0. Introduction. Asymptotic behaviour of distributions (Silva(8) and Benedetto (l)) plays a fundamental role in the analysis of singularities of generalized integral transforms. Such analysis in terms of Abelian theorems for the distributional Stieltjes transformation has been obtained by Misra (6) and Lavoine and Misra (3). To obtain his results Misra used some Abelian theorems (see Misra (6), theorems 3·1·1 and 4·1·1) for the Stieltjes transformation of functions which were essentially generalizations of the results given by Widder (9), pp. 183–185. The object of the present paper is to complete the work given by the authors in (3); Sections 2 and 3 of this paper are devoted to obtaining Abelian theorems concerning the distributional Stieltjes transformation in terms of the behaviour of this transform near the origin and at infinity.

In the solution of certain boundary-value problems by means of complex integrals or by means of the Laplace transformation, transcendental equations of the form

arise, and it is then usually necessary to show that this equation has no roots in the right half of the complex z-plane.

1. In this paper, I apply the results of (4) to a class of function spaces distinguished by their invariance under a certain family of operators. This class contains the Lp spaces (indeed, all Orlicz spaces) and the symmetric Köthe spaces of (5). My original objective was simply a more general expression of the ideas of (5), but I found myself taking up a rather different approach.

1. The object of this note is to prove the restricted Burnside conjecture for exponent 5, that is, to prove, for n = 5, the proposition:

Rn: For each positive integer k there is an integer rn, k such that every finite group of exponent n that can be generated by k elements has order at most rn, k.

We consider the Schrödinger operator α on the half-line with a periodic background potential and the Wigner–von Neumann potential of Coulomb type: csin(2ωx + δ)/(x + 1). It is known that the continuous spectrum of the operator α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator α has a subordinate solution, which can be either an eigenvalue or a “half-bound” state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.

The modular group, Γ, is the group of linear fractional transformations with integral coefficients and determinant 1. The group is generated by the transformations

which have orders 2 and 3 respectively. The transformation

is of infinite order. Abstractly, the group can be viewed as the free product of cyclic groups of order 2 and 3.

Estimators are given for the solution by Monte Carlo methods of the Fredholm integral equation of the second kind and the variances of the estimators are compared. Two integral equations arising in sequential analysis are considered in detail; some numerical examples are given.

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q-estimate with constant 1 then the corresponding space E(M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces Lq, p and the Lorentz-Schatten classes Cg, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q-estimate.

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the type

where are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the following

Theorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

It is shown that the soft X-ray emission bands from Be and A1, present in small quantities in solid solution in Cu and A1, have a form without a sharp highenergy edge, the feature most characteristic of these bands in pure metals. The diffuse form of the bands indicates (1) that all the valence electrons of the impurity atom are given up to the lattice, (2) that the impurity atom refuses to take within it those electrons of the lattice which have energies close to the maximum energy of the conduction electrons.

In [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following inequality:

He then introduced three families of Riemannian n-manifolds and two exceptional n-spaces Fn, Ln and proved the existence of a Lagrangian isometric immersion pa from into ℂPn(4) and the existence of Lagrangian isometric immersions f, l, ca, da from Fn, Ln, , into ℂHn(− 4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesie ones, these are the only Lagrangian submanifolds in ℂPn(4) and in ℂHn(− 4) which satisfy this basic equality. In this article, we obtain the explicit expressions of these Lagrangian immersions. As an application, we obtain new Lagrangian immersions of the topological n-sphere into ℂPn(4) and ℂHn(−4).

The Euler graph has vertices labelled $(n,k)$ for $n=0,1,2,\ldots$ and $k=0,1,\ldots,n$, with $k+1$ edges from $(n,k)$ to $(n+1,k)$ and $n-k+1$ edges from $(n,k)$ to $(n+1,k+1)$. The number of paths from (0,0) to $(n,k)$ is the Eulerian number $A(n,k)$, the number of permutations of $1,2,\ldots,n+1$ with exactly $n-k$ falls and k rises. We prove that the adic (Bratteli–Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford–Pettis property (DPP). As a consequence, we obtain that (c0 &[otimes ]circ;πc0)** fails the DPP. Since (c0 &[otimes ]circ;πc0)* does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if E and F are [Lscr ]1-spaces, then E &[otimes ]circ;ε has the DPP if and only if both E and F have the Schur property. Other results and examples are given.

We deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

Asymptotic approximations of Green's type to solutions of differential equations are studied, with special reference to the uniformity of the approximation given by the first term. In extension to the complex variable this is found to require substantial restrictions on the region considered. An anomaly previously noticed is traced to non-uniformity of approximation. The case where the coefficient χ0 has a simple zero and χ1 is not zero is treated by a simple method.

We say a real number x is normal to base r if the sequence is uniformly distributed modulo [0, 1]. Pollington[10] has proved the following result concerning the normality of numbers.

For any group G, x ∈ G and n ∈ ℕ (the natural numbers), let

i.e. the set of all nth roots of x in G. If G is a Hausdorff topological group, then Rn(x, G) is a closed set in G, but may otherwise be quite complicated. However, as we have observed in (4), if G is a compact Lie group, then Rn(x, G) always has a finite number of connected components, and this result has led us to wonder about the connectedness properties of Rn(x, G) for other Lie groups G. Here is the result.

The series is said to be summable (L) to sif

where sn = a0 + a1 + … + an, and it is said to be summable (l) to s if

Let X be a topological space with base-point. The algebraic K-theory AX of X is a space invented by Waldhausen in (15) for use in geometric topology. It can be defined in two ways, which I shall call geometric and ring-theoretic; Steinberger ((12)) has shown them to be equivalent.

The geometric method ((15), corollary to lemma 2·1) gives AX as the group-completion of the geometric realization of a permutative category. It follows from the machinery of May ((4), 4) or Segal (11) that AX is an infinite loop space in a well-defined way ((6)).