The following paper defines a new type of ordering of higher level on a commutative ring. This definition allows the set of all orderings of level n to be given a topology which we show is consistent with the topology of the real spectrum.

Given a bivariate function defined on some subset of the Cartesian product of two sets, it is natural to ask when that function can be decomposed as the sum of two univariate functions. In particular, is a pointwise limit of such functions itself decomposable? At first glance this might seem obviously true but, as we show, the possibilities are quite subtle. We consider the question of existence and uniqueness of such decompositions for this case and for many generalizations to multivariate functions and to cases where the sets and functions have topological or measure theoretic structure.

In the past few years, much progress have been made on several open problems in infinite dimensional Banach space theory. Here are some of the most recent results:

1. 1) The existence of boundedly complete basic sequences in a large class of Banach spaces including the ones with the so-called Radon-Nikodym property ([G-M2], [G-M4]).

2. 2) The embedding of separable reflexive Banach spaces into reflexive spaces with basis (fZl).

3. 3) The existence of long sequences of projections and hence of locally uniformly convex norms in the duals of Asplund spaces. ([F-G])

Extreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

Let Mn, c denote the free n-generator metabelian nilpotent group of class c. For m ≤ n – 2, every primitive system of m elements of Mn, c can be lifted to a primitive system of m elements of the absolutely free group Fn of rank n. The restriction on m cannot be improved.

Let E′ be the dual of a nuclear Fréchet space E and L*(t) the adjoint operator of a diffusion operator L(t) of infinitely many variables, which has a formal expression:

A weak form of the stochastic differential equation

dX(t) = dW(t) + L*(t)X(t)dt

is introduced and the existence of a unique solution is established. The solution process is a random linear functional (in the sense of I. E. Segal) on a space of generalized functionals on E′. The above is an appropriate model for the central limit theorem for an interacting system of spatially extended neurons. Applications to the latter problem are discussed.

In this paper we consider various degree two central polynomials with derivation, holding for Lie ideals in prime rings. The results give substantial generalizations of the existing ones on central and semi-centralizing derivations, and show essentially that there are no central identities of the form p(x,y) = c1xyD + C2XDy + c3yxD + C4yDx, where D is a nonzero derivation of the prime ring R.

We investigate convergence in a weighted L1 -norm of Hermite-Fejér and Hermite interpolation at the zeros of orthogonal polynomials associated with weights on the real line. The results are then applied to convergences of product integration rules. From the point of view of orthogonal polynomials, the new feature is that Freud and Erdös weights are treated simultaneously and that relatively few assumptions are placed on the weight. From the point of view of product integration, the rules exhibit convergence for highly oscillatory kernels (for example) and for functions of rapid growth at infinity.

New computations of birelative K2 groups and recent results on K3 of rings of algebraic integers are combined in generalized Mayer-Vietoris sequences for algebraic k-theory. Upper and lower bounds for SK2(ℤ G) and lower bounds for K3(ℤ G) are deduced for G a dihedral group of square-free order, and for some other closely related groups G.

The main topic of this paper is the study of the fundamental solution of the heat equation for the symmetric spaces of positive definite matrices, Pos(n,R).

Our first step is to develop a “False Abel Inverse Transform” which transforms functions of compact support on an euclidean space into integrable functions on the symmetric space. The transform is shown to satisfy the relation is the usual Laplacian with a constant drift).

Using this transform, we find explicit formulas for the heat kernel in the cases n = 2 and n = 3. These formulas allow us to give the asymptotic development for the heat kernel as t tends to infinity. Finally, we give an upper and lower bound of the same type for the heat kernel.

Let Sbe a topological semigroup, K a compact convex subset of a separated convex space Eand T: S x K → K an affine action (denoted by (s, x) → Ts(x),s ∈ S, x ∈ K) of S as continuous affine maps on K. It is shown in A. Lau and J. Wong [22] that the weakly left uniformly measurable functions WLUM(S) on S has a left invariant mean iff Shas the fixed point property for weakly measurable affine actions, i.e. affine actions such that the scalar function s → x*Ts(x) is measurable for each x ∈ K and x*∈ E* (the dual of E) with respect to the Borel sets in S. It is natural to ask for a “strongly” measurable analogue of this result. There are a number of ways to define such actions and the corresponding functions on S. In this paper, we obtained a neat analogue of this fixed point theorem by a suitable choice of strong measurability which naturally leads to another new fixed point theorem for separable actions. Also, we shall unify these and many known fixed point theorems and extend and generalise them to anti-actions of S as bounded linear operators on Banach spaces

By exploiting the known quasi-heredity of Schur algebras, the structure of basic algebras of the Schur algebras Sk(n, p) for n ≥ p over an algebraically closed field k is completely determined.