In this paper, we shall obtain two results on the class of far field patterns corresponding to the scattering of time harmonic acoustic plane waves by an inhomogeneous medium of compact support. Although the problem of characterizing the class of far field patterns is of basic importance in inverse scattering theory, very little is known about this class other than the fact that the far field patterns are entire functions of their independent (complex) variables for each positive fixed value of the wave number. In particular, the class of far field patterns is not all of L2(∂Ω) where ∂Ω is the unit sphere and this implies that the inverse scattering problem is improperly posed since the far field patterns are, in practice, determined from inexact measurements. The purpose of this paper is to show that while the class of far field patterns corresponding to the scattering of time harmonic plane waves by an inhomogeneous medium is not all of L2(∂Ω), it is dense in L2(∂Ω) for sufficiently small values of the wave number. In addition, a related result will be obtained for a special translation of the class of far field patterns. Analogous results for the scattering of time harmonic acoustic waves by a homogeneous scattering obstacle have recently been obtained by Colton [1], Colton and Kirsch [2], Colton and Monk [3, 4] and Kirsch [8].

The purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups.

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

A group G is called a Bn-group, if all its subnormal subgroups are of defect n. If all infinite subnormal subgroups of G have defect n, we say that G is an IBn group. DeGiovanni and Franciosi [2] have considered soluble IBn groups and characterized them.

In an earlier paper [4] we considered the question of whether an injective module E over a noncommutative ring R remains injective after localization with respect to a denominator set X in R. A related question is whether, given an essential extension N of an R-module M, the localization N[X–1] must be an essential extension of M[X–1]. In [1] it is shown that if R is left noetherian and X is central in R, then localization at X preserves both injectivity and essential extensions of left R-modules and, hence, preserves injective hulls and minimal injective resolutions.

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

Symmetry theory is of fundamental importance in studying systems of partial differential equations. At present algebras of classical infinitesimal symmetry transformations are known for many equations of continuum mechanics [1, 2, 4]. Methods foi finding these algebras go back to S. Lie's works written about 100 years ago. Ir particular, knowledge of symmetry algebras makes it possible to construct effectively wide classes of exact solutions for equations under consideration and via Noether's theorem to find conservation laws for Euler–Lagrange equations. The natural development of Lie's theory is the theory of “higher” symmetries and conservation laws [5].

For an inverse semigroup S we will consider the lattice of inverse subsemigroups of S, denoted L(S). A major problem in algebra has been that of finding to what extent an algebra is determined by its lattice of subalgebras. (See, for example, the survey article [9]). By a lattice isomorphism (L-isomorphism, structural isomorphism, or projectivity) of an inverse semigroup S onto another T we shall mean an isomorphism Φ of L(S) onto L(T). A mapping φ from S to T is said to induce Φ if AΦ = Aφ for all A in L(S). We say that S is strongly determined by L(S) if every lattice isomorphism of S onto T is induced by an isomorphism of S onto T.

A group G is said to be quasi-injective if, for each subgroup H of G and homomorphism θ:H→G, there is an endomorphism such that . It is of course well known that the category of groups does not possess non-trivial injective objects and so we consider groups satisfying the weaker condition of quasi-injectivity.

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.

In this paper we shall deal with radicals γ of near-rings such that for every near-ring N its radical γ(N) contains all left ideals (left invariant subgroups, respectively) I of N with I∈γ. At first, examples of such radicals will be given. Then we shall prove that these radicals are hypersolvable and have hereditary semisimple classes.

Let H be any group. We call a cardinal number r the rank r(H) of H if H can be generated by a generating system X with cardinal number r but not by a generating system Y with cardinal number s less than r. Let r(H) be the rank of H.

We call a generating system X of H a minimal generating system (M.G.S.) of H if X has the cardinal number r(H).

In this note we prove the following.

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.

In this note we discuss extensions of results [5], where transient random walks are considered, whose transition matrix is compatible with a tree-structure of the underlying discrete state space. Notation is generally as in [5], with the exception that instead of a tree T we consider an arbitrary graph Γ, which is locally finite, connected, and whose vertex set S is infinite. The edge set E is unoriented, there are no multiple edges. We consider a transition operator P, given by a stochastic matrix (p(u, v))u, v ∈ S, which gives rise to a transient Markov chain (“random walk”), related to the graph structure by:

(i) if p(u, v)>0 then [u, v] ∈ E,

(ii) there is an M>0 such that p(k)(u, v)>0 for some k=k(u, v)≦M whenever [u, v]∈E (“uniform irreducibility”).

In [1], Calderón proved that, if u is a harmonic function on Rn × ]0, ∞[, and at each point ξ of a subset E of Rn, u is bounded in some cone with vertex (ξ, 0), then u has a nontangential limit at almost every point of E × {0}. The main result of this note is a stronger version of this theorem, in which the hypotheses remain unchanged but the nontangential limits in the conclusion are replaced by limits through the more general approach regions first considered by Nagel and Stein in [7].

In any extension theory for semigroups one must determine the basic building blocks and then discover how they fit together to create more complicated semigroups. For example, in group theory the basic building blocks are simple groups. In semigroup theory however there are several natural choices. One that has received considerable attention, particularly since the seminal work on inverse semigroups by Munn ([14, 15]), is the notion of a fundamental semigroup. A semigroup is called fundamental if it cannot be [shrunk] homomorphically without collapsing some of its idempotents (see below for a precise definition).

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).

We shall concern ourselves with the class of continuous, four-parameter, one-sided probability distributions which can be characterized by the probability density function (pdf) class

It depends on the four parameters: shift c ∈ R, scale b > 0, initial shape p < 1, and terminal shape β > 0. For p ≦ 0, the definition of f(x) can be completed by setting f(c) = β/bΓ(β−1)>0 if p = 0, and f(c) = 0 if p < 0. For 0 < p < 1, f(x) remains undefined at x = c; f(x)↑ + ∞ as x↓c.

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].