Iranian Journal of Numerical Analysis and Optimization
Volume:1 Issue: 1, Winter and Spring 2008

Certain algebraic invariants of the integral group ring ZG of a group G were introduced and investigated in relation to the problem of extending the Farrell-Tate cohomology, which is defined for the class of groups of finite virtual cohomological dimension. It turns out that the finiteness of these invariants of a group G implies the existence of a generalized Farrell-Tate cohomology for G which is computed via complete resolutions.
In this article we present these algebraic invariants and their basic properties and discuss their relationship to the generalized Farrell-Tate cohomology. In addition we present the status of conjecture which claims that the finiteness of these invariants of a group G is equivalent to the existence of a finite dimensional model for EG, the classifying space for proper actions

In this article, we consider some new classes of groups, namely, Mpgroups,T0-groups, -groups, 0-groups, groups with finitely embedded involution, which were appeared at the end of twenties century. These classes of infinite groups with finiteness conditions were introduced by V.P. Shunkov.We give some review of new results on these classes of groups.

A classical result of Neumann characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups. If X is a class of groups, a group G is said to have X-conjugate classes of subgroups if G/coreG(NG(H)) 2 X for each subgroup H of G. Here we study groups which have soluble minimax conjugate classes of subgroups,
giving a description in terms of G/Z(G). We also characterize FC-groups which have soluble minimax conjugate classes of subgroups.

In this paper, we demonstrate the existence and uniqueness a semianalytical solution of an inverse heat conduction problem (IHCP) in the form : ut = uxx in the domain D = {(x, t)| 0 < x < 1, 0 < t  T}, u(x, T) = f(x), u(0, t) = g(t), and ux(0, t) = p(t), for any 0  t  T. Some numerical experiments are given in the final section.

In an earlier work we showed that for ordered fields F not isomorphic to the reals R, there are continuous 1-1 functions on [0, 1]F which map some interior point to a boundary point of the image (and so are not open). Here we show that over closed bounded intervals in the rationals Q as well as in all non-Archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 functions not mapping interior to interior. In particular, the minimal non-Archimedean ordered field Q(x), as well as ordered Laurent series fields with coefficients in an ordered field accommodate such pathological
functions.

In this paper, an extension of record models, well known as k-records is considered. Bayesian estimation as well as prediction based on k-records are presented when the underlying distribution is assumed to have a general form. The proposed procedure is applied to the Exponential, Weibull and Pareto models in one parameter case. Also, the two-parameter Exponential distribution, when both parameters are unknown, is studied in more details. Since the ordinary record values are contained in the k-records, by putting k = 1, the results for usual records can be obtained as special case.

In the linear model y=X\beta with the errors distributed as normal, we obtain generalized least square (GLS) , restricted GLS (RGLS), preliminary test (PT), Stein-type shrinkage (S) and positive-rule shrinkage (PRS) estimators for regression vector
parameter \beta when the covariance structure in known. We compare the quadratic risks of the underlying estimators and propose the dominance orders of the five estimator