We say that the functional ( 3 ) is invariant under an $\varepsilon$ -parameter group of infinitesimal transformations

$$\hat{\mathbf{y}}(t)=\mathbf{y}(t)+\varepsilon{\bm{\xi}}(t,\mathbf{y}(t))+o(\varepsilon)$$

(11)

if for any subinterval $[t_{a},t_{b}]\subseteq[a,b]$ one has

$$K_{\bar{P}}^{\alpha}\left[t\mapsto F\left\{\mathbf{y}\right\}_{P_{D},R_{I}}^{\beta,\gamma}(t)\right](t_{b})=K_{\bar{P}}^{\alpha}\left[t\mapsto F\left\{\hat{\mathbf{y}}\right\}_{P_{D},R_{I}}^{\beta,\gamma}(t)\right](t_{b}),$$

(12)

where $\bar{P}=\langle t_{a},t_{b},t_{b},1,0\rangle$ .

Consider the following terms.

$$suc(x)=1;(\breve{p};x;\breve{q})$$

and

$$pred(x)=\breve{p};ranx;q.$$

We now set:

$$q=\ln\rho.$$

An exact symplectic form on a manifold $M$ is an exact non-degenerate two-form $\omega$ . Thus

(6)

$$\omega=d\lambda$$

for some one-form $\lambda$ , which is called a Liouville one-form . The corresponding Liouville vector field $Z$ is defined by duality with respect to $\omega$ :

(7)

$$\iota_{Z}\omega=\omega(Z,\cdot)=\lambda$$

Thus the one-form $\lambda$ determines the two-form $\omega$ and the vector field $Z$ .

A Liouville domain [ 22 ] is a compact manifold with boundary, equipped with a one-form $\lambda$ , such that the two-form $\omega$ is symplectic, and the vector field $Z$ points strictly outward along the boundary.

Define an edit of a word $w=w_{1}\cdots w_{n}\in\mathcal{L}$ to be a transformation of $w$ by one of the following actions, where $u^{j}\in\mathcal{L}$ are arbitrary words and $a,a^{\prime}\in A$ are arbitrary symbols.

1. (1)

   Substitution : $w=u^{1}au^{2}\mapsto w^{\prime}=u^{1}a^{\prime}u^{2}$ .

2. (2)

   Insertion : $w=u^{1}u^{2}\mapsto w^{\prime}=u^{1}a^{\prime}u^{2}$ .

3. (3)

   Deletion : $w=u^{1}au^{2}\mapsto w^{\prime}=u^{1}u^{2}$ .

Given $v,w\in\mathcal{L}$ , define the edit distance between $v$ and $w$ to be the minimum number of edits required to transform the word $v$ into the word $w$ : we will denote this by $\hat{d}(v,w)$ .

A Boolean ring $\mathbb{B}$ (with addition denoted by “ $\dotplus$ ” and with multiplication denoted by “ $\diamond$ ” ) is a ring with multiplicative identity, denoted by “ $1$ ”, such that each element $a$ of $\mathbb{B}$ is an idempotent, i.e., such that

$$a^{2}=a.$$

It follows that $\mathbb{B}$ is a commutative ring, and that each element $a$ of $\mathbb{B}$ is its own additive inverse, i.e.,

$$a\dotplus a=0,$$

where “ $\mathrm{0}$ ” denotes the additive identity of $\mathbb{B}$ . The additive operation will often be referred to as exclusive “or” and the multiplicative operation “ $\diamond$ ” will often be referred to as logical “and” . The complement of an element $a$ of $\mathbb{B}$ , written $a^{\ast}$ , is defined as

$$a^{\ast}=1\dotplus a.$$

Let $a$ and $b$ be generic dyadic integers. Let

$$\left\{\begin{array}[c]{lll}c^{(0)}&=&a\\ &&\\ carries^{(0)}&=&b\end{array}\right.$$

and let

$$\left\{\begin{array}[c]{lll}c^{(i+1)}&=&c^{(i)}\dotplus carries^{(i)}\\ &&\\ carries^{(i+1)}&=&\mathcal{S}\left(c^{(i)}\diamond carries^{(i)}\right)\end{array}\right.$$

Then the sequences $c^{(i)}$ and $carries^{(i)}$ are convergent. The generic dyadic integer $a+b$ is defined as:

$$a+b=\lim_{i\rightarrow\infty}c^{(i)}$$

It can also be shown that

$$\lim_{i\rightarrow\infty}carries^{(i)}=0$$

Moreover, if $a$ and $b$ are generic rational integers, then $a+b$ is also a generic rational integer.

Let $e\in\mathbb{B}\left\langle\mathbf{x}\right\rangle$ . A solution of the Boolean equation

$$e=1$$

is an instantiation $\Phi$ such that

$$\Phi(e)=1\qquad.$$

Let $e\in\mathbb{B}\left\langle\mathbf{x}\right\rangle$ . The Boolean equation

$$e=1$$

is said to be scarcely satisfiable if the number of its solutions $\Phi$ is a non-zero number which is less than the number of distinct free basis elements $x_{i}$ appearing in the canonical expression for $e$ .

The mass density is given by

$$\rho(t,x)=u(t,x)^{2}.$$

(3.1)

$$j(t,x)=3u_{x}(t,x)^{2}+\frac{5}{3}u(t,x)^{6}.$$

(3.2)

Consider the following terms.

$$suc(x)=1;(\breve{p};x;\breve{q})$$

and

$$pred(x)=\breve{p};ranx;q.$$

Let $\alpha$ be a positive real number. An affine connection $\nabla$ on $M$ is called projectively compact of order $\alpha\in\mathbb{R}_{+}$ if for any $x\in\partial M$ , there is a neighborhood $U$ of $x$ in $\overline{M}$ and a defining function $\rho:U\to\mathbb{R}$ for the boundary such that the connection

(1)

$$\hat{\nabla}=\nabla+\tfrac{d\rho}{\alpha\rho}$$

on $U\cap M$ extends to all of $U$ .

A Lie group $G$ is called regular provided that, for any mapping $\varsigma:\mathbb{R}\rightarrow\mathfrak{g}$ , there exists a mapping $\theta:\mathbb{R}\rightarrow G$ with

$$\theta\left(0\right)=e$$

and

$$\theta(t+d)=\theta\left(t\right).\varsigma\left(t\right)_{d}$$

for any $t\in\mathbb{R}$ and any $d\in D$ .

A multiplicative Hom-Lie superalgebra is a triple $(\mathfrak{g},[,],\sigma)$ consisting of a $\mathbb{Z}_{2}$ -graded vector space $\mathfrak{g}$ , a bilinear map $[,]:\mathfrak{g}\times\mathfrak{g}\longrightarrow\mathfrak{g}$ and an even linear map $\sigma:\mathfrak{g}\longrightarrow\mathfrak{g}$ satisfying

	$\displaystyle\sigma[x,y]=[\sigma(x),\sigma(y)],$		(2.1)
	$\displaystyle[x,y]=-(-1)^{|x||y|}[y,x],$
	$\displaystyle(-1)^{|x||z|}[\sigma(x),[y,z]]+(-1)^{|y||x|}[\sigma(y),[z,x]]+(-1)^{|z||y|}[\sigma(z),[x,y]]=0,$		(2.2)

where $x,y$ and $z$ are homogeneous elements in $\mathfrak{g}.$