By quaternion algebra $\mathbb{H}$ with Hamiltonian product we mean a four-dimensional algebra $\langle 1,i,j,k\rangle_{\mathbb{R}}$ such that

In this algebra quaternions of $\mathbb{R}=\langle 1\rangle_{\mathbb{R}}$ are called scalars, and $V=\langle i,j,k\rangle_{\mathbb{R}}$ are called vectors. By basic quaternions we mean the following four elements of $\mathbb{H}$ : $\{1,i,j,k\}$ . We also will use the standard quaternion functions: the norm $\left|\left|a+bi+cj+dk\right|\right|=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}}$ , the real part ${\operatorname{Re}\,}(a+bi+cj+dk)=a$ , the vector part ${\operatorname{Ve}\,}(a+bi+cj+dk)=bi+cj+dk$ .

Two families $G:\left(\mathbb{R}^{n}\times\mathbb{R}^{2n},(\beta_{0},x_{0})\right)\to\mathbb{R}$ and
$\bar{G}:\left(\mathbb{R}^{n}\times\mathbb{R}^{2n},(\bar{\beta}_{0},\bar{x}_{0})\right)\to\mathbb{R}$ are $\mathcal{R}^{odd}$ -equivalent , or more precisely fibred $\mathcal{R}^{odd}$ -equivalent (cf. Definition 4.10 below) if there exist a germ of diffeomorphism $\left(\mathbb{R}^{n}\times\mathbb{R}^{2n},(\bar{\beta}_{0},\bar{x}_{0})\right)\to\left(\mathbb{R}^{n}\times\mathbb{R}^{2n},(\beta_{0},x_{0})\right)$ of the form

where $\beta(\cdot,\bar{x})\in\mathcal{D}_{n}^{odd}$ for each fixed $\bar{x}$ , such that

in the Lagrangian setting, or such that

in the Legendrian setting, for a germ of function $c:(\mathbb{R}^{2n},0)\to(\mathbb{R},0)$ .

Let $(\mathfrak{l},[\cdot,\cdot])$ be a Lie algebra; $\mathfrak{p},\mathfrak{q}\subset\mathfrak{l}$ are nonzero vector subspaces. By definition,

If $\dim(\mathfrak{p})\geq 2$ then by definition,

The vector subspace $\mathfrak{p}\subset\mathfrak{l}$ generates the Lie algebra $(\mathfrak{l},[\cdot,\cdot])$ , if $\mathfrak{l}=\mathfrak{p}_{m}$ for some natural number $m;$ the smallest number $m:=s$ with such property is called the generation degree (of the algebra $(\mathfrak{l},[\cdot,\cdot])$ by the subspace $\mathfrak{p}$ ).

For an element ${\mathfrak{f}}(z;\lambda,\mu)$ of $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]_{0}$ , consider the following properties:

• •

  (skew symmetry) ${\mathfrak{f}}(z;\lambda,\mu)={\mathfrak{f}}(-z;\mu,\lambda)$ .

• •

  (additivity) ${\mathfrak{f}}(z;\lambda+\mu,\nu)={\mathfrak{f}}(z;\lambda,\nu)+{\mathfrak{f}}(z;\mu,\nu)$ .

The set of sections of $z^{-1}\mathcal{O}_{S^{2}}[z^{-1}]_{0}$ with these properties is denoted by $\mathrm{Irr}(S)$ .

We say that $P$ is irreducible and aperiodic if

A Markov chain with an irreducible and aperiodic transition kernel is said an irreducible and aperiodic Markov chain.

Let $X$ be a smooth compact manifold. The ${\mathbb{R}}/{\mathbb{Z}}$ $K^{0}$ -theory of $X,$ denoted by $K^{0}(X,{\mathbb{R}}/{\mathbb{Z}}),$ consists of all ${\mathbb{R}}/{\mathbb{Z}}$ $K^{0}$ -cocycles with zero virtual trace in the lowest degree, modulo the ${\mathbb{R}}/{\mathbb{Z}}$ $K^{0}$ -relation. The group operation is given by the addition of ${\mathbb{R}}/{\mathbb{Z}}$ $K^{0}$ -cocycles

Suppose $\kappa$ is a regular cardinal and $\lambda$ is such that $\omega\cdot\lambda<\kappa$ . Suppose $f\in{}^{\omega\cdot\lambda}\kappa$ . Let $\mathsf{block}:{}^{\omega\cdot\lambda}\kappa\rightarrow{}^{\lambda}\kappa$ be defined by $\mathsf{block}(f)(\alpha)=\sup\{f(\omega\cdot\alpha+k):k\in\omega\}$ .

Suppose $f,g\in{}^{\omega\cdot\lambda}\kappa$ . Let $\mathsf{joint}:{}^{\omega\cdot\lambda}\kappa\times{}^{\omega\cdot\lambda}\kappa\rightarrow{}^{\lambda}\kappa$ be defined by

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$ -bocs. Then, a twisted ${\cal B}$ -module is a pair $(M,u)$ , where $M\in{\rm GMod}\hbox{-}{\cal B}$ and $u$ is a homogeneous morphism $u\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{1}(M,M)$ such that the following Maurer-Cartan equation for $u$ holds

If $(M,u)$ and $(N,v)$ are twisted ${\cal B}$ -modules, then a homogeneous morphism of twisted ${\cal B}$ -modules $f:(M,u)\smash{\mathop{\hbox to 20.0pt{\rightarrowfill}}\limits}(N,v)$ of degree $d$ is a homogeneous morphism $f:M\smash{\mathop{\hbox to 20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}$ of degree $d$ such that

We will denote by ${\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{d}((M,u),(N,v))$ the space of homogeneous morphisms of twisted ${\cal B}$ -modules of degree $d$ . Moreover, we make

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$ -bocs. Given morphisms $f,g\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}}((M,u),(N,v))$ , a homotopy $h$ from $f$ to $g$ is a morphism $h\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{-1}(M,N)$ such that

A morphism $f\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}}((M,u),(N,v))$ is null-homotopic iff there is a homotopy from $f$ to $0$ .

We denote by ${\rm TGMod}^{0}\hbox{-}{\cal B}$ the subcategory ${\rm TGMod}\hbox{-}{\cal B}$ with the same objects and only zero degree homogeneous morphisms. Thus, the notion of homotopy is an equivalence relation in the category ${\rm TGMod}^{0}\hbox{-}{\cal B}$ .

( Strong short immersion ) We call a $C^{1}$ immersion $u:\mathcal{M}\rightarrow\mathbb{R}^{m}$ strongly short if

with a non-negative function $\rho\in C(\mathcal{M})$ and symmetric tensor $h\in C(\mathcal{M};\mathbb{R}^{n\times n}_{sym})$ satisfying

( Adapted short immersion ) Given a closed subset $\Sigma\subset\mathcal{M}$ a strongly short immersion $u:\mathcal{M}\rightarrow\mathbb{R}^{m}$ is called adapted short with respect to $\Sigma$ with exponent $0<\theta<1$ if $u\in C^{1,\theta}(\mathcal{M})$ is strongly short with

such that $\Sigma=\{\rho=0\}$ ,

and there exists a constant $A\geq 1$ such that, in any chart $\Omega_{k}$

for any $x\in\Omega_{k}\setminus\Sigma$ .

A generalized Boolean algebra is a distributive lattice $L$ with designated bottom element $\bot$ satisfying

A generalized Boolean homomorphism is a lattice homomorphism $f\colon L\to M$ which preserves the bottom element. We denote the category of generalized Boolean algebras and their homomorphisms by $\mathbf{gBa}$ .

Let $(X,d)$ and $(Y,\rho)$ be pseudometric spaces. The pseudometrics $d$ and $\rho$ are combinatorially similar if there are bijections $g\colon Y\to X$ and $f\colon d(X^{2})\to\rho(Y^{2})$ such that

holds for all $x$ , $y\in Y$ . In this case, we will say that $g\colon Y\to X$ is a combinatorial similarity .

Let $\Sigma,O$ be $\mathsf{Pm}$ -sets. A separated language is an equivariant map of the form $\Sigma^{(*)}\to O$ . A separated automaton $\mathcal{A}=(Q,\delta,o,q_{0})$ consists of $Q$ , $o$ and $q_{0}$ defined as in a nominal automaton, and an equivariant transition function $\delta\colon Q\mathop{\ast}\Sigma\rightarrow Q$ .

The separated language semantics of such an automaton is given by the map $s\colon Q\mathop{\ast}\Sigma^{(*)}\to O$ , defined by

for all $x\in Q$ , $a\in\Sigma$ and $w\in\Sigma^{(*)}$ such that $x\mathop{\#}aw$ and $a\mathop{\#}w$ .

Let $s^{\flat}\colon Q\to(\Sigma^{(*)}\mathrel{-\kern-3.0pt{\ast}}O)$ be the transpose of $s$ . Then $s^{\flat}(q_{0})\colon\Sigma^{(*)}\to O$ corresponds to a separated language, this is called the separated language accepted by $\mathcal{A}$ .

Let $R$ be a ring and let $\sigma:R\to R$ be a ring endomorphism. An additive map $d:R\to R$ is called a generalized $\sigma$ -derivation (or a generalized skew derivation ) if there exists a map $\delta:R\to R$ such that

for all $x,y\in R$ .

The map $f:U\to W$ is differentiable at $u$ , if there is an operator $f^{\prime}(u)\in\mathcal{B}(V,W)$ such that

Then $f^{\prime}(u)$ is the derivative of $f$ at $u$ . If $f$ is differentiable at every point in $U$ , then we say that $f$ is differentiable . In that case, the derivative of $f$ is the map