The class of  $n$-ary polygroups is a certain subclass of $n$-ary hypergroups, a generalization of D{"o}rnte $n$-arygroups and  a generalization of polygroups. The$beta^*$-relation and the $gamma^*$-relation are the smallestequivalence relations on an $n$-ary polygroup $P$ such that$P/beta^*$ and $P/gamma^*$ are an $n$-ary group and acommutative $n$-ary group, respectively. We use the $beta^*$-...

‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this pa...

The non-commuting graph $Gamma(G)$ of a non-abelian group $G$ with the center $Z(G)$ is a graph with thevertex set $V(Gamma(G))=Gsetminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent in $Gamma(G)$if and only if $xy neq yx$. The aim of this paper is to compute the spectra of some well-known NC-graphs.

Let ( L , Γ ) $(\mathfrak {L},\Gamma )$ be an isometric boundary pair associated with a closed symmetric linear relation T in Krein space H $\mathfrak {H}$ . M $M_\Gamma$ the Weyl family corresponding to We cope two main topics. First, since need not (generalized) Nevanlinna, characterization of closure and adjoint z $M_\Gamma (z)$ for some ∈ C ∖ R $z\in \mathbb {C}\setminus {R}$ becomes nontri...

for any lacunary sequence $theta = (k_{r})$, we define the concepts of $s_{theta}-$limit point and $s_{theta}-$cluster point of a sequence of fuzzy numbers $x = (x_{k})$. we introduce the new sets  $lambda^{f}_{s_{theta}}(x)$, $gamma^{f}_{s_{theta}}(x)$ and prove some inclusion relaions between these and the sets $lambda^{f}_{s}(x)$, $gamma^{f}_{s}(x)$ introduced in ~cite{ayt:slpsfn} by aytar [...

Decay rates of the positronium molecule $\text{Ps}_{2}$ into two photons and an electron-positron pair are determined. Previous studies find that these very different, \[ \Gamma\left(\text{Ps}_{2}\to e^{+}e^{-}\right)/\Gamma\left(\text{Ps}_{2}\to\gamma\gamma\right)\simeq250\text{ (previous studies)}. \] This is puzzling since both processes have body final states same order in fine structure co...

let $r$ be a commutative ring and $m$ be an $r$-module with $t(m)$ as subset,   the set of torsion elements. the total graph of the module denoted   by $t(gamma(m))$, is the (undirected) graph with all elements of   $m$ as vertices, and for distinct elements  $n,m in m$, the   vertices $n$ and $m$ are adjacent if and only if $n+m in t(m)$. in this paper we   study the domination number of $t(ga...

A $\Gamma$-magic rectangle set $MRS_{\Gamma}(a, b; c)$ of order $abc$ is a collection $c$ arrays $(a\times b)$ whose entries are elements group $\Gamma$, each appearing once, with all row sums in every equal to constant $\omega\in \Gamma$ and column $\delta \in \Gamma$. In this paper we prove that for $\{a,b\}\neq\{2^{\alpha},2k+1\}$ where $\alpha$ $k$ some natural numbers, MRS$_{\Gamma}(a, b;c...