bluuebunny/crossref_metadata_2025_split huggingface dataset:
On a Simple Type of Integro-differential Equation
Authors: Gillespie R. P. 
Published: January 2009

In a previous paper in these Proceedings the author discussed conditions for a maximum or minimum of functions of integrals of the typeusing the methods of the Calculus of Variations. In the effort to establish a third necessary condition for a minimum—the analogue of Jacobi's condition in the ordinary variational problem—it was found that the analogue of Jacobi's Equation was an integrodifferential equation of the form
Birational transformations with isolated fundamental points
Authors: Todd J. A. 
Published: January 2009

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, thenwhere E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.
On certain infinite integrals involving Struve functions and parabolic cylinder functions
Authors: Mitra S. C. 
Published: January 2009

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.
A Plane Quartic with Eight Undulations
Authors: Edge W. L. 
Published: January 2009

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curvewhere (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points
The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.
Mean Value Theorems for Vector Valued Functions
Authors: McLeod Robert M. 
Published: January 2009

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we havefor some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to bewhere cka, b, 0 k, and .
The interaction of a contact discontinuity with sound waves according to the linearised Navier-Stokes equations
Authors: Anderson W. M. 
Published: January 2009

AbstractThe resolution of a small initial discontinuity in a gas is examined using the linearised Navier-Stokes equations. The smoothing of the resultant contact surface and sound waves due to dissipation results in small flows which interact. The problem is solved for arbitrary Prandtl number by using a Fourier transform in space and a Laplace transform in time. The Fourier transform is inverted exactly and the density perturbation is found as two asymptotic series valid for small dissipation near the contact surface and the sound waves respectively. The modifications to the structures of the contact surface and the sound waves are exhibited.
Commuting well-bounded operators on Hilbert spaces
Authors: Gillespie T. A. 
Published: January 2009

A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such thatfor every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.
Long Waves on a Rotating Earth in the Presence of a Semi-Infinite Barrier
Authors: Chambers LL. G. 
Published: January 2009

SummaryThe problem considered is that of long gravity waves approaching, from an arbitrary direction, a semi-infinite barrier, the whole system being in rotation. It is shown that the rotation gives rise to a wave in the shadow region whose amplitude depends upon the angle of incidence, but whose form is independent of it and which travels along the barrier without attenuation in that direction. The work is an extension and simplification of previous work by Crease, involving the use of methods previously developed by the author.
A Class of Integral Transforms
Authors: Wimp Jet 
Published: January 2009

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))and the generalised Mehler transform pair (7)These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).
Some Inequalities involving (r!)1/r
Authors: Minc Henryk , Sathre Leroy 
Published: January 2009

In a recent investigation of a conjecture on an upper bound for permanents of (0, 1)-matrices (2) we obtained some inequalities involving the function (r!)1/r which are of interest in themselves. Probably the most interesting of them, and certainly the hardest to prove, is the inequalitywhere ø(r = (r!)1/r. In the present paper we prove (1) and other inequalities involving the function ø(r.
Commutation Problems Involving Rings of Infinite Matrices
Authors: Patterson E. M. 
Published: January 2009

Let R be a ring and let J be the set of all integers. In the set M(R) of all mappings A: J×J→R, let addition and multiplication be defined byHere aij denotes the image of (i, j) under A and bij, cij, dij are similarly defined for the mappings B, C, D. In (2) we require A, B to be such that the sum is defined and is in R. Thus, in general, M(R) is not closed with respect to multiplication.
A Note on Griffith Cracks
Authors: Lowengrub M. 
Published: January 2009

The problem of determining the distribution of stress in the neighbourhood of a Griffith crack, denned by ∣x∣ ≦ 1, y = 0, which is subject to an internal pressure varying along the length of the crack has been solved by Sneddon and Elliott (3) and by England and Green (1). In the former paper a Fourier cosine method is used to arrive at a solution while in the latter paper the problem is reduced to an Abel integral equation by making use of integral representations of the complex potentials given in Green and Zerna (2). Neither paper deals with the calculation of the stress intensity factorwhich is extremely important to workers in fracture mechanics.
Some Fractional q-Integrals and q-Derivatives
Authors: Al-Salam Waleed A. 
Published: January 2009

A q-analogue of the integral ∣f(t)dt is defined by means ofwhich is an inverse of the q–derivativeThe present author (2) has recently obtained a q–nalogue of a formula of Cauchy, namely,where, for real or complex α and N a positive integer,
Useful Theorems on Commutative Non-associative Algebras
Authors: Losey Nora 
Published: January 2009

Recently J. M. Osborn has investigated the structure of a simple commutative non-associative algebra with unity element satisfying a polynomial identity, (4), (5) and (6). From his work it seems likely that if such an algebra is of degree three or more it is necessarily power-associative. In (4) he establishes a hierarchy of identities with the property that each identity is satisfied by an algebra satisfying no preceding identity. Following (5), (6),the next identity to consider iswhere a, c and h are elements of the ground field.
On the Sum
Authors: Mcdonagh Sean 
Published: January 2009

Erdős (1) has proved the following result:Theorem A.Every integral polynomial g(n) of degree k ≧ 3, represents for infinitely many integers n a(k-1)th power-free integer provided, in the case where k is a power of 2, there exists an integer n such that g(n)≢0 (mod 2k-1).
An estimate for trigonometrical sums over square-free integers with a constant number of prime factors
Authors: Mc Donagh Sean 
Published: January 2009

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.
Generalised Dirichlet series and Hecke's functional equation
Authors: Berndt Bruce C. 
Published: January 2009

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).
On functions and equations in distributive lattices
Authors: Rudeanu Sergiu 
Published: January 2009

SummaryIn [1], R. L. Goodstein has extended some well-known theorems on functions and equations in a Boolean algebra to the case of a distributive lattice L with 0 and 1. The purpose of this paper is to prove that most of Goodstein's theorems, as well as some additional results, are still valid in the case when L is not required to have least and greatest elements.
On the Fredholm integral equation associated with pairs of dual integral equations
Authors: Hutson V. 
Published: January 2009

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.
A procedure for summing asymptotic series
Authors: Scraton R. E. 
Published: January 2009

When a function ƒ(x) possesses an asymptotic seriesthis series provides a useful means of evaluating ƒ(x) for large values of x. The usual procedure is to sum all the terms in S(x) up to, but excluding, the term of smallest magnitude. The degree of accuracy obtained by this method cannot normally be improved by direct summation of S(x), but sometimes better accuracy can be obtained by using one of the familiar devices for accelerating the convergence of series. Simple δ2-extrapolation may be successful, and Rosser (1) and others have used the Euler transformation to some effect. The method given here provides, in suitable cases, a more effective means of evaluating ƒ(x) from the series for a wide range of values of x.
A Choquet-Deny theorem for affine functions on a Choquet simplex
Authors: Priestley H. A. 
Published: January 2009

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.
On functions of bounded boundary rotation I
Authors: Brannan D. A. 
Published: January 2009

Let Vk denote the class of functionswhich map conformally onto an image domain ƒ(U) of boundary rotation at most kπ (see (7) for the definition and basic properties of the class kπ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.
Operators with a Single Spectrum
Authors: West T. T. 
Published: January 2009

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.
A Note on the Asymptotic Expansion of a Ratio of Gamma Functions
Authors: Fields Jerry L. 
Published: January 2009

Many problems in mathematical analysis require a knowledge of the asymptotic behaviour of Γ(z + α)/Γ(z + β) for large values of |z|, where α and β are bounded quantities. Tricomi and Erdélyi in (1), gave the asymptotic expansionwhere the are the generalised Bernoulli polynomials, see (2), defined byIn this note, we show that if, instead of considering z to be the large variable, we consider a related large variable, (1) can be improved from a computational viewpoint.
On a Splitting Theorem of Gaschütz
Authors: Rose John S. 
Published: January 2009

LetGbe any finite group, andpany prime number. (All groups to be considered here are finite, and we assume this without further comment.) We denote byKp(G) the unique smallest normal subgroup ofGfor which the quotientG/Kp(G) is ap-group.G/Kp(G) is called thep-residualofG. W. Gaschütz (2, Satz 7) has proved the followingTheorem.Set K = Kp(G).If the Sylow p-subgroups of K are abelian, then G splits over K.
The “Fourier” Theory of the Cardinal Function
Authors: Whittaker J. M. 
Published: January 2009

The generalised Riesz-Fischer theorem states that ifis convergent, with 1 < p ≤ 2, thenis the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that
On Appell's Function P (θ, φ)
Authors: Humbert Pierre 
Published: January 2009

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansionwhere j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as
Certain Expansions in the Algebra of Quantum Mechanics
Authors: McCoy Neal H. 
Published: January 2009

§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the formwhere c is a real or complex scalar constant and is thus commutative with both p and q.
On the Canonical Form of a Rational Integral Function of a Matrix
Authors: Rutherford D. E. 
Published: January 2009

It is well known that the square matrix, of rank n−k + 1,which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.
Interpolated Derivatives
Authors: Spain B. 
Published: January 2009

In a previous paper [Spain, Proc. Roy. Soc. Edinburgh, Vol. LX (1940), 134], I have shown that the application of the cardinal function to the problem of interpolating the derivatives yields the resultThis formula is valid for x > a (the constant of integration), and R(n) < 0. The analytical continuation for R(n) ≥ 0. is indicated in the paper just quoted. The first term is the familiar expression for a fractional derivative, but the second term is not Riemann's complementary function. Furthermore, this result is unsatisfactory because it is impossible to perform the repeated operation of a fractional derivative of a fractional derivative.
Pythagorean Orthogonality in a Normed Linear Space
Authors: Perfect Hazel 
Published: January 2009

This note presents a proof of the following proposition:Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:Since a normed linear space is known to be Euclidean if the parallelogram law:is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).
On a Certain Matrix Product with Specified Latent Roots
Authors: Nagler H. 
Published: January 2009

Vajda's paper in this volume has suggested to the author the following problem:Let A be a n × m matrix, m≦ n, let B be a m × n matrix, and let M = I – AB, where I is the unit matrix of order n. Given A, to find B such that of the n latent roots of M′M, k are unity, and the remaining n — k are zero.
An Axisymmetric Boundary Value Problem of Mixed Type for a Half-space
Authors: Lowengrub M. , Sneddon I. N. 
Published: January 2009

In problems in the mathematical theory of elasticity related to the symmetric deformation of an infinite elastic solid with an external crack we encounter the problem of determining an axisymmetric function φ(ρ, z) which is harmonic in the half-space z>0 and satisfies the mixed boundary conditionson the plane boundary z = 0, where it is assumed that f(ρ) is continuously differentiable in [1, ∞). Further φ→0 as √(ρ2+z2)→∞.
An Integral over the Interior of a Simplex
Authors: Jack Henry 
Published: January 2009

If f(z) is analytic in a suitable domain, it is shown how the integral of f(α1x1+…+αnxn) over the interior of a simplex may be reduced to the evaluation of a contour integral, in fact to an exercise in partial fractions.The contour integral is expressed in two ways, according as the simplex is given in terms of its vertices or faces.
On Central Series
Authors: Macdonald I. D. 
Published: January 2009

Letandbe, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors Zk/Zk−1 and Γk/Γk+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers.
On the generalised Todd genus of flag bundles
Authors: Ilori S. A. 
Published: January 2009

Let V be a complex algebraic variety. Given integers a1, …, am such thatone defines a (a1, …, am)-flag as a nested systemof subspaces of Sn, the n-dimensional complex projective space. The set of all such flags is called an incomplete flag-manifold in Sn, and is denoted by W(al, …, am). Also let E be a complex n-dimensional vector bundle over V. Then we denote by E(a1, …, am−1, n; V) an associated fibre bundle of E with fibre W(a1 − 1, …, am−1 − 1, n − 1). E(a1, …, am−1 − 1, n; V) is called an incomplete flag bundle of E over V (cf. (2), (3)). In Section 10.3 and Section 14.4 of (1), the generalised Todd genus Ty(W(0, n)) and Ty(W(0, 1, …, n)) of the n-dimensional projective space W(0, n) and the flag manifold W(0, 1, …, n) (or F(n+l)) were calculated. Here we compute Ty(W(a1, …, am)) and also Ty(E(a1, …, am−1, n; V)).
Exhaustive operators and vector measures
Authors: Kalton N. J. 
Published: January 2009

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.
Some infinite Fibonacci groups
Authors: Johnson D. L. 
Published: January 2009

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.