Gelfand-Tsetlin polytopes: a story of flow & order polytopes

Definition 16 (Validity of an embedded formula) .

Global validity ( $vld$ ) of an embedded formula $\phi$ of E in HOL is defined by the equation

\text{vld}\,\lfloor\phi\rfloor=\forall z.\lfloor\phi\rfloor z

Definition A.2 (size of the each tree in the forest) .

Let

\displaystyle\tau=k/\epsilon,

assuming that $k/\epsilon\leq n/16$ . The size of each tree in the forest is

\displaystyle q=n/\tau=n\epsilon/k

Definition 4.1 .

Let $(L,A,\rho)$ be a 3-Lie-Rinehart algebra, $(R,A)$ be a 3-Lie $A$ -algebra and $\beta:L\wedge L\rightarrow Der(R)$ be an action of $(L,A,\rho)$ on $(R,A)$ . If an $A$ -linear mapping $\psi:L\rightarrow R$ has the property: for all $x,y,z\in L$ ,

(38)

\psi([x,y,z])=[\psi(x),\psi(y),\psi(z)]+\beta(x,y)\psi(z)+\beta(y,z)\psi(x)+% \beta(z,x)\psi(y),

then $\psi$ is called a 3-Lie-Rinehart derivation from $(L,A,\rho)$ to $(R,A)$ , and $Der_{\beta}(L,R)$ denotes all 3-Lie-Rinehart derivations from $(L,A,\rho)$ to $(R,A)$ .

Definition 1 .

(cf. Nomizu [ 96 ] ) A homogeneous space $M=G/H$ is reductive if the Lie algebra ${\mathfrak{g}}$ of $G$ may be decomposed into a vector space direct sum of the Lie algebra ${\mathfrak{h}}$ of $H$ and an ${\mbox{\rm{Ad}}}(H)$ -invariant subspace ${\mathfrak{m}}$ , that is

(6.15a)		${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}},~{}{\mathfrak{h}}\cap{\mathfrak{m% }}=\emptyset,$
(6.15b)		${\mbox{\rm{Ad}}}(H){\mathfrak{m}}\subset{\mathfrak{m}}.$
Condition ( 6.15b ) implies
(6.15c)		$[{\mathfrak{h}},{\mathfrak{m}}]\subset{\mathfrak{m}}$

and, conversely, if $H$ is connected, then ( 6.15c ) implies ( 6.15b ). Note that $H$ is always connected if $M$ is simply connected. The decomposition ( 6.15a ) verifying ( 6.15b ) is called a $H$ -stable decomposition .

Definition 2.1 .

A Lie algebra over a field $\mathbb{F}$ , is a vector space $\mathfrak{g}$ over $\mathbb{F}$ , equipped with a skew-symmetric bilinear map $[\,\cdot\,,\,\cdot\,]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ , satisfying the Jacobi identity :

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0,\ \ \forall\,x,y,z\in\mathfrak{g}.

The element $[x,y]\in\mathfrak{g}$ is referred to as the Lie bracket (or the Lie product) of $x$ and $y$ in $\mathfrak{g}$ .

Definition 2.2 .

A negation map on a $\mathcal{T}$ -module $\mathcal{A}$ is a semigroup isomorphism $(-):\mathcal{A}\to\mathcal{A}$ of order $\leq 2,$ written $a\mapsto(-)a$ , together with a map $(-)$ of order $\leq 2$ on $\mathcal{T}$ which also respects the $\mathcal{T}$ -action in the sense that

((-)a)b=(-)(ab)=a((-)b)

for $a\in\mathcal{T},$ $b\in\mathcal{A}.$

Definition 3.6 .

A (systemic) Morita context is a six-tuple $(\mathcal{A},\mathcal{A}^{\prime},\mathcal{M},\mathcal{M}^{\prime},\tau,\tau^{% \prime})$ where $\mathcal{A},\mathcal{A}^{\prime}$ are systems, $\mathcal{M}$ is an $\mathcal{A}-\mathcal{A}^{\prime}$ bimodule, $\mathcal{M}^{\prime}$ is an $\mathcal{A}^{\prime}-\mathcal{A}$ bimodule, and

\tau:\mathcal{M}\otimes_{\mathcal{A}^{\prime}}\mathcal{M}^{\prime}\to\mathcal{% A},\qquad\tau^{\prime}:\mathcal{M}^{\prime}\otimes_{\mathcal{A}}\mathcal{M}\to% \mathcal{A}^{\prime}

are homomorphisms, linear on each side over ${\mathcal{A}}$ and ${\mathcal{A}^{\prime}}$ respectively, which satisfy the following equations, writing $(x,x^{\prime})$ for $\tau(x,x^{\prime})$ and $[x^{\prime},x]$ for $\tau(x^{\prime},x)$ :

(i)

$(x,x^{\prime})y=x[x^{\prime},y].$
(ii)

$x^{\prime}(x,y^{\prime})=[x^{\prime},x]y^{\prime}.$

Definition 2.1 .

Let $\mathbb{F}$ be a field and $T$ an $\mathbb{F}$ -vector space endowed with a symplectic form $(\,,\,)\colon T\times T\to\mathbb{F}$ , 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 $x,y,z,u,v,w\in T$ .

Definition V.1 .

A fuzzy vector $u\in\mathcal{F}^{n}$ is skew if it cannot be written as the sum of a fuzzy vector and a non-trivial symmetric fuzzy vector; that is, if

u=v\oplus w

for some $v\in\mathcal{F}^{n}$ and $w\in\mathcal{F}^{n}_{\sf s}$ , then $w=\tilde{0}$ .

Definition V.2 .

A fuzzy vector $v\in\mathcal{F}^{n}$ is a Mareš core of a fuzzy vector $u\in\mathcal{F}^{n}$ if $v$ is skew and

u=v\oplus w

for some symmetric fuzzy vector $w\in\mathcal{F}^{n}_{\sf s}$ .

Definition VI.1 .

Fuzzy vectors $u,v\in\mathcal{F}^{n}$ are Mareš equivalent , denoted by $u\sim_{M}v$ , if there exist symmetric fuzzy vectors $w,w^{\prime}\in\mathcal{F}^{n}_{\sf s}$ such that

u\oplus w=v\oplus w^{\prime}.

Definition 3.13 .

An element $a\in L$ is said to have the left CL-property if for all $y\in C_{L}(x)$ and $x\in L$ ,

	$\displaystyle[[x,a],y]=0,$
	$\displaystyle[[a,x],y]=0.$

An element $a\in L$ is said to have the right CL-property if for all $y\in C_{L}(x)$ and $x\in L$ , we have

	$\displaystyle[[x,a],y]=0,$
	$\displaystyle[y,[a,x]]=0.$

Definition 3.1

A cone is a commutative monoid $C$ together with a scalar multiplication by nonnegative real numbers satisfying the same axioms as for vector spaces; that is, $C$ is endowed with an addition $(x,y)\mapsto x+y\colon C\times C\to C$ which is associative, commutative and admits a neutral element $0$ , and with a scalar multiplication $(r,x)\mapsto r\cdot x\colon\mathbb{R}_{+}\times C\to C$ satisfying the following axioms for all $x,y\in C$ and all $r,s\in\overline{\mathbb{R}}_{+}$ :

	$\displaystyle r\cdot(x+y)=r\cdot x+r\cdot y$	$\displaystyle(rs)\cdot x=r\cdot(s\cdot x)$	$\displaystyle 0\cdot x=0$
	$\displaystyle(r+s)\cdot x=r\cdot x+s\cdot x$	$\displaystyle 1\cdot x=x$	$\displaystyle r\cdot 0=0$

We shall often write $rx$ instead of $r\cdot x$ for $r\in\mathbb{R}_{+}$ and $x\in C$ .

A semitopological cone is a cone with a $T_{0}$ topology that makes $+$ and $\cdot$ separately continuous.

A topological cone is a cone with a $T_{0}$ topology that makes $+$ and $\cdot$ jointly continuous.

Definition 16 .

Let $T_{0}$ denote the tree consisting of a single vertex. In the following four cases we define a partial block $b$ , its length which we denote $\mathrm{length}(b)$ , and its $R_{m}$ -expression $R_{m}(b)$ . The expression $R_{m}(b)$ will be a monomial in $x_{i}$ and $(-\frac{a_{i}}{a_{m}})^{\pm 1}$ . We define $b$ to be:

1. A triple of integers

b=(i,k,h)

where $1\leq h\leq m-1$ ; $2\leq k\leq h+1$ ; and $h-(k-2)\leq i\leq m-1$ . Define

\mathrm{length}(b)=k

and

R_{m}(b)=x_{i}(-\frac{a_{m-1}}{a_{m}})^{k-2-h}(-\frac{a_{m-1-h}}{a_{m}}).

2. The triple

b=(T_{0},k,k-1)

where $k\geq 2$ . Define

\mathrm{length}(b)=k

and

R_{m}(b)=-\frac{a_{m-1-(k-1)}}{a_{m}}.

3. For $0\leq i\leq m-1$ , the $1$ -tuple

b=(i).

Define

\mathrm{length}(b)=1

and

R_{m}(b)=x_{i}.

4. The 1-tuple

b=(T_{0}).

Define

\mathrm{length}(b)=1

and

R_{m}(b)=-\frac{a_{m-1}}{a_{m}}.

Definition 2.6 .

For a $3$ -dimensional terminal quotient singularity $\mathsf{p}\in V$ of type $\frac{1}{r}(1,a,r-a)$ , we define

\operatorname{wp}(\mathsf{p})=a(r-a)

and call it the weight product of $\mathsf{p}\in V$ .

Definition C.1 .

A function $f:{\mathbb{R}^{d}}\to\mathbb{R}^{m}$ is called positively homogeneous, if for all $\alpha>0$ and all $x\in{\mathbb{R}^{d}}$ we have

f(\alpha x)=\alpha f(x)\,.

Definition 1 (K-cooperativity)

Let $\mathcal{K}\subseteq\mathbb{R}^{n}$ be a pointed, solid, convex cone. A system $\dot{x}=f(x)$ is (strictly) K-cooperative if any trajectory $(x(\cdot),\delta x(\cdot))$ of the prolonged system

\dot{x}=f(x)\qquad\dot{\delta x}=\partial f(x)\delta x,

(4)

satisfies:

0\neq\delta x(0)\in\mathcal{K}\implies\delta x(t)\in\mbox{int}(\mathcal{K})% \mbox{ for }t>0\ .\vspace{1mm}

(5)

Definition 8

We say that $(\omega,\phi)$ are continuous if for all states $s:\mathbb{N}\to\{\ast\}+\mathbb{N}$ (which encode $M,f$ ) there exists some $L$ such that for any other input state $s^{\prime}$ , if $[{s}]({L})=[{s^{\prime}}]({L})$ then

(\omega,\phi)(s)=(\omega,\phi)(s^{\prime}).

Definition 4.1 (Weak Solution) .

A weak solution of equation ( 5 ) with the initial condition $\rho_{0}\in\mathcal{M}^{+}([0,1))$ and the boundary condition $\mu\in\mathcal{M}^{+}((0,T])$ is a function $\rho:[0,T]\to\mathcal{M}^{+}([0,1))$ , such that $W\colon[0,T]\mapsto\rho_{t}([0,1))$ is integrable and such that for every $\tau\in[0,T]$ and for every $\varphi\in\Psi$ that satisfies

\varphi(t,1)=0,\mbox{ for all }t\in[0,\tau],

(37)

one has

		$\displaystyle\int_{(0,\tau]}\int_{[0,1)}\left(\partial_{t}\varphi(t,x)+\alpha(% W(t))\partial_{x}\varphi(t,x)\right)\,d\rho_{t}(x)\,dt+\int_{(0,\tau]}\varphi(% t,0)\,d\mu(t)$
		$\displaystyle-\int_{[0,1)}\varphi(\tau,x)\,d\rho_{\tau}(x)+\int_{[0,1)}\varphi% (0,x)\,d\rho_{0}(x)=0.$		(38)

Definition 2.1 .

For any non-degenerate quadratic space $(V,q)$ of dimension $m$ over $F$ , put

(v,v^{\prime})=\tfrac{1}{2}(q(v+v^{\prime})-q(v)-q(v^{\prime})).

(2.2)

Then we define the determinant $\det(V)$ and the discriminant $\operatorname{disc}(V)$ of $(V,q)$ by

	$\displaystyle\det(V)$	$\displaystyle=\det((a_{i},a_{j})_{i,j})\ \left(\mathrm{mod}(F^{\times})^{2}% \right),$
	$\displaystyle\operatorname{disc}(V)$	$\displaystyle=(-1)^{\frac{(m-1)m}{2}}\det(V)\ \left(\mathrm{mod}(F^{\times})^{% 2}\right),$

where $\{a_{i}\}$ is a basis of $V$ . These are elements of $F^{\times}/(F^{\times})^{2}$ and independent of the choice of a basis.

Definition 3.5 .

Let $\mathcal{A}\subseteq\partial X$ and $\mathcal{B}\subseteq\partial Y$ be subsets. A map $f\colon\mathcal{A}\rightarrow\mathcal{B}$ is Möbius if ${f^{4}(\mathscr{A}(X)\cap\mathcal{A}^{4})\subseteq\mathscr{A}(Y)}$ and, for each $(x,y,z,w)\in\mathscr{A}(X)\cap\mathcal{A}^{4}$ , we have

\mathrm{cr}(f(x),f(y),f(z),f(w))=\mathrm{cr}(x,y,z,w).

The latter happens if and only if $\mathrm{crt}(f(x),f(y),f(z),f(w))=\mathrm{crt}(x,y,z,w)$ for all 4-tuples ${(x,y,z,w)\in\mathscr{A}(X)\cap\mathcal{A}^{4}}$ .

Definition 2.4 .

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra. Two representations $(V,\beta,\rho)$ and $(V^{\prime},\beta^{\prime},\rho^{\prime})$ are called equivalent if there exists a linear isomorphism $\varphi:V\rightarrow V^{\prime}$ such that

(2.4)

\varphi\rho(x)=\rho^{\prime}(x)\varphi,\;\;\beta^{\prime}\varphi(x)=\varphi% \beta(x),\forall x\in\mathfrak{g}.

Definition 1

A metric $d:\mathbb{F}_{2}^{n}\times\mathbb{F}_{2}^{n}\to\mathbb{R}$ is said to be translation-invariant if

d(\mathbf{x}+\mathbf{z},\mathbf{y}+\mathbf{z})=d(\mathbf{x},\mathbf{y})

for every $\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{F}_{2}^{n}$ .

Definiton 2.3 .

A solution of $\tau_{\alpha,p}u=\lambda u$ which satisfies at a point $l\in[a,b]\backslash\{p\}$ the boundary condition

u(l)cos\theta+u^{\prime}(l)sen\theta=0,\qquad\theta\in[0,\pi)

will be denoted by $u_{l,\alpha}(\lambda)$ .

Definition 2.1 .

${}^{\@@cite[cite]{[\@@bibref{}{Scheunert}{}{}]}}$ Let $\Gamma$ be an abelian group. A bi-character on $\Gamma$ is a map $\epsilon:\Gamma\times\Gamma\rightarrow\mathbb{K}\backslash\{0\}$ satisfying

	$\displaystyle(1)~{}~{}\epsilon(\alpha,\beta)\epsilon(\beta,\alpha)=1,$
	$\displaystyle(2)~{}~{}\epsilon(\alpha,\beta+\gamma)=\epsilon(\alpha,\beta)% \epsilon(\alpha,\gamma),$
	$\displaystyle(3)~{}~{}\epsilon(\alpha+\beta,\gamma)=\epsilon(\alpha,\gamma)% \epsilon(\beta,\gamma),$

for any $\alpha,\beta\in\Gamma$ .

Definition 2.2 .

${}^{\@@cite[cite]{[\@@bibref{}{Scheunert}{}{}]}}$ A Lie color algebra is a triple $(L,[\cdot,\cdot],\varepsilon)$ consisting of a $\Gamma$ -graded space $L$ , an even bilinear mapping $[\cdot,\cdot]:L\times L\rightarrow L$ , and a bi-character $\varepsilon$ on $\Gamma$ satisfying the following conditions,

	$\displaystyle[x,y]=-\varepsilon(x,y)[y,x],$
	$\displaystyle\varepsilon(z,x)[x,[y,z]]+\varepsilon(x,y)[y,[z,x]]+\varepsilon(y% ,z)[z,[x,y]]=0,$

for any $x,y,z\in L$ .

Definition .

Let $\hbox to 7.0pt{\hrulefill}\cdot\hbox to 7.0pt{\hrulefill}\colon\mathscr{F}_{n}% \times\partial\mathscr{F}_{n}\to\partial\mathscr{F}_{n}$ be the map given by

(4.10)

x\cdot[\omega]=[x\omega],

where $x\omega\colon\mathbb{N}\to\mathscr{F}_{n}$ is given by $(x\omega)(n)=x\omega(n)$

Definition 2.1 .

For a Borel probability measure $\mu$ , we denote its Lebesgue decomposition by

\mu=\mu^{\mathrm{pp}}+\mu^{\mathrm{sc}}+\mu^{\mathrm{ac}},

(2.1)

where $\mu^{\mathrm{pp}}$ , $\mu^{\mathrm{sc}}$ , and $\mu^{\mathrm{ac}}$ are the point mass, singular continuous, and absolutely continuous parts of $\mu$ , respectively.

Definition 2.2 (Moving Frame) .

Given a smooth left Lie group action $G\times M\rightarrow M$ , a moving frame on the domain $\mathcal{U}\subset M$ is an equivariant map $\rho:\mathcal{U}\rightarrow G$ , that is

\begin{array}[]{ll}\rho(g\cdot z)=g\rho(z)&\qquad\mbox{\emph{left equivariance% }}\\ \rho(g\cdot z)=\rho(z)g^{-1}&\qquad\mbox{\emph{right equivariance}}\end{array}

Definition 3

For a graph $G=(V,E)$ , let $f(k)=|\{v:d(v)\geq k\}|$ . The smooth ccdh of a graph $G$ is a function $\varphi:\mathbb{Z}_{\geq 1}\times[0,1)\to\mathbb{R}_{\geq 0}$ defined by

\varphi(x,\epsilon)=(1-\epsilon)f(x)+\epsilon f(x+1).

For ease of notation, we may write $\varphi(x)\coloneqq\varphi(\lfloor x\rfloor,x-\lfloor x\rfloor)$ . ${}_{\Box}$

Definition 1.1

( [ 4 ] ) A BiHom-associative algebra is a 4-tuple $\left(A,\mu,\alpha,\beta\right)$ , where $A$ is a linear space and $\alpha,\beta:A\rightarrow A$ and $\mu:A\otimes A\rightarrow A$ are linear maps such that $\alpha\circ\beta=\beta\circ\alpha$ , $\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y)$ , $\beta(x\cdot y)=\beta(x)\cdot\beta(y)$ and

\displaystyle\alpha(x)\cdot(y\cdot z)=(x\cdot y)\cdot\beta(z),

(1.1)

for all $x,y,z\in A$ . The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of $A$ and condition ( 1.1 ) is called the BiHom-associativity condition.

A morphism $f:(A,\mu_{A},\alpha_{A},\beta_{A})\rightarrow(B,\mu_{B},\alpha_{B},\beta_{B})$ of BiHom-associative algebras is a linear map $f:A\rightarrow B$ such that $\alpha_{B}\circ f=f\circ\alpha_{A}$ , $\beta_{B}\circ f=f\circ\beta_{A}$ and $f\circ\mu_{A}=\mu_{B}\circ(f\otimes f)$ .

Definition 1.4

( [ 10 ] ) A left BiHom-pre-Lie algebra is a 4-tuple $(A,\mu,\alpha,\beta)$ , where $A$ is a linear space and $\mu:A\otimes A\rightarrow A$ and $\alpha,\beta:A\rightarrow A$ are linear maps satisfying $\alpha\circ\beta=\beta\circ\alpha$ , $\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y)$ , $\beta(x\cdot y)=\beta(x)\cdot\beta(y)$ and

\displaystyle\alpha\beta(x)\cdot(\alpha(y)\cdot z)-(\beta(x)\cdot\alpha(y))% \cdot\beta(z)=\alpha\beta(y)\cdot(\alpha(x)\cdot z)-(\beta(y)\cdot\alpha(x))% \cdot\beta(z),

(1.3)

for all $x,y,z\in A$ . We call $\alpha$ and $\beta$ (in this order) the structure maps of $A$ .

A morphism $f:(A,\mu,\alpha,\beta)\rightarrow(A^{\prime},\mu^{\prime},\alpha^{\prime},% \beta^{\prime})$ of left BiHom-pre-Lie algebras is a linear map $f:A\rightarrow A^{\prime}$ satisfying $f(x\cdot y)=f(x)\cdot^{\prime}f(y)$ , for all $x,y\in A$ , as well as $f\circ\alpha=\alpha^{\prime}\circ f$ and $f\circ\beta=\beta^{\prime}\circ f$ .

Definition 1.5

( [ 10 ] ) A left (respectively right) BiHom-Leibniz algebra is a 4-tuple $(L,[\cdot,\cdot],\alpha,\beta)$ , where $L$ is a linear space, $[\cdot,\cdot]:L\times L\rightarrow L$ is a bilinear map and $\alpha,\beta:L\rightarrow L$ are linear maps satisfying $\alpha\circ\beta=\beta\circ\alpha$ , $\alpha([x,y])=[\alpha(x),\alpha(y)]$ , $\beta([x,y])=[\beta(x),\beta(y)]$ and

\displaystyle[\alpha\beta(x),[y,z]]=[[\beta(x),y],\beta(z)]+[\beta(y),[\alpha(% x),z]],

(1.4)

respectively

\displaystyle[[x,y],\alpha\beta(z)]=[[x,\beta(z)],\alpha(y)]+[\alpha(x),[y,% \alpha(z)]],

(1.5)

for all $x,y,z\in L$ . We call $\alpha$ and $\beta$ (in this order) the structure maps of $L$ .

Definition 1.7

( [ 9 ] ) A BiHom-dendriform algebra is a 5-tuple $(A,\prec,\succ,\alpha,\beta)$ consisting of a linear space $A$ and linear maps $\prec,\succ:A\otimes A\rightarrow A$ and $\alpha,\beta:A\rightarrow A$ such that, for all $x,y,z\in A$ :

	$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$		(1.7)
	$\displaystyle\alpha(x\prec y)=\alpha(x)\prec\alpha(y),~{}\alpha(x\succ y)=% \alpha(x)\succ\alpha(y),$		(1.8)
	$\displaystyle\beta(x\prec y)=\beta(x)\prec\beta(y),~{}\beta(x\succ y)=\beta(x)% \succ\beta(y),$		(1.9)
	$\displaystyle(x\prec y)\prec\beta(z)=\alpha(x)\prec(y\prec z+y\succ z),$		(1.10)
	$\displaystyle(x\succ y)\prec\beta(z)=\alpha(x)\succ(y\prec z),$		(1.11)
	$\displaystyle\alpha(x)\succ(y\succ z)=(x\prec y+x\succ y)\succ\beta(z).$		(1.12)

We call $\alpha$ and $\beta$ (in this order) the structure maps of $A$ .

Definition 1.12

( [ 13 ] , [ 14 ] ) A Novikov-Poisson algebra is a triple $(A,\cdot,*)$ such that $(A,\cdot)$ is a commutative associative algebra, $(A,*)$ is a Novikov algebra and the following compatibility conditions hold, for all $x,y,z\in A$ :

	$\displaystyle(xy)\cdot z-x(y\cdot z)=(yx)\cdot z-y(x\cdot z),$		(1.17)
	$\displaystyle(x\cdot y)z=(xz)\cdot y.$		(1.18)

A morphism of Novikov-Poisson algebras from $(A,\cdot,*)$ to $(A^{\prime},\cdot^{\prime},*^{\prime})$ is a linear map $f:A\rightarrow A^{\prime}$ satisfying $f(x\cdot y)=f(x)\cdot^{\prime}f(y)$ and $f(x*y)=f(x)*^{\prime}f(y)$ , for all $x,y\in A$ .

Definition 2.1

A BiHom-Novikov algebra is a 4-tuple $(A,\mu,\alpha,\beta)$ , where $A$ is a linear space, $\mu:A\otimes A\rightarrow A$ is a linear map and $\alpha,\beta:A\rightarrow A$ are commuting linear maps (called the structure maps of $A$ ), satisfying the following conditions, for all $x,y,z\in A$ :

	$\displaystyle\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y),~{}~{}\beta(x\cdot y)=% \beta(x)\cdot\beta(y),$		(2.1)
	$\displaystyle(\beta(x)\cdot\alpha(y))\cdot\beta(z)-\alpha\beta(x)\cdot(\alpha(% y)\cdot z)=(\beta(y)\cdot\alpha(x))\cdot\beta(z)-\alpha\beta(y)\cdot(\alpha(x)% \cdot z),$		(2.2)
	$\displaystyle(x\cdot\beta(y))\cdot\alpha\beta(z)=(x\cdot\beta(z))\cdot\alpha% \beta(y).$		(2.3)

In other words, a BiHom-Novikov algebra is a left BiHom-pre-Lie algebra satisfying ( 2.3 ).

A morphism $f:(A,\mu_{A},\alpha_{A},\beta_{A})\rightarrow(B,\mu_{B},\alpha_{B},\beta_{B})$ of BiHom-Novikov algebras is a linear map $f:A\rightarrow B$ such that $\alpha_{B}\circ f=f\circ\alpha_{A}$ , $\beta_{B}\circ f=f\circ\beta_{A}$ and $f\circ\mu_{A}=\mu_{B}\circ(f\otimes f)$ .

Definition 3.1

A BiHom-Novikov-Poisson algebra is a 5-tuple $(A,\cdot,\ast,\alpha,\beta)$ such that:

(1) $(A,\cdot,\alpha,\beta)$ is a BiHom-commutative algebra;

(2) $(A,\ast,\alpha,\beta)$ is a BiHom-Novikov algebra;

(3) the following compatibility conditions hold for all $x,y,z\in A$ :

	$\displaystyle(\beta(x)\ast\alpha(y))\cdot\beta(z)-\alpha\beta(x)\ast(\alpha(y)% \cdot z)=(\beta(y)\ast\alpha(x))\cdot\beta(z)-\alpha\beta(y)\ast(\alpha(x)% \cdot z),$		(3.1)
	$\displaystyle(x\cdot\beta(y))\ast\alpha\beta(z)=(x\ast\beta(z))\cdot\alpha% \beta(y)=\alpha(x)\cdot(\beta(y)\ast\alpha(z)).$		(3.2)

The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of $A$ .

A morphism $f:(A,\cdot,\ast,\alpha,\beta)\rightarrow(A^{\prime},\cdot^{\prime},\ast^{% \prime},\alpha^{\prime},\beta^{\prime})$ of BiHom-Novikov-Poisson algebras is a map that is a morphism of BiHom-associative algebras from $(A,\cdot,\alpha,\beta)$ to $(A^{\prime},\cdot^{\prime},\alpha^{\prime},\beta^{\prime})$ and a morphism of BiHom-Novikov algebras from $(A,\ast,\alpha,\beta)$ to $(A^{\prime},\ast^{\prime},\alpha^{\prime},\beta^{\prime})$ .

Definition 1 .

The steady-state input-output relation $k$ of a dynamical system is the collection of all steady-state input-output pairs of the system. Given a steady-state input $\mathrm{u}$ and a steady-state $\mathrm{y}$ , we define

\displaystyle k(\mathrm{u})=\{\mathrm{y}:\ (\mathrm{u,y})\in k\}\text{ and }\;% k^{-1}(\mathrm{y})=\{\mathrm{u}:\ (\mathrm{u,y})\in k\}.

Definition 6.2 .

For $w\geq 1,r\geq 2$ define

\alpha(w,r)=\sup\{\kappa:\exists\kappa\text{-regular }w\text{-set system % without }r\text{ pairwise disjoint sets}\}.

It will be convenient to shorthand $\beta(w)=\alpha(w,2)$ , which can equivalently be defined as

\beta(w)=\sup\{\kappa:\exists\kappa\text{-regular intersecting }w\text{-set % system}\}.

Definition 5.2 .

An algebra $(\mathfrak{a},*)$ is called left-symmetric if the following identity holds:

\xi*(\eta*\zeta)-(\xi*\eta)*\zeta=\eta*(\xi*\zeta)-(\eta*\xi)*\zeta,\quad\mbox% {for all }\xi,\eta,\zeta\in\mathfrak{a}.

(24)

Definition 1

Let $X$ be an open subset of $\mathbb{R}^{k}$ , $k\geq 1$ . A system

\dot{x}(t)=f(t,x(t))

(23)

with $x(t)\in X$ is called a quasi-autonomous system with limiting system

\dot{y}(t)=g(y(t))

(24)

if for any compact set $K\subset X$

\int_{t_{0}}^{\infty}\sup_{x(t)\in K}||f(t,x(t))-g(x(t))||dt<\infty.

(25)

${}_{\Box}$

Definition 1.1 .

Let $f:\mathcal{R}\to N$ be a map from a Riemann surface $\mathcal{R}$ into a Riemannian manifold $N$ . Moreover, let $\gamma$ and $\rho$ be elements of $\operatorname{Aut}(\mathcal{R})$ and $\operatorname{Iso}(N)$ , respectively. Then $f$ is symmetric with respect to $(\gamma,\rho)\in\operatorname{Aut}(\mathcal{R})\times\operatorname{Iso}(N)$ if

(1.1)

f\circ\gamma=\rho\circ f

holds.

Definition 6 .

A stochastic product on $\mathcal{S}(\mathcal{H})$ is a map $(\hskip 0.853583pt\cdot\hskip 0.853583pt)\odot(\hskip 0.853583pt\cdot\hskip 0.% 853583pt)\colon\mathcal{S}(\mathcal{H})\times\mathcal{S}(\mathcal{H})% \rightarrow\mathcal{S}(\mathcal{H})$ that is convex-linear in both its arguments, i.e.,

(\alpha\hskip 0.853583pt\rho+(1-\alpha)\sigma)\odot(\epsilon\tau+(1-\epsilon)% \upsilon)=\alpha\hskip 0.853583pt\epsilon\hskip 1.707165pt\rho\odot\tau+\alpha% (1-\epsilon)\hskip 1.707165pt\rho\odot\upsilon+(1-\alpha)\epsilon\hskip 1.7071% 65pt\sigma\odot\tau+(1-\alpha)(1-\epsilon)\hskip 1.707165pt\sigma\odot\upsilon% \hskip 0.853583pt,

(34)

for all $\rho,\sigma,\tau,\upsilon\in\mathcal{S}(\mathcal{H})$ and $\alpha,\epsilon\in[0,1]$ .