In this paper we investigate the p-periodicity of the S-arithmetic groups G = GL(n, Os(K)) and G1 = SL(n, Os(K)) where Os(K) is the ring of S-integers of a number field K (cf. [12, 13]; S is a finite set of places in K including the infinite places). These groups are known to be virtually of finite (cohomological) dimension, and thus the concept of p-periodicity is defined; it refers to a rational prime p and to the p-primary component Ĥi(G, A, p) of the Farrell-Tate cohomology Ĥi(G, A) with respect to an arbitrary G-module A. We recall that Ĥi coincides with the usual cohomology Hi for all i above the virtual dimension of G, and that in the case of a finite group (i.e., a group of virtual dimension zero) the Ĥi, i ∈ℤ, are the usual Tate cohomology groups. The group G is called p-periodic if Ĥi(G, A, p) is periodic in i, for all A, and the smallest corresponding period is then simply called the p-period of G. If G has no p-torsion, the p-primary component of all its Ĥi is 0, and thus G is trivially p-periodic.

The following theorem is proved in [2[.

Let K be a finitely generated field over its prime field. Then for almost all e-tuples σ = (σ1, …, σe) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the following results hold.

In July 1982, I was asked by Prof. Jorgen Hoffmann-Jorgensen to construct an uncountable compact set K in the line which was symmetric about 0 and had the property that, for all n, the set of sums of n-tuples from K has measure 0. There are two equivalent conditions: the set of such sums should never contain an interval, or K* ≠ ℝ, where K* is the subgroup of (ℝ, +) generated by K. I did so, and the set I constructed had entropy dimension 0 (and thus also Hausdorff dimension 0). Hoffmann-Jorgensen showed that every set of entropy dimension 0 would exhibit the same behaviour. However, I did not believe that the essence of the example lay in its dimension, and I here modify my construction so that the set K has dimension 1 (and thus also entropy dimension 1), while K* ≠ ℝ, as before. By contrast, the Cantor ternary set has dimension log3(2), but the set of differences is the interval [ –1, 1], so that it does generate ℝ. It follows that the property under consideration is arithmetical rather than dimensional.

It is proved that a bounded operator on Hilbert space is the sum of two quasi-nilpotent operators if and only if it is not a non-zero scalar plus a compact operator. Necessary conditions and sufficient conditions for an operator to be the product of two quasi-nilpotent operators are given.

In this paper, a new variational formulation of the Signorini problem with friction is given in terms of the contact stresses. The method corresponds to the direct integral equation approach in classical elastostatic problems. First the displacement and mixed problems are briefly described together with some numerical results. Next the displacements are eliminated by the use of Green's function, and a constrained minimum problem with respect to the normal and tangential tractions on the contact boundary is derived. Then the resulting approximation procedure is studied and certain convergence results are proved. Finally, some remarks on the Signorini problem with Coulomb friction are presented. Numerical results illustrate the theory.

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.

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.

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

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.

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′.

We prove in this paper that on a non-singular projective variety, the χ-semistable functor is proper and the χ-stable functor is separated. This result was proved for μ-stability and μ-semistability by Langton. An essential part of our proof consists in defining a notion of stability between the μ and χ definitions and then proceeding by induction.

Order-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

A short survey is given of some recent results. The perturbations and stability of discrete spectrum and the problems of resolvent convergence of densely or non-densely defined elliptic differential operators are considered. The Courant theorem on variations of the domain is generalized. In connection with Berezanskiy's theorem on essential self-adjointness, the test for the finite velocity of propagation is extended. The Frobenius and the Krein-Heinz-Rellich factorization theorems and the Etgen Pawlowski oscillation criterion are generalized for equations of any order with operator-valued coefficients. Brusentsev's recent example of a two term fourth order differential operator with deficiency index 4 is discussed.

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.

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.

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 article deals with the uniform spaces (X, μ) such that μ is a K-analytic subset of 2X×X. G. Godefroy considered this situation for X countable, in his study of certain compact sets of measurable functions, and some of his results are extended here. We prove that the uniformity of an Eberlein compact is K-analytic, and give some applications.

A general framework is presented for the proof of the existence of classical solutions of second order elliptic equations which satisfy non-linear boundary conditions. The results obtained contain many of the known theorems for such problems and the approach used unifies the various methods of study based upon upper and lower solutions.