Dynamics of a Local Algorithm for Simulating Coulomb Interactions

Definition 5.8 .

[ HPR01 ] Let $\sigma:{\mathbb{N}}\times{\mathbb{N}}\rightarrow{\mathbb{N}}$ be any two-ary function; $\sigma$ may be nontotal and it may be many-to-one. We say that $\sigma$ is overstrong if and only if no polynomial-time computable function $f$ with $f:\{1,2\}\times{\mathbb{N}}\times{\mathbb{N}}\rightarrow{\mathbb{N}}\times{% \mathbb{N}}$ satisfies that for each $i\in\{1,2\}$ and for each $z,a\in{\mathbb{N}}$ :

((\exists b\in{\mathbb{N}})[(\sigma(a,b)=z{\;\wedge\;}i=1)\lor(\sigma(b,a)=z{% \;\wedge\;}i=2)])\,\Longrightarrow\ \sigma(f(i,z,a))=z.

Definition 2.1

A generalized tacnode is a two dimensional elliptic hypersurface singularity $(X,0)$ , such that the fundamental cycle has self intersection $(-2)$ .
In particular, $(X,0)$ is Gorenstein and by [ Lau ] , theorem $(1.3)$ , $(X,0)$ is a double point singularity, whose local analytic equation is given by

z^{2}=g(x,y),

where $g$ vanishes of order four in $0$ . The normal cone of the singularity is given by the plane $\{z=0\}$ , called the tacnodal plane.

Definition 1.1 .

A conic metric on $X$ is a Riemannian metric $g$ on $X^{\circ}$ such that in a neighborhood of any boundary component $Y$ of $X$ , there exists a boundary defining function $x$ ( $x\geq 0$ , $\{x=0\}=\partial X$ , $dx\!\!\upharpoonright_{\partial X}\neq 0$ ) in terms of which

(1.1)

g=dx^{2}+x^{2}h

where $h\in{\mathcal{C}}^{\infty}(\operatorname{Sym}^{2}T^{*}X)$ and $h\!\!\upharpoonright_{Y}$ is a metric. A conic manifold is a compact manifold with boundary endowed with a conic metric.

Definition 3.11 ( $\#$ involution)

For any index $i=1,...,M+N$ , we define $i^{\prime}$ by

i^{\prime}=\left\{\begin{array}[]{l}m^{\prime}={\widetilde{m}}+1,\ \ ({% \widetilde{m}}+1)^{\prime}=m\\ i\hskip 11.381102pt\mbox{otherwise}\end{array}\right.

(3.45)

$\#$ defined by

S_{\#}^{ij}(u)=S^{i^{\prime}j^{\prime}}(u)

(3.46)

is an order 2 automorphism of $Y(M|2n)^{+}$ .

Definition 2 .

The Homflypt skein module of $M$ , denoted by $S(M)$ , is the $k$ -module freely generated by isotopy classes of framed oriented links in $M$ including the empty link, quotient by the Homflypt skein relations given in the following figure.