In this paper, we study partial automorphisms and, more generally, injective endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks monoids $$\mathrm {IEnd}(P_n)$$ and {PAut}(P_n)$$ all $$P_n$$ with n vertices. We also describe Green’s relations calculate their cardinals.

In this paper an improved bound on the chromatic number of Pancake graph $P_n, n\geqslant 2$, is presented. The obtained using a subadditivity property graph. We also investigate equitable coloring $P_n$. An $(n-1)$-coloring based efficient dominating sets given and optimal $4$-colorings are considered for small $n$. It conjectured that $P_n$ coincides with its any $n\geqslant 2$.

There is an extensive literature on the asymptotic order of Sudler's trigonometric product $P_N (\alpha) = \prod_{n=1}^N |2 \sin (\pi n \alpha)|$ for fixed or "typical" values $\alpha$. In present paper we establish a structural result, which given $\alpha$ characterizes those $N$ $P_N(\alpha)$ attains particularly large values. This characterization relies coefficients in its Ostrowski expansi...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...

Given a probability distribution $p:=\{p_k\}_{k=1}^\infty$ on the positive integers, there are two natural ways to construct random permutation of $\mathbb{N}$. One is called $p$-biased construction and other $p$-shifted construction. For any $n\in\mathbb{N}$, amending these constructions in an obvious way yields permutations $[n]$, that distributions set $S_n$ $[n]$. In first part paper we con...

the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

Let <mml:semantics...