Let G be one of the following compact simply connected Lie groups: SU(3), Sp(2), G2. In the first two cases there is a well known stable decomposition of G as Q ∨ Sd where d = dim G and Q is a certain subspace of G. For SU(3), Q is the stunted complex quasiprojective space Σ(ℂP2/ℂP1) which fits into a cofibration sequence S3→Q→S5 with stable attaching map η:S5 → S4 For Sp(2), Q is the quaternionic quasi-projective space ℍℚ1 and fits into a cofibration sequence S3→Q→S7 with stable attaching map 2ν:S7→S4 (Here η and ν are generators of respectively.)

The Stieltjes transformation of generalized functions was investigated by Benedetto[1], Zemanian [16], Misra [10] Pandey [11], Lavoine and Misra [7], and Erdélyi [5]. The inversion theorems for the Stieltjes transform of generalized functions, in their own approaches, were given in Benedetto [1], Zemanian [16], Pandey [11], Pathak [12] (he used Pande's approach) and Erdélyi [5].

Soient K un anneau commutatif à élémént unité, P un K-module unitaire, SK(P) l'algèbre symétrique de P et le sous-K-module de SK(P) des éléments homogénes de degré m≧0. Rappelons que si P est un K-module libre de rang n + 1, l'algèbre SK(P) est isomorphe à l'algèbre des polynômes K[X0,X1,…,Xn] en les indéterminéesX0,X1,…,Xn à coefficients dans K et est le K-module libre dont une base est formée par les monômes tels que ie., Par les moôomes homogènes de degré m.

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but when

where S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1]investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.

The purpose of the present note is to answer the following question of T. A. Gillespie,learned from G. J. Murphy [4]: for which sequences{an} of complex numbers does there exist a quasinilpotent operator Q on a (separable, infinite-dimensional, complex) Hilbert space H, which has{an} as a diagonal, that is (Qen,en)=n for some orthonormal basis{en} in H?

Let Fn = 〈 a1,a2,…,an〉 denote the free group of rank n, and let θ denote the automorphism of Fn which permutes the generators cyclically, in other words:

The stochastic stability of linear systems driven by white noise has been treated by several authors e.g. Has'minskii [7], Kushner [11], Kleimann and Arnold [9], Pinsky, [14], Friedman and Pinsky [4], Itô and Nisio [8] , Mohammed [13]. Following R. Kubo [10, 1966], the Brownian movement of a molecule in a “heat bath” is modelled in [4, pp. 223–226] by an asymptotically stable linear f.d.e. which is forced by white noise.

Let B(H) be the algebra of bounded linear operators on a complex separable Hilbert space H. The problem of operator approximation is to determine how closely each operator T ∈B(H) can be approximated in the norm by operators in a subset L of B(H). This problem is initiated by P. R. Halmo [3] when heconsidered approximating operators by the positive ones. Since then, this problem has been attacked with various classes L: the class of normal operators whose spectrum is included in a fixed nonempty closed subset of the complex plane [4], the classes of unitary operators [6] and invertible operators [1]. The purpose of this paper is to study the approximation by partial isometries.

If G is a one-relator group on at least 3 generators, or is a one-relator group with torsion on at least 2 generators, then it follows from results in [1] and [6] that G has a subgroup of finite index which can be mapped homomorphically onto F2, the free group of rank 2. In the language of [2], G is equally as large as F2, written G⋍F2.

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.