bluuebunny/crossref_metadata_2025 huggingface dataset:
A table of irreducible polynomials and their exponents
Authors: Chang J. A. , Godwin H. J.
Published: November 2008

With a view to applications to electrical engineering we have recently calculated the primitive polynomials of degree 2 and 3 modulo various primes. In the course of the work we also obtained all the irreducible polynomials of these kinds and, as an extension of the tables of Church(1), Marsh(2), Peterson(3) and Lloyd(4) we now publish them. Since our primary interest was the exponent to which each polynomial belonged (see Albert(5), p. 130) we group them according to exponent.
On transversality
Authors: Hudson J. F. P.
Published: November 2008

The simplest definition of transversality in the PL or Top categories is the purely local one: Manifolds M and N are transverse in Q if, for each point x of their intersection, there is a (closed) neighbourhood U of x in Q and a (PL) homeomorphism

for suitable p, q, and r
A note on nth roots of positive operators
Authors: Jowett John H.
Published: November 2008

The existence and uniqueness of a positive self-adjoint nth. root of a positive, self-adjoint, not necessarily bounded operator on a Hilbert Space H can be readily demonstrated using the spectral representation of the transformation.
Compact semigroups in commutative Banach algebras
Authors: Kaashoek M. A. , West T. T.
Published: November 2008

A monothetic semigroup is a topological semigroup with jointly continuous multiplication which contains a dense cyclic subsemigroup. These semi-groups arise in a natural way in the study of semi-algebras. In (4) we showed that a compact monothetic semigroup in a Banach algebra can be characterized in terms of the spectral properties of a generating element. In this paper these spectral theorems are linked with the well-known structure theory of compact semigroups.
On the absolute Hausdorff summability factors of a Fourier series
Authors: Tripathy N.
Published: November 2008

Let be a given infinite series with the sequence of partial sums {sn}. Then the sequence-to-sequence Hausdorff transformation of the sequence {sn} is given by
On semi-groups of operators and the resolvents of their generators
Authors: Okikiolu G. O.
Published: November 2008

In a recent paper (4) concerned with the study of ‘fundamental operators’, we have obtained results involving the resolvents of the generators of some semi-groups of operators in Lp (− ∞, ∞). In this paper we consider those results which, under suitable conditions, can be extended to cases where the resolvents do not necessarily form a class of fundamental operators. The main results of this type, given in Theorems 2·7 and 2·8, involve inequalities between members of the semi-group and the resolvent of the generators. The semi-groups to which the main results apply include those denned by the Poisson and the Gauss–Weierstrass operators.
Forced vertical oscillations in a viscous stratified fluid
Authors: Dore B. D.
Published: November 2008

The linear fluid dynamics is considered when infinite vertical boundaries are set in oscillatory vertical motion. The case of exponential stratification with constant kinematic viscosity is explicitly studied. When the forcing frequency equals the Brunt–Vaisälä frequency for the fluid, the customary boundary layers are absent in the steady-state oscillation, however small be the kinematic viscosity; for a semi-infinite fluid the corresponding horizontal extent of the region influenced by the boundary motion is then of the order of the stratification length. The sign of the phase angle is everywhere dependent on whether the magnitude of the forcing frequency is greater than or less than that of the Brunt–Vaisäla frequency.
Integrals with a large parameter: paths of descent and conformal mapping
Authors: Ursell F.
Published: November 2008

The method of steepest descents is widely used to find the asymptotic expansion of contour integrals of the form

where N is a large positive parameter, and f(z) and g(z) are analytic functions of z. (For the sake of simplicity it will here be supposed that f(z) and g(z) are regular except at ∞, but our arguments can be extended to include singularities at other points.) In the method of steepest descents the path of integration is deformed into a new equivalent path consisting of the paths of steepest descent through a and b, together with paths of steepest descent through certain saddle points, the relevant saddle points. (Paths of descent, etc., are defined in §1 below.) Watson's Lemma then shows that the complete asymptotic expansion depends on the local behaviour of f(z) and g(z) at the end-points and at the relevant saddle-points. On the other hand to determine which saddle-points are relevant is a global (non-local) problem and requires either the tracing of paths of steepest descent all the way from saddle-points and endpoints to ∞, or some other global process. It has often been noted that in applications of Watson's Lemma the paths of steepest descent can be replaced by a wider class of paths of descent. In this way we are led to global problems like the following: Given 2points z = α and z = β where Nf(a) > Nf(β), can apath of descent be drawn from α to β? If also Ff(α) = Ff(β), can a path of steepest descent be drawn from α to β?
On topological quotient maps preserved by pullbacks or products
Authors: Day B. J. , Kelly G. M.
Published: November 2008

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:

Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?

Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)

Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?
Some formulae for Appell functions
Authors: Sharma B. L.
Published: November 2008

1. Introduction. The object of this paper is to obtain some generating functions for Appell functions with the help of fractional derivatives. The results are general in character and include as particular cases some of the results given earlier by Mexiner (3) and the author (4, 5, 6).
Massey products in K-theory
Authors: Snaith V. P.
Published: November 2008

The aim of this paper is to reformulate the work of Massey(6), and Spanier(9, 10), on higher order cohomology operations, in terms of vector bundles in such a way as to produce geometrically some higher order operations in K-theory, which we will call Massey products.
Two results on very strongly homogeneous algebras
Authors: Körner T. W.
Published: November 2008

We show that under quite general circumstances a Banach algebra R need not be closed in its tilda algebra . In answer to a query of Katznelson we show that there is a very strongly homogeneous algebra B ≠ C0(R) on R with R as carrier space such that non-analytic functions operate on B.
A generalized Vandermonde-type convolution and associated inverse series relations
Authors: Mohanty S. G. , Handa B. R.
Published: November 2008

In (1) and (2), Gould has discussed some generalized Vandermonde-type convolution identities, forCcoefficients satisfying

where

and
A Wiener-Hopf solution to the triple integral equations for the electrified disc in a coplanar gap
Authors: Spence D. A.
Published: November 2008

The axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators.

In the limit of small gap width (b − a)/b, the equation takes the form

which is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expression

for the capacity of the disc; for the opposite limit the expression

is derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.
Integer difference covers which are not k-sum covers, for k = 6, 7
Authors: Connolly Dennis
Published: November 2008

Constructions are given of sets F(k), (k = 6, 7) of residues mod q(k), (for a suitably chosen integer q(k)) such that F(k) – F(k) contains all residues, while

has a gap of an assigned number of consecutive residues.
On mass transport induced by interfacial oscillations at a single frequency
Authors: Dore B. D.
Published: November 2008

The to method of matched asymptotic expansions is employed to calculate the mass tre sport velocity due to combinations of small amplitude oscillatory waves propagatir, at a single frequency in fluid systems with density and viscosity dis-continuities. Interfacial boundary layers are considered in terms of the curvilinear coordinate system described by Longuet-Higgins(1). The order of magnitude of the mass transport velocity calculated for a general oscillatory disturbance is the same as that calculated for interfacial progressive waves by Dore(2). For standing waves, the time-averaged motion of the fluid particles forms a cellular structure in each fluid layer; the mass transport velocity due to modal interactions is associated with a similar structure.
Tail sums of convergent series of independent random variables
Authors: Barbour Andrew D.
Published: November 2008

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.
On isotropic tensors and elastic moduli
Authors: Ogden R. W.
Published: November 2008

The use of even-order isotropic tensors in non-linear elasticity theory is discussed in this paper. A notation is adopted through which these tensors can be represented conveniently so that their interdependence is clearly shown. Information about the number of independent elastic constants required is then readily available for use in an expansion of the stress to various orders in the strain relative to the undistorted configuration of the elastic material in question.

For an incompressible isotropic hyperelastic solid, it is shown that each principal component of the distortional part of the stress is expressible as a function only of the corresponding principal component of strain to the fourth order. Under certain conditions, which are not too restrictive, this result can be extended to higher orders.
Corrigendum to ‘Square Banach Spaces’
Authors: Cunningham F.
Published: November 2008

Nina M. Roy has proved (3) that every separable G-space is square, thus exposing an error in my paper (1), in which I proposed a construction of non-square G-spaces which purports to include separable examples. The mistake lies in Lemma 5, whose proof confuses the weak-* topology with the bounded weak-* topology. This note is to salvage the construction to the extent possible, which is in the nonseparable case. The context and notations of (1) are used without further comment.
On the digits of the multiples of an irrational p-adic number
Authors: Mahler Kurt
Published: November 2008

Let α be an irrational p-adic number, r an arbitrary positive integer. Our aim is to prove that there exists a rational integer X satisfying

such that every possible sequence of r digits 0, 1, …, p – 1 occurs infinitely often in the canonical p-adic series for Xα. It is clear that it suffices to prove this result for p-adic integers.
The spectral radius in a locally convex algebra
Authors: Wood A. W.
Published: November 2008

In (1), Allan introduced the concept of the spectrum of an element of a locally convex algebra, and developed a spectral theory for pseudo-complete algebras. In a commutative Banach algebra the spectral radius is a continuous seminorm, and so it is natural to investigate continuity properties of the spectral radius in various classes of locally convex algebra. Continuity is too strong a condition to be expected in any general case, and an interesting property to investigate appears to be lower semi-continuity. We shall show easily that in a commutative pseudo-complete locally m-convex algebra (in the sense of (4)) the spectral radius is lower semi-continuous. We shall then exhibit a commutative complete metrizable algebra in which lower semi-continuity fails to hold.
Theorems on factorization and primality testing
Authors: Pollard J. M.
Published: November 2008

1.Introduction. This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integernor test it for primality. One way of making these problems precise uses a multi-tape Turing machine (e.g. (1), although we require a version with an input tape). At the start of the calculationnis written in radix notation on one of the tapes, and the machine is to stop after writing out the factors in radix notation or after writing one of two symbols denoting ‘prime’ or ‘composite’. There are, of course, other definitions which could be used; but the differences between these are unimportant for our purpose.
Global results and stability of motion
Authors: Lakshmikantham V. , Leela S.
Published: November 2008

1. The proofs of many results in the theory of stability and boundedness basically depend on dividing the vicinity of some kind of invariant set (or other convenient set) into suitable subsets and then trying either to prove that solutions cannot leave such sets or to estimate the escape time. This observation makes it possible to give some global results in terms of arbitrary sets which can be employed as tools in dealing with various problems of stability and boundedness. In applications, these tools enlarge the class of useful Lyapunov like functions and also offer more flexibility.
Internal wave motion in a non-homogeneous viscous fluid of variable depth
Authors: Dore B. D.
Published: November 2008

A method of solution is given to the problem of internal wave motion in a non-homogeneous viscous fluid of variable depth. The approach is based on the inviscid theory of Keller and Mow(l) and on boundary-layer analysis. For internal progressive waves in uniform depths, it reduces essentially to the theory given by Dore(2). The present results are also applicable to surface waves when the fluid is homogeneous.
A note on some theorems on simultaneous diagonalization of two Hermitian matrices
Authors: Au-Yeung Yik-Hoi
Published: November 2008

We denote by F the field R of real numbers, the field C of complex numbers, or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be Hermitian if A = A* and unitary if AA* = In, where In is the n × n identity matrix. An n × n Hermitian matrix A is said to be positive definite (postive semi-definite resp.) if uAu* > 0(uAu* ≥ 0 resp.) for all u (╪ 0) in Fn. Here and in what follows we regard u as a 1 × n matrix and identify a 1 × 1 matrix with its single element. In the following we shall always use A and B to denote two n×n Hermitian matrices with elements in F, and we say that A and B can be diagonalized simultaneously if there exists an n×n non-singular matrix V with elements in F such that VAV* and VBV* are diagonal matrices. We shall use diag {A1, A2} to denote a diagonal block matrix with the square matrices A1 and A2 lying on its diagonal.
Manifolds of homotopy typeK(π, 1). I
Authors: Johnson F. E. A.
Published: November 2008

1. Introduction: In all that follows, by a manifold we shall mean a paracompact, Hausdorff, locally Euclidean topological space. By means of the universal covering construction, the classification problem for connected manifolds splits into two parts, namely (i) classification of simply connected manifolds, (ii) classification of covering actions of groups on simply connected manifolds.
On an unlinking theorem
Authors: Sumners D. W.
Published: November 2008

An n-link of multiplicity is a smooth embedding of the disjoint union of μ copies of Sn in Sn+2; is said to be trivial if it extends to a smooth embedding of the disjoint union of μ copies of Dn+1. Let , and Cnμ denote the wedge product of μ copies of S1 and (μ – 1) copies of Sn+1. Then clearly, if is trivial, then X ≃ Cn,μ, where ≃ denotes homotopy equivalence.
On cyclic maps
Authors: Snaith V. P.
Published: November 2008

Throughout X will denote a connected, finite C.W. complex.

Let G be a subgroup of ∑n, the symmetric group, which acts transitively on the Cartesian product, Xn, of the space X. A map f:Xn→X is G-symmetric if it commutes with the action of G. If x0εX is a base point let i:X→Xn denote the inclusion, i(x) = (x,x0, …,x0). In ((6); (7); (11)–(13)) the following problem is posed: if X is an orientable topological n-manifold, what is the set of integers which may be obtained as the degree of(f.i) where f is a G-symmetric map? The degree of(f.i) is called the James number of f. If G = ∑n(G = Zn) a G-symmetric map will be called a symmetric map (a cyclic map). If X = Sn, the n-sphere, this problem has been studied in ((6)–(8), (11)–(13))
The non-existence of standing modes in certain problems of linear elasticity
Authors: Gregory R. D.
Published: November 2008

Suppose that an elastic half-space, which contains certain surface defects, inclusions and cavities, is in free, two-dimensional, time-harmonic vibration, with the wave field at infinity ‘outgoing’ in character. It is shown that the elastic potentials representing such a ‘standing mode’ can be expressed in the form of contour integrals, for instance

U(t) being an analytic function of t. By considering the far field of these potentials, it is shown that U(t) is zero on a certain arc in the t-plane and is therefore identically zero. It follows that ø(r) is zero everywhere and this proves the non-existence of such standing modes in these configurations.

This uniqueness theorem justifies the solution given by the author (Gregory (2)) for the problem in which time harmonic stresses act on the walls of a cylindrical cavity lying beneath the surface of an elastic half-space. It is also shown that if a Rayleigh surface wave is incident on any system of surface defects, inclusions and cavities, then energy must be transferred from the surface wave to scattered outgoing body waves of both P and S types.
Representations of holomorphs of group extensions with Abelian kernels
Authors: Wehrfritz B. A. F.
Published: November 2008

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.
A proof of Sperner's lemma via Hall's theorem
Authors: Brown T. C.
Published: November 2008

Let S be a Sperner set of subsets of {1, 2, …, n}. (That is, for A, B ∈ S, if A ≠ B then A ⊄ B and B ⊄ A.) Then
Classes of unipotent elements in simple algebraic groups. I
Authors: Bala P. , Carter R. W.
Published: November 2008

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.
On products of non-negative matrices
Authors: Hajnal J.
Published: November 2008

The main theme of this paper is that under wide conditions the product of a large number of square non-negative matrices is close to a positive matrix of rank 1. (A non-negative matrix is one whose elements are non-negative real numbers.)
The departure process from a queueing system
Authors: Kelly F. P.
Published: November 2008

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution functionB(z) and finite expectation

where βλ < 1, i.e. anM/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only ifG=M(i.e.B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form ofB(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).
The maximal subgroups of M24
Authors: Curtis R. T.
Published: November 2008

In this paper, which assumes a familiarity with (1), we produce a short proof of the following theorem. The proof uses MOG techniques to extend a powerful observation of Conway (2) on intransitive subgroups of M24.
Equivalence singularity dichotomies for a class of ergodic measures
Authors: Kanter Marek
Published: November 2008

Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.
On the static field in general relativity
Authors: Islam Jamal N.
Published: November 2008

The possibility is explored that the general static solution of Einstein's vacuum field equations is given in terms of a single harmonic function, i.e. solution of the flat space Laplace equation. It is shown that coordinate conditions can be chosen such that one of Einstein's equations reduces to Laplace's equation. To examine the extent to which the metric is determined by the resulting harmonic function ø, a power series expansion is considered in which in the first non-trivial order one gets a system of eight equations for six unknowns. This is reduced to a system of three equations for three unknowns. This is some indication that the coordinate conditions are consistent and shows the sense in which, to this order, the metric is determined by ø. This system of three equations is then solved explicitly for a particular choice of ø, thereby giving an approximate non-axisymmetric solution of Einstein's vacuum equations. This solution is asymptotically flat and could represent the approximate gravitational field of a non-axisymmetric bounded rigid body. The form of a class of time-dependent metric related to the general static metric is discussed.
Consistency and Nörlund means
Authors: Maddox I. J.
Published: November 2008

Let A = (ank) be an infinite matrix of complex numbers; suppose I is the unit matrix and c is the space of convergent sequences. By (A) we denote the summability field of A, i.e. (A) = {x = (xk): Ax ∈ c}.
The uniqueness of Atkinson and Reuter's epidemic waves
Authors: Barbour Andrew D.
Published: November 2008

Atkinson and Reuter(1) consider travelling wave solutions for the deterministic epidemic, with or without removals, spreading along the line. In the case where there are no removals, they reformulate the problem in terms of the solutions X(·) to the integral equation

which satisfy X(− ∞) = − ∞, X( + ∞) = 0, X(u) < 0 for u ∈ (− ∞, ∞), where

is the left hand tail of the contact distribution, and where ʗ > 0 is the velocity of the wave corresponding to X. They show that no solution is possible unless

converges for λ > 0 sufficiently small, and that any solution X must satisfy

for some C > 0. They then prove the following existence theorems.
On the crystallographic point groups and on Cauchy symmetry
Authors: Parry Gareth P.
Published: November 2008

Elastic work functions may be restricted by material symmetry requirements and/ or by the Cauchy symmetry of the first order moduli with respect to the Green strain. This investigation enables the construction of any work function so constrained. The crystallographic point groups are considered in detail, and examples of such work functions are given.
A counterexample to R. Davidson's conjecture on line processes
Authors: Kallenberg Olav
Published: November 2008

Rollo Davidson conjectured in 1968 that every stationary second order line process in the plane which has a.s. no parallel lines is necessarily a Cox (or doubly stochastic Poisson) process. This conjecture is disproved here. An affirmative answer is further given to the question whether there exists a lattice type point process in the plane which is stationary under arbitrary area preserving affine transformations.
A new approach to subcritical instability and turbulent transitions in a simple dynamo
Authors: Robbins K. A.
Published: November 2008

The onset of turbulence in many systems appears disorganized and unpredictable. However, the present detailed study of a well-known model reveals a series of well-defined transitions from steady motion to highly non-periodic behaviour. The bifurcation structure of this simple model, which describes both a reversing disc dynamo and Benard convection in thin fluid loops, is characteristic of a class of systems with subcritical instabilities. It is found that the bifurcation curve for this instability does not have a stable branch and that non-periodic and linearly stable steady solutions coexist. The boundary between transient and non-periodic behaviour is marked by a non-uniformity in the number of oscillations between reversals. This non-uniformity, not previously observed, is a striking corroboration of the model's relationship to the geodynamo. Other features of reversal in the disc dynamo such as the presence of two time scales in the oscillations are also exhibited by geomagnetic fields. In addition to a geometric description of the transitions, a one-dimensional mapping first constructed by Lorenz for the system beyond marginal stability, is extended to the subcritical regime. This type of mapping, which can be interpreted in terms of invariant surfaces of the system, may be of value in dissecting more general systems with subcritical Hopf bifurcations.
Applications of outwardly simple line families to plane convex sets
Authors: Falconer K. J.
Published: November 2008

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.
On the weak convergence of U-statistic processes, and of the empirical process
Authors: Loynes R. M.
Published: November 2008

1. Summary and introduction

In (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.
On the static field in general relativity: II
Authors: Islam Jamal N.
Published: November 2008

Some aspects are considered of the Einstein vacuum field equations for the diagonal form of the general static metric. A power series expansion is considered in which in the first non-trivial order one gets seven equations for four unknowns. This is reduced to a single Poisson equation in which the source is given in terms of a harmonic function and some arbitrary functions of two variables. A similar reduction is carried out in detail for the next non-trivial order. The approximation scheme can be extended to arbitrarily high orders in principle. The connexion of the diagonal form with the Weyl solutions and the merits of the diagonal and non-diagonal forms are briefly discussed. A general solution of one of the field equations is presented.
The non-existence of odd primary Arf invariant elements in stable homotopy
Authors: Ravenel Douglas C.
Published: November 2008

In this paper we will show that certain elements of order p (p an odd prime) on the 2-line of the Adams-Novikov spectral sequence support non-trivial differentials and therefore do not detect elements in the stable homotopy groups of spheres. These elements are analogous to the so-called Arf invariant elements of order 2, hence the title. However, it is evident that the methods presented here do not extend to the prime 2.
The topological classification of Lorenz attractors
Authors: Rand David
Published: November 2008

The Lorenz attractor is a strange attractor which has been proposed as an explicit model for turbulence ((4), compare (5)). First studied by E. N. Lorenz as a truncation of the Navier-Stokes equations (2), it has since attracted the attention of mathematicians because of its particularly interesting dynamical properties.
A coated inclusion in an elastic medium
Authors: Walpole L. J.
Published: November 2008

Many inhomogeneous systems of practical interest are made up of an inclusion, or several inclusions, of one elastic phase bonded to a surrounding matrix of another phase, and the behaviour of such systems has been studied extensively. A thin layer of some other elastic phase intervenes between an inclusion and the matrix in some important applications, for instance, where coated nuclear fuel particles are bonded in a matrix, or where duplex cylindrical fibres are employed in fibre-reinforcement, or wherever chemical action induces a protective finish either intentionally or inevitably as when an oxide coating covers a metallic surface. Our objective is to show how to take account of the pronounced influence that even a thin coating may exert upon, for instance, the stress concentrations just outside an inclusion or cavity, or upon the overall elastic moduli of a suspension of inclusions (though discussion of this latter topic is omitted here in order that it may be explored more fully elsewhere).
The group of self-homotopy equivalences of principal three sphere bundles over the seven sphere
Authors: Rutter John W.
Published: November 2008

1. The set ℰ (X) of homotopy classes of self-homotopy equivalences of a space X forms a group under composition. In this note I complete the calculation of ℰ (X) when X is a torsion free rank 2 h-space, and give ℰ (X) up to extension for the remaining principal S3 bundles over S7.
The arithmetic plurigenera of surfaces
Authors: Wilson P. M. H.
Published: November 2008

Let S0 be a complex projective surface with only isolated Gorenstein singularities (see Introduction to (12)). By Serre's criterion ((4), p. 185) this is equivalent to saying that S0 is normal and Gorenstein. By an algebraic smooth deformation of S0, we shall mean a flat, proper morphism of varieties, ρ: say, with fibre ρ−1(y0) = S0 for some y0 ∈ Y and with the general fibre ρ−1(y) = S being a smooth surface. In the paper (12), we studied such smooth deformations of S0 and in particular the behaviour of the plurigenera Pn of the surfaces in the family. The main result of (12) was the fact that Pn(S0) ≤ Pn(S) for all positive integers n, where the choice of the particular smooth surface was irrelevant by a result of Iitaka(5). To prove the above result we introduced what were called the arithmetic plurigenera of S0, which we define again below. In this paper we shall study more closely these arithmetic quantities, and in the process answer some of the questions posed in (11).
On the centres of hereditary JBW-subalgebras of a JBW-algebra
Authors: Edwards C. M.
Published: November 2008

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions that

and

for all elements a and b in A. It follows from (1.1) and (l.2) that

for all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition that

for all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).
Inner function images of radii
Authors: Rudin Walter
Published: November 2008

The main result of the present paper is the construction of a very oscillatory inner function F in the unit disc. The existence of such an F leads to a certain extension of Fatou's classical theorem concerning the boundary behaviour of H∞-functions.
Integral inequalities for increasing functions
Authors: Ahlswede Rudolf , Daykin David E.
Published: November 2008

For numbers of increasing real functions f(x) with new integral inequalities. They generalize classical results. The proofs are short and simple being based on sequences.
A circuit covering result for matroids
Authors: Mason J. H. , Oxley J. G.
Published: November 2008

The purpose of this note is to prove the following result.

Theorem 1. let M be a connected matroid. Suppose that C is a circuit of M and p and q are elements of M. Then M has circuits cp and Cq such that .
A geometric approach to two-timing
Authors: Gilbert A. D.
Published: November 2008

This paper is concerned with a particular method used in the calculation of asymptotic expansions of solutions to problems which contain a small parameter ∈. The physical idea underlying the method is that the parameter introduces new scales (of length and/or time) into the problem, and that the solution should incorporate these (see, for example, Cole (1)). This may be done in various ways: two-timing (two-scaling, multiple-scaling) does it by demanding that the solution is a function of all relevant scales. Consider, for example, a linear oscillator whose frequency is slowly adjusted, governed by

and subject to some initial condition. Account of the slowly varying frequency is taken by the dependence of Ω on ∈t, where 0 < ∈ ≪ 1, and we assume that Ω > 1 for all arguments. The rapid oscillatory behaviour anticipated in the solution is customarily described by a phase function Θ(isin;t)/∈, a generalization of Ωt the case that Ω is constant, while slow changes due to frequency variation occur on a scale ∈t.
Manifolds with multiple knot-surgery descriptions
Authors: Brakes W. R.
Published: November 2008

Every closed connected orientable three-manifold can be constructed by surgery along an appropriately chosen link in the three-sphere,S3((10), (15)). Such alink-surgerydescription is never unique, but the equivalence between different descriptions has been explicitly identified ((8),(3)). However, the situation with regard to manifolds that can be obtained by surgery along aknotinS3is less clear ((6), p. 47): some manifolds are known to have at least two knot-surgery descriptions ((12), (14), p. 270, (1), (11)), whilst certain manifolds (e.g.S3andS2×S1) are widely suspected to have just one (see problems 1·15, 1·16 of (9)).
Calculation of Lin's Ext groups
Authors: Lin W. H. , Davis D. M. , Mahowald M. E. , Adams J. F.
Published: November 2008

The first-named author has proved interesting results about the stable homotopy and cohomotopy of spaces related to real projective space RP∞; these are presented in an accompanying paper (6). His proof is by the Adams spectral sequence, and so depends on the calculation of certain Ext groups. The object of this paper is to prove the required result about Ext groups. The proof to be given is not Lin's original proof, which involved substantial calculation; it follows an idea of the second and third authors. The version to be given incorporates modifications suggested later by the fourth author.
The distribution of sequences in Abelian groups
Authors: Dennis J. S.
Published: November 2008

1Introduction.In this paper, I give a definition of uniform distribution for sequences taking values in certain types of Abelian group—in particular, in groups of prime exponent—and one of independent distribution for sets of such sequences. In the various cases, I find conditions for the properties to hold, which are similar to Weyl's criterion for the uniform distribution of real sequences modulo 1.
Fourier series associated with the sample functions of a stochastic process
Authors: Samal G.
Published: November 2008

We consider a stochastically continuous process ω(t, α) with independent increments, whose sample functions are bounded in the unit interval 0 ≤ t ≤ 1 for almost all α. If ω(t, α) is a process with independent increments, the characteristic function of ω(t, α) is of the form exp {tψ(u)} where where F is a σ-finite measure with finite mass outside every neighbourhood of o, and a and σ are constants. There is no essential restriction in supposing ω(0, α) = 0.
On an eigenfunction expansion associated with a condition of radiation
Authors: Naylor D.
Published: November 2008

In this paper certain Bessel type eigenfunction expansions are developed by considering a non-seif-adjoint problem which involves a radiation type condition
Diffusion and convection of heat from circular cylinders with known boundary conditions
Authors: Norminton E. J.
Published: November 2008

The problem of the diffusion and convection of heat in two-dimensional, incompressible flow past a circular cylinder is considered. Solutions are found for the temperature at any point exterior to the cylinder in terms of the boundary conditions on the surface and two particular boundary conditions are considered.
A note on 3-fold branched coverings ofS3
Authors: Montesinos José María
Published: November 2008

H. Hilden(3, 4) and the author(9,10) proved independently the following result:

Theorem 1.Each closed, orientable 3-manifold is a 3-fold, dihedral covering of S3, branched over a knot.
On the Williams-Bjerknes tumour growth model: II
Authors: Bramson Maury , Griffeath David
Published: November 2008

Williams and Bjerknes introduced in 1972 a stochastic model for the spread of cancer cells. Cells, normal and abnormal (cancerous), are situated on a planar lattice. With each cellular division, one daughter cell stays put, while the other usurps the position of a neighbour; abnormal cells reproduce at a faster rate than normal cells. We are interested in the long-term behaviour of this system. We showed in ‘On the Williams-Bjerknes Tumour Growth Model I’ (1) that, provided it lives forever, the tumour will eventually contain a ball of linearly expanding radius. Here it is shown that the rate of expansion is actually linear, and that the region of infection has an asymptotic shape which is given by some (unknown) norm. To demonstrate that the tumour contains a ball of linearly expanding radius, we applied in (1) certain techniques common to the field of interacting particle systems; in particular we made use of certain auxiliary Markov chains which are imbedded in the dual processes of our model. Here different techniques are applied. We show that the first infection times of different sites are almost subadditive, and we exhibit various regularity properties of the infection times such as bounds on their moments. These properties of the Williams-Bjerknes process allow one to follow the outline prescribed by Richardson in (7), where he showed that such a process does indeed exhibit a linear growth whose shape is prescribed by a norm.
On maximal Euclidean sets avoiding certain distance configurations
Authors: Croft H. T. , Falconer K. J.
Published: November 2008

Let us say that a set points in a Euclidean space has property (2) if no 2 points X, Y ∈ S have X Y > 1. Then an easy observation is the following:

The maximum area (Lebesgue measure) of a plane set S with property(2) is ¼π, attained only when S is (essentially) a disc.
A Lagrangian Klein bottle
Authors: Hayden J. , Zeeman E. C.
Published: November 2008

Lagrangian theory (2) and elementary catastrophe theory (5) coincide locally but differ globally. Locally the singularities that occur in the two theories are the same, but globally the manifolds that occur are different. We shall show that Lagrangian manifolds are more general.
The structure of modules over polycyclic groups
Authors: Brown Kenneth A.
Published: November 2008

The structure of modules over polycyclic group rings RG is investigated using the idea of a link P ⇝ Q between prime ideals of RG. The representation theory of RG splits into two parts – the part we discuss is determined by the representation theory of certain Noetherian polynomial identity factor rings of subrings of RG.
On nth roots and infinitely divisible elements in a connected Lie group
Authors: McCrudden M.
Published: November 2008

For any group G, x ∈ G and n ∈ ℕ (the natural numbers), let

i.e. the set of all nth roots of x in G. If G is a Hausdorff topological group, then Rn(x, G) is a closed set in G, but may otherwise be quite complicated. However, as we have observed in (4), if G is a compact Lie group, then Rn(x, G) always has a finite number of connected components, and this result has led us to wonder about the connectedness properties of Rn(x, G) for other Lie groups G. Here is the result.
Stable cohomotopy and cobordism of abelian groups
Authors: Stretch C. T.
Published: November 2008

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).
Spindle-shaped drops in a viscous extensional flow
Authors: Sherwood J. D.
Published: November 2008

We study the deformation of a long slender drop of viscosity ζμ suspended in another liquid of viscosity μ. Interfacial tension causes the drop to become spherical when there is no fluid motion. When the flow is weak the drop is slightly perturbed, and this case was studied by Taylor (7). Computing the flow around an exact sphere, he used the resulting imbalance in the normal stresses to predict the perturbed drop shape. When the drop is in viscid or slightly viscous (ζ ≪ 1), and when the flow is stronger, the drop becomes long and slender. Previous slender-body analyses (Taylor (8) Buckmaster (2, 3), Acrivos & Lo(1) and Hinch & Acrivos(5)) predict pointed ends, but break down in the neighbourhood of these ends. Here we adopt an approach similar to that of Taylor (7). The zero Reynolds number flow around a spindle-shaped drop with pointed ends is computed exactly. Interfacial tension does not quite balance the hydrodynamic stress, and the resulting imbalance in the normal stresses is used to predict a more accurate representation of the drop shape.
The Variational Principle and natural transformations I. Autonomous dynamical systems
Authors: Schafir R. L.
Published: November 2008

When a system of differential equations is derivable from a variational principle, an outstanding consequence is the connection which ensues between symmetries and conservation laws. The idea behind the present work is to use this consequence as a new characterization of the variational principle itself. For general systems of equations there is a rather weak form of the connection, as will be demonstrated; but it has highly arbitrary and non-invariant features, and, for instance, there is not a formula associating a particular infinitesimal invariance with a particular constant of the motion. In terms of the modern technical concept which will be employed, the transformation is not natural. However, if the system is derivable from a variational principle, there follows a preferred transformation; and it has certain invariance properties which accord with the concept of naturality.
Approximating maps and exact C*-algebras
Authors: Archbold R. J.
Published: November 2008

Let A and E be C*-algebras, let A ⊗ B denote the minimal C*-tensor product, and let ε A *. The right slice map R: A ⊗ B → B is the unique bounded linear mapping with the property that R (a ⊗ b) = (a)b (a ε A, b ε B)(10). A triple (A, B, D), where D is a C*-subalgebra of B, is said to have the slice map property if whenever x ε A ⊗ B and R(x) D for all ε A* then x ε A ⊗ D). It is known that (A, B, D) has the slice map property whenever A is nuclear (11,13), but it appears to be still unknown whether the nuclearity of B will suffice (unless some extra condition is placed on D (l)).
Edgeworth expansion of the distribution of Stein's statistic
Authors: Hall Peter
Published: November 2008

We derive a two-term Edgeworth expansion for the distribution of Stein's double sampling statistic in the case of a non-normal parent population. The first term describes the main effect of skewness, while the second describes the main effect of kurtosis and the secondary effect of skewness. The remainder is shown to be uniformly negligible in comparison with these effects. Explicit conditions are given for the expansion to be valid. These conditions are considerably weaker than those which were imposed in an earlier derivation of the ‘exact’ distribution of Stein's statistic.