The nonlinear eigenvalue problem

for 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified is studied under various conditions on the coefficients s and q, leading to either oscillatory or non-oscillatory situations.

Let H1(U2) be the Hardy space of the bidisc as described in (3). Each function f ∈ H1(U2) has a Taylor expansion of the form . For 0<p<∞, a doubly-indexed sequence is said to be a multiplier of H1(U2) into lp if

This paper is concerned with the cases p = 2 and p = 1. Theorem 1 characterises the multipliers of into l2 and is an analogue in two variables of an old result of Hardy and Littlewood. Theorem 2 characterises the sequences (an)n≥0 such that (an+m)n,m≥0 is a multiplier of H1(U) into l1

The term “flat” is used to indicate that the minimum modulus of a function in a region is (in some sense) of the same order as the maximum modulus. Some properties concerned with this notion are described below. They came to light during an attempt to answer a question put to me by Professor Littlewood.

Basic properties of symplectic reflection algebras over an algebraically closed field k of positive characteristic are laid out. These algebras are always finite modules over their centres, in contrast to the situation in characteristic 0. For the subclass of rational Cherednik algebras, we determine the PI-degree and the Goldie rank, and show that the Azumaya and smooth loci of the centre coincide.

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

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.

We study quantum deformations of Catalan’s constant, Mahler’s measure and the double sine function. We establish quantum deformations of basic relations between these three objects.

We calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank

Throughout this paper all near-rings considered will be zero-symmetric and left distributive. All groups will be written additively, but this does not imply commutativity. The near-ring of all zero-fixing maps of a group V into itself will be denoted by Mo(V). If N is a near-ring withan identity and α ≠ 1 is an element of N such that α2 = 1, then α will be called an involution of N. Let V be a group. An involution a of Mo(V) will be called an involution on V.

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.

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.

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.

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

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.

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.

We study homogeneous Kähler structures on a non-compact Hermitian symmetric space and their lifts to homogeneous Sasakian structures on the total space of a principal line bundle over it, and we analyse the case of the complex hyperbolic space.

Consider the ordinary linear matrix differential system

ψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.

Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or $\omega$. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains which are order isomorphic to the reals.

AMS 2000 Mathematics subject classification: Primary 20M07; 20M05

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.