1 Introduction and Highlights

Definition 4.2 .

Assume that $\varphi(x,\xi)$ is a measurable-convex integrand with each $\varphi(x,\xi)\in[0,+\infty)$ and $\varphi(x,0)=0$ . For each $x\in G$ , denote the subgradient of $\varphi(x,\cdot)$ by

(26)

\displaystyle\bar{\beta}(x,\cdot)=\partial\varphi(x,\cdot).

Definition 1 .

Let ${\cal E}$ be the variety of Moufang loops with the following identities:

	$\displaystyle x^{4}=1,\,\left[x,y\right]^{2}=1,\,(x,y,z)^{2}=1,$
	$\displaystyle\left[x^{2},y\right]=1,\,\left[\left[x,y\right],t\right]=1,\,% \left[(x,y,z),t\right]=1,$
	$\displaystyle(x^{2},y,z)=1,\,(\left[x,y\right],z,t)=1,\;\;((x,y,z),t,s)=1.$		(4)

Definition 3.3 .

Let $D_{a}^{\bot}:=Sp\{\phi^{aV},\phi^{aH}\}$ be an eigenform-distribution of $\mathbb{\Phi}$ . A 1-form $\alpha\in D_{a}^{\bot}$ is said to be an integrable direction in $D_{a}^{\bot}$ if

\displaystyle d\alpha=\kappa\wedge\alpha,

(18)

for some 1-form $\kappa.$

Definition 1.1 .

We define the algebraic intersection number $\alpha\cdot\beta\in\mathbb{Z}$ by

[\delta*\beta]=(\alpha\cdot\beta)[u],

where $u$ is an embedded closed loop in $S^{2}\setminus A,$ positively oriented with respect to $x_{1},$ and hence $[u]$ is a generator of $H_{1}(S^{2}\setminus A)$ .

Definition 2.4 .

Let $G$ be an abelian group. A map $\varepsilon:G\times G\rightarrow{\bf K^{*}}$ is called a skew-symmetric bicharacter on $G$ if the following identities hold: for all $a,b,c\in G$ ,

(i)

$\varepsilon(a,b)\varepsilon(b,a)=1$ ;
(ii)

$\varepsilon(a,b+c)=\varepsilon(a,b)\varepsilon(a,c)$ ;
(iii)

$\varepsilon(a+b,c)=\varepsilon(a,c)\varepsilon(b,c)$ .

Definition 3 .

The (left) groupoid representation of $G$ is the representation $\pi$ in $L^{2}(\mathcal{\mathcal{R}},\nu)$ defined by

(\pi(g)f)((x,y))=f(g^{-1}x,y).

Definition 3.1.2 .

The Lie algebra $\mathfrak{grt}_{1}$ is spanned by the elements $(0,\psi)\in\mathfrak{tder}_{2}$ , that satisfy the following relations:

\psi(x,y)=-\psi(y,x),

(3.1)

\psi(x,y)+\psi(y,z)+\psi(z,x)=0\;\;\;\;{\rm for}\;\;\;\;x+y+z=0,

(3.2)

\psi(t^{1,2},t^{2,34})+\psi(t^{12,3},t^{3,4})=\psi(t^{2,3},t^{3,4})+\psi(t^{1,% 23},t^{23,4})+\psi(t^{1,2},t^{2,3}),

(3.3)

where the last equation takes values in the Lie algebra $\mathfrak{t}$ , and $t^{1,23}=t^{1,2}+t^{1,3}$ , etc.

Definition 3.6 .

If $\lambda\in\mathcal{L}_{n,k}^{*}$ , then define the weight of $\lambda$ by

w(\lambda)=\alpha^{\text{nrec}(\lambda)}\beta^{\text{rec}^{*}(\lambda)}r^{% \text{circ}(\lambda)}.

Definition .

(Commutative algebra) A commutative algebra $\mathcal{A}$ is a vector space equipped with an additional associative $+:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ and commutative multiplication $\cdot\,:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ such that

x\cdot(y+z)=x\cdot y+x\cdot z,\quad\lambda(x\cdot y)=(\lambda x)\cdot y

holds for all $x,$ $y$ , $z\in\mathcal{A}$ and $\lambda\in\mathbb{R}$ . An important algebra is the space $C(\mathcal{X})$ of all continuous functions $f:\mathcal{X}\rightarrow\mathbb{R}$ on the compact metric space $(\mathcal{X},\mathbf{d})$ endowed with the supremum norm

\|f\|_{\infty}:=\sup_{x\in\mathcal{X}}|f(x)|.

Definition 5

Let $\tilde{\cal CP}^{(n)}(\tilde{\Pi})$ be a set of in-plane Bézier control points of all patches which result from degree elevation of a global regular in-plane parametrisation $\tilde{\Pi}$ up to degree $n$ for every patch. Since the Bézier control points of the in-plane parametrisation always coincide along the shared edges, they are unambiguously well defined.

A determining set D is a subset of $\tilde{\cal CP}^{(n)}$ so that :

\forall P\in D,\quad Z(P)=0\Rightarrow\bar{\Psi}\ \equiv 0;

A determining set is called minimal determining set (MDS) $\tilde{\cal B}^{(n)}$ if there is no determining which size is smaller.

Definition 1 (Screening test) .

Given problem $\mathcal{P}(\lambda,\Omega,\mathbf{D},\boldsymbol{\mathrm{y}})$ with solution $\tilde{\boldsymbol{\mathrm{x}}}$ , a boolean-valued function ${T:[1\ldots K]\rightarrow\{0,1\}}$ is a screening test if and only if:

\displaystyle\forall i\in[1\ldots K],T(i)=1\Rightarrow\mathnormal{\tilde{% \boldsymbol{\mathrm{x}}}}(i)=0.

(2)

Definition 2.1 .

An algebra $(L,[\cdot,\cdot])$ over a field $F$ is called a Leibniz algebra if it is defined by the identity

[x,[y,z]]=[[x,y],z]-[[x,z],y].

Definition 2.5 .

A linear transformation $d$ of a Leibniz algebra $L$ is said to be a derivation if for any $x,y\in L$ one has

d([x,y])=[d(x),y]+[x,d(y)].

Definition 6.1 .

Let $s$ be an exponent over the finite field $L$ . We say that $s$ is a nice exponent over $L$ if the number $N(1,v)$ of solutions in $L^{2}$ of

	$\displaystyle x+y=1,$
	$\displaystyle x^{s}+y^{s}=v,$

takes at most $3$ values as $v$ runs through $L$ .

Definition 4.1

The solution $z(t,x)=(u,v,w)$ to system (1.1) is $x$ -symmetric if there exists a function $b(t)\in\mathcal{C}^{1}(\mathbb{R^{+}})$ such that

z(t,x)=z(t,2b(t)-x),\qquad\forall t\in[0,\infty),

for almost every $x\in\mathbb{R}$ , then the function $b(t)$ is called the symmetric axis of $z(t,x)$ .

Definition 3.2 .

The dual of a pair of dihedral structures $(\delta,\delta^{\prime})$ is

(\delta,\delta^{\prime})^{\vee}=(\delta^{\prime},\delta)\ .

It is well-defined on configurations, and defines an involution $\vee:\mathcal{C}_{S}\rightarrow\mathcal{C}_{S}$ .

Definition 2.26 .

Let $m\geq 2$ be an integer and $\varepsilon>0$ . We say a germ of continuous map $u:{\mathcal{O}}(S,x)\rightarrow Q$ is of class $(m,\varepsilon)$ at the point $x$ if for a smooth chart $\phi:U(u(0))\rightarrow{\mathbb{R}}^{2n}$ mapping $u(0)$ to $0$ and holomorphic polar coordinates $\sigma:[0,\infty)\times S^{1}\rightarrow S\setminus\{x\}$ around $x$ , the map

v(s,t)=\phi\circ u\circ\sigma(s,t),

defined for $s$ large, has partial derivatives up to order $m$ , which weighted by $e^{\varepsilon s}$ belong to $L^{2}([s_{0},\infty)\times S^{1},{\mathbb{R}}^{2n})$ if $s_{0}$ is sufficiently large. We say the germ is of class $m$ around a point $z\in S$ provided $u$ is of class $H^{m}_{loc}$ near $z$ .

Definition 3.6 (Offspring distribution) .

Let $r\geq 1$ and $\bm{\xi}=\left(\xi^{\left(1\right)},\ldots,\xi^{\left(r\right)}\right)$ be a family of probabilities on the $\sigma$ -algebra $\sigma\left(W_{r}\right)$ . We say that the family of probabilities $\bm{\mu}$ on the $\sigma$ -algebra $\sigma\left(\mathbb{N}^{r}\right)$ is an offspring distribution, where p is the counter map defined in Definition 3.5 and

\bm{\mu}^{\left(i\right)}=\xi^{\left(i\right)}\circ\textbf{{p}}^{-1},\ \ % \forall i=1,\ldots,r.

Definition 4.2 .

Let $G$ be a group acting on a space $X$ and let $f$ be a global function on $X$ . We say that $f$ is semi-invariant with respect to a character $\chi$ if, for any $g\in G$ ,

f(g\cdot x)=\chi(g)f(x).

Equivalently, this means that $f$ is a section of the equivariant line bundle $\operatorname{\mathcal{O}}(\chi)$ on the global quotient stack $[X/G]$ .

Definition 1.1 .

$A$ -number $a$ is left divisor of $A$ -number $b$ , if there exists $A$ -number $c$ such that

(1.1)

ac=b

∎

Definition 1.2 .

$A$ -number $a$ is right divisor of $A$ -number $b$ , if there exists $A$ -number $c$ such that

(1.2)

ca=b

∎

Definition 1.3 .

$A$ -number $a$ is divisor of $A$ -number $b$ , if there exists $A\otimes A$ -number $c$ such that

c\circ a=b

$A\otimes A$ -number $c$ is called quotient of $A$ -number $b$ divided by $A$ -number $a$ . ∎

Definition 5.1 .

Let division in the $D$ -algebra $A$ is not always defined. $A$ -number $a$ divides $A$ -number $b$ with remainder , if the following equation is true

(5.1)

c\circ a+f=b

$A\otimes A$ -number $c$ is called quotient of $A$ -number $b$ divided by $A$ -number $a$ . $A$ -number $f$ is called remainder of the division of $A$ -number $b$ by $A$ -number $a$ . ∎

Definition 5.4 .

We define canonical remainder ${\color[rgb]{.4,0,.9}b\ \mathrm{mod}\ a}$ of the division of $A$ -number $b$ by $A$ -number $a$ as selected element of the set $A\{b,a\}$ . The representation

c\circ a+(b\ \mathrm{mod}\ a)=b

of division with remainder is called canonical . ∎

Definition 1.1 .

Let $k,q,k^{\prime},q^{\prime}$ be non-negative integers where $q,q^{\prime}\neq 0$ and $\gcd(k,q)=1$ . Further, suppose that

kq^{\prime}-k^{\prime}q=1.

If $q\geq q^{\prime}\geq 1$ , then $(k^{\prime},q^{\prime})$ is said to be the lower parent of $(k,q)$ .

Definition 1

The operators $(a,b)$ are ${\mathcal{D}}$ -pseudo bosonic ( ${\mathcal{D}}$ -pb) if, for all $f\in{\mathcal{D}}$ , we have

a\,b\,f-b\,a\,f=f.

(2.1)

Definition 1.0 .

Let $\nu$ be a primitive of the volume form $\Omega$ and $\mathbf{f}:\RR^n\mapsto\RR^n$ an exact volume preserving diffeomorphism such that

\mathbf{f}^{*}{\nu}-\nu=d\lambda,

(2)

for a $n-2$ form $\lambda$ . The differential form $\lambda$ is called a generating form with respect to $\nu$ .

Definition 1.0 .

Let $\nu,\tilde{\nu}$ be two primitives of a volume form $\Omega$ , i.e. $d\nu=d\tilde{\nu}=\Omega$ and $\mathbf{f}:\RR^n\mapsto\RR^n$ an exact volume preserving diffeomorphism such that

\mathbf{f}^{*}\tilde{\nu}-\nu=d\lambda,

(3)

for a $n-2$ form $\lambda$ . The $n-2$ differential form $\lambda$ is called a generating form with respect to $(\nu,\tilde{\nu})$ .

Definition 1.2 .

Let $\mathcal{Y}$ be the class of quarter plane walks using steps from the set

\{(1,-1),(-1,0),(0,1)\}=\{\searrow,\leftarrow,\uparrow\}.

Let $\mathcal{S}$ be the class of quarter plane walks using steps from the set

\{(1,0),(1,-1),(0,-1),(-1,0),(-1,1),(0,1)\}=\{\rightarrow,\searrow,\downarrow,% \leftarrow,\nwarrow,\uparrow\}.

Definition 2.6 .

Let $(A,\alpha)$ and $(B,\beta)$ be two monoidal Hom-algebras. A left $(A,\alpha)$ , right $(B,\beta)$ Hom-bimodule consists of an object $(M,\mu)\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ together with a left $(A,\alpha)$ -Hom-action $\phi:A\otimes M\to M$ , $\phi(a\otimes m)=am$ and a right $(B,\beta)$ -Hom-action $\varphi:M\otimes B\to M$ , $\varphi(m\otimes b)=mb$ fulfilling the compatibility condition, for all $a\in A$ , $b\in B$ and $m\in M$ ,

(2.15)

(am)\beta(b)=\alpha(a)(mb).

We call a left $(A,\alpha)$ , right $(B,\beta)$ Hom-bimodule a $[(A,\alpha),(B,\beta)]$ -Hom-bimodule. Let $(M,\mu)$ and $(N,\nu)$ be two $[(A,\alpha),(B,\beta)]$ -Hom-bimodules. A morphism $f:M\to N$ is called a morphism of $[(A,\alpha),(B,\beta)]$ -Hom-bimodules if it is both left $(A,\alpha)$ -linear and right $(B,\beta)$ -linear, and satisfies the following property

(2.16)

(af(m))\beta(b)=\alpha(a)(f(m)b),

for all $a\in A$ , $b\in B$ and $m\in M$ .

Definition 1 (Rotations, Reflections and Translations) .

A matrix of the form

\left(\begin{array}[]{cc}a&-b\\ b&a\end{array}\right),\ a^{2}+b^{2}=1

is called a rotation matrix, and a matrix of the form

\left(\begin{array}[]{cc}a&b\\ b&-a\end{array}\right),\ a^{2}+b^{2}=1

is called a reflection matrix. If $\textbf{{u}}\in\mathbb{F}_{q}^{2}$ and $R$ is a rotation matrix, then a rotation about u by $R$ is an affine map of the form

{\mathcal{R}}(\textbf{{v}})=R(\textbf{{v}}-\textbf{{u}})+\textbf{{u}}.

If $\textbf{{u}}\in\mathbb{F}_{q}^{2}$ and $S$ is a reflection matrix, then a reflection about u by $S$ is an affine map of the form

{\mathcal{S}}(\textbf{{v}})=S(\textbf{{v}}-\textbf{{u}})+\textbf{{u}}.

A translation by u is an affine map of the form

{\mathcal{T}}(\textbf{{v}})=\textbf{{v}}+\textbf{{u}}.

Definition 2.2 .

Let ${\mathfrak{g}}$ be a Lie bialgebra. A left ${\mathfrak{g}}$ -crossed module $(V,{\triangleright},\alpha)$ is a left ${\mathfrak{g}}$ -module $(V,{\triangleright})$ that admits a left ${\mathfrak{g}}$ -coaction $\alpha:V\to{\mathfrak{g}}\otimes V$ such that

\alpha(x{\triangleright}v)=([x,\ ]\otimes{\rm id}+{\rm id}\otimes x{% \triangleright})\alpha(v)+\delta(x){\triangleright}v

for any $x\in{\mathfrak{g}},v\in V.$

In the case of ${\mathfrak{g}}$ is finite-dimensional, the notion of a left ${\mathfrak{g}}$ -crossed module is equivalent to a left ${\mathfrak{g}}$ -module $(V,{\triangleright})$ that admits a left ${\mathfrak{g}}^{*op}$ -action ${\triangleright}^{\prime}$ satisfying

(2.4)

\phi{}_{(1)}{\triangleright}^{\prime}v{\langle}\phi{}_{(2)},x{\rangle}+x{}_{(1% )}{\triangleright}v{\langle}\phi,x{}_{(2)}{\rangle}=x{\triangleright}(\phi{% \triangleright}^{\prime}v)-\phi{\triangleright}^{\prime}(x{\triangleright}v)

for any $x\in{\mathfrak{g}},\,\phi\in{\mathfrak{g}}^{*}$ and $v\in V,$ where the left ${\mathfrak{g}}^{*op}$ -action ${\triangleright}^{\prime}$ corresponds to the left ${\mathfrak{g}}$ -coaction $\alpha$ above via $\phi{\triangleright}^{\prime}v={\langle}\phi,v{}^{(1)}{\rangle}v{}^{(2)}$ with $\alpha(v)=v{}^{(1)}\otimes v{}^{(2)}.$

We call a left ${\mathfrak{g}}$ -module $V$ with linear map ${\triangleright}^{\prime}:{\mathfrak{g}}^{*}\otimes V\to V$ (not necessarily an action) such that ( 2.4 ) a left almost ${\mathfrak{g}}$ -crossed module .

Definition 1.3 .

We say that a connected Lie group $M$ is unimodular if

\text{trace}\left(\text{ad}(m)\right)=0,\;\forall m\in\mathfrak{m},

where $\mathfrak{m}$ denotes the Lie Algebra of $M$ .