Iranian Journal of Numerical Analysis and Optimization
Volume:2 Issue: 1, Winter and Spring 2009

If $G$ is a finite linear group of degree $n$, that is, a finite group of automorphisms of an $n$-dimensional complex vector space, or equivalently, a finite group of non-singular matrices of order $n$ with complex coefficients, we shall say that $G$ is a quasi-permutation group if the trace of every element of $G$ is a non-negative rational integer. By a quasi-permutation matrix we mean a square matrix over the complex field $C$ with non-negative integral trace. Thus every permutation matrix over $C$ is a quasi-permutation matrix. For a given finite group $G$, let $c(G)$ denote the minimal degree of a faithful representation of $G$ by quasi-permutation matrices over the complex numbers and let $r(G)$ denote the minimal degree of a faithful rational valued complex character of $G$. The purpose of this paper is to calculate $c(G)$ and $r(G)$ for the Borel and parabolic subgroups of Steinberg''s triality groups.

In this paper, we consider groups with points which were introduced by V.P. Shunkov in 1990. In Novikov-Adian''s group, Adian''s periodic products of finite groups without involutions and Olshansky''s periodic monsters every non-unit element is a point. There exist groups without points. In this article we shall prove some properties of the groups with points.

Classical control methods such as Pontryagin Maximum Principle and Bang-Bang Principle and other methods are not usually useful for solving {\it optimal control systems} (OCS) specially {\it optimal control of nonlinear systems} (OCNS).
In this paper, we introduce a new approach for solving OCNS by using some combination of atomic measures. We define a criterion for controllability of lumped nonlinear control systems and when the system is nearly null controllable, we determine controls and
states. Finally we use this criterion to solve some numerical examples.

In this paper, the variational iteration method proposed by Ji-Huan He is applied to solve both linear and nonlinear Schrodinger equations. The main property of the method is in its flexibility and ability to solve linear and nonlinear equations accurately and conveniently. In this method, general lagrange multipliers are introduced to construct correction functionals to the problems. The multipliers in the functionals can be identified optimally via the variational theory. Numerical results show that this method can readily be implemented with excellent accuracy to linear and nonlinear Schrodinger equations. This technique can be extended to higher dimensions linear and nonlinear Schrodinger equations without a series difficulty.

In some long term studies, a series of dependent and possibly truncated lifetimes may be observed. Suppose that the lifetimes have a common marginal distribution function. In left-truncation model, one observes data $(X_{i},T_{i})$ only, when $T_{i}\leq X_{i}$. Under some regularity conditions, we provide a strong representation of the $\hat{\beta}_{n}$ estimator of$\beta=P(T_{i}\leq X_{i})$, in the form of an average of random variables plus a remainder term. This representation enables us to obtain the asymptotic normality for $\hat{\beta}_{n}$.

In this paper, the geometric distribution is considered. The means, variances, and covariances of its order statistics are derived. The Fisher information in any set of order statistics in any distribution can be represented as a sum of Fisher information in at most two order statistics. It is shown that, for the geometric distribution, it can be further simplified to a sum of Fisher information in a single order statistic. Then, we derived the asymptotic Fisher information in any set of order statistics.

This paper deals with the estimation of P(X \less Y), where Y has generalized exponential distribution with parameters $\alpha$ and $\lambda$ and X has mixture generalized exponential distribution(or marginal distribution of $X_1,X_2,...,X_n$, in presence of one outlier with parameters $ \beta_1$ and $\beta_2$) such that X and Y are independent. when the scale parameter $(\lambda)$ is known the maximum likelihood estimator of $R=P(Y