Given a geodesic $\gamma$ and a constant $k$ , Define $\phi_{f_{\gamma},k}$ to be the solution to

Let $L_{f_{\gamma},k}$ to be the smallest positive number such that $\phi\left(L_{\gamma,k}\right)=0$ and define $L_{f_{\gamma},k}=\infty$ if $\phi_{f_{\gamma},k}$ is positive for all time.

A braided algebra is a triple $(V,m,\sigma)$ where $V$ is a vector space, $m:V\otimes V\rightarrow V$ is an associative operation, $\sigma:V\otimes V\rightarrow V\otimes V$ is a linear map such that on $V\otimes V\otimes V$ , the following relations are verified:

For two such algebras $\mathcal{A}=\mathcal{A}^{0}\oplus\mathcal{A}^{1}$ and $\mathcal{B}=\mathcal{B}^{0}\oplus\mathcal{B}^{1}$ the $\mathbb{Z}_{2}$ -graded tensor product is defined

if $b\in\mathcal{B}^{j}$ and $a^{\prime}\in\mathcal{A}^{k}$ . The Eq. ( 29 ) may be extended on arbitrary elements of algebras due to distributivity. It is also called the skew tensor product and denoted as $\mathbin{\widehat{\otimes}}$ Post .

Let ${\mathcal{A}}$ be a finite alphabet. The reversal operator is the operator $\sim:{\mathcal{A}}^{*}\mapsto{\mathcal{A}}^{*}$ defined by recurrence in the following way:

for all $v\in{\mathcal{A}}^{*}$ and $a\in{\mathcal{A}}$ . The fixed points of the reversal operator are called palindromes .

We call the iterated (right) palindromic closure operator the operator $\psi$ recurrently defined by the following rules:

for all $v\in{\mathcal{A}}^{*}$ and $a\in{\mathcal{A}}$ . The definition of $\psi$ may be extended to infinite words $u$ over ${\mathcal{A}}$ as $\psi(u)=\lim_{n}\psi(\operatorname{Pref}_{n}u)$ , i.e., $\psi(u)$ is the infinite word having $\psi(\operatorname{Pref}_{n}u)$ as its prefix for every $n\in\mathbb{N}$ .

Given a finite group $G$ and a natural number $n$ , write $[n]=\{1,2,\ldots,n\}$ and $B(G,n)=([n]\times G\times[n])\cup\{\vartheta\}$ . Define a binary operation (say, addition) on $B(G,n)$ by

With the above defined multiplication, $B(G,n)$ is a semigroup known as Brandt semigroup . When $G$ is the trivial group, the Brandt semigroup $B(\{e\},n)$ is aperiodic and is denoted by $B_{n}$ . For more details on Brandt semigroups, one may refer to [ 5 ] .

Let $\varphi:G\times[0,1]^{s}\to G$ be an update function. We say that $\varphi$ satisfies the ‘anywhere-to-anywhere’ condition if for all $x,y\in G$ there exists a $u\in[0,1]^{s}$ such that

Let $R$ be a ring, $\sigma$ an endomorphism of $R$ and $\delta$ an additive function, $R\to R$ , satisfying

for all $a,b\in R$ . (Such $\delta$ :s are known as $\sigma$ -derivations.) The Ore extension $R[x;\sigma,\delta]$ is the over-ring of $R$ satisfying $xr=\sigma(r)x+\delta(r)$ for all $r\in R$ and such that every element of $R[x;\sigma,\delta]$ can be written $\sum a_{i}x^{i}$ with $a_{i}\in R$ . If $\sigma=\operatorname{id}$ then $R[x;\operatorname{id}_{R},\delta]$ is called a differential operator ring .

A matched pair of algebras is a system $(A,\,X,\triangleright,\triangleleft,\leftharpoonup,\rightharpoonup\bigl{)}$ consisting of two algebras $A$ , $X$ and four bilinear maps

such that $(X,\rightharpoonup,\triangleleft)\in{}_{A}\mathcal{M}_{A}$ is an $A$ -bimodule, $(A,\triangleright,\leftharpoonup)\in{}_{X}\mathcal{M}_{X}$ is an $X$ -bimodule and the following compatibilities hold for any $a$ , $b\in A$ , $x$ , $y\in X$ :

1. (MP1)

   $a\rightharpoonup(x\,y)=(a\rightharpoonup x)\,y+(a\leftharpoonup x)\rightharpoonup y$ ;

2. (MP2)

   $(a\,b)\leftharpoonup x=a\,(b\leftharpoonup x)+a\leftharpoonup(b\rightharpoonup x)$ ;

3. (MP3)

   $x\triangleright(a\,b)=(x\triangleright a)\,b+(x\triangleleft a)\triangleright b$ ;

4. (MP4)

   $(x\,y)\triangleleft a=x\triangleleft(y\triangleright a)+x\,(y\triangleleft a)$ ;

5. (MP5)

   $a\,(x\triangleright b)+a\leftharpoonup(x\triangleleft b)=(a\leftharpoonup x)\,b+(a\rightharpoonup x)\triangleright b$ ;

6. (MP6)

   $x\triangleleft(a\leftharpoonup y)+x\,(a\rightharpoonup y)=(x\triangleright a)\rightharpoonup y+(x\triangleleft a)\,y$ ;