bluuebunny/crossref_metadata_2025 huggingface dataset:
L-ideals of M(G) determined by continuity of translation
Authors: Brown Gavin , Moran William
Published: January 2009

G denotes a locally compact abelian group and M(G) the convolution algebra of regular bounded Borel measures on G. An ideal I of M(G) closed in the usual (total variation) norm topology is called an L-ideal if μ ∈ I, ν≪ μ (ν absolutely continuous with respect to μ) implies that ν ∈ I. Here we are concerned with the L-idealsL1(G), , and M0(G) where, as usual, L1(G) denotes the set of measures absolutely continuous with respect to Haar measure, denotes the radical of L1(G) in M(G) and M0(G) denotes the set of measures whose Fourier-Stieltjes transforms vanish at infinity.
Some triple equations involving inverse finite Mellin transforms
Authors: Tweed John
Published: January 2009

In this paper we investigate some triple equations involving the inverse of the finite Mellin transform MR which is defined by the equation

This transformation is one of four which were first introduced by D. Naylor in his paper (1) and some of its properties have been listed by the author in paper (2), where its relationship to the Mellin transform is discussed in detail.
Osculating primes to curves of intersection in 4-space, and to certain curves in n-space
Authors: Dye R. H.
Published: January 2009

An irreducible curve in S4, projective 4-space, may arise as the complete intersection of three given irreducible threefolds. At a simple point P on such a curve there is an osculating solid, and we would like to have its equation. This solid, necessarily containing the tangent line to the curve at P, belongs to the net spanned by the tangent solids at P to the threefolds. We seek the appropriate linear combination of the known equations for these tangent solids.
Representations of normed algebras with minimal left ideals
Authors: Barnes Bruce A.
Published: January 2009

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.
Functions and equations in classes of distributive lattices with pseudocomplementation
Authors: Beazer R.
Published: January 2009

In (8), R. L. Goodstein gave necessary and sufficient conditions for the solvability of equations over distributive lattices with 0 and 1 together with an algorithm for computing a solution whenever one exists. In addition, the same problem was considered for a special class of equations over distributive lattices with pseudocomplementation. The validity of several of Goodstein's results for distributive lattices without 0 and 1 was pointed out by Rudeanu in (15) and (16).
Some algebraic properties of F(X) and K(X)
Authors: Alexander Freda E.
Published: January 2009

Throughout we consider operators on a reflexive Banach space X. We consider certain algebraic properties of F(X), K(X) and B(X) with the general aim of examining their dependence on the possession by X of the approximation property. B(X) (resp. K(X)) denotes the algebra of all bounded (resp. compact) operators on X and F(X) denotes the closure in B(X) of its finite rank operators. The two questions we consider are:

(1) Is K(X) equal to the set of all operators in B(X) whose right and left multiplication operators on F(X) (or on B(X)) are weakly compact?

(2) Is F(X) a dual algebra?
On rearranging maximal functions in Rn
Authors: Walker P. L.
Published: January 2009

Denote by f a positive measurable function on Rn, and by λ the distribution function of denotes the Lebesgue measure of the set specified. We shall suppose that λ(y)<∞ for each y>0, and that λ(y)→0 as y→∞. The decreasing rearrangement f* of f is defined on (0, ∞) by
Krull dimension and invertible ideals in Noetherian rings
Authors: Lenagan T. H.
Published: January 2009

In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.
Periodicity of functionals and representations of normed algebras on reflexive spaces
Authors: Young N. J.
Published: January 2009

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).
Commutative non-associative number theory
Authors: Bunder M. W.
Published: January 2009

Trevor Evans in (8) introduced postulates for a non-associative number theory similar to, but less general than, those of A. Robinson (9). Evans' number theory is also non-commutative under addition and multiplication, but an alternative equality axiom also suggested by Robinson leads to a number theory which is commutative under addition and still non-associative except in the special case:
Secondary representations for injective modules over commutative Noetherian rings
Authors: Sharp Rodney Y.
Published: January 2009

There have been several recent accounts of a theory dual to the well-known theory of primary decomposition for modules over a (non-trivial) commutative ring A with identity: see (4), (2) and (9). Here we shall follow Macdonald's terminology from (4) and refer to this dual theory as “ secondary representation theory ”. A secondary representation for an A-module M is an expression for M as a finite sum of secondary submodules; just as the zero submodule of a Noetherian A-module X has a primary decomposition in X, it turns out, as one would expect, that every Artinian A-module has a secondary representation.
On Theorems of Kawada and Wendel
Authors: Williamson J. H.
Published: January 2009

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li by

is again in L1, and
A Generalisation of a Formula due to Schubert
Authors: Brennan J. G.
Published: January 2009

Let there be given, on an algebraic curve C, of genus p, a linear series and an algebraic series of index v, both without fixed points. The number of groups of r + 1 points which are common to a set of and a set of has been shown by Schubert (1) to be

where d is the number of double points of .
A Direct Method of using the Hodograph Plane in Fluid Dynamics
Authors: Mackie A. G.
Published: January 2009

The application of the hodograph method in problems in fluid dynamics dates back to the time of Helmholtz and Kirchhoff. The underlying principle is simple. It is in effect to rewrite the governing differential equations with the roles of the original dependent and independent variables reversed. Such a procedure is not uncommon in problems depending upon ordinary differential equations. For example, if the velocity of a particle in rectilinear motion is prescribed as a function of distance from a fixed point, the problem of finding the relation between its position and the time t can be solved by one quadrature if t is regarded as the dependent variable.
The Approximate Solution of Dual Integral Equations by Variational Methods
Authors: Noble B.
Published: January 2009

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution of

such that on z = 0

Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution.
Note on the Fundamental Theorem on Irreducible Non-Negative Matrices
Authors: Schneider Hans
Published: January 2009

Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).
The Motion of a Flat Plate from Rest in a Visco-elastic Liquid
Authors: Leslie F. M.
Published: January 2009

Oldroyd (1) considered non-Newtonian liquids for which the stress tensor sik and the rate of strain tensor eik=½(υk,i+υi,k)are related as follows

Here υi denotes the velocity vector, p an isotropic pressure, gik the metric tensor, and t time; η0 is a constant having the dimensions of viscosity and λ1, λ2, μ0, ν1, ν2, are constants having the dimension of time. Covariant suffixes are written below, contravariant above and the usual summation convention is employed. A suffix i following a comma indicates covariant differentiation with respect to the space variable xi.
On the Limit of the Modulus of a Bounded Regular Function
Authors: Bowen N. A.
Published: January 2009

M. L. Cartwright has given ((2), 180–181) the following theorem, together with a neat proof of it.

Theorem C.Suppose that f(z) is regular and

in the half-strip S

of the complex plane.

Suppose also that for some constant α in α<a<β

as y→∞. Then for every δ>0.

uniformly asy→∞ for α+δ≦x≦β–δ.
Further Considerations of the Elastic Inclusion Problem
Authors: Knops R. J.
Published: January 2009

An equation is derived for the strains of an arbitrary elastic field in an infinite matrix perturbed by several inclusions. The equation is solved exactly when the shear moduli of the inclusions and matrix are identical, and also when only a single ellipsoidal inclusion perturbs a field uniform at infinity. Mean-values of the strains are then calculated for non-uniform fields perturbed either by an ellipsoid or by a system of weakly-interacting spheres.
On a Theorem of Korous
Authors: Anderson J. A.
Published: January 2009

Let

where the {lν} are numbers near to an arithmetic progression of common difference unity. Let

b being a constant. Write

so that k(z) is an integral function, and

is meromorphic with simple poles at {lν}.
A Remark on Small Values of Entire Functions
Authors: Segal S. L.
Published: January 2009

Let f(z) be an entire function, M(r) the maximum of f(z) on ∣z∣=r, and λ>1. Let Eλ=Eλ(f{z:log∣f(z)≦(1-λ)log(M∣z∣)}, and denote the density of Eλby

where m is planar measure.
Some Integral Equations Involving Hypergeometric Functions
Authors: Love E. R.
Published: January 2009

An integral equation of the first kind, with kernel involving a hypergeometric function, is discussed. Conditions sufficient for uniqueness of solutions are given, then conditions necessary for existence of solutions. Conditions sufficient for existence of solutions, only a little stricter than the necessary conditions, are given; and with them two distinct forms of explicit solution. These two forms are associated at first with different ranges of the parameters, but their validity in the complementary ranges is also discussed. Before giving the existence theory a digression is made on a subsidiary integral equation.

Corresponding theorems for another integral equation resembling the main one are deduced from some of the previous theorems. Two more equations of similar form, less closely related, will be considered in another paper. Special cases of some of these four integral equations have been considered recently by Erdélyi, Higgins, Wimp and others.
The Asymptotic Value of the Volume of a Certain Set of Matrices
Authors: Jack Henry
Published: January 2009

This paper is an appendix to a joint paper with Professor Macbeath. In (3), it was proved that the invariant measure, m(k), of the set of real n × nmatrices τ, with determinant 1 and norm satisfying ∥τ∥≦ k, had the property that
On a result in Young's quantitative substitutional analysis
Authors: Hunter D. B.
Published: January 2009

In (4) Young investigates the representation in terms of his exact seminormal units of substitutional expressions which are unchanged on premultiplication by any permutation of a given set of consecutive letters, or which are changed in sign on premultiplication by an odd permutation of those letters. He illustrates his results by deducing the forms of the matrices representing the positive and negative symmetric groups on a set of consecutive letters.
A generalization of quasi-monotone sequences
Authors: Robertson M. M.
Published: January 2009

The object of this note is to generalize the notion of quasi-monotony for sequences of real numbers and to prove corresponding generalizations of certain known theorems. First, we recall the definition of quasi-monotony.
A variant of Carathéodory's problem
Authors: Strang Gilbert
Published: January 2009

In this note we ask two questions and answer one. The questions can be combined as follows:

Does there exist a polynomial of the form

which starts with prescribed complex coefficients c0, …, cr–1; and satisfies

These differ from the classical problems of Carathéodory in one essential respect: the values of p and its first r–1 derivatives are given at the point z = 1 on the circumference of the unit circle, while in the original problem they were given at z = 0. Carathéodory's own answer was in terms of his “moment curve”, but the forms studied a few years later by Toeplitz yield a more convenient statement of the solution.
The motion of a rigid sphere embedded in an adhering elastic or viscoelastic medium
Authors: Hunter S. C.
Published: January 2009

The motion of a rigid sphere embedded in an adhering medium and subjected to an external force is analysed exactly in the context of classical elastodynamics. For the limiting case of an incompressible medium it is possible to write down a simple second order differential equation relating displacement of the ball to the external force.

The theory is generalised to the case of a viscoelastic solid and the results obtained are pertinent to recently developed experimental methods for testing the dynamical mechanical properties of very low modulus polymer gels.
Some triple series equations involving Jacobi polynomials
Authors: Lowndes J. S.
Published: January 2009

Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kind

and the triple series equations of the second kind

where f, f1, g, g1h and h1 are all known functions,

is the Jacobi polynomial (3).
Semi-simple locally compact monothetic semi-algebras
Authors: Kaashoek M. A. , West T. T.
Published: January 2009

Bonsall and Tomiuk have shown, in (3), the connection between the local compactness of a monothetic semi-algebra and the spectral properties of a generating element. This theme was developed, in (4), to give a complete characterisation of prime, strict locally compact monothetic semi-algebras in terms of the spectrum of a generator (Theorem A). Here we extend this result to the case of a semi-simple locally compact monothetic semi-algebra (Theorem B).
An exclusion theorem for tri-diagonal matrices
Authors: Jayne John W.
Published: January 2009

An n × n matrix An = (aij) is tri-diagonal if aij = 0 for |i−j|≧2. The latent roots of such matrices may be conveniently studied by forming the sequence of polynomials Ψk(λ)=|λI−Ak|, where Ak is the principal submatrix of Ak+1 obtained by deleting the last row and column of Ak+1, and then observing that these polynomials satisfy the following recurrence relation:
Fundamental solutions and surface distributions
Authors: Adams R. A. , Roach G. F.
Published: January 2009

When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formula

where u(x)∈C2(D)∩C′(D̄) is a solution of the second order self-adjoint elliptic equation

and denotes differentiation along the inward normal to ∂D at x∈∂D.
Some dual series and triple integral equations
Authors: Lowndes J. S.
Published: January 2009

In this paper we first of all solve the dual series equations

where ƒ(ρ) and g(ρ) are prescribed functions,

is the Jacobi polynomial (2).
Non-linear integral equations for Heun functions
Authors: Sleeman B. D.
Published: January 2009

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the form

Further, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.
On asymptotic behaviours of trigonometric series with δ-Quasi-monotone coefficients
Authors: Liu Ming-Chit
Published: January 2009

Let

The asymptotic behaviours of ƒ(x) and g(x), as x→+0, were first given by G. H. Hardy in (4), (5). In his papers {an}; is a monotone decreasing sequence. Further results on the asymptotic behaviours of ƒ(x) and g(x), as x→+0, for monotone coefficients have been given in (9) and (1). Recently, the results have been generalized to quasi-monotone coefficients.

This paper is concerned with asymptotic behaviours of ƒ(x) and g(x) for δ-quasi-monotone coefficients.

In what follows, we shall denote by L(x) a slowly varying function in the sense of Karamata (6), i.e.
A local mean value theorem for the complex plane
Authors: Robertson J. M.
Published: January 2009

The equation

need not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where

The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).
On compact normal semigroups
Authors: Choy S. T. L. , Dummigan B. , Duncan J.
Published: January 2009

A semigroup S is said to be normal if aS = Sa for each a in S. Thus the class of normal semigroups includes the class of groups and the class of Abelian semigroups. Given a compact semigroup S we write P(S) for the convolution semigroup of probability regular Borel measures on S. In (3), Theorem 7, Lin asserts that a compact semigroup S is normal if and only if P(S) is normal. We show in this paper that Lin's result is false. In fact, if S is the union of subsemigroups each of which has an identity element, we show that P(S) is normal if and only if S is Abelian. Thus any compact non-Abelian group contradicts Lin's result. What Lin's argument does establish is that if P(S) is normal then S is normal, and if S is normal then μP(S) = P(S)μ for each point mass measure μ.
Theorems on Strong Riesz Summability Factors
Authors: Shawyer B. L. R.
Published: January 2009

Throughout this note it will be assumed that

(i) unless otherwise stated, k; is a positive integer,

(ii) a ≧ 0; m > − 1,

(iii) q ≧ 1; q′ is the conjugate number to q, and is defined by

(iv) the functions φ(w), ψ(w) are defined in [0, ∞) with absolutely continuous kth derivatives in every interval [a, W],

(v) φ(w) is non-negative and unboundedly increasing,

(vi) λ = {λn} is an unboundedly increasing sequence with λ1 > 0.
On the Cauchy Problem for then-Dimensional Wave Equation
Authors: Young R. M.
Published: January 2009

Consider then-dimensional wave equation
A Class of Integral Equations Involving Laguerre Polynomials as Kernel
Authors: Srivastava K. N.
Published: January 2009

Recently Widder (3) has obtained inversion integrals for a convolution transform whose kernel is a Laguerre polynomial. Buschman (1) has considered convolution equations withgeneralised Laguerre polynomial kernel. His result includes that of Widder. However no attempt has so far been made towards the study of singular integral equations involving Laguerre polynomials as kernel. Here the author obtains an inversion integral for such an equation.
Infinitesimal Analysis of an arc in n-space
Authors: Sen Rabindra Nath
Published: January 2009

We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:

Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.
Some Properties of a Family of Curves on a Surface
Authors: Weatherburn C. E.
Published: January 2009

This paper shows briefly how, for a singly infinite family of curves on a given surface, the fundamental properties of the family at any point are associated with three central conies determined by the curves, in a manner resembling that in which the curvature properties of a surface at any point are associated with Dupin's indicatrix. The differential invariants employed are the two-parametric invariants for the given surface.
Apolar Triads associated with the Nodal Cubic
Authors: Saddler Wm.
Published: January 2009

Let the Nodal cubic have the equation

or,

Three points t1, t2, t3 form an apolar triad to the curve when they satisfy an equation of the form
A Note on Double Limits
Authors: Walsh C. E.
Published: January 2009

Let umn, vmn be functions of m and n, vmn being real and positive for all positive values of m and n. Suppose that either vmn increases steadily to infinity with n, or that umn both tend to zero (the latter steadily) as n → ∞, for any fixed value of m. Denote by wmn, and assume that wmn exists for every value of m, being denoted by lm. Then from Stolz' extension of a result proved by Cauchy, and an allied theorem, we have , for all values of m. It follows from Pringsheim's Theorem that if the double limit of exists, being l, then lm → l as m → ∞.
On the Reduction of Ferrers' Associated Legendre Function
Authors: Sircar Hrishikesh
Published: January 2009

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.
On the Evaluation of the definite integrals
Authors: Sircar Hrishikesh
Published: January 2009

§ 1 Introduction. The formulae

where n, q, m are positive integers, are well known. I am not aware, however, of any paper dealing with the corresponding definite integrals of products of two Ferrers' Associated Legendre Functions of different integral orders and degrees. With the help of the formulae given in my previous paper in this volume it is possible to discuss these integrals. I have found in the case of the first integral that if the orders of the two associated functions are both odd or both even and if, in each case, the degrees are both odd or both even or equal, the integral has usually a non-zero value. If, however, the orders are both odd or both even and if the degree of the function having the greater order is less than that of the other, or if the degrees are one odd and one even, the integral vanishes. Further, if one of the orders be odd and the other even, and the corresponding degrees are also even and odd (or odd and even), the integral has a non-zero value; in all other cases it vanishes. The values of the second integral can be obtained in a similar manner. An example of the methods employed and the results obtained is given in the following section.
Sets of Points Self-Conjugate with regard to a Quadric in n Dimensions
Authors: Sommerville D. M. Y.
Published: January 2009

§ 1. In space of three dimensions the properties of self-conjugate tetrads, pentads and hexads with regard to a quadric are well known (see Baker's Principles of Geometry, vol. iii). The general theorem in space of n dimensions Sn is to establish the existence of a set of n + p + 1 points A0, A1, …, An+p (0 ≤ p ≤ n−1) such that the pole, with respect to a given quadric, of the (n−1)-flat determined by any set of n of the points lies in the p-flat determined by the remaining p + 1 points.
Metrical Coordinates in Non-Euclidean Geometry
Authors: Sommerville D. M. Y.
Published: January 2009

§1. The coordinates considered are linear, i.e. in a plane the equation of a straight line, and in space the equation of a plane, is linear in the coordinates. We shall first consider point-coordinates in plane geometry, taking elliptic geometry as typical, with space-constant unity.
A Note on Resultants of Equations in a Cyclic Number System
Authors: Wilson R.
Published: January 2009

§ 1. Introduction. Little is known concerning the theory of resultants of equations other than in the complex number system. The cyclic number systems provide a simple example which is not a division algebra. In such a system with n units er. any number y ≡ y0 + y1e1 + y2e2 + … + yn−1en−1 has coefficients yr drawn from a field, and the units satisfy the product law:
On Interpolation by Iteration of Proportional Parts, without the Use of Differences
Authors: Aitken A. C.
Published: January 2009

Linear interpolation between two values of a function ua and ub can be performed, as is well known, in either of two ways. If the divided difference (ub−ua)/(b−a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formula

which is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given, but a machine is available, then the use of proportional parts in the form of the weighted average

the linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read.
A note on sequences determined by a recurrence relation
Authors: Walsh C. E.
Published: January 2009

§ 1. Copson and Ferrar have proved the following theorem:

Ifis a divergent series of positive terms, and (pn + qn) un−pnun−1 = ynalways, where yn → 0, then un → 0.
An Interpolation Series for Integral Functions
Authors: Macintyre Sheila Scott
Published: January 2009

1. The Gontcharoff interpolation series

where

has been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

the constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

and that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.
An Analogue for Strong Summability of Abel's Summability Method
Authors: Harington C. F. , Hyslop J. M.
Published: January 2009

Given a series Σan, we define , by the relation

where is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,

we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that
Generalised Young Tableaux
Authors: Wallace Andrew H.
Published: January 2009

The present note contains generalisations and new proofs of certain theorems in the theory of Young Tableaux and Invariant Matrices. For an account of Young Tableaux and their applications, and an introduction to the method of Clebsch-Aronhold symbols, reference should be made to Rutherford [1], and Turnbull [1], respectively. An invariant matrix T(A) of a given square matrix A is, as appears from the context in § 4 below, a matrix of polynomials in the elements of A, regarded as independent variables, such that T(AB) = T(A) T(B). Further details, and references to original sources, are given in Wallace [1].
A Determinantal Expansion for a Glass of Definite Integral. Part 1
Authors: Shenton L. R.
Published: January 2009

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficients

Parseval's formula gives
Generalised Algebraic Continued Fractions related to Definite Integrals
Authors: Shenton L. R.
Published: January 2009

The present paper is a continuation of the work initiated in [l]-[5]. In [5] I gave an expansion of the form

for the second order C.F. associated with

where U8, V8, W8 satisfy a fourth-order recurrence relation, there being a similar expansion for third order C.F.'s. I shall now give simple expressions for U8, V8, W8 (or related forms) in terms of χ2s(Z1), χ2s (Z2), ω2s(Z1), ω2s(Z2), where

and show that there is a remarkable relation between the recurrence formula for the first order C.F. and that satisfied by U3, V3, W3. The generalised form of these results will be stated and proved.
Simultaneous Linear Recurrence Relations with Variable Coefficients
Authors: Ursell H. D.
Published: January 2009

Our subject is a set of equations

where the uj(n)(j = 1, 2, …, k) are k “unknown” functions of the integer variable n, the zi(n) (i = 1, 2, … h) are h “known” functions of n, and the Aij(n) are hk “known” operators

which are polynomials in E, each of fixed order pij but with coefficients which may vary with n. E is the usual operator defined by

Our first task is to determine whether the equations (1) are self-consistent. Secondly, if they are self-consistent, we ask what follows from them for a given subset of the unknowns, e.g. for (uj+1, …, uk) in other words we wish to eliminate (u1, …, uj). In particular we wish to eliminate all the variables but one, say uk. We shall in fact find that either uk is arbitrary or else that it has only to satisfy a single linear recurrence relation : and the order of that relation is of interest to us. Thirdly, we ask that reduction to standard form is possible, with or without a transformation of the unknowns themselves.
A Note on Trace-Differentiation and the Ω-operator
Authors: Aitken A. C.
Published: January 2009

In theory of polarizing operators in invariants the operator

where X = [xij] is an n × n matrix of n2 independent elements xij, holds an important place. Acting upon particular scalar functions of X, namely the spur or trace of powers of X, or of polynomials or rational functions of X with scalar coefficients, it exhibits (Turnbull,. 1927, 1929, 1931) an exact analogy with results in the ordinary differentiation of the corresponding functions of one scalar variable. Turnbull denotes this operation of trace-differentiation under Ω by Ω8; and we shall follow him. Our purpose is to show how, with a suitably modified Ω, the results may be extended to the case of symmetric matrices X = X′ having ½n(n + 1) independent elements.
Note on γ-Matrices Efficient at an Isolated Point
Authors: Vermes P.
Published: January 2009

In a recent paper1 γ-matrices were constructed summing the binomial series at isolated points to its right value. It is assumed that the reader is able to refer to this paper and is familiar with its notation and terminology.

The object of this note is to give the construction of γ-matrices which sum the binomial series at an isolated point to an arbitrary given value, while inside the circle of convergence the generalised sum is necessarily the right value.
Theorems Relating to Quadratic Forms and their Discriminant Matrices
Authors: Vajda S.
Published: January 2009

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:

Let the expression be given; introduce, for c, a linear form in and obtain

If the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.
On the Φ-Invariant of Two Quadrics
Authors: Scott T.
Published: January 2009

The geometry associated with the five invariants of two quadrics is well known. There appears to be an omission, however, in the treatment of the Φ-invariant. Salmon gives the vanishing of Φ merely as a necessary condition for the possibility of the construction of a tetrahedron self-conjugate for one of the quadrics and having its six edges tangential to the other. Sommerville proves its sufficiency, and shows that this condition is poristic, giving rise to two systems of ∝ tetrahedra of the requisite type. No investigation appears to have been made of the locus of the vertices of the ∝ tetrahedra of each system and the dual problem regarding the nature of the developable surface arising from their ∝ faces.
On Symmetric, Orthogonal, and Skew-Symmetric Matrices
Authors: Hsu P. L.
Published: January 2009

Introduction and Notation. In this paper all the scalars are real and all matrices are, if not stated to be otherwise,p-rowed square matrices. The diagonal and superdiagonal elements of a symmetric matrix, and the superdiagonal elements of a skew-symmetric matrix, will be called the distinct elements of the respective matrices. Σ will denote both the set of all symmetric matrices and the ½p(p+ 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order.Kwill denote both the set of all skew-symmetric matrices and the ½p(p– 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. Any sub-set of Σ(K) will mean both the sub-set of symmetric (skew-symmetric) matrices and the set of points of Σ(K). Any point function defined in Σ(K) will be written as a function of a symmetric (skew-symmetric) matrix.Dαwill denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.
Some Remarks on Uniform Convergence
Authors: Eggleston H. G.
Published: January 2009

A useful test for uniform convergence is that first established by Buchanan and Hildebrandt [4] which is as follows.

(A) “If a sequencefn(x) of monotonic functions converges to a continuous functionf(x) in [a, b] then this convergence is uniform.”

In §1 of this paper it is shown that this test is included in a sequence of theorems, each of which establishes a type of uniform convergence. The first is a well-known topological theorem on limit sets, the second is a result on the limits of rectifiable arcs, the third is a generalisation of (A) due to Behrend [3], the fourth is (A) itself, the fifth is a one-sided version of Bendixson's test and the sixth is Bendixson's test.
Whittaker's Work on the Integral Representation of Harmonic Functions
Authors: Temple G.
Published: January 2009

It is a singular honour to be invited to deliver a lecture commemorating the work of Sir Edmund Whittaker, especially before the Edinburgh Mathematical Society, whose development owes so much to his initiative and co-operation. But when I reflect on the difficulties of the task I can only exclaim in the words of St Jerome's preface to his translation of the New Testament, “Pius labor, sed periculosa praesumptio”.
Sir Edmund Whittaker's Work on Automorphic Functions
Authors: Rankin R. A.
Published: January 2009

Although only three of Whittaker's papers (12, 13, 14) are on the theory of automorphic functions, he retained his interest in it throughout his life. His son, Dr J. M. Whittaker, informs me that his last mathematical conversation with his father was on this subject. The standard English work on automorphic functions by L. R. Ford (4), and its precursor in the Edinburgh Mathematical Tracts series, owe much to Whittaker, and it was he who suggested the term “isometric circle” of which Ford makes such elegant use in his development of the theory. Professor G. N. Watson informs me that Whittaker had expressed his willingness to contributea a chapter on automorphic functions to the revised and expanded edition of Modern Analysis which is in preparation.
The Contributions of E. T. Whittaker to Algebra and Numerical Analysis
Authors: Aitken A. C.
Published: January 2009

Whittaker's contributions to algebra are not numerous and are confined to special problems ; in one or two cases they are rediscoveries, of the kind that add a certain illumination to the original. His outlook on algebra was that of his time ; one might characterise it by referring to the spirit and content of Perron's Algebra, as contrasted with the Moderne Algebra of van der Waerden. Bred in the Cayley-Sylvester tradition of matrix algebra and the invariant theory of forms, he was expert also in the manipulation of continued fractions and determinants. His lectures to honours classes in Edinburgh always included a course on matrices, vector analysis and invariants, historical reference being made to Grassmann, Cayley and Sylvester; it is to such a course that the writer owes his first acquaintance with matrix algebra. In later years, from about 1925, Whittaker acquired an adequate knowledge, though never a marked taste, for those parts of modern abstract algebra, in particular the representation of groups and algebras, that serve a purpose in mathematical physics; his courses of research lectures, which he kept up until a late period in his tenure of the chair, bore witness of this.
Whittaker's Correlation of Physics and Philosophy
Authors: McConnell James
Published: January 2009

Whittaker's studies of the relations between physics and philosophy are contained in about forty books and papers, nearly all of which were published during the last fifteen years of his life. His approach is mainly historical and is a natural sequence to his History of the Theories of Aether and Electricity. Many of the publications were based on lectures delivered to various groups and there is naturally a certain amount of repetition.
The Sir Edmund Whittaker Memorial Prize
Authors: Fulton James
Published: January 2009

A special meeting of the Edinburgh Mathematical Society was held in the Mathematical Institute, Edinburgh, on Friday, 17th May 1957, to commemorate Sir Edmund Whittaker's work. Present at this meeting were Sir Edmund's widow, Lady Whittaker, and members of her family.

Dr J. M. Whittaker, F.E.S., on behalf of Lady Whittaker and family, asked the Society to accept the sum of £500 to be used to provide a prize which would be awarded for meritorious work in mathematics by young mathematicians in Scotland.
Enumeration of Certain Subgroups of Abelian p-Groups
Authors: Davies I. J.
Published: January 2009

The number of distinct types of Abelian group of prime-power order pn is equal to the number of partitions of n. Let (ρ) = (ρ1, ρ2, …, ρr) be a partition of n and let (μ) = (μ1, μ2, …, μs) be a partition of m, with ρ1≧ρ2≧…≧ρr and μ1≧μ2≧…≧μs, ρi≧μi, r≧s, n>m. The number of subgroups of type μ in an Abelian p-group of type (ρ) is a function of the two partitions (μ) and p, and has been determined as a polynomial in p with integer coefficients by Yeh (1), Delsarte (2) and Kinosita (3). Their results differ in form but are equivalent.
Some Aspects of Uniqueness for Solutions to Boundary Problems
Authors: Martin M. H.
Published: January 2009

The solution to the boundary problem

where r is the distance of point (x, y) from the origin, and h is a given function the arc length s along the unit circle r = 1, is not necessarily unique, Boggio (1), Weinstein (2), Stoker (3), Martin (4). Indeed if h is a positive integer m is known that the only solutions regular analytic for r≦1 are

where r, θ denote polar coordinates and A, B are arbitrary constants.
The Wiener-Pitt Phenomenon on the Half-Line
Authors: Williamson J. H.
Published: January 2009

It has been well known for many years (2) that if Fμ(t) is the Fourier-Stieltjes transform of a bounded measure μ on the real line R, which is bounded away from zero, it does not follow that [Fμ(t)]−1 is also the Fourier-Stieltjes transform of a measure. It seems of interest (as was remarked, in conversation, by J. D. Weston) to consider measures on the half-line R+ = [0, ∞[, instead of on R.
The Abstract Group G3, 7, 16
Authors: Coxeter H. S. M.
Published: January 2009

A representation of degree 3 is used in § 4 to establish the necessary and sufficient condition

for the finiteness of the group (3, p | q, r) defined by
Some Axially Symmetric Stress Distributions in Elastic Solids containing Penny-shaped Cracks
Authors: Collins W. D.
Published: January 2009

This paper concludes the investigation of axially symmetric stress distributions in elastic solids containing penny-shaped cracks, commenced in previous papers (1), (2), by considering the stress distribution in a circular beam containing a crack opened by internal pressure or by uniform tension. The method of analysis, developed in the previous papers, is to first seek a representation of the displacement at a point of the beam as a sum of two terms, one of which is a representation of the displacement due to the crack in an otherwise unbounded infinite solid whilst the second is a general representation of the displacement in an undamaged beam, and then to show that this representation satisfies the conditions on the crack and the curved surface of the beam provided an unknown function occurring in it is the solution of a certain Fredholm integral equation. This equation holds whatever the ratio of the radius of the crack to that of the beam, but is most readily solved by iteration when this ratio is small, this solution being a perturbation on that for a crack in an infinite solid.
A Cyclic Inequality and an Extension of It. I
Authors: Diananda P. H.
Published: January 2009

For positive integral n and positive real x1, … xn let

Where

and let
Numerical Values of Some Integrals involving Bessel Functions
Authors: George D. L.
Published: January 2009

In the solution of boundary value problems in mathematical physics by means of integral transforms, we often find that the solution of a particular problem can be expressed in terms of integrals of the type

where r and z are positive, and v and n are integers satisfying the convergencecondition v+n>−1.
A Pair of Dual Integral Equations Occurring in Diffraction Theory
Authors: Burlak J.
Published: January 2009

Dual integral equations of the form

where f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.