Let G be a non-abelian group. The non-commuting graph AG of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with ω(AG) ≤ 57...

The domination number γ(G), the independent domination number ι(G), the connected domination number γc(G), and the paired domination number γp(G) of a graph G (without isolated vertices, if necessary) are related by the simple inequalities γ(G) ≤ ι(G), γ(G) ≤ γc(G), and γ(G) ≤ γp(G). Very little is known about the graphs that satisfy one of these inequalities with equality. I.E. Zverovich and V...

The branching rules between simple Lie algebras and its regular (maximal) simple subal-gebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights Γ specific for each injection. The structure of Γ is easily deduc...

This article studies the relationship between simple closed geodesics and orientation reversing involutions on one-holed hyperbolic tori. 1. Simple closed geodesics on the one holed torus We consider topological surfaces of signature (g, n) (where g is the underlying genus and n is the number of simple boundary curves) with negative Euler characteristic and endowed with a hyperbolic metric such...

Let X be a complex manifold and γ a simple closed curve inX . We address the question: What conditions on γ insure the existence of a 1-dimension complex variety Σ with boundary γ in X . When X = C, an answer to this question involves the polynomial hull of gamma. When X = P, complex projective space, the projective hull γ̂ of γ, a generalization of polynomial hull, comes into play. One always h...

background: a new biological dosimeter based on methemoglobin level was developed in this study. materials and methods: methemoglobin level in erythrocytes from mice subjected to γ rays from a 60co source was detected using the methemoglobin kit. the dose range was from 0.5 to 8 gy and the dose rate was 0.5 gy/min. results: the results demonstrate that methemoglobin level increases with increas...

The normal subgroup theorem of Margulis expresses that many lattices in semi-simple Lie groups are simple up to finite error. In this talk, we give a short introduction to the normal subgroup theorem , including a brief review of the relevant notions about lattices, amenability, and property (T), as well as a short overview of the proof of the normal subgroup theorem. 1 Margulis's normal subgro...

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3 · 10 12 . That is, all zeroes β + i γ of zeta-function with 0 < ⩽ have = 1 / 2 Moreover, these are simple.

Let X=G/Γ, where G is a Lie group and Γ lattice in G, let U be an open subset of X, {gt} one-parameter subgroup G. Consider the set points X whose gt-orbit misses U; it has measure zero if flow ergodic. It been conjectured that this Hausdorff dimension strictly smaller than X. This conjecture proved when compact or simple real rank 1. In paper we prove for case G=SLm+n(R), Γ=SLm+n(Z) gt=diag(en...

Synthesis of organic compounds using nano-catalysts is more and more attention due to the numerous advantages such as cost-effectiveness, high catalytic activity, ease of product separation, recovery of the catalyst, repeated recycling potential and good stability. In this work, TiCl2/nano-γ-Al2O3, as a novel type of green heterogeneous solid acid was prepared by the immobilization of TiCl2 on ...