bluuebunny/crossref_metadata_2025_split huggingface dataset:
An Analogue for Strong Summability of Abel's Summability Method
Authors: Harington C. F. , Hyslop J. M. 
Published: January 2009

Given a series Σan, we define , by the relationwhere is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that
A Determinantal Expansion for a Glass of Definite Integral. Part 1
Authors: Shenton L. R. 
Published: January 2009

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficientsParseval's formula gives
Generalised Algebraic Continued Fractions related to Definite Integrals
Authors: Shenton L. R. 
Published: January 2009

The present paper is a continuation of the work initiated in [l]-[5]. In [5] I gave an expansion of the formfor the second order C.F. associated withwhere U8, V8, W8 satisfy a fourth-order recurrence relation, there being a similar expansion for third order C.F.'s. I shall now give simple expressions for U8, V8, W8 (or related forms) in terms of χ2s(Z1), χ2s (Z2), ω2s(Z1), ω2s(Z2), whereand show that there is a remarkable relation between the recurrence formula for the first order C.F. and that satisfied by U3, V3, W3. The generalised form of these results will be stated and proved.
Simultaneous Linear Recurrence Relations with Variable Coefficients
Authors: Ursell H. D. 
Published: January 2009

Our subject is a set of equationswhere the uj(n)(j = 1, 2, …, k) are k “unknown” functions of the integer variable n, the zi(n) (i = 1, 2, … h) are h “known” functions of n, and the Aij(n) are hk “known” operatorswhich are polynomials in E, each of fixed order pij but with coefficients which may vary with n. E is the usual operator defined byOur first task is to determine whether the equations (1) are self-consistent. Secondly, if they are self-consistent, we ask what follows from them for a given subset of the unknowns, e.g. for (uj+1, …, uk) in other words we wish to eliminate (u1, …, uj). In particular we wish to eliminate all the variables but one, say uk. We shall in fact find that either uk is arbitrary or else that it has only to satisfy a single linear recurrence relation : and the order of that relation is of interest to us. Thirdly, we ask that reduction to standard form is possible, with or without a transformation of the unknowns themselves.
A Note on Trace-Differentiation and the Ω-operator
Authors: Aitken A. C. 
Published: January 2009

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

In a recent paper1 γ-matrices were constructed summing the binomial series at isolated points to its right value. It is assumed that the reader is able to refer to this paper and is familiar with its notation and terminology.The object of this note is to give the construction of γ-matrices which sum the binomial series at an isolated point to an arbitrary given value, while inside the circle of convergence the generalised sum is necessarily the right value.
Theorems Relating to Quadratic Forms and their Discriminant Matrices
Authors: Vajda S. 
Published: January 2009

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:Let the expression be given; introduce, for c, a linear form in and obtainIf the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.
Some Remarks on Uniform Convergence
Authors: Eggleston H. G. 
Published: January 2009

A useful test for uniform convergence is that first established by Buchanan and Hildebrandt [4] which is as follows.(A) “If a sequencefn(x) of monotonic functions converges to a continuous functionf(x) in [a, b] then this convergence is uniform.”In §1 of this paper it is shown that this test is included in a sequence of theorems, each of which establishes a type of uniform convergence. The first is a well-known topological theorem on limit sets, the second is a result on the limits of rectifiable arcs, the third is a generalisation of (A) due to Behrend [3], the fourth is (A) itself, the fifth is a one-sided version of Bendixson's test and the sixth is Bendixson's test.
The Sir Edmund Whittaker Memorial Prize
Authors: Fulton James 
Published: January 2009

A special meeting of the Edinburgh Mathematical Society was held in the Mathematical Institute, Edinburgh, on Friday, 17th May 1957, to commemorate Sir Edmund Whittaker's work. Present at this meeting were Sir Edmund's widow, Lady Whittaker, and members of her family.Dr J. M. Whittaker, F.E.S., on behalf of Lady Whittaker and family, asked the Society to accept the sum of £500 to be used to provide a prize which would be awarded for meritorious work in mathematics by young mathematicians in Scotland.
Some Aspects of Uniqueness for Solutions to Boundary Problems
Authors: Martin M. H. 
Published: January 2009

The solution to the boundary problemwhere r is the distance of point (x, y) from the origin, and h is a given function the arc length s along the unit circle r = 1, is not necessarily unique, Boggio (1), Weinstein (2), Stoker (3), Martin (4). Indeed if h is a positive integer m is known that the only solutions regular analytic for r≦1 arewhere r, θ denote polar coordinates and A, B are arbitrary constants.
A Pair of Dual Integral Equations Occurring in Diffraction Theory
Authors: Burlak J. 
Published: January 2009

Dual integral equations of the formwhere f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.
On bounded sequences satisfying a linear inequality
Authors: Russell Dennis C. 
Published: January 2009

In a recent paper, E. T. Copson (2) proves the following result:Theorem C.Letki >0 (i = 1, …, m), k1 +…+ km = 1, and the real sequence (an) satisfy the inequalityIf (an) is bounded, then it must be convergent.
Passman-Zalesskii Radical of group algebras
Authors: Sinha I. 
Published: January 2009

Recently Passman (attributing the origin of the idea to Zalesskii) has defined the following ideal in a ring, (2).Definition. For a unitary ring R,N * R = {α ∈ R | αS is nilpotent for all finitely generated subrings S of R}.For a group algebra KG over a field K of characteristic p ≠ 0, he has proved the radical property:
Minimal cohesive basic sets
Authors: Goldsmith Donald L. 
Published: January 2009

A basic set (formerly basic sequence) ℬ is a set of pairs (a, b) of positive integers satisfying(1) if (a, b) ∈ ℬ, then (b, a) ∈ ℬ,(2) (a, bc) ∈ ℬ if and only if (a, b) ∈ ℬ and (a, c) ∈ ℬ,(3) (1, k ∈ ℬ, k = 1, 2, ….Some familiar examples of basic sets are, where Sk = {(1, k), (k, 1)}, = {(a, b)| a and b are relatively prime positive integers},ℒ = {(a, b)| a and b are any positive integers}.
Spectra of irreducible matrices
Authors: Minc Henryk 
Published: January 2009

A real matrix is called non-negative (positive) if all its entries are non-negative (positive). Two matrices A and B are said to be cogredient if there exists a permutation matrix Q such that QAQT = B. A square non-negative matrix is called reducible if it is cogredient to a matrix of the formwhere the blocks X and Y are square. Otherwise it is called irreducible.
On the Stepanov-almost periodic solution of a second-order operator differential equation
Authors: Rao Aribindi Satyanarayan 
Published: January 2009

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).
Solution of hypergeometric integral equations involving generalised functions
Authors: McBride Adam C. 
Published: January 2009

In a previous paper (9), we introduced the spaces Fp, μ of testing-functions and the corresponding spaces of generalised functions. For 1≦p≦∞, Fp(=Fp, 0) is the linear space of all complex-valued measurable functions φ defined on (0, ∞) which are infinitely differentiable on (0, ∞) and for which for each k = 0, 1, 2, …. In symbols,
On a singular measure of D. M. Connolly and J. H. Williamson
Authors: Ludvik P. M. 
Published: January 2009

In (1) a measure λ∈M(ℝ) is constructed and shown to satisfy the following:(i) is absolutely continuous, where denotes the measure with for all Borel sets E,(ii) λ*λ = λ2 is singular,(iii) λk = λk−1*λ is absolutely continuous for K ≧6.
An extension of a Hardy-Littlewood-Pólya inequality
Authors: Erdélyi A. 
Published: January 2009

The Hardy-Littlewood-Pólya inequality in question can be written in the formHere and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean thatdefines a locally integrable function Kf for which (1.1) holds.
Dual integral equations with a trigonometric Kernel
Authors: Singh Brij M. , Dhaliwal Ranjit S. 
Published: January 2009

In this paper, we solve the following dual integral equationswhere δ is a real positive constant and f(x) is a continuous and integrable function of x in [0, a]. The dual integral equations (1) and (2) arise in a crack problem of elasticity.
On a generalisation of a result of Ramanujan connected with the exponential series
Authors: Paris R. B. 
Published: January 2009

One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined byRamanujan (9) showed that when n is large, θn possesses the asymptotic expansionThe first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined bythen πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansionA generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by
The first hundred years (1883–1983)
Authors: Rankin R. A. 
Published: January 2009

This account of our Society is based to some extent on my Presidential address, which was given on 19 October 1977 and was devoted to the first fifty years.In the latter half of the nineteenth century there was an upsurge of interest in mathematics that resulted in the foundation of a number of mathematical societies in different countries. The London Mathematical Society (1865), the Moscow Mathematical Society (1867), the Société Mathématique de France (1873), the Edinburgh Mathematical Society (1883) and the New York (later American) Mathematical Society (1888) were all founded in this period. There had, of course, been earlier more local societies, such as the Spittalfields Mathematical Society, which flourished over a long period before becoming defunct, as well as one or two much older bodies, for example the Mathematische Gesellschaft in Hamburg (1690), which still survive.
A note an homeomorphic measures on topological groups
Authors: Morris Sidney A. , Peck Vincent C. 
Published: January 2009

The classical von Neumann–Oxtoby–Ulam Theorem states the following:Given non-atomic Borel probability measures μ, λ on In such thatthere exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set SIt is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].
Eigenvalues of the Klein-Gordon equation
Authors: Najman Branko 
Published: January 2009

Consider the Klein-Gordon equationwhere q, Aj are real valued functions on Rn, m and e positive constants. Equation (1) describes the motion of a relativistic particle of mass m and charge e in an external field described by the electrostatic potential q and the electromagnetic potential A = (Aj); units are chosen so that the speed of light is one.
A sharpening of the Berkson-Glickfeld theorem
Authors: Magyar Zoltan 
Published: January 2009

It is known that if in a Banach*-algebra with unit the following holds:then it is a C*-algebra (see [3]).We shall show that the above theorem can be sharpened in the following way: we replace the submultiplicativity of the norm by the weaker assumptionObserve that under this assumption even the existence of exp(ih) is not at all obvious, but it will be proved to be true below. Our main result is Theorem 2 which depends on Theorem 1. Our last remark is the equivalent-norm-version of the statement.
Subsimple, injective, retract
Authors: Feigelstock Shalom , Klein Aaron 
Published: January 2009

Simple and subsimple objects were introduced in [6]. It was shown that if there are enough simple objects in a category , then there is no room for injectives in . This idea was exploited in [6] and [2] to show that several classes of groups, rings and classes belonging to other categories do not possess non-trivial injectives or retracts. In this note, the above results will be strengthened by introducing a weaker condition than subsimple of [6]. As a consequence, and by employing some embedding theorems, we show that some important classes do not possess non-trivial retracts.All the categories are assumed to have a zero object.
On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold
Authors: Leung Pui-Fai 
Published: January 2009

Let Mn be an n-dimensional smooth compact Riemannian manifold. By a theorem of Nash, we can think of it as an isometrically immersed submanifold in some higher dimensional Euclidean space ℝn+m. Viewing in this way we can compare the intrinsic geometry of M to its extrinsic geometry. Classically, the Gauss equationwhere K(X,Y) denotes the sectional curvature in M corresponding to the plane spanned by the two orthonormal vectors X, Y and B denotes the second fundamental form gives one of the most important relations between the intrinsic and extrinsic geometries of M. In this note we shall prove the following.
Extensions of characters from Hall π-subgroups of π-separable groups
Authors: Isaacs I. M. 
Published: January 2009

The main result of this paper is the followingTheorem A. Let G be a π-separable finite group with Hall πsubgroup H. Suppose θεIrr(H). Then there exists a unique subgroup M, maximal with the property that it contains H and θ can be extended to a character of M.
On the structure of the set of solutions of the Darboux problem for hyperbolic equations
Authors: de Blasi F. S. , Myjak J. 
Published: January 2009

Consider the Darboux problemwhere φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:(A1) f(.,.,z) is measurable for every z ∈ Rd;(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.
The computation of the Hodgkin-Snaith operation in K*(Z × BU, Z/p)
Authors: Hegenbarth Friedrich 
Published: January 2009

This paper builds on the work of V. P. Snaith [5] (in particular Section 6) and gives a more explicit determination of the result.Recently J. E. McLure has given a satisfactory account of Dyer-Lashof operations in X-theory. For any E∞-space Y and for any r≧2 there is a mapwith specific properties (see [3], Theorem 1). Earlier, L. Hodgkin [2] and V. P. Snaith [5] constructed an operationwhere Ind(Y) = {xp\x∈Kα(Y,Z/p)} is the indeterminancy of .
An aspect of the invariant of degree 4 of the binary quintic
Authors: Edge W. L. 
Published: January 2009

A binary form of odd degree,has a quadratic covariant Г, (ab2maxbx in Aronhold's notation, and the discriminant Δ of Г is an invariant of ƒ For m = 2Δ was obtained by Cayley in 1856 [3, p. 274]; it was curiosity as to how Δ could be interpreted geometrically that triggered the writing of this note. An interpretation, in projective space [2m + 1], that does not seem to be on record, of Γ and Δ is found below. If m = 1 one has merely the Hessian and discriminant of a binary cubic whose interpretations in the geometry of the twisted cubic are widely known [5, pp. 241–2].
On normed Lie algebras with sufficiently many subalgebras of codimension I
Authors: Kissin E. V. 
Published: January 2009

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but whenwhere S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1]investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.
On the number of eigenvalues in the spectral gap of a Dirac system
Authors: Hinton D. B. , Mingarelli A. B. , Read T. T. , Shaw J. K. 
Published: January 2009

We consider the one-dimensional operator,on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).
On the other pαqβ theorem of Burnside
Authors: Bialostocki Arie 
Published: January 2009

The “other” pαqβ theorem of Burnside states the following:Theorem A.l. Let G be a group of order pαqβ, where p and q are distinct primes. If pα>qβ, then Op(G)≠1 unless(a) p is a Mersenne prime and q = 2;(b) p = 2 and q is a Fermat prime; or(c) p = 2 and q = 7.
The nonabelian tensor square of a finite split metacyclic group
Authors: Johnson D. L. 
Published: January 2009

Given any group G, its tensor square G ⊗ G is defined by the followingpresentation(see [[3]):where g,g′,h,h′ range independently over G, and gh=ghg−1. In what follows, gg′ ⊗ gh is often written in the abbreviated form g(g′ ⊗ h).
On a Generalisation of the Laplace Transformation
Authors: Erdélyi A. 
Published: January 2009

On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma hasand points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integralwhich was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.
Zeros and Periodicity of Functions of Infinite Matrices
Authors: Ibrahim Shafik Asaad 
Published: January 2009

Certain functions of infinite matrices are known to exist.† This gives rise to the following questions:1. Whether the power series of matriceshas a zero in the field ‡ of infinite matrices, and2. If f(A) exists for a certain infinite matrix A, is there an infinite matrix B such thatIn other words, is there a matrix period for f(A)?In this paper theorems concerning zeros and periodicity of functions of semi block infinite matrices § (defined below) are established.
On extension of characters from normal subgroups
Authors: Karpilovsky G. 
Published: January 2009

In what follows, character means irreducible complex character.Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.
Indicial equivalents of multiparameter definiteness conditions in finite dimensions
Authors: Binding Paul 
Published: January 2009

Let Tm, Vmn be Hermitean linear operators on complex Hilbert spaces Hm, m=1…k. A nonzero column vector satisfyingwill be called an eigenvalue. This type of problem has been studied extensively by Atkinson [2] from the viewpoint of determinantal operators on the tensor product We shall connect his work with more recent investigations [5,7] of eigenvalue indices based on minimax principles for , which can be viewed as an operator on .
On pseudo-distributive near-rings
Authors: Mason Gordon 
Published: January 2009

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfiesfor all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.