bluuebunny/crossref_metadata_2025 huggingface dataset:
On better-quasi-ordering transfinite sequences
Authors: Nash-Williams C. St. J. A.
Published: November 2008

Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.
A parallel-plate waveguide, asymmetrically bifurcated by a unidirectionally conducting half screen
Authors: Clausert H.
Published: November 2008

In a recent paper the symmetrical bifurcation of a parallel-plate waveguide by a unidirectionally conducting half screen was considered. In the present paper the analysis is both extended to cover the more general case of an asymmetrical bifurcation, and simplified by using the straightforward version of the Wiener-Hopf technique due to D. S. Jones. Some numerical values are given for the modulus of the reflexion coefficient as a function of the position of the half screen between the parallel plates.
On the method for solution of unsteady thermal boundary-layers in case of two-dimensional low-speed flows
Authors: Ðurić Milan Ð.
Published: November 2008

This paper is dedicated to the question of solving unsteady thermal boundary-layers in the case of two-dimensional low-speed flows, provided that the difference between the temperature of the stream and that of the wall is not too great (so that the density is sensibly constant) and that the change in the wall temperature Tw (x, t) takes place at the same instant as the body is set into motion. The velocity boundary-layer is uncoupled from the thermal one and can be considered separately. In paper (2) is given the method for obtaining the velocity field (u, v). The temperature field T depends on the velocity field and can be obtained only after this one. The method (2) being available for this purpose assuming the difference between the temperature of the wall Tw (x, t) and that of the main-stream T∞ in the form of

where S(x) and θ(t) are arbitrary functions of x respectively t, satisfying certain conditions required by the method.
Rods, plates and shells
Authors: Green A. E. , Laws N. , Naghdi P. M.
Published: November 2008

We discuss non-linear thermodynamical theories of rods and shells using the three-dimensional theory of classical continuum mechanics as a starting point. The three-dimensional theory is reduced to a two-dimensional theory for a shell, or plate, and a one-dimensional theory for a rod by employing an exact expansion for the displacement but an approximation for the temperature. For elastic rods and shells a method of approximation is suggested which brings the respective theories into correspondence with those of Green and Laws (1) and Green, Naghdi and Wain-wright(2).
Certain recurrence relations
Authors: Pathan M. A.
Published: November 2008

Slater ((5), p. 27) and Bose ((1), p. 202) have obtained certain recurrence relations about Whittaker and hypergeometric functions. In this note an attempt has been made to obtain generalizations of these recurrence relations.
The expansion of a wedge of gas into a vacuum
Authors: Levine Lawrence Elliott
Published: November 2008

The front from the expansion into a vacuum of an infinite wedge of polytropic gas initially at rest at constant pressure is studied in detail. It is shown how the shape of that part of the gas-vacuum interface resulting from the presence of the sharp corner changes with the wedge angle and the adiabatic index of the gas. In addition to analytic results for wedges of arbitrary angle, numerical results substantiating the analysis are given for the special case of 90°. Finally, explicit formulae for the path of a particle initially part of a 60° wedge of water allowed to collapse at time t = 0 are derived from an exact linear solution obtained by Suchkov and interpreted through the hydraulic analogy for adiabatic index 2.
Stress concentration around a loaded bolt in an axially loaded bar
Authors: Gregory R. D
Published: November 2008

In Part I of the paper, a simple theory of infinite systems of linear equations of the typeL2is presented. In Part II, these results are applied to the elastostatic problem of determining the stress distribution in an axially loaded bar, held in position by a centrally placed circular bolt. Some numerical results are obtained and these are compared with experimental observation.
The oscillations of a viscous gravitating sphere
Authors: McKenzie D. P.
Published: November 2008

By using a Laplace transformation, a general solution is obtained to the problem of the oscillations and velocity field of a viscous gravitating sphere. Lamb's oscillating solution and Darwin's exponentially decaying solution are derived as asymptotic expressions and their connexion demonstrated. Closed loops of stream lines are a remarkable feature of the flow, and the conditions for their existence are discussed. Asymptotic solutions are also obtained for the oscillations of a Maxwell sphere, and their relation to those of an elastic sphere investigated.
The transport equation of radiative transfer with isotropic scattering
Authors: Hunt G. E.
Published: November 2008

The kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.
Surface waves in rotating liquids
Authors: Rao V. Subba
Published: November 2008

Using a technique developed by Lighthill, surface waves are studied when a concentrated pressure point oscillating with a constant frequency moves along OX in a rotating frame O X Y Z on the free surface of rotating liquid bounded below by a horizontal plane. The effect of rotation is to split the coincident gravity modes of the non-rotating case and further to produce a new system of wavelets (countably infinite) which do not have any counterpart in the non-rotating case.
Unitary representations of the unitary group
Authors: Hunter D. B.
Published: November 2008

It is well known that every representation of the group Un of unitary matrices of order n × n is equivalent to a unitary representation (see e.g. Little-wood (6), ch. XI). Our object in the present paper is to discuss some properties of those representations, and to construct a specific unitary representation.
Exact solutions for fields of electric type in spherically stratified isotropic media
Authors: Sharaf A. L.
Published: November 2008

A method is given for deriving refractive index profiles which yield wave functions expressed in terms of the standard transcendental functions for fields of electric type in spherically stratified isotropic media. Some new profiles, as well as all the previously known ones, are derived by applying the method.
Special relativity and thermodynamics
Authors: Georgiou A.
Published: November 2008

It is shown that a revision is needed of certain transformation equations of the Planck–Einstein formulae. In particular it will be shown that the formulae

must be replaced by

where T and Q are respectively the absolute temperature and heat of the system. Furthermore, it will be shown that both formulae for the energy

are correct, but they refer to different cases.
Summability factors for Riesz loǵarithmic means of order one for a Fourier series
Authors: Mohapatra R. N.
Published: November 2008

Let 0 < λ1 < λ2 < … < λn → ∞ (n→∞). We write

Let ∑an be a given infinite series with the sequence {sn} for its nth partial sum. The (R, λ, 1) mean of the sequence {sn} is given by
Conformal frames and field equations in a conformal theory of ǵravitation
Authors: Islam Jamal N.
Published: November 2008

Some aspects of the field equations of the conformal theory of gravitation put forward by Hoyle and Narlikar are studied. The field equations are conformally invariant and one can use a particular conformal frame to simplify the equations, since all conformal frames are regarded as physically equivalent. However, some conformal frames may be unsuitable in some regions of space-time, and with the use of such a frame one may get an unphysical solution. The use of conformal frames and the difficulties involved are illustrated by considering a given physical situation in two different conformal frames. The physical situation is that of two isolated particles. A static solution for this situation is obtained in both frames, and it is shown that a property that is quite unphysical in one frame transforms into a physically reasonable property in the other.
Time structures for abstract polysystems
Authors: Sullivan Hugh D.
Published: November 2008

1. Introduction. Various types of multiple-valued dynamical systems have appeared in the literature over the past twenty years (2). The motivations for such generalizations are varied, but among others, control theory plays a significant role. In a recent paper (2), Bushaw has introduced an abstract approach which consolidates a broad class of these dynamical systems. The model is a quasi-ordered set (E, ρ); i.e. Ε is a non-empty set and ρ is a reflexive and transitive relation on E. E represents the ‘event’ space of some real-world system and ρ represents some class of ‘admissible’ inputs on the system; i.e. (e1, e2) ∈ ρ if and only if there is an admissible input function which ‘shifts’ the system from event e1 to event e2. ρ is called the attainability relation on E and the statement ‘ an event e2 is attainable from an event e1 ’ means that (e1, e2) ∈ ρ. Adopting the terminology of (2), we shall refer to the quasi-ordered set (E, ρ) as a polysystem.
Two examples in varieties of monoids
Authors: Isbell John R.
Published: November 2008

This note exhibits an infinite independent set of laws for monoids (viz. (xpyp)2 = (ypxp)2, p prime) and a variety of groups defined by one law (which is x2y2 = y2x2) such that the smallest containing variety of monoids is not finitely definable.
On product sets in a unimodular group
Authors: McCrudden M.
Published: November 2008

It has been pointed out to me by Professor A. M. Macbeath that the proof of Theorem 2 which is given in (2) is incorrect, in that it relies on Proposition 3·1 of (2), which is false, the mistake in its proof being the assumption that in an arbitrary compact topological space, every sequence has a convergent subsequence. However, we can still prove Theorem 2 of (2), simply by replacing section 3 of (2) by the corrected section 3 below, and replacing the proof of Proposition 4·2 of (2) by the corrected proof of Proposition 4·2 given below.
On the singular counterpart of Titchmarsh's self-reciprocal series
Authors: Billington A. E.
Published: November 2008

A result due to Titchmarsh(1) is that corresponding to the series

is the reciprocal formula
The effects of strong suction on laminar boundary layers
Authors: Buckmaster J.
Published: November 2008

The title problem is investigated with the view of clarifying the nature of the separation in an adverse pressure gradient when the suction υ0 is very strong. A linear model is investigated, but it is argued that this must contain many of the essential features of the exact (non-linear) problem. With a free-stream U = 1 − x the similarity solution u = (1 − x)f(y) is appropriate when (υ0x − y) ≫ 1, x = O(1), 1−x = O(1). There is an O(1) layer centred on y = υ0x in which more complex x-dependence first appears and then, when (y − υ0x) ≫ 1, the solution assumes its final asymptotic behaviour. The y = υ0x layer is non-uniform in x and exhibits a marked change in behaviour when (1 − x;) = O(l/υ0). The solution in this narrow rectangle is singular like (1 − x)−2 at the trailing edge. It is these singular terms that induce separation and reversed flow in an exponentially small neighbourhood of the trailing-edge.
An iterative method to calculate the phases of Coulomb functions
Authors: Martins P. De A. P.
Published: November 2008

The differential equation

defining the function of the amplitude sL is integrated by an iterative procedure. An expression for the phase

is obtained in terms of integrals of ωL and its derivatives. The method is applied to the numerical calculation of Coulomb functions and the accuracy of the results is discussed. Recurrence relations are used to obtain øL, L ≽ 1.
Comments on the classical theory of magnetic monopoles
Authors: Miller D. T.
Published: November 2008

The classical theory of electromagnetism with both electric and magnetic charges is discussed. Following Gamblin(l) we proceed from an action principle and investigate the resulting particle equations. A consistent theory in which electrically and magnetically charged particles interact amongst themselves and with each other, without any constraints, is shown to be possible, in contrast to the conclusions of Gamblin. The introduction of constraints is considered.
A further note on axially symmetric flows of elastico-viscous liquids
Authors: Jones J. R.
Published: November 2008

In earlier papers of the same main title Jones and Lewis(1) (here-after referred to as I†) and Jones (2) studied, using ‘virtual body force’ methods, the axially symmetric flows of elastico-viscous liquids caused by the rotation of surfaces of revolution of arbitrary section. Particular attention was paid in I to the (oblate and prolate) ellipsoid geometry, but, for reasons of mathematical facility, the analysis was restricted to that of small eccentricity, the interesting and degenerate scheme of maximum eccentricity being outside its range of validity. It is the main purpose of the present note to broaden the scope of the analysis of I to cover a wider class of liquids and to present an exact analytical solution (in simple closed form) for the ellipsoid geometry.
An elastostatic circle theorem
Authors: Aderogba K.
Published: November 2008

An isotropic infinite plane containing a circular inhomogeneity is subjected to an arbitrary loading condition. Assuming that the unperturbed elastic field is known, it is proved that the disturbed Papkovich potentials are expressible in terms of the potentials for the homogeneous plane. This knowledge can save considerable labour in the actual construction of the elastic field for the composite plane. The results are applied to some particular problems which include that of an arc displacement discontinuity in the composite plane. This example has as yet not been solved explicity by other methods and reveals that the displacement field at the cavity surface is independent of the elastic constants.
Collapsing K × I
Authors: Webster Dallas E.
Published: November 2008

Zeeman(5) once bravely made the following conjecture:

Conjecture. Suppose K is a contractible 2-complex and I is the unit interval [0, 1]. Then K × I collapses.

This conjecture is interesting since it trivially implies the Poincaré conjecture. For if C is a homotopy 3-cell, then it has a contractible 2-dimensional spine, so that C × I would be a collapsible 4-manifold. But then C × I would be a 4-cell, with C = C × 0 in its 3-sphere boundary, and the PL Schoenfliess theorem shows that C is a 3-cell.
On the exterior problems of acoustics
Authors: Ursell F.
Published: November 2008

The method of integral equations is the most familiar method of proving existence theorems for the Helmholtz equation of acoustics. The wave potentials are expressed as surface distributions of wave sources (for the Neumann problem) or wave dipoles (for the Dirichlet problem). By a wave source is meant the free-space wave source. The source and dipole strengths for the exterior potentials are found to be solutions of Fredholm integral equations of the second kind which are, however, singular at a certain discrete set of frequencies corresponding to eigensolutions of the interior problems. The existence of exterior solutions at the expected frequencies can still be shown, but the proof involves a detailed and complicated study of the interior solutions. It is physically evident that this difficulty arises from the method of solution and not from the nature of the problem.
Oscillations of retarded differential equations
Authors: Sficas Y. G. , Staikos V. A.
Published: November 2008

In this paper we are dealing with the oscillatory and asymptotic behaviour of the nth order, (n> 1), retarded differential equation

wheregis differentiable on the half-line (t0,∞) with (I)g(t) ≤ tfor everyt ≥ t0(II)g'(t)≥ 0 for everyt≥t0(III)= ∞.
An extension of Lévy's result concerning indecomposability of the Wishart distribution
Authors: Shanbhag D. N.
Published: November 2008

Consider A and B to be two independent and non-degenerate random variables. It is conjectured by D. G. Kendall (private communication) that the distribution of the vector (A2, 2AB, B2) is indecomposable. The purpose of the present paper is to show that this is true.
Uniform compounds and the group
Authors: Smith Anthony
Published: November 2008

The object of this note is to draw attention to a family of eight uniform compounds of regular polyhedra which arise out of an action of the group on the regular icosahedron.
The static two body problem
Authors: Zum Hagen H. Müller
Published: November 2008

The problem of whether there exist static solutions of Einstein's equations for two extended bodies surrounded by vacuum is investigated. Equilibrium conditions for the general static situation are derived and their physical meaning discussed. Special cases of the problem are solved by using these equilibrium conditions. In one of these cases, the essential assumptions are that the space time is axisymmetric and that the matter of which the two bodies are composed consists of perfect fluid. Solu tions are also obtained in which these restrictions are weakened.
A characterization of M-Summands
Authors: Evans Richard
Published: November 2008

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection if

and an L-projection if, instead

A closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.
A note on the percolation probabilities of pairs of closely similar lattices
Authors: Sykes M. F. , Rehr J. J. , Glen Maureen
Published: November 2008

The percolation probabilities of the face-centred cubic and close-packed hexagonal lattices are found not to be identical; the identity of their critical percolation probabilities remains an open question. Other pairs of closely similar lattices are discussed.
Intrinsically characterized acceleration waves in heat-conducting elastic materials
Authors: Chadwick P. , Currie P. K.
Published: November 2008

This paper is concerned with a particular class of acceleration waves in heat-conducting elastic materials, namely those waves for which the acceleration amplitude is a proper vector of both the isothermal and isentropic acoustical tensors. Acceleration waves with this characteristic property are found to fall naturally into two distinct sub-classes, and close parallels are shown to exist between the members of the sub-classes and waves for which the acceleration amplitude is either longitudinal or transverse to the direction of propagation. Particular attention is given here to waves of the quasi-longitudinal type, the quasi-transverse variety having been studied in some detail elsewhere.
The completion of a chronological space
Authors: Woodhouse N. M. J.
Published: November 2008

The problem of completing the mathematical structures abstracted from the time ordering of the events in the space-times of general relativity is motivated and solved. The new construction is compared with that of Geroch, Kronheimer and Penrose (5).
Series solution of the biharmonic equation in the rectangular domain with some applications to mechanics
Authors: Vaughan H.
Published: November 2008

A double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.

The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.

In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.
On an integral transform associated with a condition of radiation, part 2
Authors: Naylor D.
Published: November 2008

1. In this paper, the author derives a formula of inversion for the integral transform

where a > 0, k > 0. The actual formula obtained appears as equation (8) of the present paper and is deduced from an integral formula constructed in (1).
A commutant property of symmetric ternary algebras
Authors: Sullivan W. G. , Joseph A.
Published: November 2008

A commutant property of ternary algebras, defined as linear subspaces of ℬ(ℋ) (bounded operators on some Hubert space ℋ) closed under three-fold multiplication, is demonstrated through the recovery of an operator from its non- orthogonal restrictions.
Thermostatics of an elastic Cosserat plate containing a circular hole
Authors: Gürgöze İ. T.
Published: November 2008

In this paper, the general theory of a Cosserat surface given by Green, Naghdi and Wainwright(1), has been applied to the problem of a thermo-elastic Cosserat plate containing a circular hole of radius a. We assume that the major surfaces of the plate and the boundary of the hole are thermally insulated and that a uniform temperature gradient τ exists at infinity. In the limiting case, when h/a → 0, where h is the thickness of the plate, the thermal stresses at the circular hole reduce to those obtained by Florence and Goodier (4), by means of the classical plate theory. Results for the other limiting case h/a → ∞ are also given.
Estimates of critical percolation probabilities for a set of two-dimensional lattices
Authors: Neal D. G.
Published: November 2008

This paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.
Analytical considerations of barrier transmission
Authors: Heading J.
Published: November 2008

The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.
A new physical characterization of stationary and static space-times
Authors: Müller zum Hagen H.
Published: November 2008

A space time is stationary if and only if it admits a congruence of time-like curves such that the distance between any two curves is time independent. The distance between two curves A and B, which are not necessarily infinitesimally close to each other, is defined as half the trip-time of a photon, which travels from A to B and back to A.
High frequency diffraction by a circular disc
Authors: Williams W. E.
Published: November 2008

The problem considered is that of transforming the integral equations obtained by Jones (2, 3), for diffraction by a circular disc so that they can be solved by iteration for any general incident field. A method of achieving this transformation is presented for the case of axi-symmetric incident fields. The generalization of the method to non-axi-symmetric problems is also examined.
Spectra of independent power measures
Authors: Bailey W. J. , Brown G. , Moran W.
Published: November 2008

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.
The simplicial extension theorem
Authors: Sanderson B. J.
Published: November 2008

Familiarity with the category of simplicial sets and the Milnor realization functor, as in e.g. (5), is assumed. We say f: |Y| → |Z| is realized if f = |g| for some g.
Some results on the decomposability of the distribution of quadratic expressions
Authors: Shanbhag D. N.
Published: November 2008

Recently Davidson (1), Kendall (3) and Shanbhag (7) have established that if U and V are independently distributed random variables such that U is non-degenerate and P(V = 0) < 1, then the distribution of (U2, 2UV, V2) is indecomposable. This implies that the distributions of (U2, 2UV, V2) G and (U2, U) H, where G and H are non-singular real square matrices, are indecomposable. As observed by Shanbhag (7), from this it follows that the Wishart distribution Wp(1, Σ, M) with both p and rank (Σ) ≥ 2 is indecomposable. This is an extension of a result of Lévy (5) who had shown that the distribution Wp(1, Σ, 0) with Σ diagonal is indecomposable.
On the superposition of gravitational waves
Authors: Griffiths J. B.
Published: November 2008

The nature of the non-linear interaction between two gravitational waves in the general theory of relativity is considered. A new exact solution of the vacuum field equations describing this case is given. It describes two gravitational waves with both transverse and longitudinal components, propagating in opposite directions along ‘shearing’ and ‘twisting’ geodesic congruences with zero contraction
A two turning-point problem in fluid mechanics
Authors: McKee W. D.
Published: November 2008

A uniformly valid asymptotic solution is found for waves refracted by a shearing current or by bottom topography in the case where two straight caustics are present. The ratio of the amplitude of the reflected wave to that of the incident wave increases monotonically from 1/√2, when the caustics are close together, to unity when they are widely separated. A slight modification of the theory gives an expression for the frequencies of waves trapped between two caustics.
On the monotone σ-envelope of A(K)
Authors: Edwards C. M.
Published: November 2008

Let K be a compact convex set regularly embedded in a locally convex Hausdorff topological vector space E. Let Ab(K) be the GM-space (6) with unit 1K consisting of all real valued bounded of affine functions on K and let A(K) be the GM-subspace of Ab(K) consisting of continuous functions. For properties of K and A(K), see (2). The monotone σ-envelope A(K)μ of A(K) is the smallest subset of Ab(K) containing A(K) and such that the pointwise limit of every uniformly bounded monotone sequence (an) in A(K)μ also lies in A(K)μ. In (1), Alfsen shows that when K is a metrizable simplex A(K)μ is a vector lattice. In this note it is shown conversely that if K is metrizable and A(K)μ is a vector lattice, then K is a simplex.
Dualizing complexes for commutative Noetherian rings
Authors: Sharp Rodney Y.
Published: November 2008

The theory of dualizing complexes of Grothendieck and Hartshorne ((5), chapter v) has turned out to be a useful tool even in commutative algebra. For instance, Peskine and Szpiro used dualizing complexes in their (partial) solution of Bass's conjecture concerning finitely-generated (f.-g.) modules of finite injective dimension over a Noetherian local ring ((7), chapitre I, §5); and the present author first obtained the results in (9) by using dualizing complexes.
Congruence properties of the binary partition function
Authors: Hirschhorn M. D. , Loxton J. H.
Published: November 2008

We denote by b(n) the number of binary partitions of n, that is the number of partitions of n as the sum of powers of 2. As usual, two partitions are not considered to be distinct if they differ only in the order of their summands. It is convenient to define b(0) = 1.
Proof of a conjecture about the distribution of divisors of integers in residue classes
Authors: Erdös P. , Hall R. R.
Published: November 2008

Let k be a positive integer and F(x, k) denote the number of integers n < x which have a divisor in every residue class prime to k. Erdös (1) proved that for every fixed ε > 0, we have F(x, k) ˜ x when

and conjectured the following result, which we prove in this paper.
Three notes on proximity theory
Authors: Ward A. J.
Published: November 2008

It seems worthwhile to point out that a few of the theorems in (2) can, as indeed one would expect in a paper of such length and scope, be somewhat generalized or simplified. This note uses the notation of (2) and all references are to that paper.
Azumaya algebras and rings with polynomial identity
Authors: Goldie Alfred
Published: November 2008

The Formanek polynomial gives an explicit expression for the calculation of central elements in a prime ring with polynomial identity, in such a form as to provide a set of linear maps of the ring into its centre. This note makes use of properties of these maps with the object of giving a direct proof of the remarkable theorem of M. Artin on the relationship between Azumaya algebras and rings with polynomial identities. We show that such a central map can be chosen to have some of the crucial properties of the reduced trace in matrices, notably that of non-degeneracy, and our proof of the theorem rests on these properties. The method can also be used for the study of prime PI rings in general; this is illustrated by giving new proofs of theorems of Formanek and Cauchon. The changes of proof essentially amount to the elimination of the need for a change of rings to obtain a split extension in simple algebras. The main theorem is the following.
Invariant properties of wave propagation in n-dimensional space
Authors: Heading John
Published: November 2008

The paper continues earlier work on the generalization in n-dimensional space of the differential equations governing the propagation of electromagnetic waves in inhomogeneous anisotropic ionospheric plasma. Here there are investigated certain properties of the equations and their solutions that are independent of the dimension n of the space in which propagation takes place. The isotropic case is discussed, together with a special anisotropic model for which the various planes of circular polarization are geometrically related to each other.
Finite distributive quasigroups
Authors: Smith J. D. H.
Published: November 2008

The Bruck-Slaby theorem on nilpotence of commutative Moufang loops may be combined with a theorem of Belousov to give a quick elementary proof of a stronger version of Fischer's Theorem on the solubility of right multiplication groups of finite distributive quasigroups.
Associative h-spaces and algebras
Authors: Rutter John W.
Published: November 2008

1. Ganea has shown (Theorem 2·2 of (2)) that h-cogroup structures on a CW space coincide, in the homotopy category, with coalgebra structures defined with respect to the cotriple determined by SΩ. The construction used by Ganea to prove this does not dualize. It seems possible moreover that the dual statement is false. In this note I show, using the results of (4), that, for suitable spaces, there is a surjection from the set of algebra structures defined with respect to the triple determined by ΩS to the set of h-group structures, and I construct a right inverse for this surjection.
Global stability of genetic systems governed by mutation and selection. II
Authors: Moran P. A. P.
Published: November 2008

An infinite genetic population of haploid particles is considered in which selection is controlled by a single locus at which there are an infinite number of possible alleles. These alleles are arranged in an infinite sequence and mutation occurs only to nearest neighbours. This is the ‘ladder model’ of Ohta and Kimura which was put forward as a possible explanation of the distributions of electromorphs in electrophoretic observations. Following an earlier paper, conditions are obtained on the selection coefficients which ensure that a stationary stable state exists. One such model is solved explicitly. The problem, important in evolutionary theory, of the rate of approach to such stationary states starting from some other state, is also discussed briefly.
On 2-dimensional homology classes of 4-manifolds
Authors: Akbulut Selman
Published: November 2008

Let Kτ be a framed knot in S3 representing the 4-manifold which is obtained by attaching a 2-handle onto B4 along K with the framing τ (see (1)). Tristram (2) proved the following non-imbedding theorem for the generator of ).
The static–geometric analogy in the equations of thin shell structures
Authors: Calladine C. R.
Published: November 2008

The ‘static-geometric analogy’ in thin shell structures is a formal correspondence between equilibrium equations on the one hand and geometric compatibility equations on the other. It is well known as a fact, but no satisfactory explanation of its basis has been given. The paper gives an explanation for the analogy, within the framework of shallow-shell theory. The explanation is facilitated by two innovations: (i) separation of the shell surface conceptually into separate stretching (S) and bending (B) surfaces; (ii) use of change of Gaussian curvature as a prime variable. Various limitations of the analogy are pointed out, and a scheme for numerical calculation which embodies the most useful features of the analogy is outlined.
A buckling model for the set of umbilic catastrophes
Authors: Thompson J. M. T. , Gaspar Z.
Published: November 2008

Zeeman has drawn attention to the sequences in which catastrophes, or modes of instability, can be linked, and it is a common observation that sequences of catastrophes of low order are always found in the environment of catastrophes of higher order. In this paper, a simple buckling model is presented that generates in an elegant manner a complete sequence of the umbilic catastrophes as represented by the semi-symmetric branching points. The scan of a single fundamental parameter of this model is shown to trace a route through all regimes of the umbilic bracelet, giving in turn the hyperbolic, symbolic, elliptic, parabolic and hyperbolic umbilic catastrophes.
Translation invariance, compact semigroups and injective coalgebras
Authors: Tonge Andrew M.
Published: November 2008

The lack of a satisfactory notion of the dual of a compact semigroup S precludes a simple concrete characterization of the closed translation invariant subspaces of C(S). In this article we develop an abstract theory of such spaces (satisfying a weak version of the Banach approximation property) in terms of ‘injective coalgebras’. We characterize their dual spaces as those Banach algebras whose closed unit balls are compact semigroups under the weak star topology. Injective Hopf algebras are shown to be the spaces C(G), G a compact group.
Two inequalities for the complete symmetric functions
Authors: Baston V. J.
Published: November 2008

In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequality

where ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).
On the existence of an analytic set meeting each compact set in a Borel set
Authors: Mathias A. R. D. , Ostaszewski A. J. , Talagrand M.
Published: November 2008

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.
A non-uniform rate of convergence in a local limit theorem
Authors: Basu Sujit K.
Published: November 2008

Let {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as
n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.
On triangle contractive operators in Hilbert spaces
Authors: Ang Dang Dinh , Hoa Le Hoan
Published: November 2008

Let H be a finite dimensional real or complex Hilbert space. We denote by Λ(x, y, z) the area of the triangle with vertices x, y, z ∈ H. A map f: H → H is triangle contractive TC if 0 < α < 1 and for each x, y, z ∈ H either

or

and

and

We prove that if f is TC either there is a fixed point w = f(w) or a fixed line L = ⊃ f(L) We characterize the f which are TC and continuous but have no fixed point.
Unconditional bases and martingales in LP(F)
Authors: Aldous D. J.
Published: November 2008

Here we describe our results and their background: terminology (mostly standard) is denned in Section 2. Throughout, F is a separable Banach space, 1 ≤ p < ∞ and Lp(F) is the space of measurable functions [0,1] → F with P-integrable norms. Given a ‘nice’ property P for Banach spaces, we may formulate the conjecture: Lp(F) satisfies P if and only if both F and Lp (= LP(ℝ)) satisfy P. This conjecture is known to be true for various specific properties, for example the Radon–Nikodym property ((4), section 5·4); reflexivity ((4), corollary 4·1·2); super-refiexivity ((12), proposition 1·2); B-convexity ((14), p. 200); and the properties of not containing copies of c0 (6) and l1 (13). The object of this paper is to demonstrate that the conjecture is false for the property of having an unconditional basis – this answers a question in (4).
The covariance algebra of an extended covariant system
Authors: Rousseau Ronny
Published: November 2008

Let M be a von Neumann algebra acting on a Hilbert space , and let G be a locally compact group. We consider an extension of G by , the unitary group of M. If the triple satisfies an additional axiom, we say that it is an extended covariant system. We define a Hilbert space and operators , acting on . The von Neumann algebra is then the covariance algebra of the extended covariant system , denoted by .
Closed braids which are not prime knots
Authors: Morton H. R.
Published: November 2008

An element B ∈ Bn, the braid group on n strings, which can be written as

in terms of the standard generators of Bn is called a split braid. It is easy to see that the resulting closed braid is the connected sum of the closed braids Ĉ and on k + 1 and n − k strings respectively (figure 1).
On the algebra of modular forms on a congruence subgroup
Authors: Scholl A. J.
Published: November 2008

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.
Bifurcations at a spherical hole in an infinite elastoplastic medium
Authors: Bassani J. L. , Durban D. , Hutchinson J. W.
Published: November 2008

The bifurcation problem of an infinite elastoplastic medium surrounding a spherical cavity and subjected to uniform radial tension or compression at infinity is studied. The material is assumed to be incompressible, and its behaviour is modelled by both hypoelastic (flow theory) and hyperelastic (deformation theory) constitutive relations. No bifurcation was found with the flow theory. Surface bifurcation modes were discovered with the deformation theory in both tension and compression. An independent study is also presented of surface bifurcations of a semi-infinite elastoplastic material under equi-biaxial stress. The critical strain for the half-space coincides with the strain at the spherical cavity at the lowest bifurcation.
An example in the theory of solubl groups
Authors: Hawkes T. O.
Published: November 2008

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we prove

PROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.
On the shifted eigenvalue problem
Authors: Bell R. J. , Dean P.
Published: November 2008

Lanczos(1, 2) has considered the shifted eigenvalue problem where C is a (p × q) matrix of rank r, CH is its hermitean conjugate and u, v are column vectors of orders p, q respectively. In this note we extend Lanczos' work to cover the more general eigenvalue problem which arises in certain problems in solid state physics (3). In (2), the I's are unit matrices of appropriate orders and the constants a, b are real; the partitioned matrix M is thus hermitean so that its eigenvalues λ are real.
On similarity solutions of the differential equation ψzzzz + ψx = 0
Authors: Gill A. E. , Smith R. K.
Published: November 2008

Solutions of the partial differential equation ψzzzz + ψx + 0 of the type ψ = x¼nw(y), where y = zx−¼ and n is an integer, are investigated. The equation occurs as a boundary-layer approximation in certain rotating and stratified fluid flows in which the production of vorticity (due, for example, to changes in the Coriolis parameter with latitude in two-dimensional flows on a beta plane, or by the buoyant generation of vorticity in a Boussinesq fluid) is opposed by a diffusive process. The similarity functions w(y) satisfy the fourth order differential equation

.

These functions have many properties analogous to those of error functions and parabolic cylinder functions. When n is a non-negative integer, there exist polynomial solutions of the latter equation. These have analogies with Hermite polynomials, although they do not form an orthogonal set in any useful sense. The main properties of the similarity functions are listed, including their series expansions, integral representations, asymptotic expansions and recurrence relations. In particular, a pair of independent solutions, Jon(y) and Kon(y), are denned such that Jon(y) and Kon(y) are real for real y and vanish at an exponential rate as y → + ∞. The function Jon(y) also decays exponentially as y → − ∞ if n is negative, and grows algebraically as y → − ∞ if n is positive. Curves of the functions Jon(y) and Kon(y) are given for |n| ≤ 3 and 0 < y < 6, and their more useful properties are listed.
The Torres conditions are insufficient
Authors: Hillman Jonathan A.
Published: November 2008

In this note we shall show that the conditions given by G. Torres in (7) do not suffice to characterize the first Alexander polynomial of a link. We shall recall below Torres' results (for the 2-component case) and state the result of J. H. Bailey on which our argument relies, and then prove the following theorem.
On Morita's p-adic gamma function
Authors: Barsky Daniel
Published: November 2008

Y. Morita proved that, for each prime number p, one can define a p-adic continuous function Γp(x) from p to p, interpolating the sequence

where m runs through the integers m prime to p with 1 ≤ m < n. Our aim is to show how this result is related to Dwork's result on the radius of convergence of
Lattice isomorphisms of Lie algebras
Authors: Towers D. A.
Published: November 2008

Let L, M be Lie algebras over the same field F and let ℒ(L), ℒ(M) denote their lattices of subalgebras. By an L-isomorphism (lattice isomorphism) of L onto M we mean an isomorphism θ: ℒ(L) → ℒ(M); that is, θ a bijective mapping from ℒ(L) onto ℒ(M) such that

and

for all subalgebras A, B of L (where A ∪ B denotes the subalgebra of L generated by A and B). We shall write A* for θ(A), the image of A ∈ ℒ(L) under an L-isomorphism from L onto M = L*. Then we may ask how closely related are L and L*. In particular, are such important concepts as semisimplicity, solvability, nilpotency and ideals preserved by L-isomorphisms?
An intrinsic conformal Lorentz pseudodistance
Authors: Markowitz Michael J
Published: November 2008

An intrinsic pseudodistance dM is associated to the underlying causal structure of a Lorentzian manifold (M, g). An infinitesimal form for dM is defined on the bundle of null vectors over M and is used to prove that, under certain physically reasonable energy conditions, M is conformally hyperbolic, i.e. that dM is a true distance. As a corollary we obtain sufficient conditions for the conformal hyperbolicity of a globally hyperbolic space-time. Some exact solutions of the Einstein field equations are discussed and possible applications to the study of singularities are outlined.