We give an inequality for the concentration function of a sum X1 + … + Xn of independent random variables when Xv has a finite absolute moment of order kv (2 < kv ≦ 3). It is an extension of somewhat similar inequalities found earlier by Offord and by the author in the case of finite third-order absolute moments.

Throughout this article, the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. We construct an ideal called the socle-ideal of N which is antiradical in the sense that it is a direct sum of minimal left ideals and annihilates one or more of the radicals of N. We then use this socle-ideal to obtain a decomposition theorem for the s-radical JS(N) of N.

This work provides an account of the asymptotic behaviour in terms of Abelian theorems for the Mellin and inverse Mellin transforms in a distributional setting.

This note complements an earlier paper of the same title. Let G be a discontinuous group of homeomorphisms of a connected, locally path connected, Hausdorff space X, and let ∏:X → X/G denote the associated projection. We work relative to a G-invariant subgroup H of the fundamental group of X and investigate the quotient group ∏1(X/G)/∏*(H). By choosing H appropriately, we can calculate ∏1(X/G) and show that ∏1(X/G)/∏*(∏1(X)) is isomorphic to G/F, where F is the normal subgroup of G generated by those elements which have fixed points. In a final section, we give analogous results for actions of a compact Lie group.

For any integer k such that 0≦k≦m, Mk denotes the Grassmann bundle of tangent k-planes on the m-manifold M. A k-spread on M is a field Φ of tangent k-planes on Mk such that the derivative of the projection maps Φ(λ) to λ. Previous work by Douglas and others studied the local properties of such spreads. Here we develop the global theory, with special emphasis on the case in which Φ is integrable.

In the general theory of ordinary linear quasi-differential equations, the set of Shin–Zettl matrices plays an important role. This paper displays certain properties of these matrices and their behaviour under a special form of transformation. Essentially, the problems can be considered within the framework of linear algebra.

We give two generalizations of a theorem of L. Amerio and G. Prouse concerning uniqueness of the almost-periodic solution of the wave equation with a local multivalued damping term and almost-periodic forcing.

Counterexamples are given which show that these results may fail if some hypotheses of the theorems are dropped. They also show that the two generalizations are relatively independent.

A result for the Erdélyi-Kober operators, mentioned briefly by Buschman, is discussed together with a second related result. The results are proved rigorously by means of an index law for powers of certain differential operators and are shown to be valid under conditions of great generality. Mellin multipliers are used and it is shown that, in a certain sense, the index law approach is equivalent to, but independent of, the duplication formula for the gamma function. Various statements can be made concerning fractional integrals and derivatives which produce, as special cases, simple instances of the chain rule for differentiation and changes of variables in integrals.

We consider classes of functions satisfying certain simple criteria of sign and smoothness and a decomposition property. It is known that these properties are possessed by Chebysheffian B-splines and it is shown here that they are also possessed by certain trigonometric B-splines. For such a class of functions, we derive a variation-diminishing property and analyse interpolation both on a finite set of nodes and on an infinite, periodically spaced set of nodes. The results are also applied to interpolation by complex polynomial splines on the circle.

Let T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the cone

related to solubility of the multiparameter eigenvalue problem

with W(λ)−I nonpositive definite. The main result is as follows.

Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.

We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

Let Ω = ℝN or Ω be a bounded regular open set of ℝN and let γ(x, S): Ω × ℝ → ℝ be a continuous nondecreasing function in s, measurable in x, such that γ(x, 0) = 0 almost everywhere. We solve, for f ∈ L1(Ω), the problem (P): −Δu + γ(., u) = f in Ω, u = 0 on ∂Ω. (In fact, for this result, instead of assuming that γ is nondecreasing in s we need only that γ(x, s)s≧0.) We deduce an ‘almost’ necessary and sufficient condition on , in order that (P) has a solution. Roughly speaking, this condition is f = −ΔV + g, with g ∈ L1(Ω) and γ(., V) ∈ L1(Ω)

A detailed description of the totality of distributional solutions of the catenary problem for nonlinear elastic strings is given.

We extend several known properties of the Dirichlet index to the case of minimal conditions and prove that the index is invariant under positive t bounded perturbations of the associated quadratic form. The index is also shown to be minimal for fourth order operators with certain growth conditions on some of their coefficients.

We exhibit dimension-independent conditions under which the formal operator A = −Δ + a.∇ + V can be defined on such that its closure Ā in L2(Rs, dx) is quasi-m-accretive. Here, a is real so that Ā is nonselfadjoint. the method of proof is a generalized version of the argument employed in the portion of the author's thesis where term a.∇ was originally considered. Specifically, we construct exp (−tĀ) as a limit of approximating semigroups. Since the thesis appeared, Kato has also dealt with the term a. ∇ his conditions on a and V are similar to, but more general than, the conditions that appear here; in addition, he considers magnetic vector potentials. Of interest here is the semigroup method itself, the conciseness of the arguments thereby produced, and a relaxed condition on div a.

Schrödinger operators of the form T = (i grad + b(x))2 + a(x) · grad + q(x) in Rm are considered, where a, b ate real vector-valued functions and q is a scalar complex-valued function. It is shown that T is essentially quasi-m-accretive in L2(Rm) if (1 + #x2223;∣)−1a ∈ L4 + L∞, div a ∈ L∞, , and Re q ≧ 0. The proof is elementary.

This paper deals with some multiplicity results for elliptic problems with jumping nonlinearities. Our results are concerned with the case in which only one eigenvalue of the linear problem is jumped and it is simple. The main tool used is the Leray–Schauder topological degree. We consider a parametrized problem and prove the existence of two or three distinct solutions for suitable values of the parameter.

We prove that classical solutions of the perturbed wave equation in ℝn × ℝ (n = odd ≧ 3) do not satisfy Huygens' principle in the presence of symmetries. The difficulties arising from the singularities of the Riemann function (for large space dimensions) are overcome by considering a class of potentials and initial data which are radial and smooth. Our method is elementary and based on energy estimates.

For each ring R, we construct a topological space pt (R) which includes as a subspace both the classical spectrum specR and the torsion theoretic spectrum R-sp. For many rings (e.g. rings with Krull dimension), spec R is a retract of pt (R) and the retraction map θ generalizes the Gabriel correspondence for noetherian rings. There is a natural decomposition theory on MOD-R which extends the Goldman theory in the same way that the tertiary theory extends the primary theory. The map θ provides a direct comparison between this new decomposition theory and the tertiary theory. The space pt (R) is closely connected with the lattice of hereditary torsion theories on R, and for fully bounded (not necessarily noetherian) R, this connection is very tight.