bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the unitary irreducible representations of the universal covering group of the 3 + 2 deSitter group
Authors: Ehrman Joachim B. 
Published: November 2008

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.
Coupled forms of the differential equations governing radio propagation in the ionosphere. II
Authors: Budden K. G. , Clemmow P. C. 
Published: November 2008

ABSTRACTThe four first-order ‘coupled’ equations governing the propagation of electromagnetic waves in the ionosphere, previously obtained in symbolic matrix form (Clemmow and Heading (4)), are expressed explicitly in terms of the ionospheric parameters. The physical significance of the equations is illustrated by considering the energy flux in one characteristic wave when coupling and damping are neglected. Three special cases are then discussed for which second-order coupled equations are also given, namely, the cases of (a) vertical incidence with oblique magnetic field, (b) oblique incidence with vertical magnetic field, (c) horizontal magnetic field in the plane of incidence. For case (a) the second-order equations are those previously derived by Försterling(5).The form of the coupled equations is physically illuminating and, in principle, suitable for solution by successive approximations. Extensive numerical work has indeed been carried out on the second-order coupled equations in case (a) (e.g. Gibbons and Nertney(6)), and it is probable that the first-order coupled equations would prove more advantageous. The present authors, however, feel that better methods are available for purely numerical work (e.g. Budden(3)), and that the chief interest of the coupled form is that it shows the scope and limitations of the physical conception of characteristic waves.
The transverse flexure of thin elastic plates supported at several points
Authors: Bassali W. A. 
Published: November 2008

ABSTRACTThis paper depends upon the method developed by Kolossoff and Muskhelishvili for problems of plane elasticity and later extended to plate problems by Lechnitzky. Exact solutions in closed forms are obtained for the problem of a thin circular plate supported at several interior or boundary points and normally loaded over the area of an eccentric circle, the load being symmetrical with respect to the centre of the circle and the boundary of the plate being free. Explicit formulae for the deflexion, the bending and twisting moments and shearing stresses are given at any point of the plate. As limiting cases plates in the form of the infinite plane and half plane are also considered.
The focusing of radiation by a random surface when the source is at a finite distance
Authors: Longuet-Higgins M. S. 
Published: November 2008

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.
Scattering by a bounded non-linear singularity
Authors: Power E. A. , Saavedra I. 
Published: November 2008

ABSTRACTAn investigation is made of the effect of a non-linear addition to Schrödinger's equation for s-wave scattering of a beam of particles by a finite region of potential energy. The non-linear region is bounded and of an especially simple form so that an analytic solution is possible. The R-function for the problem is explicitly constructed and some related formulae such as the resonant energies and the scattering length are given. The analytical nature of R as a function of the momentum considered as a complex variable is discussed in relation to causality.
Power series with non-integral exponents which are Fourier kernels
Authors: Fox Charles 
Published: November 2008

ABSTRACTMuch is known about power series with integral exponents, and, as in the case of the Theta-functions, about series whose exponents are the squares of the natural numbers. About other types of power series little is known. I prove here that a certain power series whose exponents are the square roots of the natural numbers is a symmetrical Fourier kernel.
The electromagnetic energy and the gravitational mass of a charged particle in general relativity
Authors: Florides Petros S. 
Published: November 2008

ABSTRACTThe electromagnetic energy & of the field of a charged particle is calculated, using Møller's theory summarized in the previous paper. The contribution of & to the gravitational mass of the charged particle is discussed, by studying the behaviour of a neutral test particle in its field. The conclusion is that & gives rise to an ‘effective’ gravitational mass of the charged particle, which is equal to the (Newtonian) gravitational mass of the charged particle, plus the mass-equivalence of &. This is contrary to the currently accepted theory, that what we have called the ‘effective’ gravitational mass is equal to the ‘Newtonian’ gravitational mass of the charged particle.
Coset enumeration on digital computers
Authors: Leech John 
Published: November 2008

In 1936 Todd and Coxeter gave a method ((10), also described in (4), ch. 2) for establishing the order of a finite group defined by a set of relationssatisfied by its generators S1, S2,…, Sk. They enumerate systematically the cosets of a suitable subgroup whose order is evident from the defining relations for the whole group. They describe the method as being ‘purely mechanical’, and since that date the advent of electronic computers has led a number of people to programme the method for automatic execution. Most of this work has remained unpublished; an account is given here of work known to the author.
Some dual series equations involving Jacobi polynomials
Authors: Noble B. 
Published: November 2008

In this paper first of all we find an exact solution of the dual series equationswhereand Jn is the Jacobi polynomial defined byThe Bn are unknown coefficients which are to be determined. The functions f, g and the parameters a, λ, μ, are given. Necessary restrictions on these quantities are indicated during the analysis.
The irrotational solution of an elliptic differential equation with an unknown coefficient
Authors: cannon J. R. , Halton J. H. 
Published: November 2008

Let G be a bounded region in k-dimensional space, with boundary Γ, such that the Laplace equation,is uniquely soluble (to within an added constant) under the Neumann boundary conditionswhere ∂/∂n denotes outward normal differentiation on Γ, and it is assumed that h is a function in G ∪ ∂, and thus that g is a function on ∂. In what follows, we shall assume certain properties of the solution h: these are all well known (see, for example, Osgood(l) or Courant(2)).
Note on the generalized Mehler transform
Authors: Lowndes J. S. 
Published: November 2008

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined bywhere ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.
An infinite class of periodic solutions of ẍ + 2x3 = p(t)
Authors: Morris G. R 
Published: November 2008

An important unsolved problem in the theory of non-linear oscillations is to establish the boundedness or unboundedness of the general solution ofwhere dots denote differentiation with respect to t. When p(t) is periodic, we may seek periodic solutions. This search is interesting for its own sake, and of course leads us to special bounded solutions. In three previous papers (2, 3, 4) I have exhibited the equationas tractable: on the assumption that e(t) is even and periodic, it was shown that the equation has an infinity of periodic solutions.
Gravitational radiation in homogeneous cosmological models with space-sections of constant negative curvature
Authors: Rao J. Krishna , Vaidya P. C. 
Published: November 2008

AbstractThe propagation of type N gravitational radiation in homogeneous cosmological models with space-sections of constant negative curvature is considered. It is found that the propagation of radiation distorts the isotropy of the fluid pressure in the models permanently.
A remark about the description of free products of groups
Authors: Stallengs John 
Published: November 2008

The free product A* B of groups A and B can be described in two ways.We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .
Screening constants for relativistic wave functions
Authors: Garstang R. H. , Mayers D. F. 
Published: November 2008

AbstractFormulae for the mean radius and mean square radius of a relativistic Coulomb wave function are obtained. Screening constants for the energy, mean radius and mean square radius are defined relative to non-relativistic wave functions and the results of numerical calculations given. It is shown that if the screening constants so determined are added to the screening constants due to the presence of other electrons as found by the s.c.f. method, good agreement is obtained in a case where both effects have been considered together. The value of solving the relativistic wave equation in a Thomas-Fermi field is also shown.
Bayes procedures for deciding the Sign of a normal mean
Authors: Bather J. A. , Walker A. M. 
Published: November 2008

ABSTRACTThe decision problem discussed concerns the drift of a Wiener process with known variance per unit time. It is required to find the optimal sequential test for the sign of the drift, when sampling costs and the cost of a wrong terminal decision are specified. Initially, the drift is supposed to have a continuous prior distribution. The optimal procedure is determined by a continuation region in which a certain partial differential equation holds. The boundaries of this region can be found explicitly in a few special cases. In general, methods are developed for obtaining both inner and outer approximations to the optimal continuation region and these techniques are illustrated by application to several examples.
Large eigenvalues of a Sturm-Liouville problem
Authors: Cohn J. H. E. 
Published: November 2008

Consider the eigenvalue problemwhere q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λr (r = 0, 1, 2, …) and λn ∽ n2π2 (b − a)−2 as n → ∞.
Two more hypergeometric integral equations
Authors: Love E. R. 
Published: November 2008

SummaryThis paper is a sequel to one with a similar title to appear in the Proceedings of the Edinburgh Mathematical Society. Explicit solutions are found for two more integral equations of similar form; and also conditions necessary and sufficient for existence, and sufficient for uniqueness, of solutions. These theorems are preceded by several preparatory theorems on fractional integrals with origin ∞, including integrals of purely imaginary order.
On a summation formula for the Appell function F2
Authors: Srivastava H. M. 
Published: November 2008

1. In the course of a systematic analysis of certain problems in quantum mechanics it has been observed that their exact solutions can be expressed in terms of the Appell function F2 defined by means of (see e.g. (8), p. 211)where, as usual,and for convergence of the double series,
Note on a problem of Porte
Authors: Rousseau George 
Published: November 2008

Porte (1), p. 117, conjectures that the positive implicational propositional calculus has no finite characteristic matrix. The proof of this conjecture is a straightforward modification of Gödel's proof (2) that the intuitionistic propositional calculus has no finite characteristic matrix (see e.g. Church(3), ex. 26.12). Writing (A ∨ B) for ((A ⊃ B) ⊃ B) and Xij for (pj ⊃ pi) (i, j = 1,2,…), we define, for n > l, the formulawhere the terms associate to the left. Since provable formulae take the value n for all systems of values of the variables in the matrix {1,…,n} where x ⊃ y is n when x ≤ y and y otherwise, whereas Gn takes the value n − 1 for the system of values pi = i (i = 1,…,n), it follows that Gn is not provable. On the other hand, since A ⊢ A ∨ B and B ⊢ A ∨ B, it is easily seen that is provable whenever r ≠ s (r, s = 1,…,n). The result follows from these two remarks.
Convergence rates in the law of large numbers. II
Authors: Rohatgi V. K. 
Published: November 2008

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).
Uniform approximation by Fourier–Stieltjes coefficients
Authors: Ramirez Donald E. 
Published: November 2008

In chapter I, E = {nk} ⊂ Z is shown to be a Sidon set if and only if (**). For each x ∈ T,.Let E ⊂ Z+ be a lacunary sequence. In chapter II, it is constructively shown that the characteristic function of E is uniformly approximable by Fourier–Stieltjes coefficients; i.e. ϕE ∈ M(T)∧−.
Spherically symmetric differential equations, the rotation group, and tensor spherical functions
Authors: Burridge R. 
Published: November 2008

AbstractIn this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group. These fields may be expanded as series of functions, where,mis fixed and the matricesTl(g) form a 21+ 1 dimensional irreducible representation of.Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with givenl, nare transformed into others with the same values ofl, n. That this must be so follows from Schur's Lemma and the fact that for eachmandlthe functionsform a basis for an invariant subspace of functions onof dimension 2l+ 1 in which an irreducible representation ofacts. Explicit formulae for the results of such operations are presented.The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.
On the method of matched asymptotic expansions
Authors: Fraenkel L. E. 
Published: November 2008

AbstractThe formal method of matched expansions is applied to two further examples. The first concerns the magnetic field induced by a steady current in a thin toroidal wire. The second, which involves a non-linear ordinary differential equation of the fourth order, has been chosen to resemble the problem of flow past a circular cylinder at small Reynolds numbers. The results of the formal procedure are proved in each case to be expansions of the exact solution.
Two singular integral equations involving confluent hypergeometric functions
Authors: Prabhakar Tilak Raj 
Published: November 2008

Widder(1) obtained an inversion of the convolution transformby the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equationusing Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).
Exact solutions for vertically polarized electromagnetic waves in horizontally stratified isotropic media
Authors: Westcott B. S. 
Published: November 2008

AbstractA systematic analysis is conducted for refractive index profiles capable of yielding exact solutions in terms of hypergeometric or confluent hypergeometric functions for the electromagnetic field of vertically polarized waves propagating in horizontally stratified isotropic media. Previously known profiles emerge as special cases of the analysis. Profiles suitable for use with isotropic lossless ionospheres and of prescribed form n2 = 1 − T(z)/k2, with T(z) independent of k, do not arise in this paper.
Polynomial conformal tensors
Authors: du Plessis J. C. 
Published: November 2008

AbstractThe conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.
Some arithmetical properties ofm-ary partitions
Authors: Rödseth Öystein 
Published: November 2008

We denote bytm(n) the number of partitions of the positive integerninto non-decreasing parts which are positive or zero powers of a fixed integerm> 1 and we calltm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula fortm(n), the first term of which isMahler's result was later improved by de Bruijn (2).
The power inequality on Banach spaces
Authors: Bollobás Béla 
Published: November 2008

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined asand the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).
A steady vortex ring close to Hill's spherical vortex
Authors: Norbury J. 
Published: November 2008

AbstractThe existence of a steady vortex ring close to Hill's spherical vortex is established, and an approximate description of its boundary is given. The vorticity in the ring is proportional to the distance from the axis of symmetry. The core propagates steadily in an unbounded fluid at rest at infinity. The boundary of the vortex ring is close to an interior stream surface of Hill's vortex.
The calculation of diagonal matrix elements of two-body relative state operators between many-body oscillator states
Authors: Nash G. F. 
Published: November 2008

AbstractA general method is given for the determination of diagonal matrix elements of relative state operators in shell model states of a given supermultiplet and SU3 symmetry, for states in s, p and sd shells, and for cross-terms between these shells, enabling the excitation energies of particle–hole configurations to be evaluated.
An existence theorem for Robin's equation
Authors: Cade R. 
Published: November 2008

AbstractAn existence theorem is proved for Robin's integral equation for the density of electric charge on a closed surface, under the assumptions that the surface is convex, smooth and twice continuously differentiable. The technique is essentially Neumann's method of the arithmetic mean, used by Robin himself to show that the solution, assumed to exist, can be successively approximated by a sequence. In order to facilitate the main argument of the proof, it is assumed initially that the Gaussian curvature is everywhere positive, but this restriction is subsequently removed.