bluuebunny/crossref_metadata_2025 huggingface dataset:
The problem of a rotating incompressible disk
Authors: Clark G. L.
Published: November 2008

In this paper an investigation is made of the properties of material which is under isotropic stress p when in a strained state, and is such that the relation between the stress and the dilatation Δ is

where α is the density, assumed constant in the unstrained state, and c is the velocity of light. It is shown that in this material the waves of dilatation travel with the velocity of light and that a disk or cylinder composed of this kind of matter suffers no change in radius when it is made to rotate.

It is suggested that it is not unreasonable to attach the label incompressible to matter having these properties.
On non-linear differential equations of the second order
Authors: Cartwright M. L.
Published: November 2008

1. The problem of automatic synchronization of triode oscillators was studied by Appleton† and van der Pol‡; it gives rise to the differential equation

where α, γ, ω, E, ω1 are positive constants such that α/ω, γ/ω, (ω − ω1)/ω are small and dots denote differentiations with respect to t. When these conditions are satisfied, it is easy to see that

is an approximate solution over a limited time for any b1, b2 chosen to fit initial conditions on v and ṿ provided that v and ṿ are not too large. If it is assumed that b1 and b2 vary slowly compared with ω1t, so that can be neglected, and ḃ1, ḃ2 are comparatively small, the equations

where

are obtained. These are sufficiently accurate for the discussion of most of the physical phenomena, and have been used in this form (or in the polar-coordinate form obtained by putting b1 = b cos ø, b2 = b sin ø) by various authors §. Solutions of (1) with period 2π/ω1 are obtained approximately by putting ḃ1 = ḃ2 = 0 in (3). The steady-state solutions of (1) other than those with period 2π/ω1 correspond to periodic solutions of (3). All other solutions converge to one or other of these types of solution.
Note on some properties of maximum likelihood estimates
Authors: Diananda P. H.
Published: November 2008

The properties of (1) consistency and (2) asymptotic normality of maximum likelihood estimates of one unknown parameter, for independent observations, have been discussed rigorously by Cramér ((1), p. 500) and Doob (2). These properties were first stated by Fisher (3).
An approximate relation between the energy levels of a particle in a field of given potential energy, calculated on the relativistic and non-relativistic theories
Authors: Sewell Geoffrey L.
Published: November 2008

The relativistic Hamiltonian H for a particle in a field of potential energy V is given, in atomic units, by the formula

where any ρ commutes with any σ, any σ anticommutes with any other σ, ρ1 anticommutes with ρ3 and
A modification of the molecular orbital method: the LCMO method
Authors: Dewar M. J. S.
Published: November 2008

A modification of the MO method is described which leads to the same polynomial secular equation as does the LCAO method, but via a simpler secular determinant. The basic idea is to divide the whole molecule into parts, calculate the MO's for these by the usual method, and then use linear combinations of these MO's as trial eigen-functions in a variational treatment of the whole molecule.

Equations are developed for a number of cases, and it is shown that the computations of energy levels and MO's from them is often much simpler than from the corresponding LCAO equations.
The forces between floating bubbles and a quantitative study of the Bragg ‘Bubble Model’ of a crystal
Authors: Lomer W. M.
Published: November 2008

The object of the paper was to find the forces acting in bubble rafts and to use them for a description of the mechanical properties of these rafts. The first section is based entirely on Nicolson's work and gives expressions for the cohesive forces in the lattice. The second and third sections deal with the repulsive forces arising when a bubble is compressed uniaxially, and are really the basis of the paper. The fourth section introduces the idea of ‘interaction’ between compressions in the three close-packed directions—a conception which explains the breakdown of Cauchy's relations. The conditions of equilibrium in the lattice are investigated in § 5, and in § 6 all these forces are combined to give a complete account of the elastic properties of the rafts.

The remaining section gives a brief account of the experimental checks of the theory; it appears to be a reasonable approximation for small bubbles, and, indeed, we could certainly not expect it to apply to large ones, since for them not only are all the calculations of the attractive potentials invalid because of the large local slopes of the surface, but also the ‘flats’ and ‘caps’ become so large as to interfere with one another.

The forces are considered well enough supported to justify their further use in an investigation of the plastic behaviour of the bubble rafts—an investigation which has explained several interesting experimental facts but brought us little nearer an understanding of the low yield strains of metallic single crystals (see (2)).
On two theorems of S. Verblunsky
Authors: Delange Hubert
Published: November 2008

In connexion with moment problems, S. Verblunsky proved the following two theorems:

Theorem I. (a) If f(x) is integrable in (−∞, +∞) and satisfies 0 ≤f(x) ≤ 1, then there exists a function σ(x), bounded and non-decreasing in (−∞, +∞), such that, for ζ = ξ + iη and η ≠ 0,
On the thermal conductivity in dense stars
Authors: Mestel L.
Published: November 2008

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 Besicovitch dimension of Cartesian product sets
Authors: Eggleston H. G.
Published: November 2008

A. S. Besicovitch has defined the dimension of a point-set X in n-dimensional Euclidean space in terms of its exterior Hausdorff measure as follows (2). Let (δ, X) be any enumerable class of sets whose point-set union contains X and whose members are each of diameter less than δ. Let (δ, X) denote the class of all (δ, X).
Some probability distributions connected with recording apparatus. II
Authors: Domb C.
Published: November 2008

In a previous paper(1) a number of problems in probability were considered arising out of the finite resolving time of a recording apparatus. It was shown that, although the probability distributions themselves are complicated, their Laplace transforms are relatively simple. Recently Feather(2) has discussed a number of additional problems arising in this connexion, including the number of coincidences, and the number of k-clusters. The method developed in (1) is readily applicable to these problems, and it will be shown how complete probability distributions can be derived. An alternative type of recorder is sometimes considered (3) which remains dead as long as events succeed each other at intervals less than τ. The mathematical problem here is somewhat different, but it can again be effectively dealt with by the use of Laplace transforms.
A further note on trigonometrical inequalities
Authors: Ingham A. E.
Published: November 2008

1. The aim of this note is to prove the

Theorem. Let

where the λnare real and

and let

Then

A similar result holds for infinite seriesconverging uniformly in [−T, T].
On the expansion of a Coulomb potential in spherical harmonics
Authors: Carlson B. C. , Rushbrooke G. S.
Published: November 2008

The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,

or, in an alternative notation,
Some metrical theorems in diophantine approximation
Authors: Cassels J. W. S.
Published: November 2008

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.

It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequality

has infinitely many integer solutions q > 0. In particular, if α is any real number, the inequality

has infinitely many solutions.
An extremal property of the hypersphere
Authors: Macbeath A. M.
Published: November 2008

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.
A note on rayleigh's problem for a plate of finite width
Authors: Davies D. R.
Published: November 2008

The purpose of this note is to indicate the relation between (a) the problem of evaporation from a finite area of rectangular shape into a turbulent air stream and of the boundary layer on a flat plate of finite width, and (b) the problem of the boundary layer on a semi-infinite plate and evaporation from an infinite quadrant.
Pressure fluctuations in isotropic turbulence
Authors: Batchelor G. K.
Published: November 2008

This paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.

The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.
Note on a penny-shaped crack under shear
Authors: Segedin C. M.
Published: November 2008

Sack (3) and Sneddon (4) have considered the problem of a penny-shaped crack in an infinite medium under tension normal to the face of the crack. The same problem has been considered more recently by Green (1) by a different method. In this note we consider the case of the same shaped crack with the medium subjected to a uniform shearing stress parallel to the plane of the crack. It is found that the solution is analogous to that of the corresponding two-dimensional problem considered by Starr (5).
A generalization of the Hurwitz composition theorem to irregular power series
Authors: Eggleston H. G.
Published: November 2008

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the function

in terms of the positions of the singularities of f(z) and g(z).
The surface energies of superconductors
Authors: Pippard A. B.
Published: November 2008

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.
On the order of magnitude of Ramanujan's arithmetical function τ(n)
Authors: Pennington W. B.
Published: November 2008

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansion

This function appears in the discussion of an asymptotic formula for the function

and also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.
Periodic solutions of non-linear differential equations of the second order. V
Authors: Obi Chike
Published: November 2008

1·1. Let van der Pol's equation be taken in the form

where ε1, ε2, k1 and k2 are small, and ω ≠ 0 is a constant, rational or irrational, independent of them.
The energy loss of free magnetic poles in passing through matter
Authors: Bauer E.
Published: November 2008

Dirac has shown that in quantum theory free magnetic poles are possible only if their pole strength g is an integral multiple of ℏc/(2e) = 137e/2. The ionization produced by such particles is calculated by the classical impact parameter method and also by quantal perturbation theory; at sufficiently large velocities the rate of loss of energy of poles of unit strength is g2/e2 ≑ 5000 times as large as that of an electron of the same velocity; at low velocities the energy loss of a monopole decreases steadily without rising to a maximum as in the case of electric particles.
Congruence properties of the partition function and associated functions
Authors: Rushforth J. M.
Published: November 2008

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled

‘Properties ofp(n) and τ(n) defined by the relations

and was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties ofp(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems
Notes on a theorem of Cochran
Authors: James G. S.
Published: November 2008

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x1, …, xn are independently distributed standard normal deviates, and q1, …, qk are k real quadratic forms in the xi with ranks n1, …, nk respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q1 …, qk are independently distributed in χ2 forms with n1, …, nk degrees of freedom is that Σnj = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each qj as a sum, involving nj squares of real linear forms in the xi; it follows easily that the coefficients ci are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.
The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains
Authors: Jones D. S.
Published: November 2008

The spectrum of −∇2 (and of −∇2 + b) is investigated when the boundary conditions are given on surfaces which extend to infinity. Simple criteria are obtained for determining whether point-eigenvalues are present in the lower part of the spectrum.

Semi-infinite domains which are conical at infinity are found to possess purely continuous spectra when the boundary condition is u = 0 or ∂u/∂v = 0; the radiation condition ensures a unique solution. A counter-example shows that this is not true in general for the boundary condition ∂u/∂v + σu = 0.

Semi-infinite domains which are cylindrical at infinity have a continuous spectrum with a discrete spectrum embedded in it. An example is given.

The results are applied to the theory of surface waves. It is shown that Ursell's ‘trapping modes’ can occur in a canal of finite width when the bed has a protrusion over a finite longth but is otherwise of uniform depth. Trapping modes can also occur when the canal contains a submerged cylinder (not necessarily small in cross-section).
Some generalizations of the shear problem for isotropic incompressible materials
Authors: Adkins J. E.
Published: November 2008

In the theoretical treatment of finite elastic deformation for isotropic, incompressible materials, a series of relations may be obtained, which, in the most general case, can be reduced to four partial differential equations for the determination of four unknown quantities. For these dependent variables we may conveniently choose the three components of displacement in any given system of coordinates, and an arbitrary parameter which arises from the assumption of incompressibility. The equations thus obtained are non-linear in form, and exact solutions subject to specified boundary conditions are usually not readily obtainable by orthodox methods of approach. The successful treatment of a number of problems by Rivlin (see references), and by other workers (1,2), has depended essentially upon the imposition of a restriction upon the form of the deformation, so that by an appropriate choice of initial and final reference frames in which to specify the configuration of the elastic body, partial differential equations can be avoided.
The product of non-homogeneous linear forms. II The minimum of a class of non-homogeneous linear forms
Authors: Samet P. A.
Published: November 2008

In this paper we determine the first minimum of a class of linear forms associated with certain cubic fields that depend on a parameter.
On linearly polarized electromagnetic waves of arbitrary form
Authors: Nisbet A. , Wolf E.
Published: November 2008

Two simple laws connecting the amplitude and phase functions of a monochromatic electromagnetic wave of arbitrary form are derived, holding in the case when one of the field vectors is linearly polarized. The first is a generalized Fermat's principle which enables determination of the phase when the amplitude is known; the second expresses the propagation of the (vector) amplitude along the curves orthogonal to the co-phasal surfaces. Some other general properties of linearly polarized fields are also discussed, and illustrative examples are given.
Expansions of generalized Whittaker functions
Authors: Slater L. J.
Published: November 2008

In this paper I give a direct proof of a new expansion of the Whittaker function M(a; b; x) as a series of Bessel functions, and some generalizations of this result for the functions 2F2(x) and 4F4(x).
On semigroup algebras
Authors: Munn W. D.
Published: November 2008

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.
A note on recurrent events
Authors: Patankar V. N.
Published: November 2008

Consider a Markoff chain (by which is meant a stochastic process defined for discrete values of both variable and parameter, and whose probability dependence does not extend to more than a unit interval) with completely specified transition probability matrix P = (pij), where pij is the conditional probability that the (r + 1)th observation belongs to the state Ei, given that the rth observation belongs to the state Ei for r≥ I. We assume that there are a states E1, E2, …, Ea, where a is finite. We also assume an ergodic property for the stochastic process, that is, we consider non-periodic chains for which all the possible initial states remain permanently available. This defines what has been called the positively regular case. Also under the above assumption the moduli of all the latent roots λr of the transition probability matrix P = (pij) other than the first simple latent root λ1 = 1 are less than one (see Bartlett(1)).
The character of the equilibrium of an incompressible heavy viscous fluid of variable density
Authors: Chandrasekhar S.
Published: November 2008

This paper is devoted to a consideration of the following problem: Given a static state in which an incompressible viscous fluid is arranged in horizontal strata and the density is a function of the vertical coordinate only, to determine the initial manner of development of an infinitesimal disturbance. The mathematical problem is reduced to one in characteristic values in a fourth-order differential equation and a variational principle characterizing the solution is enunciated. The particular case of two uniform fluids of different densities (but the same kinematic viscosity) separated by a horizontal boundary is considered in some detail. The mode of maximum instability in case the upper fluid is more dense and the manner of decay in case the lower fluid is more dense are determined; and the results of the calculations are illustrated graphically. Gravity waves (obtained in the limit when the density of the upper fluid is zero) are also treated.
The character of the equilibrium of an incompressible heavy viscous fluid of variable density: an approximate theory
Authors: Hide Raymond
Published: November 2008

The equilibrium of a layer of a viscous incompressible fluid of density ρ variable in the vertical (z) direction, the fluid extending indefinitely in the horizontal directions and being confined between rigid horizontal boundaries, is examined by the usual method of studying the initial behaviour of a small disturbance. Diffusion effects are ignored. Chandra-sekhar has shown quite generally that a variational procedure for solving the resulting characteristic value problem is admissible, suggesting a method whereby an approximate solution can be obtained analytically. In this paper, by making use of appropriate approximate methods of solution, three types of density configuration, ρ(z), are studied. The first corresponds to two superposed fluids of great depth, the upper and lower fluids haying densities and coefficients of viscosity ρ2 and μ2, and ρ1 and μ1, respectively. Chandrasekhar has given an exact numerical solution to this problem for the case where μ2/ρ2 = μ1/ρ1 The results of the approximate theory, by which not all the boundary conditions can be satisfied, and the exact theory are in good agreement. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (ρ2 > ρ1) are found. The stable case (ρ2 > ρ1) is also discussed in detail; the formulae obtained for the phase and group velocities of gravity waves in deep viscous fluids are especially interesting. There is an upper limit to the natural frequency of oscillation of the system which would be well within the range of experimental investigation.

The second problem considered is that of two superposed fluids of small depth. Formulae for the phase and group velocities of ‘long gravity waves’ (which are not equal when viscous effects are included) are obtained.

Finally, a continuously stratified fluid of finite depth d in which ρ(z) = ρ0eβz is investigated. The character of the equilibrium depends on a Grashof number G = (gβd4)/(π4ν2) where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (G > 0) are obtained. In the stable case (G < 0) all modes are aperiodically damped if – G < . Otherwise, the possibility of a disturbance being propagated as a horizontal wave does arise, but only within a limited wave-length range.
The irrotational motion outside a free turbulent boundary
Authors: Phillips O. M.
Published: November 2008

In this paper there is considered the irrotational motion of an infinite fluid when the normal velocity across a plane is specified as a stationary random function of position in the plane, and a solution is obtained in terms of the specified boundary conditions. It is shown that the mean square velocity normal to the plane is equal to the sum of the mean squares of the velocities in the other two orthogonal directions. The asymptotic variations with distance normal to the plane are found for functions representing the important properties of the motion, and, in particular, the energy of the fluctuations is shown to be inversely proportional to the fourth power of the distance from the plane. The conditions postulated are shown to correspond closely to the motion outside a free turbulent boundary, and good agreement is found between the predictions of the theory and the available experimental results.
The α-dimensional measure of the graph and set of zeros of a Brownian path
Authors: Taylor S. J.
Published: November 2008

In a recent joint paper (1) with Prof. Besicovitch we announced the conjecture that for almost all one-dimensional Brownian paths, the set of zeros has dimensional number ½, and zero A½-measure. It is the purpose of this paper to give a proof of this result. In doing so we consider the graph C(ω) of a Brownian path ω as a point set in the plane, and prove that, with probability 1, C(ω) has dimensional number ¾ and zero Λ¾-measure.
A semi-infinite random walk with discrete steps
Authors: Shenton L. R.
Published: November 2008

1. A particle executes a random walk over the possible positions x = 0,1,2,…, its initial position being x = d ≥ 0. At the nth step it occupies the position x with probability pn (x | d)and is in the state (n, x). The transition from (n, x) to (n + 1, y) has the probability px, v given by
A Principle in classical mechanics with a ‘relativistic’ path-element extending the principle of least action
Authors: Schieldrop Edgar B.
Published: November 2008

1. A particle with mass m and coordinates x1x2, x3 relative to a set of rectangular axes fixed in Newtonian space is moving in a field of conservative forces with a potential energy V(x1, x2, x3) and a kinetic energy

The equations of motion, written

(representing the three equations i = l, i = 2, i = 3 in a way to be used in this paper), constitute, as they stand, a sufficient condition in order to ensure

in the sense that the Hamiltonian integral has a stationary value if the actual motion is compared with neighbouring motions with the same terminal positions and the same terminal values of the time as in the actual motion.
A class of transformations of the plane
Authors: Hedlund G. A.
Published: November 2008

LetZdenote the extended complex plane with the natural topology. That is,Zconsists of the finite complex planeZftogether with a point, denoted by ∞, any neighbourhood of which consists of the union of this point and the complement of a bounded set ofZF. LetZ* = [z │z↦Z,z╪ 0,z╪ ∞]. ThusZis homoeomorphic to a 2-sphere andZ* is homoeomorphic to a cylinder.
Correlated random walk
Authors: Gillis J.
Published: November 2008

Random walk on a d-dimensional lattice is investigated such that, at any stage, the probabilities of the step being in the various possible directions depend upon the direction of the previous step. The motion may be characterized by a generating function which is here derived. The generating function is then used to obtain some general properties of the walk. Certain special cases are considered in greater detail. The existence of recurrent points is investigated in particular, and the probability of returning to the origin after 2n steps. This latter function is evaluated asymptotically for the cases d = 1 and d = an even integer.
The cosmological principle
Authors: Littlewood D. E.
Published: November 2008

Many models of the universe have been proposed, by de Sitter, Milne, Bondi and Gold, Hoyle and others. The observed data being insufficient, the models are usually based on some simple hypothesis. The simplest is the cosmological principle, namely, that apart from local irregularities the universe presents the same general aspect at every point. Milne (5) has used a restricted form of the principle, namely, that the aspect is independent of spatial position but is dependent on the observed time from some fixed epoch in the past. Bondi and Gold(i) have proposed the ‘perfect cosmological principle’ that the aspect is completely independent of space and time.
A basis theorem for polynomial modules
Authors: Rees D.
Published: November 2008

The purpose of the present note is to prove a basis theorem for polynomial modules (Theorem 1 below). As an application, a proof of the Hilbert function theorem for homogeneous modules (Snapper (1)) is obtained. Both these proofs are elementary in the sense that they do not depend on the decomposition theory for modules over a Noether ring.
On best approximate solutions of linear matrix equations
Authors: Penrose R.
Published: November 2008

In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.
Statistical properties of a moving wave-form
Authors: Longuet-Higgins M. S.
Published: November 2008

Many of the statistical properties of a random noise function f(t) have been derived by Rice (4); for example, the mean frequency of zero crossings and the statistical distribution of maxima and minima. These distributions have been shown to apply quite accurately to physical phenomena; in particular, when f(t) represents the height of sea waves at a certain point or the pressure at a fixed point on the sea bed (1, 3, 5).
On finite groups of exponent five
Authors: Higman Graham
Published: November 2008

1. The object of this note is to prove the restricted Burnside conjecture for exponent 5, that is, to prove, for n = 5, the proposition:

Rn: For each positive integer k there is an integer rn, k such that every finite group of exponent n that can be generated by k elements has order at most rn, k.
Some properties of Wigner coefficients and hyperspherical harmonics
Authors: Stone A. P.
Published: November 2008

General shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.
Exact tests for serial correlation in vector processes
Authors: Hannan E. J.
Published: November 2008

Exact tests for serial independence in vector Markoff processes have been obtained by considering the system of regressions of the vectors observed at time 2t on the vectors observed at times (2t − 1) and (2t + 1). The tests then reduce to those obtained from canonical correlation procedures in multivariate analysis. Two particular cases are

(1) The test for serial independence of a vector of residuals from a system of regressions in which the regressors are all independent of the residuals. At the same time a test of the hypothesis that the regressor and regrediend vectors are independent is obtained.

(2) The test for serial correlation or partial serial correlation in a multiple Markoff process (autoregressive process).

An investigation of the efficiency of the test so obtained, of the hypothesis that a process is a simple Markoff process (against the alternative that the process is a second-order auto-regression) suggests that the efficiency of all of these tests will be low.
Disjunctive games with the last player losing
Authors: Grundy P. M. , Smith C. A. B.
Published: November 2008

Various authors have discussed the ‘disjunctive compound’ of games with the last player winning (Grundy(2); Guy and Smith (3); Smith (4)). We attempt here to analyse the disjunctive compound with the last player losing, though unfortunately with less complete success. The problem was explored tentatively by the first named author, but the rigorous development is due to the second.
Properties of random functions
Authors: Palmer D. S.
Published: November 2008

This paper deals with the relationships of the maxima, minima and zeros of two random functions of known autocorrelations and cross-correlation, based on the work of Rice(4). Ratcliffe(3) and Briggs and Spencer(2) discuss a similar problem in connexion with a ‘Phillips Record’ of an experiment in ionospheric reflexion. In this experiment there are two highly correlated reflected signals, their maxima coming close together, and the record shows the time lags between a maximum on one signal and a maximum on the other. Briggs and Page (1) have made an experimental study of the distribution of the differences between the positions of the maxima of two highly correlated random functions, using EDSAC to construct the functions. In §§ 3–7 the frequency distributions of intervals between successive zeros and maxima, and of the lengths of intercepts by a horizontal line, are considered. This has applications to the study of the fading of long-wave radio signals, where the tune differences between successive maxima of the amplitude have been investigated.
A study of the ‘irreversibility paradox’ for a simple statistical assembly
Authors: Temperley H. N. V.
Published: November 2008

A very simple model, consisting of N particles moving in a one-dimensional assembly divided by potential ‘humps’ into M cells, is studied. The process of passing from a quantum-mechanical description of such an assembly to the equation of diffusion type that governs it in practice is shown to consist of at least three separate steps: ‘averaging over phases’, and letting N and M become large. The effects of these steps are considered separately. Strict irreversibility in time appears after the first step, but the assembly remains ergodic until after the second step and fluctuations persist until after the third step.
On the short-wave asymptotic theory of the wave equation (∇2 + k2)ø = 0
Authors: Ursell F.
Published: November 2008

The theory of time-periodic wave problems falls into two parts. On the one hand there is the rigorous formulation in terms of differential wave equations, on the other there are approximate theories like geometrical optics. It should be possible, in principle, to deduce the latter from the former by a logical process; but this has been done only for a few simple configurations, e.g. the circle. A possible approach to the solution of the general problem is suggested here, and is applied to a typical two-dimensional acoustical example. An arbitrary closed convex curve (satisfying certain regularity conditions) is emitting short sound waves towards infinity, the normal velocity V(s) exp (− iωt) is prescribed on the curve as a function of the arc-length s, and the potential is to be found, first on the curve and then at any point in the sound field. (Only the first part of the problem is treated in detail.) The potential ø(s) exp (− iωt) on the curve satisfies all the integral equations

where G(s, s′) is any Green's function of the problem, and V(a) is prescribed. All the equations corresponding to different Green's functions have the same solution. An asymptotic and convergent short-wave solution can be found by iteration if G can be chosen explicitly so that the integral equation has a small kernel for high frequencies. At any point of the curve draw the local circle of curvature; then the explicit known solution for a source on this circle is (with slight modifications) a possible Green's function, and the equation formed with it has a small kernel and can be solved rigorously by iteration. If V(a) is independent of the frequency, the leading term in the resulting asymptotic expansion is

where c is the velocity of sound and 2πk−1 is the (short) wavelength corresponding to the frequency ω/2π. If V(a) varies rapidly, as in diffraction theory, the iterative solution still gives a convergent asymptotic expansion, but the first approximation is then practically useless in the shadow region. Diffraction problems are not treated in the present paper.

The present work appears to be the first practical and rigorous solution of a short-wave problem in optics or acoustics when a solution in closed form is not available. It is suggested that the technique (suitably combined with formal expansions) may be applicable to a wider class of radiation and diffraction problems.
On renewal theory, counter problems, and quasi-Poisson processes
Authors: Smith Walter L.
Published: November 2008

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic process

Hammersley'sx counter problem concerns the stochastic process
Summation over Feynman histories: the free particle and the harmonic oscillator
Authors: Davies H.
Published: November 2008

Using a particular parametrization of paths, the free particle and the harmonic oscillator are treated by the Feynman method of summation over histories of the system. The propagators obtained are, apart from an arbitrary phase factor, those of conventional quantum mechanics.
A relativistic form of Boltzmann's transport equation in the absence of collisions
Authors: Clemmow P. C. , Willson A. J.
Published: November 2008

With the neglect of collisions, a relativistic form of Boltzmann's transport equation is derived and its invariance under a Lorentz transformation is established.
Representation of Hilbert space operators by (nJ)-matrices
Authors: Smart D. R.
Published: November 2008

Introduction. Let be the complex separable Hilbert space. We say that the closed linear operator T, with domain dense in. , is represented by the infinite matrix H if T is the operator T˜1(H) defined† by H (with respect to some complete orthonormal set). We define an (nJ)-matrix as a Hermitian matrix H = [hij]i, j ≥ 1 for which hij = 0 when i − j > n and hij ╪ 0 when i − j = n. (Thus a Jacobi matrix is a (1J)-matrix.) If, in addition, hij = 0 when 0 < i − j < n, we call H an (nJ ┴)-matrix.
A convergent asymptotic representation for integrals
Authors: Franklin Joel , Friedman Bernard
Published: November 2008

This paper represents a new method for obtaining an asymptotic representation for integrals of the form when p is large. It is shown that if f(x) satisfies certain conditions this representation is also convergent. Numerical calculations seem to show that the first term of the representation gives a close approximation to the value of the integral for a wide range of values of p.
Some applications of Mellin transforms to the theory of bivariate statistical distributions
Authors: Fox Charles
Published: November 2008

A method is described for finding the frequency functions of bivariate random variables which are the products or ratios of other bivariate random variables. If (ξ, n) are a pair of bivariate random variables with joint frequency function f(x, y) then the method depends upon the fact that the expectation of │ ξ │r–1│η│s–1 is related to the Mellin transform of f(x, y) in two dimensions. Knowing the expectation we can then recover the frequency function by means of the inverse Mellin transform. Some examples are given to illustrate the theory.
The absorption of very high frequency sound in dielectric solids
Authors: Simons S.
Published: November 2008

A theoretical treatment is given of the absorption of longitudinally polarized sound in dielectric crystals of high symmetry, due to interaction with the thermal phonous. The Q value for the absorption is calculated in terms of the deviation from Hooke's law, the result being applicable at low temperatures when the mean free path of the phonons is greater than the wavelength of the acoustic wave. The theory is applied to cubic crystals, and in particular to solid argon, where a minimum Q value of about 108–109 is calculated. It is suggested that measurement of the absorption (at present somewhat beyond experimental techniques) would supply information on the deviation from linear elasticity, as well as providing a satisfactory verification of the basic interaction theory. The effect of isotopes, and, in more complicated structures, of optic modes, is shown to be negligible.
The transverse flexure of thin perforated elastic plates supported at several points
Authors: Bassali W. A.
Published: November 2008

Following the method outlined in a previous paper, exact solutions in finite terms are obtained for the problem of an infinitely large plate with outer edge free and an inner free circular boundary, the plate being supported at any number of interior or boundary points and normally loaded over a circle. The load considered is symmetrical with respect to the centre of the circle.
On non-dissipative Markov chains
Authors: Mauldon J. G.
Published: November 2008

Consider a Markov chain with an enumerable infinity of states, labelled 0, 1, 2, …, whose one-step transition probabilities pij are independent of time. Then

I write

and, departing slightly from the usual convention,

Then it is known ((1), pp. 324–34, or (6)) that the limits πij always exist, and that
Physical properties of some empty space-times
Authors: Joseph V.
Published: November 2008

The canonical form of the Riemann tensor is found for certain non-flat empty space-times. Some invariant physical properties of these space-times are investigated. The status of Mach's principle in general relativity theory is briefly discussed in the light of these examples.
Second order effects in the flexure of isotropic incompressible elastic cylinders
Authors: Blackburn W. S.
Published: November 2008

General methods of successive approximations for problems in the theory of large elastic deformations have been proposed by Signorini (9), Misicu (5), Rivlin (6), Green and Spratt (4) and Rivlin and Topakoglu (7). Rivlin has applied his method to the problem of torsion of cylinders as far as terms of the second order. A solution of the problem of second order effects in the torsion of incompressible cylinders of arbitrary cross-section has been given by Green and Shield (3) in terms of complex potential functions, and the complete details of the stress distribution can be obtained once certain integral equations are solved. Blackburn and Green (l) have considered the second order effects in the deformation of compressible cylinders by forces on the ends. For the problems of torsion and bending by couples they obtained details of the displacements and stress distributions in terms of two complex potential functions which satisfy a certain boundary condition. The second order torsion problem has also been discussed by Sheng(8).
The spherical harmonics with the symmetry of the icosahedral group
Authors: Cohan Norah V.
Published: November 2008

For a long time there has been some interest in obtaining spherical harmonics with the symmetry of the regular polyhedrons, particularly for electrostatic problema involving polyhedral conductors. Work on the icosahedral group has been done, among others, by Meyer (7), Laporte (6), Hodgkinson (3) and Poole (8), apart from the classic work of Klein (4). In the several approaches of these authors, only spherical harmonics for the totally symmetric representation were obtained, the most complete table being that of Laporte who obtained the spherical harmonics up tol= 21. New interest in the icosahedral group has arisen in connexion with the structureof some proteins (5), and we obtain here, by the recently developed methodof Altmann(1), expansions in spherical harmonics for ali the representations of this group. This has been done up to and includingl= 14. (For the totally symmetric representation we have also includedl= 15.) In what follows we shall use Altmann's formulae and notation.
The reflexion of long waves at a step
Authors: Bartholomeusz E. F.
Published: November 2008

A train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.
On homoeomorphism of perfect plane sets
Authors: Besicovitch A. S.
Published: November 2008

In studying the structure of perfect sets, an important method would be to establish homoeomorphisms in general classes of sets. For then the general structural features of sets could be more easily seen by considering the simplest sets of a homoeomorphic class.

In this paper we establish the homoeomorphism in two general classes of plane perfect sets (we consider only bounded sets).