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.

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

This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces $W(L_p,\ell_q)({\mathbb R}^d)$, Herz spaces and weighted versions of Feichtinger's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the $\theta$-means $\sigma_T^\theta f$ can be estimated pointwise by the Hardy–Littlewood maximal function. From this it follows that $\sigma_T^\theta f \,{\to}\, f$ a.e. for all $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$, hence $f\in L_p({\mathbb R}^d)$ for any $1\leq p\leq \infty$. Moreover, $\sigma_T^\theta f(x)$ converges to $f(x)$ at each Lebesgue point of $f\in L_1({\mathbb R}^d)$ (resp. $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$) if and only if the Fourier transform of $\theta$ is in a suitable Herz space. In case $\theta$ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for $\theta$ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.

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).

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.

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.

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.

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.

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.

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.

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%.

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

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.

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.

The discussion of momentum distribution is extended to cover electrons in double bonds and conjugated bonds of fractional order. The mean radial distribution function is obtained, and it shows that the mean momentum of the electrons is less than if they were rigidly attached to their own particular nuclei. Polar diagrams show that, if electronic mobility is high in any direction in a molecule, the momentum is generally small in that direction.

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.

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 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.