bluuebunny/crossref_metadata_2025_split huggingface dataset:
Cubic forms representing arithmetic progressions
Authors: Watson G. L. 
Published: November 2008

The following result has recently been proved by Lewis (3), Davenport (2) and Birch (1).There exists an integer n0 such that every cubic form with rational coefficients and at least n0 integral variables represents zero non-trivially.The arguments of Lewis and Birch are simple, and yield also various generalizations of this result. Davenport's proof is complicated, but it shows that the minimal n0 satisfies
The Equilibrium Stability of a System of Disk Dynamos
Authors: Lebovitz Norman R. 
Published: November 2008

ABSTRACTThe stability of the equilibrium solutions of the equations describing the behaviour of a system of coupled disk dynamos is investigated. In the absence of viscous damping, it is found that systems consisting of more than two dynamos are unstable. That a single disk dynamo is stable is known. The stability of the undamped two-dynamo system has not been ascertained. When viscous damping is present, there are two equilibrium solutions, one in which all the currents are zero, and one in which they are finite. In the zero-current case, a stability criterion is found. Stability criteria are also found in the finite-current case. Further, the existence of the finite-current equilibrium state excludes the stability of the zero-current equilibrium state.
The Fredholm Solution as the Limit for a Sum of Separable Potentials
Authors: Fairlie D. B. 
Published: November 2008

ABSTRACTThe scattering amplitude for a potential expressible as a sum of n separable potentials is obtained as the solution of a set of n simultaneous linear equations. As n→∞ the Fredholm solution for the integral form of the Schrödinger equation is recovered.
Energy transfer by gravitation in Newtonian theory
Authors: Bondi H. , McCrea W. H. 
Published: November 2008

ABSTRACTThe problem is considered as to whether, in accordance with Newtonian theory, energy can be transferred from one system to another across empty space by gravitational interaction alone. Familiar examples of apparent energy transfer by this means do not give an unambiguous answer since they involve some net change of gravitational potential energy and this is not localized in the theory. Two examples are given here of systems in which the potential energy is the same at the beginning and end of an operation that does produce a resultant energy transfer. The establishment of this result is significant as a preliminary to the discussion of energy transfer according to general relativity theory. The appendix gives a particular illustration of one of the examples that admits exact mathematical treatment.
A central limit theorem for systems of regressions
Authors: Hannan E. J. 
Published: November 2008

ABSTRACTA central limit theorem is proved for the estimates of the regression coefficients in a multiple system of regressions when the vector process generating the residuals is a linear process with coefficient matrices the sum of whose norms converges. The regressor variables are assumed to satisfy conditions, due to Grenander, which make their generalized harmonic analysis possible.
The transmission of surface waves under surface obstacles
Authors: Ursell F. 
Published: November 2008

ABSTRACTA train of surface waves (water waves under gravity) is normally incident on a cylinder with horizontal generators fixed near the free surface, and is partially transmitted and partially reflected. At a great distance behind the cylinder the wave motion tends to a regular wave train travelling towards infinity; the ratio of its amplitude to the amplitude of the incident wave is the transmission coefficient . The transmission coefficient is studied when the wavelength is short compared to the dimensions of the body; physically (though not for engineering applications) this is the most interesting range of wavelengths, which corresponds to the range of shadow formation and ray propagation in optics and acoustics. The waves are then confined to a thin layer near the free surface, and the transmission under a partially immersed obstacle is then small. In the calculation the boundary condition at the free surface is linearized, viscosity is neglected, and the motion is assumed to be irrotational.At present the transmission coefficient is known only for a few configurations, all of them relating to infinitely thin plane barriers. A method is now given which is applicable to cylinders of finite cross-section and which is worked out in detail for a half-immersed cylinder of circular cross-section. The solution of the problem is made to depend on the solution of an integral equation which is solved by iteration. Only the first two terms can be obtained with any accuracy, and it appears at first that this is not sufficient to give the leading term in the transmission coefficient at short wavelengths; this difficulty is characteristic of transmission problems. By various mathematical devices which throw light on the mechanism of wave transmission, it is, nevertheless, found possible to prove that the transmission coefficient for waves of short wavelength λ and period 2π/ω incident on a half-immersed circular cylinder of radius a is asymptotically given bywhen N = 2πα/λ = ω2α/g is large. Earlier evidence had pointed towards an exponential law. It is suggested that transmission coefficients of order N−4 are typical for obstacles having vertical tangents and finite non-zero radius of curvature at the points where they meet the horizontal mean free surface. For obstacles having both front and rear face plane vertical to a depth a, is probably of order e−2N approximately; if only one of the two faces is plane vertical, is probably of order e−N approximately. Thus is seen to depend critically on the details of the cross-section.
Stationary Cylindrically Symmetric Clusters of Particles in General Relativity
Authors: Raychaudhuri A. K. , Som M. M. 
Published: November 2008

ABSTRACTThe case of a cylindrically symmetric cluster of particles moving in circles perpendicular to the axis of symmetry is discussed on the basis of the equations of general relativity. A number of special solutions are obtained, and two interesting results on the relation between the gravitational mass and the sum of the free particle masses for these clusters are deduced.
Differential equations in Banach spaces and the extension of Lyapunov's method
Authors: Lakshmikantham V. 
Published: November 2008

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the typewhere A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the typeMlak (9) obtained some results concerning the limitations of solutions of the latter equation.
On well-quasi-ordering lower sets of finite trees
Authors: Nash-Williams C. St J. A. 
Published: November 2008

AbstractA set Q is quasi-ordered if a reflexive and transitive relation ≤ is defined on Q. It is well-quasi-ordered if it is quasi-ordered and, for every infinite sequence u1, u2,… of elements of Q, there exist i, j such that i < j and ui ≤ uj. A lower set of Q is a subset P of Q such that, if x ≤ y ∈ P, then x ∈ P. The class of lower sets of Q, quasi-ordered by ⊂, is denoted by LQ, and L2Q = L(LQ), etc. The set (, say) of all finite trees is quasi-ordered by writing T1 ≤ T2 if T1 is homeomorphic to a subtree of T2. It is proved that is well-quasi-ordered for all n.
The direction of time and the equivalence of ‘expanding’ and ‘contracting’ world-models
Authors: Schumacher D. L. 
Published: November 2008

AbstractIt is assumed initially that only the increase of entropy defines locally the sense of advance of time. This assumption, together with the feature of global coherence of statistical processes, which is provided by the cosmic line-element, imply that there is a unique way of associating the standards with the cosmic statistical processes. There is hence a connexion between the sense of divergence of geodesics of fundamental observers and the cosmic time-sense. This is in keeping with the unmodified conservation laws of gravitation and the analogous adiabatic conservation law quite generally without dependence on the explicit form of the line element. These remarks correspond exactly to consequences of the ‘elliptic’ model in the case of de Sitter space, obtained by entirely separate geometrical considerations. A remark is made with regard to the irreversibility associated with quantum measurement.
Elastico-viscous boundary-layer flows I. Two-dimensional flow near a stagnation point
Authors: Beard D. W. , Walters K. 
Published: November 2008

AbstractThe Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary-layer equations are solved numerically for the case of two-dimensional flow near a stagnation point. It is shown that the main effect of elasticity is to increase the velocity in the boundary layer and also to increase the stress on the solid boundary.
On a problem about the Epstein zeta-function
Authors: Ennola Veikko 
Published: November 2008

1. Letbe a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind isWe consider the Epstein zeta-functionthe series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.
The transport equation with plane symmetry and isotropic scattering
Authors: Smith M. G. 
Published: November 2008

AbstractThe double integral equation, which takes the place of the Milne equation in the one-dimensional problem, is derived from the governing partial differentio-integral equations. An analytical solution of the problem of a distribution of point sources on a plane, when the other boundaries are at infinity, is then found. The possibility of more complicated boundary conditions is discussed.
Inequalities for permanents and permanental minors
Authors: Brualdi R. A. , Newman M. 
Published: November 2008

Let Ωndenote the convex set of alln×ndoubly stochastic matrices: chat is, the set of alln×nmatrices with non-negative entries and row and column sums 1. IfA= (aij) is an arbitraryn×nmatrix, then thepermanentofAis the scalar valued function ofAdefined bywhere the subscriptsi1,i2, …,inrun over all permutations of 1, 2, …,n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that ifA∈ Ωnthen 0 <cn≤ per (A) ≤ 1, where the constantcndepends only onn. It is natural to inquire if per (A) is a convex function ofAforA∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I+ ½A) ≤ ½ + ½ per (A) for allA∈ Ωn. HereI=Inis the identity matrix of ordern.
Diffraction by a perfectly conducting wedge in an anisotropic plasma
Authors: Faulkner T. R. 
Published: November 2008

SummaryThe problem considered is the diffraction of an electromagnetic wave by a perfectly conducting wedge embedded in a plasma on which a uniform magnetic field is impressed. The plasma is assumed to behave as an anisotropic dielectric and the problem is reduced, by employing a contour integral representation for the solution, to solving a difference equation. Surface waves are found to be excited on the wedge and expressions are given for their amplitudes.
The scattering cross-section of an obstacle in an elastic solid for plane harmonic waves
Authors: Barratt P. J. , Collins W. D. 
Published: November 2008

AbstractIt is shown that, when a plane harmonic P or S wave is incident upon a two-or three-dimensional obstacle in an infinite elastic solid, the scattering cross-section of the obstacle can be calculated from an appropriate far-field scattering amplitude.
An inversion formula for the Varma transform
Authors: Saxena R. K. 
Published: November 2008

AbstractRecently Fox ((5)) has given an inversion formula for the transform whose kernel is xνKν(x), where Kν(x) is the modified Bessel function of the second kind, by the application of fractional integration theory. In the present paper it has been shown that the integral equationcan be thrown into the form of a Laplace integral, with the help of fractional integration, which can be solved by known methods.
Two examples in homotopy theory
Authors: Brown R. 
Published: November 2008

The examples are unconnected.According to some published footnotes, examples are known of Serre fibrations which are not Hurewicz fibrations. However, as far as I know no such example has been published. The following example, which was stimulated by a more complicated example of G. Horrocks, seems worth mentioning, since it is extremely simple and does suggest some further questions.
Analogue of a theorem of Khintchine in fields of formal power series
Authors: Cusick T. W. 
Published: November 2008

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.
The Inverse Laplace Transform of the Product of Two Whittaker Functions
Authors: Ragab F. M. 
Published: November 2008

The object of this paper is to obtain the original function of which the Laplace transform (l) is the productwhere, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equationIn § 2 it will be shown that this original function iswhere the symbol Δ(n; α) represents the set of parameters
A generalization of the Laplace transform
Authors: Dunn H. S. 
Published: November 2008

AbstractAn integral transformation is denned over a finite interval of the time domain. When the Laplace transform exists, the finite transform yields identical results. However, the finite transform is found to be considerably more general than the Laplace transform. It permits consideration of functions which are not of exponential order, leads to a simple scheme to determine system response, and is applicable to boundary-value problems.
A formal solution of certain dual integral equations involving H-functions
Authors: Saxena R. K. 
Published: November 2008

AbstractThe problem discussed is the formal solution of certain dual integral equations involving H-functions. The method followed is that of fractional integration. The given dual integral equations have been transformed, by the application of fractional integration operators, into two others with a common kernel and the problem is then reduced to solving one integral equation. In the first case the common kernel comes out to be a symmetrical Fourier kernel given earlier by Fox and the formal solution is then immediate. In the second case the common kernel is a generalized Fourier kernel and dual integral equations of this type have recently been studied by Fox.
On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method
Authors: Mohanty R. , Ray B. K. 
Published: November 2008

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us setThe sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformationThe series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, ifand is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).
Complex potentials in shallow shells of revolution
Authors: Yacoub A. R. 
Published: November 2008

AbstractBy introducing complex variable techniques in the theory of shallow shells, the problem is reduced to the solution of one fourth-order differential equation, and the displacement components and stress resultants and couples are expressed in terms of four complex potentials. Two of these complex potentials are those of thin plate theory, thus admitting of deducing the solution of a shallow shell problem from the limiting case of the circular flat plate. These results are applied to a case of non-rotational symmetry, a special case of which has practical significance in the design of tanks and boilers.
Clifford algebras in general relativity
Authors: Lord E. A. 
Published: November 2008

SummaryThe generalization of the Dirac and quaternion algebras to Riemannian spaces is outlined. The components of various elements of the algebras are interpreted as physical quantities (tensors) and their symmetries and algebraic properties are linked with the properties of the algebra. The generalization of the quaternion algebra is of particular interest in that it resolves the anomalies that arise in the usual identification of quaternions with rank two spinors. Algebraic expressions for the electromagnetic energy momentum tensor, the Ricci tensor and Einstein tensor are obtained in both E-number and quaternion form. Extension of the principles to EF-numbers yields a proof of the symmetry properties of Bel's tensor and a simple expression for its divergence.
On a conformal theory of gravitation
Authors: Islam Jamal N. 
Published: November 2008

AbstractRecently Hoyle and Narlikar(2) have put forward a conformal theory of gravitation in which they obtain an explicit solution of the field equations using certain approximations and assumptions of symmetry. To obtain the solution, the metric is assumed to have a special form. In the present paper a more general form of the metric is considered, and it is shown that many of the features of the special solution are shared by the solution in the more general case. In particular, the implications of the special solution as concerns the sign of the gravitational constant remain valid in the more general case under certain assumptions.
Creeping modes in a shadow
Authors: Ursell F. 
Published: November 2008

AbstractIn the two-dimensional theory of diffraction by smooth curves there are certain canonical problems that can be solved explicitly in series form. The series converge slowly at short wavelengths but they can be transformed by Watson's transformation into another form (the residue series or creeping-mode expansion) which has been much used in shadow regions. It is found that throughout the shadow region the first few terms of the residue series are exponentially small and decrease rapidly, and these have often been used as an estimate of the wave potential without further justification. Leppington's recent work on the shadow of an ellipse has shown, however, that in part of the shadow some of the later terms of the residue series are exponentially large. In other words, the complete residue series in part of the shadow is even more slowly convergent than the original series.In the present paper the Watson transformation is re-examined in the light of this result. The original series is expressed as the sum of a finite number of terms of the residue series and of a remainder. It is shown that throughout the shadow the remainder is at short wavelengths asymptotically of smaller order than the last term retained in the finite residue series. It follows that the residue series (which is a convergent infinite series) is also an asymptotic series, and this fact is sufficient to justify most of the usual applications. The proof is given in detail for the circle and in outline for the ellipse; it makes use of the theory of conformal mapping.
Diffraction of elastic waves by a rigid circular disc
Authors: Mal A. K. , Ang D. D. , Knopoff L. 
Published: November 2008

AbstractThe problem of the diffraction of axisymmetric harmonic elastic waves by a rigid circular disc is reduced to finding the solution of a pair of integral equations of the second kind suitable for iteration at low frequencies. Using the principle of contraction mapping, the rate of convergence of the iteration procedure is discussed and the error caused by stopping at any particular stage estimated. Detailed calculations are given for plane compressional waves at normal incidence.
A note on a source in a rotating liquid
Authors: Stewartson K. 
Published: November 2008

Consider an inviscid incompressible fluid confined within an infinite tube of radius b and assume that its motion initially consists of a uniform translation with velocity U and a uniform rotation, with angular velocity ω, about the tube axis. Suppose now that at time t = 0 a unit source of fluid situated at a point O of the axis begins to emit fluid at a uniform rate and that ultimately the flow pattern so modified becomes steady (at a fixed point as t → ∞). Under the further assumption that far upstream flow conditions are undisturbed by the source the stream function ψ of the disturbance satisfieswhere z denotes distance along the axis from O, r distance from the axis and k = 2ω/U.
On a class of functions regular in the unit disc
Authors: Hornblower R. 
Published: November 2008

MacLane(1) has introduced the class of functions with a dense set of asymptotic values on {|z| = 1} in the following sense:Definition. Let f(z) be regular and not constant in the unit disc and let ξ be a point of {|z| = 1}. Suppose that there exists a curve Γξ in {|z| < 1} and a in the closed complex plane such that f(z) → a as z → ξ along Γξ. Then we say ξ ∈ Aa.
Exact potential functions in spherically stratified media
Authors: Westcott B. S. 
Published: November 2008

AbstractA systematic analysis is made of the equation ∇. [k(r) ∇ψ] = 0 for useful profiles k(r) in spherically stratified media which enable the radial variation of the potential function ψ to be expressed exactly in terms of hypergeometric functions or confluent hypergeometric functions. Previously known profiles are shown to emerge as special cases of the present analysis.
Jacobi series which converge to zero, with applications to a class of singular partial differential equations
Authors: Wimp Jet , Colton David 
Published: November 2008

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose thatthe series being boundedly convergent for 0 ≤ x ≤ 1.