If $(\mathcal{C},\otimes,k,\sigma)$ is an additive symmetric monoidal category with an algebra modality $(S,\mu,\eta,\mathsf{m},\mathsf{u})$ , then a deriving transformation on $\mathcal{C}$ is a natural transformation $\mathsf{d}$ , with components

$$\mathsf{d}_{C}:SC\to SC\otimes C\;\;\;\;\;\;(C\in\mathcal{C})$$

such that ^{4 4} 4 Note that here, and throughout, we denote diagrammatic (left-to-right) composition by juxtaposition, whereas we denote right-to-left, non-diagrammatic composition by $\circ$ , and functions $f$ are applied on the left, parenthesized as in $f(x)$ ; however, we write composition of functors in the right-to-left, non-diagrammatic order, and functors $F$ are applied on the left, as in $FX$ . We suppress the use of the monoidal category associator and unitor isomorphisms, and we omit subscripts and whiskering on the right.

1. [d.1]

   Derivative of a constant : $\mathsf{u}\mathsf{d}=0$ ;

2. [d.2]

   Leibniz/product rule : $\mathsf{m}\mathsf{d}=[(1\otimes\mathsf{d})(\mathsf{m}\otimes 1)]+[(\mathsf{d}\otimes 1)(1\otimes\sigma)(\mathsf{m}\otimes 1)]$ ;

3. [d.3]

   Derivative of a linear function : $\eta\mathsf{d}=\mathsf{u}\otimes 1$ ;

4. [d.4]

   Chain rule : $\mu\mathsf{d}=\mathsf{d}(\mu\otimes\mathsf{d})(\mathsf{m}\otimes 1)$ ;

5. [d.5]

   Interchange ^{5 5} 5 This rule was not in the original paper [ 4 ] , but was later formally introduced in [ 5 ] , and is used in [ 9 ] . It represents the independence of order of partial differentiation. : $\mathsf{d}(\mathsf{d}\otimes 1)=\mathsf{d}(\mathsf{d}\otimes 1)(1\otimes\sigma)$ .

Such a $\mathcal{C}$ equipped with a deriving transformation $\mathsf{d}$ is called a codifferential category .

If $(\mathcal{C},\otimes,k,\sigma)$ is an additive symmetric monoidal category with an algebra modality $(S,\mu,\eta,\mathsf{m},\mathsf{u})$ , then an integral transformation on $\mathcal{C}$ is a natural transformation $\mathsf{s}$ , with components

$$\mathsf{s}_{C}:SC\otimes C\to SC\;\;\;\;\;\;(C\in\mathcal{C})$$

such that

1. [s.1]

   Integral of a constant : $(\mathsf{u}\otimes 1)\mathsf{s}=\eta$ ;

2. [s.2]

   Rota-Baxter rule : $(\mathsf{s}\otimes\mathsf{s})\mathsf{m}=[(\mathsf{s}\otimes 1\otimes 1)(\mathsf{m}\otimes 1)\mathsf{s}]+[(1\otimes 1\otimes\mathsf{s})(1\otimes\sigma)(\mathsf{m}\otimes 1)\mathsf{s}$ ];

3. [s.3]

   Interchange : $(\mathsf{s}\otimes 1)\mathsf{s}=(1\otimes\sigma)(\mathsf{s}\otimes 1)\mathsf{s}$ .

Such a $\mathcal{C}$ equipped with an integral transformation $\mathsf{s}$ is called a co-integral category .

A BiHom-Poisson superalgebra $A$ is a BiHom-Lie superalgebra $(A,[\cdot,\cdot],\phi,\psi)$ endowed with a BiHom-associative superproduct, that is, a bilinear product denoted by juxtaposition such that

$\displaystyle\phi(x)(yz)=(xy)\psi(y),$

for all $x,y,z\in A$ , and such that the BiHom-Leibniz superidentity

$\displaystyle[xy,\phi\psi(z)]=\phi(x)[y,\phi(z)]+(-1)^{|j||k|}[x,\psi(z)]\phi(y)$

holds for any $x\in A_{\overline{i}},y\in A_{\overline{j}},z\in A_{\overline{k}}$ and $\overline{i},\overline{j},\overline{k}\in\mathbb{Z}_{2}$ .

Suppose that $d$ is the exterior derivative operator, we define the Laplace-Beltrami (or Laplacian) operator $\triangle:A^{p}(M)\longrightarrow A^{p}(M)$ by

$$\triangle(\omega)=(d\circ\delta+\delta\circ d)(\omega).$$

We say a $p$ -form is harmonic if it is in the kernel of $\triangle$ .

A hom-Lie algebra over an associative, commutative, and unital ring $R$ is a triple $(M,[\cdot,\cdot],\alpha)$ where $M$ is an $R$ -module, $\alpha\colon M\to M$ a linear map called the twisting map , and $[\cdot,\cdot]\colon M\times M\to M$ a map called the hom-Lie bracket , satisfying the following axioms for all $a,b,c\in M$ and $r,s\in R$ :

	$\displaystyle[ra+sb,c]=r[a,c]+s[b,c],\quad[a,rb+sc]=r[a,b]+s[a,c],\quad\text{(bilinearity)},$
	$\displaystyle=0,\quad\text{(alternativity)},$
	$\displaystyle\left[\alpha(a),[b,c]\right]+\left[\alpha(c),[a,b]\right]+\left[\alpha(b),[c,a]\right]=0,\quad\text{(hom-Jacobi identity)}.$

In a FTvN system $({\cal V},{\cal W},\lambda)$ we say that elements $x,y\in{\cal V}$ commute if

$$\langle x,y\rangle=\langle\lambda(x),\lambda(y)\rangle.$$

Feature transformations in DNN can be characterized as the first-order partial differential equation (PDE):

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

(1)

where $\dot{{\mathbf{x}}}=\frac{\partial\mathbf{x}}{\partial t}$ , $f:T\times X\rightarrow R^{d}$ , $T\subseteq R^{+}$ , and $X\subseteq R^{d}$ . $t\in T$ is the time along the feature transformations, in which $t\geq 0$ . $\mathbf{x}\in X$ is the feature vector of dimension $d$ .

A Euclidean kernel $k$ that is a kernel over $\mathbb B_{n}$ for any $n\geqslant 1$ , is said to be $L$ -Strongly Euclidean if $k(\mathbf{x}^{(1)},\mathbf{x}^{(2)})$ can be written as:

$\displaystyle k(\mathbf{x}^{(1)},\mathbf{x}^{(2)})=g({\langle\mathbf{x}^{(1)},\mathbf{x}^{(2)}\rangle})$

(3.1)

and $g$ is $L$ -Lipschitz over the domain $[0,n]$ .

Let $\alpha=\frac{a}{b}\in\mathbb{Q}$ with $\gcd(a,b)=1$ and $b>0$ . Then we define the denominator of $\alpha$ as

$$\operatorname{denom}\alpha=b.$$

An almost contact structure on a $(2n+1)$ -dimensional smooth manifold $M$ is a triple $(\phi,\xi,\eta)$ , where $\phi$ is a $(1,1)$ -type tensor field, $\xi$ is a global vector field and $\eta$ a 1-form, such that

(2.1)

$$\phi^{2}=-\mathrm{id}+\eta\otimes\xi,\quad\eta(\xi)=1,$$

where, $\mathrm{id}$ denotes the identity mapping, which imply that $\phi(\xi)=0$ , $\eta\circ\phi=0$ and $\mathrm{rank}(\phi)=2n$ . Generally, $\xi$ is called the characteristic vector field or the Reeb vector field.

If $(M,\phi,\xi,\eta,g)$ be an almost contact metric structure, then there is a well known deformation of contact forms which is named D-homothetic deformation and is defined by

$$\bar{\eta}=a\eta,\quad\bar{\phi}=\phi,\quad\bar{\xi}=\dfrac{1}{a}\xi,\quad\bar{g}=ag+a(a-1)\eta\otimes\eta,$$

where, $a$ is a positive constant.

Let $\mathcal{R}$ be a unique factorization domain with $a,b\in\mathcal{R}$ not both zero and consider an LDE ( 45 ) for $a,b$ . If the righthand side, $c$ , in ( 45 ) is zero,

(47)

$$ax+by=0.$$

then ( 47 ) is called the homogeneous LDE for $a,b$ . If the righthand side, $c$ , in ( 45 ) is nonzero then we call ( 45 ) an inhomogeneous LDE for $a,b$ with inhomogeneity $c$ .

Let $\mathcal{R}$ be a Euclidean domain whose size function satisfies $\sigma(0)=-\infty$ and the homomorphism property,

(59)

$$\sigma(ab)=\sigma(a)+\sigma(b).$$

Let $a,b,c\in\mathcal{R}$ with $a$ and $b$ not both zero. Let $h\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\gcd(a,b)$ and assume $h\,|\,c$ . For $a\neq 0$ a solution $(x,y)$ to

(60)

$$ax+by=c$$

will be called size-reducing in $a$ if

(61)

$$\sigma(y)<\sigma(a)-\sigma(h).$$

Similarly, a solution is called size-reducing in $b\neq 0$ if

(62)

$$\sigma(x)<\sigma(b)-\sigma(h).$$

A 3-Hom-pre-Lie algebra is a triple $(L,\{\cdot,\cdot,\cdot\},\alpha)$ consisting of a vector space $L$ , with a linear map $\{\cdot,\cdot,\cdot\}:L\otimes L\otimes L\rightarrow L$ and a linear map $\alpha:L\rightarrow L$ satisfying

			$\displaystyle\{x,y,z\}=-\{y,x,z\},$		(3.3)
	$\displaystyle\{\alpha(x),\alpha(y),\{z,u,w\}\}$	$\displaystyle=$	$\displaystyle\{[x,y,z]_{L},\alpha(u),\alpha(v)\}+\{\alpha(z),[x,y,u]_{L},\alpha(v)\}$		(3.4)
			$\displaystyle+\{\alpha(z),\alpha(u),[x,y,v]_{L}\},$
	$\displaystyle\{[x,y,z]_{L},\alpha(u),\alpha(v)\}$	$\displaystyle=$	$\displaystyle\{\alpha(x),\alpha(y),[z,u,v]_{L}\}+\{\alpha(y),\alpha(z),[x,u,v]_{L}\}$		(3.5)
			$\displaystyle+\{\alpha(z),\alpha(x),[y,u,v]_{L}\},$

for any $x,y,z,u,v\in L$ and $[\cdot,\cdot,\cdot]_{C}$ is defined by

$\displaystyle[x,y,z]_{C}=\{x,y,z\}+\{y,z,x\}+\{z,x,y\}.$

(3.6)

( [ 19 ] ) Let $(L,[\cdot,\cdot,\cdot],\alpha)$ be 3-Hom-Lie algebra and $r\in L\otimes L$ . The equation

$\displaystyle[[r,r,r]]=0$

is called the 3-Lie classical Hom-Yang-Baxter equation.

A symplectic structure on a regular 3-Hom-Lie algebra $(L,[\cdot,\cdot,\cdot],\alpha)$ is a nondegenerate skew-symmetric bilinear form $\omega\in L^{\ast}\wedge L^{\ast}$ such that $\omega\circ\alpha=\omega$ satisfying the following equality

$\displaystyle\omega([x,y,z],\alpha(w))-\omega([y,z,w],\alpha(x))+\omega([z,w,x],\alpha(y))-\omega([w,x,y],\alpha(z))=0,$

(5.1)

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

Let $\rho$ and $\theta$ be real-valued functions on an $n$ -dimensional manifold $M$ with $\rho>0$ . The Madelung transform is the mapping $\Phi:(\rho,\theta)\mapsto\psi$ defined by

$$\psi=\sqrt{\rho e^{i\theta}}.$$

(8)

A lattice is an algebra $\mathbf{L}:=({\cal L},{\cal F})$ where ${\cal F}$ contains two binary operations $\vee$ and $\wedge$ (read “join” and “meet” respectively) on ${\cal L}$ that satisfy the following axiomatic identities for all $x,y,z\in{\cal L}$ ,

1. (LA-1)

   commutative laws: $x\vee y=y\vee x$ , $x\wedge y=y\wedge x$ ;

2. (LA-2)

   associative laws: $x\vee(y\vee z)=(x\vee y)\vee z$ , $x\wedge(y\wedge z)=(x\wedge y)\wedge z$ ;

3. (LA-3)

   absorption laws: $x=x\vee(x\wedge y)$ , $x=x\wedge(x\vee y)$ .