A Courant algebroid consists of the following: a vector bundle $V\to X$ over a smooth manifold, $X$ , a bilinear operator $[,]:\Gamma(V)\otimes\Gamma(V)\to\Gamma(V)$ on the space of sections of $V$ , a non-degenerate bilinear form, $\langle,\rangle$ on $V$ and an anchor map $\rho:V\to TX$ . The data $(V,\rho,[,],\langle,\rangle)$ is called a “Courant algebroid” if the conditions below hold for all $a,b,c\in\Gamma(V)$ . A Courant algebroid is called “regular” if the anchor map is of constant rank. The bracket $[,]$ can be either symmetric or skew-symmetric. ^{9 9} 9 Referred to in the literature as the “Dorfman” or “Courant” bracket respectively, though both can arise as Courant brackets above.

• •

  $[a,[b,c]]=[[a,b],c]+[b,[a,c]]$ ,

• •

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

• •

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

${}_{\blacksquare}$

The tetrad and anti-tetrad are respectively these quadrilinear operations on associative algebras:

A Weak system consists of parameters $N,s,\eta,\zeta$ , an $m$ -by- $N$ measurement matrix $\bm{\Phi}$ , and a decoding algorithm $\mathcal{D}$ , that satisfy the following property:

For any ${\mathbf{x}}\in\mathbb{R}^{N}$ that can be written as ${\mathbf{x}}=\mathbf{y}+\mathbf{z}$ , where $|\operatorname{supp}(\mathbf{y})|\leq s$ and $\left\|\mathbf{z}\right\|_{1}\leq 3/2$ , given the measurements $\bm{\Phi}{\mathbf{x}}$ and a subset $I\subseteq[N]$ such that $|I\cap\operatorname{supp}(\mathbf{y})|\geq(1-\zeta/2)|\operatorname{supp}(\mathbf{y})|$ , the decoding algorithm $\mathcal{D}$ returns $\widehat{\mathbf{x}}$ , such that ${\mathbf{x}}$ admits the following decomposition:

where $|\operatorname{supp}(\widehat{{\mathbf{x}}})|=O(s)$ , $|\operatorname{supp}(\widehat{\mathbf{y}})|\leq\zeta s$ , and $\left\|\widehat{\mathbf{z}}\right\|_{1}\leq\left\|\mathbf{z}\right\|_{1}+\eta$ . Intuitively, $\mathbf{\widehat{y}}$ and $\mathbf{\widehat{z}}$ will be the head and the tail of the residual ${\mathbf{x}}-\widehat{{\mathbf{x}}}$ , respectively.

Given $\boldsymbol{\omega}\in\wedge^{n}V,\boldsymbol{\omega}\not=\mathbf{0},$ there exists a unique linear map $\ast:\wedge^{k}V\to\wedge^{n-k}V$ that verifies the condition

for every $\mathbf{x},\mathbf{y}\in\wedge^{k}V.$

Let $x,y\in\Omega$ . Let $g:\Omega\to[0,1]$ be a function. We define the operation $\boxtimes$ as

with the $\cdot$ operation as specified in Def. 4.4 .

Assume that $\Omega\subset\mathbb{R}^{n}$ and $u\in C^{2}(\Omega;\mathbb{R}^{m})\cap L^{\infty}(\Omega;\mathbb{R}^{m})$ , are symmetric

Then $u$ is said to be a symmetric minimizer if for each bounded open symmetric lipschitz set $\Omega^{\prime}\subset\Omega$ and for each symmetric $v\in W_{0}^{1,2}(\Omega^{\prime};\mathbb{R}^{m})$ it results

A Poisson algebra is a commutative algebra $A$ equipped with a Lie bracket $\{\cdot,\cdot\}$ such that

for any $a,b,c\in A$ .

Let $R$ be a commutative ring with identity, $A$ a commutative $R$ -algebra and $L$ a Lie algebra over $R$ . The pair $(A,L)$ is called a Lie-Rinehart algebra over $A$ if $L$ is a left $A$ -module and there is an anchor map $\rho:L\rightarrow\operatorname{Der}_{R}(A)$ , which is an $A$ -module and a Lie algebra morphism, such that the following relation is satisfied,

for any $a\in A$ and $\xi,\zeta\in L$ .

A Lie coalgebra $L$ is a vector space with a $k$ -linear map $\delta:L\rightarrow\operatorname{Asym}(L\otimes L)$ such that

where $\xi:L^{\otimes 3}\rightarrow L^{\otimes 3}$ is the $k$ -linear map induced by the cyclic permutation $x\otimes y\otimes z\mapsto y\otimes z\otimes x$ .

A Lie bialgebra $L$ is a Lie algebra with a Lie coalgebra structure given by a cobracket $\delta$ such that

where

The parametrized inner-segment length function $\chi:\bm{\mathbbm{R}}\rightarrow[0,\infty)$ is defined by

A reductive decomposition [ 18 ] is a decomposition

such that

(i) A Hom-associative algebra is a triple $(A,\mu,\alpha)$ , in which $A$ is a linear space, $\alpha:A\rightarrow A$ and $\mu:A\otimes A\rightarrow A$ are linear maps, with notation $\mu(a\otimes a^{\prime})=aa^{\prime}$ , satisfying the following conditions, for all $a,a^{\prime},a^{\prime\prime}\in A$ :

We call $\alpha$ the structure map of $A$ .

A morphism $f:(A,\mu_{A},\alpha_{A})\rightarrow(B,\mu_{B},\alpha_{B})$ of Hom-associative algebras is a linear map $f:A\rightarrow B$ such that $\alpha_{B}\circ f=f\circ\alpha_{A}$ and $f\circ\mu_{A}=\mu_{B}\circ(f\otimes f)$ .
(ii) A Hom-coassociative coalgebra is a triple $(C,\Delta,\alpha)$ , in which $C$ is a linear space, $\alpha:C\rightarrow C$ and $\Delta:C\rightarrow C\otimes C$ are linear maps, satisfying the following conditions:

A morphism $g:(C,\Delta_{C},\alpha_{C})\rightarrow(D,\Delta_{D},\alpha_{D})$ of Hom-coassociative coalgebras is a linear map $g:C\rightarrow D$ such that $\alpha_{D}\circ g=g\circ\alpha_{C}$ and $(g\otimes g)\circ\Delta_{C}=\Delta_{D}\circ g$ .

Let $\mathcal{H}$ be a measurable $G$ -Hilbert bundle with bundle map $p:{\mathcal{H}}\rightarrow G^{(0)}$ . We define its fixed-point bundle as

A qualgebra $(Q,\lhd,\diamond)$ is called associative if the operation $\diamond$ is such, i.e., if for all elements of $Q$ one has

For a squandle $Q$ , a ( $\mathbb{Z}$ -valued) squandle $2$ -cocycle of $Q$ is a pair of maps $\chi:Q\times Q\rightarrow\mathbb{Z}$ , $\lambda:Q\rightarrow\mathbb{Z}$ satisfying Axioms ( 3 )-( 4 ) together with two additional ones:

The Abelian group of all squandle $2$ -cocycles of $Q$ is denoted by $Z^{2}(Q)$ .