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 = ∞.

Let K be a field, G a finite group, V a (right) KG-module. If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.

Suppose N is a normal subgroup of G. The KN-module VN is not irreducible in general, even when V is irreducible as KG-module. A part of the well-known theorem of A. H. Clifford [1, V.17.3] yields the following.

The theory of Fitting classes is, by now, a well established part of the theory of finite soluble groups. In contrast, Fitting classes have received rather scant attention in infinite groups, although some recent work of Beidleman and Karbe [2] and Beidleman, Karbe and Tomkinson [3] suggest that one can obtain results in this direction. The paper [2], cited above, in fact generalizes earlier work of Tomkinson [9] to the class of locally soluble FC-groups. The present paper is concerned with the theory of Fitting classes in a class of groups somewhat similar to the class of FC-groups, namely the class of CC-groups, introduced by Polovickiǐ in [6]. A group G is a CC-group if G/CG(xG) is a Černikov group for all x ∈ G where, as in the rest of this paper, we use the standard group theoretic notation of [7]. Recently, Alcázar and Otal [1] have shown how to generalize results of B. H. Neumann [5] to the class of CC-groups. The main purpose of the present note is to illustrate further how one can handle CC-groups, in an analogous manner to FC-groups, by using techniques similar to those used in [1] and [4].

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

In this article we study the multiparameter generalization of standard deficiency index theory. A classical result in this area states that if T is a symmetric operator in a Hilbert space then the dimension of the null space of T*−λI, λ∈ℂ, is constant for λ belonging to the upper (or lower) half-plane and further, when these two constants are equal, T admits a self-adjoint extension.

The Takai duality theorem has proved to be a fundamental tool in the theory of crossed products of C*-algebras. It was inspired by Takesaki's duality theorem for crossed products of von Neumann algebras [7], so it is not surprising that the original proof [6] depended heavily on spatial techniques. Here we shall prove Takai's theorem by exploiting the universal properties of crossed products.

If a function f analytic in Δ = {z∈ℂ:|z|<1} has a nontangential limit as z→eiθ, then limr→1−(1−r)f′(reiθ)=0 [7, p. 181). It follows that this limit is zero for almost all θ for a number of classes of functions including the set H∞ of bounded analytic functions. In this paper we prove that this result for H∞ is sharp in a strong sense.