1. Introduction. The underlying problem is to deduce the shape of a drum or plane uniform membrane from the knowledge of its spectrum of eigenvalues ωn = i√λn. It has been shown by Kac(3) that some progress is possible on establishing the leading terms of the asymptotic expansion of the trace function for small positive t. In particular, for a simply connected membrane Ω bounded by a smooth convex curve Γ for which the displacement satisfies the wave equation ∇2ø = ∂2ø/∂t2 and Dirichlet conditions on Γ

where |Ω| = area of Ω, L = length of Γ, and the constant ⅙ is determined (apart from a factor) on integrating the curvature of the boundary; moreover, if Ω is permitted to have a finite number of smooth convex holes then the constant becomes ⅙(1–r), where r = number of holes.

Let Σan be a given infinite series with partial sums sn, and rn = nan. By and we denote the nth Cesáro means of order α (α –1) of the sequences (sn) and (rn), respectively. The series Σan is said to be absolutely summable (C, a) with index k, or simply summable |C, α|k, k ≥ 1, if

Many inhomogeneous systems of practical interest are made up of an inclusion, or several inclusions, of one elastic phase bonded to a surrounding matrix of another phase, and the behaviour of such systems has been studied extensively. A thin layer of some other elastic phase intervenes between an inclusion and the matrix in some important applications, for instance, where coated nuclear fuel particles are bonded in a matrix, or where duplex cylindrical fibres are employed in fibre-reinforcement, or wherever chemical action induces a protective finish either intentionally or inevitably as when an oxide coating covers a metallic surface. Our objective is to show how to take account of the pronounced influence that even a thin coating may exert upon, for instance, the stress concentrations just outside an inclusion or cavity, or upon the overall elastic moduli of a suspension of inclusions (though discussion of this latter topic is omitted here in order that it may be explored more fully elsewhere).

The method of integral equations is the most familiar method of proving existence theorems for the Helmholtz equation of acoustics. The wave potentials are expressed as surface distributions of wave sources (for the Neumann problem) or wave dipoles (for the Dirichlet problem). By a wave source is meant the free-space wave source. The source and dipole strengths for the exterior potentials are found to be solutions of Fredholm integral equations of the second kind which are, however, singular at a certain discrete set of frequencies corresponding to eigensolutions of the interior problems. The existence of exterior solutions at the expected frequencies can still be shown, but the proof involves a detailed and complicated study of the interior solutions. It is physically evident that this difficulty arises from the method of solution and not from the nature of the problem.

1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set

and edge-set E(G(n, k)) to consist of all edges of the form

where i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)

In this paper, we shall first describe the theory of distance-regular graphs and then apply it to the classification of Moore graphs. The object of the paper is to prove that there are no Moore graphs (other than polygons) of diameter ≥ 3. An independent proof of this result has been given by Barmai and Ito(1). Taken with the result of (4), this shows that the only possible Moore graphs are the following:

A problem in fluid mechanics which has received some attention recently concerns the emergence of an incompressible Newtonian fluid jet from a uniform tube into an inviscid atmosphere. Both the axisymmetric case of a circular tube and the two-dimensional case of flow from between parallel planes are of interest. When the jet falls vertically under gravity, the motion far downstream is dominated by gravity and the expansion procedures of Clarke (3), and Kaye and Vale (10) give details of the flow in this region. When the flow near the exit is at a high Reynolds number, it is reasonable to expect the flow appropriate to that in an infinite tube to prevail right up to the exit (except, perhaps, near the point of discontinuity of the boundary conditions). With this assumption, Duda and Vrentas(5) use a numerical technique to solve for the flow in the axisymmetric jet beyond the exit, both with and without gravity acting in the axial direction. In the absence of gravity, the jet can be expected to attain a constant width some distance downstream, and at high Reynolds numbers the above assumption is sufficient to allow a mass and momentum balance to determine the contraction ratio of the jet as for the axisymmetric case, and for the two-dimensional case (see Harmon (8)). By treating the dynamics of the jet as those of a boundary layer growing on the free surface, Goren and Wronski (6) and Tillett (18) are able to examine the flow in greater detail.

The statistical operation of multiple linear regression by least squares is equivalent to the orthogonal projection of vectors of observations on a space spanned by vectors of observations; and a partial regression can be similarly represented as an oblique projection. This connexion between a statistical and a formal algebraical operation gives the main source of interest for this investigation. Its object is to develop algebraical theory which supplies terms necessary for a unified algebraical and geometrical formulation of concepts in multivariate analysis. An exposition of the statistical theory is to be made in a separate paper.

The origin of the present paper is the desire to study the asymptotic behaviour of certain tests of significance which can be based on maximum-likelihood estimators. The standard theory of such problems (e.g. Wald(4)) assumes, sometimes tacitly, that the parameter point corresponding to the null hypothesis lies inside an open set in the parameter space. Here we wish to study what happens when the true parameter point, in estimation problems, lies on the boundary of the parameter space.

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.

A copositive quadratic form is a real form which is non-negative for non-negative arguments. A completely positive quadratic form is a real form which can be written as a sum of squares of non-negative real forms. The completely positive forms are basic in the study of block designs arising in combinatorial analysis (3). The copositive forms arise in the theory of inequalities and have been considered in a paper by Mordell (4) and two papers by Diananda(1, 2).

The theory of heavy electrons recently developed by several authors may be considered to give a satisfactory account of the empirically known neutron-proton interaction. However, it now seems well established that there exists a proton-proton interaction of comparable magnitude which is not accounted for equally well. Owing to the fact that the emission of a heavy electron involves the change of a neutron into a proton or vice versa, the first approximation of this theory gives only an exchange force between unlike particles, whereas a force between like particles must be due to double transitions and thus only appears in the second approximation. It is true that the expansion in terms of the number of particles emitted is actually so badly convergent that the second order proton-proton force at distances of about 10−13 cm. is found to be not essentially smaller than the first order neutron-proton force (see FHK), but nevertheless this does not explain experimental facts, since the calculated second order force is always repulsive.

1. In the classical theory there is no difficulty in treating the effect of radiation damping on the scattering of light by a free electron in so far as it is a result of the conservation of energy. In the non-relativistic approximation the equation of motion of a free electron under the influence of a light wave is

with the periodic solution

The total energy radiated per second is then

and the total cross-section(1) is

Formula (1) differs from the Thomson formula by the factor 1/(1 +κ2). This factor becomes appreciable for energies ħν ≥ 137mc2.

Classical shallow-water theory for the propagation of long waves in running water is modified by the inclusion of the effects of the vorticity present in the main stream as the result of the action of viscosity. When this vorticity is assumed constant, a non-linear theory can be used, but for more general velocity distributions in the main stream it is necessary to linearize the problem.

In the linearized theory, a general equation is obtained connecting the wave velocity with the velocity in the undisturbed stream and this is solved in several special cases. It is shown generally that the wave velocity relative to the mean flow is always greater than the value given by the classical theory. The wave velocity relative to the bottom of the stream has two values, one less than the minimum stream velocity and the other greater than the maximum stream velocity.

The Thomas-Fermi method has been applied to molecules with tetrahedral or octahedral symmetry in the approximation in which the nuclei of the outer atoms are smeared out over the surface of a sphere. Solutions are given for the molecules CH4, CF4, CCl4, SiH4, SiF4 and SF6, and using the results presented here it will not be a difficult matter to obtain the solutions for other molecules if they should be required.

A comparison of the results for CH4 with those of Buckingham, Massey and Tibbs shows that whilst for such a light molecule the charge distribution is somewhat poor, the potential is already in good agreement with the results of the Hartree self-consistent field method. For CCl4, however, the approximation of smoothing out all the protons appears to break down. An alternative approximation in which the K and L shell electrons are compressed into the nucleus and the modified nuclear charges smoothed, has been investigated and appears to give more reasonable results.

A discussion of the energy has been given, but the method is not sufficiently accurate to be of value for energetic considerations.

With the help of certain inequalities concerning the elements of the dispersion matrix of a set of statistics, and of the information matrix, the following results have been proved. Some of these inequalities are extensions of results given by Fisher (1) in the case of a single parameter.

(i) Efficient statistics are explicit functions of the minimal set of sufficient statistics.

(ii) Functions of the minimal set of sufficient statistics, satisfying the property of uniqueness defined in the text, are best unbiased estimates. Under certain conditions estimates possessing exactly the minimum possible variance can be obtained by the method of maximum likelihood.

(iii) In large samples maximum likelihood estimates supply efficient statistics in the case of several parameters.

(iv) The importance of replacing the sample by an exhaustive set of sufficient statistics (referred to in this paper as the minimal set) as a first step in any methodological problem has been stressed by R. A. Fisher in various articles and lectures. The above discussion supplies a formal demonstration of this view so far as the problem of estimation is concerned.

Let c0, c1,…, cn,…be a sequence of real constants, and ƒ0(x), ƒ1(x),…, ƒn(x),… a sequence of functions defined, for example, in the interval (0, 1). In this paper we shall investigate some of the properties of the series

which may be obtained from the standard series

by interchanging the signs of the terms in a quite arbitrary way.

A genetical population in which different genotypes have different viabilities will undergo, by selection, changes in its genetical constitution which can be expressed in a simple mathematical form. We here consider a system of m alleles A1, A2, …, Am segregating at one locus, and we assume that the genotype AiAj has viability aij, where aij = aji ≥ 0. We assume discrete, non-overlapping generations and random mating.