Low-Rank RNN Adaptation for Context-Aware Language Modeling

Definition 4.8 .

Let $F$ be the set of reference frame configurations. We assume that there is some fixed map $\epsilon:F\to C$ , , which allows one to choose a particular element of the reference frame channel corresponding to one’s own reference frame. ⁵ ⁵ 5 For instance, in Example 4.4 , one could choose the orientation such that the $x$ -, $y$ - and $z$ -axes of the object are lined up with those of one’s own Cartesian frame. We also assume that this map is natural under a change in reference frame:

g\cdot\epsilon(f)=\epsilon(g\cdot f)

(6)

Definition C.11 .

Let $X$ be a compact Kähler manifold, let $\varphi\in\mathcal{D}^{\prime 1,1}(X)$ , we say $\varphi$ has $C^{\alpha}$ primitive if for some(hence any) smooth representative $\psi$ of the cohomology class of $\varphi$ , there is a $C^{\alpha}$ function $f$ such that

\varphi=\psi+\mathrm{d}\mathrm{d}^{\mathrm{c}}f.

Definition 2.1 .

Let $\lambda$ be a positive real number and $\rho:\mathds{R}\rightarrow\mathsf{Aut}(\mathsf{H}_{3})$ the Lie group homomorphism given by

\rho(t)(s,z)=(s,e^{it\lambda}z).

The real $\lambda$ -oscillator group of dimension $4$ , denoted by $O_{\lambda}$ , is the semidirect product of Heisenberg Lie group $\mathsf{H}_{3}$ for the group $(\mathds{R},+)$ by the representation $\rho.$

Definition 2.3 .

Given a combinatorial map $(\sigma,\alpha)$ acting on $H$ , we call $\gamma$ an automorphism if it is a permutation acting on $H$ such that $\left.\gamma\right|_{H^{\partial}}={\rm Id}_{H^{\partial}}$ and

\sigma=\gamma\sigma\gamma^{-1},\ \ \ \ \ \ \alpha=\gamma\alpha\gamma^{-1}.

Definition 6.13 .

Let $k$ be a commutative ring with unit and let $A$ be a commutative $k$ -algebra. The $A$ -module of Kähler differentials of $A$ over $k$ is the $A$ -module generated by elements $d(a)$ for $a\in A$ subject to the relations that $d$ is $k$ -linear and satisfies the Leibniz rule:

d(ab)=d(a)b+ad(b).

This $A$ -module is denoted by $\Omega^{1}_{A|k}$ .

Definition 1 (Hom-associative algebra) .

A hom-associative algebra over an associative, commutative, and unital ring $R$ , is a triple $(M,\cdot,\alpha)$ consisting of an $R$ -module $M$ , a binary operation $\cdot\colon M\times M\to M$ linear in both arguments, and a linear map $\alpha\colon M\to M$ satisfying, for all $a,b,c\in M$ ,

\alpha(a)\cdot(b\cdot c)=(a\cdot b)\cdot\alpha(c).

Definition 1.3 .

Let $(\mathcal{A},\mu,\alpha)$ and $(\mathcal{A}^{\prime},\mu^{\prime},\alpha^{\prime})~{}~{}(\hbox{resp.}~{}(% \mathcal{L},[\cdot,\cdot],\alpha)~{}\hbox{and}~{}(\mathcal{L}^{\prime},[\cdot,% \cdot]^{\prime},\alpha^{\prime}))$ be two Hom-associative $(\hbox{resp. ~{}Hom-Lie })$ algebras. A linear map $\phi:\mathcal{A}\rightarrow\mathcal{A}^{\prime}$ (resp. $\phi:\mathcal{L}\rightarrow\mathcal{L}^{\prime}$ ) is a Hom-associative (resp. Hom-Lie) algebra morphism if

\mu^{\prime}\circ(\phi\otimes\phi)=\phi\circ\mu~{}~{}(\hbox{resp. }[\cdot,% \cdot]^{\prime}\circ(\phi\otimes\phi)=\phi\circ[\cdot,\cdot])\quad\hbox{and}% \quad\phi\circ\alpha=\alpha^{\prime}\circ\phi.

Definition 1.5 .

Let $(\mathcal{A},\mu,\alpha)$ be a Hom-associative algebra. A (left) $\mathcal{A}$ -module is a triple $(M,f,\gamma)$ where $M$ is $\mathbb{K}$ -vector space and $f:M\rightarrow M$ , $\gamma:\mathcal{A}\otimes M\rightarrow M$ , are $\mathbb{K}$ -linear maps, such that the following identity is satisfied

\gamma\circ(\mu\otimes f)=\gamma\circ(\alpha\otimes\gamma).

Definition 10 (Satisfaction Indicator) .

[x\in\phi]\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{def}}}}{{=}}}1% \text{ if }x\in\phi\text{ else }0

(0.12)

Definition 6.9 .

(cf. [ CQ95 , Section 8] , [ Mes14 , Section 5] ) An $\Omega$ -connection on a right operator $B$ -module $Z$ is a completely bounded $A$ -linear map $\nabla:Z\to Z\otimes^{\mathrm{h}}_{B}\Omega$ satisfying

\nabla(zb)=\nabla(z)b+z\otimes d(b)

for all $b\in B$ and $z\in Z$ . The curvature of a connection $\nabla$ is the composite map

Z\xrightarrow{\nabla}Z\otimes^{\mathrm{h}}_{B}\Omega\xrightarrow{\mathrm{id}_{% Z}\otimes d^{1}-\nabla\otimes\mathrm{id}_{\Omega}}Z\otimes^{\mathrm{h}}_{B}% \Omega\otimes^{\mathrm{h}}_{B}\Omega,

and a flat connection is one whose curvature is zero. If $Z$ is a right Hilbert $C^{*}$ -module over $B$ , then a connection $\nabla:Z\to Z\otimes^{\mathrm{h}}_{B}\Omega$ is called Hermitian if it satisfies

(6.10)

\langle z_{1}|\nabla(z_{2})\rangle_{\Omega}-\langle z_{2}|\nabla(z_{1})\rangle% _{\Omega}^{*}=d(\langle z_{1}|z_{2}\rangle)

for all $z_{1},z_{2}\in Z$ , where we are using the pairing

\langle\cdot|\cdot\rangle_{\Omega}:Z\times(Z\otimes^{\mathrm{h}}_{B}\Omega)\to% \Omega,\qquad\langle z_{1}|z_{2}\otimes\omega\rangle_{\Omega}\coloneqq\langle z% _{1}|z_{2}\rangle\omega.

Let $\operatorname{Con^{f}}(B,A)$ denote the category whose objects $(Z,\nabla_{Z})$ are right Hilbert $C^{*}$ -modules over $B$ equipped with flat, Hermitian $\Omega(B,A)$ -connections, with morphisms $t:(W,\nabla_{W})\to(Z,\nabla_{Z})$ the adjointable maps of Hilbert $C^{*}$ -modules satisfying $\nabla_{Z}\circ t=(t\otimes\mathrm{id}_{\Omega})\circ\nabla_{W}$ .

Definition 2.6 .

Let $\mathbb{X}$ be a convex algebra. An element $z\in X$ is $\mathbb{X}$ - cancellable if

\forall x,y\in X.\,\,\forall p\in(0,1).\,\,px+\bar{p}z=py+\bar{p}z\Rightarrow x% =y.

The convex algebra $\mathbb{X}$ is cancellative if every element of $X$ is $\mathbb{X}$ -cancellable. $\diamond$

Definition 38 ( Inner derivations [ 11 ] ) .

Given an endomorphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}$ , we say that $\delta:\mathbb{F}\longrightarrow\mathbb{F}$ is an inner $\sigma$ -derivation if there exists $\gamma\in\mathbb{F}$ such that

\delta(b)=\gamma(b-\sigma(b)),

for all $b\in\mathbb{F}$ .

Definition 1.2 .

Let $A$ be an algebra and $V$ a vector space. Let $\triangleright:A\otimes V\longrightarrow V$ and $\triangleleft:V\otimes A\longrightarrow V$ be two linear maps. Then $(V,\triangleright,\triangleleft)$ is called an $A$ -bimodule if

a\triangleright(b\triangleright x)=(ab)\triangleright x,~{}~{}(x\triangleleft a% )\triangleleft b=x\triangleleft(ab),~{}~{}(a\triangleright x)\triangleleft b=a% \triangleright(x\triangleleft b),

(1.2)

where $a,b\in A$ and $x\in V$ .

An $A$ -bimodule map $f$ from $(V,\triangleright_{V},{{}_{V}\triangleleft})$ to $(W,\triangleright_{W},{{}_{W}\triangleleft})$ is a linear map $f:V\longrightarrow W$ such that $f$ is a left $A$ -module map from $(V,\triangleright_{V})$ to $(W,\triangleright_{W})$ and at the same time $f$ is a right $A$ -module map $(V,{{}_{V}\triangleleft})$ to $(W,{{}_{W}\triangleleft})$ .

Definition 3.1 .

Let $V$ be a vector space and $\mu,\nu:V\otimes V\longrightarrow V$ two linear maps such that

\mu(\mu\otimes id)=\mu(id\otimes\mu),~{}\nu(\nu\otimes id)=\nu(id\otimes\nu),

\nu(\mu\otimes id)=\mu(id\otimes\nu)=\mu(\nu\otimes id)=\nu(id\otimes\mu).

Then we call $(\mu,\nu)$ an associative compatible pair on $V$ .

Definition 2.1 .

Let $(A,\diamond)$ be a group and let $\circ$ be an associative operation on $A$ , for which there exists a function $\sigma:A\to A$ such that, for all $a,b,c\in A$ ,

a\circ(b\diamond c)=(a\circ b)\diamond\sigma(a)^{\diamond}\diamond(a\circ c).

(2.1)

A quadruple $(A,\diamond,\circ,\sigma)$ , is called a skew left truss , and the function $\sigma$ is called a cocycle .

In a symmetric way, a skew right truss is a quadruple $(A,\diamond,\circ,\sigma)$ with operations $\diamond$ , $\circ$ making $A$ into a group and a semigroup respectively, and $\sigma:A\to A$ such that

(a\diamond b)\circ c=(a\circ c)\diamond\sigma(c)^{\diamond}\diamond(b\circ c),

(2.2)

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

The adjective skew is dropped if the group $(A,\diamond)$ is abelian.

The identities ( 2.1 ) and ( 2.2 ) are refered to as left and right truss dristributive laws , respectively.

Definition 2.8 .

Take any segment region ${\rm{I}}_{k,l}\equiv[2k,\;2k+1,\;2(k+1),\cdots,2(l-1),\;2l-1,\;2l]$ with $k,l\in{\mathbb{Z}}$ $k<l$ . Assume that $g(n)\in\Upsilon_{k,l}\equiv{\Upsilon_{{\rm{I}}_{k,l}}}$ should be constant on each of the two-site edges:

g(2k)=g(2k+1)\ \text{and}\ g(2l-1)=g(2l).

(2.14)

The above requirements will be called the open SUSY boundary condition. The set of all $g(n)\in\Upsilon_{k,l}$ satisfying the open SUSY boundary condition is denoted by $\widehat{\Upsilon}_{k,l}$ .

Definition 3.1 .

Take any $k,l\in{\mathbb{Z}}$ such that $k<l$ and the finite interval ${\rm{I}}_{k,l}$ defined in ( 2.8 ). Let $f$ be a $\{-1,+1\}$ -valued sequence on the finite interval ${\rm{I}}_{k,l}$ . For any consequent triplet $\{2i-1,\;2i,\;2i+1\}\subset{\rm{I}}_{k,l}$ ( $i\in{\mathbb{Z}}$ ) assume that neither

\displaystyle f(2i-1)=-1,\ \ f(2i)=+1,\ \ f(2i+1)=-1,

(3.1)

nor

\displaystyle f(2i-1)=+1,\ \ f(2i)=-1,\ \ f(2i+1)=+1,

(3.2)

is satisfied. Furthermore assume that $f$ is constant on the left-end pair sites $\{2k,\;2k+1\}$ and on the right-end pair sites $\{2l-1,\;2l\}$ . Namely, on the left-end pair

\displaystyle f(2k)=f(2k+1)=+1\quad\text{or}\quad f(2k)=f(2k+1)=-1

(3.3)

and on the right-end pair

\displaystyle f(2l-1)=f(2l)=+1\quad\text{or}\quad f(2l-1)=f(2l)=-1.

(3.4)

The set of all $\{-1,+1\}$ -valued sequences on ${\rm{I}}_{k,l}$ satisfying all the above conditions is denoted by $\hat{\Xi}_{k,l}$ . The union of $\hat{\Xi}_{k,l}$ over all $k,l\in{\mathbb{Z}}$ ( $k<l$ ) is denoted by $\hat{\Xi}$ :

\displaystyle\hat{\Xi}:=\bigcup_{k,l\in{\mathbb{Z}}\;(k<l)}\hat{\Xi}_{k,l}.

(3.5)

Take any $p,q\in{\mathbb{Z}}$ $(p<q)$ . Let

\displaystyle\hat{\Xi}(p,q):=\bigcup_{k,l\in{\mathbb{Z}};\;p\leq k<l\leq q}% \hat{\Xi}_{k,\;l}.

(3.6)

Each $f\in\hat{\Xi}$ is called a local $\{-1,+1\}$ -sequence of conservation for the Nicolai model.

Definition 2.2 .

Let $\mathcal{A}$ and $\tilde{\mathcal{A}}$ be $\mathbb{K}$ -evolution algebras and $S=\{e_{i}:i\in V\}$ a natural basis for $\mathcal{A}$ . We say that a triple $(f,g,h)$ , where $f,g,h$ are three non-singular linear transformations from $\mathcal{A}$ into $\tilde{\mathcal{A}}$ is an isotopism if

f(u)\cdot g(v)=h(u\cdot v),\;\;\;\text{ for all }u,v\in\mathcal{A}.

In this case we say that $\mathcal{A}$ and $\tilde{\mathcal{A}}$ are isotopic. In addition, the triple is called:

(i)

a strong isotopism if $f=g$ and we say that the algebras are strongly isotopic;
(ii)

an isomorphism if $f=g=h$ and we say that the algebras are isomorphic.

Definition 19 .

The category FS of fusing strings is the strict monoidal category freely generated by one object, denoted by $*$ , and the following four morphisms:

\mu=\raisebox{-20.0pt}{\includegraphics[height=43.362pt]{cat-mu}}\hskip 28.452% 756pt\mu^{-1}=\raisebox{25.0pt}{\includegraphics[height=43.362pt, angle=180]{% cat-mu}}\hskip 28.452756pt\gamma=\raisebox{-20.0pt}{\includegraphics[height=43% .362pt]{cat-gamma}}\hskip 28.452756ptv=\raisebox{-20.0pt}{\includegraphics[hei% ght=43.362pt]{cat-v}}

where $\mu,\mu^{-1},\gamma$ and $v:*\otimes*\longrightarrow*\otimes*$ .

Definition 2.1 .

An operational tangent vector at point $m\in M$ is a linear map $\delta:C^{\infty}_{m}(M)\to\mathbb{R}$ satisfying Leibniz rule :

\delta(fg)(m)=\delta f\;g(m)+f(m)\;\delta g.

(2.1)

An operational vector field on $M$ is a collection of maps $\delta_{U}:C^{\infty}(U)\to C^{\infty}(U)$ for each open set $U\subset M$ , compatible with restrictions to open subsets and defining an operational tangent vector at every $m\in M$ .

Definition 2.4 .

Given the rule list $d$ of Equation ( 1 ) (or the prefix $e$ of Equation ( 3 )), we say that an input $\mathbf{x}\in\mathcal{X}$ is captured by the $j$ -th antecedent in $d$ (or $e$ ) if $\mathbf{x}$ satisfies $a_{j}^{(d)}$ (or $a_{j}^{(e)}$ , respectively), and for all $k\in\{0,1,...,|d|\}$ (or $k\in\{0,1,...,|e|-1\}$ , respectively) such that $\mathbf{x}$ satisfies $a_{k}^{(d)}$ (or $a_{k}^{(e)}$ , respectively), $j\leq k$ holds – in other words, $a_{j}^{(d)}$ (or $a_{j}^{(e)}$ , respectively) is the first antecedent that $\mathbf{x}$ satisfies. We define the function capt by

\text{capt}(\mathbf{x},d)=j\text{ (or }\text{capt}(\mathbf{x},e)=j\text{, % respectively)}

if $\mathbf{x}$ is captured by the $j$ -th antecedent in $d$ (or $e$ ). Moreover, given the prefix $e$ of Equation ( 3 ), we say that an input $\mathbf{x}\in\mathcal{X}$ is captured by the prefix $e$ if $\mathbf{x}$ is captured by some antecedent in $e$ , and we define $\text{capt}(\mathbf{x},e)=|e|$ if $\mathbf{x}$ is not captured by the prefix $e$ .

Definition 3.1 .

The (commutative) operations with $?$ are defined as

0*?=0,\quad x*?=?,\quad\quad y+?=?,\quad\textnormal{and }?+?=?=?*?

(3.1)

for $x\in\mathbb{F}_{q}^{*}$ and $y\in\mathbb{F}_{q}$ .

Definition 1

A vector space $\mathbb{A}$ over $\mathbb{F}$ with multiplication $\cdot:\mathbb{A}\otimes\mathbb{A}\rightarrow\mathbb{A}$ given by $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}$ such that

(\alpha\mathbf{u}+\beta\mathbf{v})\cdot\mathbf{w}=\alpha(\mathbf{u}\cdot% \mathbf{w})+\beta(\mathbf{v}\cdot\mathbf{w}),\ \ \mathbf{w}\cdot(\alpha\mathbf% {u}+\beta\mathbf{v})=\alpha(\mathbf{w}\cdot\mathbf{u})+\beta(\mathbf{w}\cdot% \mathbf{v})

whenever $\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{A}$ and $\alpha,\beta\in\mathbb{F}$ , is said to be an algebra.

Definition 10 (Assert) .

Let $p\in\Sigma^{\omega}\rightarrow\mathsf{Bool}$ be a property. The assert property transformer $\{p\}:(\Sigma^{\omega}\rightarrow\mathsf{Bool})\rightarrow(\Sigma^{\omega}% \rightarrow\mathsf{Bool})$ is defined by

\{p\}(q)=p\land q.

Definition 2.24 (Suspension action) .

Let $G$ be a Lie group which acts on an Alexandrov space $X$ . The action of $G$ on $\operatorname{Susp}(X)$ is called suspension action , if $G$ acts on $\operatorname{Susp}(X)$ as follows:

g\cdot[(x,t)]=[(gx,t)].

Definition 1 ( Matrix morphisms and vector derivations ) .

We call every ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ a matrix morphism (over $\mathbb{F}$ ), and we say that a map $\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ is a $\sigma$ -vector derivation (over $\mathbb{F}$ ) if it is additive and satisfies

\delta(ab)=\sigma(a)\delta(b)+\delta(a)b,

for all $a,b\in\mathbb{F}$ .

Definition 1.1 .

Let $p$ be a singular point of the metric $ds^{2}$ on $M^{2}$ . Then a non-zero tangent vector $\boldsymbol{v}\in T_{p}M^{2}$ is called a null vector if

(1.1)

ds^{2}(\boldsymbol{v},\boldsymbol{v})=0.

Moreover, a local coordinate neighborhood $(U;u,v)$ is called adjusted at $p\in U$ if $\partial_{v}:=\partial/\partial v$ gives a null vector of $ds^{2}$ at $p$ .

Definition 2 .

An algebra $\mathbf{L}=(L,\cdot,\backslash,/,0)$ is called a loop if for all $x,y\in L$ :

(1)

$x\backslash(x\cdot y)=y$ and $(x\cdot y)/x=y$
(2)

$x\cdot(x\backslash y)=y$ and $(y/x)\cdot x=y$
(3)

$0\cdot x=x\cdot 0=x$

Definition 2.1 ( $\Gamma$ -invariance) .

Let $\Gamma$ be an $r$ dimensional linear subspace of ${\mathbb{R}}^{d}$ . A function $f:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ is said to be $\Gamma$ -invariant if for all $x\in{\mathbb{R}}^{d}$ and $y$ belonging to the subspace $\Gamma^{\perp}$ , i.e. $M_{\Gamma}y=0$ we have that

f(x)=f(x+y)

Equivalently there exists a function $g:{\mathbb{R}}^{r}\rightarrow{\mathbb{R}}$ such that for all $x$ , $f(x)=g(M_{\Gamma}x)$ .

Definition 3 .

[ 17 ] A (right) Leibniz algebra is an non-associative algebra such that for all $x,y,z\in L$ , the following identity holds:

x(yz)=(xy)z-(xz)y.

Definition 1

The average treatment moderation effect (ATME) is defined as

\delta=

E\left[\{Y_{i}(T_{i}=1,S_{i}=1)-Y_{i}(T_{i}=0,S_{i}=1)\}-\{Y_{i}(T_{i}=1,S_{i}% =0)-Y_{i}(T_{i}=0,S_{i}=0)\}\right]=

E\left[\{Y_{i}(1,1)-Y_{i}(0,1)\}-\{Y_{i}(1,0)-Y_{i}(0,0)\}\right]

Definition 5 .

A 1-form $\omega,$ on any manifold $M$ is a map from the set of vector fields on $M$ called $\textnormal{Vect}(M)$ to $C^{\infty}(M)$ that is linear over $C^{\infty}(M).$ In other words, for any $u,v\in\textnormal{Vect}(M)$ and $g\in C^{\infty}(M)$

(1)

\omega(u+v)=\omega(u)+\omega(v)

(2)

\omega(gv)=g\omega(v)

The space of all $1-$ forms on a manifold $M$ will be denoted by $\Omega^{1}(M)$ [ 2 ] .

Definition 7 .

The exterior algebra over a vector space $V$ denoted $\Lambda V$ is the algebra generated by $V$ with the relation

v\wedge u=-u\wedge v

for vectors $u,v\in V$ where $\wedge$ is known as the wedge product [ 2 ] .

Definition 8 .

We define the differential forms on $M$ , denoted $\Omega(M)$ , to be the algebra generated by $\Omega^{1}(M)$ with the relations

\omega\wedge\mu=-\mu\wedge\omega

for all $\omega,\mu\in\Omega^{1}(M).$ Elements that are linear combinations of products of $k$ $1-$ forms are called $k-$ forms and the space is denoted by $\Omega^{k}(M)$ . Moreover,

\Omega(M)=\oplus_{k}\Omega^{k}(M)

[ 2 ] .

Definition 9 .

In particular when $k=2$ $\omega$ is a $2-$ form. If

d\omega=0,

then we say that $\omega$ is a closed $2-$ form. We say that $\omega$ is a nondegenerate $2-$ form if for any nonzero $v$ there exists $u$ such that

\omega(v,u)\neq 0

where $u,v\in T_{x}M$ [ 1 ] .

Definition 11 .

Given functions $f,g,h\in C^{\infty}(M)$ and $a,b\in\mathbb{R},$ a Poisson bracket on a manifold $M$ is a binary operation

C^{\infty}(M)\times C^{\infty}(M)\to C^{\infty}(M),

(f,g)\mapsto\{f,g\},

that satisfies the following:

(1)

Antisymmetry $\{f,g\}=-\{g,f\}$

(2)

Bilinearity

\{f,ag+bh\}=a\{f,g\}+b\{f,h\}

(3)

Jacobi Identity

\{f,\{g,h\}\}+\{\{g,h\},f\}+\{h,\{f,g\}\}=0.

(4)

Leibniz Law

\{fg,h\}=\{f,h\}g+f\{g,h\}

[ 6 ] .

Definition 4.8 .

Definition C.11 .

Definition 2.1 .

Definition 2.3 .

Definition 6.13 .

Definition 1 (Hom-associative algebra) .

Definition 1.3 .

Definition 1.5 .

Definition 10 (Satisfaction Indicator) .

Definition 6.9 .

Definition 2.6 .

Definition 38 ( Inner derivations [ 11 ] ) .

Definition 1.2 .

Definition 3.1 .

Definition 2.1 .

Definition 2.8 .

Definition 3.1 .

Definition 2.2 .

Definition 19 .

Definition 2.1 .

Definition 2.4 .

Definition 3.1 .

Definition 1

Definition 10 (Assert) .

Definition 2.24 (Suspension action) .

Definition 1 ( Matrix morphisms and vector derivations ) .

Definition 1.1 .

Definition 2 .

Definition 2.1 ( Γ Γ \Gamma roman_Γ -invariance) .

Definition 3 .

Definition 1

Definition 5 .

Definition 7 .

Definition 8 .

Definition 9 .

Definition 11 .

Definition 2.1 ( $\Gamma$ -invariance) .