In a previous paper, the author gave necessary and sufficient conditions for the algebra of a finite semigroup S over a field of suitable characteristic to be semisimple. When this algebra is semisimple, the matrix representations of S over are completely reducible.

The problem considered is the oppositely charged, circular disk condenser when the disks are very close together. An integral equation due to Love(5) is used as the governing equation of the problem. This equation is solved asymptotically for small separations by splitting the field into regions, one being an annulus containing the edges, the other being the rest of the domain, and combining these solutions. Bounds for the error in the solution of the integral equation are obtained rigorously; the error is shown to approach zero as the separation approaches zero. The capacity of the system is deduced from this solution. The problem of finding the capacity has previously been attempted by various authors, whose results have differed. The present treatment establishes which of these results is correct.

The purpose of this paper is to classify the finite simple Moufang loops. A Moufang loop M is a loop which satisfies the identity

note that the equivalent identities ((xy)z)y = x(y(zy)), x(y(xz)) = ((xy)x)z also hold, by [2], p. 115. The Moufang loop M is simple if it has no non-trivial proper homomorphic images, or equivalently, if it has no non-trivial proper normal subloops. For basic definitions and properties of Moufang loops, see [2] – in particular, the Jordan–Hölder theorem holds for finite Moufang loops ([2], p. 67). Of course if the finite simple loop M is associative, then M is a simple group, and hence is determined by the classification of finite simple groups. In [9], Paige defines, for each finite field GF(q), a finite simple Moufang loop M(q) which is not associative – M(q) is essentially the set of units in the eight-dimensional split Cayley algebra over GF(q), modulo the centre (we shall describe M(q) in much more detail in §2).

Detailed measurements have been made of the superconducting transition temperatures and critical magnetic fields of the tin isotopes of mass 116, 120 and 124. The transition temperature varies with isotopic mass according to the law Tc ∞ M−n, with n = 0·462 ± 0·014, a result very similar to that already found in mercury. The critical field curves of isotopes 116 and 124 are geometrically similar, in the sense that both many be represented by the same equation, Hc/Ho = f(T/Tc), with the same ratio Ho/Tc for both. It is deduced that the electronic specific heat in normal tin varies only slowly, if at all, with isotopic mass. The variation of Tc and Ho with M is very close to that predicted by Fröhlich and Bardeen.

Measurements of the areas of magnetization curves of superconducting colloids and films indicate that the difference between the surface energies of the normal and super-conducting states at the same temperature is much smaller than has been suggested hitherto. An attempt is made to explain the high critical fields of thin films, which led to the introduction of the surface energy, by the hypothesis that a negative interphase surface energy in films favours the persistence of superconducting threads in fields higher than those normally regarded as sufficient to destroy superconductivity. A model is proposed of the interface between normal and superconducting phases which allows the interphase surface energy to be positive in pure macroscopic specimens but negative in alloys and thin films and at lattice defects. This enables a qualitative explanation to be given of some of the details, such as supercooling, of the phase transition in pure metals and alloys.

The theory of transport phenomena in metals depends upon the solution of an integral equation for the velocity distribution function f of the conduction electrons. This integral equation is formed by equating the rate of change in f due to external fields and temperature gradients to the rate of change in f due to the mechanism which produces the resistance. If this latter rate of change is denoted by [∂f/∂t]coll it happens with some mechanisms that

where f0 is the equilibrium distribution, and ℸ is the time of relaxation which does not depend on the external fields. When equation (1) is true, the problem is comparatively simple, but in general [∂/∂t]coll is an integral operator and it is not possible to define a time of relaxation and a free path. It is known that at high temperatures, such that (Θ/T)2 can be neglected, where Θ is the Debye temperature, a free path exists; but, in general, special methods have to be used to solve the integral equation.

It is assumed that the viscosity of a liquid is due to the transfer of momentum by the irregular movement of the holes (see part I of this paper). A formula for the absolute value of the viscosity and its dependence on the temperature can be derived by the theory and is in agreement with its experiments.

In this note I obtain bounds for the least integral solutions of the equation

in terms of

For ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1), Mordell (2) and Skolem (3), but these lead only to bad estimates for general ternaries. So far as I know there have not been given estimates for n≽4. Here I generalize Thue's method to prove:

Theorem. Suppose that n ≥ 2 and that f(ɛ) represents zero. Then there is an integral solution of f(a) = 0 with

In considering the equilibrium of stars of high density the effects of the Pauli Exclusion Principle must be taken into account. For large values of the degeneracy parameter, which will be denoted by λ, an explicit formula for the partition function may be obtained, from which we may easily find the pressure and density in terms of λ. When λ ≪ 1 the relations for a Fermi-Dirac gas reduce to those for a Boltzmann gas. For λ of the order of, but less than, unity, series expansions for the relevant physical quantities can be found, but for λ of the order of, but greater than, unity, a set of numerical quadratures must be performed at intervals (in λ) close enough for interpolation purposes. The method for this is discussed in § 2 and the numerical results are given at the end of the paper.

An equation with real coefficients and given degree n being selected at random, about how many real roots may it be expected to have? The present series of papers is concerned with this question and matters arising out of it. The results we have arrived at were stated without proof in our paper I (with the same general title), which contains also some introductory remarks to which we may refer the interested reader. Here we summarize as follows.

Almost every species is, to a first approximation, in genetic equilibrium; that is to say no very drastic changes are occurring rapidly in its composition. It is a necessary condition for equilibrium that all new genes which arise at all frequently by mutation should be disadvantageous, otherwise they will spread through the population. Now each of two or more genes may be disadvantageous, but all together may be advantageous. An example of such balance has been given by Gonsalez(1). He found that, in purple-eyed Drosophila melanogaster, arc wing or axillary speck (each due to a recessive gene) shortened life, but the two together lengthened it.

The equations of propagation of electromagnetic waves in a stratified medium (i.e. a medium in which the refractive index is a function of one Cartesian coordinate only—in practice the height) are obtained first from Maxwell's equations for a material medium, and secondly from the treatment of the refracted wave as the sum of the incident wave and the wavelets scattered by the particles of the medium. The equations for the propagation in the presence of an external magnetic field are also derived by a simple extension of the second method.

The significance of a reflection coefficient for a layer of stratified medium is discussed and a general formula for the reflection coefficient is found in terms of any two independent solutions of the equations of propagation in a given stratified medium.

Three special cases are worked out, for waves with the electric field in the plane of incidence, viz.

(1) A finite, sharply bounded, medium which is “totally reflecting” at the given angle of incidence.

(2) Two media of different refractive index with a transition layer in which μ2 varies linearly from the value in one to the value in the other.

(3) A layer in which μ2 is a minimum at a certain height and increases linearly to 1 above and below, at the same rate.

For cases (2) and (3) curves are drawn showing the variation of reflection coefficient with thickness of the stratified layer.

Case (3) may be of some importance as a first approximation to the conditions in the Heaviside layer.

Definition [4]. Let A be a noetherian ring, [afr ] an ideal of A and M an A-module. M is said to be [afr ]-cofinite if M has support in V([afr ]) and ExtiA(A/[afr ], M) is a finite A-module for each i.

Remark. (a) If 0→M′→M→M″ →0 is exact and two of the modules in the sequence are [afr ]-cofinite, then so is the third one.

This has the following consequence, which will be used several times.

(b) If f[ratio ]M→N is a homomorphism between two [afr ]-cofinite modules and one of the three modules Ker f, Im f and Coker f is [afr ]-cofinite, then all three of them are [afr ]-cofinite.

Example [5, remark 1·3]. If A is local with maximal ideal [mfr ], then an A-module is [mfr ]-cofinite if and only if it is an artinian A-module.

The kth Fitting ideal of the Alexander invariant B of an arrangement [Ascr ] of n complex hyperplanes defines a characteristic subvariety, Vk([Ascr ]), of the algebraic torus ([Copf ]*)n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk([Ascr ]). For any arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides with [Rscr ]1k(A), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura [1], we conclude that all irreducible components of Vk([Ascr ]) which pass through the identity element of ([Copf ]*)n are combinatorially determined, and that [Rscr ]1k(A) is the union of a subspace arrangement in [Copf ]n, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

This paper deals with the calculation of the inter-atomic force at large distances for the cases of hydrogen and helium, and is based on the method used in calculating the polarizability of helium.

The motion of a fluid due to the rotation of an immersed plane disc about an axis perpendicular to itself is turbulent if ωa2/ν is greater than about 105, where ω is the angular velocity of the disc, a its radius, and ν the kinematic viscosity of the fluid. The torque opposing the rotation, due to the surface friction of the fluid, was calculated originally by Kármán, both when the motion of the fluid is steady and when it is turbulent.

In this paper we present results for the critical percolation probabilities of a number of two- and three-dimensional lattices. These results are based upon Monte Carlo studies of the way in which cluster-size distributions vary as the number of occupied sites in a lattice is progressively increased; the principle of the method has been described in some detail in an earlier publication (Dean (1)) in which the results of studies carried out on the ACE computer were reported. The use of a KDF 9 computer with a 32K word high-speed store has now enabled us to obtain results of a very high accuracy indeed; in those cases where our values for critical probabilities can be checked against exactly known values, they differ by no more than 0·1%.

1. Problems in elasticity which are concerned with isotropic rectangular plates have attracted the attention of many writers both from the theoretical and practical points of view. When the boundary conditions are of the simply supported type the solution of the problems is usually simple, although when double Fourier series are used the validity of such solutions is not very clearly shown in most cases. Satisfactory exact solutions for many classical problems in which the edges of the rectangular plate are clamped have only been obtained in recent years, but approximate strain energy methods often gave results which were useful for practical purposes.

1. Introduction. 1·1. Notation. Let

and

be, respectively, the upper and lower central series of a group G. By definition, Zi+1/Zi is the centre of G/Zi and Γj+1 = [Γj, G] is the commutator subgroup of Γj with G. When necessary for clearness, we write ZiG) for Zi and Γj(G) for Γj.