bluuebunny/crossref_metadata_2025_split huggingface dataset:
On induced permutation matrices and the symmetric group
Authors: Aitken A. C. 
Published: January 2009

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.
A practical method for enumerating cosets of a finite abstract group
Authors: Todd J. A. , Coxeter H. S. M. 
Published: January 2009

An important problem in finite-group theory is the determination of an abstract definition for a given group , that is, a set of relationsbetween k generating operations S1, …., Sk of , such that every other relation between S1, …., Sk is an algebraic consequence of (1).The number of groups for which abstract definitions are actually known is relatively small, but a remarkable feature of the results already obtained is the extreme simplicity of the relations (1) in the case of several groups of quite high order. This fact constitutes an additional incentive to the search for abstract definitions, and many elegant results have doubtless yet to be discovered.
On the projective geometry of paths
Authors: Haantjes J. 
Published: January 2009

An affine connection in an n-dimensional manifold Xn defines a system of paths, but conversely a connection is not defined uniquely by a system of paths. It was shown by H. Weyl that any two affine connections whose components are related by an equation of the formwhere is the unit affinor, give the same system of paths. In the geometry of a system of paths, a particular parameter on the paths, called the projective normal parameter, plays an important part. This parameter, which is invariant under a transformation of connection (1), was introduced by J. H. .C. Whitehead. It can be defined by means of a Schwarzian differential equation and it is determined up to linear fractional transformations. In § 1 this method is briefly discussed.
A series of “cut” Bessel functions
Authors: Copson E. T. , Ferrar W. L. 
Published: January 2009

In Kottler's theoretical discussion1 of the diffraction of a plane wave of monochromatic light of wave-length 2π/k by a black halfplane, the functionwhere (r, θ, z) are cylindrical coordinates, plays an important part. In particular it is necessary to have asymptotic formulae for f (r, θ), valid when r is either very large or very small compared with the wave-length.
Solutions of Laplace's equation in an n-dimensional space of constant curvature
Authors: Ruse H. S. 
Published: January 2009

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered waswhere is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-elementwas that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.
On Kilmister's Conditions for the Existence of Linear Integrals of Dynamical Systems
Authors: Boyer R. H. 
Published: January 2009

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.
On the non-existence of L2 solutions of a class of non-linear differential equations
Authors: Burlak J. 
Published: January 2009

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equationdoes not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if
Certain two-dimensional mixed boundary-value problems for wedge-shaped regions and dual integral equations
Authors: Srivastav R. P. 
Published: January 2009

Finding the distribution of stress in earth dams containing cracks is an outstanding problem of soil mechanics. Even the simplest mathematical model, viz., that of a wedge containing a plane crack which is symmetrically situated along the bisector plane of the angle of the wedge, with the plane strain assumption of the infinitesimal theory of elasticity, presents a difficult problem of solving the bi-harmonic equation subject to mixed boundary conditions. While elasticity problems related to wedge-shaped bodies have been investigated, it appears little attention has been paid to the mixed boundary-value problems.As a first step towards the solution of the mixed boundary value problem for the biharmonic equation, we discuss in this paper the solution of Laplace's equationfor wedge-shaped regions subject to mixed type of conditions on the boundary. If we assume that φ does not depend on z, the equation (1.1) is reduced to the equation
Absolute Riesz summability factors for Fourier series
Authors: Chandra Prem 
Published: January 2009

Let ∑an be a given infinite series and {λn} a non-negative, strictly increasing, monotonic sequence, tending to infinity with n. We write, for w > λ0,and, for r>0, we write is known as the Riesz sum of “ type ” λn and “ order ” r, andis called the Riesz mean of type λn and order r.
Sufficient conditions for matchings
Authors: Anderson Ian 
Published: January 2009

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).
Scalar and matrix complex nonoscillation criteria
Authors: Howard H. C. 
Published: January 2009

In his recent book on ordinary differential equations Hille (3) devotes a chapter to complex oscillation theory. Drawing upon his own work in this area and the work of Nehari, Schwarz, Taam, and others, he gives a variety of oscil-lation and nonoscillation theorems for solutions of the differential equationwhere z is a complex variable and p is regular in some appropriate domain. There are a number of results for (1.1) with an arbitrary coefficient o and some discussions for special cases of classical interest, such as the Bessel and Mathieu equations. There is a bibliography at the end of the chapter. For other recent work in this area attention is directed to papers by Herold (1, 2) Kim (4, 5) and Lavie (6) where other references are given.
A question of C. R. Hobby on regular p-groups
Authors: Macdonald I. D. 
Published: January 2009

In (2) a finite p-group G is said to be nearly regular if it has the following two properties:(i) There exists a central subgroup Z of order p and G/Z is regular.(ii) If x ∈ G and y ∈ γ2(G), then gp{x, y} is regular.
An unpleasant set in a non-locally-convex vector lattice
Authors: Pryce J. D. 
Published: January 2009

In a linear topological space E one often carries out various “ smoothing ” operations on a subset A, such as taking the convex hull co A and the closure A-. If E is also a (real) vector lattice, the solid hullis also a natural “ smoothing out ” of A. If sol A = A then A is called solid, and if E has a base of solid neighbourhoods of 0 as do all the common topological vector lattices such as C(X), Lp, Köthe spaces and so on—then E is called a locally solid space.
Convergence in partially ordered groups
Authors: Wirth Andrew 
Published: January 2009

AbstractRelative uniform limits need not be unique in a non-archimedean partially ordered group, and order convergence need not imply metric convergence in a Banach lattice. We define a new type of convergence on partially ordered groups (R-convergence), which implies both the previous ones, and does not have these defects. Further R-convergence is equivalent to relative uniform convergence on divisible directed integrally closed partially ordered groups, and to order convergence on fully ordered groups.
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 equationThis 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.
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 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 byis 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 bewhere d is the number of double points of .
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 ofsuch that on z = 0Separation 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.
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 followsHere 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 andin the half-strip Sof the complex plane.Suppose also that for some constant α in α<a<βas 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

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

SummaryAn 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.
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 formwhich starts with prescribed complex coefficients c0, …, cr–1; and satisfiesThese 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

AbstractThe 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 kindand the triple series equations of the second kindwhere f, f1, g, g1h and h1 are all known functions,is the Jacobi polynomial (3).
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 formulawhere u(x)∈C2(D)∩C′(D̄) is a solution of the second order self-adjoint elliptic equationand denotes differentiation along the inward normal to ∂D at x∈∂D.
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 formFurther, 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

LetThe 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 equationneed 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).
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.
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.
On the Evaluation of the definite integrals
Authors: Sircar Hrishikesh 
Published: January 2009

§ 1 Introduction. The formulaewhere 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.
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 formulawhich 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 averagethe 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.
An Interpolation Series for Integral Functions
Authors: Macintyre Sheila Scott 
Published: January 2009

1. The Gontcharoff interpolation serieswherehas 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 satisfiesthe 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 satisfiesand that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.