bluuebunny/crossref_metadata_2025 huggingface dataset:
On uniform interpolation sets
Authors: Earl J. P.
Published: November 2008

A well-known result in the interpolation theory of integral functions (see Whittaker (16, 17), Pólya (14), Iyer (2), Pfluger (10)) states that an integral function of at most type K < ½π of order 2 bounded at the lattice points m + in (m, n = 0, ± 1, ± 2, … ) is necessarily constant. That the value ½π cannot be increased is shown by the Weierstrass σ-function. The result has, however, been generalized in several ways; the lattice points being replaced by more general sets and the bounded-ness condition by one of restricted rate of growth.
The direct image of a vector bundle
Authors: Schwarzenberger R. L. E.
Published: November 2008

It is well known that if M is a vector bundle over a space Y, and r: X → Y is a map, then there is an ‘induced’ vector bundle r*M over X. It is also known that a vector bundle L over X determines a ‘direct image’ vector bundle r* L over Y provided that r is a finite covering map ((1), § 1).
Collapsing three-dimensional convex polyhedra
Authors: Chillingworth D. R. J.
Published: November 2008

If L is a subcomplex of a simplicial complex K, we say that L is obtained from K by an elementary simplical collapse if K − L consists of a simplex σ of some dimension d together with one ‘free’ face of σ, i.e. a face τ of dimension d − 1 which is a face of no other simplex of K except σ. Such a collapse is said to take place through σ from τ. If L can be obtained from K by a finite sequence of elementary simplicial collapses we say that K simplicially collapses (s-collapses) onto L, denoted by K ↘ L

If K is regarded as being embedded in some Euclidean space we shall for convenience of notation fail to distinguish between K and its underlying polyhedron.
Singular measures with absolutely continuous convolution squares.(Corrigendum)
Authors: Hewitt Edwin , Zuckerman Herbert S.
Published: November 2008

Mr Denis Lichtman has kindly pointed out to us that an added hypothesis is needed in Theorem (4·1). The revised theorem follows.
A Tauberian theorem for the logarithmic method of summation
Authors: Kwee B.
Published: November 2008

The series is said to be summable (L) to sif

where sn = a0 + a1 + … + an, and it is said to be summable (l) to s if
Monte Carlo estimates of critical percolation probabilities
Authors: Dean P. , Bird N. F.
Published: November 2008

In this paper we present results for the critical percolation probabilities of a number of two- and three-dimensional lattices. These results are based upon Monte Carlo studies of the way in which cluster-size distributions vary as the number of occupied sites in a lattice is progressively increased; the principle of the method has been described in some detail in an earlier publication (Dean (1)) in which the results of studies carried out on the ACE computer were reported. The use of a KDF 9 computer with a 32K word high-speed store has now enabled us to obtain results of a very high accuracy indeed; in those cases where our values for critical probabilities can be checked against exactly known values, they differ by no more than 0·1%.
Magneto-elastic coupling effects on the propagation of harmonic waves in an electrically conducting elastic plate
Authors: Purushotham C. M.
Published: November 2008

The influence of a uniform external magnetic field on the propagation of harmonic waves in a perfectly conducting elastic plate extending to infinity in vacuum, with its plane boundaries free of mechanical tractions, has been studied. With the plane of the field and the direction of wave propagation being normal to the plate, it is found that the effects of electromagnetic radiation into the adjoining free space cannot be neglected in the case of waves polarized in the plane. When the field is aligned with the direction of propagation, the maximum influence appears to be on long flexural waves, whereas if the field is normal to the plate then the effect is very much pronounced on long plate waves, and the flexural waves propagate almost with Alfvén velocity.
Central series in wreath products
Authors: Meldrum J. D. P.
Published: November 2008

In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.
Baer-invariants and extensions relative to a variety
Authors: Lue Abraham S.-T.
Published: November 2008

This paper examines the relationship between extensions in a variety and general extensions in the category of associative algebras. Our associative algebras are all unitary, over some fixed commutative ring Λ with identity, but while our discussion will be restricted to this category, it is clear that obvious analogues exist for groups, Lie algebras and Jordan algebras. (We use the notion of a bimultiplication of an associative algebra. In (2), Knopfmacher gives the definition of a bimultiplication in any variety of linear algebras.)
Non-embedding theorems for Y-spaces
Authors: Mayer K. H. , Schwarzenberger R. L. E.
Published: November 2008

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).
On topological sequence spaces
Authors: Garling D. J. H.
Published: November 2008

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called asequence space. A subsetAof ω issolidif wheneverx∈Aand |yi| ≤ |xi| for eachi, theny∈A. The theory of solid sequence spaces, topologized in a variety of ways, has been developed in considerable detail, in particular by Köthe and Toeplitz (13) and subsequently by Köthe (see, for example (12)). These results have been generalized to function spaces by Dieudonné(6), to vector-valued sequence spaces by Pietsch (18), to vector spaces with a Boolean algebra of projections by Cooper ((4), (5)), and in the real case, to partially-ordered spaces by Luxemburg and Zaanen (see, for example (14)) and Fremlin (8). This last generalization shows that many of the properties of solid sequence spaces depend upon their order structure, rather than upon their structure as sequence spaces.
Relations between arithmetical functions
Authors: Evelyn C. J. A.
Published: November 2008

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and if

then

where T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that if

where

then

Thus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.
On the slow rotation of axisymmetric solids in magnetohydrodynamics
Authors: Shail R.
Published: November 2008

The motion of a solid in a viscous conducting fluid permeated by a magnetic field has been the subject of a number of papers. Using the Stokes approximation Chester (3) first derived an expansion in powers of the Hartmann number for the drag on a sphere translating uniformly through a fluid, parallel to the applied magnetic field. In a subsequent paper (4) the same author treated the high Hartmann number limit. Later, Chang (2) considered, in the low Hartmann number regime, the motion of an axisymmetric body translating slowly along the axis of a fluid filled tube, and was able to calculate a drag formula which includes the lowest order wall effect term. An alternative treatment of this class of problems has recently been given by Williams (13), using an intergral equation formulation.
Démonstration du théorème sur la suite spectrale des fibrés au sens de Kan
Authors: Lusztig Gheorghe , Moscovici Henri
Published: November 2008

We give a relatively simple proof of the theorem on the spectral sequence of a fibre-space in the sense of Kan. The method consists in the reduction to locally trivial fibre-spaces by minimal fibrings and in the use of a very simple acyclic model theory (without degeneracies).
Nilpotent groups and compact 3-manifolds
Authors: Thomas Charles
Published: November 2008

The purpose of this paper is to give a complete list of those nilpotent groups which can be the fundamental groups of connected, closed, compact but possibly non-orientable 3-manifolds. The starting point is the following theorem of Reidmeister, which is given a neat proof in (1).
On integrals involving classical polynomials
Authors: Dhawan G. K.
Published: November 2008

We know ((6), p. 343) that

and

where α,β,α′ and β′ are parameters such that

Consider
Certain transformations of bilateral cognate trigonometrical series of the hypergeometric type
Authors: Denis R. Y.
Published: November 2008

Recently (2) I gave certain transformations involving cognate trigonometrical series of the hypergeometric type. The arguments of the ordinary bilateral hypergeometric series involved were unity. In this paper, using the same technique applied in the above paper some general transformations have been established which further generalize the previous results.
An improved lower bound for the multidimensional dimer problem
Authors: Hammersley J. M.
Published: November 2008

The dimer problem, which in the three-dimensional case is one of the classical unsolved problems of solid-state chemistry, can be formulated mathematically as follows. We define a brick to be a d-dimensional (d ≥ 2) rectangular parallelopiped with sides whose lengths are integers. An n-brick is a brick whose volume is n; and a dimer is a 2-brick. The problem is to determine the number of ways of dissecting an n-brick into dimers; and since this is only possible when n is even we confine attention hereafter to n-bricks with n even. Consider an n-brick with sides of length a1, a2, …, ad, where n = a1a2 … ad, and write a = (a1, a2, …, ad). Let fa denote the number of ways of dissecting this brick into ½n dimers. On the basis of physical and heuristic arguments chemists have known for many years that fa increases more or less exponentially with n; and recently a rigorous proof (1) of this fact has been given in the following form: if ai → ∞ for all i = 1, 2, …, d, then n−1 logfa tends to a finite limit, which we denote by λd. The principal outstanding problem for chemists is to determine the numerical value of λ3, or failing an exact determination to estimate λ3 or to find upper and lower bounds for it.
On a recent theorem of Carleson. II
Authors: Yadav B. S.
Published: November 2008

The long open question regarding the almost everywhere convergence of the Fourier series of an L2-function having been recently settled by Carleson (1), an interesting problem still awaits solution. It is to investigate whether the Fourier series of a function of class Lp (1 < p < 2), converges (or diverges) almost everywhere; for it is already known that the Fourier series of an integrable function may diverge almost everywhere. See Kolmogorov (3) and Marcinkiewicz (4) or ((8), 1, p. 305), see also Kahane(2). The object of this paper is simply to show that a suitable restriction imposed on the modulus of smoothness of an integrable function guarantees the almost everywhere convergence of its Fourier series. This will establish, in a more general form, the author's conjecture made in (5). Our proof is based on Carleson's theorem according to which the Fourier series of an L2-function converges almost everywhere.
On the general correspondence between field theories and the theories of direct interparticle action
Authors: Narlikar J. V.
Published: November 2008

It is well known that classical electrodynamics can be described both as a field theory and as a theory of direct interparticle action. In the present paper it is shown that, provided certain general conditions are satisfied, fields of arbitrary spin have their counterparts in ‘direct particle fields’. This correspondence between the two formalisms is established in the Riemannian space-time used for general relativity.
Existence theorems for H-space inverses
Authors: Sibson Robin
Published: November 2008

We define an unbased H-space to be a pair (A, m) where A is a space and

is a map such that the maps La: x → m(a, x), Ra: x → m(x, a) are homotopy equivalences for all a ∈ A. This is the same as James's definition of an H'-space in (3); we follow his notation as far as possible. (A, m) is homotopy-associative if the maps m(m × 1) and m(l × m) are homotopic. A left (right) a-inverse (a ∈ A) is a map w: A → A such that the composition m(w × 1) d (w(1 × w) d) is homotopic to ka, the constant map to a. d denotes the diagonal map a → (a, a).
On the Brunn-Minkowski coefficient of a locally compact unimodular group
Authors: McCrudden M.
Published: November 2008

Let G be a locally compact topological group, and let μ be the left Haar measure on G, with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B (G) the Borel subsets of G, and V = {μ(C):C ∈ B(G)}, then we can define ΦG: V × V → R' by

where AB denotes the product set of A and B in G. Then clearly

so that a knowledge of ΦG will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.
The boundary layer on a paraboloid of revolution
Authors: Miller D. R.
Published: November 2008

An analytical study is made of the boundary-layer approximation for the steady laminar flow of a viscous incompressible fluid past a paraboloid of revolution. The functional form of the vorticity distribution is found and shown to involve exponential decay with distance from the surface of the body, and inequalities are established concerning the displacement thickness.

The exact results are compared with various linearizations. The Oseen linearization (12) turns out to be incorrect except in the limit of vanishing Reynolds number; this result is connected with the infinite drag of any paraboloid of non-zero thickness. An alternate linearization, related to a method of Burgers (3) and allowing for the displacement effect of the body and of its boundary layer, is shown to give the correct functional form for these quantities. Numerical results are included which bear out these conclusions.
Some sets with measurable inverses
Authors: Davies Roy O.
Published: November 2008

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.
Isotopy classes of embeddings and homotopy
Authors: Wells G. S.
Published: November 2008

Haefliger's theorem (3) that the group of isotopy classes of embeddings of Sx in Sn when n − x > 2 is isomorphic to the triad group πx+1(G; G(n − x),SO), where G(n − x) is the H-space of homotopy equivalences of Sn−x−1 of degree 1, , and SO is the stable special orthogonal group, is generalized in this paper by replacing Sx and Sn by arbitrary compact connected smooth manifolds Xx, Mn without boundary. The embedding and knot problems are reduced to homotopy theory. The question of P.L. manifolds is discussed in section 4. The case Mn = Sn will be considered first; the generalization is stated in section 3.
Expansions for Kν(ξ) involving Airy's function
Authors: Donaldson James A.
Published: November 2008

In this note it is shown that the modified Bessel function of arbitrary order may be represented as the sum of two products where one product involves Airy's function and the other product involves the first-order derivative of Airy's function.
Proof of the uniqueness of the electromagnetic energy momentum tensor
Authors: Collinson C. D.
Published: November 2008

An alternative to Fock's proof of the uniqueness of the electromagnetic energy momentum tensor is presented. The proof is four-dimensional and is applicable in the curved space-time of general relativity.
Continuous data dependence for the equations of classical elastodynamics
Authors: Knops R. J. , Payne L. E.
Published: November 2008

Consider a linear elastic solid occupying a bounded regular three-dimensional region B with smooth surface ∂B. The components of displacement ui referred to cartesian axes xi are then well known to satisfy the system of governing equations

in which t denotes the time variable, x = (x1, x2, x3) denotes the position vector, ρ(x) is the non-homogeneous density, assumed positive,. i(x, t) are the Cartesian components of body force per unit mass, and cijkl(x) are the non-homogeneous elasticities, which apart from certain smoothness conditions stated later, are assumed to possess the symmetry

Throughout this paper, all suffixes range over the values 1, 2, 3 and the usual converition of summing over repeated indices is adopted. Except where it is in the interest of clarity we avoid explicit mention of the dependence of functions on their arguments.
On some applications of a theorem of Browder–Levine–Livesay
Authors: Swarup G. A. , Varadarajan K.
Published: November 2008

The main result proved in this paper can be stated as follows:

Theorem. Let Mnbe a closed l-connected topological manifold of dimension n ≥ 5. Then M × R carries a differentiable (respectively aPL) structure if and only if M × S1carries a differentiable (respectively aPL) structure.
Essential coverings of matrices
Authors: Lewin M.
Published: November 2008

Let G be a matrix. If there does not exist an infinite set of non-zero elements such that no two are on the same line (row or column), the matrix is said to be of finite term-rank. A set of lines such that each non-zero element belongs to some line of the set is said to be a covering of G.
Almost hypoelliptic differential operators with variable coefficients
Authors: Elliott Robert J.
Published: November 2008

We have called a linear partial differential operator P almost hypoelliptic in some of the variables if the distribution u is strongly regular in those variables in every open subset where this is true of Pu. In an earlier paper (l) we have characterised almost hypoelliptic operators with constant coefficients. Following techniques of Hörmander (2) and Yoshikawa(5) we obtain in this paper some a priori inequalities for almost hypoelliptic operators and derive results about the solvability of the adjoint operator P*. Our notation for various standard distribution spaces and multi-indices follows Hörmander (2).
The equality of the moduli of certain ratios occurring in the connexion formulae of solutions of some transcendental differential equations
Authors: Heading J.
Published: November 2008

The differential equations governing the propagation of waves of electric and magnetic types in a plane stratified isotropic plasma are suitably generalized, and we investigate the possibility of models for which the moduli of the reflexion coefficients are identical for the two modes. First, the models are examined without the necessity of finding general solutions, and, secondly, by using the circuit relations for the hypergeometric functions occurring in the explicit solutions.
Linked-cluster expansions for the correlation functions of lattice systems
Authors: Essam John W.
Published: November 2008

A method of deriving power series expansions for the correlation functions of lattice systems is described. The concept of lattice constants for rooted graphs is introduced and it is shown how the correlation functions can be expanded in terms of the lattice constants for connected graphs only. The weight functions in the expansions depend on correlation functions for finite systems and may, therefore, be determined by computer methods. This work is an extension of a method previously developed for the free energy. Applications to spin systems, percolation problems and the lattice gas are considered and sum rules for the weight functions are given.
Integer points on curves of genus 1
Authors: Baker A. , Coates J.
Published: November 2008

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.
On subrings of matrix rings over fields
Authors: Björk J. -E.
Published: November 2008

Let K be a commutative field and let V be an n-dimensional vector space over K. We denote by L(V) the ring of all K-linear endomorphisms of V into itself. A subring of L(V) is always assumed to contain the unit element of L(V), but it need not be a vector subspace of the K-algebra L(V). Suppose now that A is a subring of L(V). Then we may consider L(V) as a left or a right A-module. We may also consider V as a left A-module, i.e. if x ∈ V and if f ∈ A then we get the image element f(x) in V. If we choose a basis for V over K then we get the associated matrix representation of L(V). So in this way L(V) is identified with the full matrix ring Mn(K), and in this way the study of subrings of L(V) can be reduced to the study of subrings of Mn(K).
On polynomial groups
Authors: Vermani L. R.
Published: November 2008

If M is a group, Z(M) its integral group ring and AM the augmentation ideal, then following Passi we can form the Abelian groups
On the homotopy-symmetry of sphere bundles
Authors: James I. M.
Published: November 2008

1. Introduction: Let B be a space and let E be an orthogonal (m + 1)-sphere bundle over B with projection p: E → B. Consider the fibre-preserving map T: E → E which is given by the antipodal transformation in each of the fibres. Following (4) we say that E is homotopy-symmetric if there exists a fibre-preserving homotopy of T into the identity map. Since T maps the fibres with degree (– 1)m this condition requires m to be even.
Maximum-likelihood estimators with known incidental parameters
Authors: Moran P. A. P.
Published: November 2008

Suppose we have a sample ofNindependent random variablesX1, …,XNwhereXihas the distributionF(X|θ, øi). θ is ak-dimensional ‘structural’ parameter (θ(1), …, θ(k)), and the øiare scalar or vector ‘incidental’ parameters in some given space. TheXimay be scalar or vector random variables which are either discrete in which case we writef(X|θ, øi) for the probability associated with a given point, or else continuous random variables with a probability densityf(X|θ, øi). In either case we sup pose the support of the probability distribution to be fixed. We aim to estimate the true value of θ by maximum-likelihood methods.
On isotropic tensors
Authors: Jeffreys Harold
Published: November 2008

1. An isotropic tensor is one the values of whose components are unaltered by any rotation of rectangular axes (with metric σi(dxi)2). Those up to order 4 in 2 and 3 dimensions have many applications. The results suggest a general theorem for tensors of order m in n dimensions, that any isotropic tensor can be expressed as a linear combination of products of δ and є tensors, where δij = 1 if i = j and 0 otherwise, and is 0 if any two of the i1 to in are equal, 1 if i1…in is an even permutation of 1, 2, 3, …,n, and – 1 if it is an odd permutation.
On the series solution to the laminar boundary layer with stationary origin on a continuous, moving porous surface
Authors: Samuel T. D. M. A. , Hall I. M.
Published: November 2008

The similarity solution for a laminar boundary layer on a continuous, moving, porous surface is obtained using a series of exponential terms. The convergence of this series which was previously used by Ackroyd (1) is investigated by a technique due to Weyl (4) and it is summed by a novel technique.
Hausdorff measure functions
Authors: Goodey P. R.
Published: November 2008

In many contributions to the theory of Hausdorff measures, it has been the practice to place certain restrictions on the measure functions used. These restrictions usually ensure both the monotonicity and the continuity of the functions. The purpose of this work is to find conditions under which the restrictions of monotonicity and continuity may be relaxed. We shall be working mostly with sets in Euclidean space, although in the final theorem, we work in Hilbert space to show that some of our previous results do not generalize to this space. I am grateful to the referee for suggesting improvements in the proofs of the results.
Exact and approximate methods for the investigation of the propagation of waves through a system of barriers
Authors: Heading J.
Published: November 2008

There is developed the theory of the propagation of waves through a system of potential barriers. Many properties are derived and discussed, mainly by the use of matrix theory arising from the description of each barrier by a transmission matrix. The extent to which certain previously used approximate methods are applicable and valid is also examined. Two distinct kinds of resonant phenomena are discussed – first when there is no incident wave and secondly when there is no reflected wave. This latter case of perfect transmission is investigated in detail, an examination being given of both why and when it occurs and the time taken to establish such exceptional fields within the barriers.
On groups and 3-manifolds with weak dimension ≤ 1
Authors: Galewski David E.
Published: November 2008

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.
Subgroups of the modular group
Authors: Stothers W. W.
Published: November 2008

The modular group, Γ, is the group of linear fractional transformations with integral coefficients and determinant 1. The group is generated by the transformations

which have orders 2 and 3 respectively. The transformation

is of infinite order. Abstractly, the group can be viewed as the free product of cyclic groups of order 2 and 3.
Continuous images of proper analytic and proper Borel spaces
Authors: Jayne J. E.
Published: November 2008

All topological spaces considered will be completely regular Hausdorff spaces. The word space will be used to refer to such a topological space.
Atomic rings and the ascending chain condition for principal ideals
Authors: Grams Anne
Published: November 2008

LetRbe a commutative ring. We say thatRsatisfies theascending chain condition for principal ideals, or thatRhasproperty(M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals ofRterminates. Property (M) is equivalent to themaximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals ofRhas a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for ifRhas property (M) and if {Xλ} is a set of indeterminates overR, then the polynomial ringR[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).
On the K-theory of the loop space of a Lie group
Authors: Clarke Francis
Published: November 2008

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).
Schauder bases and decompositions in locally convex spaces
Authors: Webb J. H.
Published: November 2008

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j where

for each x ∈ E. The partial summation operator Sn, defined by

is a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined by

and 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.
Potential problems involving an annulus
Authors: Clements D. L. , Love E. R.
Published: November 2008

In electrostatic, elastostatic and hydrodynamic problems involving a plane annulus in space, two mixed boundary value potential problems are frequently encountered. For reasons given below, we consider here the axisymmetric cases of them; and we describe them using cylindrical polar coordinates (r, θ, z) so chosen that the annulus is a < r < b, z = 0.
Infinitesimal bundles and projective relativity
Authors: Evans Geoffrey
Published: November 2008

This paper derives some sufficient conditions for the global validity of projective relativity theory. The basic object of study is the infinitesimal bundle, which is a generalization of a principal circle bundle. A method ofconstructing infinitesimal bundles is developed. The main result is that ifspacetime is non-compact and simply connected, and has a finite homology basis, then projective relativity is globally valid.
Spectral asymmetry and Riemannian Geometry. I
Authors: Atiyah M. F. , Patodi V. K. , Singer I. M.
Published: November 2008

1. Introduction. The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula:

where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X. In particular if, near the boundary, X is isometric to the product Y x R+, the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H2(X, R) by an integral formula

where p1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p1 = (2π)−2Tr R2. It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general
J-Equivalence of group representations
Authors: Snaith V. P.
Published: November 2008

If G is a finite group of order N and ΓN is the Galois group of Q(w) over Q, where w is a primitive Nth root of unity then ΓN acts on the complex representation ring, R(G), of G. The group of co-invariants is denoted by R(G)ΓN = R(G)/W(G).
The propagation of Rayleigh waves over curved surfaces at high frequency
Authors: Gregory R. D.
Published: November 2008

A formal asymptotic theory, valid at high frequencies, is developed for the propagation of time harmonic Rayleigh surface waves over the general smooth free surface Σ of a homogeneous elastic solid. It is shown that on Σ these Rayleigh waves can be described by a system of surface rays, which are shown to be geodesics of Σ. The amplitude of the waves on Σ is shown to vary in such a way that the energy propagated along a strip of surface rays is constant. The waves are also shown to be dispersive and an explicit first-order dispersion formula is derived.
Some results on the union of Helson sets
Authors: Galanis E.
Published: November 2008

Let G be a compact abelian group and let Ĝ be its dual group. We shall denote by M(G) the set of all bounded complex valued Radon measures on G and by M(E) the elements of M(G) with support in a compact subset E of G.
A further note on the slow rotation of axisymmetric solids in magnetohydrodynamics
Authors: Shail R.
Published: November 2008

In a recent paper (3) I discussed the slow steady rotation of an insulated axisymmetric solid in a bounded viscous fluid of finite conductivity, there being a uniform magnetic field applied parallel to the axis of rotation. By reducing the small Hartmann number problem to a set of potential type Fredholm integral equations of the first kind, expressions were derived for the frictional couple G on the solid in a number of different geometrical situations. The formulae obtained were in the form of power series in the Hartmann number M together with the lowest order wall correction, the zero-order term in G being the frictional couple G0 for the non-magnetic infinite-fluid problem.
Diffeomorphisms of some smooth metastably connected manifolds
Authors: Hodgson J. P. E.
Published: November 2008

Let Mm be a closed connected smooth manifold of dimension m, and set Pm = Mm – int Dm where Dm is a disc in M. In (4), Wall has the following exact sequence

where (M), (resp. (P)) is the Δ-set of diffeomorphisms of M (resp. P) as in(l), and [M/P] is the set of diffeomorphism classes of smooth manifolds obtained by glueing a disc to the boundary of P. In this paper we obtain some results on π0((M)) for particular M, and in the following sense: Using the techniques of (2), we can determine π0((P)), so the main portion of the paper is concerned with a discussion of the kernel of α. There is a map ω: π1((P)) → Γm+1, given by the obstruction to extending h∈π1((P)) to a concordance of the identity of M to itself, and it is clear that if this map is attached at the beginning of the sequence (*) we get exactness. We will obtain (for certain M) an alternative description of the image of α in terms of those homotopy spheres which can appear as the boundaries of thickenings.
On characterization of certain probability distributions
Authors: Wang Y. H.
Published: November 2008

Introduction: LetX1, X2, …, Xnbe n (n≤ 2) independent observations on a random variableXwith distribution functionF. Also letL=L(X1, X2, …, Xn)be a linear statistic and Q = Q(X1, X2, …, Xn)be a homogeneous quadratic statistic. In this paper, we consider the problem of characterizing a class of probability distributions by the linear regression of the statisticQon the other statisticL. In section 2, we obtain a characterization of a class of probability distributions, which includes the normal and the Poisson distributions. In section 3, a class of distributions including the gamma, the binomial and the negative binomial distributions is characterized.
The numerical solution of linear stiff differential equations
Authors: Cash J. R.
Published: November 2008

A general method is given and illustrated by application to particular cases for obtaining subdominant solutions of stiff difference and differential equations, i.e. when rapidly varying solutions – transients or otherwise – are possible but are in fact excluded by the initial conditions.
Fractional parts of powers of rationals
Authors: Baker A. , Coates J.
Published: November 2008

Mahler (5) proved in 1957 that for any rational a/q, where a, q are relatively prime integers with a > q ≥ 2, and any ε > 0, there exist only finitely many positive integers n such that ∥(a/q)n∥ < e−εn; here ∥x∥ denotes the distance of x from the nearest integer taken positively. In particular there exist only finitely many n such that

and, as Mahler observed, this implies that the number g(k) occurring in Waring's problem is given by

for all but a finite number of values of k. It would plainly be of interest to establish a bound for the exceptional k and this would follow from an upper estimate for the integers n for which (1) holds. But Mahler's work was based on Ridout's generalization of Roth's theorem and, as is well known, the latter result is ineffective.
On colouring random graphs
Authors: Grimmett G. R. , McDiarmid C. J. H.
Published: November 2008

Let ωndenote a random graph with vertex set {1, 2, …,n}, such that each edge is present with a prescribed probabilityp, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωnis, with probability one,
Spectral manifolds for the perturbed Schrödinger operator in Lp (Rn)
Authors: Thompson M.
Published: November 2008

In a recent paper(14), Talenti gave a complete characterization of the spectrum of the maximal realization Hop of − Δ in Lp(Rn), for p in the range
A note on variational theorems in non-linear elastostatics
Authors: Ogden R. W.
Published: November 2008

In a recent paper Koiter (5) discussed a principle of stationary complementary energy for the finite deformation of elastic materials. The complementary energy functional he uses depends only on the components of the nominal stress, and not on the displacement field. He, incorrectly, attributes the principle to Zubov (14).
The self-intersection formula and the ‘formule-clef’
Authors: Lascu A. T. , Mumford D. , Scott D. B.
Published: November 2008

We shall consider exclusively algebraic non-singular quasi-projective irreducible varieties over an algebraically closed field. If V is such a variety will be the Chow ring of rational equivalence classes of cycles of V

and the group homomorphism defined by any proper morphism φ: V1 → V2. Also

denotes the ring homomorphism defined by φ.
Periodic solutions of certain nonlinear autonomous wave equations
Authors: Melrose Richard B. , Pemberton Malcolm
Published: November 2008

The hyperbolic boundary value problem

has, for δ = 0 and each positive integer k, t-periodic solutions of period τk(ε) = 2Φ/(k2 + ε)½. For a reasonably large class of functions, f, and, for each k, certain values of ε, one at least of these solutions persists for small δ
Splitting over nilpotent and hypercentral residuals
Authors: Hartley B. , Tomkinson M. J.
Published: November 2008

It is a well known theorem of Gaschütz (4) and Schenkman (12) that ifGis a finite group whose nilpotent residualAis Abelian, thenGsplits overAand the complements toAinGare conjugate. Following Robinson (10) we describe this situation by saying thatG splits conjugatelyoverA. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.
Two necessary conditions for embeddability of a Galton–Watson branching process
Authors: Grey D. R.
Published: November 2008

1.Introduction and statement of results. The most substantial contribution in recent years to the literature on the Galton–Watson embedding problem has been that of Karlin and McGregor (4) and (5), to which the readeris referred for a more complete bibliography of the subject. Their results and methods very much involve the analytic properties of the associated probability generating functions (p.g.f.'s), and most of the results they describe as ‘negative’ in the sense that they are sufficient conditions for non-embeddability. The two results here are similarly ‘negative’ but obtained from more direct probabilistic considerations.
Spectral asymmetry and Riemannian geometry. II
Authors: Atiyah M. F. , Patodi V. K. , Singer I. M.
Published: November 2008

In Part I of this paper (6) we proved various index theorems for manifolds with boundary including an extension of the Hirzebruch signature theorem. We now propose to investigate the geometric and topological implications of these theorems in a variety of contexts.
On characteristically simple groups
Authors: Wilson J. S.
Published: November 2008

1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.
On the duality of interaction models
Authors: Biggs Norman
Published: November 2008

Two kinds of duality arising in studies of interaction models are discussed. The first kind, which has not previously been investigated, is related to algebraic properties of the coefficient ring. The second kind is the well-known geometric duality for planar graphs. The two dualities together lead to a perfectly symmetrical relationship for a general form of partition function.
Banach algebras and absolutely summing operators
Authors: Tonge Andrew M.
Published: November 2008

If R is a Banach algebra and ø ∈ R′, the dual space, then we may define a bounded linear map by

We shall show that for suitable p the requirement that each be p-absolutely summing constrains R to be an operator algebra, or even, in certain cases, a uniform algebra. In this way we are able to give generalizations of results of Varopoulos (12) and Kaijser (4).
Stochastic bounds for the single-server queue
Authors: Köllerström J.
Published: November 2008

Various elegant properties have been found for the waiting time distributionGfor the queue GI/G/1 in statistical equilibrium, such as infinite divisibility ((1), p. 282) and that of having an exponential tail ((11), (2), p. 411, (1), p. 324). Here we derive another property which holds quite generally, provided the traffic intensity ρ < 1, and which is extremely simple, fitting in with the above results as well as yielding some useful properties in the form of upper and lower stochastic bounds forGwhich augment the bounds obtained by Kingman (5), (6), (8) and by Ross (10).
The lattice structure of C*-algebras and their duals
Authors: Green Michael D.
Published: November 2008

Let A be a *-algebra of operators on a Hilbert space H, and let Ah, A+ denote respectively the sets of self-adjoint and positive operators in A. A+ is a positive cone in Ah and it induces a partial ordering in Ah. The lattice properties of Ah were studied by R. Archbold in (1) and (2), and Chu Cho-Ho gave a different proof of some of his results in (3).
Nets of conics
Authors: Wall C. T. C.
Published: November 2008

Segre's classification of pencils of quadrics is well known, and appears in standard texts such as (1), (2). By contrast, other linear systems of quadrics do not lend them-selves to a similar exhaustive treatment. Here we discuss the simplest case, that of nets of conics. The result is intrinsically interesting, and involves some pleasant geometry. As well as deriving a list of types, and enumerating their properties, we study the elementary geometrical properties of the partition of the space Ω of nets into equivalence classes (or strata). We work over the field ℝ of real numbers, after performing preliminary calculations over ℂ.
The homology of BO and some results about the Steenrod algebra
Authors: Brown Edgar H. , Davis Donald M. , Peterson Franklin P.
Published: November 2008

Some explicit formulae for the right coaction of A* on H*BO are given. As corollaries, closed formulae for χ(ξk) and other elements in A* are given.
Deterministic epidemic waves of critical velocity
Authors: Brown K. J. , Carr J.
Published: November 2008

In (1) Atkinson and Reuter prove the existence of wave solutions with velocities strictly greater than some critical velocity ʗ0 of an equation associated with the deterministic model for the spread of an epidemic. The existence of wave solutions with velocity ʗ0 is left as an open question. In this note we show that Atkinson and Reuter's results and a simple limiting argument imply the existence of such wave solutions.
On the properties of bilinear models for the balance between genetic mutation and selection
Authors: Kingman J. F. C.
Published: November 2008

Moran (7) has shown that some models in population genetics lead to a bilinear recurrence relation which, for a finite number of alleles, has a globally stable equilibrium. When the possible alleles are infinite in number, non-trivial problems of existence and stability of the equilibrium arise, which Moran has resolved in special cases. In this paper a powerful sufficient condition is established for the existence of a globally stable equilibrium, and its consequences are explored for cases of genetical interest. A more speculative final section describes a variant of Moran's model which may possibly have some relevance to the assessment of experimental evidence for or against selective neutrality.
A circular disc containing a radial edge crack opened by a constant internal pressure
Authors: Gregory R. D.
Published: November 2008

A circular disc of radius a, made of homogeneous, isotropic, linearly elastic material, contains a radial edge crack of lengthb(0 <b< 2a). The disc is in equilibrium in a state of generalized plane stress caused by loading the faces of the crack by a constant internal pressure. The problem of determining the resulting stress field throughout the disc is solved analytically in closed form. The principal results are that the stress concentration factorat the crack tip, the total strain energyW, and the openingUat the mouth of the crack, are given exactly by

whereAis a constant whose value correct to 6 significant figures is

and,W0,U0are normalising factors defined in section 6.