The Anatomy of Star Formation in NGC 300

Definition 1

Under the PMC model, a syndrome $\sigma$ for system $G$ is defined as follows. For any two distinct nodes $u$ and $v$ with $v\in N(u)$ ,

\sigma(u,v)=\left\{\begin{array}[c]{ll}0,&\mbox{if $v$ is tested by $u$ to be % fault-free};\\ 1,&\mbox{if $v$ is tested by $u$ to be faulty}.\end{array}\right.

Definition 2

Under the MM* model, a syndrome $\sigma$ for system $G$ is defined as follows. For any three distinct nodes $u$ , $v$ and $w$ with $u,v\in N(w)$ ,

\sigma(u,v;w)=\left\{\begin{array}[c]{ll}0,&\mbox{if the test results of $u$ % and $v$ by $w$ are identical};\\ 1,&\mbox{if the test results of $u$ and $v$ by $w$ are distinct}.\end{array}\right.

Definition 4.1 .

Suppose that $A$ is a unital $C^{*}$ -algebra, $\alpha$ is an endomorphism of $A$ and $L$ is a transfer operator for $(A,\alpha)$ . We say that $L$ is faithful on an ideal $I$ of $A$ if

a\in I\mbox{ and }L(a^{*}a)=0\Longrightarrow a=0;

we say that $L$ is almost faithful on $I$ if

a\in I\mbox{ and }L({(ab)}^{*}ab)=0\text{ for all }b\in A\Longrightarrow a=0.

Definition 1.1 .

Let $A$ be a commutative $k$ -algebra where $k$ is a commutative ring. A $(k,A)$ -Lie-Rinehart algebra on $A$ is a $k$ -Lie algebra and an $A$ -module $\mathfrak{g}$ with a map $\alpha:\mathfrak{g}\rightarrow\operatorname{Der}_{k}(A)$ satisfying the following properties:

(1.1.1)		$\displaystyle\alpha(a\delta)=a\alpha(\delta)$
(1.1.2)		$\displaystyle\alpha([\delta,\eta])=[\alpha(\delta),\alpha(\eta)]$
(1.1.3)		$\displaystyle[\delta,a\eta]=a[\delta,\eta]+\alpha(\delta)(a)\eta$

for all $a\in A$ and $\delta,\eta\in\mathfrak{g}$ . Let $W$ be an $A$ -module. A $\mathfrak{g}$ - connection $\nabla$ on $W$ , is an $A$ -linear map $\nabla:\mathfrak{g}\rightarrow\operatorname{End}_{k}(W)$ which satisfies the Leibniz-property , i.e.

\nabla(\delta)(aw)=a\nabla(\delta)(w)+\alpha(\delta)(a)w

for all $a\in A$ and $w\in W$ . The $\mathfrak{g}$ -connection $\nabla$ is flat if it is a map of Lie algebras. If $\nabla$ is flat, we say that the pair $(W,\nabla)$ is a $\mathfrak{g}$ -module .

Definition 3.3 .

For any $e$ , if we are given uniformly computable enumerations of $\{X_{n,s}\}_{n\leq e,s<\omega}$ and $\{Y_{n,s}\}_{n\leq e,s<\omega}$ of c.e. sets $\{X_{n}\}_{n\leq e}$ and $\{Y_{n}\}_{n\leq e}$ , define the full $e$ -state of $x$ at stage $s$ , $\nu(e,x,s)$ , with respect to (w.r.t.) $\{X_{n,s}\}_{n\leq e,s<\omega}$ and $\{Y_{n,s}\}_{n\leq e,s<\omega}$ to be the triple

\nu(e,x,s)=\langle e,\sigma(e,x,s),\tau(e,x,s)\rangle

where

\sigma(e,x,s)=\{i\leq e:x\in X_{i,s}\}

and

\tau(e,x,s)=\{i\leq e:x\in Y_{i,s}\}.

Definition 3.4 .

For any collection of c.e. sets $\{X_{n}\}_{n\leq e}$ and $\{Y_{n}\}_{n\leq e}$ , define the final $e$ -state of $x$ , $\nu(e,x)$ , w.r.t $\{X_{n}\}_{n\leq e}$ and $\{Y_{n}\}_{n\leq e}$ to be the triple

\nu(e,x)=\langle e,\sigma(e,x),\tau(e,x)\rangle

where

\sigma(e,x)=\{i\leq e:x\in X_{i}\}

and

\tau(e,x)=\{i\leq e:x\in Y_{i}\}.

Definition 1.1 (associative algebra $A_{n}$ , algebra of chord diagrams $A(X)$ ) .

(a) For each $n\geq 2$ , let the associative algebra $A_{n}$ over the field ${\mathbb{C}}$ be generated by the symbols $t^{ij}=t^{ji}$ with $1\leq i\neq j\leq n$ and the relations

[t^{ij},t^{kl}]=0\mbox{ if }i,j,k,l\mbox{ are pairwise disjoint},\quad[t^{ij},% t^{jk}+t^{ki}]=0\mbox{ if }i,j,k\mbox{ are pairwise disjoint},

where the bracket $[\,,]:A_{n}\oplus A_{n}\to A_{n}$ is defined by $[a,b]:=ab-ba$ . Observe that the relations $[t^{ij},t^{jk}+t^{ki}]=0$ of $A_{n}$ are equivalent to

[t^{ij},t^{jk}]=[t^{jk},t^{ki}]=[t^{ki},t^{ij}]\mbox{ for all pairwise % disjoint }i,j,k\in\{1,\ldots,n\}.

The associative algebra $A_{n}$ is graded by the degree defined by $\deg(t^{ij})=1$ .

(b) Let us define the same object $A_{n}$ geometrically. Let $X$ be a 1-dimensional oriented compact manifold, possibly non-connected and with boundary. A chord diagram on $X$ is a collection of non-oriented dashed lines ( chords ) with endpoints on $X$ . Let $A(X)$ be the linear space generated by all chord diagrams on $X$ modulo the 4T relations :

The dotted arcs represent parts of the diagrams that are not shown in the figure. These parts are assumed to be the same in all four diagrams.

If $X=X_{n}$ is the disjoint union of $n$ oriented segments ( strands ), then $A(X_{n})$ can be equipped with a natural product. If in the definition of $A(X_{n})$ one allows only horizontal chords with endpoints on $n$ vertical strands, then the resulting algebra $A^{hor}(X_{n})$ is isomorphic to the algebra $A_{n}$ . Indeed, thinking of $t^{ij}$ as a horizontal chord connecting the $i$ th and $j$ th vertical strands, the relations between the $t^{ij}$ become the 4T relations:

Definition 32.1

[ 51 , p.122] Let $U_{q}(sl_{2})$ denote the unital associative $\mathbb{K}$ -algebra with generators $e,f,k,k^{-1}$ and relations

\displaystyle kk^{-1}=k^{-1}k=1,

\displaystyle ke=q^{2}ek,\qquad\qquad kf=q^{-2}fk,

\displaystyle ef-fe={{k-k^{-1}}\over{q-q^{-1}}}.

Definition 1 .

Let $\delta\in\mathbb{C}$ be a complex number. $\bf{T}_{N}(\delta)$ is an associative $\mathbb{C}$ -algebra spanned over $\mathbb{C}$ by affine diagrams $D(N\rightarrow N)$ with multiplication

\alpha\beta=\delta^{m(\alpha,\beta)}\alpha\circ\beta

for $\alpha,\beta\in D(N\rightarrow N)$ .