Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P,Q)-total (r, s)-coloring of a graph G = (V,E) is a coloring of the vertices and edges of G by s-element subsets of Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G...

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let $g$ be a finite group and $pi_{e}(g)$ be the set of element orders of $g $. let $k in pi_{e}(g)$ and $s_{k}$ be the number of elements of order $k $ in $g$. set nse($g$):=${ s_{k} | k in pi_{e}(g)}$. in this paper, it is proved if $|g|=|$ pgl$_{2}(q)|$, where $q$ is odd prime power and nse$(g)= $nse$($pgl$_{2}(q))$, then $g cong $pgl$_

The goal of this article is to give an explicit classification the possible $p$-adic Galois representations that are attached elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be imaginary quadratic field, and $\mathcal{O}_{K,f}$ order in conductor $f\geq 1$. Let curve by $\mathcal{O}_{K,f}$, such a model $p\geq 2$ prime, $G_{\mathbb{Q}(j(E))}$ absolute group ...

In the comment to the article by J. Baez and K. Krasnov (hep– th/9703112) are discussed some topics related with application of certain constructions to non-trivial principal bundles. — In chapter 2 of the article [1] is used following construction: First, define a 'transporter' from the point p to the point q to be a map from P p to P q that commutes with the right action of G on the bundle P....

Let G = (L,R; E) be a bipartite graph such that V (G) = L ∪ R, |L| = p and |R| = q. G is called (p, q)-tree if G is connected and |E(G)| = p + q − 1. Let G = (L, R;E) and H = (L′, R′;E′) be two (p, q)-tree. A bijection f : L ∪ R → L′ ∪ R′ is said to be a biplacement of G and H if f(L) = L′ and f(x)f(y) / ∈ E′ for every edge xy of G. A biplacement of G and its copy is called 2-placement of G. A ...