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.

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.

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].

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).

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.

Since the work of R. A. Fisher [2] and Kolmogorov, Petrovskij and Piskunov [7] (see [6] for further references) the problem of travelling fronts in reaction–diffusion equations has been extensively studied. For the equation

with F(0) = F(1) = 0 a travelling front is a solution

where the function of one variable φ is decreasing and satisfies φ(−∞) = 1, φ(+∞) = 0. The function φ describes the shape of the front and the constant c is the speed of propagation. There are two main types of the problem. In the all-positive case, where the function F satisfies

there is a half-line [c0, ∞), c0>0, of speeds. For each c∈[c0, ∞) there is, up to translation, a unique travelling front. Fronts for different c can be distinguished by the rate of decay towards +∞. In the threshold case, where F has the property, for some λ∈(0, 1),

there is a unique speed c0 with a travelling front, which is unique up to translation. In this case the sign of c0 is determined by the sign of the integral

Let L(H) be the algebra of all bounded linear operators on a separable complex Hubert space H. In a recent paper [7], explicit expressions for solutions of a boundary value problem in the Hubert space H, of the type

are given in terms of solutions of an algebraic operator equation

In connection with the analysis of mathematical models of real processes undergoing short time perturbations, in the last years the interest in the differential equations with impulses remarkably increased. Going back to the papers of Mil'man and Myshkis [4, 5] the investigations of this subject are now extended to different directions concerning applications in physics, biology, electronics, automatic control etc.

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operators

where λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

In this note we intend to discuss the method of A. Córdoba and R. Fefferman of using covering lemmas to control maximal functions, and make some simplifications which allow us to obtain alternative proofs of some of their results.

This note will complement and, in a certain sense, complete our earlier studies [3, 4] of the theory of inverse multiparameter eigenvalue problems for matrices. In those papers, we considered the so called “additive inverse problem” which, briefly stated for the 2-parameter case, asks for conditions on given n × n matrices A, B, C and on given points (si, ti) ∈ ℝ2, 1 ≦ i ≦ n, under which a diagonal matrix D can be found so that the 2-parameter eigenvalue problem

can be solved when (λ,μ)=(si, ti), 1 = i = n. Put another way, we look for conditions ensuring that the points (si, ti), 1 ≦ i ≦ n, belong to the eigenvalues of (1.1).

The present paper deals with the problem of asymptotic equivalence of the system with impulse effect

and

where x, y:I→Rn; g:I→Rn; f:I × Rn→Rn; bk:Rn→Rn; I = [0, ∞); Rn is the n-dimensional Euclidean space with a norm |·|; A and B are constant matrices; the moments {tk} constitute an increasing sequence 0 < t1 < … < tk < …, limk→∞tk = ∞.