1. The equation of the scroll R of tangents of the curve Γ of intersection of two quadrics in general position in [3] was found by Cayley in 1850 ((l), p. 50). Cayley also considered special positions of the quadrics but we shall not be concerned with these. was reproduced in Salmon's treatise ((8), p. 191) and Salmon, on this same page, provides an equation in covariant form and so applicable to pairs of quadrics whether in canonical form, or in general position, or not.

We shall be concerned with the Laguerre polynomials, defined by Some time ago Szegö (8) showed that for (a, b, c = 0, 1, …). It is easily deduced from this (cf. (5)) that for all A λ ≥ 3.

A k-3 cap in a three-dimensional Galois space, S3,q, is a set of k points, of which some 3, but no 4 are collinear. It is shown in (2) that for q ≥ 4 .

Every real number α admits many rational approximations in the sense that for every natural number q,

for some integer p. Perhaps surprisingly, numbers α which have k distinct rational approximations with given denominators q1,…,qk,

for a fixed ε > 0, are quite sparse; their measure in U = [0,1) is at most C(ε)/k.

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):

In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.

The basic integration theory of Radon measures on locally compact spaces, as described in (5), has been developed in various directions both in commutative and non-commutative analysis. Thus if Ω is a compact Hausdorff space, and C(Ω) denotes the space of continuous complex-valued functions on Ω, Kaplan (16) showed how topological measure theory can be performed in the second dual C(Ω)** of C(Ω). Here as usual C(Ω)* is identified with the space of Radon measures on Ω by associating with a measure μ the linear functional φμ where

Thus if f is a bounded real-valued function on Ω, there is an affine function f* defined on the set P(Ω) of Radon probability measures μ on Ω, by

and f* extends by linearity to a functional in C(Ω)**. Conversely any function x on P(Ω) determines a function x0 on Ω. by

wehere εω is the unit point of mass at ω.

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the condition

is satisfied. These may be termed the elements of thin numerical range if δ is small.

It is shown that the Wilks large sample likelihood ratio statistic λn, for testing between composite hypotheses Θ0 ⊂ Θ1 on the basis of a sample of size n, behaves as n varies like a diffusion process related to an equilibrium Ornstein-Uhlenbeck process, whenever the null hypothesis is true. This fact is used to construct large sample sequential tests based on λn, which are the same whatever the underlying distributions. In particular, the underlying distributions need not belong to an exponential family.

To make a Bayes decision we choose the infimum of an expected loss function. Catastrophe theory classifies a wide class of functions locally in terms of their critical values. Firstly we will show how this local classification relates globally to some mixtures of symmetric expected loss functions. Secondly we shall indicate how such mixtures can arise and how the above classification can be usefully applied to the qualitative study of the behaviour of a Bayes decision-maker.

For a family {Xα} of random variables over a probability space , stochastic independence can be formulated in terms of factorization properties of characteristic functions. This idea is reformulated for a family {Aα} of selfadjoint operators over a probability gage space and is shown to be inappropriate as a non-commutative generalization. Indeed, such factorization properties imply that the {Aα} mutually commute and are versions of independent random variables in the usual sense.

The probability of ever returning to the origin and the mean square displacement after n steps are studied for some lattice-valued random walks, whose successive steps constitute a Markov chain on a finite state space with transition probabilities of a simple kind, and such that the returns to the origin form a regenerative phenomenon. The case of walks on a diamond lattice with no immediate reversals is included: this example is relevant as a polymer chain building model. The numerical evaluation of the return probabilities of some three-dimensional walks is discussed and examples given.

A short solution is given for the urn problem proposed by Paul and Tatiana Ehrenfest in 1907.

In 1907 P. and T. Ehrenfest(3) proposed an urn model for the resolution of the apparent discrepancy between irreversibility and recurrence in Boltzmann's theory of gases (2). In this model it is assumed that m balls numbered 1, 2, …, m are distributed in two boxes. We perform a series of trials. In each trial we choose a number at random among 1, 2, …, m in such a way that each number has probability 1/m. If we choose j, then we transfer the ball numbered j from one box to the other. Denote by ξn the number of balls in the first box at the end of the nth trial. Initially there are ξ0 balls in the first box. If the trials are independent, then the sequence {ξn;n = 0, 1, 2,…} forms a homogeneous Markov chain with state space I = {0, 1, 2,…, m} and transition probabilities pi,i+1 = (m − i)/m for i = 0, l,…, m − 1, Pi,i−1 = i/m for i = 1, 2,…, m, and pi,k = 0 otherwise. The problem is to determine the transition probabilities

for i∈I, k∈I and n = 0,1, 2,….

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’

Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the form

where each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equation

which has a Lax representation with

(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.

In this paper, we investigate the integrability of the vector field of the initial-value problem associated with certain nonlinear wave equations. This vector field involves translations and as such is not a strongly continuous or even strongly measurable L∞-valued function. It is shown that such a vector field, although not generally Pettis integrable, does turn out to be so in an important situation. We then indicate how this result can be used to obtain pseudo-solutions of the initial-value problem.

In solid mechanics an outstanding lacuna is a general method for solving problems of plane strain for materials other than classically elastic, or visco-elastic, or rigid–plastic. Even with restriction to incrementally linear response, and to infinitesimal deformations superimposed on uniform pre-strains, there is no body of theory with the scope and power of the complex variable approach in isotropic elasticity. This lack has been highlighted recently by the growing interest in bifurcation phenomena in finitely deformed materials. So far, all analytical solutions to plane boundary-value problems with bifurcations (1,2,3,4) have relied on a separation-of-variables technique, which succeeds only for certain types of data.

This paper deals with the propagation and decay of nth-order displacement waves in linear hyperelastic rods having multiple wave speeds. The vector decay equation is integrated to find the explicit behaviour of all waves which can propagate in the rod. Examples are treated using a special form of the constitutive relations and results are given for rods of various geometries and wave multiplicities.