Introduction. In this paper, a study is made of the solutions of Lamé's differential equation as series of associated Legendre functions. The particular feature studied is the representation of the second solution corresponding to the case when the first solution is a Lamé polynomial.

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.

Let G be a connected perfect real Lie group. We show that there exists α < dim G and p ∈ $\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure on G, then the pth convolution power μ*p is absolutely continuous with respect to the Haar measure on G, with arbitrarily smooth density. As an application, we obtain that if A ⊂ G is a Borel set with Hausdorff dimension at least α, then the p-fold product set Ap contains a non-empty open set.

The purpose of this note is to complement some recent results of R. O. Davies and H.G.Eggleston.

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ and perpendicular distance t from the origin, that is H(t, θ) = {x ε Rn: x.θ = t} (Note that H(t, θ) and H(−t, −θ) are the same hyperplane.) If f(x)ε L1(Rn) we will denote by F(t, θ) the projection of f perpendicular to θ, that is the integral of f(x) over H(t, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ε L1 (Rn), F(t, θ) exists for almost all t for every θ.

The Mellin transforms of generalized hypergeometric functions are discussed in this paper, and it is shown how some of the most general integrals of the Mellin type can be deduced from them. Four general theorems are considered and a number of special cases are given in detail.

We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.

In this paper [1], two corrections must be made.

(1) In the second member of (27), m is to be replaced by m + 1; this agrees with (18), and gives the correct result.

(2) Theorem 2 is mis-stated.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

The algebraical foundation of this representation is very simple: it depends on the theorem

If a coordinate system in [m] is given, then in a general [k] of the [m] a unique set of k + 1 points can be found whose coordinates form the scheme

where h = m − k − 1, and n = (m − k)(k + 1) = kh + m.

Given a system of diagonal forms over ℚp, we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies p > (largest degree) − (smallest degree) + 1, then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms.

The notion of a Gaussian as the exponential of a quadratic is rather familiar. Such functions are of considerable importance in a number of contexts, for example within quantum theory. Thus, in the Schrödinger representation of the canonical commutation relations they alone minimize uncertainty and they appear as ground states for harmonic oscillators. Also in the complex-wave representation of a free boson field they arise as transforms of the Fock vacuum under certain Bogoliubov automorphisms.

1. In the present note we obtain the postulation of a multiple surface of S4 for primals of sufficiently high order; the method is that previously used by Campedelli to determine the postulation of a multiple curve of S3.

An action of a group, G, on a surface, F, consists of a homomorphism

ø: G → Homeo (F).

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.

Within the limitations of the classical small deflexion theory of thin plates and using complex variable methods, exact expressions are obtained in series form for the deflexion at any point of a thin isotropic circular plate simply supported along a concentric circle and subject to loading symmetrically distributed over an eccentric circular patch which lies inside the circle of support. In special and limiting cases the solutions reduce to those obtained before.

This paper presents some new results on algebraic K-theory with finite coefficients. The argument is based on a topological construction of a space $F_mK(R)$, for any ring $R$ and any integer $m\geq 2$, having the property that the ordinary homotopy theory of $F_mK(R)$ is isomorphic to the algebraic K-theory of $R$ with coefficients in $\bb {Z}/m$: $\pi_n(F_mK(R))\cong K_n(R;\bb {Z}/m)$ for $n\geq 1$. This space $F_mK(R)$ is called the $\mod m$ K-theory space of $R$. The paper is devoted to the investigation of several properties of the groups $K_n(R;\bb {Z}/m)$, for $n\in\bb {Z}$, and to some calculations of the integral homology of finite K-theory spaces.

The multiconfiguration method of Grein[3] based on the generalized Brillouin theorem is discussed and analysed, with particular emphasis on the use of molecular symmetry to reduce the amount of computing. Some results concerning the energy decrease in each iteration are also included.

In this paper we give for any r, n, 0 [les ] r [les ] n, a Quillen's model structure to the category of simplicial groups where the weak equivalences are those morphisms f[bull] such that πq(f[bull]) is an isomorphism for r [les ] q [les ] n. This is carried out by studying the cases r = 0 and n → ∞ previously and, in each one of them, we make explicit some constructions for the associated homotopy theories, such as the cylinder and path objects and the loop and suspension functors, and we also relate the simplicial homotopy relation to the homotopy relation obtained from these structures.