bluuebunny/crossref_metadata_2025 huggingface dataset:
The mean value theorem and analytic functions
Authors: Johnston Elgin H.
Published: January 2009

It is well known that the mean value theorem (MVT) does not, in general, hold for analytic functions. The most familiar example to this effect is f(z) = ez since e2πi−e0≠2πiez0 for any z0∈ℂ. On the other hand, it is easy to show that the MVT holds in ℂ if f(z) is a polynomial of degree at most 2. Thus it is natural to ask what conditions on a function f(z) analytic in a domain D are necessary and sufficient for f(z) to satisfy the MVT in D. This is one of the questions answered in this paper.
Cauchy problems for second order hyperbolic differential equations with constant coefficients
Authors: Lowndes J. S.
Published: January 2009

1. It is well known [1] that there is a one-to-one relation between solutions of the Darboux equation and the wave equation. The purpose of this paper is to show that some recent results in the fractional calculus can be used to obtain a similar connection between solutions of Darboux's equation and second order linear hyperbolic differential equations with constant coefficients.
An algebraic filtration of H*(MO;ℤ2)
Authors: Kochman Stanley O.
Published: January 2009

Let 2* denote the dual of the mod two Steenrod algebra. In [5] an algebraic filtration B*(n) of H*(BO; ℤ2) was constructed such that each B*(n) is a bipolynomial sub Hopf algebra and sub 2*-comodule of H*(BO; ℤ2). In Lemma 3.1 we prove that the Thom isomorphism determines a corresponding filtration of H*(MO;ℤ2) by polynomial subalgebras and sub 2*-comodules M*(n). Let (n) denote the subalgebra of 2 generated by Sq2k, 0 ≦ k < n, and let *(n) be its dual, a quotient Hopf algebra of 2*. In Section 3 we construct a polynomial algebra and *(n)-comodule R(n) such that M*(n)≃2*□*(n)R(n) as algebras and 2*-comodules. Here □ denotes the cotensor product defined in [9, §2]. Dually it will follow that M*(n) has a sub (n)-module and subcoalgebra T(n) such that M*(n)≃2⊗n)T(n) as coalgebras and 2-modules. We also show that M*(n) can not be realised as the homology of a spectrum for n≧4. Of course M*(0)=H*(MO;ℤ2), M*(1)=H*(MSO;ℤ2), M*(2)=H* (MSpin;ℤ2) and M*(3)=H*(MO<8>;ℤ2). Moreover, it follows from [4; Thm. 2.10, Cor. 2.11] that M*(n)=Images[H*(MO<ϕ(n)>;ℤ2)→H*(MO;ℤ2)] and M*(n) ≃ Image [H*(MO;ℤ2)→ H*(MO<ϕ(n)>;ℤ2)]. Here MO id the Thom spectrum of BO, the (k−1)-connected covering of BO, and ϕ(n)=8s + 2t where n = 4s + t, 0≦t≦3. In Section 4 we sketch the odd primary analogue—a filtration pM*(n) of H*(MUp, 0;ℤp) for p an odd prime. MUp, 0 is the Thom spectrum of the (2p-3)-connected factor of the Adams splitting [2] of BU(p).
A stochastic proof of an extension of a theorem of Rado
Authors: Øksendal Bernt
Published: January 2009

The purpose of this article is to illustrate how the theorem of Lévy about conformal invariance of Brownian motion can be used to obtain information about boundary behaviour and removable singularity sets of analytic functions. In particular, we prove a Frostman–Nevanlinna–Tsuji type result about the size of the set of asymptotic values of an analytic function at a subset of the boundary of its domain of definition (Theorem 1). Then this is used to prove the following extension of the classical Radó theorem: If φ is analytic in B\K, where B is the unit ball of ℂ;n and K is a relatively closed subset of B, and the cluster set of φ at K has zero harmonic measure w.r.t. φ(B\K)\≠Ø, then φ extends to a meromorphic function in B (Theorem 2).
Isomorphisms between radical weighted convolution algebras
Authors: Ghahramani F.
Published: January 2009

In [4] we have shown that any two semi-simple weighted convolution algebras L1(ω1) and L1(ω2) are isomorphic. In this paper, given any two radical weighted convolution algebras L1(ω1) and L1(ω2) we find necessary and sufficient conditions, in terms of ω1 and ω2, for L1(ω1) and L1(ω2) to be isomorphic.
Counting balanced signed graphs using marked graphs
Authors: Harary Frank , Kabell Jerald A.
Published: January 2009

A graph in which each line is designated as either positive or negative is called a signed graph S. The sign of a cycle in S is a product of the signs of its lines. A signed graph in which every cycle is positive is called balanced. This concept was introduced by Harary in (3) and the following characterisation of balanced signed graphs was given.
Limit circle criteria for fourth order differential operators with an oscillatory coefficient
Authors: Becker Ronald I.
Published: January 2009

A formally self-adjoint operator L is said to be of limit circle type at infinity if its highest order coefficients are zero-free and all solutions x of L(x) = 0 are square-integrable on [c, ∞) for some c. (We will drop “at infinity” in what follows.)
On the fundamental group of an almost-acyclic 2-complex
Authors: Howie James
Published: January 2009

A 2-complex K is called almost-acyclic if H2(K) = 0 and H1(K) is torsion-free. This class of complexes was introduced in a previous paper (2), and applied to a problem of J. H. C. Whitehead concerning aspherical 2-complexes. In this note, the methods developed in (2) are used to study the finitely-generated subgroups of the fundamental group of an almost-acyclic 2-complex.
On the representation of strictly continuous linear functionals
Authors: Khan Liaqat Ali , Rowlands K.
Published: January 2009

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.
Sketch of the History of Mathematics in Scotland to the end of the 18th Century: Part I.
Authors: Gibson G. A.
Published: January 2009

In the short sketch which I propose to give of the History of Mathematics in Scotland up to the end of the 18th century I must limit myself mainly to the work of the Universities. An adequate treatment of the subject would involve considerations of a general educational character that would range over the relations of the school to the University, the distribution of the various subjects of study and the place of mathematics in the educational system; but it is, of course, impossible to undertake such an extensive investigation at present, though it seems to me that an investigation, with special reference to mathematics, is greatly needed and might form the subject of a research that would be of real value as a contribution to the development of educational ideas. It would be improper, however, to omit all reference to school mathematics, since the school conditions determine, to a considerable extent, those of the University, as current discussions in Scotland clearly show, even though a sound appreciation of the relations between school and University may at times be lacking.
On the Cardinal Function of Interpolation Theory
Authors: Whittaker J. M.
Published: January 2009

1. The cardinal function is the interpolation function

which takes the values αr at the points α + rw. Its principal properties were discovered by Professor Whittaker, amongst others that

(A) When C (x) is analysed into periodic constituents by Fourier's integral-theorem, all constituents of period less than 2w are absent.
Note on the Integral Equations for the Lamé Functions
Authors: Copson E. T.
Published: January 2009

§1. The Lamé Functions of degree n (where n is a positive integer) may be defined as those solutions of the equation

which are polynomials in the elliptic functions sn x, cn x, dn x of real modulus K. Such solutions only exist for certain particular values of the constant a; there are 2n + 1 such values and 2n + 1 corresponding Lamé functions.
Sketch of the History of Mathematics in Scotland to the end of the 18th Century: Part II
Authors: Gibson G. A.
Published: January 2009

The centre of interest now shifts from St Andrews and Edinburgh to Glasgow. The troubles that afflicted Scotland during the 17th Century bore heavily on Glasgow University and more particularly on the position of Mathematics in the University; but in 1691 a distinct Professorship of Mathematics was founded, and from that date the old system of Regents disappeared from Glasgow so far as Mathematics was concerned. The first occupant of the Chair was George Sinclair, who is now chiefly remembered by the controversy in which James Gregory held up Sinclair's Treatise Ars nova et magna to ridicule. It is not fair however to take Gregory's pamphlet as a final estimate of Sinclair's contributions to science; Sinclair laid himself open to attack, but he rendered great service to the mining industry of Scotland and deserves the gratitude of posterity in spite of his many eccentricities. His contributions to mathematics however are of no importance, but during his tenure of the Chair the number of students grew rapidly and the new professorship made a good start.
Notes on the Extension of Aitken's Theorem (for Polynomial Interpolation) to the Everett Types.
Authors: Lidstone G. J.
Published: January 2009

These notes are intended to be read in connexion with Dr A. C. Aitken's paper, Proc. Edinburgh Math. Soc. (2) 1 (1929), 199-203. It is proposed to show (by a simple line of direct algebraic demonstration which is also applicable to the original formula) that Aitken's Theorem can be extended to the Everett types, i.e. the types which include two sets of terms—one set involving u (0) and the resultant of generalised operations on u (0), and the other set involving u (1) and the resultant of similar operations on u (1).
On Generating Functions
Authors: Ferrar W. L.
Published: January 2009

It is well known that the polynomial in x,

has the following properties:—

(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;

(B) it satisfies the three-term recurrence relation

(C) it is the solution of the second order differential equation

(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),

i.e. when

Several other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.
A Generalisation of a Theorem of Mercer
Authors: Copson E. T.
Published: January 2009

§ 1. It is well known that, if , the convergence of sn to a limit implies the convergence of tn to the same limit. The converse theorem, that the convergence of tn implies the convergence of sn, is false. Mercer1 proved, however, that if , then both sn and tn tend to l. This theorem has recently been extended in various directions.2 In the present note the case of Abel limits is considered.
On the “Flat” Regions of Integral Functions of Finite Order
Authors: Whittaker J. M.
Published: January 2009

The term “flat” is used to indicate that the minimum modulus of a function in a region is (in some sense) of the same order as the maximum modulus. Some properties concerned with this notion are described below. They came to light during an attempt to answer a question put to me by Professor Littlewood.
On the “elementary” solution of Laplace's Equation
Authors: Ruse H. S.
Published: January 2009

Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely

(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.
Generalised Solutions of Laplace's Equation
Authors: Ruse H. S.
Published: January 2009

The present paper contains solutions of the tensor generalisation of Laplace's Equation. The results obtained are summarised in the two theorems enunciated in § 1. They apply only to the case when the Riemannian space forming the background of the theory is flat. In the concluding paragraph a special case is considered, and it is shown that the present theory is closely connected with Whittaker's well known general solution of the ordinary Laplace's Equation.
On Professor Whittaker's solution of differential equations by definite integrals: Part I
Authors: Kermack W. O. , McCrea W. H.
Published: January 2009

In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.
Bazin's Matrix and other allied Matrices
Authors: Williamson J.
Published: January 2009

In a paper entitled “On differentiating a Matrix” H. W. Turnbull deduced some interesting and elegant results by the use of a matrix operator Ω, a matrix whose elements were the partial differential operators with respect to the elements of a square matrix x. Throughout the present paper the differential operator Ω. is used, or rather a matrix operator, which is the product of Ω and another square matrix Y.

By means of this operator in §§ 1 and 2 Bazin's matrix and Reiss's matrix are considered from the standpoint of matrices as distinct from that of determinants. Reiss's matrix is shown to be a constant times a compound of Bazin's matrix; and the latent roots of Reiss's matrix are immediately determined in terms of the latent roots of Bazin's matrix. From this result a theorem, discovered by Deruyts, is deduced as well as a more general theorem.
On One Connexion Between Levi-Civita Parallelism and Einstein's Teleparallelism
Authors: Sen R. N.
Published: January 2009

In a recent paper Einstein has proposed a new type of geometry in which it is possible to give an arbitrary vector in the field a displacement which conserves parallelism at a distance. It may be briefly stated as follows:

Let the metric ofthe space, supposed to be of n dimensions, be given by

At each point of the space we consider a set of n unit vectors, denoted by the n2 quantities phq, which form an orthogonal ennuple such that
PETER PINKERTON, M.A., D.SC, LL.D., F.R.S.E.
Authors: Dougall John
Published: January 2009

By the death of Dr Pinkerton on 22nd November 1930, at the comparatively early age of 60, we lost a fellow member who in times past had given varied and valuable service to our Society, and who, as Rector of the High School of Glasgow, occupied a distinguished position among Scottish teachers.

Peter Pinkerton was born in Kilmarnock on 8th June, 1870. He received his early education in the Academy there, and was Medallist in Classics and Mathematics. In 1886 he entered Glasgow University. In 1890 he graduated M.A. with First-class Honours in Mathematics and Natural Philosophy, and was awarded the Breadalbane and John Clark Scholarships in those subjects. For two years thereafter he attended the Royal College of Science, Dublin. He took first places in Mathematics, Mechanics, Physics and Chemistry, and second place in Botany.
James Gregory: A Study in the Early History of Interpolation
Authors: Turnbull H. W.
Published: January 2009

In a highly interesting critical account of the mathematical work of James Gregory (1638–1675), written for the Proceedings of the Edinburgh Mathematical Society, (1) 41 (1923), 2–25, by the late Professor G. A. Gibson, there occurs at p. 8 something of a mathematical puzzle. On that page a pair of formulae are quoted, which certainly are striking examples of the analytical power of Gregory, and which run as follows:—

Gibson adds that “there is another formula (Rigaud, p. 207), but it is of a very complicated character and I do not reproduce it.” It will be convenient to refer to the above pair as formulae A′ and B′, and to the more complicated but analogous series as formula C, and to the original series, from which the above were transcribed, as formulse A and B. I am indebted to Mr A. Inglis for drawing my attention to the problem.
On the Evaluation of Determinants, the Formation of their Adjugates, and the Practical Solution of Simultaneous Linear Equations
Authors: Aitken A. C.
Published: January 2009

There are various methods in existence for the practical solution of a set of simultaneous equations

Some of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.
Ueber die Endlichkeit der Invarianten binärer Formen
Authors: Weitzenböck R. , Turnbull H. W.
Published: January 2009

In der Theorie der Invarianten binärer Formen gilt der Satz, dass die projektiven Komitanten eines vollen Systems durch sukzessives Ueberschieben berechnet werden können. Hierauf beruht der Originalbeweis für die Endlichkeit, den. P. Gordan 1868 zuerst gegeben hat. Konstruktiv wird dieser Beweis aber erst dann, wenn man von vorneherein eine obere Grenze für den Grad der Komitanten eines vollen Systems in den Koeffizienten der Grundformen angeben kann. Also eine natürliche Zahl N, die die Sicherheit gibt, dass alle Komitanten eines vollen Systems gefunden sind, wenn man die Ueberschiebungen soweit berechnet hat, dass ihr Grad N nicht übersteigt.
Congruence Properties of G-Functions
Authors: Gupta Hansraj
Published: January 2009

Denoting the sum of the products of the first n natural numbers taken r at a time by the symbol G(n, r), I have shown that

with the initial values and

In general it was shown that

Defining G (x, r) by the fundamental relation:

for all values of x, we get
An addendum
Authors: Walsh C. E.
Published: January 2009

A generalised lemma used in the second of two papers enables us, as was suggested there, to extend results of the first. Thus, among others, we easily get the following:

Let tn be determined by the relation
Note on Selection from a Multivariate Normal Population
Authors: Aitken A. C.
Published: January 2009

The problem of statistical “selection” is concerned with the alteration induced in a frequency distribution in several variables by an alteration of the parameters in a subsection of the distribution. It may be illustrated by a simple trivariate case, as follows:

From a population characterised by variables x, y, z, correlated and normally distributed, with means 0, 0, 0, variances and product variances r12σ1σ2, r13σ1σ3, r23σ2σ3, a sub-population is extracted by selection in x alone, in such a way that after selection x is still normally distributed, but with mean h and variance s2 . It is required to determine the new values, in the selected population, of the means and variances of y and z, and of the product variances.
On certain new connections between Legendre and Bessel Functions
Authors: Mitra S. C.
Published: January 2009

Let n be a positive integer. Then we know that, if m>– 1,

Consider the integral

which is equal to
An Elementary Proof of the Theorems of Cauchy and Mayer
Authors: Macintyre A. J. , Wilson R.
Published: January 2009

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equation

This method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.
On a Theorem of Gauss
Authors: Gupta Hansraj
Published: January 2009

Professor Hemraj has given a proof of a part of a theorem of Gauss without using the theory of quadratic residues. Proceeding on similar lines, I have obtained a complete proof which is rather simpler and certainly more concise.
An Asymptotic Formula for a Class of Distribution Functions
Authors: Cannon E. W. , Wintner Aurel , Aitken A. C.
Published: January 2009

If x1, x2, …., xk, …. are independent random variables each of which is subjected to a distribution law σ = σ(x) independent of k and having a finite positive dispersion, then x1 + x2 + …. + xn is known to obey the Gauss law as n→ + ∞, no matter how σ (x) be chosen. There arises, however, the question whether it is nevertheless possible to determine the elementary law σ (x) from the asymptotic behaviour of the distribution law of x1 + x2 + …. + xn for very large but finite values of n. It will be shown that the answer is affirmative under very general conditions.
On the tritangent planes of a quadri-cubic space curve
Authors: Richmond H. W.
Published: January 2009

With any twisted curve of order six is associated a system of planes, usually finite in number, which touch the curve at three distinct points. The curve with its system of tritangent planes possesses properties which recall the properties of a plane quartic curve and its system of bitangent lines; and this is specially true of the sextic which is the intersection of a cubic and a quadric surface. But whereas the properties of the plane curve were discovered by geometrical methods, such methods have only recently been applied with success to the space-curve; the earliest properties were obtained by Clebsch from his Theory of Abelian Functions. In the absence of any one place to which reference can conveniently be made, an account of these properties in their geometrical aspect will be useful.
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 order

are permutation matrices. It is convenient to denote them by

where 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 relations

between 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.
Invariants of two quaternary quadrics
Authors: Brahmachari I. , Turnbull H. W.
Published: January 2009

Following the papers by H. W. Turnbull and J. Williamson, I have verified that the 122 forms, of the system of two quaternary quadrics and are actually irreducible. The original 1917 system contained 125 forms, which Williamson reduced by three. The present verification shews that no further reduction is possible. The proof was carried out as follows. I first constructed the whole system in canonical form with and for the two quadrics, and then listed the degrees in the coefficients and variables u, p, x of these concomitants. I next made Diophantine equations between these degrees for testing the supposed reducibility and found them to be impossible, except for Williamson's reduced forms.
Invariant matrices and S-functions
Authors: Zia-ud-Din M.
Published: January 2009

It is known that an induced matrix of an induced matrix is expressible as the direct sum of invariant matrices, or more generally that an invariant matrix of an invariant matrix can be expressed as a direct sum of invariant matrices. The spurs of the irreducible invariant matrices of a given matrix A = [ast], are the S-iunctions of the latent roots of A.
On the absolute summability of series by Rieszian means
Authors: Hyslop J. M.
Published: January 2009

I begin by recalling the well known definitions for summability by the methods of Cesaro and Riesz.

The series Σan is said to be summable (C, K), k> – 1, to the sum s if, as n → ∞

where

and is defined formally by the relation
Test corpuscles in general relativity
Authors: Robertson H. P.
Published: January 2009

In the general theory of relativity, as in many other branches of theoretical physics, the material and energetical content of spacetime is considered, in the first instance, as an extended field, which is specified by means of field quantities (energy-momentum-stress tensor, charge-current density, electromagnetic field strength). From this point of view corpuscles (material particles, photons) are constructs obtained by first considering the field quantities as nonvanishing only within certain world tubes, and then passing by limiting processes to the idealisation in which these world tubes are shrunk into world lines. More precisely, this passage to the corpuscular description may be thought of as accomplished by replacing the original field by successive members of a sequence of field distributions, satisfying the same field laws, which cluster more and more in the neighbourhood of the world lines, and for which in some significant sense the total measure approaches a finite limit. Each such world line, together with the limiting measures of those portions of the field quantities associated therewith, is then a corpuscle; the form of the world line determines the motion of the corpuscle, and the associated “corpuscular quantities” its physical attributes (mass or energy, momentum, charge).
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 form

where 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.
On a recurrence relation
Authors: Walsh C. E.
Published: January 2009

It is proposed here to consider the sequence un determined by the relation

where, in particular,

and initially u1 = θ1. The following is the main result to be proved.
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 function

where (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.
The general equations of biological strife in the case of historical actions
Authors: Volterra Vito
Published: January 2009

I have begun to study the laws of the struggle for life by a group of species living in the same environment in such a way that some devour others. I have used for this purpose the so called principle of encounter, considering the encounters of the individuals of the various species, and what follows by reason of the actions that the individuals exercise on one another.
On a paper by Copson and Ferrar
Authors: Erdélyi A.
Published: January 2009

Concerning your paper written in collaboration with Professor Copson, I found the expansion of

for small values of λ in the following manner: The elementary substitution tan ½φ = sin θ/(et + cos θ)yields

i.e. your equation (5.11).
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 was

where is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-element

was 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.
The place of Dirac's Equation in Five-Dimensional Riemannian Geometry
Authors: Haskey H. W.
Published: January 2009

The generalised Whittaker vector is Λμ which is prevented from vanishing by rejection of the constancy of ω, previously assumed by all writers. It is shown that (1) the null divergence of Λμ is equivalent to Dirac's equation, (2) the length of Λμ measures the probability of occurrence of the electron (3) components of Λμ are connected with the Dirac wave functions and.possible transformations of x5 are probably related to the Uncertainty Principle.
On compound permutation matrices
Authors: Aitken A. C.
Published: January 2009

In an earlier paper1 the author investigated the relation existing between the induced matrices of a group of permutation matrices and the table of group characters of the irreducible representations of the corresponding symmetric group. It was found that the traces of a particular set of induced matrices sufficed to give, by a relatively simple transformation, the complete table of characters.It was remarked also that for n > 4 the set of compound matrices of permutation matrices, on the other hand, could at most provide only part of the table; for in fact the number of compounds, n + 1. is then less than P (n), the numbe'r of partitions of n. For this reason the subject was not pursued into further detail.
The Absolute Cesaro Summability of the Successively Derived Allied Series of a Fourier Series
Authors: Mohanty R.
Published: January 2009

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series by

Then the allied series is
On Generalised Lah-Numbers
Authors: Tauber Selmo
Published: January 2009

In (1) numbers related to the Stirling numbers are defined. Later in (2) these numbers were called Lah-numbers (cf. 2, p. 43, Ex. 16). According to (1) these numbers are of importance in Mathematical Statistics. In this paper we shall generalise the method and apply it to generalised Stirling numbers as defined in (3).
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 equation

does 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 equation

for 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
Minimally strong digraphs
Authors: Geller Dennis P.
Published: January 2009

Dirac (2) and Plummer (5) independently investigated the structure of minimally 2-connected graphs G, which are characterized by the property that for any line x of G, G–x is not 2-connected. In this paper we investigate an analogous class of strongly connected digraphs D such that for any arc x, D–x is not strong. Not surprisingly, these digraphs have much in common with the minimally 2-connected graphs, and a number of theorems similar to those in (2) and (5) are proved, notably our Theorems 9 and 12.
Twisted local-product manifolds
Authors: Robertson S. A.
Published: January 2009

This note is concerned with “twisted“ analogues of the LP-structures (i.e.local-product structures) and grids of (4). To obtain these twisted structures, we modify the concepts of LP-structure and grid by removing the ordering of the local foliations involved in the definitions. The effect of this change is that global foliations need no longer exist, since the locally denned foils may now fit together to form self-intersecting immersed manifolds.
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, and

is called the Riesz mean of type λn and order r.
A generalisation of the Erdélyi-Kober operators
Authors: Lowndes J. S.
Published: January 2009

In this paper we introduce a generalised Hankel operator and generalised Erdélyi-Kober operators and deduce some relations between them. The operators are then applied to obtain solutions to some dual integral equations which have applications in diffraction theory.
A nonembedding result for complex Grassmann manifolds
Authors: Hoggar S. G.
Published: January 2009

A smooth map of a differentiable n-manifold into Euclidean (n+k)-space is called an immersion if its Jacobian has rank n at each point of M. If f is also 1-1, it is called an embedding.
Mean value properties of generalised eigenfunctions
Authors: Gerlach Eberhard
Published: January 2009

Some Hilbert spaces of continuous functions satisfying a mean value property are studied in which the generalised eigenfunctions of any selfadjoint operator again satisfy the same mean value property. Applications are made to nullspaces of some differential operators. The classes of functions involved in these applications are less general than those studied by K. Maurin (6); however, the Hilbert space norms may be arbitrary, while Maurin only considered L2-norms.
On a generalisation of monotonic sequences
Authors: Copson E. T.
Published: January 2009

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequality

then it is convergent. This I was able to prove by considering the corresponding difference equation
On commutative V*-algebras
Authors: Spain P. G.
Published: January 2009

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.
On asymptotically stable sets
Authors: Desbrow D.
Published: January 2009

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.
On elemental annihilator rings
Authors: Yue Chi Ming R.
Published: January 2009

Throughout this note A denotes a ring with identity, and “ module ” means “ left unitary module ”. In (2), C. Yohe studied elemental annihilator rings (e.a.r. for brevity). An e.a.r. is defined as a ring in which every ideal is the annihilator of an element of the ring. For example, a semi-simple, Artinian ring is an e.a.r. A is a l.e.a.r. (left elemental annihilator ring) if every left ideal is the left annihilator of an element of the ring. A r.e.a.r. (right elemental annihilator ring) is denned similarly.
Finiteness at infinity
Authors: Firby P. A.
Published: January 2009

If X is a Tychonoff topological space, and if βX is the Stone-Cech compactification of X, then βX\X will denote the complement of X in βX. If A is a subset of X, then cl [A: X] will denote the closure of A in X, and int [A: X] will denote the interior of A in X. In Isbell ((3), p. 119) a property of βX\X is called a property which X has at infinity, and it is the aim of this paper to give necessary and sufficient conditions for X to be finite at infinity. Since βX is T1 we can say that if X is finite at infinity, then βX\X is closed in βX. So we lose nothing by restricting our attention to locally compact, Tychonoff spaces, and for the remainder of the paper X will denote such a space.
On the structure of certain basic sequences associated with an arithmetic function
Authors: Goldsmith Donald L.
Published: January 2009

We have previously studied in some detail the multiplicative properties of a given arithmetic function f with respect to a fixed basic sequence (see, for example, (1), (2)). We investigate here the structure of M(f), the collection of all basic sequences such that f is multiplicative with respect to , and in particular we focus our attention on the maximal members of M(f). Our principal result will be a proof that each maximal member of M(f) contains the same set of type II primitive pairs. Moreover, we will give a simple criterion for determining, in terms of the behaviour of f, whether or not a particular primitive pair (p, p) is in any (and therefore every) maximal member of M(f).
Thin sets in
Authors: Tweddle I.
Published: January 2009

In (4) J. F. C. Kingman and A. P. Robertson introduced the idea of thin sets in certain ℒ1 spaces. Thin sets are extreme cases of sets which are not total, and so the problem naturally arises of partitioning a measure space relative to a given set of integrable functions in such a way that on each element of the partition, the set of functions is either thin or total in a sense which is made precise below. In the present note, such partitions are obtained in §2 for finite or totally σ-finite measure spaces. In §3 the basic ideas are reformulated in terms of Radon measures on locally compact spaces, leading to an extension of the results of §2 in this context.
Eigenvalues of complex tridiagonal matrices
Authors: Gibson P. M.
Published: January 2009

Results of Arscott (1) and Jayne (3) on real matrices are generalized to obtain bounds for the real parts of the eigenvalues of certain complex tridiagonal matrices, and bounds for the imaginary parts of the eigenvalues of other tridiagonal matrices are given. It is shown that analogous results hold for zeros of the permanent of certain characteristic matrices.
Pseudo-rings of infinite matrices
Authors: Jump K.
Published: January 2009

Patterson (4) introduced the concept of a pseudo-ring and considered the pseudo-ring of infinite matrices over a ring. In this paper we shall generalize and improve the work of Patterson, using certain additions to the general theory of pseudo-rings which have recently been introduced (1). We shall follow the conventions and notations used in (1) and (4).
On partitions ofnintoksummands
Authors: Gupta Hansraj
Published: January 2009

In his recent paper on partitions (1), Jakub Intrator proved that the numberp(n,k) of partitions ofninto exactlyksummands, 1 <k≦n, is given by a polynomial of degree exactlyk− 1 inn, the first [(k+1)/2] coefficients of which (starting with the coefficient of the highest degree term), are independent ofnand the rest depend on the residue ofnmodulo the least common multiple of the integers 1, 2, 3, …,k. He even showed (ignoring the casek= 3) that the [(k+3)/2]-th coefficient in the polynomial depends only on the parity ofnand is not the same forneven andnodd.
On the algebra generated by a Hermitian operator
Authors: Dowson H. R.
Published: January 2009

The purpose of this note is to solve a problem of Dr A. M. Sinclair. Denote by Aw(I, T) the algebra with identity generated by a bounded linear operator T in the weak operator topology. We prove the following result.
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 condition

is both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).
Countable-codimensional subspaces of locally convex spaces
Authors: Webb J. H.
Published: January 2009

A barrel in a locally convex Hausdorff space E[τ] is a closed absolutely convex absorbent set. A σ-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled (countably barrelled) if every barrel (σ-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (σ-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2).
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 equation

where 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 hull

is 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

Relative 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.