not to have been investigated in detail.
\begin{defn}
Let $G=AB$ be a product of subgroups $A$ and $B$. We say that $G$ is a \emph{weak normal product} of $A$ and $B$ if
\begin{itemize}
\item[(a)] $A$ is normal in $G$.
\item[(b)] $B$ permutes with all the maximal subgroups of Sylow subgroups of $A$.
\end{itemize}
\end{defn}
As an important first step in the study of weak normal products $G = AB$, and motivated by the mutually permutable case, we analyse the situation $A \cap B=1$. In this case they are semidirect products of $A$ and $B$.
\begin{defn} Let the group $G = AB$ be the weak normal product of $A$ and $B$ with $A$ normal in $G$. We say that $ G $ is a \emph{weak direct product} of $A$ and $B$ if $A \cap B = 1$. In this case we write $G = [A]B$.
\end{defn}
We study these products when the factors are supersoluble and widely supersoluble and analyse the behaviour of the residuals associated to these classes of groups. Recall that a widely supersoluble group, or $w$-\emph{supersoluble} group for short, is defined as a group $ G $ such that every Sylow subgroup of $G$ is $\mathbb{P}$-subnormal in $G$ (a subgroup $H$ of a group $G$ is $\mathbb{P}$-subnormal in $G$ whenever either $H=G$ or there exists a chain of subgroups $H=H_{0} \leqslant H_{1} \leqslant \cdots \leqslant H_{n-1} \leqslant H_{n}=G$, such that $| H_{i} {:} H_{i-1}|$ is a prime for every $i=1, \dots, n$).
The class of $w$-supersoluble groups, denoted $w\mathfrak{U} $, is a subgroup-closed saturated formation containing the subgroup-closed saturated formation $\mathfrak{U} $ of all supersoluble groups. Moreover $w$-supersoluble groups have a Sylow tower of supersoluble type (see \cite[Corollary]{VVT10}). \\
Our first aim is to show that the saturated formations of all supersoluble groups and w-supersoluble groups are closed under the formation of weak direct products.
\begin{athm}\label{A}
Let $ G =[A]B $ be a weak direct product of $ A $ and $ B $. If $ A $ and $ B $ belong to $ \mathfrak{U} $, then $ G $ is also supersoluble.
\end{athm}
\begin{CorA}\label{Gwsupersoluble}
Let $ G =[A]B $ be a weak direct product of $ A $ and $ B $. If $ A $ and $ B $ belong to $ w\mathfrak{U} $, then $ G $ is w-supersoluble.
\end{CorA}
Our second aim is to show that the product of the supersoluble (respectively, w-supersoluble) residuals of the factors of weak direct products is just the supersoluble (respectively, w-supersoluble) residual of the group.
\begin{athm}\label{B}
Suppose that $ \mathfrak{F} =\mathfrak{U}$ or $ \mathfrak{F} =w\mathfrak{U}$. Let $ G =[A]B $ be a weak direct product of $ A $ and $ B $. Then
\begin{center}
$ G^{\mathfrak{F}}=A^{\mathfrak{F}}B^{\mathfrak{F}} $.
\end{center}
\end{athm}
We now analyse the behaviour of weak normal products with respect to the formations of all supersoluble and w-supersoluble groups. Our next result shows that Baer's theorem can be generalised in this new direction:
\begin{athm}\label{C}
Let $ G=AB $ be a weak normal product of A and B. If $ G'$ is nilpotent, $A$ is normal in $G$, and $ A, B\in \mathfrak{U} $, then $ G \in \mathfrak{U}$.
\end{athm}
As a corollary, we obtain the result for $w \mathfrak{U} $-groups.
\begin{CorB}\label{G'nilpotentwsupersoluble}
Let the group $G=AB$ be a weak normal product of $ w\mathfrak{U} $-subgroups $A$ and $B$. If $G'$ is nilpotent and $A$ is normal in $G$, then $G$ belongs to $w \mathfrak{U} $.
\end{CorB}
Our second objective is to investigate the residuals of weak normal products. Unfortunately, it does not follow that $ G^{\mathfrak{U}}=A^{\mathfrak{U}}B^{\mathfrak{U}} $ when $ G $ is a weak normal product as the following example shows:
\begin{example}\label{counterweaknormal}
Let
\begin{align*}A=\langle g_2,g_4,g_5,g_6,g_7\mid \mbox{}&g_2^3=g_4^3=g_5^3=g_6^3=g_7^3=1,\\
&g_4^{g_2}=g_4g_6, g_5^{g_2}=g_5g_7, g_6^{g_2}=g_6, g_7^{g_2}=g_7,\\
&g_5^{g_4}=g_5, g_6^{g_4}=g_6, g_7^{g_4}=g_7, \\
&g_6^{g_5}=g_6, g_7^{g_5}=g_7,\\& g_7^{g_6}=g_7\rangle.
\end{align*}
Let $Q=\langle b\rangle\cong C_4$ act on $A$ via
\begin{align*}
g_2^b&=g_2,&g_4^{b}&=g_4g_5,&g_5^b&=g_4g_5^2,&g_6^{b}&=g_6g_7,&g_7^{b}&=g_6g_7^2.
\end{align*}
Let $G=[A]Q$ be the corresponding semidirect product.
Note that $A'=\Phi(A)=\langle g_6, g_7\rangle$. Let $A_0=\langle g_4, g_5\rangle$. Then $A_0$ is not a normal subgroup of $A$, but is normalised by $Q$. Let $B=A_0\langle b\rangle$, then $Core_G(B)=1$. Furthermore, $B$ permutes with the $13$ maximal subgroups of $A$. The supersoluble residual of $G$ is $\langle g_4, g_5, g_6,g_7\rangle$, giving a quotient isomorphic to $C_{12}$. Consequently $ G^{\mathfrak{U}}\neq A^{\mathfrak{U}}B^{\mathfrak{U}} $. This group corresponds to \texttt{SmallGroup(972, 406)} of GAP.
\end{example}
We prove the following:
\begin{athm}\label{D}
Let the group $G=AB$ be a product of the subgroups $A$ and $B$. Assume that $A$ is a normal subgroup of $G$ and every Sylow subgroup of $ B $ permutes with every maximal subgroup of every Sylow subgroup of $ A $. If $G'$ is nilpotent, then $ G^{\mathfrak{U}}=A^{\mathfrak{U}}B^{\mathfrak{U}} $.
\end{athm}
\begin{CorC}
Let the group $G=AB$ be a product of the subgroups $A$ and $B$. Assume that $A$ is a normal subgroup of $G$ and every Sylow subgroup of $ B $ permutes with every maximal subgroup of every Sylow subgroup of $ A $. If $G'$ is nilpotent, then $ G^{w \mathfrak{U}}=A^{w \mathfrak{U}}B^{w \mathfrak{U}} $.
\end{CorC}
Denote by $ \mathfrak{N} $ the class of all nilpotent groups. A nice result of Monakhov \cite[Theorem 1]{M18} states that if $ G=AB $ is the mutually permutable product of the supersoluble subgroups $ A $ and $ B $, then $ G^{\mathfrak{U}}=(G')^{\mathfrak{N}}=[A, B]^{\mathfrak{N}} $. We prove an analogue of this result for weak normal products.
\begin{CorD}\label{U-residual}
Let $G=AB$ be a weak normal product of the supersoluble subgroups $A$ and $B$. If $A$ is normal in $G$, we have that $G^{\mathfrak{U}}=(G')^{\mathfrak{N}}=[A,B]^{\mathfrak{N}}$.
\end{CorD}
\section{Preliminary Results}
It is easy to see that factor groups of weak normal products are also weak normal products. For weak direct products we have the following:
\begin{lemma}\label{factor}
Let $ G =[A]B $ be a weak direct product of $ A $ and $ B $.
\begin{itemize}
\item[(a)] If $ N $ is a normal subgroup of $ G $ such that $ N\leqslant A $ or $ N\leqslant B $, then $ G/N=[AN/N](BN/N) $ is a weak direct product of $AN/N$ and $BN/N$.
\item[(b)] If $ K $ is a subgroup of $ B $, then $ [A]K $ is a weak direct product of $ A $ and $ K $.
\end{itemize}
\end{lemma}
\begin{proof} (a) Let $H/N$ be a Sylow $p$-subgroup of $AN/N$. Then $H/N=PN/N$, where $P$ is a Sylow $p$-subgroup of $A$. Let $K/N$ be a maximal subgroup of $H/N$. Then $K=K\cap PN=N(P\cap K)$ and $K/N=N(P\cap K)/N$. Thus $$p=|PN/N:(P\cap K)N/N|=\frac{|P||N|}{|P\cap N|} \frac{|P\cap K\cap N|}{|P\cap K| |N|}=|P: P\cap K|.$$ Hence $P\cap K$ is a maximal subgroup of $P$. Then $B$ permutes with $P\cap K$ and so $BN/N$ permutes with $K/N$. Therefore $ G/N=[AN/N](BN/N) $ is a weak direct product of $AN/N$ and $BN/N$.
(b) Let $K$ be any proper subgroup of $B$ and $H$ be any maximal subgroup of a Sylow subgroup of $A$. By the hypotheses, we have $HB = BH$ and so $ H = H(A \cap B) = A \cap HB $. Since $ A $ is normal in $G$, it follows that $H$ is normal in $HB $ and so $ B$ normalizes $H$. Hence $ K $ permutes with $ H $. Therefore $ [A]K $ is a weak direct product of $ A $ and $ K $.
\end{proof}
Our second lemma contains some of the properties of $\mathbb{P}$-subnormal subgroups.
\begin{lemma}\label{subnormal}\cite[Lemma 1.4]{VVT10}
Let $G$ be a soluble group and $H$ and $K$ two subgroups of $G$. The following properties hold:
\begin{itemize}
\item[(i)] If $H$ is $\mathbb{P}$-subnormal in $G$ and $N$ is normal in $G$ then $HN/N$ is $\mathbb{P}$-subnormal $G/N$.
\item[(ii)] If $N$ is normal in $G$ and $HN/N$ is $\mathbb{P}$-subnormal in $G/N$ then $HN$ is $\mathbb{P}$-subnormal in $G$.
\item[(iii)] If $H$ is $\mathbb{P}$-subnormal in $K$ and $K$ is $\mathbb{P}$-subnormal in $G$ then $H$ is $\mathbb{P}$-subnormal in $G$.
\end{itemize}
\end{lemma}
\section{Supersoluble and w-supersoluble residuals }
We start this section by proving Theorem A.
\begin{proof}[\textbf{Proof of Theorem A}]
Assume that the result is false and let $G$ be a counterexample of minimal order. Clearly, $G$ is soluble and $A$ and $B$ are proper subgroups of $G$. Let $N$ be a minimal normal subgroup of $G$ contained in $A$, then applying Lemma~\ref{factor}(a), $G/N=[A/N] (BN/N)$ is a weak direct product of $A/N$ and $BN/N$.
By the minimality of $G$, $G/N \in \mathfrak{U}$. Since the class of all supersoluble groups is a saturated formation, there exists a unique minimal normal subgroup $N$ of $G$ contained in $A$, $N$ a $p$-group for some prime $p$, $|N| > p$, and $\Phi(A) = 1$. Since $A$ is supersoluble, $A$ has a normal Sylow subgroup and since $N$ is the unique minimal normal subgroup of $G$ contained in $A$, it follows that $\textbf{F}(A)$ is a $p$-group and $\textbf{F}(A)$ is an elementary abelian Sylow $p$-subgroup of $A$.
Assume that $A$ is not a $p$-group. Then $\textbf{F}(A)$ is a completely reducible $A$-module and so $ \textbf{F}(A) = N \times Z$, for some $A$-module $Z$. Let $L$ be a minimal normal subgroup of $A$ contained in $N$. Then $N = L \times D$, for some $A$-module $D$. Then $\textbf{F}(A) = L \times DZ$, and $ DZ $ is a maximal subgroup of $\textbf{F}(A)$ because $L$ is of prime order. Therefore $ E = DZ$ permutes with $B$. Hence $ DZ=A\cap (DZ)B $ and so $ DZ $ is normalised by $B$. Since $DZ$ is also normalised by $A$, it follows that $DZ$ is a normal subgroup of $G$. The minimality of $N$ forces $D = 1$ and so $N$ is of prime order, which is a contradiction. Consequently, $A$ is an elementary abelian $p$-group. Note that $A$ cannot be cyclic since $|N| > p$. Let $1 \neq X$ be a maximal subgroup of $A$. Arguing as above, we have that $X$ is normal in $XB$ so that $X$ is normalised by $B$. Hence $X$ is normal in $G$ because $A$ is abelian. Therefore $N$ is contained in $X$ and so $N \leq \Phi(A)= 1$, our final contradiction.
\end{proof}
\begin{proof}[\textbf{Proof of Corollary A}]
Assume, by way of contradiction, that the result fails, and let $G$ be a counterexample of least order. Clearly $G$ is soluble and $A$ and $B$ are proper subgroups of $G$. Since the class of all w-supersoluble groups is a saturated formation, we can argue as in Theorem~\ref{A} to conclude that there exists a unique minimal normal subgroup $N$ of $G$ contained in $A$, $N$, and $N$ is a $p$-group for some prime $p$. Moreover, $\Phi(A) = 1$, and $A_p = \textbf{F}(A)$ is the Sylow $p$-subgroup of $A$. By the minimality of $G$, $G/N$ is $w$-supersoluble. Let $P$ be a Sylow $p$-subgroup of $G$. Then $P/N$ is $\mathbb{P}$-subnormal in $G/N$. By Lemma \ref{subnormal}(ii), $P$ is $\mathbb{P}$-subnormal in $G$. Suppose that for every prime $q \neq p$ dividing $|G|$ and every Sylow $q$-subgroup $B_{q}$ of $B$ we have that $AB_{q}$ is a proper subgroup of $G$. Let $A_{q}$ be a Sylow $q$-subgroup of $A$ such that $G_{q}=A_{q}B_{q}$ is a Sylow $q$-subgroup of $G$. Since $G/N$ is $w$-supersoluble, it follows that $G_{q}N$ is $\mathbb{P}$-subnormal in $G$. By Lemma \ref{factor}(b), $AB_{q}$ satisfies the hypotheses of the theorem. Hence $AB_{q}$ is $w$-supersoluble by the choice of $G$. Thus $G_{q}N \leq AB_{q}$ is $w$-supersoluble. Consequently, $G_{q}$ is $\mathbb{P}$-subnormal in $G_{q}N$ which is $\mathbb{P}$-subnormal in $G$. Applying Lemma \ref{subnormal}(iii), $G_{q}$ is $\mathbb{P}$-subnormal in $G$. Therefore the Sylow subgroups of $G$ are $\mathbb{P}$-subnormal in $G$ and so $G$ is $w$-supersoluble, a contradiction. Thus we may assume there exists $q \neq p$ such that $G=AB_{q}$. Let $T=A_{p}G_{q}=(A_{p}A_{q})B_{q}$. Since $A$ is normal in $G$, we have that $A_{q}$ is normal in $G_{q}$ and then $A_{p}A_{q}$ is normalized by $B_{q}$. Moreover, $A_{p}A_{q}$ is a $w$-supersoluble metanilpotent subgroup of $G$. By \cite[Theorem 2.13(1)]{VVT10}, $A_{p}A_{q}$ is supersoluble. It is clear that $T$ is a weak direct product of the supersoluble subgroups $A_{p}A_{q}$ and $B_{q}$. Applying Theorem~\ref{A}, it follows that $T$ is supersoluble. Therefore $T$ is $w$-supersoluble. But $G_{q}N \leq T$ which is $w$-supersoluble. Thus $G_{q}$ is $\mathbb{P}$-subnormal in $G_{q}N$ which is $\mathbb{P}$-subnormal in $G$. Again the application of Lemma \ref{subnormal}(iii) yields $G_{q}$ is $\mathbb{P}$-subnormal in $G$. If $G_{r}$ is a Sylow $r$-subgroup of $G$ for some prime $r \neq p, q$, then $G_{r}$ is contained in $A$ and so $G_{r}$ is $\mathbb{P}$-subnormal in $A$. Since $A$ is also $\mathbb{P}$-subnormal in $G$, we have that $G_{r}$ is $\mathbb{P}$-subnormal in $G$. Consequently, every Sylow subgroup of $G$ is $\mathbb{P}$-subnormal in $G$ and $G$ is w-supersoluble. This final contradiction completes the proof of the corollary.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem B}]
Suppose that the result is not true and let $ G $ be a minimal counterexample. Then\\ \\
\textit{(i) $ A \in \mathfrak{F} $, $ B^{\mathfrak{F}}\not= 1 $, $ Core_{G}(B)=1 $ and $ G^{\mathfrak{F}}=B^{\mathfrak{F}}N $ for every minimal normal subgroup $ N $ of $ G $ such that $ N\leqslant A $.}
Let $ N $ be a minimal normal subgroup of $ G $ such that $ N\leqslant A $ or $ N\leqslant B $. Then $ G/N=[AN/N](BN/N) $ is a weak direct product of $AN/N$ and $BN/N$ by Lemma~\ref{factor}(a). The minimal choice of $G$ implies that $ G^{\mathfrak{F}}N/N=(A^{\mathfrak{F}}N/N) (B^{\mathfrak{F}}N/N) $, that is, $ G^{\mathfrak{F}}N=A^{\mathfrak{F}}B^{\mathfrak{F}}N $. Since $ G/G^{\mathfrak{F}}\in \mathfrak{F} $, $ AG^{\mathfrak{F}}/G^{\mathfrak{F}} $ and $ BG^{\mathfrak{F}}/G^{\mathfrak{F}} $ also belong to $ \mathfrak{F} $ and then $ A^{\mathfrak{F}}\leqslant G^{\mathfrak{F}} $ and $ B^{\mathfrak{F}}\leqslant G^{\mathfrak{F}} $. If $ G^{\mathfrak{F}}\cap N= 1 $, then $ G^{\mathfrak{F}}=A^{\mathfrak{F}}B^{\mathfrak{F}}(G^{\mathfrak{F}}\cap N)=A^{\mathfrak{F}}B^{\mathfrak{F}} $, a contradiction. Hence $ G^{\mathfrak{F}}=A^{\mathfrak{F}}B^{\mathfrak{F}} N $ for every minimal normal subgroup $ N $ of $ G $ such that $ N\leqslant A $ or $ N\leqslant B $. If $A^{\mathfrak{F}} \neq 1$, then there exists a minimal normal subgroup $N$ of $G$ contained in $A^{\mathfrak{F}}$ because $A^{\mathfrak{F}}$ is normal in $G$. This contradiction yields $A \in \mathfrak{F}$ and $ G^{\mathfrak{F}}=B^{\mathfrak{F}}N $ for every minimal normal subgroup $ N $ of $ G $ such that $ N\leqslant A $ or $ N\leqslant B $. If $ B \in \mathfrak{F} $, then $ G\in \mathfrak{F} $ by Theorem~\ref{A} and Corollary~A, contrary to assumption. Hence $ B^{\mathfrak{F}} \neq 1 $.
Suppose that $Core_{G}(B) \neq 1$. Let $N$ be a minimal normal subgroup of $G$ contained in $B$ and let $R$ be a minimal normal subgroup of $G$ contained in $A$. Then $G^{\mathfrak{F}}=B^{\mathfrak{F}}N \cap B^{\mathfrak{F}}R \leq B \cap B^{\mathfrak{F}}R=B^{\mathfrak{F}}(B \cap R)=B^{\mathfrak{F}}$, a contradiction. Consequently we have that $Core_{G}(B)=1$. \\ \\
\textit{(ii) $ \mathbf{F}(A)$ is a Sylow $p$-subgroup of $A$, where $ p $ is the largest prime dividing $ |A| $ .}
Since $ A \in \mathfrak{F} $, it follows that $A$ is a Sylow tower group of supersoluble type. In particular, $ 1 \neq \mathbf{O}_{p}(A)$ is the Sylow $p$-subgroup of $A$, where $ p $ is the largest prime dividing $ |A| $. If $ \mathbf{F}(A) $ is not a $ p $-group, then $1 \neq \mathbf{O}_{q}(A)\leqslant \mathbf{O}_{q}(G)$. Let $ N_{1} $ be a minimal normal subgroup of $ G $ contained in $\mathbf{O}_{p}(A) $ and let $ N_{2} $ be a minimal normal subgroup of $ G $ contained in $\mathbf{O}_{q}(A) $. Then $ G^{\mathfrak{F}}=B^{\mathfrak{F}}N_{1}=B^{\mathfrak{F}}N_{2}$, which is a contradiction since $ B^{\mathfrak{F}}\cap N_{1}=B^{\mathfrak{F}}\cap N_{2}=1 $. Therefore $ \mathbf{F}(A) = \mathbf{O}_{p}(A)$ is the Sylow $p$-subgroup of $A$.\\ \\
\textit{(iii) $ G $ is soluble, $ AK $ belongs to $ \mathfrak{F} $ for every proper subgroup $ K $ of $ B $; in particular, $B$ is a minimal non-supersoluble group and $ B^{\mathfrak{F}} $ is a $ q $-subgroup of $ B $ for some prime $ q $.}
Suppose that $ K $ is a proper subgroup of $ B $. By Lemma \ref{factor}, $AK$ satisfies the hypotheses of the theorem and so $ (AK)^{\mathfrak{F}}=K^{\mathfrak{F}} $ by the minimal choice of $G$. Since $ (AK^{x})^{\mathfrak{F}}=(K^{x})^{\mathfrak {F}}=(K^{\mathfrak{F}})^{x} $ for any $ x\in B $, it follows that $ A $ normalizes $ (K^{\mathfrak{F}})^{x} $. Thus $ A $ normalizes $ \langle (K^{\mathfrak{F}})^{x} \mid x\in B \rangle $. Then $ \langle (K^{\mathfrak{F}})^{x} \mid x\in B \rangle \lhd G $, contrary to $ Core_{G}(B)=1 $. Hence $ (K^{\mathfrak{F}})^{x}=1 $. Consequently, $ AK $ belongs to $ \mathfrak{F} $. This shows that $ B $ is $ \mathfrak{F} $-critical and by \cite[Theorem 2.9]{VVT10}, we have that $ B $ is a minimal non-supersoluble group. By \cite[Theorem~10]{BBER07}, we have that $ B^{\mathfrak{F}} $ is a $ q $-group for some prime $ q $. In particular, $B$ and then $G$ are soluble.\\ \\
\textit{(iv) $ G^{\mathfrak{F}}=B^{\mathfrak{F}}\times N $ is an elementary abelian $ p $-group.}
Applying (iii), it follows that $B^{\mathfrak{F}}$ is a $q$-group for some prime $q$. Let $N$ be a minimal normal subgroup of $G$ contained in $A$. Then $G^{\mathfrak{F}}=B^{\mathfrak{F}}N$ by (i), and $N$ is a $p$-group by (ii).
Suppose that $B^{\mathfrak{F}}$ is a normal subgroup of $G^{\mathfrak{F}}$. Then $G^{\mathfrak{F}}/B^{\mathfrak{F}}$ is an elementary abelian $p$-group. Consequently, the residual $X$ of $G^{\mathfrak{F}}$ associated to the formation of all elementary abelian $p$-groups is a normal subgroup of $G$ contained in $B$. Hence $X \leq Core_G(B) = 1$ and $G^{\mathfrak{F}}$ is an elementary abelian $p$-group.
Assume that $p \neq q$. Let $N$ be a minimal normal subgroup of $G$ contained in $A$. Then $G^{\mathfrak{F}}=B^{\mathfrak{F}}N$ and $N$ is a $p$-group by (ii). Hence $B^{\mathfrak{F}}$ is a Sylow $q$-subgroup of $ G^{\mathfrak{F}}=B^{\mathfrak{F}}N $. Applying Frattini's argument, we have that $G=G^{\mathfrak{F}}N_{G}(B^{\mathfrak{F}})=NN_{G}(B^{\mathfrak{F}})$. Since $Core_{G}(B)=1$, it follows that $N_{G}(B^{\mathfrak{F}})$ is a proper subgroup of $G$. Hence $N$ is not contained in $\Phi(G)$ for each minimal normal subgroup $N$ of $G$ contained in $A$. If $\Phi(A) \neq 1$, a minimal normal subgroup of $G$ must be contained in $\Phi(A) \leq \Phi(G)$, a contradiction. Therefore $\Phi(A)=1$. Let $N$ be a minimal normal subgroup of $G$ contained in $A$. Then $N=N_{1} \times N_{2} \times \cdots \times N_{r}$ is a direct product of minimal normal subgroups of $A$, and there exists $i \in \{1,2, \dots, r \}$ such that $N_{i}$ is not contained in $ \Phi(A)$. Suppose $i=1$. Let $M$ be a maximal subgroup of $A$ such that $A=N_{1}M$ and $N_{1} \cap M=1$. Assume first that $A$ is a $p$-group. Then $BM$ is a subgroup of $G$, and $M=BM \cap A$ is a normal subgroup of $BM$. Hence $M$ is normalized by $B$ and so $M$ is a normal subgroup of $G$. Now $N=N_{1}(M \cap N)$. But $M \cap N$ is normal in $G$. The minimality of $N$ yields $N=N_{1}$ and then $|N|=p$. Thus $G/C_{G}(N)$ is abelian. Hence $G^{\mathfrak{F}}$ centralises $N$, and $B^{\mathfrak{F}}$ is a normal subgroup in $G^{\mathfrak{F}}$ and so $G^{\mathfrak{F}}$ is an elementary abelian $p$-group. This contradiction implies that $A$ is not a $p$-group. Then $T= \mathbf{F}(A)B$ is a proper subgroup of $G$ which is a weak direct product of $ \mathbf{F}(A)$ and $B$. By the minimality of $G$, $T^{\mathfrak{F}}=B^{\mathfrak{F}}$. Then $B^{\mathfrak{F}}$ is a normal subgroup of $G^{\mathfrak{F}}$ and so $G^{\mathfrak{F}}$ is an elementary abelian $p$-group, a contradiction which shows that $p = q$. Then $B^{\mathfrak{F}}$ is a subnormal subgroup of $G$. By \cite[Lemma~A.14.3]{DH92}, $N$ normalises $B^{\mathfrak{F}}$ and therefore $B^{\mathfrak{F}}$ is a normal subgroup of the elementary abelian $p$-group $G^{\mathfrak{F}}$.\\ \\
\textit{(v) Final contradiction.}
By \cite[IV, 5.18]{DH92}, since $ B^{\mathfrak{F}} $ is abelian, there exists an $ \mathfrak{F} $-projector $ K $ of $ B $ such that $ B=B^{\mathfrak{F}}K $ and $ K\cap B^{\mathfrak{F}}=1 $. Consider the subgroup $Z=AK$ of $G$. Applying (iii), $Z$ belongs to $ \mathfrak{F} $ and $G=B^{\mathfrak{F}}Z=F(G)Z$. By \cite[III, 3.23(b)]{DH92}, there exists a unique $\mathfrak{F}$-projector of $G$ containing $Z$, $E$ say. Hence $G=B^{\mathfrak{F}}Z=G^{\mathfrak{F}}E$ and $G^{\mathfrak{F}} \cap E=1$ by (iii) and \cite[IV, 5.18]{DH92}. In particular, $B^{\mathfrak{F}} \cap Z=1$. Now $|Z||B^{\mathfrak{F}}|=|E| |G^{\mathfrak{F}} |=|E||B^{\mathfrak{F}}| |N |$. Hence $|Z |=|E| |N| $. This implies $Z=E$ and then $B^{\mathfrak{F}}=G^{\mathfrak{F}}$, a contradiction.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem C}]
Assume that the result is false and let $G$ be a minimal counterexample. Then every proper epimorphic image of $G$ is supersoluble, and hence $G$ has exactly one minimal normal subgroup $N$ which is not contained in the Frattini subgroup of $G$. Since $G$ is soluble, it follows that $N$ is abelian, $N=C_{G}(N)=F(G)$ and there exists a core-free maximal subgroup of $G$ such that $G=NM$ and $N \cap M = 1$. Let $p$ be the prime dividing $|N|$. Then $|N| > p$. Since $1 \neq G'$ is nilpotent, we have that $G'=N$ and $M$ is abelian. But $\mathbf{O}_{p}(M) = 1$ by \cite[Lemma~A.13.6]{DH92}. Hence $M$ is a $p'$-group and $N$ is the Sylow $p$-subgroup of $G$. Since $B \neq G$, we have that $N \leq A$. Note that $N = C_{A}(N) = \mathbf{O}_{p'p}(A)$. Therefore $A/\mathbf{O}_{p'p}(A)=A/\mathbf{O}_{p}(A)= A/N$ is abelian of exponent dividing $p-1$ because $A$ is supersoluble. Assume $BN$ is a proper subgroup of $G$. Then, by the minimality of $G$, $BN$ is supersoluble, and so $B_{p'} \cong BN/\mathbf{O}_{p'p}(BN)$ is abelian of exponent dividing $p-1$. Consequently $M$ is abelian of exponent dividing $p-1$. Since $N$ is an irreducible and faithful module for $M$, we have that $N$ has order $p$ by \cite[Theorem~B.9.8]{DH92}, a contradiction. Hence $G=BN$. Now $B \cap N$ is a normal subgroup of $G$ contained in $N$. Thus $B \cap N=1$, and $G=BN$ is the weak direct product of $B$ and $N$. By Theorem \ref{A}, $G$ is supersoluble. This contradiction proves the theorem.
\end{proof}
\begin{proof}[\textbf{Proof of Corollary B}]
Note that since $ G' $ is nilpotent, $ A $ and $ B $ are metanilpotent. By \cite[Theorem 2.11]{VVT10}, $ A $ and $ B $ are supersoluble. By Theorem~\ref{C}, $ G $ is supersoluble and hence $ G\in w \mathfrak{U} $.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem D}]
Suppose the theorem is not true and let $ (G, A, B) $ be a counterexample with $ |G| + |A| + |B| $ as small as possible. Let $ N $ be a minimal normal subgroup of $ G $. It is easy to check that $ G/N $ satisfies the hypotheses of the theorem. By the minimality of $ G $, we have that $ G^{\mathfrak{U}}N=A^{\mathfrak{U}}B^{\mathfrak{U}}N $. Hence $ G^{\mathfrak{U}}=A^{\mathfrak{U}}B^{\mathfrak{U}}(G^{\mathfrak{U}}\cap N) $. Consequently, $ Soc(G) $ is contained in $ G^{\mathfrak{U}} $ and $ G^{\mathfrak{U}}=A^{\mathfrak{U}}B^{\mathfrak{U}}N $ for every minimal normal subgroup $ N $ of $ G $. Since $ G^{\mathfrak{U}}$ is contained in $G'$, we have that $ G^{\mathfrak{U}}$ is nilpotent.
Note that $A^{\mathfrak{U}}$ is a normal subgroup of $G$. If $ A^{\mathfrak{U}}\not= 1 $, then there exists a minimal normal subgroup $N$ of $G$ such that $N \leq A^{\mathfrak{U}} $ and so $ G^{\mathfrak{U}}=A^{\mathfrak{U}}B^{\mathfrak{U}}N=A^{\mathfrak{U}}B^{\mathfrak{U}} $, a contradiction. Hence we may assume that $ A $ is supersoluble and that $ G^{\mathfrak{U}}=B^{\mathfrak{U}}N $ for every minimal normal subgroup $N$ of $G$. If $B$ were supersoluble, then $G$ would be supersoluble by Theorem~\ref{C}, which is a contradiction. Hence $ B^{\mathfrak{U}} \neq 1$. Furthermore, $ B^{\mathfrak{U}} $ cannot contain a normal subgroup of $ G $. Hence $ Core_{G}(B^{\mathfrak{U}})=1 $. Let $p$ be the largest prime dividing $ |A| $. Since $A$ is a Sylow tower group of supersoluble type, $A$ has a normal Sylow $p$-subgroup, $ A_{p} $ say, which is also normal in $G$. Hence $G$ has a minimal normal subgroup $ N $ of $ G $ which is a $ p $-group. Since $G^{\mathfrak{U}}$ is nilpotent, we have that $B^{\mathfrak{U}}$ is a subnormal subgroup of $G$. By \cite[Lemma~A.14.3]{DH92}, $B^{\mathfrak{U}}$ is normalized by $N$. Thus $B^{\mathfrak{U}}$ is a normal subgroup of $G^{\mathfrak{U}}$ and $ G^{\mathfrak{U}}/B^{\mathfrak{U}} $ is an elementary abelian $ p $-group. Consequently $ B^{\mathfrak{U}} $ contains the residual $X$ of $ G^{\mathfrak{U}}$ associated to the formation of all elementary abelian $p$-groups. Since $X$ is a normal subgroup of $G$, it follows that $X \leq Core_{G}(B^{\mathfrak{U}})=1$. Hence $ G^{\mathfrak{U}} $ is an elementary abelian $ p $-group.
Since $ Soc(G) $ is contained in $ G^{\mathfrak{U}} $, $ \textbf{O}_{p'}(G)=1 $ and hence $ \mathbf{F}(G)=\textbf{O}_{p}(G) $. Therefore $ G' \leq \mathbf{F}(G) $ is a $ p $-group and $ \mathbf{F}(G) $ is the unique Sylow $ p $-subgroup of $ G $. Moreover the Hall $p'$-subgroups of $G$ are abelian (note that $G$ is soluble). Assume $ A_{p}B < G $. Then $ A_{p}B $ satisfies the hypotheses of the theorem. By the choice of $ G $, we have that $ (A_{p}B)^{\mathfrak{U}}=B^{\mathfrak{U}} $. Note that $G' \leq A_{p}B $. Hence $A_{p}B$ is a normal subgroup of $ G $. This implies that $ B^{\mathfrak{U}} $ is normal in $ G $, a contradiction. Hence $ G=A_{p}B$ and $A_{p}$ and $B$ satisfy the hypotheses of the theorem. If $A \neq A_{p}$, the choice of $(G, A, B)$ implies that $ G^{\mathfrak{U}}=B^{\mathfrak{U}} $, a contradiction. Consequently we have that $A=A_{p}$.
Write $T=AB_{p'}$. By Theorem~\ref{C}, $T$ is supersoluble. Moreover, since $G=F(G)B_{p'}$, it follows that every minimal normal subgroup $N$ of $G$ contained in $T$ is a minimal normal subgroup of $T$. Thus $|N|=p$. Consequently, $N$ is ${\mathfrak{U}}$-central in $G$. By \cite[V, 3.2]{DH92}, $N$ is contained in every supersoluble normaliser of $G$. Let $E$ be one of them. Then $G=G^{\mathfrak{U}}E$ and $G^{\mathfrak{U}} \cap E=1$. However, $N \leq G^{\mathfrak{U}} \cap E=1$. This final contradiction proves the theorem.
\end{proof}
\begin{proof}[\textbf{Proof of Corollary C}]
Since $\mathfrak{U}\subseteq w\mathfrak{U}$, we have $G^{w\mathfrak{U}} \leq G^{\mathfrak{U}} \leq G'$. Then $G/G^{w\mathfrak{U}}$ is a metanilpotent w-supersoluble group. Applying \cite[Theorem 2.11]{VVT10}, we have that $G/G^{w\mathfrak{U}}$ is supersoluble. Hence $G^{\mathfrak{U}} \leq G^{w \mathfrak{U}}$, and therefore $G^{\mathfrak{U}} = G^{w \mathfrak{U}}$ and the same is true for $A$ and $B$. Therefore, by Theorem~\ref{D}, $ G^{w \mathfrak{U}}=A^{w \mathfrak{U}}B^{w \mathfrak{U}} $, as desired.
\end{proof}
\section{An analogue of Monakhov's result}
The following two results are the key to prove Corollary~D.
\begin{lemma} \cite[Theorem A]{BBPA18}\\
Let the group $G=HK$ be the product of the subgroups $H$ and $K$. Assume that $H$ permutes with every maximal subgroup of $K$ and $K$ permutes with every maximal subgroup of $H$. If $H$ is supersoluble, $K$ is nilpotent and $K$ is $\delta$-permutable in $H$, where $\delta$ is a complete set of Sylow subgroups of $H$, then $G$ is supersoluble.
\end{lemma}
\begin{proposition}\label{subnormalderived}
Let $G=AB$, be a weak normal product of $A$ and $B$ with $A$ and $B$ supersoluble and $A$ normal in $G$. Then $B'$ is a subnormal subgroup of $G$.
\end{proposition}
\begin{proof}
Assume the result is not true and let $G$ be a counterexample of minimal order with $|A|$ as small as possible. Let $p$ be the largest prime dividing the order of $A$. Then $A$ has a normal Sylow $p$-subgroup $A_{p}$ which is also a normal subgroup of $G$. Let $N$ be a minimal normal subgroup of $G$ such that $N \leq A_{p}$. It is clear that $A_{p}B$ satisfies the hypotheses of the theorem. Assume that $A_{p}B$ is a proper subgroup of $G$. By the minimality of $G$, $B'$ is a subnormal subgroup of $A_{p}B$. Hence $B' \leq F(A_{p}B)$. By Lemma~\ref{factor}(a), $G/N$ is a weak normal product of $A/N$ and $BN/N$. By the minimality of $G$ we have that $B'N$ is a subnormal subgroup of $G$. Since $N \leq F(A_{p}B)$, it follows that $B'N \leq F(A_{p'}B)$. Hence $B'N$ is a subnormal nilpotent subgroup of $G$. Consequently $B'N \leq F(G)$. Thus $B'$ is a subnormal subgroup of $G$, a contradiction. Hence we may assume that $G=A_{p}B$. The minimality of $|A|$ implies $A=A_{p}$. Applying now the above Lemma, we conclude that $G$ is supersoluble and therefore $G'$ is nilpotent. Hence $B'$ is subnormal in $G$. This final contradiction proves the proposition.
\end{proof}
\begin{proof}[\textbf{Proof of Corollary D}]
Arguing as in \cite[Theorem 1]{M18}, we obtain $G^{\mathfrak{U}}=(G')^{\mathfrak{N}}$. Moreover, by \cite[Lemma 1(3)]{M18} we have that $G'=A'B'[A,B]=(A')^{G}(B')^{G}[A,B]$. Since $A$ is a normal subgroup of $G$, then $A'$ is a subnormal subgroup of $G$. Also the application of Proposition \ref{subnormalderived} yields $B'$ is subnormal in $G$ and both $A'$ and $B'$ are nilpotent. Hence $(A')^{G}(B')^{G}$ is a normal nilpotent subgroup of $G$. By \cite[II, Lemma~II.2.12]{DH92}, $(G')^{\mathfrak{N}}=((A')^{G}(B')^{G})^{\mathfrak{N}}[A,B]^{\mathfrak{N}}=[A,B]^{\mathfrak{N}}$, as desired.
\end{proof}
\section{Acknowledgements}
These results are part of the R+D+i project supported by the Grant
PGC2018-095140-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by ``ERDF A way of making Europe'' and
the Grant PROMETEO/2017/057
funded by GVA/10.13039/501100003359.
| 154,582
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\chapter{Fundamentals of Linear Algebra}
\label{App:math_formulas}
This appendix aims at defining some of the fundamental topics in linear Algebra. It specifically explains the notion of trace, rank, determinant, eigenvalues, and singular values of a matrix. Moreover, some of the useful properties of such entities are included and the relation between them is also provided.
\section{Properties of the Trace Operator}
\label{s:trace_properties}
The trace of a square matrix $\V{A}$ is defined as the sum of the elements of its main diagonal. Let $\V{A}$ be a square matrix defined as $\V{A}_{M\!\times\! M}=\{a_{mn}\},\:m,n\in\{1,...,M\}$, then the trace of $\V{A}$ is
\begin{equation}
\tr(\V{A})=\sum\limits_{m=1}^{M}{a_{mm}}.
\end{equation}
Based on the above definition, the following properties trivially hold. Let $\V{A}$ and $\V{B}$ be square matrices of the same dimension, and let $a$ be a complex scalar, then
\begin{eqnarray*}
\tr(a)&=&a\\
\tr(\V{A}+\V{B})&=&\tr(\V{A})+\tr(\V{B})\\
\tr(\V{A}^T)&=&\tr(\V{A})\\
\tr(\V{A^*})&=&\tr(\V{A})^*
\end{eqnarray*}
From the above properties, the following relationships also hold;
\begin{eqnarray*}
\tr(\V{A}\hr)&=&\tr(\V{A}^T)^*=\tr(\V{A})^*\\
\tr(\V{A}+\V{A}\hr)&=&\tr(\V{A})+\tr(\V{A})^*=2\,\Re\{\tr(\V{A})\}\\
\end{eqnarray*}
Moreover, the trace operator is invariant under cyclic permutations, i.e.
\begin{equation}
\label{eq:tr_permutation}
\tr(\V{ABC})=\tr(\V{CAB})=\tr(\V{BCA})
\end{equation}
\section{Properties of the Determinant Operator}
\label{s:Determinant}
The determinant is an operator that operates on a square matrix and results in a scalar. This section summarizes some useful properties of the determinant function. Let $\V{A}$ and $\V{B}$ be any two square matrices then the following holds;
\begin{eqnarray*}
\det(\V{AB})&=&\det(\V{A})\det(\V{B})\\
\det(\V{A}^{-1})&=&(\det(\V{A}))^{-1}\\
\det(\V{A}^{T})&=&\det(\V{A})\\
\det(\V{A}\hr)&=&\det((\V{A}^T)^*)=(\det(\V{A}))^*
\label{eq:Det_properties}
\end{eqnarray*}
Furthermore the determinant of diagonal matrices is the product of the elements along the main diagonal. So if $\V{A}$ is an $M\!\times\!M$ diagonal matrix of diagonal entries $a_m$, $m\in\{1,...,M\}$, then
\begin{equation*}
\det(\V{A})=\prod\limits_{m=1}^{M}{a_m}
\end{equation*}
An important information that a determinant of a matrix provides, is whether or not the matrix has an inverse. Explicitly, a matrix is invertible if and only if its determinant is non-zero.\\
Another very useful theorem that has been extensively used in this thesis to evaluate certain determinants is known as Sylvester's determinant theorem and it states the following;
\begin{theorem}
Let $\V{A}$ be an $M\!\times\!N$ matrix and $\V{B}$ an $N\!\times\!M$ matrix, then
\begin{equation*}
\det(\V{I}_M+\V{AB})=\det(\V{I}_{N}+\V{BA})
\end{equation*}
\end{theorem}
\section{Rank of a Matrix}
\label{rank}
The rank of an $M\!\times\!N$ matrix $\V{A}$ is defined as the number of linearly independent rows or columns of $\V{A}$ which is at most the minimum of both dimensions $M$ and $N$, namely
\begin{equation}
\rank(\V{A})\leq\min(M,N).
\end{equation}
A matrix whose rank is the maximum achievable is said to have full rank, otherwise it is rank deficient. Full rank matrices have non-zero determinant and therefore are invertible, where as rank deficient matrices have zero determinant and therefore are non-invertible (singular).\\
For a diagonal matrix $\V{A}$, if there exist $r$ non-zero elements on the main diagonal, then there exist $r$ linearly independent vectors in $\V{A}$, therefore $r$ is the rank of $\V{A}$. Consequently, the rank of a diagonal matrix is the number of non-zero elements on its main diagonal.\\
Another useful property of the rank operator is that it is invariant under multiplication from any side by a full rank square matrix. Let $\V{U}$ and $\V{V}$ be full rank matrices of dimension $M\!\times\!M$ and $N\!\times\!N$, respectively, and let $\V{A}$ be an arbitrary $M\!\times\!N$ matrix, then the following holds
\begin{equation}
\label{eq:rank_unitary}
\rank(\V{UA})=\rank(\V{AV})=\rank(\V{A})
\end{equation}
\section{Eigenvalue Decomposition and Singular Value Decomposition}
\label{s:EVD_SVD}
This section provides the basic theory of eigenvalues and singular values as one of the important concepts in the field of linear Algebra. Two well-known matrix decompositions are defined, namely the eigenvalue decomposition and the singular value decomposition. The section also includes some of the useful properties of eigenvalues and singular values that have been used in this thesis.
\subsection{Eigenvalue Decomposition}
\label{ss:EVD}
When a square matrix $\V{A}$ acts on vector $\V{x}$, it may change its magnitude or its direction or both. If only the magnitude of $\V{x}$ is changed by a factor $\lambda$, this can be described as
\begin{equation}
\label{eq:Eigen_value_problem}
\V{A}\V{x}=\lambda\V{x}
\end{equation}
where $\lambda$ is in general positive or negative complex scalar, so the direction of $\V{x}$ might be reversed in case $\lambda$ is negative. The special vectors which keep their direction unchanged (or possibly reversed) after being acted upon by matrix $\V{A}$ are known as the eigenvectors of $\V{A}$, and the special factors of magnitude change of the corresponding eigenvectors are known as the eigenvalues of $\V{A}$. On the other hand, if $\V{A}$ acts on a non-eigenvector $\V{x}$, the output vector $\V{Ax}$ points in a direction other than that of $\V{x}$ or $-\V{x}$.\\
"Eigen" is a German prefix that means "own" or "Characteristic". This indicates that eigenvectors and eigenvalues are of characteristic and unique nature to a matrix. An $M\!\times\!M$ matrix $\V{A}$ can have at most $M$ non-zero eigenvalues, and it can be decomposed as
\begin{equation}
\label{eq:EVD}
\V{A}=\V{X\Lambda X}^{-1}
\end{equation}
where $\V{X}$ is an $M\!\times\!M$ matrix whose columns are the eigenvectors of $\V{A}$, and $\V{\Lambda}$ is a diagonal matrix whose diagonal contains the corresponding eigenvalues of $\V{A}$, namely $\lambda_i,\, i\in\{1,...,M\}$. Such a decomposition is known as Eigen-Value Decomposition (EVD) and is defined only for square matrices. Nevertheless, not all square matrices can be eigen-decomposed. Only matrices whose all eigenvectors are linearly independent can be eigen-decomposed, because otherwise matrix $\V{X}$ will be rank deficient, and therefore will not be invertible. It is obvious to see that for diagonal matrices, $\V{X}$ is an identity matrix, and therefore the elements on the main diagonal of $\V{A}$ are themselves the eigenvalues.\\
Using the properties of the determinant operator defined in Section \ref{s:Determinant}. The following theorem holds;
\begin{theorem}
\label{theo:det_EV}
Let $\V{A}$ be an $M\!\times\!M$ matrix which is eigen-decomposable. Then the determinant of $\V{A}$ is the product of its eigenvalues.
\end{theorem}
\begin{proof}
\begin{equation}
\begin{aligned}
\V{A}&=\V{X\Lambda X}^{-1}\\
\det(\V{A})&=\det(\V{X\Lambda X}^{-1})=\det(\V{X})\det(\V{\Lambda})\det(\V{X}^{-1})\\
&=\det(\V{X})\det(\V{\Lambda})\frac{1}{\det(\V{X})}=\det(\V{\Lambda})=\prod\limits_{i=1}^{M}{\lambda_i}
\end{aligned}
\label{eq:det_properties}
\end{equation}
\end{proof}
with $\lambda_i$ being the $i^{\text{th}}$ eigenvalue of $\V{A}$.\\
Using the properties of the trace operator defined in Section \ref{s:trace_properties}. The following theorem holds;
\begin{theorem}
Let $\V{A}$ be an $M\!\times\!M$ matrix which is eigen-decomposable. Then the trace of $\V{A}$ is the sum of its eigenvalues.
\end{theorem}
\begin{proof}
\begin{eqnarray*}
\V{A}&=&\V{X\Lambda X}^{-1}\\
\tr(\V{A})&=&\tr(\V{X\Lambda X}^{-1})=\tr(\V{\Lambda}\V{X}^{-1}\V{X})=\tr(\V{\Lambda})=\sum\limits_{i=1}^{M}{\lambda_i}
\end{eqnarray*}
\end{proof}
The next theorem relates the eigenvalues of a matrix to its rank.
\begin{theorem}
\label{theo:rank_EV}
The rank of a matrix is the number of its non-zero eigenvalues.
\end{theorem}
\begin{proof}
Since both $\V{X}$ and $\V{X}^{-1}$ are full rank square matrices by the definition of EVD. Then it follows from (\ref{eq:rank_unitary}) that
\begin{equation}
\rank(\V{A})=\rank(\V{X\Lambda X}^{-1})=\rank(\V{\Lambda})
\end{equation}
Since the rank of $\V{\Lambda}$ is the number of non-zero elements (eigenvalues) on its main diagonal, therefore it follows that the rank of a matrix $\V{A}$ is the number of its non-zero eigenvalues.
\end{proof}
\subsection{Singular Value Decomposition}
\label{ss:SVD}
In this subsection, another useful matrix decomposition is defined. For this we need to first define the unitary property of matrices.
\begin{definition}
A square $M\!\times\!M$ matrix $\V{A}$ is said to be unitary if its inverse is the same as its conjugate transpose, namely
\begin{equation*}
\V{A}^{-1}=\V{A}\hr.
\end{equation*} Therefore,
\begin{equation*}
\V{U}\hr\V{U}=\V{UU}\hr=\V{U}^{-1}\V{U}=\V{I}_{M}.
\end{equation*}
\end{definition}
Singular Value Decomposition (SVD) is a matrix factorization which applies not only for square matrices as EVD but also for rectangular matrices. Let $\V{A}$ be $\in\mathbb{C}^{M\!\times\!N}$, SVD is defined as
\begin{equation}
\V{A}=\V{U\Sigma V}\hr,
\label{eq:SVD}
\end{equation}
where $\V{U}\in\mathbb{C}^{M\!\times\!M}$, $\V{V}\in\mathbb{C}^{N\!\times\!N}$ and both are unitary matrices. The columns of $\V{U}$ are orthonormal basis vectors known as left-singular vectors, where as the columns of $\V{V}$ are orthonormal basis vectors known as right-singular vectors. $\V{\Sigma}$ is an $M\!\times\!N$ diagonal matrix whose diagonal entries are known as the singular values of $\V{A}$, and they are non-negative real numbers. Matrix $\V{A}$ has at most $\min(M,N)$ singular values, which are denoted as $\sigma_i,\:i\in\{1,...,\min(M,N)\}$. Unlike EVD, any matrix can be singular-decomposed.\\
There exist some relationships between the singular values and the eigenvalues of matrices. Here only one relation will be shown in the next theorem. However, before describing such a relation, it is useful to first define the notion of \emph{similar} matrices as follows
\begin{definition}
Two $M\!\times\! M$ matrices $\V{A}$ and $\V{B}$ are said to be similar if
\begin{equation*}
\V{B}=\V{QA}\V{Q}^{-1}
\end{equation*}
for some invertible $M\!\times\!M$ matrix $\V{Q}$. Similar matrices have the same rank, determinant, trace and same eigenvalues.
\end{definition}
This can be easily verified using respectively the rank property in (\ref{eq:rank_unitary}), the first two determinant properties in (\ref{eq:det_properties}), the trace property in (\ref{eq:tr_permutation}), and the definition of the EVD, namely
\begin{eqnarray*}
\V{A}&=&\V{X\Lambda X}^{-1}\\
\V{B}&=&\V{QA}\V{Q}^{-1}=\underbrace{\V{QX}}_{\V{X}'}\V{\Lambda}\underbrace{\V{X}^{-1}\V{Q}^{-1}}_{\V{X}'^{-1}}.
\end{eqnarray*}
Now it is possible to define the desired relation between eigenvalues and singular values as shown in the following theorem.
\begin{theorem}
\label{theo:EV_SV2}
The eigenvalues of matrix $\V{A}\hr\V{A}$ or $\V{AA}\hr$ are the square of the singular values of matrix $\V{A}$
\end{theorem}
\begin{proof}
Let $\V{A}$ be an $M\!\times\!M$ matrix
\begin{eqnarray*}
\V{A}&=&\V{U\Sigma V}\hr\\
\V{A}\hr\V{A}&=&\V{V\Sigma}\hr\underbrace{\V{U}\hr\V{U}}_{\V{I}_M}\V{\Sigma V}\hr\\
&=&\V{V}(\V{\Sigma}\hr\V{\Sigma})\V{V}\hr\\
&=&\V{V}(\V{\Sigma}\hr\V{\Sigma})\V{V}^{-1}
\end{eqnarray*}
where $\V{V}\hr=\V{V}^{-1}$ since $\V{V}$ is unitary. In the last line we see that matrices $\V{A}\hr\V{A}$ and $\V{\Sigma}\hr\V{\Sigma}$ are \emph{similar} matrices, and therefore they have the same eigenvalues, namely
\begin{equation*}
\lambda_m(\V{A}\hr\V{A})=\lambda_m(\V{\Sigma}\hr\V{\Sigma})\:\:\:\forall\: m\in\{1,...,M\}
\end{equation*}
Furthermore, since $\V{\Sigma}\hr\V{\Sigma}$ is a diagonal matrix, therefore its eigenvalues are the elements on the main diagonal which are the square of the singular values of $\V{A}$. This concludes the theorem that the eigenvalues of $\V{A}\hr\V{A}$ are the same as the square of the singular values of $\V{A}$, i.e.
\begin{equation}
\label{eq:EV_SV2}
\lambda_m(\V{A}\hr\V{A})=\sigma_m^2(\V{A})\:\:\:\forall\: m\in\{1,...,M\}
\end{equation}
A similar proof holds for $\V{AA}\hr$.
\end{proof}
In the following theorem, the relation between the rank of a matrix and its singular values is shown.
\begin{theorem}
\label{theo:rank_SV}
The rank of matrix $\V{A}$ is the number of non-zero singular values of $\V{A}$.
\end{theorem}
\begin{proof}
Using the SVD of $\V{A}$,
\begin{eqnarray*}
\V{A}&=&\V{U\Sigma V}\hr\\
\rank(\V{A})&=&\rank(\V{U\Sigma V}\hr)
\end{eqnarray*}
Since both $\V{U}$ and $\V{V}\hr$ are unitary and therefore full rank square matrices, then it follows from (\ref{eq:rank_unitary}) that
\begin{equation*}
\rank(\V{A})=\rank(\V{\Sigma})
\end{equation*}
Since the rank of $\V{\Sigma}$ is the number of non-zero elements (singular values) on its main diagonal, therefore it follows that the rank of a matrix is the number of its non-zero singular values.
\end{proof}
Next is another theorem that uses the SVD to prove the rank of some matrix.
\begin{theorem}
\label{theo:rank_AA_A}
The rank of matrix $\V{A}\hr\V{A}$ is the same as the rank of matrix $\V{A}$, i.e.
\begin{equation}
\label{eq:rank_AA_A}
\rank(\V{A}\hr\V{A})=\rank(\V{A})
\end{equation}
\end{theorem}
\begin{proof}
if $\V{A}$ has the SVD as $\V{A}=\V{U\Sigma V}\hr$, then
\begin{eqnarray*}
\V{A}&=&\V{U\Sigma V}\hr\\
\V{A}\hr\V{A}&=&\V{V\Sigma}\hr\underbrace{\V{U}\hr\V{U}}_{\V{I}_M}\V{\Sigma V}\hr\\
&=&\V{V}(\V{\Sigma}\hr\V{\Sigma})\V{V}\hr\\
&=&\V{V}(\V{\Sigma}\hr\V{\Sigma})\V{V}^{-1}
\end{eqnarray*}
where $\V{V}\hr=\V{V}^{-1}$ since $\V{V}$ is unitary. In the last line we see that matrices $\V{A}\hr\V{A}$ and $\V{\Sigma}\hr\V{\Sigma}$ are \emph{similar} matrices, and therefore they have the same rank. Now, the rank of $\V{\Sigma}\hr\V{\Sigma}$ is the number of non-zero square singular values of $\V{A}$ which is the same as the number of non-zero singular values of $\V{A}$ which is from theorem \ref{theo:rank_SV} the rank of $\V{A}$. Therefore the rank of $\V{A}\hr\V{A}$ equals the rank of $\V{A}$.
\end{proof}
\abbrev{EVD}{Eigen-Value Decomposition}
\abbrev{SVD}{Singular Value Decomposition}
| 142,202
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TITLE: Taylor series at boundary of domain
QUESTION [3 upvotes]: Is it possible to compute the Taylor series expansion of a function at an edge of its domain? If so, what are the conditions for it to hold? For example: Does
$f: \mathbb{R}^+ \to [1\,;+\infty), x \mapsto \exp(\sqrt x)$
have a Taylor series expansion at $x=0$?
REPLY [0 votes]: No. The function $f$ is not differentiable at $x=0$, since
$\frac{f(1/n)-f(0)}{1/n}=nf(1/n) \to \infty $ as $n \to \infty$.
REPLY [0 votes]: No, the function $f$ has not a Taylor expansion at $0$ because $f$ is not differentiable at zero. However $f$ has an expansion (not polynomial) in a right neighbourhood of zero. Since $f$ is the composition of $e^x$, which has a Taylor expansion at $0$, and $\sqrt{x}$, which is continuous at $0$, we have that
$$f(x)=\exp(\sqrt x)=\sum_{k=0}^n\frac{(\sqrt x)^k}{k!}+o((\sqrt x)^n).$$
Recall that a Taylor expansion at $0$ is a sum of non-negative integer powers of $x$.
| 133,061
|
TITLE: What are the current research directions in the geometric theory of dynamical systems?
QUESTION [6 upvotes]: By geometric theory of dynamical systems, I mean the kind found in the book by Palis, or papers like this one. In other words, dynamics on manifolds, but not specifically hyperbolic dynamical systems.
What are some recommended papers, survey articles, lecture notes, or books to read to explore this topic further? I really like this flavour of dynamics and would like to know what the modern research directions/questions are.
Thanks in advance!
REPLY [4 votes]: I think the book by Bonatti, Diaz and Viana: "Dynamics beyond uniform hyperbolicity' can give you a nice overview of one possible point of view. https://link.springer.com/book/10.1007/b138174
The book by Katok-Hasselblatt and their Handbook contains a lot of other points of view.
REPLY [3 votes]: Perhaps not the type of dynamical system in which you are interested, but
an interesting example:
Schwartz, Richard Evan. "The Farthest Point Map on the Regular Octahedron." Experimental Mathematics (2021): 1-12.
Preliminary version: arXiv abs.
The farthest point map associates to each point $p$ on the surface
the set of points $\mathcal{F}_p$ that are furthest from $p$, with distance
measured by shortest paths (geodesics segments).
Of special interest are the points $p$ for which $\mathcal{F}_p$
is a single point.
Even on the regular octahedron, the dynamics are quite intricate, but
calculable.
| 49,508
|
Language: English
Category: Sex /
Visit Website
A guide to sex and sex issues for the Age of Google
Episode 2 in the Sex and Politics themed 7 in 7 challenge via the Freak Network. Today I discuss:
More about the challenge
Sex related reasons to vote including this Palin related issue about rape victims paying for their rape kits
Corrections from yesterday's podcast regarding the case about sodomy in Texas and sextoys in Texas
ZOMG! The Debate!
VOTE VOTE VOTE!!!!
*nudge nudge wink wink* at Nobilis about his political erotic short
Need to contact me? Send me lists??? postmodern.sexgeek@gmail.com
Direct download of the podcast is here!
[ Fri, 3 Oct 2008 04:52:00 GMT ]
| 115,806
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TITLE: Conjecture: $a^n+b^n+c^n\ge x^n+y^n+z^n$
QUESTION [7 upvotes]: let $a,b,c,x,y,z>0$ such $x+y+z=a+b+c,abc=xyz$,and $a>\max\{x,y,z\}$,
I conjecture $$a^n+b^n+c^n\ge x^n+y^n+z^n,\forall n\in N^{+}$$
Maybe this kind of thing has been studied, like I found something relevant, but I didn't find the same one.[Schur convexity and Schur multiplicative convexity for a class of symmetric function]
I found sometime maybe The sum of squared logarithms conjecture
REPLY [7 votes]: A triple $(a,b,c)$ with $a+b+c=s$, $abc=p$, and $ab+bc+ca=t$ is the triple of roots of $X^3-sX^2+tX-p=0$, i.e., of $-X^2+sX+p/X=t$. Fix $s$ and $p$; the left-hand part has just two local extrema on the right semiaxis. Let $a(t)>b(t)>c(t)$ be the three roots (defined for all values of $t$ when they exist and are sdistinct). Looking at the graph, one can easily see that $a(t)$ and $c(t)$ decrease as $t$ grows, while $b(t)$ grows as well.
Now it suffices to prove that $(a^n+b^n+c^n)'=n(a^{n-1}a'+b^{n-1}b'+c^{n-1}c')\leq 0$ (the derivative is taken with respect to $t$). We know that $a'+b'+c'=a'bc+b'ca+c'ab=0$, hence $a'=\lambda a(b-c)$, $b'=\lambda b(c-a)$, and $c'=\lambda c(a-b)$, where $\lambda<0$ as $a'<0$. Hence the required inequality reads $a^n(b-c)+b^n(c-a)+c^n(a-b)\geq0$, or
$$
a(b-c)(a^{n-1}-b^{n-1})\geq c(a-b)(b^{n-1}-c^{n-1}),
$$
which follows from
$$
\frac{a^{n-1}-b^{n-1}}{a-b}\geq\frac{b^{n-1}-c^{n-1}}{b-c}.
$$
The last inequality holds for all (not neccessarily integer) $n\geq 2$, e.g., by Lagrange's mean value theorem.
REPLY [5 votes]: Yes, it is true. Without loss of generality, $a\geq b \geq c$. Let $a+b+c=s,\ abc=p$ then $b=\frac{(s \ - \ a) \ + \ d}{2}, \ c=\frac{(s \ - \ a) \ - \ d}{2}$ with $d=b-c=\sqrt{(s-a)^2- \frac{4p}{a}}$.
If we consider $b,c$ as variables in $a,s,p$, we find that
$$
\begin{eqnarray}
\frac{\partial (a^n+b^n+c^n)}{\partial a} &=& na^{n-1}+nb^{n-1}\frac{1}{2}\left(-1+\frac{\partial d}{\partial a}\right)+nc^{n-1}\frac{1}{2}\left(-1-\frac{\partial d}{\partial a}\right)\\
&=& \frac{n}{2}\left(\left(2a^{n-1}-b^{n-1}-c^{n-1}\right)+\frac{\partial d}{\partial a}\left(b^{n-1}-c^{n-1}\right) \right).
\end{eqnarray}
$$
Hence it is enough to prove the following:
$$
\frac{\partial d}{\partial a}+\frac{2a^{n-1}-b^{n-1}-c^{n-1}}{b^{n-1}-c^{n-1}}\geq 0
$$
We have
$$
\frac{\partial d}{\partial a}=\frac{a-s+2p/a^2}{d}=\frac{-b-c+2bc/a}{b-c}=\frac{2bc(b^{n-2}+b^{n-3}c+\dots)/a-(b+c)(b^{n-2}+b^{n-3}c+\dots)}{b^{n-1}-c^{n-1}} \
$$
Hence,
$$
\frac{\partial d}{\partial a}+\frac{2a^{n-1}-b^{n-1}-c^{n-1}}{b^{n-1}-c^{n-1}}
=\frac{2bc(b^{n-2}+b^{n-3}c+\cdots)/a+2a^{n-1}-(b+c)(b^{n-2}+b^{n-3}c+\dots)-(b^{n-1}+c^{n-1})}{b^{n-1}-c^{n-1}}
$$
Taking the derivative with respect to $a$ yields $$\frac{\partial^2 d}{\partial a^2} = \frac{2(n-1)a^{n-2} - \frac{2bc}{a^2(b^{n-2} \ + \ b^{n-3}c \ + \ \cdots)}}{b^{n-1} - c^{n-1}}$$ which is greater than or equal to zero. Hence the expression above is minimal if we choose $a$ to be minimal, i.e. $a=b$. In this case, we have: $$\frac{\partial d}{\partial a}=-1=-\frac{2a^{n-1}-b^{n-1}-c^{n-1}}{b^{n-1}-c^{n-1}}$$.
| 38,069
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With the long-awaited resumption of Parliament, my Private Member’s legislation, Bill C-222, An Act to amend the Expropriation Act (protection of private property),is moving forward in Parliament.
Bill C-222, which amends the Expropriation Act, is intended to provide some protections from government taking people’s property without compensation. Given the absence of property rights in the Canadian Constitution, landowners must look to expropriation legislation to protect their rights.
People should be secure in their homes and the best way for that to happen is through clear, enforced ownership rules. Things have not been going very well for liberty-minded Canadians lately, with globalization and the rise of authoritarianism in Canada. The lockdowns have made it convenient to replace free market economics with state control. Is this the hidden agenda of the new generation of radicalized Liberals?
The reality is countries with stronger property rights are more economically advanced.
Other interesting effects regarding property rights come into play. If you don’t have clear title to your home, you may not have a strong incentive to improve it. Rental housing turning into slums comes to mind.
If owning your home is more costly or difficult, you may end up renting. Making property rights more secure and easier to exercise seems likely to encourage people to maintain their homes.
With record sales, and high prices for real estate, Trudeau and his new “green” Finance Minister have taken notice, and not in a good way.
Our new green Finance Minister was looking at ways to raise taxes by taxing principal residences.
Canadians will have to wait and see if a New Federal Home Equity Tax currently under consideration, will be implemented. Without a doubt, something is being planned. The billions borrowed by the government during the pandemic mean tax increases in the future.
Certain factors are working against homeowners. Left-wing or socialist parties, into which the Liberal Party has evolved under Trudeau, do not believe in property rights. To quote a recent observation in the Canadian media about the current federal government,
“the comments of those who are advising this government on housing wealth and inequality have revealed an attitude that many Canadians have “won the lottery” with the value of homes increasing so much, and that the glorification of home ownership is a “regressive canard”.
The decision by the Liberal Government to require each of us to declare our principal residence on tax returns is information being collected for a reason. The homes of Canadians represent their largest asset for most people. A cash-starved government would like to “unlock” the value in your home. Only by unlocking the value will the federal government then be in a position to tax it. The question which needs to be asked by taxpayers is not when principal residences will be taxed, but how.
The most obvious change is removing the exemption on capital gains, including principal residences. The problem with that change is it requires the homeowners to sell their homes. That is where a Federal Home Equity Tax comes in. If homeowners are required to pay a home equity tax on top of property taxes, the Liberal government reaps the financial benefit immediately.
If that happens, many Canadians, particularly those on fixed retirement incomes, like our seniors, will be forced to sell their homes.
A Home equity tax is not the same as a capital gains tax. It is a broader tax on something called imputed rent. Imputed rent is an estimate of the rent an owner-occupied property would earn if the owner were paying rent rather than owning the property. Those earnings would be added to a taxpayer’s taxable income.
The other method to steal value from private property, is by “defacto confiscation,” which is why I brought forward my Private Member’s legislation, Bill C-222.
Chances are you missed the recent announcement by Trudeau about land.
Ahead of the United Nations meeting planned for Kunming, China, Trudeau has pledged to place restrictions on 30 per cent of our land by 2030.
Of the 10 largest countries in the world by land mass, only Canada has signed on to the United Nations scheme to set aside 30 per cent of its land and water by 2030.
Not China. Not Russia. Not the United States. Not Brazil, Australia, India, Argentina, Kazakhstan or Algeria.
Long on talk, short on details, nowhere in his announcement was mention made of compensation to private property owners. They will suffer the loss of the right to enjoy their land they pay property taxes on, when their property is included in the 30 per cent land grab.
This is very similar to the May 1st announcement banning thousands of firearms. Conspicuously missing in that charade of promising a buy-back is an actual dollar figure.
Confiscation without compensation.
Earlier this year I introduced Bill C-222, An Act to Amend the Expropriation Act (private property rights) to protect you from government policy that reduces the value of your private property without paying for it.
Should government be legally required to provide fair compensation when it steals value from private property owners?
That is now not the case.
Only by electing a majority Conservative government, will my Private Member’s Bill have any chance of becoming law.
| 128,539
|
\section{Proof of Theorem~\texorpdfstring{\ref{thm:mainlb}}{1.1} from Theorem~\texorpdfstring{\ref{thm:genlowhc}}{1.2} and Theorem~\texorpdfstring{\ref{thm:mainrank}}{1.3}}\label{sec:crt}
We prove a slightly stronger consequence, namely, that there is an algorithm that counts the satisfying assignments of a given a CNF-formula on $n$ variables in $O^*((2-\eps)^n)$ time for some $\eps>0$.
Let $n$ be the number of variables of the given CNF-formula $\varphi$. The Chinese Remainder Theorem (CRT) tells us that given the number of satisfying assignments of $\varphi$ modulo primes $p_1,\ldots,p_\ell$, we can compute the number of satisfying solutions of $\varphi$ as long as $\prod^\ell_{i=1}p_i \geq 2^n$. By the Prime Number Theorem~\cite[p.~494, Eq.~(22.19.3)]{HardyWright08}, there are at least $r/\log_2 r$ primes between $r$ and $2r$, and thus
\[
\prod_{r\, \leq \, p \text{ prime} \, \leq \, 2r } p \geq r^{r / \log r} \geq 2^{\Omega(r)}.
\]
It follows that for counting the number of satisfying assignments of a given CNF-formula, it is sufficient to count the number of satisfying assignments modulo $p$ for any $p=\Theta(n)$.
We do this using Lemma~\ref{modbound} combined with the algorithm for \CHCP. For fixed $t$ we have that $\rank_p(\MC_t)=\rank(\MC_t)$ for large enough $p$ (which can for example be shown by upper bounding the determinant of $\MC_t$ by $t!$). The assumed algorithm for \CHCP also counts the number of Hamiltonian cycles modulo $p$. By Theorem~\ref{thm:mainrank} we have $\lim_{p \rightarrow \infty}r_p=4$ and Theorem~\ref{thm:mainlb} follows.
| 110,708
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TITLE: $\mathbb{F}_2 \times \mathbb{F}_2$ is not subgroup separable
QUESTION [4 upvotes]: I read that $\mathbb{F}_2 \times \mathbb{F}_2$ is not subgroup separable (ie. for every finitely generated subgroup $H$ and $g \notin H$, there exists a finite index subgroup $K$ such that $H \subset K$ and $g \notin K$). Do you know an argument justifying the assertion?
We already know that $$\text{Subgroup separable} \Rightarrow \text{Residually finite} \Rightarrow \text{Hopfian},$$
is $\mathbb{F}_2 \times \mathbb{F}_2$ residually finite or Hopfian?
REPLY [5 votes]: Question 1: To prove that $F_2\times F_2$ is not subgroup seperable, the idea is as follows: take a surjective homomorphism $\phi:F_2\rightarrow G$ where $G$ is not residually finite (for example, take $G=\langle a, b; a^{-1}b^3ab^2\rangle$) and consider the diagonal subgroup $\Delta\leq G\times G$. Then $\Delta$ is not seperable in $G\times G$, and hence $Q=(\phi\times\phi)^{-1}(\Delta)$ is not seperable in $F_2\times F_2$ (this is not obvious, but it only uses surjections). Moreover, $Q$ is finitely generated (it is generated by the elements $(a, a)$, $(b, b)$, $(a^{-1}b^3ab^2, 1)$ and $(1, a^{-1}b^3ab^2)$).
A similar idea can be used to prove that the subgroup membership problem is insoluble for $F_2\times F_2$. The trick here is to take the group $G$ to be finitely presented with insoluble word problem. Also, I believe that the subgroup $Q=(\phi\times\phi)^{-1}$ is called a Mihailova subgroup, after K. A. Mihailova, The occurrence problem for direct products of groups Dokl. Acad. Nauk SSRR 119 (1958), 1103-1105.
Question 2: Recall that a group $G$ is residually finite if for every non-trivial $g\in G$ there exists a homomorphism $\phi_g: G\rightarrow Q_g$ where $Q_g$ is finite and where $\phi_g(g)\neq_{Q_g}1$. That is, for every element there exists a map onto a finite group where the element does not die.
Theorem: The group $F_2\times F_2$ is residually finite.
Proof: Suppose $1\neq(u, v)\in F_2\times F_2$, and without loss of generality suppose $u$ is non-trivial in $F_2$. Then $(u, v)$ does not die under the map $(u, v)\mapsto u$, which clearly extends to the following homomorphism
$$
\begin{align*}
\phi_{(u, v)}:
F_2\times F_2&\rightarrow F_2\\
(w_1, w_2)&\mapsto w_1
\end{align*}
$$
As $F_2$ is residually finite, there exists a map $\phi_u:F_2\rightarrow Q_u$ where $Q_u$ is some finite group and $\phi_u(u)$ is non-trivial. Then $\phi_{(u, v)}\circ\phi_u: F_2\times F_2\rightarrow Q_u$ is a homomorphism to a finite group and $\phi_{(u, v)}\circ\phi_u(u, v)$ is non-trivial. Hence, we can conclude that $F_2\times F_2$ is residually finite.
The only fact that we used here was that $F_2$ was residually finite. This means that the following, more general result holds (and the proof is essentially identical).
Theorem: If $H$ and $K$ are residually finite then so is $H\times K$.
| 121,443
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John Terry Praises Wondolowski’s Movment Off the Ball Posted on June 26, 2019 at June 26, 2019 by admin 68 0 There are a lot of people who don’t really think Chris Wondolowski deserves all of the attention he’s getting these days. Sure, he’s scored a ton of goal in the MLS, but he hasn’t scored against tougher competition. And, he’s not scoring Thierry Henry or Robbie Keane type goals. Wondo isn’t smooth player like those two. Wondo’s more of a pure in the box goalscorer. There’s nothing wrong with that but… Wondolowski is maybe in the same category as a player like England’s Gary Lineker, one of those classic goal poachers. But what Wondo doesn’t get enough credit for is his movement off the ball and constant running. It’s something John Terry witnessed first hand and complimented Wondolowski on when Chelsea faced off against the MLS All-Star team about a year ago: Here’s the video of Terry talking to Wondolowski – comes at about the 17:50 mark: And Wondolowski had this to say about Terry’s comments: .” (MLS) Wondolowski is excellent at timing his runs too, and making those runs throughout the game. Moreover, he’s tireless in his running and desire to get ahead of his defender just enough so he can score. It’s almost as though that now that he’s gotten a taste of what it feels like to score a goal game after game, he has to do it again, and again, and agian. He’s a goal scoring addict, which isn’t a bad thing at all for the San Jose Earthquakes or maybe the United States national time. The thing is though, for the U.S. national team, Wondolowski has only scored goals against teams like Belize and Cuba, not say Mexico or Costa Rica. But maybe he should get a chance to score goals against better teams. His coach, Jurgen Klinsmann, wasn’t such a bad forward in his day, so I’m guess he’ll know if Wondolowski is the right man for the job. Simply stated though, if Wondo keeps making runs like he does he’ll probably score against whoever he faces up with. He’s a goal scoring addict. And, John Terry of all people might have something to do with giving Wondo a surge in confidence. Coerver Coaching: Get FREE Soccer Training tips. Signup Now!
| 187,165
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(1:00pm) WONR skirts the French River, approaching West Thompson Road, on its run south. 43 cars
Photographed by Bruce Macdonald, June 17, 2022.Added to the photo archive by Bruce Macdonald, June 17, 2022.Railroad: Providence & Worcester.
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Outstanding!
Posted by Alan Marshall on 2022-06-17 16:56:48
Thanks Al!
Posted by Bruce Macdonald on 2022-06-18 13:41:14
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Photo of Nicole Richie and Joel Madden at the Sephora Project Launch Party
Nicole and Joel Take Time For Charity
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Nic.
Read More Joel MaddenNicole Richie
Nicole Richie
he´s hot,very hot,and sunglasses are cool I like them,and more on him
Yikes! She is looking very thin again.
Joel's so cute :) Nicole looks so thin
I love them as a couple. :) They're so cute.
:sleepy:
What a pretty girl.
i'm always impressed by how giving this couple is. usually you see that it's the older more established celebs that get so involved but these 2 really have made the effort to help and it's something wonderful to commend them on. besides - they both look great here too!
I wish she would change her style. She looks like a Rachel Zoe clone. And Rachel Zoe is not pretty.
um he looks like elton john.
they both look like they've lost weight. Hope Nicole's not back on the "H" I will say that Nicole got the better of the two brothers. Paris' BF looks like a joke, a complete joke! Short and ugly who tries to put on a bulldog face every time someone takes his pic. Paris never gets the great ones :( Poor Paris.
Gorgeous :)
Eek, she looks so thin!
ok, he needs to NEVER put another hat on his head again. he is so attractive without it.
Why so serious, Joel? :D They look cute together, I have to admit.
He looks TOTALLY different!
haha. It looks like he is trying to impress GQ magazine or something.
Yea, WTH with the sunglasses?!
He's not wearing a hat! Yay! He's so cute, but I don't like his sunglasses.
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\begin{document}
\begin{abstract}{Conjugations in space $L^2$ of the unit circle commuting with multiplication by $z$ or intertwining multiplications by $z$ and $\bar z$ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces.}
\end{abstract}
\keywords{conjugation, $C$--symmetric operator, Hardy space, model space, invariant subspaces for the unilateral shift, truncated Toeplitz operator.}
\maketitle
\section{Introduction}
Let $\h$ be a complex Hilbert space and denote by $\bh$ the algebra of all bounded linear operators on $\h$. A {\it conjugation} $C$ in $\h$ is an antilinear isometric involution, i.e., $C^2=id_{\h}$ and
\begin{equation}\label{e1}
\langle Cg,Ch \rangle=\langle h,g\rangle \quad \text{ for } g,h\in\h.
\end{equation}
Conjugations have recently been intensively studied and the roots of this subject comes from physics.
An operator $A\in\bh$ is called {\it $C$--symmetric} if $CAC=A^*$ (or equivalently $AC=CA^*$).
A strong motivation to study conjugations comes from the study of complex symmetric operators, i.e., those operators that are $C$--symmetric with respect to some conjugation $C$. For references see for instance \cite{GP,GPP, GP2, CKP, CFT, KoLee18}. Hence obtaining the full description of conjugations with certain properties is of great interest.
Let $\mathbb{T}$ denote the unit circle, and let $m$ be the normalized Lebesgue measure on $\mathbb{T}$. Consider the spaces $L^2=L^2(\mathbb{T,}m)$, $L^{\infty}=L^\infty(\mathbb{T},m)$, the classical Hardy space $H^2$ on the unit disc $\mathbb{D}$ identified with a subspace of $L^2$, and the Hardy space $H^\infty$ of all analytic and bounded functions in $\mathbb{D}$ identified with a subspace of $L^\infty$.
Denote by $M_\varphi$ the operator defined on $L^2$ of multiplication by a function $\varphi\in L^\infty$.
The most natural conjugation in $L^2$ is $J$ defined by $Jf=\bar f$, for $f\in L^2$. This conjugation has two natural properties: the operator $M_z$ is $J$--symmetric, i.e., $M_z J=J M_{\bar z}$, and
$J$ maps an analytic function into a co-analytic one, i.e., $J H^2=\overline{H^2}$.
Another natural conjugation in $L^2$ is $J^{\star}f=f^{\#}$ with $f^{\#}(z)\overset{df}{=}\overline{ f(\bar z)}.$ The conjugation $J^{\star}$ has a different behaviour: it commutes with multiplication by $z$ ($M_z J^\star=J^\star M_{ z}$) and leaves analytic functions invariant, $J^\star H^2\subset{H^2}$.
The map $J^{\star}$ appears for example in connection with Hankel operators (see \cite[pp. 146--147]{MAR}). Its connection with model spaces was studied in \cite{CGW} (Lemma 4.4, see also \cite[p. 37]{berc}). Hence a natural question is to characterize conjugations with respect to these properties. The first step was done in \cite{CKP} where all conjugations in $L^2$ with respect to which the operator $M_z$ is $C$--symmetric were characterized, see Theorem \ref{c11}.
In Section 2 we give a characterization of all
conjugations which commute with $M_z$, Theorem \ref{c1}. In Section 3, using the above characterizations we show that there are no conjugations in $L^2$ leaving $H^2$ invariant, with respect to which the operator $M_z$ is $C$--symmetric. We also show that $J^{\star}$ is the only conjugation commuting with $M_z$ and leaving $H^2$ invariant.
Beurling's theorem makes subspaces of $H^2 $ of the type $\theta H^2$ ($\theta$ inner function, i.e., $\theta\in H^\infty$, $|\theta|=1$ a.e. on $\mathbb{T}$) exceptionally interesting, as the only invariant subspaces for the unilateral shift $S$, $Sf(z)=zf(z)$ for $f\in H^2$. On the other hand, model spaces (subspaces of the type $K_\theta=H^2\ominus \theta H^2$), which are invariant for the adjoint of the unilateral shift, are important in model theory, \cite{NF}.
In \cite{CKP} all conjugations $C$ {with respect to which} the operator $M_z$ is $C$--symmetric and mapping a model space $K_\alpha$ into another {model space} $K_\theta$ were characterized, with the assumption that $\alpha$ divides $\theta$ ($\alpha\leqslant\theta$). {Recall that for $\alpha$ and $\theta$ inner, $\alpha\leqslant\theta$ means that $\theta/\alpha$ is also an inner function.} In what follows we will show that the result holds without the assumption $\alpha\leqslant\theta$.
In Section 4 conjugations {commuting with $M_z$ and} preserving model spaces are described. Section 5 is devoted to conjugations between $S$--invariant subspaces (i.e., subspaces of the form $\theta H^2$ with $\theta$ {an} inner function). In the last section we deal with conjugations commuting with the truncated shift $A^\theta_z$ {($A^\theta_z=P_{\theta}M_{z|K_{\theta}}$ where $P_{\theta}$ is the orthogonal projection from $L^2$ onto $K_{\theta}$)} or
conjugations such that $A^\theta_z$ is $C$--symmetric with respect to them.
\section{$M_z$ and $M_z$--commuting conjugations in $L^2$.}
Denote by $J$ the conjugation in $L^2$ defined as $Jf=\bar f$, for $f\in L^2$. This conjugation has the following obvious properties:
\begin{proposition}\label{J}
\begin{enumerate}
\item $M_z J=J M_{\bar z}$;
\item $M_\varphi J=JM_{\bar\varphi}$ for all $\varphi\in L^\infty$;
\item $J H^2=\overline{H^2}$.
\end{enumerate}
\end{proposition}
Let us consider all conjugations $C$ in $L^2$ satisfying the condition
\begin{equation}\label{mz}
M_z C=C M_{\bar z}.
\end{equation}
Such conjugations were studied in \cite{CKP} and are called {\it $M_z$--conjugations}. The following theorem characterizes all $M_z$--conjuga\-tions in $L^2$.
\begin{theorem}[\cite{CKP}]\label{c11}
Let $C$ be a conjugation in $L^2$. Then the following are equivalent:
\begin{enumerate}
\item $M_z C=C M_{\bar z}$,
\item $M_\varphi C=C M_{\bar\varphi}$ for all $\varphi\in L^\infty$,
\item there is $\psi\in L^\infty$, with $|\psi|=1$, such that $C=M_\psi J$,
\item there is $\psi^\prime\in L^\infty$, with $|\psi^\prime|=1$, such that $C=JM_{\psi^\prime}$.
\end{enumerate}
\end{theorem}
Another natural conjugation in $L^2$ is defined as
\begin{equation}\label{djstar}J^{\star}f=f^{\#}\ \text{ with }\ f^{\#}(z)\overset{df}{=}\overline{ f(\bar z)}.\end{equation}
The basic properties of $J^{\star}$ are the following:
\begin{proposition}\label{Jstar}
\begin{enumerate}
\item $M_z J^{\star}=J^{\star} M_{ z}$;
\item $M_{\bar z} J^{\star}=J^{\star} M_{\bar z}$;
\item $M_\varphi J^{\star}=J^{\star}M_{\varphi^{\#}}$ for all $\varphi\in L^\infty$;
\item $J^\star H^2=H^2$.
\end{enumerate}
\end{proposition}
In the context of the
Proposition \ref{Jstar} it seems natural to consider all conjugations $C$ in $L^2$
commuting with $M_z$, i.e.
\begin{equation}\label{eq1}M_z C=C M_{ z}.
\end{equation}
Such conjugations will be called {\it $M_z$--commuting}. In what follows we will often deal with functions
$f\in L^2$ such that $f(z)=f(\bar z)$ {a.e. on $\mathbb{T}$}, which will be called {\it symmetric}. Observe that if $f$ is symmetric and $f\in H^2$, then we also have $f\in\overline{H^2}$ and so it is a constant function.
\begin{theorem}\label{c1}
Let $C$ be a conjugation in $L^2$. Then the following are equivalent:
\begin{enumerate}\item $ M_z C=C M_{z}$ ,
\item $M_\varphi C=C M_{\varphi^{\#}}$ for all $\varphi\in L^\infty$,
\item there is a symmetric unimodular function $\psi\in L^\infty$
such that $C=M_\psi J^{\star}$,
\item there is a symmetric unimodular function $\psi^\prime\in L^\infty$
such that $C=J^{\star}\,M_{\psi^\prime}$.
\end{enumerate}
\end{theorem}
\begin{proof} We will show that $(1)\!\Rightarrow \!(3)$. \!The other implications are straightforward.
Assume that $M_{ z}C =CM_z$. Then $M_z CJ^{\star}=CM_{
z}J^{\star}=CJ^{\star}M_z$. It follows that the linear operator
$CJ^{\star}$ commutes with $M_z$. By \cite[Theorem 3.2]{RR}, there is $\psi\in L^\infty$
such that $CJ^{\star}=M_{\psi}$. Hence
$C=M_{\psi}J^{\star}=J^{\star}M_{\psi^{\#}}$.
By \eqref{e1} for any $f,g\in L^2$ we have
\begin{align*}\int g\bar{f}\,dm&=\langle g, f\rangle=
\langle Cf,Cg\rangle\\
&=\langle \psi f^{\#},\psi g^{\#}\rangle=\int|\psi(z)|^2\,\overline{f(\bar z)}g(\bar z)\,dm(z)\\
&=\int|\psi(\bar z)|^2\,\overline{f( z)}g( z)\,dm(z).
\end{align*}
Hence $|\psi|=1$ a.e. on $\mathbb{T}$. On the other hand, since $C^2=I_{L^2}$, for all $f\in L^2$ we have
$$f=C^2 f=M_\psi J^{\star} M_\psi J^{\star} f=M_\psi J^{\star}({\psi f^{\#}})=\psi\psi^{\#} f,$$
which implies that $\psi\psi^{\#}=1$ a.e. on $\mathbb{T}$. Therefore $\psi$ is symmetric, i.e., $\psi(z)=\psi(\bar z)$ a.e. on $\mathbb{T}$.
\end{proof}
\section{Conjugations preserving $H^2$.}
In the previous section all $M_z$--conjugations $C$ in $L^2$, i.e., such that
\begin{equation*}\label{orr}
M_{ z} C=C M_{\bar z}
\end{equation*} or $M_z$-- commuting conjugations, i.e. such that
\begin{equation*}\label{orr1}
M_{z} C=C M_{ z}
\end{equation*}
were characterized.
Let us now consider the question which of them preserve $H^2$. Clearly, if $C$ is a conjugation in $L^2$ and $C(H^2)\subset H^2$, then $C(H^2)= H^2$. Since $J^\star$ preserves $H^2$, it can be considered as a conjugation in $H^2$.
The following result shows that $J^\star$ is in that sense unique.
\begin{corollary}\label{co1}
Let $C$ be an $M_z$--commuting conjugation in $L^2$. If $C(H^2)\subset H^2$, then $C=\lambda J^{\star}$ for some $\lambda\in\mathbb{T}$.
\end{corollary}
\begin{proof}
By Theorem \ref{c1} we have that $C=M_{\psi}J^{\star}$ for some $\psi\in L^{\infty}$ with $|\psi|=1$ and $\psi(z)=\psi(\overline{z})$ a.e. on $\mathbb{T}$. Since $C$ preserves $H^2$, we have
$$\psi=M_{\psi}J^{\star}(1)=C(1)\in L^{\infty}\cap H^2=H^{\infty}.$$
Thus $\psi$ is analytic. Since it is symmetric, it is also co-analytic. Hence $\psi$ must be a constant function, so $\psi=\lambda\in\mathbb{C}$ and $|\lambda|=|\psi|=1$.
\end{proof}
\begin{corollary}
There are no $M_z$--conjugations in $L^2$ which preserve $H^2$.
\end{corollary}
\begin{proof}
If $C$ is an $M_z$--conjugation in $L^2$, then by Theorem \ref{c11} it follows that $C=M_{\psi}J$ for some $\psi\in L^{\infty}$ with $|\psi|=1$. As in the proof of Corollary \ref{co1} the assumption $C(H^2)\subset H^2$ implies that $\psi\in H^{\infty}$, which in turn means that $\psi$ is an inner function. Moreover, for $n=0,1,2,\dots$ we have
$$0=\langle C z^{n+1},\overline{z}\rangle=
\langle \psi\overline{z}^{n+1},\overline{z}\rangle=\langle \psi,z^n\rangle=0.$$
So $\psi=0$ which is a contradiction.
\end{proof}
The following example shows that not all conjugations in $L^2$ satisfy
either \eqref{mz} or \eqref{eq1}.
\begin{example}There is a set of naturally defined conjugations. For $k,l\in\mathbb{Z}$, $k<l$, define $C_{k,l}\ \colon\ L^2\rightarrow L^2$ by
\begin{equation}\label{ckl}
C_{k,l}\Big(\sum_{n\in\mathbb{Z}}a_n z^n\Big)=\overline{a}_l z^k+\overline{a}_k z^l+\sum_{n\notin\{k,l\}}\overline{a}_n z^n,
\end{equation}
where $\{z^n\}$ is the standard basis in $L^2$. Then \eqref{mz} and \eqref{eq1} are not satisfied since
$$M_{ z} C_{k,l}(z^k)=M_z(z^l)=z^{l+1},\quad C_{k,l} M_{\bar
z}(z^k)=C_{k,l} (z^{k-1})=z^{k-1}$$ and
$$C_{k,l} M_{z}(z^k)=C_{k,l} (z^{k+1})=\begin{cases}z^{k+1}&\text{if}\ k+1\neq l,\\z^k&\text{if}\ k+1=l.\end{cases}$$
Note that, on the other hand, $C_{k,l}$ preserves $H^2$ whenever $k\geqslant 0$ or
$l<0$.
\end{example}
\section{Conjugations preserving model spaces}
There is another class of conjugations in $L^2$ which appear naturally in connection with model spaces.
For a nonconstant inner function $\theta$, denote by $K_\theta$ the so called {\it model space} of the form $H^2\ominus\theta H^2$.
The conjugation $C_\theta$ defined in $L^2$ by
$$C_\theta f=\theta \bar z\bar f$$
has the important property that it preserves the model space $K_\theta$, i.e., $C_\theta K_\theta=K_\theta$. Thus $C_\theta$ can be considered as a conjugation in $K_\theta$.
Such conjugations are important in connection with truncated Toeplitz operators (see for instance \cite{GMR}).
Here we present several simple properties of such conjugations, which we will use later.
\begin{proposition}\label{ppp} Let $\alpha,\beta,\gamma$ be nonconstant inner functions. Then
\begin{enumerate}
\item $C_\beta C_\alpha=M_{\beta\bar \alpha}$,
\item $M_{\gamma} C_\alpha M_{\bar\gamma}$ is a conjugation in $L^2$,
\item $C_\beta\, M_\gamma = M_{\bar\gamma}\,C_\beta$.
\end{enumerate}
\end{proposition}
Now we will discuss relations between $M_z$--conjugations and model spaces.
The theorem below was proved in \cite[Theorem 4.2]{CKP} with the additional assumption that $\alpha\leqslant \theta$. As we prove here, this assumption is not necessary.
\begin{theorem}\label{t1}
Let $\alpha, \gamma, \theta$ be inner functions ($\alpha,\theta$ nonconstant).
Let $C$ be a conjugation in $L^2$ such that $M_z C=C M_{\bar z}$. Assume that $C(\gamma\kda)\subset\kdt$. Then there is an inner function $\beta$ such that $C=C_\beta$, with $\gamma\alpha\leqslant\beta\leqslant\gamma \theta$ and $\alpha\leqslant \theta$.
\end{theorem}
\begin{proof} Recall the standard notation for the reproducing kernel function at $0$ in $\kda$, namely, $k_0^\alpha=1-\overline{\alpha(0)}{\alpha}$ and its conjugate $\tilde k_0^\alpha=C_\alpha k_0^\alpha=\bar z(\alpha-\alpha(0))$.
By Theorem \ref{c11} we know that $C=M_\psi J$ for some function $\psi\in L^\infty$, $|\psi|=1$. Hence
$$\kdt \ni C (\gamma\tilde k_0^\alpha)= M_\psi J (\gamma\tilde k_0^\alpha)=\psi \overline{\gamma\bar z(\alpha-\alpha(0))}=\bar\gamma\bar\alpha z\psi(1-\overline{\alpha(0)}\alpha).$$
Thus there is $h\in \kdt$ such that $h=\bar\gamma\bar\alpha z\psi(1-\overline{\alpha(0)}\alpha)$.
Since $(1-\overline{\alpha(0)}\alpha)^{-1}$ is a bounded analytic function, we have
$$\bar \gamma\bar\alpha z\psi=h(1-\overline{\alpha(0)}\alpha)^{-1}\in H^2.$$
Since $\bar\gamma\bar\alpha z\psi\in H^2$ and $|\bar \gamma\bar\alpha z\psi|=1$ a.e. on $\mathbb{T}$, it has to be an inner function. Moreover $\beta=z\psi$ has to be inner and divisible by $\gamma\alpha$, i.e., $\gamma\alpha\leqslant\beta$.
On the other hand, we have similarly
$$\kdt\ni C_\theta C (\gamma k_0^\alpha) =C_\theta(\psi \overline{\gamma (1-\overline{\alpha(0)}\alpha)}=\theta \gamma\bar z\bar\psi (1-\overline{\alpha(0)}\alpha),$$
and $\theta\gamma\bar\beta={\theta}{\gamma} \bar z\bar\psi\in H^2$. Hence $\beta$ divides $ {\theta}{\gamma}$, i.e.,
$\beta\leqslant \gamma\theta$. It is clear that $C=C_\beta$. Finally, we have $\alpha\leqslant\theta$ as a consequence of $\gamma\alpha\leqslant\beta\leqslant \gamma\theta$.
\end{proof}
Note that if $\alpha\leqslant \theta$ and $C=C_\beta$ for some inner $\beta$ with $\gamma\alpha\leqslant\beta\leqslant\gamma \theta$, then $K_{\alpha}\subset K_{\tfrac{\beta}{\gamma}}\subset K_{\theta}$, $C_{\beta}M_{\gamma}=C_{\tfrac{\beta}{\gamma}}$ and
$$C(\gamma K_{\alpha})=C_{\beta}M_{\gamma}(K_{\alpha})=C_{\tfrac{\beta}{\gamma}}(K_{\alpha}) \subset K_{\tfrac{\beta}{\gamma}}\subset K_{\theta}.$$ Hence the implication in Theorem \ref{t1} is actually an equivalence.
The corollary bellow strengthens \cite[Proposition 4.5]{CKP}.
\begin{corollary}\label{bbb} Let $\alpha, \theta$ be nonconstant inner functions, and
let $C$ be a conjugation in $L^2$ such that $M_z C=C M_{\bar z}$. Assume that $C(\kda)\subset\kdt$. Then $\alpha\leqslant \theta$ and there is an inner function $\beta$ such that $C=C_\beta$, with $\alpha\leqslant\beta\leqslant \theta$.
\end{corollary}
Let us turn to discussing the relations between $M_z$--commuting conjugations and model spaces.
The following proposition describes some more properties of $J^\star$.
\begin{proposition} Let $\alpha$ be an inner function. Then
\begin{enumerate}
\item $J^\star (\alpha H^2)=\alpha^{\#}H^2$;
\item $J^\star(K_\alpha)=K_{\alpha^{\#}}$;
\item $J^{\star}C_\alpha =C_{\alpha^{\#}}J^{\star}$.
\end{enumerate}
\end{proposition}
\begin{proof}
The condition (1) is clear, (2) and (3) were proved in \cite[Lemma 4.4]{CGW}.
\end{proof}
Hence the conjugation $J^{\star}$ has a nice behaviour in connection with model
spaces, namely $J^{\star}(\kda)=K_{\alpha^{\#}}$. Theorem \ref{bb1} below
says that the conjugation $J^{\star}$ is, in some sense the only $M_z$--commuting conjugation with this property.
We start with the following:
\begin{proposition}\label{bb} Let $\alpha, \gamma, \theta$ be inner functions ($\alpha,\theta$ nonconstant).
Let $C$ be an $M_z$--commuting conjugation in $L^2$. Assume
that $C(\gamma\kda)\subset\kdt$. Then $\alpha\leqslant
\theta^{\#}$ and there is an inner function $\beta$ with
$\gamma\alpha\leqslant\beta\leqslant\gamma \theta^{\#}$ such that
$C=J^{\star} M_{\frac{\beta}{\gamma \alpha} \bar\gamma}$.
\end{proposition}
\begin{proof} Observe that since $C$ is an $M_z$--commuting conjugation, taking antilinear adjoints and applying \cite[Proposition 2.1]{CKP}
we get $M_{\bar z} C=C M_{\bar z}$. Since by Proposition \ref{ppp} the antilinear operator ${M_{\gamma}} C_\alpha {M_{\overline{\gamma}}}$ is a conjugation, then
$J^{\star}C{M_{\gamma}} C_\alpha {M_{\overline{\gamma}}}$ is also a conjugation.
Note also that ${M_{{\gamma}}} C_\alpha
{M_{\overline{\gamma}}}M_z = M_{\bar z}{M_{{\gamma}}} C_\alpha {M_{\overline{\gamma}}}$. Hence
\begin{equation}
J^{\star}C {M_{{\gamma}}} C_\alpha {M_{\overline{\gamma}}} M_z= M_{\bar z}J^{\star}C{M_{{\gamma}}}C_\alpha{M_{\overline{\gamma}}}.
\end{equation}
On the other hand,
\begin{displaymath}\begin{split}
J^{\star}C{M_{{\gamma}}} C_\alpha{M_{\overline{\gamma}}}(\gamma\kda)&\subset
J^{\star}C{M_{{\gamma}}} C_\alpha(\kda)\\
& \subset
J^{\star}C(\gamma\kda)\subset J^{\star}(K_{\theta})\subset
K_{\theta^{\#}}.
\end{split}
\end{displaymath}
By Theorem \ref{t1} there is an inner function $\beta$
such that $ J^{\star}C {M_{{\gamma}}}C_\alpha {M_{\overline{\gamma}}}=C_\beta$, with
$\gamma\alpha\leqslant\beta\leqslant\gamma \theta^{\#}$ and
$\alpha\leqslant \theta^{\#}$. Hence $C=J^{\star} C_\beta{M_{{\gamma}}}
C_\alpha {M_{\overline{\gamma}}}$. Therefore
\[C=J^{\star} M_{\frac{\beta}{\gamma \alpha} \bar\gamma}
\]
\end{proof}
{As in Theorem \ref{t1} the implication in Proposition \ref{bb} can be reversed. Indeed, if $\alpha\leqslant \theta^{\#}$ and $C=J^{\star} M_{\frac{\beta}{\gamma \alpha} \overline{\gamma}}$ for some inner function $\beta$ with $\gamma\alpha\leqslant\beta\leqslant\gamma \theta^{\#}$, then $K_{\alpha}\subset K_{\tfrac{\beta}{\gamma}}\subset K_{\theta^{\#}}$ and
$$C(\gamma K_{\alpha})=J^{\star} M_{\frac{\beta}{\gamma} \overline{\alpha}}(K_{\alpha})=J^{\star} C_{\frac{\beta}{\gamma}}C_{\alpha}(K_{\alpha})\subset J^{\star}(K_{\tfrac{\beta}{\gamma}})\subset J^{\star}( K_{\theta^{\#}})= K_{\theta}.$$}
\begin{theorem}\label{bb1} Let $\alpha, \theta$ be nonconstant inner functions, and let $C$ be an $M_z$--commuting conjugation in $L^2$, i.e., $M_z C=C M_{ z}$. Assume
that $C(\kda)\subset\kdt$. Then $\alpha\leqslant \theta^{\#}$ and
$C=\lambda J^{\star} $ with $\lambda\in \mathbb{T}$.
\end{theorem}
\begin{corollary}\label{ct}
Let $C$ be an $M_z$--commuting conjugation in $L^2$\!. \!Assume that there is some nonconstant inner function $\theta$ such
that $C(\kdt)\subset K_{\theta^{\#}}$. Then $C=\lambda J^{\star}$ with $\lambda\in \mathbb{T}$.
\end{corollary}
\begin{proof}[Proof of Theorem \ref{bb1} ] By Proposition \ref{bb} there is an inner function $\beta$ with $\alpha\leqslant\beta\leqslant \theta^{\#}$ such that $C=J^{\star} M_{\frac{\beta}{ \alpha} }$. The function $\frac{\beta}{ \alpha}$ is inner and by Theorem \ref{c1} it is symmetric. As observed before it follows that it is constant. Hence $C=J^{\star} $ up to multiplication by a constant of modulus $1$.
\end{proof}
\section{Conjugations preserving $S$-invariant subspaces of $H^2$}
Beurling's theorem says that all invariant subspaces for the unilateral shift $S$ are of the form $\theta H^2$ with $\theta$ inner.
We will now investigate conjugations in $L^2$ which preserve subspaces of this form. Since $C_{\theta}$ transforms $\theta H^2$ onto $\overline{z H^2}$, the operator $$C_{\theta}J^{\star}C_{\theta}=M_{\theta}J^{\star}M_{\overline{\theta}}$$
is an example of such a conjugation. Note that
$$(C_{\theta}J^{\star}C_{\theta})M_z=M_z(C_{\theta}J^{\star}C_{\theta}).$$
Let $\alpha$, $\theta$ be two inner functions. Then the operator $C_{\theta}J^{\star}C_{\alpha}\ \colon\ L^2\rightarrow L^2$ is an antilinear isometry which maps $\alpha H^2$ onto $\theta H^2$ and commutes with $M_z$. This operator however does not have to be an involution.
\begin{lemma}\label{L9}
Let $\alpha$, $\theta$ be two inner functions. The operator $C_{\theta}J^{\star}C_{\alpha}$ is an involution (and hence a conjugation in $L^2$) if and only if the function
$\theta\overline{\alpha^{\#}}$ is symmetric
(or equivalently $\alpha\alpha^{\#}=\theta\theta^{\#}$).
\end{lemma}
\begin{proof}
Note that by Proposition \ref{ppp},
$$(C_{\theta}J^{\star}C_{\alpha})(C_{\theta}J^{\star}C_{\alpha})=C_{\theta}C_{\alpha^{\#}}C_{\theta^{\#}}C_{\alpha}=M_{\theta\overline{\alpha^{\#}}}M_{\theta^{\#}\overline{\alpha}}=M_{\theta\overline{\alpha^{\#}}\theta^{\#}\overline{\alpha}}.$$
Therefore $C_{\theta}J^{\star}C_{\alpha}$ is an involution if and only if
$$\theta\overline{\alpha^{\#}}\theta^{\#}\overline{\alpha}=1\quad\text{a.e. on }\mathbb{T}, \text{ i.e., } \theta\theta^{\#}=\alpha\alpha^{\#},$$
which means that
$$(\theta\overline{\alpha^{\#}})(z)
=(\overline{\theta^{\#}}\alpha)(z)=\theta(\overline{z})\overline{\alpha^{\#}(\bar z)}=(\theta\overline{\alpha^{\#}})(\overline{z})\quad\text{a.e. on }\mathbb{T}.
$$
\end{proof}
The theorem bellow characterizes all $M_z$--commuting conjugations mapping one $S$--invariant subspaces into another $S$--invariant subspace. \begin{theorem}\label{THMC}
Let $\theta$ and $\alpha$ be two inner functions and let $C$ be a conjugation in $L^2$ such that $CM_z=M_zC$. Then $C(\alpha H^2)\subset \theta H^2$ if and only if $\theta\theta^{\#}\leqslant \alpha\alpha^{\#}$ and $C=C_{\beta}J^{\star}C_{\alpha}$, where $\beta$ is an inner function such that $\theta\leqslant \beta$, $\beta\beta^{\#}=\alpha{\alpha^{\#}}$. Moreover, in that case $C(\alpha H^2)=\beta H^2$.
\end{theorem}
Let $\alpha$ be a fixed inner function. By Lemma \ref{L9}, for each inner function $\beta$ with $\beta\beta^{\#}=\alpha\alpha^{\#}$ there exists an $M_z$--commuting conjugation $C$ which maps $\alpha H^2$ onto $\beta H^2$, namely $C=C_{\beta}J^{\star}C_{\alpha}$. On the other hand, if $\beta$ is an inner function and there exists an $M_z$--commuting conjugation $C$ which maps $\alpha H^2$ onto $\beta H^2$, then by Theorem \ref{THMC}, $\beta\beta^{\#}\leqslant\alpha{\alpha^{\#}}$ and $C=C_{\gamma}J^{\star}C_{\alpha}$ for some inner function $\gamma$ such that $\beta\leqslant \gamma$, $\gamma\gamma^{\#}=\alpha\alpha^{\#}$. In particular, $C(\alpha H^2)=\gamma H^2=\beta H^2$ and so $\gamma$ is a constant multiple of $\beta$, $\beta\beta^{\#}=\alpha\alpha^{\#}$.
It follows from the above that Lemma 5.3 characterizes all possible spaces of type $\beta H^2$ such that for a given $S$--invariant subspace $\alpha H^2$ there is an $M_z$--commuting conjugation mapping $\alpha H^2$ onto $\beta H^2$.
\begin{lemma}
Let $\alpha$ be a nonconstant inner function. Then \begin{multline}\qquad\{\beta: \beta \text{ is inner, }\alpha\alpha^{\#}= \beta\beta^{\#} \}\\= \{\lambda\, uv^{\#}: u, v \text{ are inner, } \alpha=uv, \lambda\in\mathbb{T}\}.\qquad \end{multline}
\end{lemma}
For two inner functions $\alpha$ and $\beta$ denote by $\alpha \wedge\beta$ the greatest common divisor of $\alpha$ and $\beta$. We will write $\alpha \wedge\beta=1$ if the only common divisor of $\alpha$ and $\beta$ is a constant function.
\begin{proof}
Note that for $\alpha=uv$ and $\beta=\lambda uv^{\#}$ we have $\alpha\alpha^{\#}=\beta\beta^{\#}$, hence one inclusion is proved.
For the other inclusion let $u=\alpha\wedge\beta$ and we can write $\alpha=uv$ and $\beta=u v_1$. From the condition $\alpha\alpha^{\#}=\beta\beta^{\#}$ it follows that
$$uvu^{\#}v^{\#}=uv_1u^{\#}v_1^{\#}.$$
Hence
$vv^{\#}=v_1v_1^{\#}$. Since $v\wedge v_1=1$, we have that $v$ divides $v_1^{\#}$ and $v_1$ divides $v^{\#}$ and vice--versa. Thus we can take $v_1=\lambda v^{\#}$ with $\lambda\in\mathbb{T}$, and so $\beta= \lambda uv^{\#}$.
\end{proof}
\begin{proof}[Proof \!of Theorem \ref{THMC}]
Assume firstly that $CM_z=M_zC$ and $C(\alpha H^2)\subset \theta H^2$. By Theorem \ref{c1}, $C=M_{\psi}J^{\star}$ for some unimodular symmetric function $\psi\in L^{\infty}$. In particular,
$$\psi\alpha^{\#}=M_{\psi}J^{\star}(\alpha)=C(\alpha)\in \theta H^2,$$
and there exists $u\in H^2$ such that $\psi\alpha^{\#}=\theta u$. Note that $u$ must be inner and so $\psi=\beta \overline{\alpha^{\#}}$ with $\beta=\theta u$, $\theta\leqslant\beta$. Clearly $\beta\overline{\alpha^{\#}}$ is symmetric, i.e., $\beta\beta^{\#}=\alpha\alpha^{\#}$. Hence $\theta\theta^{\#}\leqslant\alpha\alpha^{\#}$.
Assume now that $\theta\theta^{\#}\leqslant\alpha\alpha^{\#}$, and let $\alpha= \alpha_1\cdot (\alpha\wedge\theta)$ and $\theta=\theta_1\cdot (\alpha\wedge\theta)$. Since $(\alpha\wedge\theta)^{\#}=\alpha^{\#}\wedge\theta^{\#}$, we get $\alpha^{\#}= \alpha_1^{\#}\cdot (\alpha^{\#}\wedge\theta^{\#})$ and $\theta^{\#}=\theta_1^{\#}\cdot (\alpha^{\#}\wedge\theta^{\#})$. Note also that $\theta_1\theta_1^{\#}\leqslant \alpha_1\alpha_1^{\#}$ and $\theta_1\wedge\alpha_1=1$, so $\theta_1\leqslant \alpha_1^{\#}$.
Thus
$$\frac{\alpha\alpha^{\#}}{\theta\theta^{\#}}=\frac{\alpha_1\alpha_1^{\#}}{\theta_1 \theta_1^{\#}}=\frac{\alpha_1^{\#}}{\theta_1}\frac{\alpha_1}{\theta_1^{\#}}=u u^{\#},$$
where $u=\frac{\alpha_1^{\#}}{\theta_1}$ is an inner function. Now we may take $\beta=\theta u$.
Since $\theta\leqslant\beta$ and $\beta\beta^{\#}=\alpha{\alpha^{\#}}$, by Lemma \ref{L9} and by Proposition \ref{ppp}, $C=M_{\beta \overline{\alpha^{\#}}}J^{\star}=C_{\beta}J^{\star}C_{\alpha}$ is a conjugation which maps $\alpha H^2$ onto $\beta H^2\subset \theta H^2$.
\end{proof}
\begin{corollary}
Let $\theta$ be an inner function and let $C$ be an $M_z$--commuting conjugation in $L^2$. Then
\begin{enumerate}
\item $C(\theta H^2)\subset \theta H^2$ if and only if $C=\lambda C_{\theta}J^{\star}C_{\theta}$ with $\lambda\in \mathbb{T}$;
\item $C(\theta H^2)\subset \theta^{\#} H^2$ if and only if $C=\lambda J^{\star}$ with $\lambda\in \mathbb{T}$.
\end{enumerate}
\end{corollary}
\begin{proof}
By Theorem \ref{THMC}, $C(\theta H^2)\subset \theta H^2$ if and only if there exists an inner function $\beta$ such that
$\theta\leqslant\beta$ and $\beta\beta^{\#}=\theta{\theta^{\#}}$. This is only possible if $\beta$ is constant multiple of $\theta$ and (1) is proved. The proof of (2) is similar.
\end{proof}
Note that by Theorem \ref{THMC} (Lemma \ref{L9}, actually) if $\theta\overline{\alpha^{\#}}$ is symmetric, then there exists an $M_z$--commuting conjugation from $\alpha H^2$ into $\theta H^2$. The following example shows that in that case there may be no such conjugation between the corresponding model spaces $K_{\alpha}$ and $K_{\theta}$.
\begin{example}
Fix $a,b\in\mathbb{D}$ such that $a\neq b$, $a\neq \overline{a}$ and $b\neq \overline{b}$, and put
$$\alpha(z)=\tfrac{a-z}{1-\overline{a}z}\ \tfrac{b-z}{1-\overline{b}z}\qquad\text{and}\qquad \theta(z)=\tfrac{a-z}{1-\overline{a}z}\ \tfrac{\overline{b}-z}{1-{b}z}.$$
Then
$$\alpha^{\#}(z)=\tfrac{\overline{a}-z}{1-{a}z}\ \tfrac{\overline{b}-z}{1-{b}z}\qquad\text{and}\qquad \theta^{\#}(z)=\tfrac{\overline{a}-z}{1-{a}z}\ \tfrac{{b}-z}{1-\overline{b}z}$$
and so $\alpha\alpha^{\#}=\theta\theta^{\#}$.
Thus there exists an $M_z$--commuting conjugation from $\alpha H^2$ onto $\theta H^2$. In this case however neither $\alpha\leqslant\theta^{\#}$ nor $\theta\leqslant\alpha^{\#}$, so by Theorem \ref{bb1} no $M_z$--commuting conjugation between $K_{\alpha}$ and $K_{\theta}$ exists. Here also neither $\alpha\leqslant\theta$ nor $\theta\leqslant\alpha$, and so by Theorem \ref{t1} no $M_z$--conjugation between $K_{\alpha}$ and $K_{\theta}$ exists.
\end{example}
Finally, consider $M_z$--conjugations preserving $S$--invariant subspaces.
\begin{proposition}\label{THMC1}
Let $\theta$ and $\alpha$ be two inner functions.
There are no $M_z$--conjugations in $L^2$ which map $\alpha H^2$ into $\theta H^2$
\end{proposition}
\begin{proof}
If $C$ was such a conjugation, then by Theorem \ref{c11}, $C=M_{\psi}J$ for some unimodular function $\psi\in L^{\infty}$ and, in particular,
$$C(\alpha)=\psi\overline{\alpha}=\theta g\quad\text{for some }g\in H^2.$$ Clearly $g$ must be an inner function and $\psi =\alpha\theta g$. Then, for every $h\in H^2$,
$$C(\alpha h)=\alpha\theta g\overline{\alpha h}=\theta g\overline{ h}\in \theta H^2,$$
and so $g\overline{h}\in H^2$. It follows that $g=0$ and $C(\alpha)=0$ which is a contradiction.
\end{proof}
\section{Conjugations and truncated Toeplitz operators}
For $\varphi\in L^2$ define the {\it truncated Toeplitz operator} $A^\theta_\varphi$ by
$$A^\theta_\varphi f=P_\theta (\varphi f), \text{ for } f\in H^\infty\cap \kdt,$$
were $P_\theta\colon L^2\to \kdt$ is the orthogonal projection (see \cite{Sarason}). The operator $A^\theta_\varphi$ is closed and densely defined, and if it is bounded, it admits a unique bounded extension to $\kdt$. The set of all bounded truncated Toeplitz operators on $\kdt$ is denoted by $\mathcal{T}(\theta)$. Note that $A^\theta_\varphi\in \mathcal{T}(\theta)$ for $\varphi\in L^\infty$. It is known that every operator from $\mathcal{T}(\theta)$ is $C_{\theta}$--symmetric (see \cite[Lemma 2.1]{Sarason}).
Observe that if $k\geqslant 0$, then the conjugation $C_{k,l}$ defined by \eqref{ckl} satisfies neither
$M_zC_{k,l}=C_{k,l}M_z$ nor $SC_{k,l}=C_{k,l}S$.
However, for $0\leqslant n<k$ and
$\theta(z)=z^n$,
$$C_{k,l}(K_\theta)=K_\theta\quad\text{and}\quad
A_z^{\theta}C_{k,l}=C_{k,l}A_z^{\theta}$$ (since here
$C_{k,l|K_{\theta}}=J^{\star}_{|K_{\theta}}$ and
$\theta^{\#}=\theta$).
Theorem below characterizes conjugations intertwining truncated shifts $ A_z^{\theta}$ and $A_z^{\theta^{\#}}$.
\begin{theorem}\label{c3}
Let $\theta$ be a nonconstant inner function and
let $C$ be a conjugation in $L^2$ such that $C(K_\theta)\subset K_{\theta^{\#}}$.
Then the following are equivalent:
\begin{enumerate}
\item $A_{\varphi^{\#}}^{\theta^{\#}} C=C A_{\varphi}^{\theta}$ on $K_{\theta}$ for all $\varphi\in H^\infty$,
\item $ A_z^{\theta^{\#}} C=C A_z^{\theta}$ on $K_{\theta}$,
\item there is a function $\psi\in H^{\infty}$ such that
$C_{|K_{\theta}}= J^{\star}A_{\psi}^{\theta}$ and $A_{\psi}^{\theta}$ is an isometry,
\item there is a function $\psi'\in H^{\infty}$ such that
$C_{|K_{\theta}}= A_{\psi'}^{\theta^{\#}}J^{\star}_{|K_{\theta}}$ and $A_{\psi'}^{\theta^{\#}}$ is an isometry.
\end{enumerate}
\end{theorem}
\begin{proof}
We will only prove that $(2)\Rightarrow (3)$.
Since $J^{\star}(K_{\theta^{\#}})=K_\theta$
and $J^{\star}A_{\varphi^{\#}}^{\theta^{\#}} = A_{\varphi}^{\theta}J^{\star}$ for all $\varphi\in H^{\infty}$ (see \cite[Lemma 4.5]{CGW}), we have
$$J^{\star}CA_z^{\theta}=J^{\star}A_z^{\theta^{\#}}C=A_z^{\theta}J^{\star}C$$
on $K_{\theta}$ and so $J^{\star}C_{|K_{\theta}}=A_{\psi}^{\theta}$ for some $\psi\in H^{\infty}$ (\cite[Theorem 14.38]{FM}). Hence $A_{\psi}^{\theta}$ is an isometry and
$$C_{|K_{\theta}}=J^{\star}A_{\psi}^{\theta}.$$
\end{proof}
It is much more restrictive if $\theta^{\#}=\theta$.
\begin{proposition}
Let $\theta$ be an inner function such that $\theta^{\#}=\theta$ and
let $C$ be a conjugation in $K_\theta$. Then the following are equivalent:
\begin{enumerate}
\item $A_{\varphi^{\#}}^{\theta}C=CA_{\varphi}^{\theta}$ for all $\varphi\in H^\infty$,
\item $ A_z^{\theta} C= CA_z^{\theta}$,
\item
$C=\lambda J^{\star}_{|K_{\theta}}$ with $\lambda\in\mathbb{T}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Implications $(1)\Rightarrow (2)$ and $(3)\Rightarrow (1)$ are clear.
To prove $(2)\Rightarrow (3)$ apply Theorem \ref{c3} to the conjugation $\tilde{C}$ in $L^2$ defined by $$\tilde{C}=C\oplus C_{\theta}\ \colon\ K_{\theta}\oplus (K_{\theta})^{\perp}\rightarrow K_{\theta}\oplus (K_{\theta})^{\perp}.$$ It follows that $\tilde{C}_{|K_{\theta}}=C=J^{\star}A_{\psi}^{\theta}$ for $\psi\in H^{\infty}$ such that $A_{\psi}^{\theta}$ is an isometry. Since $C(K_{\theta})=K_{\theta}$ and $J^{\star}(K_{\theta})=K_{\theta^{\#}}=K_{\theta}$, we see that $A_{\psi}^{\theta}=J^{\star}C$ maps $K_{\theta}$ onto $K_{\theta}$ and is in fact unitary. Thus we have
$$A_{\overline{\psi}}^{\theta}A_{\psi}^{\theta}=A_{\psi}^{\theta}A_{\overline{\psi}}^{\theta}=I_{K_{\theta}}.$$
On the other hand, $C^2=I_{K_{\theta}}$ so
$$C^2=J^{\star}A_{\psi}^{\theta}J^{\star}A_{\psi}^{\theta}=A_{\psi^{\#}}^{\theta}A_{\psi}^{\theta}=A_{\psi}^{\theta}A_{\psi^{\#}}^{\theta}=I_{K_{\theta}}.$$
Hence $A_{\overline{\psi}}^{\theta}=A_{\psi^{\#}}^{\theta}$ and
$A_{\overline{\psi}-\psi^{\#}}^{\theta}=0$, which gives
$\overline{\psi}-\psi^{\#}\in \overline{\theta H^2}+\theta H^2$ (see \cite{Sarason}). In
other words, $\overline{\psi}-\psi^{\#}=\overline{\theta h_1}+\theta
h_2$ for some functions $h_1, h_2\in H^2$. Thus there exists a
constant $\lambda$ such that $${\psi}-\theta
h_1=\overline{\psi^{\#}+\theta h_2}=\overline{\lambda}.$$ We now have
$$A_{\psi}^{\theta}=A_{\theta
h_1+\overline{\lambda}}^{\theta}=\overline{\lambda} I_{K_{\theta}}.$$
Moreover
$\lambda\in\mathbb{T}$, since $A_{\psi}^{\theta}$ is unitary. Hence
\[C=J^{\star}A_{\psi}^{\theta}={\lambda} J^{\star}_{|K_{\theta}}.\]
\end{proof}
Now we characterize conjugations intertwining the truncated shifts $ A_z^{\theta}$ and $A_{\bar z}^{\theta}$.
\begin{theorem}\label{c4}
Let $\theta$ be an inner function and let $C$ be a conjugation in
$K_\theta$. Then the following are
equivalent:
\begin{enumerate}
\item $A_{\varphi}^{\theta} C=C A_{\overline{\varphi}}^{{\theta}}$ for all $\varphi\in H^\infty$,
\item $ A_z^{\theta} C=C A_{\bar z}^{\theta}$ ,
\item there is a function $\psi\in H^\infty$ such that
$C=A_\psi^{\theta}C_{\theta}$ and $A_\psi^{\theta}$
is unitary,
\item there is a function $\psi^\prime\in H^\infty$ such that
$C=C_{\theta}A_{\overline{\psi^\prime}}^{\theta}$ and $A_{\psi^\prime}^{\theta}$
is unitary.
\end{enumerate}
\end{theorem}
\begin{proof}
Let us start with $(2)\Rightarrow (3)$.
Since $A_z^{\theta}$ is $C_{\theta}$--symmetric,
$$A_z^{\theta} CC_{\theta}=CA_{\bar
z}^{\theta}C_{\theta}=CC_{\theta}A_z^{\theta}.$$ Hence, by \cite[Proposition 1.21]{berc},
$CC_{\theta}=A_\psi^{\theta}$ for some $\psi\in H^\infty$. Clearly,
$A_\psi^{\theta}$ is unitary and
$$C=A_\psi^{\theta}C_{\theta}=C_{\theta}(A_\psi^{\theta})^{*}=C_{\theta}A_{\overline{\psi}}^{\theta}.$$
To prove that $(4)\Rightarrow (1)$ note that,
since $A_{\overline{\psi^\prime}}^{\theta}$ and
$A_{\overline{\varphi}}^{\theta}$ commute, we have
$$A_{\varphi}^{\theta}C=A_{\varphi}^{\theta}C_{\theta}A_{\overline{\psi^\prime}}^{\theta}
=C_{\theta}A_{\overline{\varphi}}^{\theta}A_{\overline{\psi^\prime}}^{\theta}=C_{\theta}A_{\overline{\psi^\prime}}^{\theta}A_{\overline{\varphi}}^{\theta}=CA_{\overline{\varphi}}^{\theta}=C(A_{{\varphi}}^{\theta})^{*}.$$
All other implications are straightforward.
\end{proof}
\begin{corollary}
If $C$ is a conjugation in $K_\theta$
and every $A\in \mathcal{T}(\theta)$ is $C$--symmetric, then
$C=A_\psi^{\theta}C_{\theta}$ for some $\psi\in
H^\infty$ such that $A_\psi^{\theta}$ is unitary.
\end{corollary}
For a complete description of unitary operators from $\mathcal{T}(\theta)$ see \cite[Proposition 6.5]{SED}.
| 8,873
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ICE Steps Up OTC Conversion To October
IntercontinentalExchange has move up plans to transition cleared over-the-counter energy swaps and options to futures from January to Oct. 15 because of customer demand.
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Hiring someone who’s wrong for your business can be costly.
Anecdotal evidence suggests that time starved businesses often concentrate on filling a gap right now versus finding the ‘right candidate’ who will add value and strengthen the business.
In this case, the future under-performance of an employee can often be traced back to the interview process where a candidate’s skills and knowledge compared to the job you need done is incorrectly matched… and whose fault is the future under-performance in that case?
Considering the impact an under or poor performing employee can have on a small to medium sized business (e.g. loss of business, negativity in fellow workers or worse, future legal ramifications of not handling the performance matter ‘fairly’) it’s important to get it right.
But how do you know who the right candidate is?
- Clearly identifying what the successful candidate will ‘look like’ (e.g. professional background, experience, knowledge, passions and values, alignment to key tasks and similar characteristics in your other high performing employees).
- Consistency in the process – asking the same questions of all candidates so benchmarking can be determined.
- Recording the interviews so notes can be compared (and relied upon to show fairness and process if ever needed).
- Actually talking to referees and treating this as a mini interview as well.
- Oh, and lastly, never be afraid to push for the truth or examples of previous work during the interview… or even ending an interview if there is any hint of deceit. Better to do that at the time rather than undertake a performance management process with the candidate as a bona fide employee of the company.
If you rely upon an outsourced recruitment function, then much of this should already be taken care of. If not, maybe have a chat to them next time and ask why not?
Taking the time to map out these critical areas before you post your job ad will put you on track to identifying the ‘right candidate’.
Time remains a critical commodity for small / medium sized businesses and this will often dictate how much pre-planning and targeted interviewing will take place. But trust us on this, failing to do any planning will result in poor performers getting into the business.
For this reason, seeking initial advice or even outsourcing to HR and Recruitment consultancy can help avoid the poor performers from even getting through the door.
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MUSIC
ALBUM REVIEW
David Gray
Life In Slow Motion
***½
by Preston Jones on September 12, 2005
The sonic equivalent of throwing open the drapes and letting the light in, the follow-up to David Gray's underrated 2002 long-player A New Day at Midnight is a revivifying expansion of his traditionally spare sound. Teaming with producer Marius de Vries, Gray largely discards the austere, deadly serious singer-songwriter shtick in favor of a lush, cinematic canvas upon which he reliably spins his literate, melodic rock. Traces of the earlier, more electronically inclined White Ladder are apparent but few; Gray generally shies away from techniques relied upon before, content to plow new ground. Life In Slow Motion finds the English troubadour in a somewhat sunnier mood than on its introspective, autumnal predecessor. Lead single "The One I Love" playfully cribs from Bruce Springsteen's "Hungry Heart," while "Lately" eerily channels Van Morrison, but regardless of the sonic homages, Gray's trademarked dense narratives of the human heart still captivate. Working with de Vries and his trusted percussionist Clune, Gray seems unfettered by the trappings of his past—the one-two punch of the achingly beautiful "Ain't No Love" and the sprightly "Hospital Food" are but a few of the album's highlights. Winnowing the disc down to 10 tracks leaves little room for filler and if any is present, it's probably the overly earnest "Nos Da Caraid." Much of Life In Slow Motion feels contemplative, but not morosely so—the devastating repetition of "Slow Motion" captures a relationship crumbling before your eyes while "Now & Always" is a bittersweet love note to a special someone. The bracing, deliberate pace of the album (hey, it's an album title that fits) may frustrate those in search of another "Babylon," but, unhurried and indifferent, Gray fashions this compelling, low-key collection of work that ranks among one of the more worthwhile records released in a while.
- Label: ATO/RCA
-
| 286,285
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