In our article [8] we examined asymptotic mean square stability for linear retarded f.d.e.'s which are perturbed by white noise. It is shown in [8] and [10] that if the deterministic linear retarded f.d.e. is asymptotically stable, then so is the perturbed stochastic f.d.e.

This paper answers two questions posed by P. M. Neumann and K. Hickin on embeddings of infinite permutation groups. We first give some definitions that would make their questions understood.

Strongly prime rings were introduced by Handelman and Lawrence [5] and in [2] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special class in the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincides with the intersection of all the strongly prime ideals of N.

We consider the Dirichlet problem

where g is continuous and h ∈ L2(0, π).

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].

These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .

Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.

We will assume throughout this paper that polynomials are nonconstant. Let P be any complex polynomial and let p denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and multiplication is defined by fg = f ο P ο g for all f,g∈p. The near-ring p is referred to as a laminated near-ring and P is referred to as the laminating element or laminator. In [1] the problem was posed of determining Aut p the automorphism group of p. It was shown that exactly three infinite groups occur as automorphism groups of the laminated near-rings p and for each of the three groups those polynomials P were characterized such that Aut p is isomorphic to that particular group. The infinite groups turn out to be GL(2), the full linear group of all 2×2 nonsingular real matrices and two of its subgroups.

It is characteristic for the development of many processes that in certain moments they change their state in jumps. Systems with impulse effect provide an adequate mathematical model of such processes. The investigation of these systems begins with the paper of Millman and Myshkis [7] and afterwards the number of publications dedicated to this problem rapidly increases.

Let K be a field of characteristic zero and let Δ ={δ1,…,δn} be a set of commuting K-derivations of the commutative Noetherian K-algebra R. Let S = R[X1,…,Xn] be the corresponding ring of differential operators, so [Xi, r] = Xir − rXi=δi(r and [Xi, Xj]=0, for 1≦i, j≦n. Let M be a maximal ideal of R with R/M of finite dimension over K. The purpose of this note is to describe the groups

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

Let ξ be an irrational number with simple continued fraction expansion ξ= [a0;a1,a2,…], Pn/qn be its nth convergent, . The following two theorems were proved by Müller [9] and rediscovered by Bagemihl and McLaughlin [1]:

Theorem 1.For n>1,

One can define “Laurent” and “Hankel” operators relative to groups more general than the circle group . We do that here, derive some of their properties, and compute their spectra, using a concrete realization of a crossed product C*-algebra by the two-element group ℤ2.

We recall the following definition (see [1]):

A finite group G is said to be a Frobenius–Wielandt group provided that there exists a proper subgroup H of G and a proper normal subgroup N of H such that H∩Hg≦N if g∈G–H. Then H/N is said to be the complement of (G, H, N) (see [1] for more details and notation).

We obtain a series representation for ζ(s) valid in Res> – k(<0). This representation is obtained by a sequence of regrouping of the series 1–2−s + 3−s–4−s + …( = (1—21−s)ζ(s), Re s > 0). We can obtain asymptotic relations like

as an application of our series representation for ζ(s).

Let (M, g, J) be an almost Hermitian manifold. More precisely, M is a ∞ differentiable manifold of dimension 2n, J is an almost complex structure on M, i.e. it is a tensor field of type (1, 1) such that

for any X∈(M), ((M) is the Lie algebra of ∞ vector fields on M), and g is a Riemannian metric compatible with J, i.e.

The purpose of this note is to provide a simplified proof of the following result due to Crawley-Boevey, Kropholler and Linnell.

Let G be a group and K a field. We shall denote by U(KG) the group of units of the group ring of G over K. Also, if X is a group, T(X) will denote the torsion subset of X, i.e., the set of all elements of finite order in X.

Group theoretical properties of U(KG) have been studied intensively in recent years and it has been found that some conditions about U(KG) imply that T = T(G) must be a subgroup of G and that every idempotent of KT must be central in KG.

Let R be an associative ring with 1 and G a finitely generated torsion-free abelian group. In this note, we classify all R-automorphisms of the group ring RG. The special case where G is infinite cyclic was previously settled in [8], and our interest in this problem was rekindled by the recent paper of Mehrvarz and Wallace [7], who carried out the classification in the case where R contains a nilpotent prime ideal.

We are concerned with the problem of bifurcation of solutions of a non-linear multiparameter problem at a simple eigenvalue of the linearised problem.

Let X and Y be real Banach spaces, and let A, Bi, i = 1, …, n∈B(X, Y). Let : Rn × X → Y be a non-linear mapping. We consider the equation

where

and λ=(λ1, λ2,…,λn) ∈ Rn is an n-tuple of spectral parameters.

We show that if X is a Banach space of infinite dimension and μh is a Hausdorff measure, where h is continuous, then there exists a measurable set K ⊂ X such that 0<μh(K)< + ∞. We also characterize the normed spaces in which the unit ball can be covered by a sequence of balls whose radii rn < 1 converge to zero as n → ∞.