let $cr_n(f_p)$ denote the algebra of $n times n$ circulant‎ ‎matrices over $f_p$‎, ‎the finite field of order $p$ a prime‎. ‎the‎ ‎order of the unit groups $mathcal{u}(cr_3(f_p))$‎, ‎$mathcal{u}(cr_4(f_p))$ and $mathcal{u}(cr_5(f_p))$ of algebras of‎ ‎circulant matrices over $f_p$ are computed‎.

let $mathcal{r}$ be a  commutative ring with identity, let $a$ and $b$ be two $mathcal{r}$-algebras and $varphi:blongrightarrow a$ be an $mathcal{r}$-additive algebra homomorphism. we introduce a new algebra $atimes_varphi b$, and give some basic properties of this algebra. generalized $2$-cocycle derivations on $atimes_varphi b$ are studied. accordingly, $atimes_varphi b$ is considered from th...

Motivated by the terminal Wiener index‎, ‎we define the Ashwini index $mathcal{A}$ of trees as‎ begin{eqnarray*}‎ % ‎nonumber to remove numbering (before each equation)‎ ‎mathcal{A}(T) &=& sumlimits_{1leq i

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Si...

We consider the quasilinear Kirchhoff's problem$$ u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+f(u)=0 ,;; x in {mathcal{R}}^{N}, ;; t geq 0,$$with the initial conditions  $ u(x,0) = u_0 (x)$  and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3, ;  f(u)=|u|^{a}u$ and $(phi (x))^{-1} in L^{N/2}({mathcal{R}}^{N})cap L^{infty}({mathcal{R}}^{N} )$ is a positive function. The purpose of our work is to ...

‎generalizing the notion of character amenability for banach‎ ‎algebras‎, ‎we study the concept of $varphi$-connes amenability of‎ ‎a dual banach algebra $mathcal{a}$ with predual $mathcal{a}_*$‎, ‎where $varphi$ is a homomorphism from $mathcal{a}$ onto $bbb c$‎ ‎that lies in $mathcal{a}_*$‎. ‎several characterizations of‎ ‎$varphi$-connes amenability are given‎. ‎we also prove that the‎ ‎follo...

let $g$ be a group and $mathcal{n}$ be the class of all nilpotent groups. a subset $a$ of $g$ is said to be nonnilpotent if for any two distinct elements $a$ and $b$ in $a$, $langle a, brangle notin mathcal{n}$. if, for any other nonnilpotent subset $b$ in $g$, $|a|geq |b|$, then $a$ is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by $ome...

A permutation with no fixed points is called a derangement.The subset $mathcal{D}$ of a permutation group is derangement if all elements of $mathcal{D}$ are derangement.Let $G$ be a permutation group, a derangementgraph is one with vertex set $G$ and derangement set $mathcal{D}$ as connecting set. In this paper, we determine the spectrum of derangement graphs of order a product of three primes.