The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

In this paper, we investigate the polynomial numerical index \(n^{(k)}(l_p),\) symmetric multilinear \(n_s^{(k)}(l_p),\) and \(n_m^{(k)}(l_p)\) of \(l_p\) spaces, for \(1\leq p\leq \infty.\) First prove that \(n_{s}^{(k)}(l_1)=n_{m}^{(k)}(l_1)=1,\) every \(k\geq 2.\) We show \(1 \lt p \infty,\) \(n_I^{(k)}(l_p^{j+1})\leq n_I^{(k)}(l_p^j),\) \(j\in \mathbb{N}\) \(n_I^{(k)}(l_p)=\lim_{j\to \infty...

We obtain a lower bound for $$\#\{x/2<p_n\leq x \colon\, \, p_n\equiv\dots\equiv p_{n+m}\equiv a\pmod{q}$$ , $$p_{n+m} - p_n\leq y\}$$ where $$p_n$$ is the $$n$$ th prime.

the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

let $g$ be a finite $p$-soluble group‎, ‎and $p$ a sylow $p$-sub-group of $g$‎. ‎it is proved‎ ‎that if all elements of $p$ of order $p$ (or of order ${}leq 4$ for $p=2$) are‎ ‎contained in the $k$-th term of the upper central series of $p$‎, ‎then the $p$-length of‎ ‎$g$ is at most $2m+1$‎, ‎where $m$ is the greatest integer such that‎ $p^m-p^{m-1}leq k$‎, ‎and the exponent of the image of $p$...

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M$ $R$-module with Cosupport in $\mathrm{V}(\mathfrak{a})$. We show that is $\mathfrak{a}$-coartinian if only $\mathrm{Ext}_{R}^{i}(R/\mathfrak{a},M)$ artinian for all $0\leq i\leq \mathrm{cd}(\mathfrak{a},M)$, which provides computable finitely many steps to examine $\mathfrak{a}$-coartinianness. also consider the duality...

Abstract We give an upper bound for the minimum s with property that every sufficiently large integer can be represented as sum of positive k th powers integers, each which is three cubes cases $2\leq k\leq 4.$

In any graph $G$, the domination number $\gamma(G)$ is at most independence $\alpha(G)$. The \emph{Inverse Domination Conjecture} says that, in isolate-free there exists pair of vertex-disjoint dominating sets $D, D'$ with $|D|=\gamma(G)$ and $|D'| \leq \alpha(G)$. Here we prove that this statement true if upper bound $\alpha(G)$ replaced by $\frac{3}{2}\alpha(G) - 1$ (and $G$ not a clique). We...