bluuebunny/crossref_metadata_2025_split huggingface dataset:
Bounded representations of the positive values of an indefinite quadratic form
Authors: Lawton B. 
Published: November 2008

Letbe a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfyingwhere F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.
Further computations of quadrupole line strengths
Authors: Garstang R. H. 
Published: November 2008

ABSTRACTThe methods introduced into the theory of atomic spectra by Racah are applied to the calculation of electric quadrupole line strengths for transitions involving d and s electrons. The transitions considered are of the types dn−dn−1s, dn−1s−dn−1s and dn−1s−dn−2s2. Sum rules and rules for almost closed shells are obtained. Detailed numerical results are given for various transitions in and between the configurations d9 and d8s, and d7, d6s and d5s2.
On two-dimensional waves in an elastic half-space
Authors: Craggs J. W. 
Published: November 2008

ABSTRACTTwo-dimensional elastic waves in a half-space 0 ≤ r < ∞, 0 ≤ θ ≤ π are examined under the assumption of dynamic similarity, so that the stresses depend only on r/t, θ. Analytic solutions are given for constant surface traction on θ = 0, 0 < r/t < V, where V is constant, the rest of the surface being unloaded, and for a concentrated load at r = 0.Numerical results are quoted for the particular case V → ∞, corresponding to a load on half the bounding plane.
The interaction of nuclear multipole moments with an external charge distribution in elliptic coordinates
Authors: Stephen M. J. 
Published: November 2008

ABSTRACTThe calculation of the interaction between a point multipole moment and an external charge distribution is considered in elliptic coordinates. It is complicated by the singularity at the point moment. The results obtained are useful in calculating hyperfine interactions in diatomic molecules where the elliptic coordinates are the most natural system of coordinates. The results lead to an extension of Heine's expansion of (λ + μ)−1.
A note on the convergence of multi-point Taylor's series
Authors: Fine M. , Head J. W. 
Published: November 2008

Series which behave rather like Taylor's series at more than, one point are described and their convergence is discussed briefly.Letwhere z, zs are complex numbers and zs are the points, (the base points)about which a function is to be expanded. They need not be distinct. p{z) is a polynomial of degree n. in z. Assume first that z, zs are real and the zs are distinct. Then polynomials pn-1,t(z) may be chosen so that the serieshas contact of order N at the points zs with any specified real function which has N derivatives at these points. This is proved merely by rearranging the simplest polynomial in z which has contact of order N with/(z) at z = zs. The term given by t = 0 in (2) allows the series to have the values f(zs) at z = zs; adding the term for t = 1 does not alter the values of the series at z = zs; it allows the series to have a derivative with values f′(zs) at z = z; similarly adding the term for t = 2 does not alter the value of the series or its first derivative at z = zs, it allows the series to have a second derivative with values f″{zs) at z = zs; and so on. If m of the zs coincide then f(z) has mN + m− 1 derivatives there and contact is of order mN + m− 1. If z, zs are complex the result is seen to be valid subject to the obvious generalization of ‘ contact of the ith order’ to complex variables and functions.
Inclusion of sets of regular summability matrices
Authors: Baker J. W. , Petersen G. M. 
Published: November 2008

1. In this paper we wish to discuss some problems which arise from a paper by Lorentz and Zeller; see (5). If {μn} is a fixed sequence monotonically increasing to infinity, and every sequence {sn} summed by both of the regular matrices A = (amn) and B = (bmn) and satisfying sn = O{μn) is summed to the same value by both matrices, the matrices are called (μn)-consistent. The two matrices are called consistent if they are (μn)-consistent for all {μn}, μn↗∞; they are b-consistent if the bounded sequences summed by both are summed to the same value by both. The matrix A is said to be (μn)-stronger than the matrix B, if every sequence {μn} that is B summable and satisfying sn = O(μn) is also A summable. The matrix A is stronger than B if every B summable sequence is A summable; A is b-stronger if every bounded B summable sequence is A summable. The symbol A -lim x denotes the value to which the sequence x = {xn} is summed by A; Am(x) is the transformationand A(x) is the sequence {Am(x)}. Let {A(i)}i ∈ I be any family, infinite or finite, of regular summability matrices. This family is called simultaneously consistent if, given any finite subset of I, say F, and any set of sequences {x(i)i ∈ F such that A(i) sums x(i) for each i in F, and such that is the null sequence, then .
On a class of infinite products
Authors: Verblunsky S. 
Published: November 2008

1.Introduction.In this note, we consider some estimates for the modulus of infinite products of the formwhere thelvare near to an arithmetic progression of common difference unity. In a subsequent note, an application will be given of these estimates to the study of a class of Cauchy exponential series which includes, as a special case, the series of Korous(l).
On a property of verbal and marginal subgroups
Authors: Stroud P. W. 
Published: November 2008

Let G be a group, and let φ (x1, …, xn) be any word in the variables x1, …, xn. We denote by φ(G) and φ*(G) respectively the corresponding verbal and marginal subgroups of G, which are defined as follows.(i) φ(G)is the subgroup generated by all elements of the form φ(a1; …, an) with ai in G(for i = 1, 2, …, n);(ii) an element b of G lies in φ*(G) if and only iffor all choices of a1, …, an in G and i= 1, 2, …,n.
On multilattice groups
Authors: McAlister D. B. 
Published: November 2008

Multilattice groups have been introduced by Benado ((1)) and considered by Vaida ((9)), who has initiated the study of the structure of a class of these groups. The purpose of this paper is to further the investigation begun by Vaida.Basic in the work contained here is the setHconsisting of the differences between the minimal upper bounds of pairs of elements of a multilattice groupG. By means of conditions imposed onGand based onH, we determine the structure of the congruence relations on a multilattice group and study direct sums and lexicographic products and extensions of such groups (sections 2, 3).
On a Sturm-Liouville expansion in series of Bessel functions
Authors: Naylor D. 
Published: November 2008

1. The determination of potentials and wave functions defined in regions bounded by the natural coordinate surfaces of a cylindrical polar coordinate system leads to eigenvalue problems connected with Bessel's differential equationWe shall consider the annulus 0 < a ≤ r ≤ b and take first the boundary condition to be that of vanishing on both surfaces. It is then convenient to introduce the functionThis function will automatically satisfy the condition of vanishing for r = a. It will also vanish on r = b if u and k are related by the condition that ψ(u, k, b) = 0. If we regard u as being fixed and real then there will be an infinite number of zeros kn(u)(n = 1,2,…).
On the least non-residue of a quartic polynomial
Authors: Williams Kenneth S. 
Published: November 2008

Let p be a prime and let f(x) be a quartic polynomial with integral coefficients. I consider the problem of estimating the least non-negative non-residue k of f(x) (modp) (I omit the mod p hereafter), for large primes p, so f(x) ≡ r has a solution forbut not for r = k. The same problem for cubics has been considered by Mordell ((1)), who showed thatas p → ∞, where the constant implied in the O-symbol is independent of the coefficients of the cubic. In fact a more detailed examination of Mordell's proof gives the better estimateIt is the purpose of this paper to show that this same estimate also holds for quartic polynomials.
On the Uniqueness and Boundedness of Solutions of Hyperbolic Differential Equations
Authors: Lakshmikantham V. 
Published: November 2008

Consider a characteristic initial value problem of partial differential equationswhere the functions E (x) and F (y) are real valued, uniformly Lipschitz continuous on 0 ≤ x ≤ a, 0 ≤ y ≤ b, respectively. Suppose f (x, y, u, p, q) is a real-valued and continuous function defined on 0 ≤ ≤ b. By a solution of (1), we mean a real-valued continuous function u (x, y), having partial derivatives ux (x, y), uy (x, y) and ux, y (x, y) in the domain 0 ≤ x ≤ a, 0 ≤ y ≤ b almost everywhere.
Bessel function Jν(z) of complex order and its zeros
Authors: Hall Laurence S. 
Published: November 2008

AbstractMethods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.
On the summability |R, log ω, 1| of Fourier series and an associated series
Authors: Mohanty R. , Ray B. K. 
Published: November 2008

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series is summable |R, λ, r|, where r > 0, ifwhere A is a fixed positive number(3).
On the laminar compressible boundary layer with stationary origin on a moving flat wall
Authors: Ackroyd J. A. D. 
Published: November 2008

SummaryThe growth of the laminar compressible boundary layer on a moving flat wall is considered analytically for the case of zero velocity in the free stream outside the boundary layer. The results of this analysis are compared with other published results for the cases in which the free stream has some finite velocity. In all the cases considered in the present paper, the boundary layer is taken to originate at some stationary point on the moving wall. This type of boundary-layer flow occurs behind moving shock waves and it is argued that the case of particular interest in the present paper, that of the stationary gas outside the boundary layer, provides bounding values of such parameters as displacement and momentum thicknesses for shock-induced laminar boundary-layer flows.
Elastic wave fields generated by scalar wave functions
Authors: Chadwick P. , Trowbridge E. A. 
Published: November 2008

AbstractIn this paper we obtain a representation of elastic wave fields (i.e. solutions of the equations of motion of classical elastokinetics) in terms of three functions satisfying scalar wave equations. Although the form of the representation implies no restriction upon the choice of coordinate system or upon the shape of the elastic body, it is found that the result can only be applied advantageously to initial-boundary-value problems having spherical polar coordinates as a natural frame of reference. The representation is shown to be complete in the sense that every (sufficiently smooth) elastic wave field in a homogeneous, isotropic body bounded by two concentric spheres can be expressed in the given form.Under conditions of axial symmetry the representation generates wave fields which can be decomposed into poloidal and toroidal constituents, the former arising from two scalar wave functions and comprising both dilatational and rotational waves, and the latter being associated with a single scalar wave function and a state of pure shear of the elastic solid. Finally, the representation is used to obtain a formal solution describing the elastic pulse generated in an infinite body by the application of time-dependent tractions to the surface of a spherical cavity.
The refraction of sound pulses
Authors: Grimshaw Roger 
Published: November 2008

AbstractThe problem considered is that of the diffraction of sound pulses by a plane interface separating two stratified media in each of which the refractive index increases with distance from the interface. Two types of diffraction phenomena occur, glancing ray diffraction and total reflection. The solution is obtained as a series of ‘pulse propagation modes’, using a double transform technique. The main results are some formulae for the pulse field near a diffracted wavefront.
Products in homotopy groups with certain prime coefficients
Authors: Green Paul 
Published: November 2008

Let p be a prime and n ≥ 3. Then there is a simply connected CW-complex, , unique up to homotopy type, such thatLet X be a CW-complex. Write , the group of pointed homotopy classes of pointed maps from to X. The group structure derives from the fact that, under the restriction on n, is a suspension.
A mini-gap theorem for Fourier series
Authors: Hayman W. K. 
Published: November 2008

Suppose thatbelongs to L2( − π, π). If most of the coefficients vanish then f (x) cannot be too small in a certain interval without being small generally. More precisely Ingham ((2), Theorem 1) has proved the followingTHEOREM A. Suppose that f (x) is given by (1·1) and that an = 0, except for a sequence n = nν, where nν+1 − nν ≥ C. Then given ∈ > 0 there exists a constant A (∈), such that we have for any real x1
Integral equations and relations for Lamé functions and ellipsoidal wave functions
Authors: Sleeman B. D. 
Published: November 2008

AbstractNon-linear integral equations and relations, whose nuclei in all cases is the ‘potential’ Green's function, satisfied by Lamé polynomials and Lamé functions of the second kind are discussed. For these functions certain techniques of analysis are described and these find their natural generalization in ellipsoidal wave-function theory. Here similar integral equations are constructed for ellipsoidal wave functions of the first and third kinds, the nucleus in each case now being the ‘free space’ Green's function. The presence of ellipsoidal wave functions of the second kind is noted for the first time. Certain possible generalizations of the techniques and ideas involved in this paper are also discussed.
Exact solutions for electromagnetic wave propagation in spherically stratified isotropic media
Authors: Westcott B. S. 
Published: November 2008

AbstractFollowing Heading (4) a systematic search is made for useful refractive index profiles in spherically stratified isotropic media which are capable of yielding wave functions expressed in terms of the standard transcendental functions. Previously known profiles are shown to emerge as special cases of the analysis. Subject to the criteria adopted all possible profiles are derived.
On better-quasi-ordering transfinite sequences
Authors: Nash-Williams C. St. J. A. 
Published: November 2008

AbstractLet Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.
A parallel-plate waveguide, asymmetrically bifurcated by a unidirectionally conducting half screen
Authors: Clausert H. 
Published: November 2008

AbstractIn a recent paper the symmetrical bifurcation of a parallel-plate waveguide by a unidirectionally conducting half screen was considered. In the present paper the analysis is both extended to cover the more general case of an asymmetrical bifurcation, and simplified by using the straightforward version of the Wiener-Hopf technique due to D. S. Jones. Some numerical values are given for the modulus of the reflexion coefficient as a function of the position of the half screen between the parallel plates.
On the method for solution of unsteady thermal boundary-layers in case of two-dimensional low-speed flows
Authors: Ðurić Milan Ð. 
Published: November 2008

AbstractThis paper is dedicated to the question of solving unsteady thermal boundary-layers in the case of two-dimensional low-speed flows, provided that the difference between the temperature of the stream and that of the wall is not too great (so that the density is sensibly constant) and that the change in the wall temperature Tw (x, t) takes place at the same instant as the body is set into motion. The velocity boundary-layer is uncoupled from the thermal one and can be considered separately. In paper (2) is given the method for obtaining the velocity field (u, v). The temperature field T depends on the velocity field and can be obtained only after this one. The method (2) being available for this purpose assuming the difference between the temperature of the wall Tw (x, t) and that of the main-stream T∞ in the form ofwhere S(x) and θ(t) are arbitrary functions of x respectively t, satisfying certain conditions required by the method.
Rods, plates and shells
Authors: Green A. E. , Laws N. , Naghdi P. M. 
Published: November 2008

AbstractWe discuss non-linear thermodynamical theories of rods and shells using the three-dimensional theory of classical continuum mechanics as a starting point. The three-dimensional theory is reduced to a two-dimensional theory for a shell, or plate, and a one-dimensional theory for a rod by employing an exact expansion for the displacement but an approximation for the temperature. For elastic rods and shells a method of approximation is suggested which brings the respective theories into correspondence with those of Green and Laws (1) and Green, Naghdi and Wain-wright(2).
The expansion of a wedge of gas into a vacuum
Authors: Levine Lawrence Elliott 
Published: November 2008

AbstractThe front from the expansion into a vacuum of an infinite wedge of polytropic gas initially at rest at constant pressure is studied in detail. It is shown how the shape of that part of the gas-vacuum interface resulting from the presence of the sharp corner changes with the wedge angle and the adiabatic index of the gas. In addition to analytic results for wedges of arbitrary angle, numerical results substantiating the analysis are given for the special case of 90°. Finally, explicit formulae for the path of a particle initially part of a 60° wedge of water allowed to collapse at time t = 0 are derived from an exact linear solution obtained by Suchkov and interpreted through the hydraulic analogy for adiabatic index 2.
Stress concentration around a loaded bolt in an axially loaded bar
Authors: Gregory R. D 
Published: November 2008

AbstractIn Part I of the paper, a simple theory of infinite systems of linear equations of the typeL2is presented. In Part II, these results are applied to the elastostatic problem of determining the stress distribution in an axially loaded bar, held in position by a centrally placed circular bolt. Some numerical results are obtained and these are compared with experimental observation.
The oscillations of a viscous gravitating sphere
Authors: McKenzie D. P. 
Published: November 2008

AbstractBy using a Laplace transformation, a general solution is obtained to the problem of the oscillations and velocity field of a viscous gravitating sphere. Lamb's oscillating solution and Darwin's exponentially decaying solution are derived as asymptotic expressions and their connexion demonstrated. Closed loops of stream lines are a remarkable feature of the flow, and the conditions for their existence are discussed. Asymptotic solutions are also obtained for the oscillations of a Maxwell sphere, and their relation to those of an elastic sphere investigated.
The transport equation of radiative transfer with isotropic scattering
Authors: Hunt G. E. 
Published: November 2008

AbstractThe kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.
Surface waves in rotating liquids
Authors: Rao V. Subba 
Published: November 2008

AbstractUsing a technique developed by Lighthill, surface waves are studied when a concentrated pressure point oscillating with a constant frequency moves along OX in a rotating frame O X Y Z on the free surface of rotating liquid bounded below by a horizontal plane. The effect of rotation is to split the coincident gravity modes of the non-rotating case and further to produce a new system of wavelets (countably infinite) which do not have any counterpart in the non-rotating case.
Special relativity and thermodynamics
Authors: Georgiou A. 
Published: November 2008

AbstractIt is shown that a revision is needed of certain transformation equations of the Planck–Einstein formulae. In particular it will be shown that the formulaemust be replaced bywhere T and Q are respectively the absolute temperature and heat of the system. Furthermore, it will be shown that both formulae for the energyare correct, but they refer to different cases.
Summability factors for Riesz loǵarithmic means of order one for a Fourier series
Authors: Mohapatra R. N. 
Published: November 2008

Let 0 < λ1 < λ2 < … < λn → ∞ (n→∞). We writeLet ∑an be a given infinite series with the sequence {sn} for its nth partial sum. The (R, λ, 1) mean of the sequence {sn} is given by
Conformal frames and field equations in a conformal theory of ǵravitation
Authors: Islam Jamal N. 
Published: November 2008

AbstractSome aspects of the field equations of the conformal theory of gravitation put forward by Hoyle and Narlikar are studied. The field equations are conformally invariant and one can use a particular conformal frame to simplify the equations, since all conformal frames are regarded as physically equivalent. However, some conformal frames may be unsuitable in some regions of space-time, and with the use of such a frame one may get an unphysical solution. The use of conformal frames and the difficulties involved are illustrated by considering a given physical situation in two different conformal frames. The physical situation is that of two isolated particles. A static solution for this situation is obtained in both frames, and it is shown that a property that is quite unphysical in one frame transforms into a physically reasonable property in the other.