The comparison theorem is an algebraic theorem, which corresponds to the following topological situation. A map between two fibre spaces induces a homomorphism between the two corresponding spectral sequences relating the homologies of the base, fibre and fibre space; if the map induces an isomorphism on any two of these quantities, then it also does on the third. J. C. Moore (l) proved the theorem using a mapping cylinder, which requires that the spectral sequences arise from filtered differential groups. The present proof assumes only the existence of the spectral sequences from the E2 terms onwards. Moreover, we generalize the theorem, assuming that the homomorphisms on the two given quantities are isomorphic only up to given dimensions, and deducing the same for the third quantity. For algebraic translucence we use a simpler hypothesis (property (iii)) than Moore's concerning the E2 terms. However, if his hypothesis is assumed, together with the existence of E1 terms, then his theorem can be deduced as a corollary. The analogous result for cohomology is similar, and is stated at the end.

We develop some of the basic theory of quaternionic hyperbolic geometry. We give necessary criteria for groups of quaternionic hyperbolic motions to be discrete. We give lower bounds on the volumes of cusped quaternionic hyperbolic manifolds.

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

Ramanujan's notebooks were the theme of a lecture (20) given by G. N. Watson to the London Mathematical Society in 1931. At that time, he and B. M. Wilson were collaborating on a critical edition; Watson was anticipating that the task might take a further 5 years. Unhappily, Wilson died, and the task of editing the notebooks has never been completed.

1. An isotropic tensor is one the values of whose components are unaltered by any rotation of rectangular axes (with metric σi(dxi)2). Those up to order 4 in 2 and 3 dimensions have many applications. The results suggest a general theorem for tensors of order m in n dimensions, that any isotropic tensor can be expressed as a linear combination of products of δ and є tensors, where δij = 1 if i = j and 0 otherwise, and is 0 if any two of the i1 to in are equal, 1 if i1…in is an even permutation of 1, 2, 3, …,n, and – 1 if it is an odd permutation.

We recall that a locale is a complete lattice L satisfying a ∧ (∨bα) = ∨(a ∧ bα), for all a ∈ L, and {bα} ⊆ L. Examples of locales include the lattices (X) of open subsets of topological spaces X. Following Joyal and Tierney [7], a morphism f: L → M of locales is a ∨-, ∧-, and τ-preserving map. Such functions are sometimes called ‘frame homomorphisms’, in which case the right adjoint f*:M → L (which exists since f preserves ∨) is then called a ‘morphism of locales’.

It is well known that a single neutron may cause a nuclear reaction chain of considerable magnitude, if it moves in a medium in which the number of secondary neutrons which are produced by neutron impact is, on the average, greater than the number of absorbed neutrons. From recent experiments it would appear that this condition might be satisfied in the case of uranium.

Rosseland proposed a theory of the production of characteristic X-rays. He calculated the number of atoms from which on the average an electron with definite energy will remove an electron from a particular core orbit, but neglected the velocity of the core electron in its orbit and the increased velocity of the impinging electron due to the atomic field. Taking these into account considerably alters his formula.

Throughout, ‘link’ means ‘oriented link in S3’, and ‘polynomial’ means ‘Laurent polynomial’. Let L be a link; then denotes the oriented polynomial of L (see [2, 8]), written in Morton's variables (see [5]), and denotes the semi-oriented polynomial of L (see [3]). Let L have components K1, …, Kn. The total linking of L, denoted t(L), is the sum of the linking numbers lk (Ki, Kj) with 1 ≤ i < j ≤ n. Let f be a framing of L (that is, f assigns an integer to each Ki); then f(L) denotes the total framing of L (that is, the sum of these integers), and A(L, f) denotes the naturally associated union of annuli embedded in S3 Define the framed polynomial {L, f} (v, z) to be (–1)n plus (v−1–v)z−1 times the sum, over all non-empty sublinks K of L, of (where K has k components with 1 ≤ k ≤ n).

The object of this note is to show that elementary probability considerations suggest a very natural way of measuring the non-convexity of a set in euclidean space or, more generally, in a real Hilbert space . In particular they give a proof, much simpler and under less restrictive conditions, of results due to Shapley, Folkman and Starr which are of importance in Mathematical Economics ((1),(2)).

The intensities of X-ray spectral lines resulting from transitions between the K, L and M shells have been calculated using screened relativistic wave functions. The relativistic modifications are shown to decrease the Lβ3β4 doublet ratio but do not alter the Kα1α2 and Kβ1β3 ratios, in agreement with experiment. The calculated intensities of the forbidden lines arising from quadrupole transitions are in good agreement with experiment. Lines due to transitions arising from the magnetic dipole moment of the electron are found to be very weak, but the LI → K transition might just be observable for heavy elements.

The absolute simplicial approximation theorem, which dates back to Alexander (l), states that there is a simplicial approximation g to any given continuous map f between two finite simplicial complexes (see for instance (2), p. 37 or (3), p. 86). The relative theorem given here permits us to leave f unchanged on any subcomplex, on which f happens to be already simplicial.

In a recent paper (1), the author has discussed how certain ergodic properties of a doubly infinite non-negative matrix T = {tij} are related to those of its finite (n × n) corner truncations, (n)T = {(n)tij}.

The well-known methods of classical mechanics, based on the use of a Lagrangian or Hamiltonian function, are adequate for the treatment of nearly all dynamical systems met with in practice. There are, however, a few exceptional cases to which the ordinary methods are not immediately applicable. For example, the ordinary Hamiltonian method cannot be used when the momenta pr, defined in terms of the Lagrangian function L by the usual formulae pr = ∂L/∂qr, are not independent functions of the velocities. A practical case of this kind is provided by the electromagnetic field, considered as a dynamical system with an infinite number of degrees of freedom, since the momentum conjugate to the scalar potential at any point vanishes identically. Again, for the very simple example of the relativistic motion of a particle of zero rest-mass in field-free space, the Lagrangian function vanishes and the usual Lagrangian method is not applicable.

A circular disc of radius a, made of homogeneous, isotropic, linearly elastic material, contains a radial edge crack of length b(0 < b < 2a). The disc is in equilibrium in a state of generalized plane stress caused by loading the faces of the crack by a constant internal pressure. The problem of determining the resulting stress field throughout the disc is solved analytically in closed form. The principal results are that the stress concentration factor at the crack tip, the total strain energy W, and the opening U at the mouth of the crack, are given exactly by

where A is a constant whose value correct to 6 significant figures is

and , W0, U0 are normalising factors defined in section 6.

Let X1, …, Xn be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let Sn(rn) denote their sum when the rn largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of Sn(rn), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when rn = r these conditions are the same, but when rn → ∞the former holds more generally.

In 1936 Todd and Coxeter gave a method ((10), also described in (4), ch. 2) for establishing the order of a finite group defined by a set of relations

satisfied by its generators S1, S2,…, Sk. They enumerate systematically the cosets of a suitable subgroup whose order is evident from the defining relations for the whole group. They describe the method as being ‘purely mechanical’, and since that date the advent of electronic computers has led a number of people to programme the method for automatic execution. Most of this work has remained unpublished; an account is given here of work known to the author.

1. Introduction. In a recent contribution to these Proceedings Alladi Rama-krishnan (23) has discussed a number of problems which arise when the development of a cascade shower of cosmic rays is considered from the standpoint of the theory of stochastic processes. As has often been remarked (see, for example, the important study by Niels Arley (1)), there is a close formal analogy between such physical phenomena and the growth of biological populations. In particular, if the distance of penetration (t) is identified with the time, and the energy (E) of a particle in the shower is replaced by the age (x) of an individual in the population, there emerges an obvious analogy between stochastic fluctuations in the energy spectrum of the shower and similar fluctuations in the age distribution of the population.

A language-theoretic definition of word hyperbolic semigroup is given, which coincides with the original definition for a group. Word problems of semigroups are also considered.

In this paper the mixed problem for an isotropic, elastic half-space is considered. Boundary conditions are prescribed interior and exterior to a circular region of unit radius, and the state of stress is assumed to be axially symmetric. Several authors have treated this problem. Mossakovskii(1) considered a punch adhering to and indenting an elastic half-space. Has solution was obtained by introducing certain operators that transformed the half-space problem into a problem in plane potential theory. The method of linear relationship was used to solve this auxiliary problem and inverse operators returned the plane to the half-space. The general case of a circular, rigid punch adhering to a half-space was treated by Ufliand (2,3) and a solution was obtained through the use of toroidal coordinates and the Mehler-Fok integral transforms.