Heat transfer between elastic solids with randomly rough surfaces

Definition 2.16 .

A $B$ -module $M$ is called a Poisson module if it is endowed with a Lie algebra action $\{{,\}}:B\otimes M\to M$ that satisfies the Leibniz rules:

\{{a,bm\}}=\{{a,b\}}m+b\{{a,m\}},\ \{{ab,m\}}=a\{{b,m\}}+b\{{a,m\}}.

Definition 4.2

An $\left(\varepsilon\right)$ -almost paracontact metric structure $(\varphi,\xi,\eta,g,\varepsilon)$ is called an $(\varepsilon)$ - $s$ -paracontact metric structure if

\nabla\xi=\varepsilon\varphi.

(4.6)

A manifold equipped with an $(\varepsilon)$ - $s$ -paracontact structure is said to be $(\varepsilon)$ - $s$ -paracontact metric manifold .

Definition 2.3 .

Let $\pi$ be a group. A $\pi$ -module is a pair $(A,\phi)$ where $A$ is an abelian group and $\phi\colon\pi\rightarrow Aut(A)$ a group homomorphism. A map of $\pi$ -modules $f\colon(A,\phi)\rightarrow(A^{\prime},\phi^{\prime})$ is a group homomorphism $f\colon A\rightarrow A^{\prime}$ such that

f(\phi(x)a)=\phi^{\prime}(x)f(a)

for all $x\in\pi$ and $a\in A$ . The category of $\pi$ -modules is denoted by $\pi$ -mod.

Definition 28

A $G$ -Hilbert $A$ -module is a Hilbert $A$ -module ${\cal H}$ with a given continuous action

	$\displaystyle G\times{\cal H}$	$\displaystyle\rightarrow$	$\displaystyle{\cal H}$
	$\displaystyle(g,v)$	$\displaystyle\mapsto$	$\displaystyle gv$

such that

1.

$g(u+v)=gu+gv$ ,
2.

$g(ua)=(gu)(ga)$ ,
3.

$\langle gu,gv\rangle=g\langle u,v\rangle$ ,

for all $g\in G$ , $u,v\in{\cal H}$ , $a\in A$ .

Definition 3.1 .

The quaternions ( $\mathbb{H}$ for short after Hamilton, their discoverer) are the elements of the vector space ${\mathbb{R}}^{4}$ with basis $\{1,i,j,k\}$ . Addition is defined by vector addition, and multiplication follows the rules for scalar multiplication by real numbers, and:

i^{2}=j^{2}=k^{2}=-1,

ij=k,\quad jk=i,\quad ki=j,

and

ji=-k,\quad kj=-i,\quad ik=-j.

Definition 3

The non-negative function $E$ is a pseudo-measure for the LOCC $\mathcal{C}_{\mathcal{X}}$ , if:

	$\displaystyle{\rm(i^{\prime})}\quad\sigma\mbox{ separable}\quad\Rightarrow% \quad E(\sigma)=0,$
	$\displaystyle{\rm(ii)}\quad\forall\Lambda\in\mathcal{C}_{\mathcal{X}}:E(\rho)% \geq E\left(\frac{\Lambda(\rho)}{{\rm tr}\,\Lambda(\rho)}\right).$

Definition 2.2 .

Suppose $G$ is a locally compact Hausdorff groupoid and $H$ is a closed subgroupoid with a Haar system. Let $X=s^{-1}(H^{(0)})$ . We define the imprimitivity groupoid $G^{H}$ to be the quotient of $X*X=\{(\gamma,\eta):s(\gamma)=s(\eta)\}$ by $H$ where $H$ acts diagonally via right translation. We give $G^{H}$ a groupoid structure with the operations

[\gamma,\eta][\eta,\zeta]=[\gamma,\zeta],\quad\text{and}\quad[\gamma,\eta]^{-1% }=[\eta,\gamma].

Definition 2.1 .

A binary Hermitian form over $F$ is a map $\phi:F^{2}\to\mathbb{Q}$ of the form

\phi(x,y)=ax\bar{x}+bx\bar{y}+\bar{b}\bar{x}y+cy\bar{y},

where $a,c\in\mathbb{Q}$ and $b\in F$ such that $\phi$ is positive definite.