Variable gravity Universe

Definition 4.4 .

A crossed module of Lie algebras $(\mathfrak{g},\,[.,.]^{\mathfrak{g}})$ and $(\mathfrak{h},\,[.,.]^{\mathfrak{h}})$ is a homomorphism $\partial:\mathfrak{g}\to\mathfrak{h}$ together with an action by derivation of $\mathfrak{h}$ on $\mathfrak{g}$ , that is, a linear map $\chi:\mathfrak{h}\to Hom(\mathfrak{g},\mathfrak{g})$ such that

(34)

\partial(\chi(h)g)=[h,\partial(g)]^{\mathfrak{h}},\,\,\,\,\mbox{\rm for all}\,% \,\,g\in\mathfrak{g},\,h\in\mathfrak{h}

and

(35)

\chi(\partial(g_{1}))g_{2}=[g_{1},g_{2}]^{\mathfrak{g}},\,\,\,\,\mbox{\rm for % all}\,\,\,g_{1},g_{2}\in\mathfrak{g}.

Definition 3.3 .

Let $\omega$ and $\omega^{\prime}$ denote two paths. The initial point (in $X$ ) of a nonconstant path $\omega$ is defined by

{\rm in}(\omega)=\omega({\rm ld}(\omega)).

The terminal point of a nonconstant path $\omega$ is defined by

{\rm ter}(\omega)=\omega({\rm ud}(\omega)).

The initial and terminal points of a constant path $\omega$ taking the constant value $x\in X$ is defined by

{\rm in}(\omega)={\rm ter}(\omega)=x.

The composition $\omega\cdot\omega^{\prime}$ of paths $\omega$ and $\omega^{\prime}$ exists if ${\rm ter}(\omega)={\rm in}(\omega^{\prime})$ and is defined as follows:

$(\omega\cdot\omega^{\prime})$ $=\left\{\begin{array}[]{lll}\omega&\mbox{if $\omega^{\prime}$ is constant }\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\omega^{\prime}&\mbox{if $\omega$ is % constant.}\end{array}\right.$

If $\omega$ and $\omega^{\prime}$ are nonconstant then

$(\omega\cdot\omega^{\prime})(n)$ $=\left\{\begin{array}[]{lll}\omega^{\prime}(n)&\mbox{for all $n\leq{\rm ud}~{}% \omega^{\prime}$,}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\omega(n-{\rm ud}~{}\omega^{\prime}+{\rm ld% }~{}\omega)&\mbox{for all $n\geq{\rm ud}~{}\omega^{\prime}$.}\end{array}\right.$

It is clear that the composition law $\cdot$ on paths is associative.

Definition 3.12 .

If $\omega$ is a path, define the inverse path $\omega^{-1}$ by

\omega^{-1}(n)=\omega(-n).

Definition 2.1 .

Let $m$ be a positive integer. A permutation, $f$ , of the integers mod $m$ i.e., of the set $\mathbf{Z}/m\mathbf{Z}$ , is called a $($ n $m$ - $)$ coloring automorphism if for any two integers $a$ and $b$ , the coloring condition is preserved i.e.,

f(a\ast b)=f(a)\ast f(b)\qquad\text{ mod }m

where $a\ast b:=2b-a$ .

Definition 2.3 .

We define $t_{0}$ to be the point of maxima of the function $g_{n}(t)$ , i.e., $t_{0}$ is the solution of the equation

(6)

\varphi^{\prime}(t)t=n.

Definition 2 .

A $(n,k,d,t)$ -LDLSC is a pair containing an encoder, $f:\mathcal{X}^{n}\mapsto\mathcal{Y}^{k}$ , and a decoder, $g:\mathcal{Y}^{k}\mapsto\mathcal{X}^{n}$ , where the decoder is $t$ -local and the distortion is bounded, $\mathbb{E}[d(X^{n},g(f(X^{n})))]\leq d$ .
Let

	$\displaystyle k^{*}(n,d,t)=$
	$\displaystyle\min\{k\text{~{}such that~{}}\exists(n,k,d,t)-\text{~{} LDLSC~{}}\},$		(14)

and

R^{*}(d,t)=\limsup_{n\to\infty}\frac{k^{*}(n,d,t)}{n}.

(15)

Definition 2.5 .

A Lie ring $L$ is an additive Abelian group $L$ together with an operation $[\cdot,\cdot]:L\times L\longrightarrow L$ satisfying the following identities

1.

$[x,x]=0$
2.

$[y,x]=-[x,y]$
3.

$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$

for all $x,y,z\in L$ . The Property 3. above is called the Jacobi identity. We call $[,]$ the Lie bracket or the Lie commutator.

Definition 2.1 .

An arithmetical function $f\colon{\mathbb{N}}\to{\mathbb{C}}$ is multiplicative if

f(mn)=f(m)f(n)

(2.1)

for all $m,n\in{\mathbb{N}}$ with $(m,n)=1$ .

Definition 2.2 .

An arithmetical function $f\colon{\mathbb{N}}\to{\mathbb{C}}$ is quasimultiplicative if there exists a nonzero constant $c$ such that

c\,f(mn)=f(m)f(n)

(2.3)

for all $m,n\in{\mathbb{N}}$ with $(m,n)=1$ .

Definition 4.13 .

Functional ( 1 ) is said to be invariant under an $\varepsilon$ -parameter family of infinitesimal transformations

\bar{u}(t)=u(t)+\varepsilon\xi(t,u(t))+o(\varepsilon)

(11)

with $\xi\in C^{1}\left(\bar{\Delta}_{n};\mathbb{R}^{N}\right)$ such that $B_{P_{t_{i}}^{1}}^{\alpha_{i}}\xi$ and $K_{P_{t_{i}}^{2}}^{\beta_{i}}\xi$ exist and are continuous on $\bar{\Delta}_{n}$ , $i\in\{1,\dots,n\}$ , if

\int\limits_{\Delta_{n}^{*}}F\left(t,u(t),\nabla_{B_{P^{1}}}^{\bm{\alpha}}u(t)% ,\nabla_{K_{P^{2}}}^{\bm{\beta}}u(t)\right)dt=\int\limits_{\Delta_{n}^{*}}F% \left(t,\bar{u}(t),\nabla_{B_{P^{1}}}^{\bm{\alpha}}\bar{u}(t),\nabla_{K_{P^{2}% }}^{\bm{\beta}}\bar{u}(t)\right)dt

for any $\Delta_{n}^{*}\subseteq\Delta_{n}$ .

Definition 5 .

An execution $\gamma:T\rightarrow D$ is periodic if there exists $s\in T$ and $\tau>0$ such that

\displaystyle\gamma(s)=\gamma(s+\tau).

(1)

If there is no smaller positive number $\tau$ such that ( 1 ) holds, then $\tau$ is called the period of $\gamma$ , and we will say $\gamma$ is a $\tau$ –periodic orbit .

Definition 2.2

þ A general frame is a triple $\mathcal{W}=\langle W,R,A\rangle$ , where $\langle W,R\rangle$ is a Kripke frame, and $A$ is a family of subsets of $W$ which is closed under Boolean operations, and under the operation

\Box X=\{w\in W:\forall v\in W\,(w\mathrel{R}v\Rightarrow v\in X)\}.

Sets $X\in A$ are called admissible . A model $\mathcal{M}=\langle W,R,{\vDash}\rangle$ is based on $\mathcal{W}$ if the set

\{w\in W:\mathcal{M},w\vDash p\}

is admissible for every variable $p$ (which implies the same holds for all formulas). A formula is valid in $\mathcal{W}$ if it holds in all models based on $\mathcal{W}$ , and the notions of $L$ -frames, soundness, and completeness are defined accordingly. A Kripke frame $\langle W,R\rangle$ can be identified with the general frame $\langle W,R,\mathcal{P}(W)\rangle$ .

Definition 2.1 ( [ 21 ] ) .

The data $(E,[\,\,,\,\,],\rho,\langle\,\,,\,\,\rangle)$ is a Courant algebroid if the following conditions hold for all $a,b,c\in\Gamma(E)$ :

•

$[a,[b,c]]=[[a,b],c]+[b,[a,c]]$
•

$[a,b]+[b,a]=2\,d\langle a,b\rangle$
•

$\rho(a)\langle b,c\rangle=\langle[a,b],c\rangle+\langle b,[a,c]\rangle$ .

We call $[\,\,,\,\,]$ the Dorfman bracket , $\rho$ the anchor and $\langle\,\,,\,\,\rangle$ the pairing of $E$ .

Definition 3.6 .

Let $G$ be a smooth affine group scheme equipped with a lift of Frobenius $\phi_{G}$ and let $\alpha\in L_{\delta}(G)$ . Then the equality

l\delta(u)=\alpha

will be referred to as a $\delta_{G}$ -linear equation with unknown $u\in G$ .

Definition 4.4 .

Let $w$ be a quasi-valuation on a ring $R$ . An element $c\in R$ is called left stable with respect to $w$ if

w(cr)=w(c)+w(r)

for every $r\in R$ . Analogously, one defines the notion right stable .

Definition 2.2

Let $G$ and $H$ be groups acting on the sets $X$ and $Y$ from the left respectively. Then the wreath product $H\wr_{X}G$ acts on the set $Y\times X$ by defining

(\alpha;g)\cdot(y,x)=((\alpha(g(x)))(y),g(x))

for any $x\in X$ , $y\in Y$ , $g\in G$ , $\alpha\in A$ . It is routine to verify that $((\alpha;g_{1})\cdot(\beta;g_{2}))\cdot(y,x)=(\alpha;g_{1})\cdot((\beta;g_{2})% \cdot(y,x))$ for any $x\in X$ , $y\in Y$ , $\alpha,\beta\in A$ , $g_{1},g_{2}\in G$ .

Definition A.1 .

A torsor is a set $G$ with a map $G^{3}\to G$ , $(x,y,z)\mapsto(xyz)$ satisfying the following algebraic identities:

(PA)

para-associative identity : $((xuv)wz)=(x(wuv)z)=(xu(vwz)))$ .
(IP)

idempotency identity $(xxy)=y=(yxx)$ .

The opposite torsor is $G$ with $(xyz)^{opp}=(zyx)$ , and a torsor is called commutative if $G=G^{opp}$ , i.e., it satisfies the identity

(C)

$(xyz)=(zyx)$ .

Categorial notions are defined in the obvious way. In every torsor, left-, right- and middle multiplication operators are the maps $G\to G$ defined by equation ( 0.1 ).

Definition 2.13

Let $\rho^{\prime}$ and $\rho^{\prime\prime}$ be two permutations of the sets $\{1,\dots,i\}$ , $\{i+1,\dots,i+j\}$ , respectively. We define the permutation $\rho^{\prime-1}\cup\rho^{\prime\prime-1}$ of $\{1,\dots,i+j\}$ , which acts on $\{1,\dots,i\}$ as $\rho^{\prime-1}$ and on $\{i+1,\dots,i+j\}$ as $\rho^{\prime\prime-1}$ . We define the set of shuffles of two given permutations, denoted by $sh(\rho^{\prime},\rho^{\prime\prime})$ , as the set of all permutations $\rho$ of the set $\{1,2,\dots,i+j\}$ such that $\rho^{-1}$ is the composition of a shuffle of sets $\tau\in sh(i,j)$ (see Definition 1.4 ) and with $\rho^{\prime-1}\cup\rho^{\prime\prime-1}$ . That is,

\rho^{-1}=\tau\circ(\rho^{\prime-1}\cup\rho^{\prime\prime-1}).