PivotCompress: Compression by Sorting

Definition 1.4 .

(PZ-equivalence) Two open projections $p,q\in P_{o}(A^{**})$ are said to be PZ-equivalent, $p\sim_{\text{PZ}}q$ in symbols, if there exists a partial isometry $v\in A^{**}$ such that

p=v^{*}v,\qquad q=vv^{*},

and

vA_{p}\subset A,\qquad v^{*}A_{q}\subset A.

Definition 1 .

A function $g:\rho\rightarrow F^{\ast}$ is transitive if and only if

g\left(i,j\right)g\left(j,k\right)=g\left(i,k\right)

for all $\left(i,j\right),\left(j,k\right)\in\rho.$

Definition 2.2 .

An affine permutation of order $n$ is a bijection $f:\mathbb{Z}\rightarrow\mathbb{Z}$ which satisfies the condition

(2.4)

f(i+n)=f(i)+n

for all $i\in\mathbb{Z}.$ The affine permutations of order $n$ form a group, which we denote $\widetilde{S}_{n}.$

DEFINITION 2.3 .

The subvariety $\mathbf{DMSH_{1}}$ of level 1 of $\mathbf{DMSH}$ is defined by the identity: $x\land x^{\prime*}\land x^{2{\rm(}}{{}^{\prime*{\rm{)}}}}\approx x\land x^{% \prime*},$ or equivalently, by the identity:

(L1)

$(x\land x^{\prime*})^{\prime*}\approx x\land x^{\prime*}$ .

Definition 3.1 .

A vector $w$ in a vector space $V$ (over ${\mathbb{C}}$ ) on which a non-degenerate bi-linear form $\langle~{},~{}\rangle$ is defined, is called isotropic with respect to that bi-linear form if

\langle w,w\rangle=0.

(Note that the form is assumed to be bi-linear, not sesqui -linear.)

Definition 2.2 .

The point equivalence transformation of a class of PDEs ( 2.5 ) is an invertible transformation of the independent and dependent variables of the form

\begin{array}[]{ll}\bar{x}=\phi(x,u),\bar{u}=\psi(x,u),\\ \end{array}

(2.11)

that maps every equation of the class into an equation of the same family, viz.

\begin{array}[]{ll}E_{\alpha}(\bar{x},\bar{u},...,\bar{u}_{(k)})=0,&\alpha=1,.% ..,m.\end{array}

(2.12)

Definition \thechapter .3 .

An assignment $s$ on $\mathfrak{M}$ is a function $s\colon\mathrm{Var}\to\left|\mathfrak{M}\right|$ .

If $x\in\mathrm{Var}$ and $m\in\left|\mathfrak{M}\right|$ , $s[x/m]$ is the assignment defined by

s[x/m](y)=\begin{cases}m&\text{if $y=x$}\\ s(y)&\text{otherwise}\end{cases}

Definition 2.2 ( [ 13 ] ) .

Let $(A,\mu,\alpha)$ and $(A^{\prime},\mu^{\prime},\alpha^{\prime})$ be two Hom-alternative algebras. A linear map $f:A\to A^{\prime}$ is said to be a morphism of Hom-alternative algebras if

\mu^{\prime}\circ(f\otimes f)=f\circ\mu\text{ and }f\circ\alpha=\alpha^{\prime% }\circ f.

(3)

Definition 2.2 .

A locally defined $p$ -form $\lambda$ of $J^{k}(E)$ , $p\geq 1$ , is called holonomic if $\imath_{X}\lambda=0$ for any holonomic vector field $X$ . A local $0$ -form (i.e., a $\mathcal{C}^{\infty}$ function on an open set) is called holonomic if it vanishes identically.

If $\alpha$ , $\alpha^{\prime}$ are $p$ -forms on the same open subset $\mathcal{U}\subset J^{k}(E)$ , they are called variationally equivalent if

\alpha-\alpha^{\prime}=\lambda+d\mu

for some holonomic $p$ -form $\lambda$ and some holonomic $(p-1)$ -form $\mu$ .

Definition 4.8 .

Suppose that $D,\widetilde{D}$ are closed subsets of $\mathbb{C}$ . A degree $1$ map (in $z$ and $\bar{z}$ separately) on $D$ is a function $f$ from $D$ to $\mathbb{C}$ of the form

(1)

f(z)=\alpha+\beta z+\gamma\bar{z}+\delta z\bar{z}\text{}

for some $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ such that moreover for some $\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime}\in\mathbb{C}$ ,

(2)

f(z)\overline{f(z)}=\alpha^{\prime}+\beta^{\prime}z+\gamma^{\prime}z+\delta^{% \prime}z\bar{z}\text{}

for every $z\in D$ . A degree $1$ homeomorphism from $D$ to $\widetilde{D}$ is a homeomorphism $\varphi:D\rightarrow\widetilde{D}$ such that $\varphi$ and $\varphi^{-1}$ are degree $1$ maps. The closed subsets $D$ and $\widetilde{D}$ of $\mathbb{C}$ are degree $1$ homeomorphic if there is a degree $1$ homeomorphism from $D$ to $\widetilde{D}$ .

Definition 2.1 .

[ 11 ] A right Leibniz algebra $L$ is a vector space over a field $\mathbb{K}$ endowed with a bilinear product $[\cdot,\cdot]$ satisfying the Leibniz identity

[[y,z],x]=[[y,x],z]+[y,[z,x]],

for all $x,y,z\in L$ .

Definition 2.2 .

[ 7 ] A Leibniz triple system is a vector space $T$ endowed with a trilinear operation $\{\cdot,\cdot,\cdot\}:T\times T\times T\rightarrow T$ satisfying

	$\displaystyle\{a,\{b,c,d\},e\}\!=\!\{\{a,b,c\},d,e\}\!-\!\{\{a,c,b\},d,e\}\!-% \!\{\{a,d,b\},c,e\}\!+\!\{\{a,d,c\},b,e\},$		(2.1)
	$\displaystyle\{a,b,\{c,d,e\}\}\!=\!\{\{a,b,c\},d,e\}\!-\!\{\{a,b,d\},c,e\}\!-% \!\{\{a,b,e\},c,d\}\!+\!\{\{a,b,e\},d,c\},$		(2.2)

for all $a,b,c,d,e\in T$ .

Definition 2.2 (Metropolis-Hastings kernel) .

Given the current state $u_{k}$ , a candidate state $u^{\prime}$ is drawn from the proposal distribution $q(u_{k},\cdot)$ . Define the pair of measures

\begin{array}[]{rll}\nu(du,du^{\prime})&=&q(u,du^{\prime})\mu(du)\\ \nu^{\bot}(du,du^{\prime})&=&q(u^{\prime},du)\mu(du^{\prime}).\end{array}

(5)

Then the next state of the Markov chain is set to $u_{k+1}=u^{\prime}$ with probability

\alpha(u_{k},u^{\prime})={\rm min}\left\{1,\frac{d\nu^{\bot}}{d\nu}(u_{k},u^{% \prime})\right\};

(6)

otherwise, with probability $1-\alpha(u_{k},u^{\prime})$ , it is set to $u_{k+1}=u_{k}$ .

Definition 2.2 .

[ 4 ] A map $\phi:G\times G\times G\longrightarrow\mathbb{K}^{\times}$ is called a cocycle if

	$\displaystyle\phi(h,k,l)\phi(g,hk,l)\phi(g,h,k)=\phi(g,h,kl)\phi(gh,k,l),$		(2.1)
	$\displaystyle\phi(g,e,h)=1,$		(2.2)

hold for any $g,h,k,l\in G$ , where $e$ is the identity of $G$ .

Definition 5.5 .

Take an automorphism system $\sigma:G\rightarrow Aut(\mathbb{K})$ , where we consider the field $\mathbb{K}$ as the associative algebra $B$ on the natural way. A quasicrossed mapping $\delta:G\times G\rightarrow\mathbb{K}^{\times}$ (see Definition 4.7 ) is called a coboundary if there is a function $u:G\rightarrow\mathbb{K}^{\times}$ such that

\begin{split}\displaystyle\delta(g,h)=u(g)\sigma(g)(u(h))u(gh)^{-1},\end{split}

for any $g,h\in G$ .

Definition .

We say that there exists a quantum homomorphism from a graph $X$ to a graph $Y$ , and write $X\xrightarrow{q}Y$ , if there exist positive semidefinite matrices $\rho_{xy}$ for $x\in V(X),y\in V(Y)$ and $\rho$ satisfying

	$\displaystyle\langle\rho,\rho\rangle=1$
	$\displaystyle\sum_{y\in V(Y)}\rho_{xy}=\rho\text{ for all }x\in V(X)$
	$\displaystyle\langle\rho_{xy},\rho_{x^{\prime}y^{\prime}}\rangle=0\text{ if }(% x=x^{\prime}\ \&\ y\neq y^{\prime})\text{ or }(x\sim x^{\prime}\ \&\ y\not\sim y% ^{\prime})$

Definition .

We say that there exists a entanglement-assisted homomorphism from a graph $X$ to a graph $Y$ , and write $X\xrightarrow{*}Y$ , if there exist positive semidefinite matrices $\rho_{xy}$ for $x\in V(X),y\in V(Y)$ and $\rho$ satisfying

	$\displaystyle\langle\rho,\rho\rangle=1$
	$\displaystyle\sum_{y\in V(Y)}\rho_{xy}=\rho\text{ for all }x\in V(X)$
	$\displaystyle\langle\rho_{xy},\rho_{x^{\prime}y^{\prime}}\rangle=0\text{ if }(% x\sim x^{\prime}\ \&\ y\not\sim y^{\prime})$

Definition 3.1 .

Define

{\bf q}(t)=\left[\begin{array}[]{c}{\bf y}(t)\\ {\bf z}(t)\end{array}\right]

We will use the notation $B$ to denote the matrix

B=\left[\begin{array}[]{cc}\alpha M^{\prime}&-(\alpha-1)M^{\prime}\\ I&0\end{array}\right]

where $\alpha=2-2/(9U+1)$ ; note that in the previous section this would have been denoted $B(1)$ . Finally, we will use ${\bf g}(t)$ to denote the vector that stacks up the subgradients $g_{i}(t),i=1,\ldots,n$ .

Definition 311 .

(Multiplier semigroup) Let $(S,*)$ be an involutive semigroup. A multiplier of $S$ is a pair $(\lambda,\rho)$ of mappings $\lambda,\rho\colon S\to S$ satisfying the following conditions:

(M1)

$a\lambda(b)=\rho(a)b$ ,
(M2)

$\lambda(ab)=\lambda(a)b$ , and
(M3)

$\rho(ab)=a\rho(b).$

The map $\lambda$ is called the left action of the multiplier and $\rho$ is called the right action of the multiplier. For $m=(\lambda,\rho)$ and $s\in S$ we write

ms:=\lambda(s)\quad\mbox{ and }\quad sm:=\rho(s).

We write $M(S)$ for the set of all multipliers of $S$ and turn it into an involutive semigroup by

(\lambda,\rho)(\lambda^{\prime},\rho^{\prime}):=(\lambda\circ\lambda^{\prime},% \rho^{\prime}\circ\rho)\quad\mbox{ and }\quad(\lambda,\rho)^{*}:=(\rho^{*},% \lambda^{*})

with $\lambda^{*}(a):=\lambda(a^{*})^{*}$ and $\rho^{*}(a)=\rho(a^{*})^{*}$ (see [ Jo64 ] ).

Definition 3.3 .

Let $k=2m+1$ and $l=2n+1$ . For $\rho:\pi_{1}(k,l)\rightarrow\mathrm{SL}_{2}(\mathbb{C})$ define

x=\operatorname{\mathrm{t}r}\rho(a),~{}y=\operatorname{\mathrm{t}r}\rho(b)\ % \text{and}\ z=\operatorname{\mathrm{t}r}\rho(ab^{-1})

and for a word $u$ in $F_{a,b}$ define the polynomial $P_{u}(x,y,z)=\operatorname{\mathrm{t}r}\rho(u)\in\mathbb{C}[x,y,z]$ . Further, let $t=P_{w_{k}}$ and

\varphi(x,y,z)=P_{rab}-P_{\overleftarrow{r}ab}.

Definition 5.1 .

A map $\mu:\mathcal{P}(\mathbb{N})\rightarrow[0,\infty]$ is a submeasure supported by $\mathbb{N}$ if for $A,B\subseteq\mathbb{N}$

	$\displaystyle\mu(\emptyset)=0$
	$\displaystyle\mu(A)\leq\mu(A\cup B)\leq\mu(A)+\mu(B).$

It is lower semicontinuous if for all $A\subseteq\mathbb{N}$ we have

\mu(A)=\lim_{n\rightarrow\infty}\mu(A\cap[1,n]).

Definition 1 .

A dressing field is a map ${\bar{u}}:\mathcal{P}\rightarrow H$ with equivariance property $\mathcal{R}^{*}_{h}{\bar{u}}=h^{-1}{\bar{u}}$ and on which the action of the gauge group $\mathcal{H}$ is thus

\displaystyle{\bar{u}}^{\bar{\gamma}}={\bar{\gamma}}^{-1}{\bar{u}}.

(4)

Definition 11 .

An automorphism of a $*$ -algebra $\mathcal{A}$ is a linear invertible map $\alpha:\mathcal{A}\to\mathcal{A}$ that satisfies

\alpha(ab)=\alpha(a)\alpha(b),\qquad\alpha(a^{*})=\alpha(a)^{*}.

We denote the group of automorphisms of the $*$ -algebra $\mathcal{A}$ by $\operatorname{Aut}(\mathcal{A})$ .

An automorphism $\alpha$ is called inner if it is of the form $\alpha(a)=uau^{*}$ for some element $u\in\mathcal{U}(\mathcal{A})$ where

\mathcal{U}(\mathcal{A})=\{u\in\mathcal{A}:uu^{*}=u^{*}u=1\}

is the group of unitary elements in $\mathcal{A}$ . The group of inner automorphisms is denoted by $\operatorname{Inn}(\mathcal{A})$ .

The group of outer automorphisms of $\mathcal{A}$ is defined by the quotient

\displaystyle\operatorname{Out}(\mathcal{A}):=\operatorname{Aut}(\mathcal{A})/% \operatorname{Inn}(\mathcal{A}).

Definition 21 .

We denote by $\mathcal{A}^{\textup{op}}$ the opposite algebra: it is identical to $\mathcal{A}$ as a vector space, but with opposite product:

a^{\textup{op}}b^{\textup{op}}=(ba)^{\textup{op}}.

Here $a^{\textup{op}},b^{\textup{op}}$ are the elements in $\mathcal{A}^{\textup{op}}$ corresponding to $a,b\in\mathcal{A}$ , respectively.

Definition 1.2 .

Given a primitive $n$ th root $\sqrt[n]{\theta}$ of $\theta\in k$ and $\sigma\in G_{k}$ , then $\kappa(\theta)\sigma$ is the element of $A$ such that

\sigma\sqrt[n]{\theta}=\zeta^{\kappa(\theta)\sigma}\sqrt[n]{\theta}.

Definition 2.1 .

[ 5 ] A right Leibniz algebra $L$ is a vector space over a field $\mathbb{K}$ endowed with a bilinear product $[\cdot,\cdot]$ satisfying the Leibniz identity

[[y,z],x]=[[y,x],z]+[y,[z,x]],

for all $x,y,z\in L$ .

Definition 2.2 .

[ 1 ] A Leibniz triple system is a vector space $T$ endowed with a trilinear operation $\{\cdot,\cdot,\cdot\}:T\times T\times T\rightarrow T$ satisfying

	$\displaystyle\{a,\{b,c,d\},e\}\!=\!\{\{a,b,c\},d,e\}\!-\!\{\{a,c,b\},d,e\}\!-% \!\{\{a,d,b\},c,e\}\!+\!\{\{a,d,c\},b,e\},$		(2.1)
	$\displaystyle\{a,b,\{c,d,e\}\}\!=\!\{\{a,b,c\},d,e\}\!-\!\{\{a,b,d\},c,e\}\!-% \!\{\{a,b,e\},c,d\}\!+\!\{\{a,b,e\},d,c\},$		(2.2)

for all $a,b,c,d,e\in T$ .

Definition 2.1 (Class of symbols)

For any $p,q\in\mathbb{N}$ , define ${\mathscr{P}}_{p,q}$ to be the space of homogeneous complex-valued polynomials on ${\mathscr{Z}}$ such that $b\in{\mathscr{P}}_{p,q}$ if and only if there exists a (unique) bounded operator $\tilde{b}\in{\mathscr{L}}(\vee^{p}{\mathscr{Z}},\vee^{q}{\mathscr{Z}})$ such that for all $z\in{\mathscr{Z}}$ :

\displaystyle b(z)={\langle}z^{\otimes q},\tilde{b}\,z^{\otimes p}{\rangle}\,.

(9)

Definition 2.3 (Schensted’s column algorithm) .

Let $a\in\mathrm{I}$ be a column and let $x\in A$ .

x\cdot a=\begin{cases}xa,\,\mbox{if $xa$ is a column;}\\ a^{\prime}\cdot y,\,\mbox{otherwise}\end{cases}

where $y$ is the rightmost letter in $a$ and is larger than or equal to $x$ , and $a^{\prime}$ is obtained from $a$ by replacing $y$ with $x$ . We say that an element $y$ is connected to $x$ or simply that elements $y$ , $x$ are connected. And we will use the notation

x\rightleftarrows y:=\begin{cases}1,\mbox{ iff $x$ is connected to $y$,}\\ 0,\mbox{ otherwise.}\end{cases}

Definition 4.6 .

Let $(M,c)$ be a conformal manifold of dimension at least 3, a Möbius surface or a Laplace curve. A regular curve $\gamma:I\rightarrow M$ is a parametrized conformal , resp. Möbius geodesic if any adapted Weyl structure $\nabla$ (i.e., for which $\nabla_{\dot{\gamma}}\dot{\gamma}=0$ ) satifies

h^{\nabla}(\dot{\gamma})=0

(23)

on $I$ . It is an unparametrized conformal geodesic if the section of $\nu$ induced by $h^{\nabla}(\dot{\gamma})$ vanishes identically.

Definition 111 .

Let $X$ be a topological space, and let $a,b,c\in H^{i}(X,R)$ be cohomology classes such that

a\cup b=b\cup c=0,

where $R$ is any ring of coefficients.

Then the triple Massey product $<a,b,c>$ is defined as follows: take cocycle representatives $a^{\prime},b^{\prime},c^{\prime}$ for $a,b,c,$ respectively, and cochains $x,y\in C^{2i-1}(X,R)$ such that

dx=a^{\prime}\cup b^{\prime},dy=b^{\prime}\cup c^{\prime}.

Then $<a,b,c>$ is the class of

a^{\prime}\cup y+(-1)^{i+1}x\cup c^{\prime}

inside

H^{3i-1}(X,R)/(a\cup H^{2i-1}(X,R)+H^{2i-1}(X,R)\cup c).

Definition 1.2 .

$\Delta$ is involutive if there is a 1-form $\eta$ such that $ker(\eta)=\Delta$ and

\eta\wedge d\eta=0.

Definition 2.2 .

A hom-Leibniz algebra is a triple $(\mathfrak{g},[\cdot,\cdot],\alpha)$ consisting of a vector space $\mathfrak{g}$ , a bilinear map (bracket) $[\cdot,\cdot]:\wedge^{2}\mathfrak{g}\longrightarrow\mathfrak{g}$ and an algebra endomorphism $\alpha:\mathfrak{g}\,\rightarrow\,\mathfrak{g}$ satisfying the following hom-Leibniz rule

[\alpha(x),[y,z]]=[[x,y],\alpha(z)]+[\alpha(y),[x,z]],\quad\forall x,y,z\in% \mathfrak{g}.

(3)

Definition 3.5 (Minimal Acceleration Cost) .

For all measures ${\boldsymbol{\mu}}^{1},{\boldsymbol{\mu}}^{2}\in{\mathscr{P}}_{2}({\mathbf{R}}% ^{2d})$ let ${\mathrm{ADM}}({\boldsymbol{\mu}}^{1},{\boldsymbol{\mu}}^{2})$ denote the set of transport plans ${\boldsymbol{\omega}}\in{\mathscr{P}}({\mathbf{R}}^{4d})$ with

({\mathbbm{p}}^{1},{\mathbbm{p}}^{2})\#{\boldsymbol{\omega}}={\boldsymbol{\mu}% }^{1}\quad\text{and}\quad({\mathbbm{p}}^{3},{\mathbbm{p}}^{4})\#{\boldsymbol{% \omega}}={\boldsymbol{\mu}}^{2}.

The minimal acceleration cost is the functional ${\mathrm{A}}_{\tau}$ defined by

{\mathrm{A}}_{\tau}({\boldsymbol{\mu}}^{1},{\boldsymbol{\mu}}^{2})^{2}:=\inf% \Bigg{\{}\int_{{\mathbf{R}}^{4d}}a_{\tau}({\boldsymbol{x}}^{1},{\boldsymbol{x}% }^{2})^{2}\,{\boldsymbol{\omega}}(d{\boldsymbol{x}}^{1},d{\boldsymbol{x}}^{2})% \colon{\boldsymbol{\omega}}\in{\mathrm{ADM}}({\boldsymbol{\mu}}^{1},{% \boldsymbol{\mu}}^{2})\Bigg{\}}.

(3.4)