Evolution of initially localized perturbations in stratified ionized disks

Definition 2.6 .

Let $\sigma$ be a 2-complex. Define $w(e)$ , weight of an edge $e\in\sigma$ , to be

w(e)=\left\{\begin{array}[]{ll}1&\mbox{if $e\subset\partial\sigma$}\\ 2&\mbox{otherwise}\end{array}\right.

Define $w(\sigma)$ , the weight of $\sigma$ , to be the total weight of all edges of $\sigma$ .
For $\psi$ a 1-complex. Define $w(\psi)$ to be the total number of edges of $\psi$ .
From the definition, it is clear that $w(\sigma)\geq w(\partial\sigma)$ .

Definition 6.3 .

Suppose that $R$ is an algebra, $\sigma$ an automorphism of $R$ and $\delta$ a $\sigma$ -derivation of $R$ , that is, a linear map $\delta:R\to R$ such that

\delta(rs)=\delta(r)s+\sigma(r)\delta(s)

for all $r,s\in R$ . Then the Ore extension $R[t]$ is the free left $R$ -module on the set $\{t^{n}\,|\,n\geq 0\}$ , with multiplication defined by

tr=\sigma(r)t+\delta(r).

Definition 4.1 .

Let $R$ be a partial E-ring, and $M$ an $R$ -module. (There is no exponential structure on $M$ ; it is just a module in the usual sense.) A derivation from $R$ to $M$ is a map $R\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}M$ such that for each $a,b\in R$ ,

•

$\partial(a+b)=\partial a+\partial b$ , and
•

$\partial(ab)=a\partial b+b\partial a$

It is an exponential derivation or E-derivation iff also for each $a\in A(R)$ we have $\partial(\exp(a))=\exp(a)\partial a$ .

Write $\operatorname{Der}(R,M)$ for the set of all derivations from $R$ to $M$ , and $\operatorname{EDer}(R,M)$ for the set of all E-derivations from $R$ to $M$ . For any subset $C$ of $R$ , we write $\operatorname{Der}(R/C,M)$ and $\operatorname{EDer}(R/C,M)$ for the sets of derivations (E-derivations) which vanish on $C$ . It is easy to see that these are $R$ -modules.

Definition 3.3 .

We can turn a one-component chord diagram with a base point into an arrow diagram according to the following rule. Starting from the base point we travel along the diagram with doubled chords. During this journey we pass both copies of each chord in opposite directions. Choose an arrow on each chord which corresponds to the direction of the first passage of the copies of the chord. Here is an example.

\raisebox{-9.0pt}[20.0pt][15.0pt]{\begin{picture}(25.0,15.0)(0.0,0.0)\put(0.0,% 0.0){\includegraphics[width=25.0pt]{cd22ch4.eps}} \end{picture}}\quad\raisebox{2.0pt}[15.0pt][15.0pt]{\begin{picture}(25.0,15.0)% (0.0,0.0)\put(0.0,0.0){\includegraphics[width=25.0pt]{totor.eps}} \end{picture}}\quad\raisebox{-9.0pt}[20.0pt][15.0pt]{\begin{picture}(25.0,15.0% )(0.0,0.0)\put(0.0,0.0){\includegraphics[width=25.0pt]{cd22par.eps}} \end{picture}}\quad\raisebox{2.0pt}[0.0pt][0.0pt]{\begin{picture}(25.0,15.0)(0% .0,0.0)\put(0.0,0.0){\includegraphics[width=25.0pt]{totor.eps}} \end{picture}}\quad\raisebox{-9.0pt}[20.0pt][15.0pt]{\begin{picture}(25.0,15.0% )(0.0,0.0)\put(0.0,0.0){\includegraphics[width=25.0pt]{cd22arw.eps}} \end{picture}}\ .

Definition 5.12 .

Assume that $M,M^{\prime}$ are monoids acting on sets $X$ and $X^{\prime}$ , respectively. A morphism $\varphi{:}\,M{\to}M^{\prime}$ and a map $\psi{:}\,X{\to}X^{\prime}$ are called compatible if

(5.6)

\psi(x\mathbin{\scriptscriptstyle\bullet}a)=\psi(x)\mathbin{\scriptscriptstyle% \bullet}\varphi(a)

holds whenever $x\mathbin{\scriptscriptstyle\bullet}a$ is defined. Then, we denote by $[\varphi,\psi]$ the functor of $\mathcal{C}(M,X)$ to $\mathcal{C}(M^{\prime},X^{\prime})$ that coincides with $\psi$ on objects and maps $(x,a,y)$ to $(\psi(x),\varphi(a),\psi(y))$ .

Definition 6 .

An affine connection $\nabla$ is compatible with $\left(\mathcal{H},\left<\cdot,\cdot\right>\right)$ , and thus its associated co-metric $g$ , if

(16)

\nabla g=0

Definition 6 (Skew $k$ -linear succession) .

For $\pi=(\varepsilon,\sigma)\in G_{\ell,n}$ , the value $\sigma(i)$ $(1\leq i\leq n)$ is a skew $k$ -linear succession $(k\geq 1)$ of $\pi$ at position $i$ if

\pi(i)=\pi(i-1)+k,

where, by convention, $\sigma(0)=0$ and $\varepsilon(0)=1$ .

Definition 1.1

A quandle $(Q,\ast)$ is a set $Q$ with a binary operation $\ast:Q\times Q\longrightarrow Q$ that satisfies the following axioms $:$

$\mathbf{(i)}$ For all $q\in\ Q,$ $q\ast\ q=q$

$\mathbf{(ii)}$ $Q$ is a self distributive set with $\ast$ as an operation, i.e. for all $p,q,r\in\ Q,$

(p\ast q)\ast r=(p\ast r)\ast(q\ast r)

and finally,

$\mathbf{(iii)}$ For each $p,q\in\ Q$ there is a unique $r\in\ Q$ such that $p=r\ast q$

Definition 1.1 .

A (left) Leibniz algebra is an algebra $A$ for which all the left multiplications are derivations, that is,

a(bc)=(ab)c+b(ac)

for all $a,b,c\in A$ .

Definition 1.7 .

A bimodule of a Leibniz algebra $A$ is a vector space $M$ with two bilinear compositions $am,ma$ for $a\in A$ and $m\in M$ such that

\begin{split}\displaystyle a(bm)&\displaystyle=(ab)m+b(am)\\ \displaystyle a(mb)&\displaystyle=(am)b+m(ab)\\ \displaystyle m(ab)&\displaystyle=(ma)b+a(mb)\\ \end{split}

for all $a,b\in A$ and $m\in M$ .

Definition 3.4 .

(a) A Lie bialgebra $(\mathfrak{g},[\cdot,\cdot],\delta)$ is called quasitriangular if there exists a classical $r$ -matrix $r\in\mathfrak{g}\otimes\mathfrak{g}$ such that for all $g\in\mathfrak{g}$

\delta(g)=[r,1\otimes g+g\otimes 1]\ .

(b) $(\mathfrak{g},[\cdot,\cdot],\delta)$ is called triangular if there exists a skew-symmetric classical $r$ -matrix $r\in\Lambda^{2}\mathfrak{g}$ such that for all $g\in\mathfrak{g}$

\delta(g)=[r,1\otimes g+g\otimes 1]\ .

(c) A Lie bialgebra $(\mathfrak{g},[\cdot\cdot],\delta)$ is called coboundary, if there exists $r\in\Lambda^{2}\mathfrak{g}$ such that for all $g\in\mathfrak{g}$

\delta(g)=[r,1\otimes g+g\otimes 1]\ .

Definition 1 .

The Schroedinger algebra is the six–dimensional $*$ –Lie algebra generated by $b$ , $b^{\dagger}$ , $b^{2}$ , ${b^{\dagger}}^{2}$ , $b^{\dagger}\,b$ and $1$ where $b^{\dagger}$ , $b$ and $1$ are the generators of a Boson Heisenberg algebra with

(2.1)

[b,b^{\dagger}]=1\qquad;\qquad{(b^{\dagger})}^{*}=b

Definition 2 .

For $\lambda\in\mathbb{C}$ let $y(\lambda)=e^{\lambda\,b}$ . The Heisenberg Fock space $\mathcal{F}$ is the Hilbert space completion of the linear span of the exponential vectors $\{y(\lambda)\,;\,\lambda\in\mathbb{C}\}$ with respect to the inner product

(2.8)

\langle y(\lambda),y(\mu)\rangle=e^{\bar{\lambda}\,\mu}

Definition .

Let $y\in G_{j}$ for some $j$ and let $\zeta$ be the unique point in $\tilde{C}_{j}^{+}$ with

{\mathbf{x}}^{\sharp}(\zeta)=y

(5.13)

We define $\Theta(\,\cdot\,;y)$ to be the character automorphic function

\Theta(z;y)=\biggl{[}\biggl{(}\frac{{\mathbf{x}}(z)-{\mathbf{x}}(\zeta)}{{% \mathbf{x}}(z)-{\mathbf{x}}(\zeta_{j})}\biggr{)}\,\eta(z)\eta(0)^{-1}\biggr{]}% ^{1/2}

(5.14)

and denote by ${\mathfrak{A}}(y)\in\Gamma^{*}$ its character. Moreover, we define ${\mathfrak{A}}(\infty)$ to be the character of $B(z)$ .

Definition 2.11 .

We say that two multimodal maps $f,g:I\rightarrow I$ are topologically conjugate or simply conjugate if there is a homeomorphism $h:I\rightarrow I$ such that

h\circ f=g\circ h.

Definition 1.6 .

Let $m\geq 2$ be an integer and $\varepsilon>0$ . The germ of a continuous map $u:{\mathcal{O}}(S,z)\rightarrow Q$ is called of class $(m,\varepsilon)$ at the point $z$ if for a smooth chart $\varphi:U(u(0))\rightarrow{\mathbb{R}}^{2n}$ of $Q$ mapping $u(0)$ to $0$ and for holomorphic polar coordinates $\sigma:[0,\infty)\times S^{1}\rightarrow S\setminus\{z\}$ around $z$ , the map

v(s,t)=\varphi\circ u\circ\sigma(s,t),

which is defined for $s$ large, has partial derivatives up to order $m$ , which if weighted by $e^{\varepsilon s}$ , belong to $L^{2}([R,\infty)\times S^{1},{\mathbb{R}}^{2n})$ if $R$ is sufficiently large. The germ is of called of class $m$ around the point $z\in S$ , if $u$ belongs to the class $H^{m}_{\text{loc}}$ near $z$ .

Definition 1 .

A faithful self-similar action $(G,\mathsf{X})$ is a faithful action of a group $G$ on the set $\mathsf{X}^{*}$ such that for every $g\in G$ and $x\in\mathsf{X}$ there exist $h\in G$ such that

g(xw)=g(x)h(w)

for all $w\in\mathsf{X}^{*}$ .

Definition 2.3.1 .

A semilinear (or semivector ) space on ${\mathbb{R}}_{+}$ is a triple $(X,+,\cdot)$ such that $(X,+)$ is an Abelian semigroup with neutral element $0\in X$ and $\cdot$ is a function ${\mathbb{R}}_{+}\times X\to X$ which satisfies for all $x,y\in X$ and $a,b\in{\mathbb{R}}_{+}$ :

(i)

$a\cdot(b\cdot x)=(ab)\cdot x$ ,
(ii)

$(a+b)\cdot x=(a\cdot x)+(b\cdot x)$ ,
(iii)

$a\cdot(x+y)=(a\cdot x)+(a\cdot y)$ , and
(iv)

$1\cdot x=x$ .

Whenever an element $x\in X$ admits an inverse it can be shown to be unique and is denoted $-x$ . If we replace in the above definition ${\mathbb{R}}_{+}$ with ${\mathbb{R}}$ and “semigroup” with “group” we obtain an ordinary vector (or linear) space. $\blacktriangle$

Definition 2.6.1 ( [ KuVa94 , Vi99 ] ) .

Let $(X,d)$ be a quasi-metric space. The quasi-metric $d$ is called a weightable quasi-metric if there exists a function $w:X\to{\mathbb{R}}_{+}$ , called the weight function or simply the weight , satisfying for every $x,y\in X$

d(x,y)+w(x)=d(y,x)+w(y).

In this case we call $d$ weightable by $w$ .

A quasi-metric $d$ is co-weightable if its conjugate quasi-metric ${d}^{\ast}$ is weightable. The weight function $w$ by which ${d}^{\ast}$ is weightable is called the co-weight of $d$ and $d$ is co-weightable by $w$ .

A triple $(X,d,w)$ where $(X,d)$ is a quasi-metric space and $w$ a function $X\to{\mathbb{R}}_{+}$ is called a weighted quasi-metric space if $(X,d)$ is weightable by $w$ and a co-weighted quasi-metric space if $(X,d)$ is co-weightable by $w$ .

In all the above, if the weight function $w$ takes values in ${\mathbb{R}}$ instead of ${\mathbb{R}}_{+}$ , the prefix generalised is added to the definitions. $\blacktriangle$

Definition 2.6.9 .

Let $(X,\rho)$ be a metric space. A bundle over $(X,\rho)$ [ Vi99 ] is the weighted quasi-metric space $(X\times{\mathbb{R}}_{+},d,w)$ where

d((x,\xi),(y,\eta))=\rho(x,y)+\xi-\eta

and

w((x,\xi))=2\xi.

$\blacktriangle$

Definition Definition 4.2

The Dynkin diagram $G(D_{6})$ of type $D_{6}$ is a graph with vertex set $V(D_{6})=\{1^{\prime},1,2,3,4,5\}$ . Two vertices $i$ and $j$ in the subset $\{1,2,3,4,5\}$ are adjacent (connected by an edge) if and only if $|i-j|=1$ , and the vertex $1^{\prime}$ is adjacent only to $2$ .

If $i$ and $j$ are any two vertices of $V(D_{6})$ , we define the integer

m(i,j)=\begin{cases}1&\text{ if $i=j$,}\cr 2&\text{ if $i$ is not adjacent to % $j$, and}\cr 3&\text{ if $i$ is adjacent to $j$.}\cr\end{cases}

The Coxeter group $W=W(D_{6})$ of type $D_{6}$ is the group given by the presentation

\langle s_{1^{\prime}},s_{1},s_{2},s_{3},s_{4},s_{5}\ |\ (s_{i}s_{j})^{m(i,j)}% =1\rangle.

Definition 1.3.1 .

If $u,v\in\hbox{\bf E}_{1}^{3}$ , we call the (Lorentzian) vector product of $u$ and $v$ to the unique vector denoted by $u\times v$ that satisfies

\langle u\times v,w\rangle=\mbox{det }(u,v,w),

(1.2)

where $\mbox{det}(u,v,w)$ is the determinant of the matrix obtained by putting by columns the coordinates of the three vectors $u$ , $v$ and $w$ .

Definition 2.6 .

Definition 6.3 .

Definition 4.1 .

Definition 3.3 .

Definition 5.12 .

Definition 6 .

Definition 6 (Skew k 𝑘 k italic_k -linear succession) .

Definition 1.1

Definition 1.1 .

Definition 1.7 .

Definition 3.4 .

Definition 1 .

Definition 2 .

Definition .

Definition 2.11 .

Definition 1.6 .

Definition 1 .

Definition 2.3.1 .

Definition 2.6.1 ( [ KuVa94 , Vi99 ] ) .

Definition 2.6.9 .

Definition Definition 4.2

Definition 1.3.1 .

Definition 6 (Skew $k$ -linear succession) .