Connectivity in the Astronomy Digital Library

Definition 2.2 .

The Hyperbolic algebra $R\{\theta,\xi\}$ is defined to be the $R-$ algebra generated by $x,y$ subject to the following relations:

xy=\xi,\quad yx=\theta^{-1}(\xi)

and

xa=\theta(a)x,\quad ya=\theta^{-1}(a)y

for any $a\in R$ . $R\{\theta,\xi\}$ is called a Hyperbolic algebra over $R$ .

Definition 8

The symplectic inner product of vectors $(a,b),(a^{\prime},b^{\prime})\in\mathbb{F}_{q}^{2m}$ is given by

(a,b)\odot(a^{\prime},b^{\prime})=a\cdot b^{\prime}\oplus a^{\prime}\cdot b.

(1)

Definition 2.2 .

A Poisson (or Hamiltonian ⁴ ⁴ 4 The expression Hamiltonian manifold is often used for the generalization of Poisson structure in the case of infinite dimension manifolds. ) structure on a $C^{\infty}$ manifold $M$ is a skew-symmetric bilinear mapping $(f,g)\mapsto\{f,g\,\}$ on the space $C^{\infty}(M)$ , which satisfies the Jacobi identity

(3)

\{\{f,g\,\},h\,\}+\{\{g,h\,\},f\,\}+\{\{h,f\,\},g\,\}=0,

as well as the Leibnitz identity

(4)

\{f,gh\,\}=\{f,g\,\}h+g\{f,h\,\}.

Definition 2

We say that the function $a\in L^{1}(0,T)$ is completely positive on $[0,T]$ , if for any $\mu\geq 0$ , the solutions of the equations

s(t)\!+\!\mu(a\star s)(t)=1\quad\mbox{and}\quad r(t)\!+\!\mu(a\star r)(t)=a(t)

(3)

satisfy $s(t)\geq 0$ and $r(t)\geq 0$ on $[0,T]$ .

Definition 2.1 .

We denote by $R\{\theta,\xi\}$ the corresponding $R-$ algebra generated by $x,y$ subject to the following relations:

xy=\xi,\quad yx=\theta^{-1}(\xi)

and

xa=\theta(a)x,\quad ya=\theta^{-1}(a)y

for any $a\in R$ . And $R\{\theta,\xi\}$ is called a Hyperbolic algebra over $R$ .

Definition 8.10 .

The dual braid relations with respect to $W$ and $c$ are all the formal relations of the form

rr^{\prime}=r^{\prime}r^{\prime\prime},

where $r,r^{\prime},r^{\prime\prime}\in R$ are such that $r\neq r^{\prime}$ , $rr^{\prime}\in[1,c]$ , and the relation $rr^{\prime}=r^{\prime}r^{\prime\prime}$ holds in $W$ .

Definition 4 (Pre-valuation)

A pre-valuation on an orthomodular lattice ${\mathcal{L}}$ is a function $\nu:{\mathcal{L}}\rightarrow\left\{{0,1}\right\}$ which verifies:

	$\displaystyle\nu(\top)=1$
	$\displaystyle{\forall\,{x,y\in{\mathcal{L}}},\ }\bigl{(}x\,\bot\,y\Rightarrow% \nu(x\vee y)\geq\nu(x)+\nu(y)\bigr{)}$
	$\displaystyle{\forall\,{x,y\in{\mathcal{L}}},\ }\bigl{(}x\hbox{\ and\ }y\hbox{% \ compatible}\Rightarrow\nu(x\wedge y)=\nu(x)\times\nu(y)\bigr{)}$

Definition 2.2 (Density Matrices)

The set of density matrices (states) on a Hilbert space $\mathcal{H}$ fulfilling

	$\displaystyle\rho=\rho^{{\dagger}}$		selfadjoint
	$\displaystyle\rho\geq 0$		positive
	$\displaystyle tr[\rho]=1$		normalised

will be denoted by $S(\mathcal{H})$ . $S(\mathcal{H})$ is a closed convex subset of the bounded linear operators $\mathcal{B}(\mathcal{H})$ on $\mathcal{H}$ . Note, that all states are by definition trace-class operators.

Definition 6.1 .

A complex geodesic is the projection of a solution of the Hamiltonian system ( 4.3 ) with the non-standard boundary conditions

x(0)=0,\quad x(1)=x,\quad z(0)=0,\quad z(1)=z,\quad\text{and}

\theta_{m}=-i\tau_{m},\quad m=1,2,3,

on the $(x,z)$ -space.

Definition 2.2 .

The Hyperbolic algebra $R\{\theta,\,\xi\}$ is defined to be the $R-$ algebra generated by $x,\,y$ subject to the following relations:

xy=\xi,\quad yx=\theta^{-1}(\xi)

and

xa=\theta(a)x,\quad ya=\theta^{-1}(a)y

for any $a\in R$ . And $R\{\theta,\,\xi\}$ is called a Hyperbolic algebra over $R$ .