Bulletin of Iranian Mathematical Society
Volume:39 Issue: 4, 2013

In this paper, by using of Frobenius theorem a relation between Lie subalgebras of the Lie algebra of a top space T and Lie subgroups of T(as a Lie group) is determined. As a result we can consider these spaces by their Lie algebras. We show that a top space with the finite number of identity elements is a C^{∞} principal fiber bundle, by this method we can characterize top spaces.

Let $X$ be a reflexive Banach space, $T:Xto X$ be a nonexpansive mapping with $C=Fix(T)neqemptyset$ and $F:Xto X$ be $delta$-strongly accretive and $lambda$- strictly pseudocotractive with $delta+lambda>1$. In this paper, we present modified hybrid steepest-descent methods, involving sequential errors and functional errors with functions admitting a center, which generate convergent sequences to the unique solution of the variational inequality $VI^*(F, C)$. We also present similar results for a strongly monotone and Lipschitzian operator in the context of a Hilbert space and apply the results for solving a minimization problem.

In this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially Cohen-Macaulay.

Let Z2 = {0, 1} and G = (V, E) be a graph. A labeling f: V → Z2 induces an edge labeling f*: E →Z2 defined by f*(uv) = f(u).f (v). For i ε Z2 let vf (i) = v(i) = card{v ε V: f(v) = i} and ef (i) = (i) = {e ε E: f*(e) = i}. A labeling f is said to be Vertex-friendly if | v(0) − v(1) |≤ 1. The vertex balance index set is defined by {| ef (0) − ef (1) |: f is vertex-friendly}. In this paper we completely determine the vertex balance index set of Kn, Km,n, Cn×P2 and Complete binary tree.

Let A be a unital R-algebra and M be a unital A-bimodule. It is shown that every Jordan derivation of the trivial extension of A by M, under some conditions, is the sum of a derivation and an antiderivation.

Let G = (V,E) be a locally finite graph, i.e. a graph in which every vertex has finitely many adjacent vertices. In this paper, we associate a topology to G, called graphic topology of G and we show that it is an Alexandroff topology, i.e. a topology in which intersec- tion of every family of open sets is open. Then we investigate some properties of this topology. Our motivation is to give an elementary step toward investigation of some properties of locally finite graphs by their corresponding topology which we introduce in this paper.

‎Suppose $n$ is a fixed positive integer‎. ‎We introduce the relative n-th non-commuting graph $Gamma^{n} _{H,G}$‎, ‎associated to the non-abelian subgroup $H$ of group $G$‎. ‎The vertex set is $Gsetminus C^n_{H,G}$ in which $C^n_{H,G} = {xin G‎: ‎[x,y^{n}]=1 mbox{~and~} [x {n},y]=1mbox{~for~all~} yin H}$‎. ‎Moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $H$ and $xy^{n} eq y^{n}x$ or $x^{n}y eq yx^{n}$‎. ‎In fact‎, ‎the relative n-th commutativity degree‎, ‎$P_{n}(H,G)$ the probability that n-th power of an element of the subgroup $H$ commutes with another random element of the group $G$ and the non-commuting graph were the keys to construct such a graph‎. ‎It is proved that two isoclinic non-abelian groups have isomorphic graphs under special conditions‎.

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. The total domination number $gamma _{t}(G)$ of a graph $G$ is the minimum cardinality of a total dominating set, which is a set of vertices such that every vertex of $G$ is adjacent to one vertex of it. A graph is $K_{r}$-covered if every vertex of it is contained in a clique $K_{r}$. Cockayne et al. in [Total domination in $K_{r}$ covered graphs, Ars Combin. textbf{71} (2004) 289-303] conjectured that the total domination number of every $K_{r}$-covered graph with $n$ vertices and no $K_{r}$-component is at most $frac{2n}{r+1}.$ This conjecture has been proved only for $3leq rleq 6$. In this paper, we prove this conjecture for a big family of $K_{r}$-covered graphs.

The reverse degree distance of a connected graph $G$ is defined in discrete mathematical chemistry as [r (G)=2(n-1)md-sum_{uin V(G)}d_G(u)D_G(u),] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $G$, respectively, $d_G(u)$ is the degree of vertex $u$, $D_G(u)$ is the sum of distance between vertex $u$ and all other vertices of $G$, and $V(G)$ is the vertex set of $G$. We determine the unicyclic graphs of given girth, number of pendant vertices and maximum degree, respectively, with maximum reverse degree distances. We also determine the unicyclic graphs of given number of vertices, girth and diameter with minimum degree distance.

In this paper, we propose a new method for the numerical solution of two-dimensional linear and nonlinear Volterra integral equations of the first and second kinds, which avoids from using starting values. An existence and uniqueness theorem is proved and convergence is verified by using an appropriate variety of the Gronwall inequality. Application of the method is demonstrated for solving the useful telegraph equation.

In this paper, we introduce $p$-semilinear transformations for linear algebras over a field ${bf F}$ of positive characteristic $p$, discuss initially the elementary properties of $p$-semilinear transformations, make use of it to give some characterizations of linear algebras over a field ${bf F}$ of positive characteristic $p$. Moreover, we find a one-to-one correspondence between $p$-semilinear transformations and matrices, and we prove a result which is closely related to the well-known Jordan-Chevalley decomposition of an element.

n this paper, we propose a generalized iterative method for finding a common element of the set of fixed points of a single nonexpannsive mapping and the set of solutions of two variational inequalities with inverse strongly monotone mappings and strictly pseudo-contractive of Browder-Petryshyn type mapping. Our results improve and extend the results announced by many others.

The purpose of this paper is to propose a composite iterative scheme for approximating a common solution for a finite family of m-accretive operators in a strictly convex Banach space having a uniformly Gateaux differentiable norm. As a consequence, the strong convergence of the scheme for a common fixed point of a finite family of pseudocontractive mappings is also obtained.

In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.