Journal of Mathematical Extension
Volume:5 Issue: 1, Winter 2011

The main purpose of this paper is to study a general norm on extension of a Hilbert’s type linear operator in the continuous and discrete form. In addition to expressing the norm of a Hilbert’s type linear operator T : L2(0,1) ! L2(0,1), a more general case with  > 0, for the continuous form has been studied. By putting  = 1 a norm of extension of Hilbert’s integral linear operator is obtained. Similar results have been expressed for series when 0 <  6 2.

In this paper we introduce a new concept of autosoluble groups, which is in a way a generalized version of the notion of soluble groups. Using the autocommutators, a new series will be constructed, which is some how a generalization of the derived series of a given group G. We then determine the structure of such groups, when the generalized series are terminated.

This paper presents the application of the Homotopy Anal- ysis Method (HAM) and Homotopy Perturbation Method (HPM) for solving systems of integral equations. HAM and HPM are two ana- lytical methods to solve linear and nonlinear equations which can be used to obtain the numerical solution. The HAM contains the auxiliary parameter h, provide us with a simple way to adjust and control the convergence region of solution series. The results show that HAM is a very e±cient method and that HPM is a special case of HAM.

The theory of integral equation is one of the major topics of applied mathematics. The main purpose of this paper is to introduce a numerical method based on the interpolation for approximating the solution of the second kind linear Fredholm integral equation. In this case, the divided di®erences method is applied. At last, two numerical examples are presented to show the accuracy of the proposed method.

The topic of ”Homorooty” (for integer numbers) has been introduced and studied in [2]. There are some applications of the homorooty in studying and solving some Diophantine equations and systems, as an interesting and useful elementary method. As a continuation of the Homorooty, we consider it for arbitrary rings and will study its properties in different rings, especially UFD and homorooty rings (which will be introduced). At last we shall state some applications of homorooty in studying some equations over homorooty rings.

In this paper, we will obtain an efficient computable upper bound for approximate solution of linear Fredholm integral equations obtained by Adomian decomposition method. Numerical examples are presented to show the effectiveness of the upper bounds.

Let R = n>0Rn be a graded Noetherian ring with local base ring R0 and let R+ = n>1Rn. Let M and N be finitely generated graded R-modules. In this paper we extend some of the known results about ordinary local cohomology modules HiR +(M) to generalized local cohomology modules HiR +(M,N). Indeed, among other things, we prove that certain submodules and factor modules of HiR +(M,N) are Artinian for some i. Also we obtain some results on the asymptotic behaviour of the n-th graded components HiR +(M,N)n of HiR +(M,N) for n −! −1.

In this paper, we consider different types of generalized vector variational-like inequalities and study the relationships between their solutions. We study the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for the Clarke’s subdifferential of non-differentiable locally Lipschitz functions and prove the existence of their solutions. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities.

The asymptotic distribution for the ratio of sample pro- portions in two independent bernoulli populations is introduced. The presented method can be used to derive the asymptotic con¯dence in- terval and hypothesis testing for the ratio of population proportions. The performance of the new interval is comparable with similar con¯- dence intervals in the large sample cases. Then the simulation study is provided to compare our con¯dence interval with some other meth- ods. The proposed con¯dence set has a good coverage probability with a shorter length.

We give sufficient conditions for the boundedness of the analytic projection on the set of multipliers of the Banach weighted Hardy spaces. This presents the sufficient conditions to a problem that has considered by A. L. Shields.