bluuebunny/crossref_metadata_2025 huggingface dataset:
Bemerkungen zur Theorie der Vielverbände. IV (Über die Möbius'sche Funktion)
Authors: Benado Mihail
Published: November 2008

Einleitung.Man verdankt Richard Dedekind (siehe etwa(1)) das erste Beispiel einer Umkehrungsformel einer endlichen Reihe; dabei spielt, wie man weiβ, die sogenannte Möbius'sche Funktion (der Zahlentheorie) eine groβe Rolle. Auch die sogenanntenAufzählungsmethodender Gruppentheorie, usw., die mit der soeben erwähnten Dedekindschen Umkehrungsformel eng verbunden sind, stützen sich, wie hauptsächlich von Herren Louis Weisner(2) und Philip Hall (3) gezeigt wurde, auf eine passende Verallgemeinerung der gewöhnlichen Möbius'schen Funktion, die für alle teilweise geordneten Mengen einen Sinn hat, deren Quotienten endlich sind. Siehe auch Ward (4).
The determination of units in totally real cubic fields
Authors: Godwin H. J.
Published: November 2008

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.
Mutual Magnetic Interactions and Oscillator Strengths in the First Spectrum of Oxygen
Authors: Garstang R. H.
Published: November 2008

The effects of the spin-other-orbit and spin-spin interactions on the energy levels of the 2p3 3s configuration of oxygen are investigated, and they are found to make an appreciable contribution to the fine structure. The interactions are shown to have a negligible effect on the strengths of all the stronger lines of the 2p4-2p3 3s array, and the more important of the weaker transitions are likewise little affected. The value to be adopted for the radial integral involved in the absolute oscillator strengths is discussed, and numerical values of the oscillator strengths are given.
Completely simple and inverse semigroups
Authors: McFadden R. , Schneider Hans
Published: November 2008

The purpose of this paper is to investigate the structure of certain types of semigroups. Rees(6),(7) has determined the structure of a completely simple semigroup, and has shown that such a system may be realized as a type of matrix semigroup. Clifford (2) and Schwarz (8) have found conditions, namely, the existence of minimal left and minimal right ideals, under which a simple semigroup is completely simple, and have made a more detailed study of such semigroups. Preston (4), (5) has studied inverse semigroups, in which each non-zero element has a unique relative inverse, and has also considered inverse semigroups which contain minimal right or left ideals. In the present paper we obtain a set of conditions on a simple semigroup, each of which is equivalent to the semigroup being both completely simple and inverse. Section 2 defines the terms used and gives a brief resume of the main results which have already been proved. Section 3 is devoted to our present considerations.
Some consequences of bond alternation in long polyenes
Authors: Salem L.
Published: November 2008

The effect of a twofold bond alternation on the bond ordersandand on the π-electron contributionto the diamagnetic susceptibility of a cyclic even hydrocarbon polyeneis investigated. Power series expansions and asymptotic expressions are given forand χπin terms of the parametert2(0 ≤t2< 1) which characterizes the amount of bond alternation. It is found that bond alternation reduces both bond order and χπstrongly, the effect being more drastic the greater the bond alternation (t2→0).
Tensor operators under semi-simple groups
Authors: Stone A. P.
Published: November 2008

Tensor operators under any group are defined and the theory is developed for semi-simple continuous groups. Coupled tensor operators are introduced and the matrix elements of tensor operators are expressed in terms of the coupling coefficients. The structure of generalized Casimir operators is investigated.
Eigenvalues and eigenfunctions of finite-difference operators
Authors: Bickley W. G. , McNamee John
Published: November 2008

Numerical solution of differential and integral equations is concerned mainly with the determination of the wanted function at a finite number of discrete points which are, in general, uniformly spaced. A first approximation to the solution can be obtained if the given differential or integral system is replaced by a difference system. Any differential or integral operator can be expressed as an infinite series of difference operators and the difference system of the first approximation is obtained by neglecting all but the first few terms of the infinite expansions. We shall distinguish two processes for improving the approximation: the first uses a tabular interval of moderate length but the approximation to the given continuous system is improved by including in the difference system as many terms of the infinite expansions as are necessary or practicable; the second process uses an unvarying difference system of low accuracy, but the tabular interval is reduced in successive approximations, the process being continued until two successive approximations agree within the accuracy required. We regard these processes as essentially distinct. If the solutions obtained by the two processes approach limits, these limits need not coincide.
Self reciprocal properties of certain functions
Authors: Rao V. V. L. N.
Published: November 2008

The object of this note is to study the properties of some functions self reciprocal in the Hankel transform. I denote a function f(x) as Rμ, if it is self reciprocal for Hankel transforms of order μ so that it is given by

where Jμ(x) is a Bessel function of order μ. If μ = ½, f(x) is denoted by Rs while f(x) is written as Rc when μ = − ½.
Energy considerations in the instability of a current-vortex sheet
Authors: Michael D. H.
Published: November 2008

In the normal procedure for establishing linearized stability criteria a mean steady state which we denote by U is perturbed by a small additional disturbance u in the Eulerian sense. In the linearization it is usual to regard the disturbance as resolved into Fourier components which are additive so far as the first order goes. The linearized problem will then determine a time amplification factor eiσr so that the linearized disturbance is of the form uexp{i(αx + σt)} and the whole motion U + uexp{i(αx + σt)}. As usual it is implied that the real part of complex functions such as uexp{i(αx + σt)} be taken. In cases of instability (imaginary part of σ negative), where the magnitude of the disturbance increases with the time, we are prompted to ask how energy is supplied to the disturbance. This question is especially relevant to many of the classical problems of fluid motion instability in which the systems are self-contained in the sense that energy is neither supplied nor withdrawn by an outside agency. If such a system is non-dissipative we expect that the total energy of the organized motion and the electromagnetic field, if present, shall be conserved, whilst such energy in a dissipative system will be gradually reduced by heat losses. In either case energy which is taken into a growing disturbance must be provided from the mean energy stored in the system in its initial state. One of the simplest cases of instability to examine from this point of view is the Helmholtz instability of a plane in viscid vortex sheet. This system is self-contained and conservative in the energy of the organized motion, and moreover the only form of energy with which we have to deal is kinetic energy. In the first part of this note we examine how disturbance kinetic energy is provided in this case by the break up of the organized motion of the mean stream. In the second part we consider what happens to the energy balance of a current-vortex sheet in the non-dissipative case. There are here two forms of energy, kinetic and magnetic, and it is shown that when the interface is unstable the overall magnetic energy increases at the expense of the kinetic energy.
On covering a regular polygon with a triangle
Authors: Eggleston H. G.
Published: November 2008

In this note we consider a particular type of covering problem. Let T be given a plane set and Tθ be the set obtained from T by a rotation about some point in the plane through an angle θ in the clockwise sense. If a set K is such that for every θ there is a translation which transforms Tθ into a subset of K then we say that K is a rotation cover of T. The problem considered here is to determine for fixed n, l the triangle of least area which is a rotation cover for an n-sided regular polygon of side length l. In each case the solution is an equilateral triangle the altitude of which is given by

This solution is unique.
Sequential Monte Carlo
Authors: Halton J. H.
Published: November 2008

This paper defines the concept of sequential Monte Carlo and outlines the principal modes of approach which may be expected to yield useful sequential processes. Three workable sequential processes, derived from a non-sequential method of J. von Neumann and S. M. Ulam for solving systems of linear algebraic equations, are described and analysed in detail.
On a Measure of Asymmetry of Convex Bodies
Authors: Asplund E. , Grosswald E. , Grünbaum B.
Published: November 2008

In the present note we discuss some properties of a ‘measure of asymmetry’ of convex bodies in n-dimensional Euclidean space. Various measures of asymmetry have been treated in the literature (see, for example, (1), (6); references to most of the relevant results may be found in (4)). The measure introduced here has the somewhat surprising property that for n ≥ 3 the n-simplex is not the most asymmetric convex body in En. It seems to be the only measure of asymmetry for which this fact is known.
A Matrix Factorization Problem in the Theory of Random Variables Defined on a Finite Markov Chain
Authors: Miller H. D.
Published: November 2008

Let {kr} (r = 0, 1, 2, …; 1 ≤ kr ≤ h) be a positively regular, finite Markov chain with transition matrix P = (pjk). For each possible transition j → k let gjk(x)(− ∞ ≤ x ≤ ∞) be a given distribution function. The sequence of random variables {ξr} is defined where ξr has the distribution gjk(x) if the rth transition takes the chain from state j to state k. It is supposed that each distribution gjk(x) admits a two-sided Laplace-Stieltjes transform in a real t-interval surrounding t = 0. Let P(t) denote the matrix {Pjkmjk(t)}. It is shown, using probability arguments, that I − sP(t) admits a Wiener-Hopf type of factorization in two ways for suitable values of s where the plus-factors are non-singular, bounded and have regular elements in a right half of the complex t-plane and the minus-factors have similar properties in an overlapping left half-plane (Theorem 1).
A Note on Characteristic Functions Which Vanish Identically in an Interval
Authors: Smith Walter L.
Published: November 2008

Some years ago, in connexion with some unpublished work in the theory of queues, the question arose as to whether the characteristic function of a non-negative random variable could vanish identically in an interval. The purpose of this note is to show that such a thing is impossible.
Copositive and completely positive quadratic forms
Authors: Hall Marshall , Newman Morris
Published: November 2008

A copositive quadratic form is a real form which is non-negative for non-negative arguments. A completely positive quadratic form is a real form which can be written as a sum of squares of non-negative real forms. The completely positive forms are basic in the study of block designs arising in combinatorial analysis (3). The copositive forms arise in the theory of inequalities and have been considered in a paper by Mordell (4) and two papers by Diananda(1, 2).
Planetary equations in terms of vectorial elements
Authors: Allan R. R. , Ward G. N.
Published: November 2008

The planetary equations for natural elements, which do not depend upon the choice of a particular frame of reference, are derived by using the theory of Poisson brackets. These natural elements include vectorial elements which lead to the introduction of vector and dyadic Poisson brackets. The use of vectorial elements can also lead to redundancy in the set of elements; this idea is extended to show that as many redundant elements as desired can be used and that the evaluation of the derivatives of the disturbing function can be simplified thereby.
Relative simplicial approximation
Authors: Zeeman E. C.
Published: November 2008

The absolute simplicial approximation theorem, which dates back to Alexander (l), states that there is a simplicial approximation g to any given continuous map f between two finite simplicial complexes (see for instance (2), p. 37 or (3), p. 86). The relative theorem given here permits us to leave f unchanged on any subcomplex, on which f happens to be already simplicial.
The development of a shock from standing waves of finite amplitude in an isentropic fluid
Authors: Weiss N. O.
Published: November 2008

Numerical integration of the partial differential equations shows that rigid boundaries promote the development of a shock from a finite-amplitude disturbance. A simple step-by-step method agrees with one that also utilizes the characteristics.
Distributions of the direction and collinearity factors in discriminant analysis
Authors: Kshirsagar A. M.
Published: November 2008

Exact tests devised by Bartlett and Williams for the adequacy of a single discriminant function in the case of several groups are applicable to a wide range of problems in multivariate analysis. The appropriate likelihood criterion is factorized into direction and collinearity factors; and the exact distribution of these factors under the conditions of only one non-zero canonical correlation was first given by Bartlett in the general case of any numbers of dependent and independent variables, making use of geometrical arguments.

In this paper the factors are expressed directly in terms of the matrices occurring in the multivariate analysis of variance table; and the results are derived by an analytical method making use of matrix transformations and rectangular coordinates.
A theorem on the continuity of homomorphisms of locally compact groups
Authors: Varopoulos N. Th.
Published: November 2008

0. Introduction. When f: T1 → T2 is a mapping between two topological spaces we shall say that f is sequentially continuous if and only if in T1 implies that in T2.
The formulation of quantum mechanics in terms of phase space functions
Authors: Fairlie D. B.
Published: November 2008

A relationship between the Hamiltonian of a system and its distribution function in phase space is sought which will guarantee that the average energy is the weighted mean of the Hamiltonian over phase space. This relationship is shown to imply the existence of a wave function satisfying the Schrödinger equation, and dictates the possible forms of time-dependence of the distribution function.
The stability of Couette flow of certain anisotropic fluids
Authors: Leslie F. M.
Published: November 2008

The stability of the flow between concentric, rotating cylinders is discussed when the gap is small and the cylinders are rotating in the same direction for a class of anisotropic fluids in which the fluid has a preferred direction. An important conclusion of the analysis is that a steady-state solution of the equations has previously been considered unstable on false grounds.
Short surface waves in the presence of a finite dock. I
Authors: Holford R. L.
Published: November 2008

A long thin plate of finite width (a finite dock) extends over part of the otherwise free surface of a semi-infinite body of water under gravity and is given a forced heaving motion of small amplitude about this position. The solution of the boundary-value problem for the velocity potential of the resulting forced (two-dimensional) surface-wave motion may be reduced to that of an integral equation of the second kind for the value of the potential on the dock by means of an appropriate Green's function. There is an infinite number of choices for such a Green's function, and we show below how to construct one for which the kernel of the corresponding integral equation tends to zero with the ratio wavelength/dock-width. The integral equation may then be solved by iteration to calculate the short-wave asymptotics of the original wave-making problem.
On matrix methods for the solution of partial differential equations
Authors: Wadsworth M. , Wragg A.
Published: November 2008

Bickley and McNamee (1) describe techniques for obtaining the solution of finite difference equations, arising from partial differential equations, making extensive use of matrix methods. In all cases solutions are obtained by solving algebraic equations as distinct from differential equations. For example, in order to solve

the second space derivative is replaced by finite differences and the time derivative is replaced either by substituting the backward finite difference form or by using the Laplace transformation.
Boundary problems for additive processes defined on a finite Markov chain
Authors: Keilson J. , Wishart D. M. G.
Published: November 2008

In a previous paper (3), to which this is a sequel, a central limit theorem was presented for the homogeneous additive processes defined on a finite Markov chain, a class of processes treated extensively by Miller (4). A typical homogeneous process {R(t), X(t)} takes its values in the space

and is described by a vector distribution F(x, t) with components

and an increment matrix distribution B(x) governing the transitions. The present paper treats the motion of the process in the same space when its homogeneity is modified by the presence of a set of boundary states in x. Such bounded processes have many applications to the theory of queues, dams, and inventories. Indeed, this paper and its predecessor were motivated initially by a desire to discuss queuing systems with many servers and many service phases. We will treat both absorbing boundaries and associated passage time densities, and reflecting boundaries. For the latter our main objective is an asymptotic discussion of the tails of the ergodic distribution.
Diffraction at high frequencies by a circular disc
Authors: Jones D. S.
Published: November 2008

It is shown how to convert the integral equation for the pressure distribution on a circular disc in an axisymmetric sound wave into a singular integral equation of the first kind. This singular integral equation has a simple kernel and can be transformed easily into an integral equation of the second kind which is especially useful at high frequencies. Although a direct iteration on this last integral equation fails, an indirect method is devised which ensures that each iterate is of lower order than the preceding at high frequencies. The form of the general term in the iteration is given together with the first terms of its asymptotic behaviour. It is relatively simple to estimate the error caused by stopping at any particular iterate.

Detailed calculations are made for a plane wave striking the disc at normal incidence.
On a transformation of use in multiple scattering problems
Authors: Millar R. F.
Published: November 2008

In a recent communication ((5)), Skal'skaya has considered the interesting two-dimensional problem of the diffraction of a plane E-polarized electromagnetic wave by two perfectly conducting half-planes, inclined with respect to each other. For simplicity, a symmetric case was studied; thus the current densities on the two half-planes were identical, and it was necessary to determine only one function. By expressing the scattered field in terms of the induced current density, and by imposing the (Dirichlet) boundary condition satisfied on the half-planes, Skal'skaya derived an integral equation of the first kind for the unknown density. This was transformed into an equation of the second kind (a more suitable starting point for iterative procedures) by an application of the Kontorovich–Lebedev integral transform. The free term in the new equation was the solution for an isolated half-plane and, by iteration, an approximate solution was obtained.
The reduced dual of a direct product of groups
Authors: Wulfsohn Aubrey
Published: November 2008

If one of the locally compact groups H and K is of type I, the reduced dual of their product is shown to be homeomorphic to the product of the reduced duals.
A spatially homogeneous gravitational field
Authors: Joseph V.
Published: November 2008

A solution of Einstein's vacuum field equations, apparently new, is exhibited. The metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V. The canonical form of the Riemann tensor, which is of Petrov Type I, is determined.
An extension of Banach's mapping theorem, with applications to problems concerning common representatives
Authors: Perfect Hazel , Pym J. S.
Published: November 2008

In this paper, we give an extension (Theorem 1) of the following well-known result of Banach ((l)): If X, Y are sets and Θ: X → Y, ψ: Y → X are injective mappings, then there exist partitions X = X1 ∪ X2, Y = Y1 ∪ Y2 such that Θ(X1) = Y1 and ψ(Y2) = X2. Here, as is usual, we say that X = X1 ∪ X2 is a partition of the set X = X1 ∩ X2 = π. Our theorem is applied in sections 2 and 3 to problems concerned with the existence of common representatives for two families of sets.
The analyticity of cross-product Bessel function zeros
Authors: Cochran James Alan
Published: November 2008

Introduction. In this paper we consider the two cross-product combinations of Bessel functions

where δ = (k− 1)zand (') denotes differentiation with respect to the argument. HereJνandYνdesignate respectively the Bessel functions of the first and second kind of order ν.
Supertranslations in flat space-time
Authors: Crampin M. , Foster J.
Published: November 2008

This paper contains the derivation of a metric for flat space-time associated with the supertranslations of the Bondi-Metzner group, and a description of the geometry of the null hypersurfaces on which the metric is based.
A periodicity theorem in homological algebra
Authors: Adams J. F.
Published: November 2008

Introduction. In (1–3,6) it is shown that homological algebra can be applied to stable homotopy-theory. In this application, we deal with A -modules, where A is the mod p Steenrod algebra. To obtain a concrete geometrical result by this method usually involves work of two distinct sorts. To illustrate this, we consider the spectral sequence of (1,2):

Here each group Extss, t which occurs in the E2 term can be effectively computed; the process is purely algebraic. However, no such effective method is given for computing the differentials dr in the spectral sequence, or for determining the group extensions by which is built up from the E∞ term; these are topological problems.
Insufficiency of the hyperspace
Authors: Isbell John R.
Published: November 2008

I asserted in (1), p. 35, that the topology of the hyperspace H(μX)of a uniform space μX determines the uniformity μ, and sketched a proof. D. Hammond Smith has shown (2) that the proof indicated is wrong, even for metric spaces. However, Smith proved the assertion for metric spaces and some others, and left open the question whether it was true in general. This note gives a counter example. I should note that I began with a transfinite construction based on Smith's example; whether that would have worked or not, the following construction is simpler.
A hypergeometric transformation associated with the Appell function F4
Authors: Srivastava H. M.
Published: November 2008

Recently in these proceedings (see (5), p. 175) by applying Goldstein's form of Parseval's relation ((4), p. 105) to the operational pairs ((3), p. 284)

where R[(b ± c)2] > 0 and ((3), p. 293)

where R(½ − λ + μ) > 0, it was proved that if

then

provided R(½ − λ + μ + ν) > 0 and R[(b ± c)2] > R(a), 0.
Lower bound energy calculations for
Authors: Walmsley Mary , Coulson C. A.
Published: November 2008

Two different calculations are made of lower bounds for the electronic energy of . In the first the method of truncated Hamiltonians due to Bazley and Fox is adapted in such a way that the nuclear charge rather than the energy becomes the eigenvalue. Lower bounds are calculated for the energies of the six lowest σg and six lowest σu states, as well as of the three lowest of both πg and πu symmetries. This approach gives better convergence than when the energy is used as eigenvalue. In the second calculation the method of Temple and Kato is shown to give a satisfactory value for the energy of the ground state, provided that some necessary knowledge of the energy of the first-excited state is available.
On Primitive Prime Factors of an-bn
Authors: Schinzel A.
Published: November 2008

Let a, b be relatively prime integers with |a| > |b| > 0. For any integer n > 0, let π n denote the nth cyclotomic polynomial, denned by

where ζn is a primitive nth root of unity.
The Monte Carlo method in quantum statistical mechanics
Authors: Handscomb D. C.
Published: November 2008

This paper describes, in general terms, a Monte Carlo method for estimating statistical parameters of quantum-mechanical systems. In this method, we construct a Markov chain of transitions between finite sequences of indices, and obtain these parameters in terms of parameters of the limit distribution. This is an extension of the method of Metropolis for classical systems, and may have equally wide application.
Note on an electrified circular disk situated inside a coaxial infinite hollow cylinder
Authors: Sneddon I. N.
Published: November 2008

In a recent paper Collins (l) has shown that Copson's method of solution of the problem of determining the electrostatic field due to an electrified disk (2) can be extended to determine the surface charge density induced on an electrified disk when it is situated inside an earthed coaxial infinitely long hollow cylinder. In Collins's solution the problem is reduced to that of solving a Fredholm integral equation of the second kind. The object of the present note is to show that Collins's solution can be obtained more simply by making use of a technique in the theory of dual integral equations developed recently by the author (3) and Copson(4).
Locally flatPLsubmanifolds with codimension two
Authors: Wall C. T. C.
Published: November 2008

We refer the reader to the IHES notes of Zeeman (14) for basic facts aboutPL(or piecewise-linear) manifolds. IfMmis a locally flatPL-submanifold ofQm+2, our object will be to study the normal structure ofMinQ: one of our main results is:

There exists a PL-bundle over M, with fibre a 2-simplex, which is PL-homeomorphic to a neighbourhood of M in Q; moreover, the bundle and homeomorphism are unique up to equivalence. We also make an application to smoothing theory.
On the convergence factor of a Fourier series
Authors: Mohanty R.
Published: November 2008

Definition A. Let 0 < λ0 < λ1 < λ2 < …, λn → ∞ and suppose thatis a given infinite series write

where λm ≤ ω > λm+1. Also write for k > 0,

.
Further results for the solutions of certain third and fourth order differential equations
Authors: Harrow Martin
Published: November 2008

In this paper we investigate certain generalized forms of third and fourth order non-homogeneous differential equations. Ezeilo (l) derived interesting results for the problem

.

We show similar results for:

,

and

.
A result on hypergeometric functions
Authors: Bhattacharya Samir Kumar
Published: November 2008

Summary. The main result of the present paper is contained in the following: Theorem ℜ α ≠ −1, −2, −3,…

where2F1[.,.;.;.] and 1f1[.;.;.] (l).
On the structure of causal spaces
Authors: Kronheimer E. H. , Penrose R.
Published: November 2008

The paper examines the structure obtained by abstracting from the conventional (manifold) representation of relativistic space-time the concept of an event-set equipped with two partial orderings, whose counterparts are the notions ‘causally precedes’ and ‘chronologically precedes in the history of some observer’.
A note on ideals of homological dimension one in local domains
Authors: Burch Lindsay
Published: November 2008

In this Note I shall give a characterization of ideals of homological dimension one in a local domain, and deduce a result on ideals in a regular local ring of dimension two.
A generalization of a boundedness theorem for a certain third-order differential equation
Authors: Ezeilo J. O. C.
Published: November 2008

1. This paper investigates the boundedness, as t → ∞, of the solutions of the real differential equation

where α is a constant and φ2, φ3, ψ depend on the arguments shown with φ2, φ′3, ψ continuous.
A direct iteration method of obtaining latent roots and vectors of a symmetric matrix
Authors: Jennings A.
Published: November 2008

This paper describes a method whereby iteration can be carried out simultaneously with two or more trial vectors in order to obtain the latent roots of largest modulus of a symmetric matrix and their corresponding latent vectors. The convergence of any particular vector is automatically accelerated by making use of the current approximations to those vectors subdominant to it, the dominant latent vectors being obtained quite rapidly. The method is particularly suitable where a few of the dominant latent roots and vectors are required of a large symmetric matrix.
A generalization of the Peter-Weyl theorem
Authors: Wood G. V.
Published: November 2008

The Peter-Weyl theorem states that for a compact topological group G, the set of finite dimensional (almost invariant) functions on G is uniformly dense in the set of continuous functions on G. In this paper, we consider the question: which subalgebras of C(G) inherit this property?
Systems of representatives with repetition
Authors: Mirsky L.
Published: November 2008

The part of combinatorial analysis which deals with systems of representatives draws on a variety of different sources. Thus the theory of graphs, the theory of flows in networks, linear programming, and integral linear programming all provide relevant methods and arguments. Nevertheless, it is in all likelihood possible to give a unified set-theoretic account of the subject, based in the main on Hall's criterion for the existence of transversals and on a certain generalization of the Schröder–Bernstein theorem. This approach has the added advantage of greater flexibility since it enables us sometimes to cope with transfinite situations where other methods may be only applicable to finite cases.
Oscillations of a rigid sphere embedded in an infinite elastic solid
Authors: Chadwick P. , Trowbridge E. A.
Published: November 2008

A detailed study is made of angular oscillations of small amplitude about a fixed axis of a rigid sphere embedded in an infinite elastic solid. Three modes of vibration of the sphere are considered: steady oscillations arising from the application of a periodic torque; forced oscillations produced by an arbitrary time-dependent torque; and free oscillations excited by an impulsive torque. Due to the transfer of energy to the surrounding material by the radiation of an elastic shear wave, free oscillations of the sphere are damped, the principal parameter affecting the damping being the density contrast between the sphere and its surroundings. Illustrative numerical results, referring to steady and free oscillations of the sphere, are presented in graphical form.
Oscillations of a rigid sphere embedded in an infinite elastic solid
Authors: Chadwick P. , Trowbridge E. A.
Published: November 2008

In this paper, which is a continuation of (1), we study steady (i.e. time-harmonic) and transient rectilinear oscillations of small amplitude of a rigid sphere embedded in an infinite elastic solid. Two types of transient motions are considered, forced oscillations in which the sphere is subject to a prescribed time-dependent force, and free oscillations in which the sphere is set in motion by an impulsive force. For each mode of vibration of the sphere the character of the solution is determined by two parameters, the density contrast between the sphere and its surroundings and a parameter related to the Poisson's ratio of the elastic solid. Numerical results referring to transient rectilinear oscillations are presented in graphical form.
Infinite sets of identities for classical polynomials and Bessel functions
Authors: Abdelkader Mostafa A.
Published: November 2008

An identity for a rational function is used, in conjunction with generating functions for a certain class of polynomials, to derive infinite sets of identities for these polynomials. Identities are given for the polynomials of Legendre, Gegenbauer, Hermite, Tchebycheff and L. Carlitz. A novel type of classification for these polynomials is indicated. Next, infinite sequences of identities for various Bessel functions and for power functions are derived. New identities for trigonometric functions are also given, as well as brief miscellanea on: functions resembling Dirac's delta function, integral transforms and a functional equation.
Some solutions of Einstein's equations in general relativity
Authors: Marek J. J. J.
Published: November 2008

A new class of axi-symmetric stationary solutions of Einstein's empty space field equations is obtained. Non-existence of solutions of certain other classes is proved.
Absolute summability of infinite series on a scale of Abel type summability methods
Authors: Mishra Babban Prasad
Published: November 2008

Suppose that λ > − 1 and that

It is easy to show that

With Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the series

is convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.
Finite approximations to infinite non-negative matrices, II: refinements and applications
Authors: Seneta E.
Published: November 2008

In a recent paper (1), the author has discussed how certain ergodic properties of a doubly infinite non-negative matrix T = {tij} are related to those of its finite (n × n) corner truncations, (n)T = {(n)tij}.
On the linear exponential family
Authors: Wani J. K.
Published: November 2008

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given by

where a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by
Subplanes of the Hughes plane of order 9
Authors: Denniston R. H. F.
Published: November 2008

The literature of finite projective planes consists largely of general investigations, taking in as many as possible of these systems at once. However, the geometry in a specific finite plane may well be an amusing, and not entirely trivial, field of study on its own. Some papers (5, 9, 13) have in fact appeared on the geometry of the translation plane of order 9: but the Hughes plane of the same order has comparatively been neglected. The object of the present paper is to make a beginning with the study of this plane from a synthetic point of view.
An algorithm for the construction of 3-manifolds from 2-complexes
Authors: Neuwirth L.
Published: November 2008

In order to study 3-manifolds with some particular group as fundamental group, one likes to have examples. There is an absence in the literature of means of constructing such examples. In this paper, theorems are proved which give necessary and sufficient conditions for the canonical 2-complex which corresponds to a group presentation to be a spine (2-dimensional skeleton in a cell decomposition with one 3-cell) of a connected closed† orientable 3-manifold. By enumerating all presentations of a group given by a particular presentation, two of the theorems provide an effective algorithm which permits one to attempt in a systematic way to construct 3-manifolds with a given group as fundamental group. Since every closed 3-manifold has a spine which corresponds to a group presentation, if the group is the fundamental group of an orientable 3-manifold, then the algorithm will eventually yield a 3-manifold, if not, then one is out of luck. There is no way out of this since Stallings has shown (1) that no algorithm exists which can, for every finitely presented group G, answer the question: Is G the fundamental group of a 3-manifold?
Cubic polynomials with the same residues (mod p)
Authors: McCann K. , Williams K. S.
Published: November 2008

Some recent work by the authors (1) on the distribution of the residues of a cubic polynomial modulo an odd prime p led to the conjecture that, in general, two cubic polynomials with integer coefficients possessing the same residues modulo p (not necessarily occurring to the same multiplicity) are equivalent, that is are related by a linear transformation modulo p. The purpose of the present paper is to prove this conjecture. We establish the following theorem.
Maxwell fields satisfying Huygens's principle
Authors: Künzle H. P.
Published: November 2008

It is shown that Huygens's principle holds for the solutions of Maxwell's equations for p-forms of all degrees in a gravitational plane wave space, while the solutions of the wave equation for 1, 2, and 3-forms, however, may have tails.
The expansion of water-wave potentials at great distances
Authors: Ursell F.
Published: November 2008

The wave function φ(x, y) satisfies the equation

in the infinite region

and satisfies the boundary condition Kφ + φy = 0 on the two line segments (y = 0, −∞ < x < −R) and (y = 0, R < x < ∞). We are not concerned with the properties of φ(x, y) inside the circle r = R. It is shown that outside the circle r = R the wave function φ(x, y) can be written as the sum of a wave source, a wave dipole, regular waves and wave-free potentials. (These basic functions are defined in section 2.) The proof makes use of analytic continuation into the image region (r > R, y ≤ 0) cut along the negative y-axis.

Two methods are given for this step. The first method uses Schwarz's symmetry principle to define the auxiliary wave function Φ = Kφ + φy throughout r > R, and φ can then be found by solving differential equations in the y or x direction. The second method uses Green's theorem. Applications in the theory of water waves are suggested.
Linearized plane strain of a mixture of two elastic solids. II
Authors: Steel T. R.
Published: November 2008

In a recent paper (1), we derived the solutions to the equilibrium equations for linearized plane strain of an isotropic mixture of two elastic solids. The solutions were given in terms of four complex potential functions. Here we examine the properties of these solutions and evaluate the forces and couples across an arc in the mixture.
Separation axioms for topological ordered spaces
Authors: McCartan S. D.
Published: November 2008

A topological ordered space (X, , <) is a set X endowed with both a topology and a partial order <, and is usually denoted by (X, ), it being understood that the symbol ≤ is used to denote all partial orders.
Generalization of Hölder's and Minkowski's inequalities
Authors: Daykin D. E. , Eliezer C. J.
Published: November 2008

In a recent paper (1) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfying

for every pair of unequal values x1 and x2). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a1, …,an; bl, …, bn, the function

which had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (σab)2 when x = 0 and (σa2) (Σb2) when x = 1, and the property of convexity gives Cauchy's inequality.
An integral involving the G-function and Kampé de Fériet function
Authors: Shanker O
Published: November 2008

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.
On stochastic theories of quantum mechanics
Authors: Gilson J. G.
Published: November 2008

Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
Steady longitudinal motion of an insulating cylinder in a conducting fluid
Authors: Waechter R. T.
Published: November 2008

A mathematical analogue between a special class of problems in steady magnetohydrodynamics and appropriate problems in the theory of the scattering of sound pulses is established and exploited. Rigorous asymptotic expansions for large Hartmann numberMare obtained for the exact solution of the prototype problem, the circular cylinder. Asymptotic expansions for largeMin the case of a cylinder whose cross-section is an arbitrary closed convex curve are obtained as a straightforward application of the theory of geometrical optics and Keller's geometrical theory of diffraction.
Semigroup laws in varieties of solvable groups
Authors: Lewin Jacques , Lewin Tekla
Published: November 2008

Let be a variety of groups. We say that the variety (the group G) has a semigroup law if u(x1, …, xn) = ν(x1, …, xn) is a law in (in G), with u and ν different words in the free semigroup freely generated by x1,…,xn. It seems very difficult to determine under what conditions a variety has a semigroup law. All we can say in general is that if does have a semigroup law, then it is in fact characterized by its semigroup laws (Section III).
On the general quantum theory of measurement
Authors: Fine Arthur I.
Published: November 2008

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.
The decay of oscillations of a non-homogeneous fluid within a container
Authors: Dore B. D.
Published: November 2008

The theory of laminar boundary layers is employed to determine the rate of decay of small amplitude oscillations of a non-homogeneous viscous fluid which completely fills a stationary rectangular container. An application of Green's formula is invoked and an expression is obtained in terms of the (general) density and viscosity distributions.
Steady rotation of a body of revolution in a conducting fluid
Authors: Waechter R. T.
Published: November 2008

We investigate the steady rotation of an insulating body of revolution in an unbounded electrically conducting fluid permeated by a uniform axial applied magnetic field. The assumptions of a small magnetic Reynolds number (Rm ≪ 1, i.e. the weakly conducting situation) and negligible inertia forces compared with the magnetic forces (R/M2 ≪ 1) permit us to suppress the inflow at the poles and outflow at the equator, which normally occurs for a non-conducting viscous fluid ((12), pp. 436–439). Thus in the case of the sphere, we find an exact solution of the reduced equations in terms of an infinite series of Legendre polynomials of order 1 with coefficients which are the ratios of modified spherical Bessel functions. This is the canonical problem by which results for arbitrary bodies of revolution are obtained.