We prove that for F: [0,∞) × ℝn → K (ℝn) a Lipschitzian multifunction with compact values, the set of derivatives of solutions of the Cauchy problem

is a retract of

An algebraic equation

determines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.

One of the many interesting problems discussed by Ramanujan is concerned with the effect of truncating at its maximum term nn/n! the exponential series for en, where n is a positive integer. When n is large, the sum of the first n terms is, roughly speaking, half the sum of the whole series.

It is an obvious remark that the Mathieu functions, being the harmonic functions of the elliptic cylinder, must be closely related to the Bessel functions, the harmonic functions of the circular cylinder. Reference has been made to some aspects of this relationship in two earlier communications, to which the present paper may be regarded as a sequel.

Let $R$ be a commutative ring. Let $M$ respectively $A$ denote a Noetherian respectively Artinian $R$-module, and $\mathfrak{a}$ a finitely generated ideal of $R$. The main result of this note is that the sequence of sets $(\mathrm{Att}_R\mathrm{Tor}_1^R((R/\mathfrak{a}^{n}),A))_{n\in\mathbb{N}}$ is ultimately constant. As a consequence, whenever $R$ is Noetherian, we show that $\mathrm{Ass}_R\mathrm{Ext}_R^1((R/\mathfrak{a}^{n}),M)$ is ultimately constant for large $n$, which is an affirmative answer to the question that was posed by Melkersson and Schenzel in the case $i=1$.

AMS 2000 Mathematics subject classification: Primary 13E05; 13E10

In [4], Maxson studied the properties of a ring R whose only ring endomorphisms φ: R → R are the trivial ones, namely the identity map, idR, and the map 0R given by φ(R) = 0. We shall say that any such ring is rigid, slightly extending the definition used in [4] by dropping the restriction that R2 ≠ 0. Maxson's most detailed results concerned the structure of rigid artinian rings, and our main aim is to complete this part of his investigation by establishing the following

Theorem. Let R(≠0) be a left-artinian ring. Then R is rigid if and only if

(i) , the ring of integers modulo a prime power pk,

(ii) R ≅ N2, the null ring on a cyclic group of order 2, or

(iii) R is a rigid field of characteristic zero.

The purposes of the following note are these:—(1) To show the relation between Whittaker's Cardinal Function and Fourier's Repeated Integral; (2) to give a new derivation of Fourier's Integral Formula; and (3) to extend the notion of the Fourier Integral to the case in which the variables involved are complex.

Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by letting

where α(x, y) is a non-zero element of K, subject to the condition that

which is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and ifG has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.

Classically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.

Ten papers deal with the theory of relativity, and all are concerned in some way with electromagnetism. Those who know Whittaker through his Modern Analysis and Analytical Dynamics will recognise in these papers the same mastery over complicated situations which enabled him to disdain the support of notational refinements, that same elegance, brevity and persuasive charm which make difficult arguments seem easier than they really are. The new element which emerges is the strongly geometrical approach; but he remains true to the Lagrange tradition and draws no diagrams of space-time, although these must surely have been before his mind's eye and would have helped his readers.

In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric function

in the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.

If the elements of a symmetric matrix lie in the real field it is well known that the roots of its characteristic equation are real. This implies that the discriminant of that equation (i.e. the product of the squared differences of the roots) is a polynomial in the elements which is non-negative and the same must be true for the leading coefficients of all the other Sturm functions associated with the characteristic equation. One would expect that it should be possible to express them as a sum of squares. Conversely, such an expression would establish the reality of the roots.

Let E be an ordinary elliptic curve over a finite field q of q elements. We improve a bound on bilinear additive character sums over points on E, and obtain its analogue for bilinear multiplicative character sums. We apply these bounds to some variants of the sum-product problem on E.

Let A(z) be a transcendental entire function and f1, f2 be linearly independent solutions of

We prove that if A(z) has Nevanlinna deficiency δ(0, A) = 1, then the exponent of convergence of E: = flf2 is infinite. The theorems that we prove here are similar to those in Bank, Laine and Langley [3].

The topology of certain weighted inductive limits of Fréchet spaces of holomorphic functions on the unit disc can be described by means of weighted sup-seminorms in case the weights are radial and satisfy certain natural assumptions due to Lusky; in the sense of Shields and Williams the weights have to be normal. It turns out that no assumption on the (double) sequence of normal weights is necessary for the topological projective description in the case of o-growth conditions. For O-growth conditions, we give a necessary and sufficient condition (in terms of associated weights) for projective description in the case of (LB)-spaces and normal weights. This last result is related to a theorem of Mattila, Saksman and Taskinen.

AMS 2000 Mathematics subject classification: Primary 46E10. Secondary 30H05; 46A13; 46M40

When a function ƒ(x) possesses an asymptotic series

this series provides a useful means of evaluating ƒ(x) for large values of x. The usual procedure is to sum all the terms in S(x) up to, but excluding, the term of smallest magnitude. The degree of accuracy obtained by this method cannot normally be improved by direct summation of S(x), but sometimes better accuracy can be obtained by using one of the familiar devices for accelerating the convergence of series. Simple δ2-extrapolation may be successful, and Rosser (1) and others have used the Euler transformation to some effect. The method given here provides, in suitable cases, a more effective means of evaluating ƒ(x) from the series for a wide range of values of x.

This paper studies questions connected with when the Rees algebra of an ideal or the formring of an ideal is Gorenstein. The main results are for ideals of small analytic deviation, and for m-primary ideals of a regular local ring (R, m). The general point proved is that the Gorenstein property forces (and is sometimes equivalent to) lowering the reduction number of the ideal by one from the value predicted if one only assumes the Rees algebra or formring is Cohen–Macaulay.

Let {pn} be a positive sequence. The Nörlund transformation (N, pn) maps the sequence {Sn} into the sequence {tn} by means of the equation

where

Let G be a finite group generated by n elements and defined by m relations, then G has a presentation, G = {x1, …, xn | R1, …, Rm} = F/R where F is free on generators x1, …, xn and R is the normal closure in F of R1, …, Rm. The deficiency of this presentation is n − m. Since G is finite the deficiency is non-positive and the deficiency of G is the maximal over the deficiencies of all presentations for G.

The problem of determining the distribution of stress in the neighbourhood of a Griffith crack, denned by ∣x∣ ≦ 1, y = 0, which is subject to an internal pressure varying along the length of the crack has been solved by Sneddon and Elliott (3) and by England and Green (1). In the former paper a Fourier cosine method is used to arrive at a solution while in the latter paper the problem is reduced to an Abel integral equation by making use of integral representations of the complex potentials given in Green and Zerna (2). Neither paper deals with the calculation of the stress intensity factor

which is extremely important to workers in fracture mechanics.