(CARMA(p,q) process)
A CARMA(p,q) process $y(t)$ , $0\leq q<p$ , is defined as the stationary solution of

$$y=\mathbf{c}\;\mathbf{u}(t),$$

(39)

with the linear differential equation for the state vector $\mathbf{u}(t)\in\mathds{R}^{p}$

$$\mathbf{\dot{u}}(t)=\mathrm{\mathbf{A}}\;\mathbf{u}(t)+\mathbf{b}\;\xi(t),$$

(40)

where $\xi(t)$ is white noise with $E\{\xi(t)\}=0$ and $E\{\xi(t)\xi(t+\tau)\}=\sigma^{2}\delta(\tau)$ , $\;\sigma\in\mathds{R}$ , $\delta(\cdot)$ is the Dirac function,

$\displaystyle\mathrm{\mathbf{A}}=\left(\begin{matrix}-a_{1}&1&0&\cdots&0\\ -a_{2}&0&1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&0\\ -a_{p-1}&0&\cdots&0&1\\ -a_{p}&0&\cdots&\cdots&0\end{matrix}\right),\;\;\;\mathbf{b}=\left(\begin{array}[]{c}b_{p-1}\\ b_{p-2}\\ \vdots\\ b_{1}\\ b_{0}\\ \end{array}\right),\;\;\;\mathbf{c}=\left(\begin{array}[]{c}1\\ 0\\ \vdots\\ 0\\ \end{array}\right),$

(41)

and $b_{j}=0$ if $j>q$ .

Let $P_{1}$ and $P_{2}$ be two planes in $\mathbb{R}^{3}$ that intersect with each other. Let $\phi\in(0,\pi)$ be one of the associated intersecting dihedral angle of $P_{1}$ and $P_{2}$ . Set

$$\phi=\alpha\cdot\pi,\ \ \alpha\in(0,1).$$

Then, $\phi$ is said to be an irrational dihedral angle if $\alpha$ is an irrational number; and it is said to be a rational dihedral angle of degree $q$ if $\alpha=p/q$ with $p,q\in\mathbb{N}$ and irreducible.

A polynomial $f\in R$ is called semi-invariant with respect to $G$ , if there exists a linear character $\eta$ of $G$ such that

$$g\cdot f=\eta(g)\cdot f$$

for all $g\in G$ .

Let $T$ be a real vector space endowed with a non-zero alternating form $(\,,\,)\colon T\times T\to\mathbb{R}$ , and a triple product $[\,,\,,\,]\colon T\times T\times T\to T$ . It is said that $(T,[\,,\,,\,],(\,,\,))$ is a symplectic triple system if satisfies

(1)		$\displaystyle[x,y,z]=[y,x,z],$
(2)		$\displaystyle-[x,z,y]=(x,z)y-(x,y)z+2(y,z)x,$
(3)		$\displaystyle]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]],$
(4)		$\displaystyle([x,y,u],v)=-(u,[x,y,v]),$

for any $x,y,z,u,v,w\in T$ .

Given $i:\mathsf{B}\to\mathsf{C}$ complete homomorphism of complete boolean algebras, $i$ extends to a map $\hat{i}:V^{\mathsf{B}}\to V^{\mathsf{C}}$ defined by transfinite recursion by

$$\hat{i}(\tau)=\left\{\langle\hat{i}(\sigma),i(b)\rangle:\,\langle\sigma,b\rangle\in\tau\right\}.$$

Given $\tau_{1},\dots,\tau_{n}\in V^{\mathsf{B}}$ , $\phi(\tau_{1},\dots,\tau_{n})$ is generically absolute for $i$ if

$$i(\left\llbracket\phi(\tau_{1},\dots,\tau_{n}\right\rrbracket_{\mathsf{B}})=\left\llbracket\phi(\hat{i}(\tau_{1}),\dots,\hat{i}(\tau_{n})\right\rrbracket_{\mathsf{C}}.$$

For a central extension $\xymatrix{1\ar[r]&A\ar[r]&\tilde{G}\ar[r]^{\pi}&G\ar[r]&1}$ , a cochain $\tau:\tilde{G}\to A$ that satisfies the condition

$\displaystyle\tau(a\tilde{g})=a+\tau(\tilde{g})$

for all $\tilde{g}\in\tilde{G}$ and $a\in A$ is called a connection cochain .

We say that a function clone $\mathcal{C}$ has local pseudo-Siggers operations if for every finite $A\subseteq C$ there exists a $6$ -ary $s\in\mathcal{C}$ and unary $\alpha,\beta\in\mathcal{C}$ satisfying

$$\alpha s(x,y,x,z,y,z)=\beta s(y,x,z,x,z,y)$$

for all $x,y,z\in A$ .

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ , $g:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $h:\mathbb{R}^{n}\to\mathbb{R}^{m}$ be continuously differentiable. We say $f$ is a Lyapunov function of,

$$\dot{x}=g(x),$$

(17)

if $(\nabla f^{\top}g)(\mathbb{R}^{n})\subset\mathbb{R}_{-}$ , and $h$ is a conserved quantity if $(\nabla h^{\top}g)(\mathbb{R}^{n})=\{0\}$ . Further, we define the stationary set $\mathcal{C}=(\nabla f^{\top}g,h)^{-1}(\{(0,0)\})$ , namely critical points on the constraint manifold $\mathcal{M}=h^{-1}(\{0\})$ , it is a closed set as continuous preimage of a closed set. We complete the definition of a Lyapunov function by requiring that,

$$(\nabla f^{\top}g)^{-1}(\{0\})\subset\mathcal{C}.$$

Let $\mathrm{Pairs}(k,n)$ denote the set of elements $\beta$ where each element $\beta$ is a multi-set of $k$ pairs of integers:

$$\beta=\{\beta(1),\beta(2),...,\beta(k)\}$$

(6)

where

$$\beta(i)=\{\beta(i,1),\beta(i,2)\}$$

such that $\beta(i,1)<\beta(i,2)$ and each $\beta(i,j)\in\{1,2,...,n\}$ . Note that $\beta$ is a set: even though we have used an ordering of the pairs $\beta(1),...,\beta(k)$ in the notation of ( 6 ), another ordering would result in the same element $\beta$ .

Let $|\beta|$ denote

$$|\beta|=\#\text{ distinct numbers that appear as $\beta(i,1)$ or $\beta(i,2)$}.$$

For $\beta\in\mathrm{Pairs}(k,n)$ , define

$$\mathcal{D}(\beta)=\prod_{i=1}^{k}(z_{\beta(i,1)}-z_{\beta(i,2)})$$

The Kauffman skein $\mathcal{D}(M)$ of a 3-manifold $M$ is the vector space spanned by framed unoriented links in $M$ , modulo the Kauffman skein relations:

(2.1)

$\displaystyle\vbox{\hbox{\includegraphics[height=56.905512pt]{poscross.eps}}}\,\,-\,\,\vbox{\hbox{\includegraphics[height=56.905512pt]{negcross.eps}}}\quad=\quad(s-s^{-1})\left(\vbox{\hbox{\includegraphics[height=56.905512pt]{idresolution.eps}}}\right.\,\,-\,\,\left.\vbox{\hbox{\includegraphics[height=56.905512pt]{capcupresolution.eps}}}\right),$

(2.2)

$\displaystyle\vbox{\hbox{\includegraphics[height=56.905512pt, keepaspectratio]{invvh.eps}}}\quad=\quad v^{-1}\,\,\vbox{\hbox{\includegraphics[height=56.905512pt, keepaspectratio]{frameresolution.eps}}}\,\,,\qquad\qquad\vbox{\hbox{\includegraphics[height=56.905512pt, keepaspectratio]{vh.eps}}}\quad=\quad v\,\,\vbox{\hbox{\includegraphics[height=56.905512pt, keepaspectratio]{frameresolution.eps}}}$

Suppose $\Sigma$ to be a smooth spacelike hypersurface of a spacetime $\mathcal{M}$ . A closed orientable two-dimensional surface $\sigma_{0}$ in $\Sigma$ is a dynamically transversely trapping surface (DTTS) if and only if there exists a timelike hypersurface $S$ in $\mathcal{M}$ that intersects $\Sigma$ precisely at $\sigma_{0}$ and satisfies the following three conditions at arbitrary points on $\sigma_{0}$ :

	$\displaystyle\bar{k}=0,$		(the momentarily non-expanding condition);		(21)
	$\displaystyle\mathrm{max}\left(\bar{K}_{ab}k^{a}k^{b}\right)=0,$		(the marginally transversely trapping condition);		(22)
	$\displaystyle{}^{(3)}\bar{\pounds}_{\bar{n}}\bar{k}\leq 0,$		(the accelerated contraction condition),		(23)

where $k^{a}$ is arbitrary future-directed null vectors tangent to $S$ and the quantity $\pounds_{\bar{n}}\bar{k}$ is evaluated with a time coordinate in $S$ whose lapse function is constant on $\sigma_{0}$ .

[ 13 , Proposition 6.46] , [ 12 ] The Iwasawa decomposition of the complex vector space $\mathfrak{g}$ is

$$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}.$$

The Iwasawa decomposition of $G$ [ 13 , Theorem 6.46] is

$$G=KAN.$$

Let $G$ be $O(p,q)$ $p+q=2n+1$ or $Sp_{2n}(\mathbb{R})$ , let $\mathfrak{g}_{0}$ be its Lie algebra, with complexification $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ . Conjugation $\bar{}:\mathfrak{g}\to\mathfrak{g}$ is defined by the real form $\mathfrak{g}_{0}$ . Define the star operation as the conjugate linear map ${}^{*}:\mathfrak{g}\to\mathfrak{g}$ such that:

$$g^{*}=-\bar{g}\text{ for all }g\in\mathfrak{g}.$$

Define the operation ${}^{\bullet}:\mathfrak{g}\to\mathfrak{g}$ by:

$$p^{\bullet}=\bar{p}\text{ for all }p\in\mathfrak{p}.$$

$$k^{\bullet}=-\bar{k}\text{ for all }k\in\mathfrak{k}.$$

A $\lambda$ -lattice is an algebra $(L,\vee,\wedge)$ of type $(2,2)$ satisfying the following identities:

1. (i)

   $x\vee y\approx y\vee x$ , $x\wedge y\approx y\wedge x$ ( commutativity ) ,

2. (ii)

   $x\vee((x\vee y)\vee z)\approx(x\vee y)\vee z$ , $x\wedge((x\wedge y)\wedge z)\approx(x\wedge y)\wedge z$ ( weak associativity ) ,

3. (iii)

   $x\vee(x\wedge y)\approx x$ , $x\wedge(x\vee y)\approx x$ ( absorption )

We set

$$n=n(p)=\begin{cases}4&\text{ if }p=2;\\ p&\text{ otherwise}.\end{cases}$$

We write $I=I(p)$ for the monomial ideal on the set of $2n+1$ variables

$$X=X(p)=\{v_{0},\dots,v_{n},e_{1,2},e_{2,3},\dots,e_{n-1,n},e_{n,1}\}$$

generated by the set of monomials $\{m_{0},m_{1},\dots,m_{n}\},$ where $m=\prod_{y\in X}y$ and

	$$\displaystyle m_{0}=\frac{m}{v_{0}},\qquad m_{1}=\frac{v_{0}m}{e_{n,1}v_{1}e_{1,2}},\qquad m_{n}=\frac{v_{0}m}{e_{n-1,n}v_{n}e_{n,1}},$$
	$$\displaystyle\text{ and }\qquad m_{i}=\frac{v_{0}m}{e_{i-1,i}v_{i}e_{i,i+1}}\qquad\text{ for }i=2,\dots,n-1.$$

Also, for convenience of notation, for each $e_{i,j}\in X(p)$ we set $e_{j,i}=e_{i,j}$ .