Coupled stationary bifurcations in nonlinear operator equations for functions, which are defined on a real interval with non-flux boundary conditions at the ends, are analysed in the framework of imperfect bifurcation theory. The bifurcation equations resulting from a Lyapunov–Schmidt reduction possess a natural structure which can be obtained by taking real parts of a diagonal action in ℂ2 of the symmetry group 0(2). A complete unfolding theory is developed and bifurcation equations are classified up to codimension two. Structurally stable bifurcation diagrams are given and their dependence on the wave numbers of the unstable modes is clarified.

We are concerned in this paper with polynomials in n variables xi over the field 2 of two elements. The Steenrod squares Sqk are linear operators which act on the variables by the rules

and on the product uv of homogeneous polynomials u and v by the Cartan formula

In the classical theory of representations of a finite group by matrices over a field , the concept of the group algebra (group ring) over is of fundamental importance. The chief property of such an algebra is that it is semi-simple, provided that the characteristic of is zero or a prime not dividing the order of the group. As a consequence of this, the representations of the algebra, and hence of the group, are completely reducible.

It is shown that, when a plane harmonic P or S wave is incident upon a two-or three-dimensional obstacle in an infinite elastic solid, the scattering cross-section of the obstacle can be calculated from an appropriate far-field scattering amplitude.

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.

1. We consider a small cube of the solid, of edge l, in the actual strained position. If the stresses were removed and the cube cooled to the absolute zero, the displacements of its particles would be (−u, −v, −w). Then the particles may be considered to have displacements (u, v, w) from a standard position; and the cube has strain components of the usual forms

The geometrical or vectorial representation of a sample as a vector with n mutually perpendicular components corresponding to the n observations in the sample was introduced into statistics by Fisher (1), and led to the solution of many theoretical problems of statistical distributions. Subsequently Fisher (2) gave an alternative algebraic method applicable to finding distributions in connection with regression and the analysis of variance. The actual use of symbolic vector notation in deducing some of the properties of the sample vector—besides tending to stress the complementary character of these two different methods of proof, and the common principle underlying the analysis of any sample into its components—indicates also some points in connection with the assumption of the normal law in which it seems almost essential to retain the geometrical side of this vector representation. Moreover, it will be seen that the vector theory used here in reviewing briefly the analysis of a sample of one dependent variate can readily be extended to cover the case of correlated variates.

Formulae are found for the number of configurations of particles on two-and three-dimensional lattices when each particle (a) occupies two closest neighbour sites, and (b) consists of three groups which occupy three sites on the lattice in such a way that adjacent groups in the molecule occupy closest neighbour sites on the lattice. Bethe's method is used to determine the equilibrium conditions of the corresponding order-disorder problem, and the number of configurations is determined from these equilibrium conditions. For the case in which the molecules occupy two closest neighbour sites on the surface the determination of the number of configurations from geometrical considerations is discussed.

It is found that for molecules which occupy two closest neighbour sites the number of configurations of particles for a square lattice, a simple cubic lattice and a body-centred cubic lattice respectively are and For molecules which occupy three sites on the lattice the corresponding results are and when the molecules are perfectly flexible and and when the molecules are completely inflexible, Ns being the total number of sites in the lattice.

The author wishes to thank Dr J. K. Roberts for suggesting this work; the geometrical treatment given in § 3 was developed from manuscript notes prepared by him. The problem arose during an investigation of the vapour pressure equations of solutions in which the solute molecules are chain molecules consisting of a large number of groups, undertaken as part of the programme of fundamental research of the British Rubber Producers' Research Association, whom the author wishes to thank for the grant of a Research Scholarship.

If H is a Moufang loop, and x ∈ H, there are defined permutations of H, L(x):y ↦ xy and R(x): y ↦ yx. The group Gr (H), generated by these permutations for all choices of x, is called the multiplication group of H. It has a close connexion with the structure of H, as shown, for instance, in the papers of Albert(1). The purpose of this paper is to investigate the correspondence between groups and loops, so that group theoretic results may be applied to determine the structure of Moufang loops.

1. The properties of partitions of numbers extensively investigated by Hardy and Ramanujan (1) have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy-Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar (2) for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose-Einstein, and Fermi-Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper.

The general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ is

The applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.

Expressions are found for the changes caused by slow selection in populations whose characters are determined by incompletely dominant, multiple, or polyploid factors, and for the equilibria attained in certain of these cases.

Given a locally convex Hausdorff linear topological space, we construct and examine the following topologies in the space:

the finest locally convex topology with the same convergent sequences as the initial topology, and the finest locally convex topology with the same precompact sets as the initial topology.

The notion of associated prime ideal and the related one of primary decomposition are classical. In a dual way one defines attached prime ideals and secondary representation. This theory is developed in the appendix to §6 in Matsumura[5] and in Macdonald[3].

When γ-radiation passes through a gas it ionizes by means of fast electrons which produce clusters of secondary ionization at intervals along their paths. At high pressures a considerable amount of recombination of ions takes place in these clusters; in the present paper a theory is described which enables the proportion of ions escaping recombination to be calculated as a function of the gas pressure and collecting field. A review of the available experimental data concerning the variation of ionization current with pressure and collecting field is given, and it is shown that the predictions of the cluster recombination theory are in satisfactory agreement with these experimental data.

Jaffé has given a theory of initial recombination valid for a columnar distribution of ionization, and has shown that this theory is in agreement with experiments in which α-particles produce the ionization. A number of authors have applied Jaffé's equations to ionization produced by fast electrons regardless of the initial localization of the ions in clusters. It is shown in the present paper that such measure of agreement with experiment as is obtained by this procedure is only obtained at the expense of assigning incorrect values to certain known constants. Also in the case of X- and γ-rays the columnar theory predicts a variation of the proportion of recombination with the wave-length of the radiation which is much too rapid. It is further shown that the method based on Jaffé's equations which is used by Clay and his colleagues to extrapolate experimental ionization currents at finite collecting fields to saturation currents at infinite fields is liable to systematic error in a direction likely to lead to the deduction of a spurious wall effect or the exaggeration of an existing wall effect.

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the function

is also in LUC (G). Hence we may define a product

for n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.

We obtain an expression for the dimension of a mixing repeller of a non-conformal mapping analogous to the well-known Bowen-Ruelle formula for conformal repellers. The dimension is given in terms of a generalized pressure defined in the context of the thermodynamic formalism. In the course of the paper we develop a subadditive version of the thermodynamic formalism that is suited to our needs and also obtain a ‘bounded distortion’ principle applicable to the non-conformal situation.

In applying the theory of infinite Markov chains to practical examples, it is important to know how the ergodic properties defined by the infinite stochastic or substochastic matrix under consideration are related to those of the n × n (n = 1, 2, 3, …) truncated corner sub-matrices. In particular, it is of interest whether the relevant eigenvalues and eigenvectors of the truncated matrices in some sense approximate to corresponding quantities for the infinite matrix as n → ∞.

The thickness of a graph is a measure of its nonplanarity and has applications in the theory of printed circuits. To determine the thickness of an arbitrary graph is a seemingly intractable problem. This is made precise in this paper where we answer an open problem of Garey and Johnson (2) by proving that it is NP-complete to decide whether a graph has thickness two.

We consider the steady co-current flow of two immiscible viscous liquids in a horizontal pipe, the fluid interface being ripple-free and plane. It is shown that if the cross-section of the duct is symmetric with respect to the interface, the velocity distribution may be expressed in terms of two separate pipe-flow solutions. One corresponds to the flow of a single fluid occupying the whole of the pipe, and the second to a similar flow in a pipe whose cross-section coincides with that of the region occupied by one fluid in the two-phase motion. Flow rates and other quantities of engineering interest are evaluated, and several particular flows are discussed.