bluuebunny/crossref_metadata_2025 huggingface dataset:
Rectilinear plastic flow of a Bingham solid
Authors: Oldroyd J. G.
Published: November 2008

A short discussion of steady rectilinear plastic flow of Bingham material between rigid cylinders of general shape in relative motion parallel to their generators was given in a paper of the same main title(1). It was shown that, for sufficiently great relative velocity V of the boundaries, plastic flow in the whole region would be expected, as for a Newtonian liquid. But for slower relative motion, part of the material might remain elastically deformed (i.e. stressed below the yield point) with a limited region of plastic flow surrounding the inner boundary. In the paper referred to, the problem was illustrated by the case of eccentric circular cylindrical boundaries. The choice of circular boundaries introduced a great simplification when the region of plastic flow did not extend to the outer boundary, namely axial symmetry in the flow pattern in the plastic region. It is the main purpose of the present paper to illustrate how the methods already given for determining velocity distributions may be modified to suit more general problems, by consideration of confocal elliptic cylinders as boundaries.
Field theories with high derivatives
Authors: Chang T. S.
Published: November 2008

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.

After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative of

with respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.
Note on Peng's treatment of the divergency difficulties in quantized field theories
Authors: Touschek B.
Published: November 2008

The purpose of this note is to show that Peng's (1) method of approximation is not practicable in the case of quadrilinear interaction (i.e. Fermi's (2) original theory of β-decay), and that it does not remove the infinite self-energy of a nucleon in interaction with a charged scalar meson field. The two examples do not, however, provide a serious argument against Peng's theory, since both refer to field theories which have been abandoned for physical reasons. The principal failure of Peng's method to cope with quadrilinear interactions could even be interpreted in such a way as to mean that interactions of this sort do not exist—a statement which would be equivalent to the assumption that Fermi particles can only interact by means of a Bose field. (2n + 1) linear interaction terms must contain one Bose quantity at least, since Fermi field quantities are not observable and therefore can only enter the interaction term in bilinear form.
On surfaces of minimum area
Authors: Besicovitch A. S.
Published: November 2008

This paper is devoted to the classical problem of surfaces of minimum area subtending a given closed contour.
The statistical distribution of the length of a rubber molecule
Authors: Moran P. A. P.
Published: November 2008

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the form

where σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.
Notes on Fourier series (V): Summability (R1)
Authors: Hardy G. H. , Rogosinski W. W.
Published: November 2008

1·1. There is a familiar method of summation of series usually called the method (R, 1), or sometimes Lebesgue's method. If sn = u0 + u1 + … + un and, as it will be convenient to suppose throughout, u0 = 0, then

when h → + 0§ the convergence of the series for small positive h is presupposed. The method is not 'regular'; there are convergent series not summable (R, 1).
Inverse probability and sufficient statistics
Authors: Huzurbazar V. S.
Published: November 2008

1. It is an interesting fact that in many problems of statistical estimation the results given by the theory of inverse probability (as modified by Jeffreys) are indistinguishable from those given by the methods of ‘fiducial probability’ or ‘confidence intervals’. The derivation of some of the important inverse distributions by Jeffreys(3) arouses one's curiosity. It seems that when this agreement is noticed there are usually sufficient statistics for parameters in the distribution. The object of this note is to throw some light, in general terms, on the similarity in form between the posterior probability-density function of the parameters and the probability-density function of the distribution when it admits sufficient statistics. For convenience the following notation in Jeffreys's probability logic is used below:

P(q | p) is the probability of a proposition q on data p.
Effect of tension on magnetization curves
Authors: Stewart K. H.
Published: November 2008

It has recently been shown (4, 5) that in order to give a true account of the magnetization processes in a ferromagnetic it is necessary to consider the influence of the demagnetizing field of a finite specimen not merely in a quantitative way, affecting the magnitude of the field acting on the specimen, but also in a qualitative way, affecting the actual arrangement of the domains of the ferromagnetic. It is the purpose of this paper to give an account of the process of magnetization in a specimen which has a tension applied in the same direction as the magnetic field. It will be shown that by considering fully the effects of demagnetizing fields results are obtained which agree well with experiment and remove some discrepancies of the older domain theory.
Some calculations on the γ-ray dose produced by a 14 Mev. betasynchrotron using an internal target
Authors: Allen-Williams D. J. , Appleyard R. K.
Published: November 2008

Relations are derived between the dose in r./min. delivered by the γ-ray beam from a betasynchrotron and the mean electron current which is incident on the internal target. Comparison with experiment shows that multiple passage of electrons through the target probably plays an important part in production of the γ-ray beam. Some possible effects connected with the multiple passage of electrons are discussed.
On differentiation and Denjoy-behaviour of functions of two real variables
Authors: Mossaheb G. H.
Published: November 2008

The study of differential properties of functions of two real variables in the light of the modern theory of functions of real variables was started by H. Rademacher in 1919. Although much work has since been done on the subject, and a great many general results have been obtained, a number of questions have remained open, some of which will be discussed in the present work.
A comparison of some absolute methods of measuring fast neutron flux
Authors: Allen K. W. , Livesey D. L. , Wilkinson D. H.
Published: November 2008

The absolute measurement of fast neutron flux presents several difficult problems. Few methods have yet been described in the literature, although the experimental techniques developed by several authors for the detection of fast neutrons (Baldinger, Huber and Staub(7), Barshall and Kanner(9), Amaldi, Bocciarelli, Ferretti and Trabacchi (3), Gray (19), Barshall and Battat(8)) may easily be adapted to this type of measurement. It is, however, most important to have available methods of measuring fast neutron flux to permit the determination of cross-sections for nuclear processes induced by fast neutrons, and several such methods have been developed in the Cavendish Laboratory in recent years. They are the subjects of separate papers (Bretscher and French (13), Kinsey, Cohen and Dainty (21), Allen (l), Allen and Wilkinson (2)). The main purpose of the present paper is to describe the results of experiments carried out to compare these methods in order to test the validity of the assumptions implicit in the individual methods.
Note on the evaluation of complex determinants
Authors: Goodwin E. T.
Published: November 2008

A square matrix A, of order n, having complex coefficients can be inverted without the aid of any operations involving complex numbers. This can be done if the coefficients ars + iαrs are replaced by their matrix equivalents and the resulting 2n × 2n real matrix A1 is inverted. The inverse will be a 2n × 2n matrix of similar form in which the subsidiary 2 × 2 matrices can be replaced by the equivalent complex numbers, thus yielding the inverse of A. It is the purpose of this note to show that a similar technique can be employed to evaluate det A.
On the product of three homogeneous linear forms
Authors: Chalk J. H. H. , Rogers C. A.
Published: November 2008

Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by
Differential forms on a Kähler manifold
Authors: Hodge W. V. D.
Published: November 2008

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.
Diffraction by a wave-guide of finite length
Authors: Jones D. S.
Published: November 2008

When the electric intensities on two parallel planes, of which the two perfectly conducting sides of a wave-guide of finite length and infinite width are portions, are taken as unknowns, the problem of the diffraction of a plane harmonic electromagnetic wave polarized parallel to the edges of the guide leads to two integral equations. By means of the Laplace transform these equations are converted into others suitable for solution by successive substitutions. The series thus obtained is too complex for practical purposes, and so an approximate solution is found for the case when the length of the guide is large compared with the wavelength. Finally, there is a brief discussion of the difference between the distant fields when l is large and when l is infinite.
On the choice of standard solutions to Weber's equation
Authors: Miller J. C. P.
Published: November 2008

In this paper the principles for the choice of a pair of standard solutions of a homogeneous linear differential equation of the second order, described in an earlier paper (2), are applied to Weber's equation
Electronic bound states at the surface of a metal
Authors: Baldock G. R.
Published: November 2008

The conditions under which bound states associated with atoms in the surface of a metal may exist are investigated, using the tight-binding approximation. These states arise as a result of modifications in the parameters of certain atoms. The modifications required to produce (a) bound states associated with all the atoms in the surface (surface states) and (b) bound states associated with particular small groups of atoms are found for the simple cubic lattice. It is also shown that most of the simpler crystal structures do not exhibit surface states without such modifications; in the graphite and diamond lattices, however, surface states exist solely by virtue of the lattice geometry.
On measure of sum sets II. The sum-theorem for the torus
Authors: Macbeath A. M.
Published: November 2008

This note is concerned with the r-dimensional torus Tr, whose points x are r-tuples (x1, x2, …, xr), where the xi are not numbers but residue-classes modulo 1. The addition of two elements of Tr follows the vector law
The stability of the surface of a cavitation bubble
Authors: Binnie A. M.
Published: November 2008

A theoretical examination is made of the stability of a spherical interface between two inviscid liquids accelerated radially from rest. If surface tension be neglected, it is found that the interface is unstable or stable according as the acceleration is directed towards or away from the liquid of greater density. Surface tension has a marked stabilizing effect, and numerical results are given for the minimum outward acceleration required to destroy the equilibrium of the surface of a cavity formed in water. These results are suggested as a possible explanation of the behaviour of cavitation bubbles seen to form near a fixed object under test in a water tunnel.
The roots of J−(l + ½) (λη)Jl + ½ (λ) − Jl + ½ (λη) J−(l + ½)(λ) = 0
Authors: Chandrasekhar S. , Elbert Donna
Published: November 2008

Spherical Bessel functions which vanish at x = 1 and x = η (where η is an assigned positive constant less than 1) occur in the solution of many problems in applied mathematics. Such functions can be expressed in terms of Bessel's functions of the half odd integral orders in the form

where λ is a root of the equation
On the dynamical theory of the rotation of the earth. II. The effect of precession on the motion of the liquid core
Authors: Bondi H. , Lyttleton R. A.
Published: November 2008

In an earlier paper of the same general title (1) the possibility that the core of the Earth, in view of its supposed liquid nature, does not partake of the rigid-body motion of the outer shell was discussed with particular reference to the secular diminution of the angular velocity. In addition to this small rate of change of the magnitude of the angular velocity vector of the shell there occur changes in its direction consisting of the precession and nutation, but all the rates of change therein involved are small. The secular retardation takes place with extreme slowness, the nutations involve deviations of the axis with small angular amplitudes, while the precession, though of large angular amplitude, is of very long period compared with the rotation period of the Earth. Accordingly, it may be supposed that the effects of these various changes in the angular velocity can be considered separately in their relation to the motion within the core, and it is the object of this paper to give an account of our investigation into what may be termed for brevity the precession problem. It should perhaps be stated at the outset that the work does not constitute a solution of the problem, which our studies have led us to believe is one of the utmost mathematical difficulty presenting features of an exceptional character in hydro-dynamic theory. After first obtaining the equations of steady motion applicable to the interior, and those applicable to the boundary layer, the solution of the latter equations has been obtained; but in respect of the former equations we have been able to carry the question of the interior motion only as far as showing that no motion representable everywhere by analytic functions and consistent with the boundary conditions is possible. The investigation strongly suggests that no steady-state motion of a permanent character is possible for the interior, though the precise nature of the motion that actually occurs poses a problem of special interest from a hydrodynamic standpoint, but it is one to which we are not able to arrive at any definite answer at present. Without making any progress with the problem thus produced, the paper nevertheless makes clear the inherent difficulties of the problem and also serves to emphasize the inadequacy of any simplified mode of attack assuming classical fluid and resembling, for example, Poincaré's method for the nutation problem adopted by Lamb (3). Thus despite its incompleteness it seemed worth while to publish some account of such progress with these highly interesting questions as we have been able to make.
The diffraction of electro-magnetic waves by a semi-infinite circular wave guide
Authors: Pearson J. D.
Published: November 2008

An exact solution is obtained for the diffraction of a plane harmonic electromagnetic wave incident on a semi-infinite open-ended perfectly conducting circular wave guide.
Long waves in running water
Authors: Burns J. C.
Published: November 2008

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.
Reductions of ideals in local rings
Authors: Northcott D. G. , Rees D.
Published: November 2008

This paper contains some contributions to the analytic theory of ideals. The central concept is that of areductionwhich is defined as follows: ifandare ideals and⊆, thenis called a reduction ofifn=n+1for all large values ofn. The usefulness of the concept depends mainly on two facts. First, it defines a relationship between two ideals which is preserved under homomorphisms and ring extensions; secondly, what we may term the reduction process gets rid of superfluous elements of an ideal without disturbing the algebraic multiplicities associated with it. For example, the process when applied to a primary ideal belonging to the maximal ideal of a local ring gives rise to a system of parameters having the same multiplicity; but the methods work almost equally well for an arbitrary ideal and bring to light some interesting facts which are rather obscured in the special case. The concept seems to be suitable for a variety of applications. The present paper contains one instance which is a generalized form of the associative law for multiplicities (see § 8), and the authors hope to give other illustrations in a separate paper.
Elimination of waiting time in automatic computers with delay-type stores
Authors: Freedman A. L. , Wilkes M. V.
Published: November 2008

Introduction. Many automatic computers employ magnetic drums or ultrasonic delay lines as their main store. A part of the time taken by a computation on such a machine is spent in waiting for words to become available from the store. A number of methods and devices have been used or suggested to reduce this waste of time. This paper attempts to evaluate these methods by determining on the one hand the amount of waiting time eliminated and on the other hand the design or programming complications introduced. No quantitative method for determining the amount of eliminated waiting time was known. Available estimates were based on the feeling of the programmer and varied accordingly. It was therefore necessary in the first place to develop a process by which the amount of waiting time eliminated by any given method could be determined. This process is set out while being applied to the methods discussed. The methods themselves are not new, except that closer analysis has shown that the method adopted with the ACE pilot model and known as ‘optimum programming’ is in fact not a single method but a combination of several methods with different features. Also, it appeared that, contrary to an opinion expressed elsewhere (2), there is a simple way of staggering locations in the store even with words of two lengths.
The expected number of clumps when convex laminae are placed at random and with random orientation on a plane area
Authors: Mack C.
Published: November 2008

As Armitage(1) has noted, this is a problem of some importance in dust particle counting where overlapping particles are usually counted as single particles owing to their small size and the difficulty of detecting overlapping. In his paper Armitage(1) obtained approximate formulae and also expansions for the expected number of clumps (as he called a self-overlapping group) (i) when the particles were all circular laminae (though of different sizes) and (ii) when the particles were similar rectangular laminae randomly oriented. His expansions were accurate to the first two terms, but he only gave approximate values for the third terms. However, by using a method similar to that of the author (5,6) in treating aggregates in random placings of points and to that of Hammersley (4) in treating one-dimensional problems analogous to those dealt with here, this paper gives exact formulae for any mixture of laminae of different shapes and sizes, the only restrictions being (i) that they are all convex (though they need not be symmetrical), (ii) that they are randomly oriented with no preferred direction, and (iii) that there are a large number of each kind. With these restrictions we shall show in addition that, surprisingly, the only relevant geometrical properties of the laminae are (i) their areas and (ii) their perimeter lengths, their shapes apart from this being immaterial.
The translations of linear sets of fractional dimensions
Authors: Moran P. A. P.
Published: November 2008

If E is a linear set with a positive Lebesgue measure, and E(δ) is its translation through a distance δ, it is obvious that there always exist non-zero values of δ for which E. E(δ) has a positive measure. In the present note I show that for α-dimensional linear sets (0 < α < 1), and α-dimensional measure, the analogous result is not true. This is a further illustration of the deep structural difference between α-dimensional sets for α < 1 and for α = 1.
On Ingham summability and summability by Lambert series
Authors: Pennington W. B.
Published: November 2008

In his paper ‘Some Tauberian theorems connected with the prime number theorem’, Ingham(10) discusses the method of summation of the series defined by
The bound state representation
Authors: Hamilton J. , Salam A.
Published: November 2008

Eigenvalues and eigenvectors of the total Hamiltonian in field theory are determined by the method of direct diagonalization, using canonical transformations. The difficulties of this method are discussed in detail.
Non σ-finite closed subsets of analytic sets
Authors: Davies Roy O.
Published: November 2008

A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.
A singular integral equation in the theory of meson-nucleon scattering
Authors: Dalitz R. H. , Sundaresen M. K. , Bethe H. A. , Mott N. F.
Published: November 2008

The Tamm-Dancoff theory of meson-nucleon scattering (pseudoscalar coupling) leads to a series of singular integral equations. The asymptotic behaviour of the solutions of these equations is obtained here for all scattering states. For attractive states there is a critical coupling constant beyond which no normalizable solutions exist for these equations; for repulsive states another critical coupling constant appears, beyond which the solutions oscillate infinitely often but are still normalizable. It is concluded that the renormalization procedures proposed previously (3) are consistent and successfully define finite vertex-functions, and that the equations obtained for these vertex-functions have satisfactory properties for numerical work.
Motion of a sphere through a conducting fluid in the presence of a strong magnetic field
Authors: Stewartson K.
Published: November 2008

The steady motion of a perfectly conducting sphere in an inviscid conducting fluid in the presence of a strong magnetic field is discussed. It is shown that if the fluid velocity is ultimately steady then it is two-dimensional, and a cylinder of fluid whose generators are parallel to the direction of the field moves with the sphere as if solid. The streamlines outside are straight lines if the sphere moves in the direction of the field but have to execute sharp turns if it moves at right angles to the field. The motion to be expected in practice is discussed using an analogy.
Elements of an improved linear perturbation theory of steady supersonic flow with axial symmetry
Authors: Jones C. W.
Published: November 2008

The usual linearized theory neglects small terms, of second and higher orders of the velocity perturbation, in the coefficients of the equation for the potential. In this paper these terms are approximated by constant mean values and the familiar formulae of the first approximation are then generalized. The characteristic lines of this improved linear theory thereby differ from the Mach lines of the undisturbed flow, though still being straight and parallel. The new formulae thus give a better calculation of the velocity field near bodies not slender enough for the ordinary theory to be accurate. A numerical example, for a 10° cone in a stream of Mach number about 3, illustrates this fact, and shows that a shock of constant strength may be fitted to the improved solution.

Although the improved theory as it stands must fail at large distances from the axis of symmetry, just as the first approximation fails, the tentative suggestion is made that a technique due to G. B. Whitham may be applied to the former, as to the latter, perhaps leading to an improved description of the whole field of flow.
The penetration depth in impure superconducting tin
Authors: Chambers R. G. , Pippard A. B.
Published: November 2008

A new method is described for measuring the surface impedance of metals at low temperatures and at radio-frequencies. Using this method, the surface impedance of normal tin and the penetration depth λ in superconducting tin have been studied at about 9 Mc/s frequency as a function of impurity content. The measured surface impedances agree well with the values expected, and the penetration depths increase with impurity content, in confirmation of Pippard's observations (14) at 9400 Mc/s. The variation of λ with reduced temperature t is well represented by the equation λ/λ0 = (1 − t4)−½, in impure as well as in pure tin. The absolute value of λ found for pure tin is about 4 × 10−6 cm., significantly lower than the values previously found.
Finite-by-nilpotent groups
Authors: Hall P.
Published: November 2008

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.
Bending of an elastically restrained circular plate under a linearly varying load over an eccentric circle
Authors: Bassali W. A.
Published: November 2008

The complex potentials and deflexion at any point of a thin circular plate with a normal linearly varying load over an eccentric circle are determined under a general boundary condition including the usual clamped and hinged boundaries.
Matrix representations of semigroups
Authors: Munn W. D.
Published: November 2008

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.
Analysis of tangential properties of curves of infinite length
Authors: Besicovitch A. S.
Published: November 2008

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.
Some formulae for the product of hypergeometric functions
Authors: Ragab F. M.
Published: November 2008

The formula to be established is

where α > ½, β > ½and b is real and positive. In § 3 we shall generalize this by proving

where α > (p−l)/p, β > (p−l)/p, b > 0 and p = 2,3,….
On the velocities of the maxima in a moving wave-form
Authors: Longuet-Higgins M. S.
Published: November 2008

In a recent paper (1) expressions were found for the statistical distributions of the velocities of the zero-crossings and of the maxima and minima of a random moving wave-form. In the present note it will be shown that these distributions are to some extent arbitrary, depending on how the points of observation are selected.
Bipartite partitions
Authors: Nanda V. S.
Published: November 2008

Asymptotic theory of partitions of unipartite numbers has been extensively investigated after the classic work of Hardy and Ramanujan on the subject. The more complicated problem of bipartite partitions has been only recently broached. Auluck(1) has deduced asymptotic formulae for p(m, n), the number of partitions of the bipartite number (m, n), both for m fixed and n → ∞, and for both m and n tending to infinity in such a way that m/n = 0(1) and n/m = 0(1). It will be shown here that the formula obtained for the first case is actually valid for m = 0(n¼). This proves the conjecture made earlier by the author (3).
The existence of subgroups of given order in finite groups
Authors: McLain D. H.
Published: November 2008

1. Discussion of results

1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.
Some properties of skew-symmetric elements of a ring
Authors: Barnett I. A. , Abian S.
Published: November 2008

Let R* be a ring with unit element I. For every A ∈ R*, let the mapping A → A′ be an involutory anti-automorphism, i.e. a one-to-one mapping of R* onto itself with the following properties:

Let C* denote the set of all symmetric elements of R*, i.e. C′ = C for all C ∈ C*.

An element A ∈ R* is said to be skew-symmetric if A′ + A = 0 (the zero element of the ring).
Causal amplitudes and the Yang-Feldman formalism
Authors: Polkinghorne J. C.
Published: November 2008

The Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.
Norms of summation methods
Authors: Petersen G. M.
Published: November 2008

We shall understand the norm, h(A) of a regular matrix A = (amn) to be

The method has a norm, ‖ ‖, ‖ ‖ ≥ 1, given by ‖ ‖ = inf h(A) where the inf is taken over all the matrix methods equivalent to for bounded sequences (b-equivalent). These definitions are due essentially to Brudno(1) though his definition of
The Spectral resolution of Watson transforms
Authors: Akutowicz Edwin J.
Published: November 2008

If φ belongs to L2(0, ∞), the Watson transform W φ of φ is defined by

where k(u)/u belongs to L2(0, ∞) and

The operator W is unitary in L2(0, ∞), and its inverse is given by
Some theorems on fractional integrals
Authors: Flett T. M.
Published: November 2008

1. Let φ(z) be regular in │z│ < 1, let

and for any real α let

where (in)−α=n−αexp(— ½απi). The function φαis an adaptation to power series of the Weyl fractional integral of order a of α function of a real variable.† It exists and is regular in │z│ < 1 for all α. Let also
Diffraction by an imperfectly conducting right-angled wedge
Authors: Williams W. E.
Published: November 2008

A new and exact solution is given for the diffraction of the field of an electric line current by an imperfectly conducting wedge of exterior angle . Previous attempts at solving this problem assumed that the solution could be expressed as a series in inverse powers of the conductivity of the wedge, but this approach can only be applied to wedges of exterior angle less than π as the solution obtained is singular at the edge of the wedge for angles exceeding π. It is shown that in the present case no such expansion is uniformly valid; at large distances from the wedge, however, the present solution agrees with that obtained by assuming an expansion in inverse powers of the conductivity, even though this latter solution is singular at the edge, to order (conductivity of the wedge)−1; the next term in the expansion of the present exact solution is O(conductivity). The case of an incident plane wave may be obtained by letting the source tend to infinity.
Note on the probability distribution of a small number of vectors
Authors: Vincenz S. A. , Bruckshaw J. McG.
Published: November 2008

In this note the problem of random flights is considered for the case of a small number of vectors, each vector possessing the same constant magnitude. A formula is developed which permits the calculation of the probability function of the sum of N + 1 such vectors if the function of the sum of N vectors is known. The formulae for the probability distributions up to N = 8 are quoted and some of their practical applications mentioned.
The Number of Zeros of a Polynomial in a Half-Plane
Authors: Talbot A.
Published: November 2008

The determination of the number of zeros of a complex polynomial in a half-plane, in particular in the upper and lower, or right and left, half-planes, has been the subject of numerous papers, and a full discussion, with many references, is given in Marden (l) and Wall (2), where the basis for the determination is a continued-fraction expansion, or H.C.F. algorithm, in terms of which the number of zeros in one of the half-planes can be written down at once. In addition, determinantal formulae for the relevant elements of the algorithm can be obtained, and these lead to determinantal criteria for the number of zeros, including that of Hurwitz (3) for the right and left half-planes.
The numerical integration of ordinary differential equations possessing exponential type solutions
Authors: Dennis S. C. R.
Published: November 2008

Although there are many variations of finite-difference methods of obtaining approximate numerical solutions to ordinary differential equations they share the common feature that they tend to treat an equation of a given type as a standard problem and take no account of any special characteristics the wanted solution may have. We here suggest an alternative procedure when the wanted solution exhibits exponential characteristics. In essence the idea is that if a differential equation has an exponential type solution y(x) it is useful to solve numerically, instead of the equation for y, the equation for u = logey. The error-building and stability characteristics are then those of u rather than y and consequently the accuracy of the solutions may be improved. Although there is nothing basically new in this, of course, the point that we demonstrate is that the differential equation in y can be solved numerically in such a manner that the transformation from y to u is not actually carried out, i.e. we retain the original dependent variable but take account of the exponential variation by modifying the integration formula. Consider for example, in the usual notation, the first-order equation

with a given initial condition y(x0) = y0. If x0, x1, …, xr, xn is a set of pivotal values of x;, usually assumed equally spaced so that xr+1 − xr = h, the usual approach replaces (1) by the formula

which, once the integral is expressed in terms of pivotal values of f using a difference series, represents a step-by-step formula for constructing successive values of y.
An extension of the Pitman–Koopman Theorem
Authors: Jeffreys Harold
Published: November 2008

The theorems of Pitman and Koopman on laws of chance that yield sufficient statistics are extended to laws where the parameter or the observable, or both, may admit only a discrete set of values. The method depends on making changes, not necessarily small, within the values permitted for the variables, but it is still necessary to assume a solubility condition.
Eigenvalues of Composite Matrices
Authors: Friedman Bernard
Published: November 2008

Because of the symmetry of the problems encountered in the applications of matrices to physics, electrical engineering and numerical analysis, it frequently turns out that the matrix to be considered is a composite matrix, that is, a matrix whose elements are matrices. For example, the matrix may be where A1 and A2 are square matrices of order n which need not commute. It is easy to prove that the eigenvalues of this matrix of order 2n are the eigenvalues of the two matrices, A1+A2 and A1−A2 of order n.
On Riesz and Riemann Summability
Authors: Burkill H.
Published: November 2008

Several relations between the Cesàro and the Riemann methods of summation are known. For instance, Verblunsky(3) has shown that a series summable (C, k − δ), where k is a positive integer, is also summable (R,k+ 1) and Kuttner (2) has proved that, for k = 1, 2, summability (R, k) implies summability (C, k + δ). In this paper we consider Riesz's typical means and a generalized Riemann summability both of which are intimately connected with almost periodic functions. The result we establish is similar to Verblunsky's, except that we start from a Riesz mean of integral order k*.
Waves on a Sloping Beach
Authors: Williams W. E.
Published: November 2008

A method employed by the author for a class of boundary-value problems arising in diffraction theory is applied to the problem of the oscillations of water waves on a sloping beach. The two cases of long and short waves are considered. The approach is valid for all values of the parameters and particular cases do not have to be treated separately. It is shown that the present approach enables certain solutions to be obtained which cannot arise in earlier solutions of the problem.
On a class of non-measurable groups
Authors: Cole A. J. , Świerczkowski S.
Published: November 2008

A group G will be called measurable if there exists a real-valued, non-negative set function μ(A) defined on all subsets A ⊂ G which satisfies .
A definite integral involving associated Legendre function of the first kind
Authors: Saxena R. K.
Published: November 2008

Recently Ragab ((3), p. 538) evaluated an integral involving the product of an associated Legendre function and an E-function. In this note we derive an integral which is more general than the one given by Ragab.
The physical interpretation of entity-structure
Authors: Newman D. J.
Published: November 2008

Several recent papers (Bastin and Kilmister(1),(2); Newman(3),(4)) have made the role of algebraic structures in physics the subject of detailed discussion. The general purpose of this work has not been to discover new physical laws or predict new results, but rather to give the theoretical physicist some further insight into the significance of familiar concepts. This is, of course, being done in the hope that it will lead to new refinements in our theoretical techniques.
Note on an electrified circular disk situated inside an earthed coaxial infinite hollow cylinder
Authors: Collins W. D.
Published: November 2008

Copson(1) has shown that the problem of determining the surface density of electric charge induced on a thin circular disk maintained at a constant potential in an external electrostatic field can be solved by two applications of the known solution of Abel's integral equation. This note shows that Copson's method can be extended to determine the surface charge density induced on the disk when situated inside an earthed coaxial infinitely long hollow cylinder. Whilst this problem is not solved in closed form, it is shown to be governed by a Fredholm integral equation of the second kind, which can be solved approximately by iteration when the radius of the cylinder is large compared with that of the disk.
Two-dimensional elastic inclusion problems
Authors: Jaswon M. A. , Bhargava R. D.
Published: November 2008

An account is given of Eshelby's point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.
On a theorem of L. J. Mordell
Authors: Diananda P. H.
Published: November 2008

In a recent paper (3) Mordell proved the following theorem.

Theorem 1. Let xn + i = xi ≥ 0 for all i and x1 + … + xn = 1. Then
On diagonal values of probability vectors of infinitely many components
Authors: Besicovitch A. S.
Published: November 2008

Given † probability vector μ(X) = (μ1(X), … μk (X)) of a finite number of components on a Borel class of sets X, we say that μ(X0) has a diagonal value α if μi(X0) = α for all i = 1, 2,…,K. J. Neyman(l), (2), (3) has proved that in the class of Borel sets of real numbers any non-atomic vector μ(X) takes all diagonal values. A. Liapounoff has studied the full range of values of k-dimensional vector-valued measures and in two papers (4) he has proved that the range is closed and in the case of non-atomic measures the range is also convex. He also gave an example showing that neither of these results holds in the case of vectors of infinitely many components. A simplified proof of Liapounoff's results has been given by P. R. Halmos (5). In the present paper I study the range of values of probability vectors of infinitely many components. Various types of conditions are studied which are sufficient to imply that, for each ε > 0, 0 ≤ α ≤ 1, it is possible to find a set X such that
Entropy, Markov processes and Boltzmann's H-theorem
Authors: Moran P. A. P.
Published: November 2008

Consider a Markov chain with a finite number of states E1,…,En and a transition matrix, (Pij), of probabilities of transition from state Ej to state Ei which is such that after some fixed finite number of steps, any state is accessible from any initial state. Such a chain is said to be completely regular and it is easy to show that starting from any initial state, or distribution of states, the probabilities, , that the system is in state i after t steps, will converge to non-zero quantities, , which are independent of the initial state. This condition of accessibility corresponds, in statistical mechanics, to the condition of ‘metrical transitivity’.
The use of algebraic structures in physics
Authors: Kilmister C. W. , Newman D. J.
Published: November 2008

The hypothesis of the existence of abstract structures in physics, which has been made in recent papers, is made precise. Difficulties in previous formulations are removed and insight into Maxwell electrodynamics is obtained. General relativity is related to other field theories which deal with flux from sources, and new field equations for general relativity are proposed.
A short proof of MacMahon's ‘Master Theorem’
Authors: Good I. J.
Published: November 2008

MacMahon (2) makes much use of the following result, which he describes as the “Master Theorem”.

Let

and let m1, m2, … mn be non-integers. Then the coefficient ofinis equal to the coefficient ofin
A Note on the Vague Topology for Measures
Authors: Wulfsohn Aubrey
Published: November 2008

We refer for general background to N. Bourbaki, Intégration, chapters iv and v. We consider a locally compact Hausdorff space R and denote the set of continuous functions with compact support by The Riesz-Markov theorem shows that there is a 1−1 correspondence between the set of regular Borel measures on R and the set of positive elements of the topological dual of . Let {μn}, μ be regular Borel probability measures on R. The sequence of measures {μn} is said to converge vaguely to μ if, for all . Thus the vague topology is that of simple convergence in . We shall call a μ-measurable set μ-quarrable if its boundary is a μ-null set.