bluuebunny/crossref_metadata_2025 huggingface dataset:
Non-repetitive sequences
Authors: Pleasants P. A. B.
Published: November 2008

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.
Some Tauberian theorems on a scale of Abel type summability methods
Authors: Mishra Babban Prasad
Published: November 2008

Suppose that λ > − 1 and that
Compact operators and Banach algebras
Authors: Duncan J.
Published: November 2008

1. Introduction. Let X be a complex Banach space and let (X) denote the Banach algebra of all bounded linear operators on X. In this paper we study subalgebras of (X) that contain non-zero compact operators and that are Banach algebras with respect to some norm dominating the operator norm. Our main aim is to give conditions for the existence of minimal one-sided ideals in . Barnes ((l) Theorem 2·2) has shown that if every element of a semi-simple Banach algebra has countable spectrum then the algebra has minimal one-sided ideals. If, in particular, is a uniformly closed sub-algebra of the compact operators on X, then every element of has countable spectrum by the Riesz–Schauder theory and so has minimal one-sided ideals. We extend this latter result by showing that if is a semi-simple uniformly closed subalgebra of (X) that contains some non-zero compact operator, then has minimal one-sided ideals. The proof depends heavily on the fact (see e.g. Bonsall (3)) that the spectral projection at a non-zero eigenvalue of a compact operator T belongs to the least closed subalgebra of (X) that contains T. The uniform norm is quite critical for Bonsall's result, but we are able to give a mild generalization which leads to conditions for the existence of minimal one-sided ideals in subalgebras of (X, Y,〈,〉) where (X, Y,〈,〉) are Banach spaces in normed duality.
Functions which operate on positive definite functions
Authors: Rider Daniel
Published: November 2008

1. Introduction. Let P be a collection of functions mapping some set into the complex plane C and let δ be a subset of C. A function F: δ → C is said to operate on P provided the composition F o φ belongs to P whenever φ ∈ P and the range of φ is contained in δ. The basic problem is to characterize the functions which operate.
Some computations in the modular representation ring of a finite group
Authors: Pietro John Santa
Published: November 2008

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).
Summability methods which include the Riesz typical means. II
Authors: Russell B. C.
Published: November 2008

By making use of a convergence-factor theorem of Bosanquet(3), Cooke((4), Theorem I) gave conditions for a regular sequence-to-sequence summability matrix B to be at least as strong as Cesàro summability (C, κ) (κ > 0), namely:

Theorem C. Let κ > 0. In order that the T-matrix B = (bρμ) shall satisfy B ⊇ (C, κ) it is necessary and sufficient that

If 0 < κ ≤ 1 then (2) alone is necessary and sufficient.
General relativity from gauge invariance
Authors: Lord Eric A.
Published: November 2008

The free field equations for particles with spin are invariant under a group SL(2, c) whose transformations correspond to changes of representation of the twocomponent spinor algebra. The generalization of the equations which extends this invariance to a guage invariance in the Yang–Mills sense necessitates the introduction of auxiliary fields (which are also necessary to maintain Lorentz covariance). These fields can be interpreted as the potentials of a spin-2 field, just as the auxiliary fields for the charge gauge group are the potentials of a spin-l field (electromagnetism); this spin-2 field is then self-interacting. The Bargmann–Wigner formulation of the linear spin-2 field, when modified by the proposed self-interaction, yields a non-linear theory of a spin-2 field which is shown to be identical with Einstein's gravitational theory. With this interpretation the auxiliary fields take on an extra role of Yang–Mills field for the general coordinate transformation group – that is, they are the components of the affine connexion.
Local systems and Sylow subgroups in locally finite groups. I
Authors: Rae A.
Published: November 2008

1. Introduction. By a local system for a group G we shall mean a collection Σ of subgroups of G such that for every finite subset of G there is a member of Σ containing it. If is a class of groups G is locally if G has a local system of subgroups.
Codimension and analytic spread
Authors: Burch Lindsay
Published: November 2008

In this paper, I shall establish the sufficiency of certain conditions on an ideal A of a local ring Q, and on a set {g1 …,gk} of elements of Q generating a proper ideal G, for the ideals A and G to be analytically disjoint. Hence I shall establish an upper bound for the analytic spread of A.

The maximal ideal of Q will be denoted throughout by M, and it will be assumed that the field Q/M is infinite.
Multipliers on some sequence spaces
Authors: Kurtz J. C.
Published: November 2008

Let ω be the space of all (complex) sequences. If E, F are subspaces of ω and if A is any (infinite) normal matrix, we set

and

If A is the matrix of a sequence–sequence summability transform, Ā and Â shall denote the series–sequence and series–series forms of the transform, respectively. The multiplier spaces M(c(Ā), c(Ā)), M(lp(Â), l1(Â)) and M(lp(Â), c(Ā)) are characterized (1 ≤ p < ∞). Partial results are given for the spaces M(c(Ā), lp(Â)) and M(lp(Â), lp(Â)).
On a theorem of Posner
Authors: Awtar Ram
Published: November 2008

Posner (1) proved the following result:

Theorem. Let R be a prime ring and d a derivation of R such that, for all a∈R, ad(a)–d(a) a∈Z (centre of R). Then, if d is not a zero derivation, R is commutative.

The purpose of this note is to give a short proof of this theorem. We state the following known result ((1), Lemma 1).
On the development of a solitary wave moving over an uneven bottom
Authors: Johnson R. S.
Published: November 2008

The numerical and experimental results given in Madsen and Mei(16) are predicted using asymptotic methods and some knowledge of the Korteweg-de Vries (K-dV) equation. This is accomplished by first deriving, using formal asymptotic expansions, the K-dV equation valid over a variable depth. The depth is chosen, in the first instance, to slowly vary on the same scale as the initial (small) amplitude of the motion. The appropriate form of the Kd-V equation is then

whereH(X,ξ)describes the surface profile andd(σX)is the changing depth. The rest of the paper is devoted to a study of this equation.
Low Reynolds number flow past a porous spherical shell
Authors: Jones I. P.
Published: November 2008

This paper is concerned with the flow of viscous fluids around and through porous bodies. Previous boundary conditions that have been used are discussed and a generalization of a boundary condition adopted by Beavers and Joseph, for plane boundaries, is proposed for curved surfaces. Using this condition the problem of slow viscous flow past a spherical shell is solved and several special limiting cases are considered.
Transnormal hypersurfaces
Authors: Bolton J.
Published: November 2008

Let M be a C∞ manifold smoothly embedded as a convex hypersurface of Euclidean n-space n. M has constant width if the distance between each pair of parallel supporting hyperplanes in n is equal. This is equivalent to the condition that any chord of M normal to M at one end-point be normal to M at both end-points.
Characterization for the queueing system M/G/∞
Authors: Shanbhag D. N.
Published: November 2008

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.
Zeros of generalized hypergeometric functions
Authors: D'Adda A. , D'Auria R.
Published: November 2008

In this paper we derive the conditions which have to be satisfied in order to obtain some classes of zeros of the generalized hypergeometric series of the type

These conditions read:
On the eigenvectors of the energy-momentum tensor in the Einstein–Maxwell theory
Authors: Woolley M. L.
Published: November 2008

It is shown that in the four-dimensional, source-free Einstein–Maxwell theory, four distinct orthogonal eigenvectors of the energy-momentum tensor may be obtained formally by the action of four projection operators on an almost arbitrary vector.

The extremal field, of the unified theory of Maxwell, Einstein and Rainich, is obtained explicitly in terms of eigenvectors of the energy-momentum tensor and the energy-momentum tensor itself, expressed in terms of eigenvectors, is seen to take on two equivalent forms. From this, a two-parameter group of Lorentz rotations, which leave the tetrad of eigenvectors unchanged, is deduced.
A characterization theorem for realizations of sl(2)
Authors: Joseph A.
Published: November 2008

Let A1 denote the algebra generated by one pair of canonical operators. It is shown that the realizations of sl(2) in A1 divide into two non-empty classes determined by the spectrum of the semisimple element of sl(2) in A1.
A source-sink flow in a rotating fluid
Authors: Johnson J. A.
Published: November 2008

The flow from a line source to a line sink in a multiply connected region is discussed. It is shown that the transport from source to sink passes through the interior region and the Stewartson layers as well as through the Ekman layers.
On forced three-dimensional surface waves in a channel in the presence of surface tension
Authors: Rhodes-Robinson P. F.
Published: November 2008

In this paper wave-maker theory including the effect of surface tension is determined for three-dimensional motion of water in a semi-infinite rectangular channel with outgoing surface wave modes allowed for at infinity; the motion is generated by a harmonically oscillating vertical plane wave-maker at the end of the channel and the cases of both infinite and finite constant depth are treated. The solution of the boundary-value problem for the velocity potential is more complicated in the presence of surface tension due mainly to the additional effect of the channel walls at which the normal free surface slopes are prescribed—as also is the slope at the wave-maker—to ensure uniqueness. The simpler three-dimensional solution for a semi-infinite region—obtained long ago by Sir Thomas Havelock in the absence of surface tension for the case of infinite depth—is also noted.
Abstract orthogonality and orthocomplementation
Authors: Cattaneo Gianpiero , Manià Alessandro
Published: November 2008

The notion of orthogonality is axiomatically defined on a poset. Various notions of orthocomplementation are distinguished and conditions are given in order to induce an orthocomplementation from an orthogonality and vice versa. Subsequently ⊥-modular ⊥-poset are defined and the set of morphisms between two posets with orthogonality is briefly discussed. Given the notions of additive monoid and of positive semi-ring, an orthogonality relation is introduced on the set of idempotent elements of a positive semi-ring. Finally, the obtained results are applied to the set of idempotent and absorbent endomorphisms of an additive monoid.
On orthogonality of Riesz products
Authors: Brown Gavin , Moran William
Published: November 2008

A typical Riesz product on the circle is the weak* limit

where – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).
Some theorems on strong regularity of Nörlund methods
Authors: Kuttner B.
Published: November 2008

1. Given a series

we will throughout write {sn} for the sequence of its partial sums; that is to say,

We will write p, q to denote the sequences {pn}, {qn}; we will write throughout

and will take P−1 = p−1 = 0.
Analytical theory of non-linear oscillations
Authors: Obi Chike
Published: November 2008

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equation

where g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).
Spatial numerical range of an operator
Authors: Tillekeratne K.
Published: November 2008

0. Introduction. Let X be a normed space and let T be an operator on X. Let S(X) denote its unit sphere, {x ∈ X: ∥x∥ = 1}, B(X) = {x ∈ X: ∥x∥ ≤ 1} its unit ball, X′ its dual and ℬ(X) the normed algebra of bounded linear operators on X. Let II be the subset of the Cartesian product X × X′ defined by
On the elasticity and stability of perfect crystals at finite strain
Authors: Hill R.
Published: November 2008

The concept of ideal strength of perfect crystals as an instability phenomenon is critically reviewed in the context of generalized moduli associated with arbitrary measures of stress and strain. Further aspects of elastic response to finite strain are discussed in relation to the classical model of an atomic lattice.
Regular neighbourhoods and mapping cylinders
Authors: Kearton C.
Published: November 2008

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.
A diophantine problem on groups. III
Authors: Baker R. C.
Published: November 2008

The following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequences

are uniformly distributed on the circle, for almost all x in G.
The downstream solution for steady viscous flow past a paraboloid
Authors: Miller D. R.
Published: November 2008

The flow of a viscous incompressible fluid past a paraboloid of revolution is described by matching a series of boundary-layer approximations valid far downstream to a series of potential flow solutions valid far from the solid surface. The development parallels that of Goldstein (3) and Murray (10) for the flat plate, becoming identical for infinitely large Reynolds number; it is found that logarithmic terms must be introduced at the third stage of the matching, and that these produce constants in the downstream solution which remain indeterminate. These terms result from interaction between inner and outer solutions, rather than from appearance of a complementary function of the inner equation.
Functional Massey products and homological algebra
Authors: Snaith V.P.
Published: November 2008

0. Introduction. In (8, 9) certain higher order operations, called K-theory Massey Products, were introduced and developed. These operations were designed to investigate the Kunneth formula spectral sequence in equivariant K-theory, constructed in (4). In that application the important feature of Massey products was that they gave operations on certain Tor-algebras which were well-behaved with respect to the algebraic coboundary, δa.
Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients
Authors: Askey Richard , Gasper George
Published: November 2008

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the types

and their duals

hold, where pn(x) and qn(x) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e−pxLn(x).
The localizability of particles in de Sitter space
Authors: Hannabuss K. C.
Published: November 2008

Motivated by the Iwasawa decomposition and its geometrical interpretation, two new decompositions of the de Sitter group are obtained. The first is applied to construct the representations of the de Sitter group in a form immediately comparable with those of the Poincaré group. In particular they act on functions over an hyperboloid like the momentum hyperboloid of the Poincaré group, although they require both positive and negative mass shells of that hyperboloid. Using the second decomposition it is shown that the representations of the de Sitter group are localizable in the sense of Mackey and Wightman. Position operators are exhibited.
A note on the image system for a stokeslet in a no-slip boundary
Authors: Blake J. R.
Published: November 2008

The velocity and pressure fields for Stokes's flow due to a point force (‘stokeslet’) in the vicinity of a stationary plane boundary are analysed, using Fourier transforms, to obtain the image system required to satisfy the no-slip condition on the boundary. The image system, which is illustrated by diagrams, is found to consist of a stokeslet equal in magnitude but opposite in sign to the initial stokeslet, a stokes-doublet and a source-doublet, the displacement axes for the doublets being in the original direction of the force. The influence of the wall on the near and far fields is discussed. In the far field it is found that a stokeslet aligned parallel to the wall produces a stokes-doublet far-field, whereas a stokeslet normal to the wall produces a combination of a source-doublet and a stokes-quadrupole far-field. Although results can be alternatively derived by the method of Lorentz (7) using a reciprocal theorem, the present method yields much more clearly the form of the image system.
The propagation of magneto-thermo-viscoelastic plane waves in a parallel union of the Kelvin and Maxwell bodies
Authors: Chandrasekhariah D. S.
Published: November 2008

The propagation of plane waves in a viscoelastic body representing a parallel union of the Kelvin and Maxwell bodies placed in a magneto-thermal field is investigated. It is shown that the longitudinal component of the wave is in general coupled with a transverse component and the wave travels in two families. In particular if the primary magnetic field is either parallel or perpendicular to the direction of wave propagation, the three components of the wave travel unlinked, with either the longitudinal component or the transverse components unaffected by the presence of the electromagnetic field. If the electrical conductivity of the solid is infinite the effect of the primary magnetic field is to increase the values of the material constants. The effect of wave propagation on magnetic permeability is equivalent to an anisotropic rescaling of the primary magnetic field. Some of the results obtained in the earlier works are obtained as particular cases of the more general results derived here.
On the absolute summability and convergence of Fourier series and associated series
Authors: Nayak M. K.
Published: November 2008

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x by

We denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the type

We write

The object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)
The topologies of separate continuity. II
Authors: Knight C. J. , Moran W. , Pym J. S.
Published: November 2008

In (6) we studied a topology T on the product set X × Y of two topological spaces X and Y which was defined by the requirement that each mapping from X × Y which was continuous in each variable separately was also continuous in T; we called (X × Y, T) the tensor product of X and Y, and denoted it by X ⊗ Y. Theorem (3·2) of (6) indicated that X ⊗ Y was rarely completely regular; as complete regularity is of importance in analytic problems, we consider here a ‘completely regular tensor product’ . Roughly speaking, gives a tensor product in the category of completely regular topological spaces. The categorical properties of are discussed in section 5.
On m-ary partitions
Authors: Gupta Hansraj
Published: November 2008

1. Let g(n) = g(n; m) denote the number of partitions of a natural number n into powers of a given natural number m (> 2). Then the generating function of g(n) is

(1) where we take g(0) = 1.
On the homeotopy group of a non-orientable surface
Authors: Birman Joan S. , Chillingworth D. R. J.
Published: November 2008

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).
Coin tossing and powers of singular measures
Authors: Brown Gavin , Moran William
Published: November 2008

We shall be concerned with the probability distributions which arise from naive coin tossing experiments and their repetitions. In each case our main question is whether the resulting measure is singular with respect to Lebesgue measure or absolutely continuous. It is a remarkable fact that althoughthese distributions were defined and studied some forty years ago (see (17)for an interesting account), they already provide examples necessary to recent studies of the convolution measure algebra of a locally compact abeliangroup. In order to verify this one is obliged to make substantial improvements on the classical results but, at least at the technical level, this requires no modern apparatus. Accordingly, we shall restrict attention to the circle. Before doing this, however, we summarize the motivational background from abstract harmonic analysis and indicate the applications of our results to that area.
Combinatorial applications of an inequality from statistical mechanics
Authors: Seymour P. D. , Welsh D. J. A.
Published: November 2008

The main part of this paper shows how an inequality of statistical mechanics has several applications in combinatorial theory.

The inequality (known as the FKG inequality) was derived by Fortuin, Kasteleyn and Ginibre(4) in their work on the statistical mechanics of Ising ferromagnets. We first show how it leads to new properties of log supermodular functions, Bernstein polynomials, and log convex sequences.
Linear forms in algebraic points of Abelian functions. I
Authors: Masser D. W.
Published: November 2008

Let Ω be a Riemann matrix whose 2n columns are vectors of Cn. It is well-known (e.g. (10)) that the field of meromorphic functions on Cn with these vectors among their periods is of transcendence degree n over C. More precisely, this field can be written as C(A, B) where A = (A1, …, An) is a vector of algebraically independent functions of the variable z = (z1, …, zn) and B is algebraic over C(A). We shall assume that B is in fact integral of degree d over the ring C[A]. Since the derivatives ∂f/∂zi of a periodic function f(z) are also periodic, the field is mapped into itself by the differential operators ∂/∂zi. Thus there exists a function C(z) in C[A] such that these operators map the ring C[A, B, C−1] into itself.
Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties
Authors: Brown Ronald , Higgins Philip J. , Morris Sidney A.
Published: November 2008

It is well-known ((2), Theorem 9·11) that any closed subgroup of Rn is isomorphic (topologically and algebraically) to Ra × Zb, where a, b are suitable non-negative integers. For an infinite product of copies of R, it is also known that any locally compact (hence closed) subgroup is a product of copies R and Z, and that any connected subgroup is a product of copies of R (see (7), (3), respectively). Some information is also given in (3) on closed subgroups of products of copies of R and T, where T = R/Z is the circle group.
The law of large numbers for additive arithmetic functions
Authors: Elliott P. D. T. A.
Published: November 2008

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequencies

We shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.
A look back at Ramanujan's Notebooks
Authors: Birch B. J.
Published: November 2008

Ramanujan's notebooks were the theme of a lecture (20) given by G. N. Watson to the London Mathematical Society in 1931. At that time, he and B. M. Wilson were collaborating on a critical edition; Watson was anticipating that the task might take a further 5 years. Unhappily, Wilson died, and the task of editing the notebooks has never been completed.
Lower dimensional absolute continuity properties of quasiconformal mappings
Authors: Gehring F. W.
Published: November 2008

Let Rn denote Euclidean n-space. For p ∈ [0, ∞), the normalized Hausdorff p-dimensional outer measure of a set E ⊂ Rn is defined as

where the infimum is taken over all countable coverings of E by sets Ej with

and

(See, for example (4).) Then ℋP (E) measures the p-dimensional size of E. For example, ℋ0 (E) ≤ ∞ if and only if E is finite, in which case ℋ0(E) = card (E). Next if E is an arc or Jordan curve, then ℋ1(E) ≤ ∞ if and only if E is rectifiable, in which case ℋ1(E) = length (E). Finally if p is an integer in [1, n] and if E lies in a p-dimensional hyperplane T in Rn, then

where mp denotes the Lebesgue p-dimensional outer measure in T. Thus as p varies from 0 to n, the measures ℋp interpolate in a natural way between the counting measure and Lebesgue outer measure in Rn.
Partial isometries which are sums of shifts
Authors: Fishel B.
Published: November 2008

Let denote a Hilbert space (real or complex), with inner product (|). In order to present our notation, we recall that if is a vector subspace of (and ‘vector sub-spaces’ will always be closed), with orthocomplement , a partial isometry V with initial domain is a linear operator in which preserves length, and so inner-product, in and is zero in is the final domain of V, and it is easy to verify that V*, the adjoint operator, is also a partial isometry, with initial domain and final domain .
Non-linear angle-sum relations for polyhedral cones and polytopes
Authors: McMullen P.
Published: November 2008

It is shown that the internal and external angles at the faces of a polyhedral cone satisfy various bilinear relations. The first two of these are related to the Gauss–Bonnet and Steiner parallel formulae for spherical polytopes, while the third is completely new. However, the proofs are basically combinatorial in nature, rather than differential geometric, as in the more classical treatments. These relations lead to inversion formulae, analogous to Euler-type relations, for certain functions defined on polytopes and polyhedral cones. As a result, various new relations involving quermassintegrals and Grassmann angles are found; there is also an application to lattice polytopes.
A remark on the K-theory of BГq
Authors: Whitley Alan
Published: November 2008

1. In (4) Bott and Heitsch discussed certain classifying spaces of importance in the study of foliations of various types. For example they studied the space BГq which helps to classify real foliations of codimension q and proved that a canonical map

induces an injection in integral cohomology for q ≥ 2. Similar results hold for classifying spaces BUГq and BSГq related to complex and oriented foliations respectively. The object of this paper is to demonstrate that the canonical maps referred to above also induce injections in integral K-theory.
A new formula for cumulants
Authors: Good I. J.
Published: November 2008

Let θ be ann-dimensional random vector with components θ1, θ2, …, θnand let itscumulants

be defined as usual by the identity

wherex1,x2, …,xnare purely imaginary variables,Edenotes expectation, and

Let θ(1), θ(2), …, θ(R) be independent and identically distributed (i.i.d.) random vectors each with the same distribution as θ, whereR=r1+r2+ … +rn= |r|, theorderof the cumulantKr. Let θ* have the components
Gap Tauberian theorem for generalized Abel summability
Authors: Krishnan V. K.
Published: November 2008

Let α > − 1. For a given series write

it being tacitly assumed that the series defining a(t) and A(t) converge for all t > 0. Σan is said to be summable (Aα) to A if
A characterization of the Poisson process using forward recurrence times
Authors: Isham Valerie , Shanbhag D. N. , Westcott M.
Published: November 2008

Consider a renewal process on the nonnegative real line with non-arithmetic distribution function F(x). Denote by V(x; t) the distribution function of the forward recurrence time from t, t ≤ 0. If t is chosen at random with distribution function Ф(t), the corresponding unconditional forward recurrence time has distribution function
Linear forms in algebraic points of Abelian functions. II
Authors: Masser D. W.
Published: November 2008

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.
A class of approximate exterior rotating solutions of Einstein's equations
Authors: Islam Jamal N.
Published: November 2008

A class of approximate stationary vacuum solutions are obtained by expanding the metric in powers of a certain parameter, and by solving explicitly the first few orders in terms of two harmonic functions ø and ψ. These solutions, to the order considered, reduce to the Weyl solutions when ψ = 0 and yield the Papapetrou solutions when ø = 0, and contain a subclass that is asymptotically flat and has non-zero mass. The physical interpretation and the possibility of proceeding further are briefly discussed.
A model for the two-way propagation of water waves in a channel
Authors: Bona Jerry L. , Smith Ronald
Published: November 2008

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equations

This system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.
Applications of Leray–Schauder degree theory to problems of hydrodynamic stability
Authors: Benjamin T. Brooke
Published: November 2008

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.
On semifinite AW*-algebras
Authors: Wright J. D. Maitland
Published: November 2008

Kaplansky (8, 9, 10) introduced the concept of an AW*-algebra and, by purely algebraic methods, succeeded in extending the ‘Type Theory’ of Murray and von Neumann from W*-algebras to these more general algebras. Every W*-algebra is an AW*-algebra but the converse is false, for there exist commutative AW*-algebras which are not W*-algebras.
On a Gaussian process related to multivariate probability density estimation
Authors: Silverman B. W.
Published: November 2008

The multivariate Gaussian process with the same variance/covariance structure as the multivariate kernel density estimator in Euclidean space of dimension d is considered. An exact result is obtained for the limit in probability of the maximum of the normalized process. In addition weak and strong bounds are placed on the asymptotic behaviour of the maximum of the process over a multidimensional interval which is allowed to increase as the sample size increases. All the bounds obtained on the process are

Only the uniform continuity of the underlying density is assumed; the conditions on the kernel are also mild.
The number of solutions of xp = 1 in a finite group
Authors: Laffey Thomas J.
Published: November 2008

Let G be a finite group, p a prime divisor of |G| and suppose that G is not a p-group. In this note, we show that the number of elements x ∈ G such that xp = 1 is at most (p|G|)/(p + 1). This answers a question posed by D. MacHale. When G is a Frobenius group of order p(p + 1), p a Mersenne prime, the above bound is attained.
Spinors in Hilbert space
Authors: Plymen R. J.
Published: November 2008

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:

The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).
Modifications of Coulombic interactions by polarizable atoms
Authors: Judd B. R.
Published: November 2008

The analysis of the interactions between the electrons of different atoms is complicated by many effects. Unless the atoms are free and well separated, account has to be taken of covalency, overlap, exchange, and superexchange. But, although such effects are often important, it is sometimes useful to regard the electrons as being located on separate atoms. Two electrons, labelled 1 and 2, on respective atoms A and B, then interact principally through the Coulomb energy e2/r12. Neighbouring atoms L, M, N, …, which we suppose are spherically symmetric, make their presence felt by acting as polarizable spheres. The electric fields thereby produced modify the simple Coulombic expression for the energy of interaction of the two electrons. It is the purpose of this article to examine such modifications in detail.
On eigenstresses in a semi-infinite solid
Authors: Aderogba K.
Published: November 2008

A uniform eigenstrain is prescribed within a spherical subregion of an isotropic linearly elastic half-space. Combining an application of potential theory with the stress-function approach of Papkovitch and Neuber, and starting with Eshelby's well-known solution for the homogeneous infinite solid, it is shown that the residual problem of potential to be solved is the determination of Boussinesq's first and second three-dimensional logarithmic potentials of volume distributions. Although explicit results are supplied only for the case of a spherical inclusion, the dependence of the solution on the infinite solid solution holds good for an arbitrarily shaped transforming inclusion. This can be established on the basis of the principle of superposition, considering that an arbitrary volume is essentially made up from an infinite spectrum of spherical regions.
Orlicz sequence spaces without local convexity
Authors: Kalton N. J.
Published: November 2008

In this paper we continue the study of Orlicz sequence spaces initiated by Lindberg (5) and Lindenstrauss and Tzafriri (7), (8) and (9). Our main concern is to investigate features of the theory which occur when the restriction of local convexity is lifted. It is clear that some results will hold with identical proofs, at least when the space is locally bounded. However, we are chiefly interested in the differences which arise. We always assume that the Orlicz function F satisfies the Δ2-condition.
The ideal structure of the Stone-Čech compactification of a group
Authors: Baker J. W. , Milnes P.
Published: November 2008

Let S be both a topological space and a semigroup. For s ∈ S, define the maps λs and ρs of S to S by

We shall say that S is a right-topological (resp. left-topological) semigroup if ρs (resp. λs) is continuous for each s in S. We denote by Λ(S) the set

this is a subsemigroup of S.
Some combinatorial results involving Young diagrams
Authors: James G. D.
Published: November 2008

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.
The injective tensor product of ℒp-algebras
Authors: Carne T. K. , Tonge A. M.
Published: November 2008

When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.
Tensor products of compact convex sets and Banach algebras
Authors: Batty C. J. K.
Published: November 2008

Alfsen and Andersen(2) defined the centre of the complete order-unit space A(K) associated with a compact convex set K to be the set of functions in A(K) which multiply with A(K) pointwise on the extreme boundary of K, thereby generalizing the concept of centres of C*-algebras. It is therefore possible to extend this definition to include the space A (K; B) of continuous affine functions of K into a Banach algebra B. Such spaces arise in the theory of weak tensor products E ⊗λB of B with a Banach space E, which may be embedded in A(K; B) where K is the unit ball of E* in the weak* topology. Andersen and Atkinson(4) considered multipliers in A(K; B) and showed that if B is unital, then the multipliers are precisely those functions which are continuous in the facial topology on the extreme boundary. It is shown here that this result extends to non-unital Banach algebras with trivial left annihilator.
The variances of a sequence
Authors: Maddox I. J.
Published: November 2008

In statistical problems, use is often made of the variance s2 of a finite sequence of numbers x1, x2, …, xn, where

x¯ being the arithmetic mean of x1, x2, …, xn. The object of this note is to introduce a generalized variance for an infinite sequence x = (xk) and to examine some of its properties.
On non-uniqueness of the order of saturation
Authors: Kuttner B. , Sahney B.
Published: November 2008

1. Let X be a normed linear space of periodic functions (of period 2π), which includes the class of trigonometric polynomials. We restrict ourselves throughout to functions f ∈ X. Let D) = (dn, k) be the matrix of a regular sequence-to-sequence transformation. (We could also consider sequence-to-function transformations, in which case n is replaced by a continuous variable.) We suppose that, for all n,

Let {Ln(x)} be the D transform of the Fourier series of f(x). If f(x) is a constant, it follows from (1) that, for all n, Ln(x) = f(x). However, it is often found that, roughly speaking, except in this trivial case, Ln(x) cannot tend to f(x) (in the topology given by the norm) with more than a certain degree of rapidity. This leads to the concept of ‘saturation’, which was first introduced by Favard (1) and Zamanski (8) and which has since been investigated by many authors (see (2), (3), (4), (5), (6), (7)). This is defined as follows.
On length functions and normal forms in groups
Authors: Hurley Bernard
Published: November 2008

1. Introduction. Abstract length functions were first considered axiomatically in Lyndon (4). Length functions satisfying a less restrictive set of axioms were studied by Chiswell(1). Groups with a normal form structure (NFS-groups) were studied in Hurley (3) using a slightly different axiomatization to that used in this paper. Length functions naturally arise from normal form structures. In this paper we show that the length functions arising from normal form structures can be characterized axiomatic-ally by a set of axioms that consists of those of (1) together with one further axiom.
Splitting of certain spaces CX
Authors: Cohen F. R. , May J. P. , Taylor L. R.
Published: November 2008

There are a number of theorems to the effect that spaces of the form ΩnσnX split stably into wedges of simpler spaces when X is connected (by which we mean pathwise connected). The proofs generally proceed by exploitation of combinatorially manageable approximations for ΩnσnX.
The modular curve X0(39) and rational isogeny
Authors: Kenku M. A.
Published: November 2008

Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.
On harmonic functions which take integer values on integer lattices
Authors: Armitage D. H.
Published: November 2008

Throughout this paper α denotes a number larger than 1 and C denotes the complex plane. Pó1ya(15),(16) (see also Hardy(10) and Landau(13)) proved the following results.
Locally convex syntopogenous spaces
Authors: Burgess D. C. J. , Fitzpatrick M.
Published: November 2008

In (1) the authors introduced the idea of a syntopogenous pre-ordered space and, in (2), discussed the more general notion of a convex syntopogenous space. The former generalizes Nachbin's ‘uniform preordered space’ (4) and Singal and Lal's ‘proximity preordered space’ (7); the latter does the same for Nachbin's ‘convex topological space’ (4) and Redfield's ‘nearly uniform ordered space’ (5).
On the Dixmier property of certain algebras
Authors: Archbold R. J.
Published: November 2008

1. Introduction. The main result of this paper is that the simple C*-algebra (5) has the Dixmier property. As a corollary of the proof we obtain that the relative commutant of the Choi algebra (4) in is trivial.
Contractive inhomogeneous products of non-negative matrices
Authors: Cohen Joel E.
Published: November 2008

1. Introduction. Hajnal (5) showed that under wide conditions a sequence of products H(1, q) = Mq…M1q = 1,2,…, of square non-negative matrices Mq approaches a sequence of positive matrices of rank 1. We call a product H(1,q) inhomogeneous if its factors M1,…, Mq are not necessarily all equal to one another. When the matrices Mq are members of an ‘ergodic set’, and x and y are positive vectors, the projective distance d(H(1,q)x, H(1,q)y) decays at least exponentially fast as q increases. An important condition on an ergodic set is that any product of g members from the set be a matrix in which all elements are (strictly) positive, where g is some fixed positive integer.