bluuebunny/crossref_metadata_2025 huggingface dataset:
Transonic flow past finite wedges
Authors: Mackie A. G. , Pack D. C.
Published: November 2008

The solution for the flow of an incompressible fluid past an infinitely long wedge with a finite sloping edge (a finite wedge) is generalized by the hodograph method. In the flow thus obtained the axis of symmetry and a sloping edge of the wedge are again part of one streamline. It becomes possible to describe the flow of an ideal gas past a finite wedge if the hypothesis is made that the first singularity on this streamline, along the sloping edge, corresponds to the shoulder of the wedge. For a given wedge, with gradually increasing velocity at infinity upstream, the singularity appears at first at subsonic velocity. Beyond a certain critical velocity at infinity the singularity is always associated with the speed of sound. The hypothesis thus implies that put forward by Maccoll(9) and supported by Busemann(l). A qualitative examination shows that the solution reproduces experimentally known features of the flow of compressible fluid past a finite wedge.
On positive stochastic matrices with real characteristic roots
Authors: Perfect Hazel
Published: November 2008

Suleῐmanova (3) describes a geometrical technique* for discussing the conditions under which a given set of n + 1 numbers can be a set of characteristic roots of a positive stochastic matrix of order n + 1. Suleῐmanova states that, using this technique, it is possible to prove that a set of n + 1 real numbers 1, λ1, λ2, …, λn, where | λi | < 1 for i = 1, 2, …, n, is a set of characteristic roots of a positive stochastic matrix provided that the sum of the moduli of the negative numbers of the set is less than unity. No indication of the method to be employed is given in (3), and it would seem that the problem is a difficult one. In the present paper I give a full discussion of how Suleῐmanova's technique may be applied to solve the problem for third-order matrices, and simply an indication of the solution for fourth-order matrices; and in, the latter case I point out that Suleῐmanova's result may be replaced by a stronger one. For second-order matrices the problem is almost trivial. As a preliminary I prove two general theorems.
The anomaly of convex bodies
Authors: Rankin R. A.
Published: November 2008

I write X for the point (x1, x2, …, xn) of n-dimensional Euclidean space Rn. The coordinates x1, x2, …, xn are real numbers. The origin (0, 0,…, 0) is denoted by O. If t is a real number, tX denotes the point (tx1, tx2, …, txn); in particular, − X is the point (−x1, −x2,…, −xn). Also X + Y denotes the point {x1 + y1, x2 + y2, …, xn + yn).
On the region defined by | xy | ≤ 1, x2 + y2 ≤ t
Authors: Ollerenshaw Kathleen
Published: November 2008

In this paper I find the critical lattices of the region Kt defined by the inequalities | xy | ≤ 1, x2 + y2 ≤ t (t > 0). I deduce in §9 the arithmetic minima of a pair of binary quadratic forms

where f1 is indefinite, f2 is positive definite, and f1, f2 are harmonically related, that is to say

and
Bounded-weak topologies and completeness in vector spaces
Authors: Roberts G. T.
Published: November 2008

Suppose that X is any vector space on which it is possible to recognize a class of sets of such a nature that it is natural to call them ‘bounded’. (Precise conditions for such a class of sets are given in § 2.) Let L be any vector space of linear functionals on X which map each ‘bounded’ set into a bounded set. We say that a filter is boundedly-weakly convergent if it is convergent in the weak topology of the linear system [X, L] and contains a ‘bounded’ set. If M is a vector space of boundedly-weakly continuous linear functionals on X which includes L, we say that a subset S of M is limited if 〈 , f〉 converges uniformly for f ε S whenever is a boundedly-weakly convergent filter.
On eigenvalues and inverse singular values of compact linear operators in Hilbert space
Authors: Silberstein J. P. O. , Smithies F.
Published: November 2008

1·1. In this paper we shall be concerned with the equations

where K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.
Inhomogeneous lattices
Authors: Swinnerton-Dyer H. P. F.
Published: November 2008

1. Mahler (1) has obtained a number of valuable existence theorems for homogeneous problems in the geometry of numbers. It appears that no analogous results for inhomogeneous problems have yet been published; and although none of them present great difficulty, it is convenient for applications to have them in print.
Researches on the equation (E) x¨+ (∈1 + ∈2x) ẋ + x + ∈3x2 = 0
Authors: Obi Chike
Published: November 2008

1·1. The differential equation (E) of the title, which occurred in connexion with a problem of electrical engineering, was mentioned to the author by Miss M. L. Cart-wright. It is of some general interest in that, as will be shown, it may have a non-trivial periodic solution (p.s.), although the damping changes sign only once.
Some formulae of P. Stein and others concerning trigonometrical sums
Authors: Slater N. B.
Published: November 2008

Let

where the frequencies vr are strictly positive and the amplitudes ar are non-zero; and let

be the number of zeros of f(t) – a for 0 ≤ t < T. In physical problems, where f(t) may represent, for example, a general coordinate in a vibrating dynamical system or a set of alternating currents in a cable, formulae are required for the frequency with which f(t) passes through a given value a; that is, for asymptotic or average values of Ga/T. The purpose here is to collect such formulae, and to sketch their background and relations in a way which may suggest extensions. The writer is particularly indebted to Prof. P. Stein for a manuscript containing new results (SI-V in §§2 and 5 below) and for his permission to publish them without his original proofs; these proofs were extensions of the method he gave (6) for the case n = 2.
A note on commutators in free products
Authors: Griffiths H. B.
Published: November 2008

In (1), Higman introduces the unrestricted free product of a set of groups, and gives it a natural topology. When this set is an infinite sequence of free cyclic groups he denotes by F their unrestricted free product; we shall denote by K the product for a general sequence {Gn} and, throughout the paper, we assume that each Gn is non-trivial and countable. Higman proves as an incidental result that the commutator subgroup [F, F] is not closed in the topological group F, and the first object of this note is to generalize this result to K. From this, the more interesting deduction immediately follows that K is never equal to L = [K, K]. Indeed, we prove in fact that the cardinal of L and its index in K are both c, the cardinal of the continuum.
Asymptotic formulae in the theory of partitions
Authors: Haselgrove C. B. , Temperley H. N. V.
Published: November 2008

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.
The central limit theorem for m-dependent variables asymptotically stationary to second order
Authors: Diananda P. H.
Published: November 2008

In a recent paper (3) the Lindeberg-Lévy theorem (2) was extended for certain types of stationary dependent variables. In the present paper mainly the same basic ideas as were used in (3) are employed to give central limit theorems for m-dependent scalar variables (a) stationary to second order and (b) asymptotically stationary to second order, the sufficient condition in each case being akin to the Linde-berg condition ((1), p. 57) for independent variables. The analogue of the main theorem for vector variables is given. An extension of the Lindeberg-Lévy theorem is derived.
Diffraction by two parallel planes of finite length
Authors: Williams W. E.
Published: November 2008

The diffraction of a plane harmonic electromagnetic wave by two parallel, perfectly conducting planes of infinite width and finite length is considered. The wave is assumed to be polarized perpendicular to the edges of the plane, and thus the problem can also be regarded as that of diffraction of a plane sound wave. Two integral equations are obtained, and the equations are solved approximately when the length of the planes is large compared with the wave-length. The resonance of the system is also considered and an equation obtained for the resonant lengths. This equation is solved by an approximation depending on the assumption that twice the product of the wave number (k) and the resonant length (l) is always large.
Note on the Green's function of an elastic plate
Authors: Dean W. R.
Published: November 2008

1. An expression in finite terms is found in §§2,3 for the small displacement of a thin elastic plate due to the application of a transverse force at an arbitrary point; the plate is infinite and is bounded internally by an ellipse along which it is clamped. The limiting case in which the ellipse degenerates to a straight line is considered in §4, and the other limiting case of a circular boundary in §5.
On the motion of a flat plate at high speed in a viscous compressible fluid. I. Impulsive motion
Authors: Stewartson K.
Published: November 2008

The effect of giving an infinite plate, immersed in a compressible fluid, an impulsive velocity U in its own plane is examined on the assumption that M = U/a0 ≫ 1 where a0 is the velocity of sound in the undisturbed fluid. To begin with the flow field is very complex but very soon after the motion has begun a shock wave moves at right angles to the plate gradually weakening and eventually degenerating into a sound wave. Attention is concentrated on the flow field when there is a distinct strong shock wave, it being assumed, and justified a posteriori, that the flow behind it may be divided into an inviscid region and a boundary layer region. The equation of the shock wave and a complete description of the flow behind it is found when log M, where ν0 is the coefficient of kinematic viscosity of the undisturbed air. An approximate method is also given to enable the solution to be joined up to Van Dyke's (10) which is valid when .
A generalized inverse for matrices
Authors: Penrose R.
Published: November 2008

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.
Contributions to the theory of stokes waves
Authors: De S. C.
Published: November 2008

The well-known Stokes theory (9, 10) of waves of permanent form in water of finite depth has been extended to the fifth order of approximation. The solutions have been first obtained in the form of equations for the space coordinates x and y as functions of the velocity potential Φ and stream function ψ. Expressions for the complex potential W in terms of the complex variable z ( = x + iy), the form of the wave profile, and the square of the wave velocity have been obtained to the fifth order.

Expressions for the three physical quantities Q, R and S, where Q is the volume flow rate per unit span, R is the energy per unit mass (i.e. g times the total head, measuring heights from the bottom and pressures from atmospheric) and S is the momentum flow rate per unit spaa, corrected for pressure forces and divided by density, have been obtained to the fifth order. The values for the dimensionless quantities r = R/Rc and s = S/Sc, where Rc and Sc refer to the values of R and S for a critical stream of volume flow Q, are tabulated for certain values of the ratios mean depth to wavelength and amplitude to wavelength. The values of r and s thus obtained have been used to calculate the ratios of mean depth to wavelength and of wave height to wavelength according to the cnoidal wave theory as recently presented by Benjamin and Light-hill(1), and the results are found to be in satisfactory agreement with that from Stokes's theory for waves longer than six times the depth.

The (r, s) diagram introduced in the recent work of Benjamin and Lighthill(1) has been further considered, and the unshaded part of the diagram referred to in that paper has been mapped with a network of curves for constant values of the ratios of mean depth to wavelength and of wave height to wavelength (Fig. 2). The third barrier to the existence of steady flows, corresponding to ‘waves of greatest height’ referred to in that paper, has also been indicated in Fig. 2.
The growth of secondary circulation in frictionless flow
Authors: Hawthorne W. R.
Published: November 2008

The appearance of a component of vorticity in the direction of flow materially alters the pattern of flow of a fluid in three dimensions. Expressions are obtained for this secondary vorticity in an inviscid compressible fluid flowing under the action of body forces. They are applied to examples such as a liquid under gravity and gas flow behind a curved shock. In compressible gas flow with varying temperature but constant stagnation pressure no secondary circulation appears. In a perfect gas atmosphere it is shown that secondary circulation may appear because of nou-adiabatic lapse rates as well as wind-speed gradients. It is also shown that in a liquid with a density gradient, gravitational effects can give rise to secondary vorticity components.
Jacobians of linear systems on an algebraic variety
Authors: Monk D.
Published: November 2008

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and let

The purpose of this note is to prove that the Jacobian of these systems is given by the equivalence

where Kh is Eger's operator, defined as follows:
The numerical evaluation of a class of integrals
Authors: Moran P. A. P.
Published: November 2008

Suppose thatx1, …,xnare jointly distributed in a multivariate normal distribution,f(x1…,xn), with a given correlation matrix (ρij). Then the evaluation of

is a problem which has been much discussed and has many applications (see David (1)). Whenn= 3
A method of improving Energy-level calculations for ‘series’ electrons
Authors: Douglas A. S.
Published: November 2008

A method is described for improving the approximate calculation of energy levels for atomic configurations consisting of a set of closed shells and a single ‘series’ electron. This involves the inclusion of a ‘polarization’ potential in the Hartree-Slater-Fock equation, caused by the series electron. The form of potential found in two cases investigated, namely, Si+3 and Ca+, is described and discussed.
Distributions defined as limits I. Distributions as derivatives; continuity
Authors: Ravetz J. R.
Published: November 2008

The theory of distributions of L. Schwartz (3) provides a unified and rigorous foundation for special methods used in various branches of mathematics. Schwartz's treatment is on the most general level, and presupposes an understanding of modern abstract analysis. Several alternative approaches to distributions have been developed, all of them ‘elementary’ in one sense or another. We follow here the approach of Mikusiński (2) and Temple (4), in which distributions are defined as generalized limits of sequences of continuous functions. We find that, with this approach, it is possible to prove the basic theorem: every distribution is (locally) a derivative. The property of continuity of a distribution does not enter into the arguments establishing this result, but instead follows from it. Hence we are able to reduce the ‘regular sequence’ definition of a distribution to its simplest form. In a later paper we shall study convolution products of distributions, defined in the natural manner by regular sequences.
Limiting properties of Mathieu functions
Authors: Jaswon M. A.
Published: November 2008

A new analysis is made of the coefficients appearing in periodic Mathieu functions, and this is applied to establish the limiting properties of certain solutions of the equation
The numerical solution of linear differential equations in Chebyshev series
Authors: Clenshaw C. W.
Published: November 2008

This paper describes a method for computing the coefficients in the Chebyshev expansion of a solution of an ordinary linear differential equation. The method is valid when the solution required is bounded and possesses a finite number of maxima and minima in the finite range of integration. The essence of the method is that an expansion in Chebyshev polynomials is assumed for the highest derivative occurring in the equation; the coefficients are then determined by integrating this series, substituting in the original equation and equating coefficients.

Comparison is made with the Fourier series method of Dennis and Foots, and with the polynomial approximation method of Lanczos. Examples are given of the application of the method to some first and second order equations, including one eigenvalue problem.
The torque on an infinite strip exposed to plane sound waves
Authors: Levine Harold
Published: November 2008

The mean torque on an infinite fixed strip due to plane harmonic sound waves is calculated, employing simple acoustical theory as a basis and neglecting viscosity. In the range of wavelength λ small compared to the strip width δ, the torque per unit length is

where α denotes the angle between the incident wave normal and that of the strip and ρο the equilibrium density of the medium. This oscillatory character of the torque in its dependence on δ/λ is not to be found at long wavelengths where, as is well known, the torque acts only in the sense of aligning the strip broadside to the incident wave normal.
Causal products in quantum field theory
Authors: Polkinghorne J. C.
Published: November 2008

A new type of product of field operators is considered and the equations that it satisfies obtained.
The norm of iterations of regular matrices
Authors: Petersen G. M.
Published: November 2008

By an iteration of a regular summation matrix B = (bmn) with a regular matrix A = (amn), or B.A, we shall mean the summation method defined by the matrix B operating on the sequence {tn} of A transforms of a sequence {sn}. Thus the sequence {τn} of B. A is given by
Exceptional sets with respect to order of integral functions of two variables
Authors: Maude Ronald
Published: November 2008

1. Introduction. The definition of the order of an integral function of one complex variable extends immediately to integral functions of several complex variables to give us their total order. In the case of integral functions of two complex variables, Lelong(5) has shown that the order on any analytic plane through the origin is equal to the total order except for an exceptional set of planes of exterior capacity 0 on which the order is less than the total order.
The Hilbert function of two ideals
Authors: Bhattacharya P. B.
Published: November 2008

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.
Random processes in genetics
Authors: Moran P. A. P.
Published: November 2008

Wright's model of a random process in genetica is modified by supposing that births and deaths occur individually at random so that the generations are no longer simultaneous. An exact solution is then obtained for the distribution of the number of a-genes in a haploid organism when mutation is occurring in both directions. When there is no mutation, the rate of approach to homozygosity is found in more complicated models with two sexes and diploid individuate. This rate is twice that occurring in Wright's models.
Elliptical whirl of flooded journal bearings
Authors: Halton J. H.
Published: November 2008

The paper studies the problem of elliptical whirl of perfectly alined, finite, flooded, journal bearings, with incompressible isoviscous lubricant; it bases itself on Reynolds's equation. Explicit solutions are obtained, using the long (L/D≫ 1) and short (D/L ≫ 1) bearing approximations.

The physical situations corresponding to the solutions obtained are those of unloaded and constant-loaded bearings, with elastic, massive, frictional suspensions. The agreement with empirical findings is generally good, and the paper is published in the hope that further experimental research will be stimulated by it.
Quadratic equations in several variables
Authors: Birch B. J. , Davenport H.
Published: November 2008

Let

be a quadratic form in n variables (n ≥ 3) with integral coefficients. One of us gave recently (2) a simple proof of a result of Cassels (l), in the following form: if the equation f = 0 is properly soluble in integers, then it has a solution satisfying

where γn–1 denotes Hermite's constant.
Tangential properties of Fréchet surfaces
Authors: Eggleston H. G.
Published: November 2008

The first two papers of Reifenberg ((4), (5)) under the general heading ‘Parametric Surfaces’ contain a detailed and profound study of the tangential properties of these surfaces. Since their publication one of the fundamental problems in the subject, that of obtaining a convenient representation for the surface, has been solved by Cesari (1). In this paper we obtain direct proofs of Reifenberg's results from Cesari's theorem. Whereas Reifenberg had to contend with both topological and real-variable problems combined the effect of Cesari's theorem is to remove the topological difficulties and to leave a straightforward real variable problem. The definition of approximate tangential plane used here is not the same as either of the two employed by Reifenberg, but the differences between it and one of Reifenberg's definitions ((5), definition 2) are not very important.
On prime-power groups with two generators
Authors: Blackburn N.
Published: November 2008

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be written

For any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.
Series expansions and general transforms
Authors: Miller J. B.
Published: November 2008

A number of writers have noticed Fourier sine or cosine transform relations between certain pairs of series. For example, Duffin(4) has shown that in a variety of circumstances

are a pair of sine transforms, and Guinand(7) that

are cosine transforms, using L2 theory. Other examples are given in (2) and (5); and in (6) power series and general transforms are considered.
The interaction of light with neutral systems
Authors: Power E. A. , Shail R.
Published: November 2008

A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian

between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.
A thin circular plate normally and uniformly loaded over a concentric elliptic patch
Authors: Bassali W. A. , Nassif M.
Published: November 2008

This paper is concerned with the small transverse displacement of a thin circular plate elastically restrained along the circular boundary and subject to uniform normal pressure distributed over the area of an ellipse concentric with the plate. The method of complex potentials is used and the general assumptions relating to the small deflexion theory of thin plates are adopted. Explicit formulae for the bending and twisting moments and shearing forces are given at any point on the boundary and at the centre of the plate.
Bending of a thin circular plate under hydrostatic pressure over a concentric ellipse
Authors: Bassali W. A.
Published: November 2008

An exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.
A note on the sums of certain bilateral hypergeometric series
Authors: Shukla H. S.
Published: November 2008

Some years ago M. Jackson(5) obtained the sum of a particular 3H3 series which generalized the theorems of Whipple and Watson on sums of a 3F2 series, and later she(6) deduced the sum of a particular well-poised 6H6( – 1) series. In this note sums of certain particular bilateral hypergeometric series of the same type are given. They are believed to be new. In the sequel, the sum of a particular 8Ψ8 series has been obtained. The existence of a particular case of this sum was pointed out earlier by Slater and Lakin (11).
A Note on the Use of Logical Computers to Determine the most Efficient Method of using Factory Machines
Authors: Rose Alan
Published: November 2008

In this note we shall give two extensions of the results of (2) concerning the economic use of factory machines. Familiarity with (2) will be assumed and the same notation will be used. We shall show, first, that the number of decision elements required for the solution of the problem can be reduced considerably, and secondly, that the computer will also give the solutions to certain related problems.*
Flexural problems of circular ring plates and sectorial plates. II
Authors: Bassali W. A. , Gorgui M. A.
Published: November 2008

In a previous paper solutions have been derived for the deflexion of a circular annular plate under various combinations of boundary conditions and subject to certain line loadings spread along the circumference of a concentric circle. These solutions, which were used in the analysis of circular ring plates and sectorial plates acted upon by a concentrated load or a concentrated couple at any point, are utilized in this paper to obtain solutions appropriate to lateral loadings of the typedistributed over the entire circular ring plate. Solutions corresponding to the case φ(r) =rmare then explicitly found, and the principle of superposition can be applied to get the solution when the load on the annulus can be expanded into a Laurent-Fourierseries in the polar coordinatesr, θ. The method of images is used to deduce solutions for an isotropic plate bounded by two arcs of concentric circles and two radii, and loaded by the normal pressurep=p0rmwhen the plate is simply supported along the straight edges and is subject to various conditions along the circular edges.
The Interaction of Longitudinally Polarized Vibrations in an Isotropic Dielectric
Authors: Simons S.
Published: November 2008

A theoretical treatment is given of the relaxation time for the absorption of longitudinally polarized vibrations in an isotropie medium. Employing standard phonon interaction theory, the relaxation time is obtained in terms of certain integrals, depending parametrically on the ratio of transverse to longitudinal sound velocity. These integrals are evaluated numerically over the range which this ratio takes in practice. The results are applied to a consideration of solid argon, sodium chloride and germanium.
Modular orthogonal groups in four dimensions
Authors: Morris Ifor
Published: November 2008

Littlewood has shown(1) that any rotation in 4 dimensions can be expressed as the resultant of a right-handed and a left-handed rotation, and that every right-handed rotation commutes with every left-handed rotation. It follows that the rotation group in 4 dimensions is expressible in some way as the product of two groups of right-handed and left-handed rotations, respectively.
Pseudo-inverses in Semigroups
Authors: Munn W. D.
Published: November 2008

Drazin (2) has recently introduced the concept of a pseudo-invertible element of an associative ring or semigroup. In this note we first show that such an element of a semigroup S may be characterized by the fact that some power of it lies in a subgroup of S. The main result (Theorem 2) provides a new criterion for complete simplicity: a simple semigroup is completely simple if and only if all its elements are pseudo-invertible. This is applied to the principal factors of a general semigroup and two previously known results are obtained as corollaries.
An asymptotic expansion for Bessel functions derived with respect to their order
Authors: Sim A. C.
Published: November 2008

Formulae for Bessel derivatives with respect to order have generally been obtained by differentiating integral forms of the Bessel functions. In this note an asymptotic expansion will be derived from the differential equation satisfied by the functions, and it will be obtained in a form which terminates when the order is half an odd integer. Previous discussion of these functions has been restricted to the order one half. (See, for example, Ansell and Fisher (1) and Oberhettinger(2).)
Unsymmetrical flow patterns past a finite wedge profile in a High subsonic stream
Authors: Helliwell J. B.
Published: November 2008

The flow pattern past a thin wedge-like profile set at a small angle of attack in a gas flowing with high subsonic or sonic velocity is discussed within the order of the transonic approximation. In the model considered the flow has a stagnation point at the nose of the wedge and breaks away, with velocity equal to that of sound, from the shoulders. The velocity is subsonic throughout the whole field of flow. The solution of the boundary-value problem for the wedge in a channel is formulated as a pair of dual integral equations. The complete solution is given for the wedge in a free stream and the dimensions of the profile, together with the lift coefficient, are computed as functions of the transonic similarity parameter.
The effects of surface tension and viscosity on the stability of two superposed fluids
Authors: Reid W. H.
Published: November 2008

The effect of surface tension on the stability of two superposed fluids can be described in a universal way by a non-dimensional ‘surface tension number’ S which provides a measure of the relative importance of surface tension and viscosity. When both fluids extend to infinity, the problem can be reduced to the finding of the roots of a quartic equation. The character of these roots is first analysed so as to obtain all possible modes of stability or instability. Two illustrative cases are then considered in further detail: an unstable case for which the density of the lower fluid is zero and a stable case for which the density of the upper fluid is zero, the latter case corresponding to gravity waves. Finally, the variational principle derived by Chandrasekhar for problems of this type is critically discussed and it is shown to be of less usefulness than had been thought, especially in those cases where periodic modes exist.
Testing the independence of a system of axioms, using a logical computer
Authors: Foxley Eric
Published: November 2008

The object of this paper is to show how a well-known method of demonstrating the independence of one member of a system of axioms can be incorporated into a systematic method of testing for independence, suitable for solution on a logical computer of the type now being built at the University of Nottingham.
A short proof of the Farahat-Mirsky refinement of Birkhoff's theorem on doubly-stochastic matrices
Authors: Hammersley J. M.
Published: November 2008

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.
Some self-reciprocal functions and kernels
Authors: Lakshmikanth V.
Published: November 2008

The aim of this note is to find out some self-reciprocal functions and kernels for Fourier-Bessel integral transforms. Following Hardy and Titchmarsh(i), we shall denote by Rp the class of functions which satisfy the homogeneous integral equation

where Jp(x) is a Bessel function of order p ≥ − ½. For particular values of p = ½, − ½, we write Rs and Rc irrespectively.
A geometrical model for the real number field
Authors: Greve W.
Published: November 2008

Recently Cunningham and Valentine gave in (3) an axiomatic description of the one-dimensional real affine space in terms of its order structure and the (abstract) group of affine transformations It is the purpose of the present note to show that the system of axioms in (3) (cf. (L. 1)–(L. 5) of this note) leads in a natural way to a model of the real number field. Our method is suggested by a result of Hall ((4), p. 382), namely, that an infinite doubly transitive Frobenius group is isomorphic to the group of affine transformations in a near-field, provided that there is at most one transformation displacing all points and taking a given point a into a given point b. The salient point of our investigation is the redundancy of the latter condition in the case where the underlying space is endowed with a certain linear order structure which is invariant under the transformations of the given group.
The determination of the class-numbers of totally real cubic fields
Authors: Godwin H. J.
Published: November 2008

In a previous paper (2) it was shown how the work of finding the units of a totally real cubic field could be facilitated by consideration of the sum of squares of differences between a number and its conjugates. In the present paper it is shown that the same ideas can be helpful in the calculation of class-numbers, and a list of the fields with class-number greater than unity and discriminant less than 20,000 is given.
On homotopy Abelian H-spaces
Authors: Stasheff James
Published: November 2008

We consider H-spaces which are Abelian ‘up to homotopy’ (cf. (9)). We analyse this condition in terms of the homotopy of an associated space called ‘the projective plane’. The analysis is applied to show that ΩCP(3) is homotopy Abelian.
A uniqueness theorem concerning gravity fields
Authors: Smith Russell A.
Published: November 2008

It is well known that a uniform spherical shell produces the same gravitational field outside itself as that of a concentric uniform solid sphere with the same total mass. A given external field may therefore be produced by a variety of mass systems even under the restriction that the bodies of the system should have a prescribed uniform density. It can be asked what further conditions on the system are sufficient to ensure its uniqueness. The problem is of interest in connexion with the geological interpretation of local anomalies in the earth's gravitational field and work has already been done on it by several authors, (3), (6), (7), (8), (9). The results proved so far have all been restricted to systems consisting of a single body and the purpose of the present paper is to extend one of these results to include more general mass systems. The example of the spherical shell, mentioned above, might suggest that the absence of cavities in the bodies is a sufficient additional condition to ensure uniqueness. This is not so however. Celmiņš (3) gives an example of two different mass systems which have the same external field, each system consisting of three bodies of the same uniform density without any cavities in them.
Two problems on convex bodies
Authors: Croft H. T.
Published: November 2008

We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).

Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?
Applications of Møller's theory on energy and its localization in general relativity
Authors: Florides Petros S.
Published: November 2008

Møller's theory on the concept of energy and its localization in general relativity is applied to calculate the energy of the interior and exterior Schwarzschild fields. It is found that the energy is equal to the (Newtonian) gravitational mass of the material system associated with the fields and that all the energy resides wholly in the material system.
On a Genetics Model of Moran
Authors: Karlin Samuel , McGregor James
Published: November 2008

Wright (14), Feller (2) and others have proposed various stochastic models of genetics to study the fluctuations of gene frequency under the influence of mutation and selection. One of their simplest models has the following structure. There are a fixed numberNof gametes each of which may be of two typesaorA. The processX(n),n= 0, 1, 2, … which is assumed to have stationary transitions, is said to be in statejwhen there arejgametes of typea, andN − jof typeA. Let γ1denote the probability that immediately after formation an a gamete mutates into anAgamete and letγ2denote the probability of anAgamete mutating into an a gamete. Each of the gametes of the next generation is independently formed by making a random selection from the gametes of the present generation. The probability of a particular gamete in the next generation being of type a is thenand of typeAwherejrepresents the state of the process. Roughly speaking, the chance of a mating resulting in a gamete of a prescribed kind for the next generation is proportional to the fraction of these gametes present in this generation allowing for mutation effects.
Transition Integrals for Si IV and CaII
Authors: Douglas A. S. , Garstang R. H.
Published: November 2008

Atomic wave functions calculated using the Hartree—Fock equation and a polarization potential are used to obtain mean square radii of atomic orbitals for the ions Si8+ and Ca+, and transition integrals for the Si IV and Ca II spectra. A table of radial wave functions for Ca+ is given. Our results are compared with those obtained from other wave functions, and in one case with an experimental value, and generally good agreement is obtained.
Some Cyclic and Other Inequalities
Authors: Diananda P. H.
Published: November 2008

In a recent paper (2) Rankin has proved that for all positive integral n where As stated by Rankin, his proof is not simple although it is based on elementary ideas. Also, as noted by him, no other proof is available for any inequality of the form with λ a positive constant independent of n. It is the main purpose of this note to give a simple proof for the following result which implies the truth of (3) with λ = ¼, though not the truth of (1).
Factorization in Non-Commutative Power Series Rings
Authors: Cohn P. M.
Published: November 2008

The study of the divisibility of polynomials in a single variable over a field is essentially based on the division algorithm, which may be stated as follows: (A) Given polynomials a, b such that there exist polynomials q and a1, such that† .
On certain types of isolated s-ple points on algebraic primals in Sd
Authors: Herszberg J.
Published: November 2008

Singular points on irreducible primals were investigated briefly by C. Segre(8), where the author classified multiple points by the nature of the nodal tangent cone. For surfaces the problem of classification was investigated by, amongst others, Du Val(1) and a complete classification of isolated double points of surfaces lying on non-singular threefolds was given by Kirby(5). In (3) we classified certain types of double points on algebraic primals in Sn. An isolated double point which after a finite number of resolutions gave rise to at most a finite number of isolated double points was called a double point of rank zero. We found that the only isolated double points of rank zero are those which are analogous to the binodes, unodes and exceptional unodes (2) of surfaces.
On magneto-thermo-elastic plane waves
Authors: Paria G.
Published: November 2008

The propagation of plane waves in a thermo-elastic body placed in a magnetic field has been investigated. It is shown that the problem can be reduced to one of pure thermo-elasticity, provided that the longitudinal elastic wave velocity is suitably increased in the case of infinite electrical conductivity. For finite electrical conductivity and low frequency, however, the thermo-elastic coupling factor is to be increased.
On a conjecture of Barnes and Swinnerton-Dyer
Authors: Godwin H. J.
Published: November 2008

The second minimum of the form x2 − 23y2 is evaluated and shown to be non-isolated, thus exhibiting a possibility conjectured by Barnes and Swinnerton-Dyer in 1952.
On non-strictly simple groups
Authors: Hall P.
Published: November 2008

Ascending series. LetGbe a group and λ an ordinal number. An ascending series ofGoftypeλ is a set of subgroupsGαofG, defined for all α ≤ λ, and such that (i)G0= 1,Gλ=G; (ii)Gαis a proper normal subgroup ofGα+1for all α < λ (iii)for all limit ordinals μ ≤ λ.
The spectral sequence of an extraordinary cohomology theory
Authors: Maunder C. R. F.
Published: November 2008

Given an ‘extraordinary cohomology theory’, that is, a cohomology theory satisfying all the axioms of Eilenberg and Steenrod (7) except the dimension axiom, it is well known that there exists a spectral sequence relating the ordinary cohomology of a space with the extraordinary theory (see, for example, (3) in the case of K*(X)). Obviously, it would be useful to know the differentials in this spectral sequence, and it is the purpose of this paper to identify them in terms of cohomology operations defined by certain k-invariants. We shall make use of E. H. Brown's recent theorem (5) on the representability of extraordinary cohomology theories, to construct a second spectral sequence, in which the differentials are readily identifiable, which we shall prove is isomorphic to the usual one.
Basic series corresponding to a class of hypergeometric polynomials
Authors: Fields Jerry L. , Wimp Jet
Published: November 2008

Letgn(z) be a polynomial of degreeninz. Then there exist constants Πk, nsuch that

.
On space-times with bounded empty regions
Authors: Marder L.
Published: November 2008

If two empty regions of space-time are entirely separated by matter, little information of the nature of one region can be gained by knowledge of the other. Examples of a hollow Schwarzschild sphere and a hollow static torus are considered. In the first of these, it is shown that the size of the flat interior is not at all limited by the smallness of the Schwarzschild mass constant, the outer spherical boundary, or the condition that the sphere be constructed only of matter of positive density with positive stress-energy scalar.
Fundamental figures, in four and six dimensions, over GF(2)
Authors: Edge W. L.
Published: November 2008

Introduction. This paper falls into three parts.

In §§ 1–5 it is explained how, when the base field of the geometry is GF(2), there are figures of n + 2 interlocking polygons in [n], every two polygons sharing a vertex. When n is even these ½(n + 1) (n + 2) vertices lie in an [n − 1], and two of them are conjugate in a certain null polarity when, and only when, they do not belong to the same polygon.
Application of the Laplace transform method to wave motions involving strong discontinuities
Authors: Chadwick P. , Powdrill B.
Published: November 2008

Formulae are obtained for the Laplace transforms of the first and second partial derivatives of a function of time and position which, together with its first partial derivatives, is discontinuous on a moving surface. These formulae are then used in a discussion of the application of the Laplace transform method to the solution of mixed initial and boundary-value problems in linear thermoelasticity. Particular attention is given to thermoelastic disturbances involving plane shock waves, and it is pointed out that no properly posed formulation of problems of this type has yet been found.
Non-linear interactions, causality and the R-matrix
Authors: Power E. A. , Saavedra I.
Published: November 2008

The problem of causality in non-relativistic scattering where a bounded region of non-linear interaction occurs is re-examined in the context of a paper by Ebel. In the model considered it is shown that the acausal behaviour cannot be accounted for in the straightforward manner that is possible for an energy-dependent but linear potential which gives rise to a similar R-matrix.
Note on the continuity of lacunary structure in Taylor's interpolation series
Authors: Read G. A.
Published: November 2008

In this note we shall consider the series

where lim sup |an| = 1 and
A note on diffraction by a right-angled wedge
Authors: Williams W. E.
Published: November 2008

It has been shown in recent years ((5)–(8), (10)) that it is possible to obtain closed form solutions for the time harmonic wave equation when a linear combination of the wave function and its normal derivative is prescribed on the surface of a wedge. Boundary-value problems of this type occur in the problem of diffraction by a highly conducting wedge or by a wedge whose surfaces are thinly coated with dielectric. In certain circumstances such surfaces can support surface waves and one important aspect of the solution of the boundary-value problem is the determination of the amplitude of the surface wave excited.