bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Principle in classical mechanics with a ‘relativistic’ path-element extending the principle of least action
Authors: Schieldrop Edgar B. 
Published: November 2008

1. A particle with mass m and coordinates x1x2, x3 relative to a set of rectangular axes fixed in Newtonian space is moving in a field of conservative forces with a potential energy V(x1, x2, x3) and a kinetic energyThe equations of motion, written(representing the three equations i = l, i = 2, i = 3 in a way to be used in this paper), constitute, as they stand, a sufficient condition in order to ensurein the sense that the Hamiltonian integral has a stationary value if the actual motion is compared with neighbouring motions with the same terminal positions and the same terminal values of the time as in the actual motion.
Correlated random walk
Authors: Gillis J. 
Published: November 2008

ABSTRACTRandom walk on a d-dimensional lattice is investigated such that, at any stage, the probabilities of the step being in the various possible directions depend upon the direction of the previous step. The motion may be characterized by a generating function which is here derived. The generating function is then used to obtain some general properties of the walk. Certain special cases are considered in greater detail. The existence of recurrent points is investigated in particular, and the probability of returning to the origin after 2n steps. This latter function is evaluated asymptotically for the cases d = 1 and d = an even integer.
On finite groups of exponent five
Authors: Higman Graham 
Published: November 2008

1. The object of this note is to prove the restricted Burnside conjecture for exponent 5, that is, to prove, for n = 5, the proposition:Rn: For each positive integer k there is an integer rn, k such that every finite group of exponent n that can be generated by k elements has order at most rn, k.
Exact tests for serial correlation in vector processes
Authors: Hannan E. J. 
Published: November 2008

ABSTRACTExact tests for serial independence in vector Markoff processes have been obtained by considering the system of regressions of the vectors observed at time 2t on the vectors observed at times (2t − 1) and (2t + 1). The tests then reduce to those obtained from canonical correlation procedures in multivariate analysis. Two particular cases are(1) The test for serial independence of a vector of residuals from a system of regressions in which the regressors are all independent of the residuals. At the same time a test of the hypothesis that the regressor and regrediend vectors are independent is obtained.(2) The test for serial correlation or partial serial correlation in a multiple Markoff process (autoregressive process).An investigation of the efficiency of the test so obtained, of the hypothesis that a process is a simple Markoff process (against the alternative that the process is a second-order auto-regression) suggests that the efficiency of all of these tests will be low.
Properties of random functions
Authors: Palmer D. S. 
Published: November 2008

ABSTRACTThis paper deals with the relationships of the maxima, minima and zeros of two random functions of known autocorrelations and cross-correlation, based on the work of Rice(4). Ratcliffe(3) and Briggs and Spencer(2) discuss a similar problem in connexion with a ‘Phillips Record’ of an experiment in ionospheric reflexion. In this experiment there are two highly correlated reflected signals, their maxima coming close together, and the record shows the time lags between a maximum on one signal and a maximum on the other. Briggs and Page (1) have made an experimental study of the distribution of the differences between the positions of the maxima of two highly correlated random functions, using EDSAC to construct the functions. In §§ 3–7 the frequency distributions of intervals between successive zeros and maxima, and of the lengths of intercepts by a horizontal line, are considered. This has applications to the study of the fading of long-wave radio signals, where the tune differences between successive maxima of the amplitude have been investigated.
A study of the ‘irreversibility paradox’ for a simple statistical assembly
Authors: Temperley H. N. V. 
Published: November 2008

ABSTRACTA very simple model, consisting of N particles moving in a one-dimensional assembly divided by potential ‘humps’ into M cells, is studied. The process of passing from a quantum-mechanical description of such an assembly to the equation of diffusion type that governs it in practice is shown to consist of at least three separate steps: ‘averaging over phases’, and letting N and M become large. The effects of these steps are considered separately. Strict irreversibility in time appears after the first step, but the assembly remains ergodic until after the second step and fluctuations persist until after the third step.
On the short-wave asymptotic theory of the wave equation (∇2 + k2)ø = 0
Authors: Ursell F. 
Published: November 2008

ABSTRACTThe theory of time-periodic wave problems falls into two parts. On the one hand there is the rigorous formulation in terms of differential wave equations, on the other there are approximate theories like geometrical optics. It should be possible, in principle, to deduce the latter from the former by a logical process; but this has been done only for a few simple configurations, e.g. the circle. A possible approach to the solution of the general problem is suggested here, and is applied to a typical two-dimensional acoustical example. An arbitrary closed convex curve (satisfying certain regularity conditions) is emitting short sound waves towards infinity, the normal velocity V(s) exp (− iωt) is prescribed on the curve as a function of the arc-length s, and the potential is to be found, first on the curve and then at any point in the sound field. (Only the first part of the problem is treated in detail.) The potential ø(s) exp (− iωt) on the curve satisfies all the integral equationswhere G(s, s′) is any Green's function of the problem, and V(a) is prescribed. All the equations corresponding to different Green's functions have the same solution. An asymptotic and convergent short-wave solution can be found by iteration if G can be chosen explicitly so that the integral equation has a small kernel for high frequencies. At any point of the curve draw the local circle of curvature; then the explicit known solution for a source on this circle is (with slight modifications) a possible Green's function, and the equation formed with it has a small kernel and can be solved rigorously by iteration. If V(a) is independent of the frequency, the leading term in the resulting asymptotic expansion iswhere c is the velocity of sound and 2πk−1 is the (short) wavelength corresponding to the frequency ω/2π. If V(a) varies rapidly, as in diffraction theory, the iterative solution still gives a convergent asymptotic expansion, but the first approximation is then practically useless in the shadow region. Diffraction problems are not treated in the present paper.The present work appears to be the first practical and rigorous solution of a short-wave problem in optics or acoustics when a solution in closed form is not available. It is suggested that the technique (suitably combined with formal expansions) may be applicable to a wider class of radiation and diffraction problems.
On renewal theory, counter problems, and quasi-Poisson processes
Authors: Smith Walter L. 
Published: November 2008

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic processHammersley'sx counter problem concerns the stochastic process
A convergent asymptotic representation for integrals
Authors: Franklin Joel , Friedman Bernard 
Published: November 2008

ABSTRACTThis paper represents a new method for obtaining an asymptotic representation for integrals of the form when p is large. It is shown that if f(x) satisfies certain conditions this representation is also convergent. Numerical calculations seem to show that the first term of the representation gives a close approximation to the value of the integral for a wide range of values of p.
Some applications of Mellin transforms to the theory of bivariate statistical distributions
Authors: Fox Charles 
Published: November 2008

ABSTRACTA method is described for finding the frequency functions of bivariate random variables which are the products or ratios of other bivariate random variables. If (ξ, n) are a pair of bivariate random variables with joint frequency function f(x, y) then the method depends upon the fact that the expectation of │ ξ │r–1│η│s–1 is related to the Mellin transform of f(x, y) in two dimensions. Knowing the expectation we can then recover the frequency function by means of the inverse Mellin transform. Some examples are given to illustrate the theory.
The absorption of very high frequency sound in dielectric solids
Authors: Simons S. 
Published: November 2008

ABSTRACTA theoretical treatment is given of the absorption of longitudinally polarized sound in dielectric crystals of high symmetry, due to interaction with the thermal phonous. The Q value for the absorption is calculated in terms of the deviation from Hooke's law, the result being applicable at low temperatures when the mean free path of the phonons is greater than the wavelength of the acoustic wave. The theory is applied to cubic crystals, and in particular to solid argon, where a minimum Q value of about 108–109 is calculated. It is suggested that measurement of the absorption (at present somewhat beyond experimental techniques) would supply information on the deviation from linear elasticity, as well as providing a satisfactory verification of the basic interaction theory. The effect of isotopes, and, in more complicated structures, of optic modes, is shown to be negligible.
The transverse flexure of thin perforated elastic plates supported at several points
Authors: Bassali W. A. 
Published: November 2008

ABSTRACTFollowing the method outlined in a previous paper, exact solutions in finite terms are obtained for the problem of an infinitely large plate with outer edge free and an inner free circular boundary, the plate being supported at any number of interior or boundary points and normally loaded over a circle. The load considered is symmetrical with respect to the centre of the circle.
On non-dissipative Markov chains
Authors: Mauldon J. G. 
Published: November 2008

Consider a Markov chain with an enumerable infinity of states, labelled 0, 1, 2, …, whose one-step transition probabilities pij are independent of time. ThenI write and, departing slightly from the usual convention,Then it is known ((1), pp. 324–34, or (6)) that the limits πij always exist, and that
The reflexion of long waves at a step
Authors: Bartholomeusz E. F. 
Published: November 2008

ABSTRACTA train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.
On homoeomorphism of perfect plane sets
Authors: Besicovitch A. S. 
Published: November 2008

In studying the structure of perfect sets, an important method would be to establish homoeomorphisms in general classes of sets. For then the general structural features of sets could be more easily seen by considering the simplest sets of a homoeomorphic class.In this paper we establish the homoeomorphism in two general classes of plane perfect sets (we consider only bounded sets).
Adiabatic second-order energy derivatives in quantum mechanics
Authors: Brown W. Byers , Higgins H. C. Longuet 
Published: November 2008

ABSTRACTThe general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ isThe applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.
Self-consistent fields without exchange for Pr3+ and Tm3+
Authors: Ridley E. Cicely 
Published: November 2008

ABSTRACTSelf-consistent fields without exchange have been calculated for the ground state of Pr3+ and Tm3+. The spin-orbit coupling constant ζ(4f) is found to have the values 785 and 2742 cm−1 in Pr3+ and Tm3+, respectively. The corresponding values of are 29·4 and 77·5 Å−3. Values of the Slater integrals F2(4f, 4f), F4(4f, 4f) and F6(4f, 4f) are also given for each structure.
The propagation of shock waves of constant strength
Authors: Mackie A. G. , Weir D. G. 
Published: November 2008

ABSTRACTConditions are examined under which plane shook waves of constant strength can propagate through a gas in a given homentropic motion. Since the entropy change across the shock is taken to be constant homentropic flow exists behind the shock also. The motion behind the shock is determined by solving a Cauchy problem with data given on the back of the shock. The theory is illustrated by two examples, one of which generalizes a result obtained by Copson.
Generalizations of Wald's fundamental identity of sequential analysis to Markov chains
Authors: Tweedie M. C. K. 
Published: November 2008

ABSTRACTAn extension of Bellman's analogue of Wald's identity is proved for Markov processes with a finite number of states, in which (a) the diagonal matrices of the state functions and the transition matrices may be heterogeneous but are so related that the product of the two matrices which apply at any stage has at least one modal vector in common with all of the corresponding products which apply at other stages, and (b) the probability distribution of the duration of any one realization of the process satisfies certain conditions. The condition (a) is called equimodality. The conditions for (b) are satisfied when there is asymptotically a sufficiently abrupt exponential upper bound to the probability distribution of the duration, and extends simply to validate analogues involving latent roots other than the one which appears in Bellman's formula. A certain generalization of Wald's stopping rule produces a bound of this kind.
The Stokes phenomenon and certain nth-order differential equations
Authors: Heading J. 
Published: November 2008

In papers I(1) and II(2), the author has investigated the Stokes phenomenon associated with the arbitrary constants that occur in the asymptotic expressions of solutions of the nth-order differential equationand has provided rules for tracing a given linear combination of n such asymptotic expressions across certain defined lines, designated as Stokes lines, anti-Stokes lines and branch-cuts. The object of these investigations is to apply the results to the approximate solution of more complicated linear differential equations of the nth order, and to this end it is an advantage to develop the theory using matrix techniques. In the present paper, the equations are recast into a special matrix form, and it is shown how the results obtained in II may be used in a systematic manner.
Bending of Curvilinear and Rectilinear Polygonal Plates Symmetrically Loaded Over a Concentric Circle
Authors: Bassali W. A. , Hanna N. O. M. 
Published: November 2008

ABSTRACTComplex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours withnsides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in thez-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function. Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.
Vertex generated waves outside metallic wedges
Authors: Williams W. E. , Rosenhead L. 
Published: November 2008

ABSTRACTA study is made of the waves generated by a magnetic line source placed at the vertex of a wedge of high conductivity and arbitrary angle. The boundary-value problem is reduced to the solution of a difference equation and an exact solution obtained. The method is also applied to the case of dielectric coated wedges where the surface reactance and resistance are arbitrary, and the propagation of surface waves along such surfaces is considered briefly. The forms of the solution for large and small values of the surface impedance are obtained and show complete agreement with the known results available for a right-angled wedge and a plane.
Two classical lattice point problems
Authors: Gangadharan K. S. , Ingham A. E. 
Published: November 2008

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.
Error bounds for the Liouville–Green (or WKB) approximation
Authors: Olver F. W. J. 
Published: November 2008

ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.
A method of evaluating the complex zeros of polynomials using polar coordinates
Authors: Bodmer W. F. 
Published: November 2008

Methods for finding complex roots numerically may broadly be classified into the following three types.(a) Direct methods. These require no knowledge of an initial approximation. The best known are Graeffe's ‘root-squaring’ method (see, for example, Whittaker and Robinson(17)) and Bernoulli's method extended to complex roots (Aitken(1)). Fry (8) gives an interesting method for finding the characteristic roots of a matrix, originally due to Duncan and Collar (5), (6), which contains the above-mentioned methods as special cases.
Thermo-elastic Rayleigh waves in a prestressed medium
Authors: Flavin J. N. 
Published: November 2008

ABSTRACTWe consider the propagation of thermo-elastic Rayleigh waves in a half space, which has been subjected to large uniform extensions, at constant temperature, in three mutually perpendicular directions, one being perpendicular to the surface, and the extension ratios in the other two directions being equal. Rayleigh waves, assuming (a) isentropic conditions and (b) isothermal conditions, are studied, and the relation of the approximation (a) to the exact thermo-elastic theory is considered.
On the numerical evaluation of some definite integrals
Authors: Barrett W. 
Published: November 2008

AbstractAn account is given of the application of Gaussian quadrature formulae to the numerical evaluation of certain definite integrals appearing in a recent paper by Craggs (2). Suitable changes of variable are used to effect a substantial reduction in the order of formula required. In comparing methods, use is made of an expression for the remainder as a contour integral, whose value is estimated with the aid of certain asymptotic formulae.
Studies in harmonic analysis
Authors: Varopoulos N. Th. 
Published: November 2008

In this paper we shall be mainly concerned with the following three apparently widely differing questions.(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?