Bulletin of Iranian Mathematical Society
Volume:39 Issue: 5, 2013

In this paper, we introduce the notions of C-numerical range and C-spectrum of matrix polynomials. Some algebraic and geometrical properties are investigated. We also study the relationship between the C-numerical range of a matrix polynomial and the joint C-numerical range of its coefficients.

To demonstrate more visibly the close relation between the continuity and integrability, a new proof for the Banach-Zarecki theorem is presented on the basis of the Radon-Nikodym theorem which emphasizes on measure-type properties of the Lebesgue integral. The Banach-Zarecki theorem says that a real-valued function $F$ is absolutely continuous on a finite closed interval if and only if it is continuous and of bounded variation when it satisfies Lusin''s condition. In the present proof indeed a more general result is obtained for the Jordan decomposition of $F$.

Employing a three critical points theorem, we prove the existence of multiple solutions for a class of Neumann two-point boundary value Sturm-Liouville type equations. Using a local minimum theorem for differentiable functionals the existence of at least one non-trivial solution is also ensured.

In this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite Hamiltonian group. The coverings with three proper subgroups and the principal series of such a group are also counted. Finally, we give a complete description of the lattice of characteristic subgroups of a finite Hamiltonian group.

In this paper, two inverse problems of Stephen kind with local (Dirichlet) boundary conditions are investigated. In the first problem only a part of boundary is unknown and in the second problem, the whole of boundary is unknown. For the both of problems, at first, analytic expressions for unknown boundary are presented, then by using these analytic expressions for unknown boundaries and boundary conditions of main problem, analytic solution of unknown function of main inverse problem is calculated.

Let A and B be n × m matrices. The matrix B is said to be g-row majorized (respectively g-column majorized) by A, if every row (respectively column) of B, is g-majorized by the corresponding row (respectively column) of A. In this paper all kinds of g-majorization are studied on Mn,m, and the possible structure of their linear preservers will be found. Also all linear operators T: Mn,m ---> Mn,m preserving (or strongly preserving) g-row or g-column majorization will be characterized.

We find an explicit expression of the Tutte polynomial of an $n$-fan. We also find a formula of the Tutte polynomial of an $n$-wheel in terms of the Tutte polynomial of $n$-fans. Finally, we give an alternative expression of the Tutte polynomial of an $n$-wheel and then prove the explicit formula for the Tutte polynomial of an $n$-wheel.

Let $L$ be a lattice in $ZZ^n$ of dimension $m$. We prove that there exist integer constants $D$ and $M$ which are basis-independent such that the total degree of any Graver element of $L$ is not greater than $m(n-m+1)MD$. The case $M=1$ occurs precisely when $L$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. As a corollary, we show that the Castelnuovo-Mumford regularity of the corresponding lattice ideal $I_L$ is not greater than $ rac{1}{2}m(n-1)(n-m+1)MD$.

A module M is called epi-retractable if every submodule of M is a homomorphic image of M. Dually, a module M is called co-epi-retractable if it contains a copy of each of its factor modules. In special case, a ring R is called co-pli (resp. co-pri) if RR (resp. RR) is co-epi-retractable. It is proved that if R is a left principal right duo ring, then every left ideal of R is an epi-retractable R-module. A co-pli strongly prime ring R is a simple ring. A left self-injective co-pli ring R is left Noetherian if and only if R is a left perfect ring. It is shown that a cogenerator ring R is a pli ring if and only if it is a co-pri ring. Moreover, if R is a left perfect ring such that every projective R-module is co-epi-retractable, then R is a quasi-Frobenius ring.

Let R be a commutative ring with identity and M be a unitary R-module. Let : S(M) −! S(M) [{;} be a function, where S(M) is the set of submodules of M. Suppose n  2 is a positive integer. A proper submodule P of M is called (n − 1, n) − -prime, if whenever a1,. .. , an−1 2 R and x 2 M and a1. .. an−1x 2 P(P), then there exists i 2 {1,. .. , n − 1} such that a1. .. ai−1ai+1. .. an−1x 2 P or a1. .. an−1 2 (P: M). In this paper we study (n − 1, n) − -prime submodules (n  2). A number of results concerning (n−1, n)−-prime submodules are given. Modules with the property that for some , every proper submodule is (n−1, n)−- prime, are characterized and we show that under some assumptions (n−1, n)-prime submodules and (n − 1, n) − m-prime submodules coincide (n,m  2).

A finite group G is said to be a POS-group if for each x in G the cardinality of the set {y in G | o(y) = o(x)} is a divisor of the order of G. In this paper we study the structure of POS-groups with some cyclic Sylow subgroups.

In the present paper, we consider biflatness of certain classes of semigroup algebras. Indeed, we give a necessary condition for a band semigroup algebra to be biflat and show that this condition is not sufficient. Also, for a certain class of inverse semigroups S, we show that the biflatness of ell^{1}(S)^{primeprime} is equivalent to the biprojectivity of ell^{1}(S).

Some necessary and sufficient conditions are given for the existence of a G-positive (G-repositive) solution to adjointable operator equations $AX=C,AXA^{left(astright)}=C$ and $AXB=C$ over Hilbert $C^{ast}$-modules, respectively. Moreover, the expressions of these general G-positive (G-repositive) solutions are also derived. Some of the findings of this paper extend some known results in the literature.

In this paper, we present some common fixed point theorems for two generalized nonexpansive multivalued mappings in CAT(0) spaces as well as in UCED Banach spaces. Moreover, we prove the existence of fixed points for generalized nonexpansive multivalued mappings in complete geodesic metric spaces with convex metric for which the asymptotic center of a bounded sequence in a bounded closed convex subset is nonempty and singleton. The results obtained in this paper extend and improve some recent results.

We obtain the expression of Ricci tensor for a $GCR$-lightlike submanifold of indefinite complex space form and discuss its properties on a totally geodesic $GCR$-lightlike submanifold of an indefinite complex space form. Moreover, we have proved that every proper totally umbilical $GCR$-lightlike submanifold of an indefinite Kaehler manifold is a totally geodesic $GCR$-lightlike submanifold.

We study the limiting distribution of the degree of a given node in a scaled attachment random recursive tree, a generalized random recursive tree, which is introduced by Devroye et. al (2011). In a scaled attachment random recursive tree, every node $i$ is attached to the node labeled $lfloor iX_i floor$ where $X_0$, $ldots$, $X_n$ is a sequence of i.i.d. random variables, with support in [0, 1) and distribution function $F$. By imposing a condition on $F$, we show that the degree of a given node is asymptotically normal.

The non-split extension group $overline{G} = 5^3{^.}L(3,5)$ is a subgroup of order 46500000 and of index 1113229656 in Ly. The group $overline{G}$ in turn has L(3,5) and $5^2{:}2.A_5$ as inertia factors. The group $5^2{:}2.A_5$ is of order 3 000 and is of index 124 in L(3,5). The aim of this paper is to compute the Fischer-Clifford matrices of $overline{G}$, which together with associated partial character tables of the inertia factor groups, are used to compute a full character table of $overline{G}$. A partial projective character table corresponding to $5^2{:}2A_5$ is required, hence we have to compute the Schur multiplier and projective character table of $5^2{:}2A_5$.