In this paper we prove the existence of classical solutions for all t ≧ 0 for parabolic equations u′ + A(t)u = –f(u, ∇y, …, ∇2m–2 u) of arbitrary order. 2m is the order of the elliptic principal part. f must satisfy some monotonicity and growth conditions. Moreover, similar results are also valid for semilinear elliptic equations.

In this paper we study the propagation of weak discontinuities in quasi-linear hyperbolic systems of equations with discontinuous coefficients when one or more speeds of propagation o f the discontinuity wave is coincident with the speed of propagation of the strong discontinuity.

In this paper, the electrostatic potential of a point charge in a Reisser-Nordström gravitational field is found in closed form by using the theory of Hadamard's elementary solution of a partial differential equation of elliptic type.

Let X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ , the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

This paper is concerned with integral inequalities of the form

where p, q are real-valued coefficients, with p and w non-negative, on the compact interval [a, b] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

§1. Let pi denote the i-th prime number, and let

The prime number theorem implies Er ≤ r. Bombieri and Davenport [1] showed that

improving earlier results of Erdős, Rankin and Ricci. Their basic result (corresponding to Lemma 1 below) counts pairs of primes differing by 2n with some weight t(n).

Let K/k be a normal extension of algebraic number fields whose Galois group G is a Frobenius group. Then K/k is said to be a Frobenius extension. Most of the structure of the unit group and of the ideal class group of K is determined by that; of the subfields fixed by the Frobenius kernel N and by a complement F. Here this is investigated when G is a maximal or metacyclic Frobenius group. In particular, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when G is a dihedral group of order 2n for an odd integer n. A. Scholz, taking n = p = 3, was the first to consider this problem.

An arithmetic function f(n) is said to be additive, if it satisfies the relation f(ab) = f(a) + f(b), for every pair of coprime integers a and b; and stronglya dditive if, in addition, f(pm) = f(p) for every prime-power pm.

In this paper we continue our study of the values taken by Euler's ø-function begun in [1]–[3]. Let ør(n) be the iterated ø-function, that is ør(n) = ø{ør-1(n} where ø1 = ø1. Let

The question, whether a given element of a C*-algebra has an image of rank one in some faithful representation, was studied in [3]. Such elements were characterised there by the property of being “single” (as defined below). As was pointed out in [3], Section 5, this criterion fails for general Banach algebras and the purpose of this paper is to provide a stronger condition giving the required representation property for any semi-simple Banach algebra.

It is well-known that the σ -algebra of Borel subsets of a metric space coincides with the smallest family of sets which contains the open sets and is closed under countable intersections and countable disjoint unions «3, Th.3, p. 348». A deeper and less known result of Sierpiński is that for separable metric spaces the family of open sets may be replaced by the family of closed sets in the above result «16, p. 272–275» (and «17, p. 51» for the real line). This paper gives an in depth analysis of these and related generation processes. Several abstract formulations, generalizations and limiting examples are given.

An incident sound field is scattered by a semi-infinite rigid screen with periodically arranged slits or circular apertures and an approximate solution is sought when the slit (or aperture) width is small and the wavelength is large compared with the separation. An integral equation formulation is used to show that the scattering properties of the screens are equivalent to those of a homogeneous compliant plate. The effective compliance is estimated and is found to be essentially uniform over the plate, with corrections close to the edges.

Let Xt be a Lévy process in Rd, d dimensional euclidean space. That is Xt is a strong Markov process whose transition function, P(t, x, A), satisfies:

In [4] we have given a simple method of estimating trigonometrical sums over prime numbers. Here we show how the argument can be adapted in order to give estimates for the distribution of αp modulo 1 which are sharper than those obtained by I. M. Vinogradov [5], [6]. Vinogradov uses the sieve of Eratosthenes to relate the sum

to the bilinear form

the function μ being the Mobius function. When d1 … ds is small compared with N this can be treated in a fairly straightforward manner. However, in order to treat the terms with d1 …ds close to N, Vinogradov has to introduce an argument of a rather recondite combinatorial nature.

I discuss various necessary and sufficient conditions for a K-analytic space to be Souslin. In particular, I show that if the continuum hypothesis is true, then there is a non-Souslin K-analytic space in which every compact set is metrizable; while if Martin's Axiom is true and the continuum hypothesis is false, this is impossible.

We let K(a1a2, ak) denote the continuant formed from the positive integers a1 …, ak; that is,

Of course, K(a1, …, an) is the denominator of the continued fraction

We use the convention that K (empty set) = 1.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.