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)

An analogue of the group algebra of a finite group is defined for any finite projective plane, in the case where the characteristic of the scalar field divides the order of the plane.

We study the evaluation of multilinear forms under arbitrary rearrangements of the entries of increasing n-tuples x(1), x(2),…, x(m), and we show that the difference of two such multilinear forms under certain circumstances can be written as a sum of obviously definite forms.

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

A new proof is given that all regular biordered sets come from regular semigroups.

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 prove existence and uniqueness of solutions of i(∂ψ/∂t) = (−Δ+x1g(t)+q(x))ψ, ψ(x, s) = ψs (x) in ℝ3 for potentials q(x) including the Coulomb case. Existence and completeness of the wave operators is established for g(t) periodic with zero mean and q(x) short-range, smooth in the x1 direction. We characterize scattering and bound states in terms of the period operator.

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.

Functionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.

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.

The implication of Dirac's large numbers hypothesis (LNH) that there are two cosmological spacetime metrics, gravitational (E) and atomic (A), is used to formulate the gravitational laws for a general mass system in atomic scale units within such a cosmology. The metric is constrained to be asymptotic to the cosmological A metric at large distance. The gravitational laws are illustrated in application to the case of a single spherical mass immersed in the smoothed out expanding universe. The condition is determined for such a metric to apply approximately just outside a typical member of a cosmic distribution of such masses. Conversely, the condition is given when the influence of the universe as a whole can be neglected outside such a mass. In the latter situation, which applies in particular to stars, a Schwarzschild-type metric is derived which incorporates variable G in accordance with the LNH. The dynamics of freely moving particles and photons in such a metric are examined according to the theory and observational tests are formulated.

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.

The simple 2q-knots, q ≧ 5, for which contains no ℤ-torsion, have been classified in terms of Hermitian duality pairings on their homology and homotopy modules. In this paper, a necessary and sufficient condition is given for such a knot to be doubly-null-concordant.

Critical Rayleigh numbers are obtained for the onset of convection when the Maxwell–Cattaneo heat flux law is employed. It is found that convection is possible in both heated above and below cases.

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.

Page 56: In the estimate , C1 = CSn/(2πσ)n, h = (2πσ)−1 where σ depends on y. Therefore, {fy(t)} may not be bounded in as y → 0, y ∈ C′.