bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Hardy's Theory of m-Functions
Authors: Copson E. T. 
Published: January 2009

§1. The Cardinal Function of Interpolation Theory is the functionwhich takes the values an at the points x = n. Ferrar has recently provedTheorem1. If are convergent, C(x)is an m-function3forThis means that C(x) is a solution of the intergral equationFerrar's proof deals with functions of a real variable and involves some rather difficult double limit considerations.
On the Number of Ways of Colouring a Map
Authors: Birkhoff George D. 
Published: January 2009

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the followingTheorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then(1)This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.
On Professor Whittaker's solution of differential equations by definite integrals: Part II
Authors: Kermack W. O. , McCrea W. H. 
Published: January 2009

In Part I it has been shown that, given a contact transformation, two equationscan be derived which lead to the compatible differential equationsIt will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is thatregarded as an equation in the non-commutative variables q, p which themselves satisfy the conditionWe shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.
Professor GEORGE A. GIBSON, M.A., LL.D., F.R.S.E.
Authors: Bell R. J. T. 
Published: January 2009

George Alexander Gibson was born at Greenlaw, Berwickshire, in 1858. His father was a man of rare capacity and character who wrote a history of Greenlaw and taught himself Latin so as to be able to obtain the material for his history from original documents.After attending an elementary school at Greenlaw, Gibson enrolled in 1874 as a student at the University of Glasgow, where he gained the highest places in all the classes of the Arts curriculum. At this period of his life his health was not robust and it threatened to terminate his university studies. Consequently, in 1881, he took the degree of M.A. without honours. In the following year he was able to sit the honours examination in Mathematics and Natural Philosophy, and he so distinguished himself that he was awarded the Ewing Fellowship. When the degree of D.Sc. was afterwards insti tuted and the regulation required that a candidate for it should have an honours degree in Arts or Science, his ordinary degree precluded Gibson from becoming a candidate for the doctorate which his published work would doubtless have gained for him.
Sets of Semi-Commutative Matrices
Authors: Williamson J. 
Published: January 2009

In a paper entitled “Sets of anticommuting matrices” Eddington proved that if El, E2, …., Eqform a set of q four-rowed square matrices satisfying the relations,,where E is the unit matrix, then the maximum value of q is five. Later Newman showed that this result is a particular case of the general theorem that ifE1, E2, …., Eqform a set of q t-rowed square matrices satisfying (1), where t = 2Pτ and τ is odd, then the maximum value of q is 2p + 1.
Gauss' Theorem in a General Space-time
Authors: Ruse H. S. 
Published: January 2009

In classical mechanics Gauss' Theorem for a gravitational field states that, if S is a closed surface and N the component of gravitational force along the outward normal, thenwhere β is the Newtonian constant of gravitation and M is the total mass inside S. This result has recently been extended to general relativity by E. T. Whittaker,1 who, however, considered only the case of a statical gravitational field, the line-element of which is given by2where the coefficients U and αμν are independent of t. It is not immediately clear from his work whether the results are extensible to more general space-times.
On a function which is self-reciprocal in the Hankel transform
Authors: Phillips E. G. 
Published: January 2009

It is well-known that if v≧–½ andthen ø (x) is said to be self-reciprocal in the Hankel transform and may be described as Rv. If v = ±½(1) reduces to the Fourier sine or cosine transform. Functions of these two classes may be described as Rg and Re.In a list of pairs of reciprocal functions, G. A. Campbell gives the example thatis Rc, but there does not seem to be any explicit reference to this function in the literature. It suggests that there is a corresponding function which is Rv.
Functions with Dominant Polar Singularities on the Circle of Convergence
Authors: Wilson R. 
Published: January 2009

1. Introduction. In an earlier paper1 the functionwas considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the formwhere the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.
The Linear j-Differential Equation
Authors: Ingram W. H. 
Published: January 2009

The basic reciprocity of j-differential and LM-integralfor bounded functions f(x) with simple discontinuities but continuous on the left at each point and for g(x) in the somewhat restricted class B of functions of bounded variation and also left-continuous, was established in (2) and (3); the dot here indicates the lower product of and (jg, g+ (x+)dx), with , and the integral indicated is the RJDS-integral, equivalent to (LM) .
A Generalisation of a Result of Abel with an Application to Tree Enumerations
Authors: Robertson M. M. 
Published: January 2009

We prove the following theorem, which was established by Abel (1) for the case u = 1.Theorem. If u, k are positive integers and x, 1, , u, are real numbers, thenwhere the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .
The radical of the group algebra of a sub-group, of a polycyclic group and of a restricted SN-group
Authors: Wallace D. A. R. 
Published: January 2009

Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by lettingwhere α(x, y) is a non-zero element of K, subject to the condition thatwhich is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and if
G has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.
Characterizations of linear differential systems with a regular singular point
Authors: Harris W. A. 
Published: January 2009

The linear differential systemwhere w is a vector with n components and A is an n by n matrix is said to have z = 0 as a regular singular point if there exists a fundamental matrix of the formsuch that S is holomorphic at z = 0 and R is a constant matrix ((1), p. 111; (2), p. 73). For such systems A will have at most a pole at z = 0 and we may writewhere p is an integer, Ã is holomorphic at z = 0, and Ã(0) ≠ 0. However, the converse is not true. When p ≦ − 1, A is holomorphic at z = 0, and every fundamental matrix is holomorphic at z = 0. If p ≧ 1, the non-negative integer p is called (after Poincaré) the rank of the singularity and there is a significant difference between the cases p = 0 and p ≧ 1. If p = 0 the linear differential system (1) is known to have z = 0 as a regular singular point ((1), p. 111) ; whereas, if p ≧ 1, z = 0 may or may not be a regular singular point.
Inequalities for certain cyclic sums II
Authors: Boarder J. C. , Daykin D. E. 
Published: January 2009

Let k ≧ 2 be an integer and each of ν1, ν2, …, νk and δ1, δ2, …, δk be 0 or 1. Then given any positive integer M and non-negative reals a1, a2, …, aM we putwhereand
A result on the convolution of distributions
Authors: Fisher B. 
Published: January 2009

In a recent paper (1), Jones extended the definition of the convolution of two distributions to cover certain pairs of distributions which could not be convolved in the sense of the previous definition. The convolution ω1 * ω2 of two distributions ω1 and ω2 was defined as the limit of the sequence ω1n * ω2n, provided the limit ω exists in the sense thatfor all fine functions φ in the terminology of Jones (2) whereand τ is an infinitely differentiate function satisfying the following conditions:
An Integral Transform Solution of the Differential Equation for the Transverse Motion of an Elastic Beam
Authors: Fulton J. 
Published: January 2009

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution.It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions.
An Algebraic Problem in Control Theory
Authors: Wouk Arthur 
Published: January 2009

In the series of papers in the early forties summarised in (1), (2), A. I. Lur'e showed how to utilise Liapunov's second or direct method in the investigation of the stability of linear automatic control systems with a single nonlinear actuator. His approach consists of1. the transformation of the original system of differential equations via the so-called Lur'e transformation into canonical coordinates in which the construction of the Liapunov function is direct, and2. the conversion of the differential problem into a purely algebraic problem.We will be concerned here with the questions of the existence and construction of the Lur's transformation.
Asymptotic Solutions of Integral Equations with Convolution Kernels
Authors: Hutson V. 
Published: January 2009

SummaryThe equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.
B# Modular Annihilator Algebras
Authors: Duncan J. 
Published: January 2009

The structure of various classes of annihilator algebras has been known for some time. Bonsall and Goldie (1) considered semi-simple Banach algebras with the properties(i)r(L)≡{x:xε,yx=0(yεL)} ≠(0) for each proper closed left ideal L of ,(ii)l(K)≡{x:xε,xy=0(yεK)}≠(0) for each proper closed left ideal K of ,
On Certain Integral Equations of Convolution Type with Bessel-Function Kernels
Authors: Srivastav R. P. 
Published: January 2009

In this paper we first obtain elementary solutions of the integral equationsandUsing these solutions we then define operators of fractional integration. These operators may be regarded as a generalisation of the operators of fractional integration introduced by Sneddon (1) as a modification of Erdé1yi–Kober operators. In fact Erdélyi–Kober–Sneddon operators may be obtained by multiplying both sides of the equations by α-1½β and considering the limiting case α→0. We employ these operators to find a generalisation of the Mellin transform.
Derivation of Solutions of the Klein-Gordon Equation from Solutions of the Wave Equation
Authors: Chambers LL. G. 
Published: January 2009

The Klein–Gordon equationΩ being a constant of dimensions [time]-1 and c being a constant velocity, appears in nuclear physics (1) and, when the Laplacian operator is twodimensional, in the theory of long gravity waves on a rotating earth (2). If Ω is zero it reduces to the wave equation
Homomorphism theories for universal algebras
Authors: Hoehnke Hans-Jürgen 
Published: January 2009

It is well-known that a homomorphism ø(A→B) between groups A and B induces a homomorphism ø*(ZA→ZB) between the corresponding group rings ZA and ZB over the ring of integers Z. The identical congruence O on B and the unit element eB of B can be characterised by the equations x–y = 0 and x–eB = 0 (x,y ∈ B) respectively. Similarly the congruence Γø corresponding to ø and the corresponding normal subgroup of A areand {x∈A1 = A,(x–eA)ø = 0} respectively.
A note on logarithmic summability (L):Corrigendum
Authors: Sitaraman Y. 
Published: January 2009

I am indebted to Dr. B. Kuttner for having kindly drawn my attention to certain corrections necessary in my paper with the above title, appearing in these Proceedings, Vol. 15 (Series II), Part 1, June 1966, pages 47–55.The major corrections are as follows. On pages 48-9, Lemma 1 should appear without the factor log (1 – z)−1 of the denominator in the integrand at the end of the lemma; the formula in the proof of Lemma 1 is incorrect, its correct form being
On partitions of N points
Authors: Watson Donald 
Published: January 2009

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:(i) Find the number, ηk(N), of regions into which a k-dimensional space is partitioned by a set of N (k- l)-dimensional hyperplanes.(ii) Find the number, vk(N), of distinct partitions of a given set of N points in a k-dimensional space E that can be induced by (k- 1)-dimensional hyperplanes.
Further Arithmetical Functions in Finite Fields
Authors: Cohen Stephen D. 
Published: January 2009

In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisationwhere P1, …, Pr are distinct irreducible polynomials (i.e. primes).
Two Enumerative Proofs of an Identity of Jacobi
Authors: Sudler C. 
Published: January 2009

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namelyHere, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.
Note on the Elementary Divisors of Some Related Determinants
Authors: Aitken A. C. 
Published: January 2009

The matrices considered in the following note are non-singular, and are related to a given matrix A having elements taken from the field of positive and negative integers and zero. The invariant properties of such a matrix A, under multiplications by matrices of determinant equal to unity, can be formulated, as is well known, in terms of the “elementary divisors” of the determinant |A|. Thus if A is of the nth order, and p is a prime occurring in |A| to the power hn, in the H.C.F. of the first minors of |A| to the power hn−1, in the H.C.F. of the second minors to the power hn−2, and so on, h0 by convention being zero, then the first differences of the h's,are invariant under the transformations considered, and it is known that r ≥ er−1. The numberswhere the product includes all prime factors of |A|, are called the elementary divisors of |A| and
Further Notes on the Stieltjes Integral
Authors: Hyslop J. 
Published: January 2009

The definition here used of the Stieltjes Integral is the same as that of a previous note, viz.:—Let f (x), φ (x) be two real functions defined in (a, b) a finite interval on the axis of the real variable x. Let Δ1, Δ2, …, Δn, be a finite set of sub-intervals which together make up (a, b). Δrφ denotes the increment of φ (x) in Δr. Let ξr be any point of Δr, and form the sum
The Composition of Linear Differential Systems
Authors: Whittaker J. M. 
Published: January 2009

A common method of solving a linear differential equation consists in expressing the differential operator as a product of factors. The possibility of doing so has been studied extensively by Vessiot, following the work of Picard and Drach, on the lines of the Galois theory of algebraic equations. The analogous process of resolving a, linear differential system, consisting of an equation together with boundary conditions, into two or more systems of lower order does not seem to have been investigated. Such a resolution is not always possible, even in cases where the differential equation can be factorised. Thus the systemis equivalent to the systemsbut the systemcannot be so resolved.
Factorial Moments and Frequencies of Charlier's Type B
Authors: Campbell J. T. 
Published: January 2009

When a given frequency distribution is to be graduated, it is customary to express the constants of the fitted curve in terms of the moments of the frequency distribution. The rth moment of the distribution, in which the relative frequency of a measure x is φ(x) or, in the case of a continuous variable, the differential of frequency is φ(x)dx, is defined in the respective cases bythe summation or integration being taken over the whole range of possible values of x. In the present paper we make use of another kind of moment, the factorial moment, which has already been considered by several writers, and which is specially suited to the case when the frequencies of the distribution are given for discrete, equidistant values of the variable. The (r + 1)th factorial moment, for the case where x, measured from some arbitrary origin, can increase by increments h, 2h, 3h,.…, will be defined to bewhere the summation extends over all possible values of x; it will be denoted by m(r+1). By a suitable choice of scale the increments of x may be taken as equal to unity in any given case.
On the Absolute Summability (A) of Infinite Series
Authors: Fekete M 
Published: January 2009

§1. A serieshas been defined by J. M. Whittaker to be absolutely summable (A), ifis convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.
A Note on the Capelli Operators associated with a Symmetric Matrix
Authors: Wallace Andrew H. 
Published: January 2009

In the Proceedings of the Edinburgh Mathematical Society, 1948, there appear two papers by Lars Gårding and Turnbull respectively (Gårding [1], Turnbull [2]) which formulate the theory of Cayley and Capelli operators associated with symmetric matrices. Turnbull derives the modification, appropriate to symmetric matrices, of Capelli's Theorem, which states that (taking a third order operator for the sake of ease in writing)where the symbol (xyz)ijk stands for the determinantwith a similar meaning for the determinantal differential operator, while the symbols are polarisations (Capelli [1]; cf. Turnbull [1], p. 116). Gårding's theorem deals with the effect of such modified Capelli operators on powers of the determinant of the symmetric matrix in question. The subject of this note is an alternative derivation of the modified Capelli theorem and of Gårding's theorem.
A Type of Alternant
Authors: Ponting F. W. 
Published: January 2009

We definewhere ap ≠ aq when p ≠ q. If N = Σλi, then the partition (λ1, λ2, …, λn) of N with λ1 ≥ λ2 ≥ … ≥ λn is denoted by (λ) and we setAll partitions will be in descending order and the usual notation for repeated parts will be used.
On a Set of Conform-Invariant Equations of the Gravitational Field
Authors: Buchdahl H. A. 
Published: January 2009

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact