bluuebunny/crossref_metadata_2025 huggingface dataset:
The focusing of radiation by a random surface when the source is at a finite distance
Authors: Longuet-Higgins M. S.
Published: November 2008

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantity

where J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.
Scattering by a bounded non-linear singularity
Authors: Power E. A. , Saavedra I.
Published: November 2008

An investigation is made of the effect of a non-linear addition to Schrödinger's equation for s-wave scattering of a beam of particles by a finite region of potential energy. The non-linear region is bounded and of an especially simple form so that an analytic solution is possible. The R-function for the problem is explicitly constructed and some related formulae such as the resonant energies and the scattering length are given. The analytical nature of R as a function of the momentum considered as a complex variable is discussed in relation to causality.
Power series with non-integral exponents which are Fourier kernels
Authors: Fox Charles
Published: November 2008

Much is known about power series with integral exponents, and, as in the case of the Theta-functions, about series whose exponents are the squares of the natural numbers. About other types of power series little is known. I prove here that a certain power series whose exponents are the square roots of the natural numbers is a symmetrical Fourier kernel.
Note on an equation of boundary-layer theory
Authors: Coppel W. A.
Published: November 2008

The differential equation

which is of importance in boundary-layer theory, has recently been discussed in detail for all non-negative values of the parameter λ(1). I wish to point out here that when λ = − 1 the equation can be solved completely in terms of known functions.
The Enriques threefold
Authors: Tyrrell J. A.
Published: November 2008

It is a classical and well-established result that a unirational algebraic variety of dimension ≪ 2 is necessarily rational. It is also generally agreed, though not so well established, that this result is no longer true for varieties of dimension three. In this connexion, the critical example has been the Enriques threefold E, asserted by Roth (1) to be unirational but not rational. The purpose of this note is to point out a fallacy in the proof of the non-rationality of E and, incidentally, to resolve a difficulty raised by Serre (2) in which E would appear to feature as a counter-example to a well-established general theorem. As far as the author is aware, there are no other examples of nonrational unirational threefolds, so that the question of their existence is still open.
The electromagnetic energy and the gravitational mass of a charged particle in general relativity
Authors: Florides Petros S.
Published: November 2008

The electromagnetic energy & of the field of a charged particle is calculated, using Møller's theory summarized in the previous paper. The contribution of & to the gravitational mass of the charged particle is discussed, by studying the behaviour of a neutral test particle in its field. The conclusion is that & gives rise to an ‘effective’ gravitational mass of the charged particle, which is equal to the (Newtonian) gravitational mass of the charged particle, plus the mass-equivalence of &. This is contrary to the currently accepted theory, that what we have called the ‘effective’ gravitational mass is equal to the ‘Newtonian’ gravitational mass of the charged particle.
On the Thom Isomorphism Theorem
Authors: Cockcroft W. H.
Published: November 2008

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).
Expansions of Generalized Hypergeometric Functions in Series of Products of Generalized Whittaker Functions
Authors: Ragab F. M.
Published: November 2008

In a previous paper (l) in this journal L. J. Slater gave expansions of the generalized Whittaker functions . She gave this name to the generalized hypergeometric function in the sense that it is a generalization of the well-known Whittaker function . In this paper series of products of generalized Whittaker functions will be evaluated in terms of such functions or in terms of generalized hypergeometric functions pFp(x). These expansions are These formulae will be proved in § 2 and particular cases will be given in § 3.
Coset enumeration on digital computers
Authors: Leech John
Published: November 2008

In 1936 Todd and Coxeter gave a method ((10), also described in (4), ch. 2) for establishing the order of a finite group defined by a set of relations

satisfied by its generators S1, S2,…, Sk. They enumerate systematically the cosets of a suitable subgroup whose order is evident from the defining relations for the whole group. They describe the method as being ‘purely mechanical’, and since that date the advent of electronic computers has led a number of people to programme the method for automatic execution. Most of this work has remained unpublished; an account is given here of work known to the author.
Some formulae for the G -function
Authors: Saxena R. K.
Published: November 2008

The object of the present note is to evaluate some integrals involving Meijer's G-function, in which the argument of the G-function contains a factor where m and n are positive integers and t is the variable of integration. Two different forms of the general result have been obtained, one for m > n and the other for m < n. The value of the corresponding integral when m = n is also obtained. For the definition, properties and the behaviour of the G-function, see (2), §§ 5·3, 5·31 and (5), § 18.
Some dual series equations involving Jacobi polynomials
Authors: Noble B.
Published: November 2008

In this paper first of all we find an exact solution of the dual series equations

where

and Jn is the Jacobi polynomial defined by

The Bn are unknown coefficients which are to be determined. The functions f, g and the parameters a, λ, μ, are given. Necessary restrictions on these quantities are indicated during the analysis.
A new Monte Carlo method for percolation problems on a lattice
Authors: Dean P.
Published: November 2008

A new and general Monte Carlo technique is described for solving some well-known percolation and cluster-size problems on regular lattice networks. The method has been applied to ten two-dimensional structures.
On group representations and invariant stochastic processes
Authors: Mclaren A. D.
Published: November 2008

The theory of group representations is explored with regard to statistical applications. Ease of analysis depends rather critically on the amount of symmetry present and this point is examined in detail. ‘ Stationarity’ assumptions are considered for a finite number of variates, with examples of their use in experimental situations.
Packings, free products and residually finite groups
Authors: Macbeath A. M.
Published: November 2008

In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake, though it does not appear to lead to Gruenberg's other results in this connexion. I have to thank Mr W. J. Harvey and Mr C. Maclachlan for checking a first draft of this paper and pointing out a few errors.
On axially symmetric waves
Authors: Craggs J. W.
Published: November 2008

The method of Part I(3) is applied to problems involving boundary conditions on an equatorial plane.
On the nilpotence of nil ideals
Authors: Macrae R. E.
Published: November 2008

One of the classical results of ring theory is due to Levitzki ((3), p. 199) and asserts the equivalence of the notions of nil and nilpotent one-sided ideals in a ring satisfying the ascending chain condition on left ideals. Recent, distinctive proofs of this result have been given by Goldie (1) and Herstein (2). We present below yet a fourth way of proving Levitzki's theorem.
The irrotational solution of an elliptic differential equation with an unknown coefficient
Authors: cannon J. R. , Halton J. H.
Published: November 2008

Let G be a bounded region in k-dimensional space, with boundary Γ, such that the Laplace equation,

is uniquely soluble (to within an added constant) under the Neumann boundary conditions

where ∂/∂n denotes outward normal differentiation on Γ, and it is assumed that h is a function in G ∪ ∂, and thus that g is a function on ∂. In what follows, we shall assume certain properties of the solution h: these are all well known (see, for example, Osgood(l) or Courant(2)).
Presentations of groupoids, with applications to groups
Authors: Higgins P. J.
Published: November 2008

1. Introduction. In (7), a theory was developed which dealt with certain classes of partial algebras in which the domains of the operators were determined by an ‘operator scheme’. This device allows the use of methods normally available only for full abstract algebras. In particular, one can present suitable partial algebras by means of generators and relations.
Note on the generalized Mehler transform
Authors: Lowndes J. S.
Published: November 2008

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined by

where ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.
On non-expansive mappings of Banach spaces
Authors: Edelstein Michael
Published: November 2008

Let f: X → X be a mapping of the metric space X into itself. We call f non-expansive if
Complete functors in homology II. The exact homology sequence
Authors: Kelly G. M.
Published: November 2008

This is the second paper of a series, and supposes familiarity with the results and the notations of the first(l), which we refer to as I.
Weak Cauchy sequences in normed linear spaces
Authors: Garling D. J. H.
Published: November 2008

It follows from the Krein-Milman theorem that (c0) is not isomorphic to the dual of a Banach space. Using a technique due to Banach ((4), page 194) we shall extend this result to show that if a subspace of (c0) is isomorphic to the dual of a normed linear space, then it is finite dimensional (Proposition 1). Using this result, we shall show that if E is a normed linear space, the unit ball of which is contained in the closed absolutely convex cover of a weak Cauchy sequence, then Eis finite dimensional (Proposition 2). This result has applications to the Banach-Dieudonné theorem, and to the theory of two-norm spaces.
A theorem on the Bohr compactification of a locally compact Abelian group
Authors: Varopoulos N. Th.
Published: November 2008

Notations. If G denotes an Abelian group and L a locally compact group topology on G, G(L) will denote the resulting topological group, and will denote the character group with the usual Pontrjagin topology (which is locally compact). Rn will denote the real n-dimensional vector group. Finally, for any set X, |X| will denote its cardinal number.
An infinite class of periodic solutions of ẍ + 2x3 = p(t)
Authors: Morris G. R
Published: November 2008

An important unsolved problem in the theory of non-linear oscillations is to establish the boundedness or unboundedness of the general solution of

where dots denote differentiation with respect to t. When p(t) is periodic, we may seek periodic solutions. This search is interesting for its own sake, and of course leads us to special bounded solutions. In three previous papers (2, 3, 4) I have exhibited the equation

as tractable: on the assumption that e(t) is even and periodic, it was shown that the equation has an infinity of periodic solutions.
Gravitational radiation in homogeneous cosmological models with space-sections of constant negative curvature
Authors: Rao J. Krishna , Vaidya P. C.
Published: November 2008

The propagation of type N gravitational radiation in homogeneous cosmological models with space-sections of constant negative curvature is considered. It is found that the propagation of radiation distorts the isotropy of the fluid pressure in the models permanently.
Notes on general topology
Authors: Ward A. J.
Published: November 2008

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.
Certain two-dimensional problems of stress distribution in wedge-shaped elastic solids under discontinuous load
Authors: Srivastav R. P. , Narain Prem
Published: November 2008

In this paper we consider the problem of distribution of stress in an infinite wedge of homogeneous elastic isotropic solid under the usual assumptions pertaining to plane strain in classical (infinitesimal) theory of elasticity. The wedge is supposed to occupy the region 0 ≤ ρ ≤ ∞, − α ≤ θ ≤ α (in plane polar coordinates) where the pole is taken on the apex of the wedge with the line bisecting the wedge angle as the initial line. We report here the investigations of two types of stress fields: (i) the stress field which is set up by the application of known pressure to inner surfaces of a crack situated on the bisector of the wedge angle, and (ii) the stress field generated by the indentation of the plane faces of the wedge by the rigid punch. The corresponding boundary-value problems are shown to be equivalent to the problem of solving dual integral equations involving inverse Mellin-transforms. Similar equations have been discussed by Srivastav ((l)) in a recent paper. The method of (1) is easily modified to reduce the dual equations to single Fredholm equations of the second kind which are best solved numerically. The boundary-value problems discussed here are of the mixed type and appear to have been considered only recently by Matczynski ((2)) who has investigated a contact problem. He identifies the problem with a Wiener-Hopf integral equation and solves the integral equation approximately using a method due to Koiter ((3)). The method presented here seems to have the advantage that it requires no elaborate tools of analysis which are necessary for resolving the problem of Wiener-Hopf and numerical results may be obtained comparatively easily.
Hyperspaces of a uniformizable space
Authors: Smith D. Hammond
Published: November 2008

If X is a uniformizable space any uniform structure compatible with its topology will give a uniform structure on the collection H(X) of all non-empty closed sets of X ((1), Section 2, Exercises 7, 8, definition 1·6): for each entourage U ∈ we define an entourage U on H(X) by putting U = {(A, B); A ⊂ U(B) and B ⊂ U(A)}. It has been conjectured ((6), p. 35, Exercise 17) that if 1 and 2 are distinct uniform structures on X, 1 and 2 induce indifferent topologies on H(X). Unfortunately the method proposed in (6) fails, even for metrizable structures, as we shall show by an example. We shall, however, establish a number of partial results tending to support the conjecture; in particular we shall establish the result when both the uniform structures are metrizable, or when one is precompact.
Carleman's formula for subharmonic functions
Authors: Thompson C. L.
Published: November 2008

The purpose of this paper is to prove theorems for subharmonic functions analogous to those stated below for regular functions. We make the following definition: D(∈, R) is the domain in the Euclidean plane defined by
The hydromagnetics of an ellipsoid moving in an aligned field
Authors: Ludford G. S. S. , Singh M. P.
Published: November 2008

This note supplements our paper ((4)) on the hydromagnetics of an ellipsoid moving in a cross-field. Familiarity with this paper will be assumed. Together they account for creeping motion in an arbitrary direction when the applied magnetic field is parallel to a principal axis.
A remark about the description of free products of groups
Authors: Stallengs John
Published: November 2008

The free product A* B of groups A and B can be described in two ways.

We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .
Bounds for the class of nilpotent wreath products
Authors: Scruton Teresa
Published: November 2008

Introduction. In his paper ((1)), Baumslag has shown that the wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group, the prime p being the same for both groups. Liebeck ((3)) has obtained the exact nilpotency class of A wr B when A and B are Abelian. Let A be an Abelian p -group of exponent pn and let B be a direct product of cyclic groups, whose orders are pβ1, …, pβn, with β1 ≤ β2 ≤ … βn. Then A wr B has nilpotency class .
The constrained quantum mechanical harmonic oscillator
Authors: Dean P.
Published: November 2008

The effect of restricting the displacements of a quantum mechanical harmonic oscillator to semi-infinite and finite regions is studied. It is shown how the eigenvalue problem may be solved, both graphically, and from certain series expansions.
Some formulae for Jacobi polynomials
Authors: Manocha H. L. , Sharma B. L.
Published: November 2008

Introduction. The object of this paper is to prove some formulae for Jacobi polynomials. The formulae (6), (7), (9), (10) and (13) are believed to be new.
Factorizations of the residual likelihood criterion in discriminant analysis
Authors: Radcliffe J.
Published: November 2008

Certain exact tests were developed by Williams (1952) to deal with the goodness of fit of a single hypothetical discriminant function. Bartlett (1951) generalized these results by the use of the geometric method to any number of dependent and independent variables. Bartlett's paper is divided into two parts. The first deals with an approximate factorization of the residual likelihood criterion into an effect due to the difference between the hypothetical and sample functions, and an effect due to non-collinearity. A method is given for constructing confidence intervals from the first factor. The second part of the paper gives two possible exact factorizations of the likelihood criterion, expressing the results in terms of the sample canonical variables. Kshirsagar (1964a) has expressed these results in terms of the original variables and given an analytic proof of the distribution of the factors. Williams (1955, 1961) has outlined a generalization of these results to several discriminant functions and given the result for one of the possible factorizations.
Screening constants for relativistic wave functions
Authors: Garstang R. H. , Mayers D. F.
Published: November 2008

Formulae for the mean radius and mean square radius of a relativistic Coulomb wave function are obtained. Screening constants for the energy, mean radius and mean square radius are defined relative to non-relativistic wave functions and the results of numerical calculations given. It is shown that if the screening constants so determined are added to the screening constants due to the presence of other electrons as found by the s.c.f. method, good agreement is obtained in a case where both effects have been considered together. The value of solving the relativistic wave equation in a Thomas-Fermi field is also shown.
Bayes procedures for deciding the Sign of a normal mean
Authors: Bather J. A. , Walker A. M.
Published: November 2008

The decision problem discussed concerns the drift of a Wiener process with known variance per unit time. It is required to find the optimal sequential test for the sign of the drift, when sampling costs and the cost of a wrong terminal decision are specified. Initially, the drift is supposed to have a continuous prior distribution. The optimal procedure is determined by a continuation region in which a certain partial differential equation holds. The boundaries of this region can be found explicitly in a few special cases. In general, methods are developed for obtaining both inner and outer approximations to the optimal continuation region and these techniques are illustrated by application to several examples.
A non-singular electromagnetic field
Authors: Misra Markandeya
Published: November 2008

In an earlier paper (2) the author and Radhakrishna have, under certain conditions, obtained the following result.
On quadrics through five real points
Authors: Denniston R. H. F.
Published: November 2008

Let Q1,…, Q5 be five fixed points (no four coplanar) of the real projective space S3: let s be a variable quadric surface through these points. The set of all such quadrics can be represented by the points of a real S4, in which there is a quartic primal that represents cones. The geometry of this threefold is well known in the complex case, but has hardly been considered at all in the real case: and one object of the present paper is to describe the real threefold and determine its homology groups.
Large eigenvalues of a Sturm-Liouville problem
Authors: Cohn J. H. E.
Published: November 2008

Consider the eigenvalue problem

where q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λr (r = 0, 1, 2, …) and λn ∽ n2π2 (b − a)−2 as n → ∞.
Additive set functions and the theory of probability
Authors: Kingman J. F. C.
Published: November 2008

This paper is a contribution to the study of set functions which, although additive, fail to be measures by not being countably additive. It is motivated by the suggestion (to be found, for instance, in (2), section 2.3) that such functions might be used instead of measures in the theory of probability, and by speculation about the effect which the adoption of such a suggestion would have on the theory of stochastic processes.
Two more hypergeometric integral equations
Authors: Love E. R.
Published: November 2008

This paper is a sequel to one with a similar title to appear in the Proceedings of the Edinburgh Mathematical Society. Explicit solutions are found for two more integral equations of similar form; and also conditions necessary and sufficient for existence, and sufficient for uniqueness, of solutions. These theorems are preceded by several preparatory theorems on fractional integrals with origin ∞, including integrals of purely imaginary order.
On a summation formula for the Appell function F2
Authors: Srivastava H. M.
Published: November 2008

1. In the course of a systematic analysis of certain problems in quantum mechanics it has been observed that their exact solutions can be expressed in terms of the Appell function F2 defined by means of (see e.g. (8), p. 211)

where, as usual,

and for convergence of the double series,
An addition theorem for spherical harmonics
Authors: Tikochinsky Y.
Published: November 2008

An addition theorem, expressing the spherical harmonics in terms of the spherical harmonics and , is derived.
Seiches over parabolic bottoms
Authors: Laitone E. V.
Published: November 2008

The solutions are derived for the shallow water standing waves of small amplitude that can form in channels or lakes of varying breadth with a concave parabolic bottom. In addition explicit solutions are given for standing waves in a ring-type lake with a parabolic bottom and a central circular island that has vertical walls.
Note on a problem of Porte
Authors: Rousseau George
Published: November 2008

Porte (1), p. 117, conjectures that the positive implicational propositional calculus has no finite characteristic matrix. The proof of this conjecture is a straightforward modification of Gödel's proof (2) that the intuitionistic propositional calculus has no finite characteristic matrix (see e.g. Church(3), ex. 26.12). Writing (A ∨ B) for ((A ⊃ B) ⊃ B) and Xij for (pj ⊃ pi) (i, j = 1,2,…), we define, for n > l, the formula

where the terms associate to the left. Since provable formulae take the value n for all systems of values of the variables in the matrix {1,…,n} where x ⊃ y is n when x ≤ y and y otherwise, whereas Gn takes the value n − 1 for the system of values pi = i (i = 1,…,n), it follows that Gn is not provable. On the other hand, since A ⊢ A ∨ B and B ⊢ A ∨ B, it is easily seen that is provable whenever r ≠ s (r, s = 1,…,n). The result follows from these two remarks.
An automorphism of the symplectic group Sp4(2n)
Authors: Duncan Anne
Published: November 2008

In his discussion (3) of automorphisms of finite simple linear groups, Steinberg shows that the group denoted by B2(K) admits graph automorphisms when K is a perfect field of characteristic 2. The group B2(K) is identified in (2) as the symplectic group Sp4(K) when K is finite and of characteristic 2, and it is the object of this note to give, by means of this identification, an explicit description of the graph automorphisms of B2(2n).
Alexander's theorem for real-compactness
Authors: Hayes Allan
Published: November 2008

Alexander's theorem (5) states that a topological space is compact if there is a sub-base, , for its closed sets such that every subclass of with the finite intersection property has a non-empty intersection. An analysis and extension of this is given here which has applications, inter alia, to problems concerning real-compactness (2).
Application of Δ and E operators to evaluate certain integrals
Authors: Agrawal B. M.
Published: November 2008

In a recent paper Ragab and Simary (6) have deduced some integrals involving the products of generalized Whittaker functions. We feel that some of the integrals are not correctly evaluated. Indeed the integral (2) should be symmetrical in α and β, as we see by changing the variables by s = − t in the integral, while it is not symmetrical. A similar remark will follow in the case of the integrals (3) and (6).
Solutions of a non-linear differential equation. II
Authors: Billigheimer C. E.
Published: November 2008

We consider in this paper the solution behaviour as s → 0 of the equation

where the primes indicate differentiation with respect to s, and a, b, c are constants.
On differentiable manifolds with degenerate metrics
Authors: Crampin Michael
Published: November 2008

A metric on a differentiable manifold induces a bilinear form on the tangent space at each point of the manifold. The set of tangent vectors orthogonal, with respect to this bilinear form, to the whole tangent space is a vector subspace of the tangent space: the metric is called non-degenerate or degenerate at the point according as the subspace is or is not empty. This paper is concerned with the geometry of manifolds having everywhere degenerate metrics.
Stability of wedges and semi-algebras
Authors: Brown Gavin
Published: November 2008

We obtain a simple new proof of Bonsall's characterization theorem for type 1 semi-algebras of continuous functions. An extension of our methods produces several new results on stability properties of wedges.
A note on the queueing system GI/G/∞
Authors: Shanbhag D. N.
Published: November 2008

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).
Convergence rates in the law of large numbers. II
Authors: Rohatgi V. K.
Published: November 2008

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense that

Set qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)

where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence of

to zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).
Free surface instability in electrohydrodynamics
Authors: Michael D. H.
Published: November 2008

The study of electrostatic effects on the motion of fluids goes back to the work of Rayleigh (l) who discussed the effect of surface charges on the vibration of spherical drops, and Bassett (2) who studied the effect of a radial electric field on the stability of a jet. The subject has recently been taken up again by Melcher (3) and Crowley (4), and the author became interested in the subject through the monograph of the former author in which a wide variety of wave problems are described.
Uniform approximation by Fourier–Stieltjes coefficients
Authors: Ramirez Donald E.
Published: November 2008

In chapter I, E = {nk} ⊂ Z is shown to be a Sidon set if and only if (**). For each x ∈ T,

.

Let E ⊂ Z+ be a lacunary sequence. In chapter II, it is constructively shown that the characteristic function of E is uniformly approximable by Fourier–Stieltjes coefficients; i.e. ϕE ∈ M(T)∧−.
Stable subspaces of L1 + L∞
Authors: Fremlin D. H.
Published: November 2008

1. In this paper, I apply the results of (4) to a class of function spaces distinguished by their invariance under a certain family of operators. This class contains the Lp spaces (indeed, all Orlicz spaces) and the symmetric Köthe spaces of (5). My original objective was simply a more general expression of the ideas of (5), but I found myself taking up a rather different approach.
An inner measure associated with a von Neumann algebra
Authors: de Barra G.
Published: November 2008

An ‘inner measure’ analogous to Lebesgue inner measure and associated with a von Neumann algebra is constructed on the linear sets of a Hilbert space. We supposed to be of finite type and countably decomposable, as these restrictions will be necessary for some of the results obtained. As remarked by Dye in (2), Type II1 algebras have a structure analogous to that of a finite non-atomic measure space, the trace corresponding to the measure. We define a class of ‘measurable sets’ and obtain some of its properties. The development of the theory is indicated also from the starting point of a function-valued dimension function. The idea for the construction of such an inner measure comes from a remark in (3), paragraph 16·2, and the terminology of (3) and of (5) is used throughout.
Gravitational induction
Authors: Levy H.
Published: November 2008

Near-field transfer of gravitational energy for quasi-static axisymmetric systems is investigated by a perturbation method; a relativistic analogue of Bondi's Newtonian Poynting vector is defined. It is concluded that a quasi-static axisymmetric system can lose energy only in the presence of a receiver.
Almost periodicity and B-equicontinuity in topological dynamics
Authors: Yang Jeong Sheng
Published: November 2008

In the previous paper(8), we considered a property of families of functions we termed. ‘B-equicontinuity’. It was shown that B-equicontinuity is stronger than the usual equicontinuity, and is weaker than the equicontinuity defined by Bartle (3). In this paper we consider the concept of B-equicontinuity on topological transformation groups. The net characterization of equicontinuity obtained in (8) is used in discussion. It is proved in (1) that if (X, T, π) is almost periodic, the transition group {πt|t ∈ T} is equicontinuous. One might wonder whether this conclusion can be strengthened to say that {πt|t ∈ T} is B-equicontinuous; we show here by an example that this is not true and a partial solution to this problem is given. Some relations between almost periodicity and B -equicontinuity are also discussed.
Spherically symmetric differential equations, the rotation group, and tensor spherical functions
Authors: Burridge R.
Published: November 2008

In this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group. These fields may be expanded as series of functions, where,mis fixed and the matricesTl(g) form a 21+ 1 dimensional irreducible representation of.

Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with givenl, nare transformed into others with the same values ofl, n. That this must be so follows from Schur's Lemma and the fact that for eachmandlthe functionsform a basis for an invariant subspace of functions onof dimension 2l+ 1 in which an irreducible representation ofacts. Explicit formulae for the results of such operations are presented.

The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).

This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.
On the method of matched asymptotic expansions
Authors: Fraenkel L. E.
Published: November 2008

The formal method of matched expansions is applied to two further examples. The first concerns the magnetic field induced by a steady current in a thin toroidal wire. The second, which involves a non-linear ordinary differential equation of the fourth order, has been chosen to resemble the problem of flow past a circular cylinder at small Reynolds numbers. The results of the formal procedure are proved in each case to be expansions of the exact solution.
On the note emitted from a mug while mixing instant coffee
Authors: Farrell W. E. , McKenzie D. P. , Parker R. L.
Published: November 2008

We have recently observed a curious phenomenon, undoubtedly well known, while mixing instant coffee in a mug. If the bottom of the mug was tapped repeatedly with the spoon as the powder was stirred into the water, the note emitted could be heard to rise in pitch by over an octave. This rise occurred in a matter of seconds, about the same time as it took the mug to heat up. Further experimentation showed that almost any powder could produce the effect in cold or hot water, but water alone would not. Vigorous stirring did not seem essential. A full mug showed as pronounced an effect as a half-filled one.
On asymmetrical slow viscous flows caused by the motion of two equal spheres almost in contact
Authors: O'Neill M. E.
Published: November 2008

An exact solution is given for the slow viscous flow caused by the translation of two equal spheres in contact with equal velocities perpendicular to their line of centres. An asymptotic theory is presented for solving the problem when two equal spheres of radius a almost in contact rotate with equal and opposite angular velocities. The problem when the spheres touch is shown not to be well posed; the forces acting on the spheres are shown to be O(1) and the couples of the form α log∈ + β where α and β are independent of the minimum clearance 2∈α between the spheres and have been determined explicitly. The relevance of the results to the free settling of two spheres in a viscous fluid under the influence of gravity is discussed.
Symmetry adapted functions for double point groups. I. Non-cubic point groups
Authors: Cracknell Arthur P.
Published: November 2008

Basis functions are derived and tabulated for the double-valued representations of the triclinic, monoclinic, orthorhombic, trigonal, tetragonal and hexagonal crystallographic point groups using the method of projection operators.
Two singular integral equations involving confluent hypergeometric functions
Authors: Prabhakar Tilak Raj
Published: November 2008

Widder(1) obtained an inversion of the convolution transform

by the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equation

using Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).
On the extendibility of Weyl coordinates
Authors: Hagen H. Müller Zum
Published: November 2008

Axially symmetric static fields produced by one extended body are considered. The metric outside the body is the Weyl metric which can be written in so-called Weyl coordinates in the form ds2 = a2(dr2 + dz2) + b2dϕ2 – c2dt2 (a, b, c functions of (r, z) and r > 0) with b2 = r2c−2. Criteria are found under which these Weyl coordinates can be extended through the whole of the body so that ds2 keeps the form above (although in general b2 ǂ r2c−2 inside the body). In the spherically symmetric (⇒ axially symmetric) case it is shown that the Weyl coordinates cover the whole of spacetime only if one allows r < 0 in a certain part of the body.
-normalizers and -covering subgroups
Authors: Prentice Mary Jane
Published: November 2008

In (1), Carter and Hawkes define the -normalizers of a finite soluble group for any saturated formation . These subgroups are conjugate, invariant under homomorphisms of the group, cover and avoid the chief factors of the group and may be characterized by means of the maximal chains of subgroups connecting them to the group. The first aim of the present paper is to generalize both the -normalizers and the relative system normalizers (Hall (5)) of a finite soluble group G. We choose an arbitrary normal subgroup X(p) of G for each prime p dividing the order of G, forming a normal system = {X(p)} of G. Each of these normal systems of G yields a conjugacy class of subgroups, called the -normalizers of G, which possess the above properties of -normalizers. However, they do not satisfy all the properties of the -normalizers unless is a so-called integrated normal system of G.
A characterization of quotient algebras of Ll(G)
Authors: Ramirez Donald E.
Published: November 2008

Let G be a locally compact Abelian group; Γ the dual group of G; C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; CB(Γ) the continuous, bounded functions on Γ; M (G) the algebra of bounded Borel measures on G; L1(G) the algebra of absolutely continuous measures; and M(G)∩ the algebra of Fourier–Stieltjes transforms.
Magnetohydrodynamic Stokes flows for rotating solids of revolution
Authors: Ranger K. B.
Published: November 2008

Riley (1) has considered a magnetohydrodyriamic Stokes flow in which the modifications to the well-known Stokes flow past a fixed sphere were investigated for the case in which there is a magnetic pole situated at its centre. Again, Barthel and Lykoudis (2) and Ludford and Murray (3) have considered the low Reynolds number flow and inviscid flow respectively, past a sphere in the presence of a magnetic field due to a dipole situated at the centre of the sphere and directed parallel to the uniform stream. For the viscous problem, the velocity field and induced magnetic field were calculated for small values of the Hartmann number M and an expression found for the total drag on the sphere. Tamada and Sone (5) have also considered the same problem for a viscous conducting fluid at small Reynolds numbers and calculated the solution for small values of M while an asymptotic solution was found for large values of M. For small values of M they show that, since the Lorentz force retards the fluid motion, the viscous forces are reduced and hence the drag on the sphere is also reduced. However, the magnetic stresses contribute a force on the boundary which actually results in a net increase for the total drag on the sphere.
Exact solutions for vertically polarized electromagnetic waves in horizontally stratified isotropic media
Authors: Westcott B. S.
Published: November 2008

A systematic analysis is conducted for refractive index profiles capable of yielding exact solutions in terms of hypergeometric or confluent hypergeometric functions for the electromagnetic field of vertically polarized waves propagating in horizontally stratified isotropic media. Previously known profiles emerge as special cases of the analysis. Profiles suitable for use with isotropic lossless ionospheres and of prescribed form n2 = 1 − T(z)/k2, with T(z) independent of k, do not arise in this paper.
The structure of unique factorization rings
Authors: Fletcher C. R.
Published: November 2008

1. Introduction. In (1) we proved that the direct sum of a finite number of unique factorization rings is a unique factorization ring (UFR), and in particular that the direct sum of a finite number of unique factorization domains (UFD's) is a UFR. The converse, however, does not hold i.e. not every UFR can be expressed as a direct sum of UFD's. Here we investigate the structure of UFR's and show that every UFR is a finite direct sum of UFD's and of special UFR's. There is thus a relationship with the structure theorem for principal ideal rings ((2), p. 245).
A method for Fourier transforms
Authors: Kaufman R.
Published: November 2008

0. Our aim in this note is to obtain theorems about measures, perfect sets, and transforms in the group G of p-adic integers (for a prime p), related to recent developments in compact Abelian groups (6, 7) and again to Fourier analysis in Euclidean space (2–5).
Polynomial conformal tensors
Authors: du Plessis J. C.
Published: November 2008

The conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.
Some arithmetical properties ofm-ary partitions
Authors: Rödseth Öystein
Published: November 2008

We denote bytm(n) the number of partitions of the positive integerninto non-decreasing parts which are positive or zero powers of a fixed integerm> 1 and we calltm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula fortm(n), the first term of which is

Mahler's result was later improved by de Bruijn (2).
Internal labelling: the classical groups
Authors: Sharp R. T.
Published: November 2008

The internal labelling problems for the group-subgroup combinations Sp (2n) ⊃ Sp (2n − 2) × Sp (2) and O(n) ⊃ O(n − 1) are solved by the method of elementary multiplets.
On scalar-type spectral operators
Authors: Spain P. G.
Published: November 2008

The purpose of this paper is to give two characterizations of scalar-type spectral operators.
The power inequality on Banach spaces
Authors: Bollobás Béla
Published: November 2008

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined as

and the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).
A steady vortex ring close to Hill's spherical vortex
Authors: Norbury J.
Published: November 2008

The existence of a steady vortex ring close to Hill's spherical vortex is established, and an approximate description of its boundary is given. The vorticity in the ring is proportional to the distance from the axis of symmetry. The core propagates steadily in an unbounded fluid at rest at infinity. The boundary of the vortex ring is close to an interior stream surface of Hill's vortex.
Sur l'ensemble des points fixes des actions de S1
Authors: Sebastiani Marcos
Published: November 2008

A special type of S1-actions on homotopy spheres is defined and the properties of the fixed point set are studied.
The calculation of diagonal matrix elements of two-body relative state operators between many-body oscillator states
Authors: Nash G. F.
Published: November 2008

A general method is given for the determination of diagonal matrix elements of relative state operators in shell model states of a given supermultiplet and SU3 symmetry, for states in s, p and sd shells, and for cross-terms between these shells, enabling the excitation energies of particle–hole configurations to be evaluated.
An existence theorem for Robin's equation
Authors: Cade R.
Published: November 2008

An existence theorem is proved for Robin's integral equation for the density of electric charge on a closed surface, under the assumptions that the surface is convex, smooth and twice continuously differentiable. The technique is essentially Neumann's method of the arithmetic mean, used by Robin himself to show that the solution, assumed to exist, can be successively approximated by a sequence. In order to facilitate the main argument of the proof, it is assumed initially that the Gaussian curvature is everywhere positive, but this restriction is subsequently removed.