bluuebunny/crossref_metadata_2025 huggingface dataset:
General series involving H-functions
Authors: Jain R. N.
Published: November 2008

MacRobert (4–7) and Ragab(8) have summed many infinite and finite series of E-functions by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration. Verma (9) has given two general expansions involving E-functions from which, in addition to some new results, all the expansions given by MacRobert and Ragab can be deduced. Proceeding similarly, we have studied general summations involving H-functions. The H-function is the most generalized form of the hypergeometric function. It contains a vast number of well-known analytic functions as special cases and also an important class of symmetrical Fourier kernels of a very general nature. The H-function is defined as (2)

where 0 ≤ n ≤ p, 1 ≤ m ≤ q, αj, βj are positive numbers and aj, bj may be complex numbers.
Degrees of greatest common divisors of invariant factors of two regular polynomial matrices
Authors: Barnett S.
Published: November 2008

In a previous paper it was shown that a necessary and sufficient condition for two regular polynomial matrices T and U to have relatively prime determinants is that a certain matrix R be non-singular. This generalization of the resultant of two scalar polynomials is extended to give an expression for the sum of the degrees of the greatest common divisors of all pairs of invariant factors of T and U in terms of the rank of R. Further, if rank R exceeds a given value then specified pairs of factors are relatively prime.
Some cobordism invariants for links
Authors: Tristram A. G.
Published: November 2008

The purpose of this paper is to obtain some necessary conditions for a link in Euclidean 3-space to be spanned by a locally unknotted surface of given type in one half of 4-space. In particular necessary conditions for two links to be cobordant are proved.
On the evaluation of certain lattice Green's functions
Authors: Byrnes J. S. , Podgor S. , Zachary W. W.
Published: November 2008

New methods are presented for the evaluation of lattice Green's functions for the body-centred cubic lattice.
The scattering of surface waves by a vertical plane barrier
Authors: Jarvis R. J. , Taylor B. S.
Published: November 2008

In this paper we use a method due to Williams(1) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.
A generalization of Rado's theorem on independent transversals
Authors: Perfect Hazel
Published: November 2008

The growth of transversal theory has been greatly enriched by the study of abstract independence. In this development, Rado's theorem on independent transversals (4) has played a prominent role, and recent work (e.g. by Welsh(7)) suggests that its applications are by no means exhausted. Rado's theorem can be extended in many directions, and my object is to describe a further generalization (Theorem 2 below) which does not seem to have been noted in the literature.
A note on simplex spaces
Authors: Ng Kung-Fu
Published: November 2008

Let X be a real Banach space which is partially ordered by a closed cone C. Let X* denote the Banach dual space, and let

X is called a simplex space (2) if (X*, C*,‖·‖) is an AL-space. We proved in (3) that X is a simplex space if and only if X is an approximate order-unit-normed space and has the Riesz decomposition property. When the given norm on X is an order-unit norm, some representation theorems are obtained by Effros ((2), theorem 4·8) and Sterrmer ((6), theorem 4·3). In this note, we shall study the representation problem for more general simplex spaces. One of our aims is to prove that X is equivalent to C0(ω) if and only if X has the Urysohn property (for definitions, see below).
Derivation of correction terms for general quadrature formulae
Authors: Grant J. A.
Published: November 2008

In this paper, it will be shown how it is possible, using Bernoulli polynomials of the first kind, to derive correction terms for any quadrature approximation in the form of a weighted sum over a finite interval. The terms derived will involve either the difference or the sum of derivatives taken at the ends of the range of integration, and they will be described as being of either Type A or Type B respectively. There will also be some discussion of the uses of such terms.
Weyl ǵroups and finite Chevalley ǵroups
Authors: Carter R. W.
Published: November 2008

In his fundamental paper (1) Chevalley showed how to associate with each complex simple Lie algebra L and each field K a group G = L(K) which is (in all but four exceptional cases) simple. If K is a finite field GF(q), G is a finite group of order

where l is the rank of L, m is the number of positive roots of L and d is a certain integer determined by L and K. The integers m1, m2,…,m1 are determined by L only and satisfy the condition
Boundary layers in micropolar liquids
Authors: Willson A. J.
Published: November 2008

Consideration is given to the steady flow of a micropolar liquid near a rigid boundary. It is shown that while micro-inertia itself is not important, nevertheless, some properties of the microstructure do play a vital rôle in determining the structure of the boundary layer. The Kármán—Polhausen method is used to provide an estimate of the shear stress at the boundary, and detailed calculations are given for flow near stagnation. The important significance of the standard length of the micropolar liquid is demonstrated for this flow, as for other flows in this medium.
On monotonicity of the relaxation functions of viscoelastic materials
Authors: Day W. A.
Published: November 2008

This note is concerned with one-dimensional viscoelastic materials. Experiments show that for linear materials the relaxation functions are monotone decreasing functions of the time. This monotonic property has not, as far as I am aware, been characterized and in this note I provide a characterization formulated in terms of an assertion about the work done on the material over certain closed paths in strain space. More explicitly, consider any path in strain space which starts from equilibrium and arrives at the strain e at time t. We can now take the material around a closed path by either retracing the given path immediately or we can do it by holding the strain fixed at value e for a time r and then retracing the given path. Let us call the total work done on the material on the first closed path w(0) and on the second w(r). In general, because of the memory of the material, w(t) ≠ w(0). If it is true that w(r) ≥ w(0), whatever value r ≥ 0 has and whatever the initial strain path starting from equilibrium, we say that work is always increased by delay on retraced paths. The characterization proved here is that if G(·) is the relaxation function, with equilibrium elastic modulus G(∞), then G(·) −G(∞) is completely monotone if and only if work is always increased by delay on retraced paths.†
The Rikitake two-disc dynamo system
Authors: Cook A. E. , Roberts P. H.
Published: November 2008

It is shown that the solution of the Rikitake two-discdynamo system may be described by an orbiting point which, for sufficiently large time, lies arbitrarily close to a limit surface of bounded area. Reversal of the field currents of the dynamo coils are shown to occur in the juxtaposed regions of the two sheets of the limit surface. The two singular points of the system are shown, by Liapounov's direct method, to be unstable foci. An asymptotic theory is developed for the case of small difference in dynamo velocities; the theory is applied to cases of small dissipation in which orbits tend to become nearly periodic with a single oscillation between reversals.
Convergence of weighted sums of independent random variables
Authors: Rohatgi V. K.
Published: November 2008

1. Introduction. Let {Xk: k ≥ 1} be a sequence of independent, but not necessarily identically distributed, random variables. Suppose that the random variables Xn are uniformly bounded by a random variable X in the sense that

(1) P(|Xn| ≥x) ≤ P(|X| ≥ x)for all x > 0. Write qn(x) = P(|Xn| ≥ x) and q(x) = P(|X| ≥ x).
On hearing the shape of a drum: an extension to higher dimensions
Authors: Waechter R. T.
Published: November 2008

The inverse eigenvalue problem for vibrating membranes (4), may also be examined in three or more dimensions. Let us suppose that λn are the eigen values of the problem

where Ω is a closed convex region or body in En and S is the bounding surface of Ω. The basic problem is to determine the precise shape of Ω on being given the spectrum of eigenvalues λn. In analogy with the membrane problem, it is clear that the trace function may be constructed in identical fashion; thus

where G(r, r', t) is the Green's function of the diffusion equation

and satisfies the Dirichiet condition G(r, r', t) = 0, r∈S, and the initial condition G(r, r', t) → δ(r–r') as t → 0.
A note on Watson transforms
Authors: De Snoo H. S. V.
Published: November 2008

In (6) Plancherel considered linear, bounded transforms which satisfy

where the linear, bounded transform is defined by

Among other things he proved: if W1, W2 and W3 are linear, bounded transforms satisfying (1), then
The polygon-circle paradox and convergence in thin plate theory
Authors: Murray N. W.
Published: November 2008

The solution for a simply supported many-sided polygonal plate does not agree with that for the corresponding circular plate. This paper describes the earlier work of Rao and Rajaiah on polygonal plates and then explains why best convergence of series solutions occurs when the boundary conditions are defined as
An open mapping theorem
Authors: Ng Kung-Fu
Published: November 2008

Let (E, τ) be a complete, semi-metrizable topological vector space. Let p be a pseudo-norm (not to be confused with a semi-norm, cf. (8), p. 18) inducing the topology τ. For each positive real number r, let

Let f be a continuous linear function from E into a topological vector space F. The open mapping theorem of Banach may be stated as follows: If f is nearly open, that is, if the closure of each f(Vr) is a neighbourhood of O in F then whenever β > α > O; in particular, each f(Vr) is a neighbourhood of O. We note that f, identifying with its graph, is a closed linear subspace of the product space E × F. In this paper, we shall employ techniques developed by Kelley (6) and Baker (1) to extend the theorem to the case where f is taken to be a closed cone in E × F. The generalized theorem throws some light onto the duality theory of ordered spaces. In particular, the theorem of Andô–Ellis is generalized to (not assumed, a priori to be complete) normed vector spaces.
Uniformities with totally ordered bases have paracompact topologies
Authors: Hayes Allan
Published: November 2008

This result does not appear to be in the literature, though the method of proof is related to those used in (1) and, particularly, (2) to prove A. H. Stone's theorem that metrics have paracompact topologies.
A note on a class of integral inequalities
Authors: Horgan Cornelius O.
Published: November 2008

A unified variational approach to a class of second-order integral in-equalities is presented. A special case recently considered in a different manner by Anderson, Arthurs and Hall (1) is recovered.
Probability and the elementary symmetric functions
Authors: Smith J. Denmead
Published: November 2008

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.
The numerical range of functions and best approximation
Authors: Harris Lawrence A.
Published: November 2008

In this note, we state general conditions which imply that the numerical range of a function mapping a setSinto a normed linear spaceYis the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.
A theorem in infinite products
Authors: Price W. L. V.
Published: November 2008

Certain infinite products occur in potential problems involving plane boundaries and elements, such as line charges, which produce logarithmic potentials. In this paper, the author derives a theorem which is used to evaluate a particular product which arose in the calculation of the electric potential distribution in a flat, square sheet of conducting material when an electric current enters and leaves respectively at line-contact electrodes attached to adjacent corners of the square.
On the structure of non-Euclidean crystallographic groups
Authors: Singerman David
Published: November 2008

Let denote the group of transformations of the upper-half complex plane U onto itself of the form

If, on U, we introduce the Riemannian metric ds = |dz| y−1 (z = x + iy), then U becomes a model of the hyperbolic plane and its group of isometries. The set of elements of type I, the orientation-preserving isometries form a subgroup of index two in , which we denote by .
Module invariants and root numbers for quaternion fields of degree 41r
Authors: Fröhlich A.
Published: November 2008

1. The results. Let l be an odd prime, r ≥ 1, and let

be the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.
The Stokes phenomenon and generalized contiguous transformations of some generalized hypergeometric functions
Authors: Heading J.
Published: November 2008

Previous investigations by the author into the Stokes phenomenon pertaining to solutions of the differential equation dnu/dzn = (–1)nzmu are extended in order to find when different equations have the same set of Stokes multipliers, with perhaps a series of zeros being additionally allowed. The reason for periodic cycles to exist (with n fixed and m varying), with the same Stokes multipliers regained after a complete cycle, is traced to certain transformation properties of the equations. Within the first cycle (with n fixed and m varying) further remarkable identities exist between the Stokes multipliers, and this also is traced to special transformations between the equations. Relations are found toexist when values of m are chosen so that the highest common factors of the two integers n and n/(n + m) are identical. Finally, a transformation of the independent variable is deduced whereby the set of Stokes multipliers for an equation of order n is identical (apart from the additional zeros) to that for an equation of lower order. A hierarchy of equations is thrown up, whereby certain basic equations are transformed to yield more advanced equations of higher order and different m.
Dynamic elastic moduli of a suspension of imperfectly bonded spheres
Authors: Mal A. K. , Bose S. K.
Published: November 2008

An isotropic elastic material containing a random distribution of identical spherical particles of another elastic material is considered. The bonding between the spheres and the matrix is imperfect, so that slip may occur at interfaces when stress is applied to the medium. The shear stresses at the interface is assumed to be proportional to the amount of slip. The velocity and attenuation of the average harmonic elastic waves propagating through such a medium are calculated. The results are valid to the lowest order in frequency for wave lengths long compared with the radius of the sphere. The dynamic elastic moduli are obtained from these results and are compared with available results for welded contact. The variations in thePandSwave velocities for propagation across earthquake faults is discussed.
A footnote on the mystic hexagram
Authors: Edge W. L.
Published: November 2008

1. Cremona's projection of 15 lines on a nodal cubic surface onto a plane π is a striking example of the resolution of a highly complicated figure into its constituent simplicities, for it explains, almost at a glance, so many properties of Pascal's mystic hexagram. These may appropriately be called ((12); (10), p. 162) the Veronese properties and have been described on several occasions ((12); (6); (10); (1), pp. 219–236; (2), pp. 349–355).
Extensions of self-dual orders
Authors: Singer Michael
Published: November 2008

We give a new criterion for an order in a commutative split algebra over the quotient field of a discrete valuation ring R to be self-dual. We derive a result on the ideal classes whose order is an extension by R of a self-dual order. This yields a wide generalization of a theorem of Faddeev(1). We can also answer a question arising from a paper of Fröhlich(2).
A generalized inverse for arbitrary operators between Hilbert spaces
Authors: Erdelyi I.
Published: November 2008

A function analytic approach to the generalized inversion problem, for arbitrary operators between Hilbert spaces, is investigated in the present paper.

The generalized inverse is defined as the inverse of the largest invertible restriction of the given operator and it is extended by zero to the orthogonal complement of the range of the invertible restriction. This domain-dense operator satisfies, on some restricted manifolds, the defining properties of the generalized inverse in some special cases, and it provides the least extremal solution of possibly inconsistent linear equations.
Cobordism of homology manifolds
Authors: Martin N.
Published: November 2008

It is the object of this paper to prove a transversality theorem for homology manifolds which is strong enough to be then able to prove a Thom theorem for the cobordism ring of homology manifolds. The new bundle theory necessary for this follows on naturally from that developed in ‘Homology Cobordism Bundles’(2).
Realizing symplectic bordism classes
Authors: Ray Nigel
Published: November 2008

This is the second of two papers elaborating (4), and is concerned with the more geometrical aspects of the symplectic bordism ring in low dimensions. As such, it is a natural sequel to (2). However, the reader who does not care for the algebra therein need only consult (3), in which appears all the notation and prerequisites for this paper.
Operators on the generalized entire sequences
Authors: Maddox I. J.
Published: November 2008

Let p = (pk) and q = (qk) be any sequences of strictly positive numbers. Denote by c0(p) the set of all complex sequences x = (xk) such that . The Köthe–Toeplitz dual of c0(p) was determined in Theorem 6(4). In the case in which p∈l∞ (the space of all bounded sequences), c0(p) becomes a locally convex FK space under the paranorm where M = max (1, sup pk). It has been recently proved by Lascarides(3) that the following three properties are equivalent:

(a) p∈c0(i.e.pk → 0),

(b) c0(p) is perfect (in the sense of Köthe-Toeplitz duality),

(c) c0(p) has the Schur property (i.e. weak and strong sequential convergence are equivalent).
Topological bifurcation for the double cusp polynomial
Authors: Godwin A. N.
Published: November 2008

In his work on elementary catastrophes Zeeman(1) has considered what he has named as the double cusp catastrophe. This catastrophe is defined by the unfolding of the two variable polynomialx4+y4. Using Mather's results (2) on stability of singular germs ofC∞maps we can find an expression for the unfolding. The eight dimensional unfolding can then be considered as a polynomial in two variables with eight parameters.
Generalized metric spaces are paracompact
Authors: Souppouris D. J.
Published: November 2008

A generalized metric space is a triple (X, d, G), where X is a set, G an ordered Abelian group and d: X x X → G is a function which satisfies the metric axioms. The generalized metric d defines a topology on X in the same way as an ordinary metric does for a metric space. These spaces are studied by Stevenson & Ton(5) and Sikorski (4) amongst others. A survey of these spaces is the subject of an M.Phil. thesis by the author (6). Generalized metric spaces satisfy most of the separation properties of ordinary metric spaces (T2, regular, etc.). We prove here that these spaces are also paracompact. The proof is a generalization of the proof given by Ruffin (2) for metric spaces. The result given here follows, in fact, from a theorem proved by Hayes (1). He proved that uniformities with totally ordered bases have paracompact topologies. It is known (Stevenson & Ton (5); Shock (3); Souppouris (6), chapter 5) that these spaces are identical to generalized metric spaces. The proof given here uses metric space language.
Lattice bounding, Radonifying and summing mappings
Authors: Garling D. J. H.
Published: November 2008

There are several known results concerning Radonifying and summing properties of linear mappings in terms of their transposed operators. Thus Schwartz has shown that a sufficient condition for a mapping to be p-Radonifying is that its transposed mapping should be p-decomposing, and that this condition is also necessary if the image space has good topological properties. But as Schwartz observes, these properties are seldom observed by biduals of Banach spaces with their weak-topologies, which arise when 0 ≤ p ≤ 1. This suggests that it would be worth looking for a weaker condition than ‘p-decomposing’.
Some remarks on a paper by Vizing on critical graphs
Authors: Fiorini Stanley
Published: November 2008

In papers (7, 8), Vizing studied graphs which are critical with respect to edge colourings. In particular, he obtained bounds on the number of edges and on the circuit length of such graphs in terms of its maximum valency. The object of this paper is to improve these bounds by obtaining others which also take into account the order of the graph.
Some extremal properties of convex sets
Authors: Lillington J. N.
Published: November 2008

In this paper we shall suppose that all convex sets are compact convex subsets of Euclidean space En. We shall be concerned in producing upper and lower bounds for the ‘total edge lengths’ of simplices which are contained in or contain arbitrary convex sets in terms of the inradii and circumradii of these sets. However, before proceeding further, we shall introduce some notation and give some motivation for this work.
Sufficient conditions for regularity, recurrence and ergodicity of Markov processes
Authors: Tweedie Richard L.
Published: November 2008

Two sets of conditions are found on the Q-matrix of an irreducible Markov process on a countably infinite state space which ensure that Q is regular; the first set also implies that the Markov process is ergodic, the second that it is recurrent. If the process is not irreducible, the conditions still imply regularity, and then either non-dissipativity or ‘ultimate recurrence’ (reducible analogues of ergodicity and recurrence) of the process. Conditions sufficient for ergodicity or recurrence of the process in the non-regular and regular case are also given. The results parallel (and use) results for discrete time Markov chains, and the known discrete time recurrence condition is extended to the reducible case. The conditions are illustrated by a competition process example.
Integrability of singular distributions on Banach manifolds
Authors: Chillingworth David , Stefan Peter
Published: November 2008

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity. In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.
Derangements and Laguerre polynomials
Authors: Even S. , Gillis J.
Published: November 2008

Given a set consisting of n1, objects of type 1, n2 of type 2, …, nk of type k, we denote by the number of possible derangements of the set i.e. permutations in which no object occupies a site originally occupied by an object of the same type. A formula is found for in terms of Laguerre polynomials, and some of its implications are considered.
Hilbert space methods in the theory of Jordan algebras. II
Authors: Devapakkiam C. Viola , Rema P. S.
Published: November 2008

In this paper we consider the classification problem for separable special simple J*-algebras (cf. (8)). We show, using a result of Ancochea, that if is the (finite-dimensional) Jordan algebra of all complex n × n matrices and ø a Jordan isomorphism of onto a special J*-algebra J then An can be given the structure of an H*-algebra such that ø is a *-preserving isomorphism of the J*-algebra onto J. This result enables us to construct explicitly a canonical basis for a finite-dimensional simple special J*-algebra isomorphic to a Jordan algebra of type I from which we also obtain canonical bases for special simple finite-dimensional J*-algebras isomorphic to Jordan algebras of type II and III.
Borsuk shape and a generalization of Grothendieck's definition of pro-category
Authors: Deleanu Aristide , Hilton Peter
Published: November 2008

Following LeVan (6), we were led in (3) to investigate a purely categorical formulation of shape theory, which had hitherto been confined to certain rather circumscribed areas of topology. However, several authors (6, 7, 8, 9, 10, 11) had noted that, within this topological context, shape theory could be related to the process of expressing objects of the larger category T as filtered limits of objects of the given full subcategory P.
The generation of surface waves over a sloping beach by an oscillating line source
Authors: Morris Clare A. N.
Published: November 2008

A line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.
The Stone-Čech compactification of a topological semigroup
Authors: Baker J. W. , Butcher R. J.
Published: November 2008

1. Introduction. Throughout this paper we shall take S to be a separately continuous, completely regular and Hausdorff, topological semigroup. We denote by Cb(S) the space of continuous and bounded complex-valued functions on S, and by βS the Stone-Čech compactification of S.
An empirical form of the square root law and the central limit theorem
Authors: Freedman David
Published: November 2008

Consider a long, finite sequence xl, x2, …, xN of repeated measurements on the same physical quantity. The usual model is that the data consists of observed values on a sequence X1, X2, …, XN of independent and identically distributed random variables. It is customary to require that the Xi have a finite mean and variance: µ = E(Xi) is the ‘true value’ of the quantity being measured; and

measures the variability in the measuring process.
Point processes on spaces of flats and other homogeneous spaces
Authors: Papangelou F.
Published: November 2008

The present paper is primarily concerned with the structure of certain random sets of flats in n-dimensional Euclidean space Rn. To be more specific, let be the space of k-dimensional affine subspaces of Rn (to be called k-flats below), equipped with its natural topology ((14), p. 132). is a topological homogeneous space under the action of the group of Euclidean displacements in Rn and our purpose is to give conditions under which a stochastic point process on , stationary under , is a doubly stochastic Poisson process. The results will be seen to apply equally well to abstract spaces sharing some of the structural properties of .
Some points in βN
Authors: Kunen Kenneth
Published: November 2008

Assuming either the continuum hypothesis or Martin's axiom, we show that in the space βN – N, there are: (a) points which are not P-points, but which are also not limit points of any countable set, and (b) a countable set of points dense in itself such that each of the points is not a limit point of any countable discrete set. Our method is to construct such points in the Stone space of a measure algebra, and then embed that Stone space into βN – N. We also, by a similar use of measures, establish the independence of the existence of selective ultrafilters by showing that there are none in the random real model.
On Legendre transformations and elementary catastrophes
Authors: Sewell M. J.
Published: November 2008

The purpose of these remarks is to use elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and certain elementary catastrophes. Multi-valued Legendre transforms appear in subjects such as non-linear elasticity and non-convex optimization.
On the Banach–Saks property
Authors: Partington J. R.
Published: November 2008

A Banach space X is said to have the Banach–Saks property (BS) if every bounded sequence (xn) in X has a subsequence (), which is (C, 1) convergent in norm to a point x in X; that is,

Kakutani (7) showed that all uniformly convex spaces are (BS); moreover, all (BS) spaces are reflexive. It is further known that both these implicationsare strict: see, for example, Baernstein (1) and Diestel (4).
The left Hilbert algebra associated to a semi-direct product
Authors: Rousseau R.
Published: November 2008

Let A and G be locally compact groups and α a continuous action of G on A, and let denote the semi-direct product of A and G. Then we prove that the left Hilbert algebra of continuous functions with compact support, has the same achieved left Hilbert algebra, as the crossed product of K(A)" by the associated action α̃ of G on . As a consequence we obtain that the canonical weight on is the dual weight of the canonical weight on K(A)".
On the Lp convergence of sums of independent random variables
Authors: Hall Peter
Published: November 2008

Let {Xnj, 1 ≤ j ≤ kn} be independent random variables with zero means and satisfying . Let p ≥ 1. We prove that

if and only if, for all ε > 0,

and we use this result to obtain necessary and sufficient conditions for the Lp convergence of sums of non-negative, independent random variables.
On the existence theory for irrotational water waves
Authors: Keady G. , Norbury J.
Published: November 2008

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.

Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

where g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.
On the minimal prime ideals of a tensor product of two fields
Authors: Vámos P.
Published: November 2008

Let F be a field, L a commutative F-algebra and K an extension field of F. An important area of commutative algebra is the study of the passage from L to the k-algebra K ⊗FL, i.e. the investigation of the behaviour of the ideals of L under ‘extension of scalars’. In most problems of this kind one finds that the problem is reduced to the case when the algebra L is itself an extension field of F. It is for this reason that tensor products of fields play an important role (see, for example, (2), chap, viii, (3), (5), (9) and (12), vol. I).
Injective modules for group algebras of locally finite groups
Authors: Musson I. M.
Published: November 2008

Two recent results relate the existence of injective modules for group algebras which are ‘small’ in some sense to the structure of the group.

(1) The trivial kG-module is injective if and only if G is a locally finite group with no elements of order p = char k (9).

(2) If (G) is a countable group, then every irreducible kG-module is injective if and only if G is a locally finite p′ group which is abelian-by-finite (9) and (11)
The spectra associated to ℐ-monoids
Authors: May J. P.
Published: November 2008

In this final sequel to (9), I shall prove a general consistency statement which seems to me to complete the foundations of infinite loop space theory. In particular, this result will specialize to yield the last step of the proof of the following theorem about the stable classifying spaces of geometric topology.
Instability theorems for certain fifth-order differential equations
Authors: Ezeilo J. O. C.
Published: November 2008

1. Consider the constant-coefficient fifth-order differential equation:

It is known from the general theory that the trivial solution of (1·1) is unstable if, and only if, the associated (auxiliary) equation:

has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients a1, a2,…, a5. For example, if

it is clear from a consideration of the fact that the sum of the roots of (1·2) equals ( – a1) that at least one root of (1·2) has a positive real part for arbitrary values of a2,…, a5. A similar consideration, combined with the fact that the product of the roots of (1·2) equals ( – a5) will show that at least one root of (1·2) has a positive real part if

for arbitrary a2, a3 and a4. The condition a1 = 0 here in (1·4) is however superfluous when

for then X(0) = a5 < 0 and X(R) > 0 if R > 0 is sufficiently large thus showing that there is a positive real root of (1·2) subject to (1·5) and for arbitrary a1, a2, a3 and a4.
Representations and automorphisms of some wreath products
Authors: Gow Roderick
Published: November 2008

In this paper we study the faithful irreducible representations of certain families of wreath products and show how a particular class of such representations is related to the automorphisms of these groups. Our methods enable us to calculate the orders of the automorphism groups, although they do not deal with their structure. The wreath products that we consider here include most of the Sylow p-subgroups of the classical groups, which are described in (1) and (4).
Boundaries for polydisc algebras in infinite dimensions
Authors: Globevnik J.
Published: November 2008

Let AB be the algebra of all bounded continuous functions on the closed unit ball B of c0, analytic on the open unit ball, with sup norm, and let AU be the sub-algebra of AB of those functions which are uniformly continuous on B. Call a set S ⊂ B a boundary of AB (AU) if

for every f ∈ AB (f ∈AU, respectively). In the paper we study the boundaries of AB and AU. We give a complete description of the boundaries of AU and present some necessary and some sufficient conditions for a set to be a boundary of AB. We also give some examples of boundaries.
Special functions, infinite divisibility and transcendental equations
Authors: Ismail Mourad E. H. , May C. Ping
Published: November 2008

We establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.
Polygons and gravitons
Authors: Hitchin N. J.
Published: November 2008

1. There has been much interest recently in ‘instantons’. These correspond, in differential geometric terms, to connexions in principal bundles over Euclidean 4-space whose curvature satisfies the Yang-Mills equations. The connexion is also required to approach the trivial connexion at infinity. The self-dual solutions have been classified by converting the problem into one of algebraic geometry using the ideas of R. Penrose and R. Ward
Abelian theorems for the distributional Stieltjes transformation
Authors: Lavoine Jean , Misra O. P.
Published: November 2008

0. Introduction. Asymptotic behaviour of distributions (Silva(8) and Benedetto (l)) plays a fundamental role in the analysis of singularities of generalized integral transforms. Such analysis in terms of Abelian theorems for the distributional Stieltjes transformation has been obtained by Misra (6) and Lavoine and Misra (3). To obtain his results Misra used some Abelian theorems (see Misra (6), theorems 3·1·1 and 4·1·1) for the Stieltjes transformation of functions which were essentially generalizations of the results given by Widder (9), pp. 183–185. The object of the present paper is to complete the work given by the authors in (3); Sections 2 and 3 of this paper are devoted to obtaining Abelian theorems concerning the distributional Stieltjes transformation in terms of the behaviour of this transform near the origin and at infinity.
Total positivity and the embedding problem for Markov chains
Authors: Frydman Halina , Singer Burton
Published: November 2008

We prove that the class of transition matrices for the finite state time-inhomogeneous birth and death processes coincides with the class of non-singular totally positive stochastic matrices. Thus we obtain a complete solution to the embedding problem for this class of Markov chains.
Prime knots and concordance
Authors: Kirby Robion C. , Lickorish W. B. Raymond
Published: November 2008

This paper proves that any knot is concordant to a prime knot; it thus solves Problem 13 of (3). In doing so it makes an exploration of a fairly general method of proving that a knot is a prime. Throughout, the word ‘knot’ means a knot of S1 in S3 (orientations being here irrelevant); occasionally reference will be made to the idea of a knotted arc spanning a 3-ball.
Subgroups of the unimodular group defined by a congruence
Authors: Rankin R. A.
Published: November 2008

In an earlier paper I considered the problem of determining those subgroups of SL (2, ℤ) consisting of all matrices

satisfying a linear congruence
The projective characters of the Mathieu group M12 and of its automorphism group
Authors: Humphreys J. F.
Published: November 2008

In (1), Burgoyne and Fong have shown that the Schur multiplier of the Mathieu group M12 is of order 2. It is shown in Theorem 2·4 that the Schur multiplier of Aut M12, the automorphism group of M12, is also of order 2. It is therefore possible to choose a complex 2-cocycle α of Aut M12, taking only the values 1 and − 1, such that the cohomology class of α is of order 2 and the cohomology class of the restriction of α to M12 is of order 2. The characters of the irreducible α-projective representations of Aut M12 are calculated in § 2.
On bundle completion of parallelizable manifolds
Authors: Dodson C. T. J. , Sulley L. J.
Published: November 2008

In (5), Schmidt devised a procedure for completing a space-time M by adjoining a boundary ∂M called the b-boundary. Bosshard(1) and Johnson(4) have shown that certain Friedmann and Schwarzschild space-times have non-Hausdorff b-completions.

In (2), (3), the first author considered a modification of the b-completion which gives a Hausdorff completion in the Friedmann case. The modification used a parallelization of the space-time and can be given in terms of the linear connexion determined by the parallelization. We show that the completion obtained is the Cauchy completion in a particular Riemannian metric on the manifold and so is always Hausdorff.
Graphs with even girth and small excess
Authors: Biggs N. L. , Ito T.
Published: November 2008

Let G be a regular graph with even girth g = 2r ≥ 4 and valency k ≥ 3. It is well known, and easy to prove, that G must have at least n0 = n0(g, k) vertices, where
The continuous singular spectrum of the Schrödinger operator
Authors: Schechter Martin
Published: November 2008

We give sufficient conditions on the potentialV(x) which ensure that the Schrödinger operator (1 · 1) of quantum mechanics has no singular continuous spectrum This generalizes previous results.
Ergodic theorems for semifinite von Neumann algebras: II
Authors: Yeadon F. J.
Published: November 2008

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averages

converge in norm.
Transitivity for the modular group
Authors: Sheingorn Mark
Published: November 2008

Let Γ be a Fuchsian group of the first kind acting on the upper half plane H+. Let be a Ford fundamental region for Γ in H+. Let ξ be a real number (a limit point) and let L( = Lξ) = {ξ + iy|0 ≤ y < 1}. L can be broken into successive intervals each one of which can be mapped by an element of Γ into . Since L is a hyperbolic line (h-line), this gives us a set of h-arcs in which we will call the image.
Catastrophe theory and equations of state: conditions for a butterfly singularity
Authors: Stewart Ian N.
Published: November 2008

Parametrized equations of state of the form

where s = (s1, … sr) ∈ Rr, x ∈ R, and F is smooth, are common in applied science. Often there exist multiple states x for given s, which may bifurcate as s varies.
Attenuation of elastic waves in a cracked, fluid-saturated solid
Authors: Chatterjee A. K. , Mal A. K. , Knopoff L. , Hudson J. A.
Published: November 2008

The problem of the determination of the overall dynamic elastic moduli of an elastic solid permeated by uniformly distributed penny-shaped cracks is considered. The cracks are assumed to be filled with a viscoelastic material. The orientations of the cracks may be either parallel or perfectly random. The overall velocities as well as the specific attenuation coefficients of plane harmonic compressional and shear waves are calculated for low frequencies and dilute concentration of the cracks.
Sequences characterizing the sine function
Authors: Burkill H.
Published: November 2008

Recently J. Roe(1) established the striking characterization of the sine and cosine functions given in Theorem 1 (below). His proof was an ingenious application of Fourier transforms of generalized functions.
A quadratic mapping with invariant cubic curve
Authors: Penrose Roger , Smith Cedric A. B.
Published: November 2008

In the projective plane, if H is a harmonic homology (linear transformation with H2 = I), and G a general inversion (quadratic transformation projectively equivalent to an inversion), then under a certain condition there is a pencil of cubics each of which is invariant under G, H separately. These are related to transformations discovered by Mandel, Todd and Lyness. As a near converse, we find that, given a Pascal configuration, there is a quadratic Cremona transformation under which each cubic passing through the vertices of the configuration is invariant. As a by-product, parametric expressions are found for elliptic functions of a fifth of a period.
The cohomology of finite H-spaces as U(M) algebras: I
Authors: Kane Richard
Published: November 2008

By an H-space (X, μ) we will mean a topological space X having the homotopy type of a connected CW complex of finite type together with a basepoint preserving map μ: x × X → X with two sided homotopy unit. Let p be a prime and let /p be the integers reduced modp. Given an H-space (X, μ) then H*(X;/p) is a commutative associative Hopf algebra over the Steenrod algebra A*(p) and H*(X;/p) is iso- morphic, as an algebra, to a tensor product [⊗ , where each algebra At is generated by a single element ai (see Theorem 7.11 of (24)). The decomposition Ai is called a Borel decomposition and the elements {ai} are called the Borel generators of the decomposition. The decomposition ⊗ Ai and the resulting generators {at} are far from unique. Many choices are possible. Since A*(p) acts on H*(X;/p) an obvious restriction would be to choose the Borel decomposition to be compatible with this action. We would like the /p module generated by the Borel generators and their iterated pth powers to be invariant under the action of A * (p). More precisely we would like H*(X;/p) to be the enveloping algebra U(M) of an unstable Steenrod module M (see § 2). If H*(X;/p) admits such a choice then it is called a U(M) algebra. The fact that H*(X; /p) is a U(M) algebra has applications in homotopy theory. In particular there exist unstable Adams spectral sequences which can be used to calculate the homotopy groups of X (see (21) and (7)). However, the question of U(M) structures for modp cohomology seems of most interest simply as a classification device for finite H-spaces.
Cohen–Macaulay properties for balanced big Cohen–Macaulay modules
Authors: Sharp R. Y.
Published: November 2008

Let A be a (commutative, Noetherian) local ring (with identity) and let a1,…, an be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay.A-module with respect to a1,…, an if a1,…, an is an M-sequence, that is if M ‡ = (a1,…, an) M and, for each i = 1,…, n,

One of the main open problems in commutative algebra at the present time is that of establishing the existence of a big Cohen–Macaulay module with respect to a specified s.o.p. in an arbitrary local ring. The work and writings of Hochster, such as (5), show that, if the existence of such modules could be established, then several conjectures in commutative algebra, some of which are quite long-standing, would be settled. Moreover, Hochster has established the existence of such big Cohen–Macaulay modules whenever the local ring A concerned contains a field as a subring, or has dimension not exceeding 2: see ((5), chapters 4, 5) and (4).