bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Bose-Einstein Condensation
Authors: Landsberg P. T. 
Published: November 2008

ABSTRACTThis paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of statesN(E) ∝Enfor high quantum numbers, a condition for condensation beingn> 0. These results are obtained on the basis of the following assumptions: (i) For large volumesV(a) all energy levels behave likeV−θ, and (b) there exists a finite integerMsuch that it is justifiable to put for thejth energy levelEj=c V−θand to use the continuous spectrum approximation, wheneverj ≥ Mc θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.
The inhomogeneous minima of complex cubic norm forms
Authors: Swinnerton-Dyer H. P. F. 
Published: November 2008

1. Let K1, K2, K3 be three conjugate* cubic algebraic number fields, of which we shall take K1 to be real and K2, K3 to be complex conjugate. Let ωi1, ωi2, ωi3 the conjugate basis for the integers of Ki. Writeso that ξi runs through the integers of Ki as the xj run independently through the rational integers, and the ξi (i = 1, 2, 3) are conjugate. The determinant of the ξi, regarded as linear forms in the xj, is ± √d, where d is the common discriminant of the fields.
An extension of the Ahlfors distortion theorem
Authors: Huckemann F. 
Published: November 2008

1. The conformal mapping of a strip domain in the z-plane on to a parallel strip— parallel, say, to the real axis of the ζ ( = ξ + iμ)-plane—brings about a certain distortion. More precisely: consider a cross-cut on the line ℜz = c joining the two sides of the frontier of the strip domain (in these introductory remarks we suppose for simplicity that there is only one such cross-cut on that line), and denote by ξ1(c) and ξ2(c) the lower and upper bounds of ξ on the image in the ζ-plane. The theorem of Ahlfors (1), now classical, states thatprovided thatwhere a is the width of the parallel strip and θ(c) the length of the cross-cut.
A note on valuations associated with a local domain
Authors: Rees D. 
Published: November 2008

The purpose of the present note is to prove the following two theorems:Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.
The laminar boundary-layer equation: a method of solution by means of an automatic computer
Authors: Leigh D. C. F. 
Published: November 2008

ABSTRACTA method, very suitable for use with an automatic computer, of solving the Hartree-Womersley approximation to the incompressible boundary-layer equation is developed. It is based on an iterative process and the Choleski method of solving a simultaneous set of linear algebraic equations. The programming of this method for an automatic computer is discussed. Tables of a solution of the boundary-layer equation in a region upstream of the separation point are given. In the upstream neighbourhood of separation this solution is compared with Goldstein's asymptotic solution and the agreement is good.
The correction of observational data for instrumental band width
Authors: Kahn F. D. 
Published: November 2008

ABSTRACTThe readings Φ(x) of an instrument with the apparatus functiong(x) are related to the true valuesf(x)of the observed function by the equationA three-point interpolation operator can be chosen to simulate the spreading action of a class of functionsg(x). Under certain conditions the interpolation operator has an exact inverse which, when applied to Φ(x), gives a function which is a better approximation tof(x). The accuracy obtained is discussed, and the effect of the method on the standard errors in the measurement is found.
The ergodic properties of non-homogeneous finite Markov chains
Authors: Hajnal J. , Bartlett M. S. 
Published: November 2008

ABSTRACTThe paper deals with the behaviour of finite non-homogeneous Markov chains having regular transition matrices. Two types of ergodic behaviour are distinguished, and sufficient conditions are given for each type. Chains with commutative transition matrices are specially investigated. By means of expansions in terms of the characteristic roots several types of behaviour of chains which are not ergodic are studied.
Green's functions for thin isotropic plates containing holes
Authors: Bassali W. A. , Dawoud R. H. 
Published: November 2008

ABSTRACTThis paper is concerned with the small transverse displacement of an infinite thin plane isotropic plate due to the application of a transverse force applied at an arbitrary point of the plate. The plate has its outer edge free, and is clamped along and bounded internally by a closed curve that can be mapped onto the unit circle by means of a polynomial. Three polynomials are considered and in each of these cases the deflexion is obtained in finite terms. Circular and elliptic holes as well as curvilinear polygonal holes are included as special cases.
On whether elastic wave surfaces possess cuspidal edges
Authors: Musgrave M. J. P. 
Published: November 2008

ABSTRACTConsideration of the direction of the normal at any point on a continuous sheet of a surface yields a sufficient condition for the existence of parabolic points on that sheet. This condition has been used to derive some simple inequalities between elastic constants, whose fulfilment determines the existence of parabolic points upon the inverse surfaces of media of orthorhombic, tetragonal, cubic or hexagonal symmetry; in virtue of the polar reciprocal relation between inverse and wave surfaces, the existence of cusp points on the latter is thereby simultaneously established. It is also pointed out that conditions for external conical refraction may prevail in hexagonal media.
On the numerical solution of a class of partial differential equations
Authors: Douglas A. S. 
Published: November 2008

ABSTRACTFor a suitable choice of E*, the solution as t becomes large of the equationwhere Y is given independent of t over the space boundaries, tends to the ground state solution of the wave equationwith the same boundary conditions on P as on Y. As a preliminary to using this relation to solve wave equations in more than one variable, the solution of the equationhas been studied. Methods of numerical solution are discussed, and the convergence of these is examined. Some practical experiments using an electronic computing machine are described.
Problems concerning the bending of isotropic thin elastic plates subject to various distributions of normal pressures
Authors: Bassali W. A. , Swinnerton-Dyer H. P. F. 
Published: November 2008

ABSTRACTWithin the limitations of the small-deflexion plate theory, complex variable methods are used in this paper to obtain an exact solution for the problem of a thin circular plate supported at several interior or boundary points, and subjected to a certain normal loading spread over the area of an eccentric circle, the boundary of the plate being free. The load considered includes as a special case a linearly varying load over the circle and, as the radius of the loaded circle tends to zero, this load can be made to tend to a couple nucleus at its centre. As limiting cases the procedure adopted provides us with solutions appropriate to a circular plate, an infinitely large plate and a half-plane having free boundaries and acted upon by any normal system of concentrated forces and concentrated couples in equilibrium. Formulae for the moments, shears and deflexions relating to special examples are worked out in detail.
Transverse bending of infinite and semi-infinite thin elastic plates. III
Authors: Bassali W. A. , Nassif M. , Swinnerton-Dyer H. P. F. 
Published: November 2008

ABSTRACTWithin the restrictions of the classical plate theory, complex variable methods are used in this paper to develop an exact expression for the transverse displacement of an infinitely large isotropic plate having a free outer boundary and elastically restrained at an inner circular boundary, the plate being subjected to a general type of loading distributed over the area of a circle. The limiting case of a half-plane clamped along the straight edge and acted upon normally by the same loading is also considered.
A uniqueness theorem for the exponential series of Herglotz
Authors: Verblunsky S. 
Published: November 2008

If {λ;n}, {bn} are sequences of complex numbers, and we consider the series ∑bn exp (−λnx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing(if λn = 0 the corresponding term is ½bnx2) where the series is supposed is to be uniformly convergent in (0, 1), we havefor 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φv(x)} of integrable functions with the property thatthen, on multiplying (1) by φv(x) and integrating over (0,1), we obtain a formula for bv in terms of F(x). On integrating by parts twice, bv will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑bn exp (−λnx).
An extension of the Newtonian law of gravitation
Authors: Bastin J. A. 
Published: November 2008

ABSTRACTAn attempt is made to relate such diverse phenomena as kinetic energy, rest mass energy, the size of the universe, the velocity of propagation both of light and gravitation, and the recession of the galaxies. The correlation is made by considering an extension of the Newtonian gravitational law which covers a particular simple case when the two attracting bodies are in relative motion. The extension is treated as postulatory, although in the last section using the idea of gravitational flux, the assumed gravitational law will be shown to be the simplest of a number of possible extensions to the Newtonian law. The paper implies a new approach to special relativity, and it is therefore hoped to treat aspects of the work in greater detail subsequently.
An infinite Legendre integral transform and its inverse
Authors: Clemmow P. C. 
Published: November 2008

ABSTRACTA consideration of Legendre's equation, of order zero, expressed in terms of the angular variable θ, leads heuristically, but methodically, to an infinite integral representation of the delta function in terms of two independent non-periodic solutions Ev−½(θ), E−v−½(θ). This result yields an integral transform and inverse formula, where in the former the kernel is E−v−½(θ) sin θ and the integration over θ runs from − ∞ to ∞. By rigorous contour integral methods, the inverse is verified when the original function is a rectangular pulse, and the standard series expansion in terms of Legendre polynomials is recovered when the original function is even and of period 2π in θ. Analogies with the theory of Fourier transforms are emphasized.
Landau damping for non-Maxwellian distributions
Authors: Hudson John F. P. 
Published: November 2008

ABSTRACTIt has been shown by Landau(8) that in a uniform plasma with a Maxwelliau velocity distribution longitudinal waves are damped. Penrose(9) has considered more general distributions and developed stability criteria. In particular, if the velocity distribution function has only one maximum, then the plasma is stable. In this paper the values of the Landau damping for some stable single maximum distributions are compared in order to assess the sensitivity of Landau damping to the form of the distribution function, and to investigate the usefulness of approximating to the Landau damping in a Maxwellian plasma by using an algebraically simpler velocity distribution function.It is shown also that, for at least some velocity distributions having a maximum particle velocity, the behaviour of the plasma can no longer be described in terms of exponential damping, since there will be longer lasting perturbations with phase velocity equal to the maximum particle velocity.
Absorption Probabilities for Sums of Random Variables Defined on a Finite Markov Chain
Authors: Miller H. D. 
Published: November 2008

SummaryThis paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.
A Class of Conserved Tensors in an Arbitrary Gravitational Field
Authors: Collinson C. D. 
Published: November 2008

ABSTRACTFour-index tensors with all possible terms either (a) quadratic in the Riemann curvature tensor, or (b) linear in its second derivatives, with coefficients constructed from products of metric tensors, are enumerated for the case of Riemannian space-time. All linear combinations of these which have zero divergence, and which might therefore be associated with the super-energy in gravitational theory, are found by elementary methods. The only independent such combination reduces in empty space-time to the tensor previously found by Bel and Robinson.
Diffraction by a Screen Causing Large Random Phase Fluctuations
Authors: Mercier R. P. 
Published: November 2008

ABSTRACTThe propagation of a scalar wave is studied, after it has suffered changes of phase in its passage through an irregular refracting medium, such as the ionosphere. The rms phase fluctuations may be large; i.e. greater than one radian. The paper is in three parts. In the first, the receiver is at a great distance from the screen (Fraunhofer diffraction), and a formula is found for all even order moments of the amplitude, and for the correlation function of the even powers of the amplitude. The probability distribution of the amplitude is proved to be a Rice distribution. The multiple integrals expressing the moments of the amplitude reduce to sums of correlations of the original phase fluctuations amongst points over the screen, which are evaluated using combinatorial arguments.In the second part the variance of the square of the amplitude is calculated numerically for points near the screen (Fresnel diffraction). When the rms phase fluctuation is greater than one the variance is maximum near the point where the irregularities in the screen would bring the rays to a focus.In the third part radio star scintillation and scatter propagation are discussed in the light of the theory.
On the Stability and Boundedness of Differential Systems
Authors: Lakshmikantham V. 
Published: November 2008

Consider the differential systemswhere A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.
On axially symmetric waves
Authors: Craggs J. W. 
Published: November 2008

AbstractThe wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.
Power-free groups
Authors: Geddes A. 
Published: November 2008

1. Introduction. Consider a set P with elements a, b, c,… and a certain distinguished element e. Assume that P satisfies the following conditions.(I) With each ordered pair (a, b) of distinct elements a and b of P there is associated a unique element of P (to be denoted by ab and called the product of a and b). Further, with the pair (e, e) there is associated a unique element ee of P.(II)(ab)c = a(bc) for all elements a, b, c in P for which both sides are defined.(III) ea = a = ae for all elements a ∈ P.(IV) To each element a ∈ P, there corresponds an element a′ ∈ P such that
A differential equation for undamped forced non-linear oscillations. III
Authors: Morris G. R. 
Published: November 2008

The most general differential equation to which the dynamical description of the title applies iswhere dots denote differentiation with respect to t. The essential problem for this equation is to determine the behaviour of solutions as t → ∞. When we attack this problem, the most obvious question is whether, under reasonable conditions on p(t), every solution is bounded as t → ∞ this question is open except when g(x) is linear. In the special case when p(t) is periodic, (1·1) may have periodic solutions; it is clear that any such solution is bounded, and it is worth mentioning that finding periodic solutions is the easiest way of finding particular bounded ones. So long as the bounded-ness problem is unsolved, there is a special interest in finding a large class of particular bounded solutions: if we know such a class then, although we cannot say whether the general solution is bounded or not, we can make the imprecise comment that either the general solution is in fact bounded or the structure of the whole set of solutions is quite complicated.
Ignorable coordinates and steady motion in classical mechanics
Authors: Kilmister C. W. , Pirani F. A. E. 
Published: November 2008

AbstractIt is shown, for a classical dynamical system with a Lagrangian, that the existence of an ignorable coordinate is equivalent to the vanishing of a certain Lie derivative. On this covariant description is based a new definition of steady motion. A definition given earlier by Synge is criticized.
A general theory of unsteady compressible boundary layers with and without suction or injection
Authors: Sarma G. N. 
Published: November 2008

AbstractThe theory of unsteady two-dimensional compressible boundary layers is studied in two parts. Part I deals with the solutions when the main stream is steady and the body is in an arbitrary motion while Part II deals with the solutions when the main stream is steady and the body is at rest with an arbitrary unsteady suction or injection perturbing about a zero mean. General methods of obtaining solutions are given when the coefficient of viscosity is proportional to temperature and the Prandtl number is unity. The main feature of the work in this paper lies in subjecting the equations to a transformation similar to that of Howarth and in linearization similar to that of Lighthill. Following the work of Sarma in incompressible flows two types of solutions are developed, one for large times and the other for small times. General expressions for the skin friction, the pressure ratio, the heat transfer and the difference between the normal coordinates of compressible and incompressible flows are obtained.
Some expansions of generalized Whittaker functions
Authors: Srivastava H. M. 
Published: November 2008

In the usual notation letwhere (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.Dr L. J. Slater ((3), page 628) gave an expansion of the formfor all values of n and t, real or complex.
A note on the temperature distribution in a channel bounded by two co-axial cylinders
Authors: Bhatnagar P. L. , Tikekar V. G. 
Published: November 2008

If we neglect the viscous dissipation, the temperature distribution in an incompressible fluid flowing in the annulus between two coaxial cylinders with radii r1 and r2 (> r1) is determined by the equationInitial conditionsBoundary conditionswhere ∂Q/∂t is a function of time alone representing the rate of heat generation per unit volume per unit time in the fluid.
On integral operators with kernels involving Bessel functions
Authors: Okikiolu G. O. 
Published: November 2008

AbstractBy representing integral operators defined by kernels involving Bessel functions as compositions of two operators, we determine their mapping properties and, in one case, derive an inversion process. The operators considered are of the formwhere τ(t) is given respectively by and and in the last three cases the integrals converge in norm.
Permanent of the product of doubly stochastic matrices
Authors: Brualdi R. A. 
Published: November 2008

If A = [aij] is an n × n matrix, the permanent of A is the scalar valued function of A defined bywhere the summation extends over all permutations (i1, i2, …, in) of the integers 1, 2, …, n. If we assume that A is a non-negative matrix (that is, that A has non-negative entries) then we come upon an extremely interesting situation. Much work has been done in finding significant upper bounds for the permanent and permanental minors of A (see, e.g. (2, 4–7) in (5) an excellent bibliography is given). If B = [bij] is another n × n non-negative matrix, then we may form the product AB and consider the permanent of this matrix. Unlike the determinant, the permanent is not a multiplicative function. In our circumstances here, however, it is easy to verify and indeed a proof was written down in (1) that
An inequality involving Radon-Nikodym derivatives
Authors: Kingman J. F. C. 
Published: November 2008

In (2) a simple proof was given of the following inequality of Atkinson, Watterson and Moran (l), a slight generalization of a result of Scheuer and Mandel ((4), see also (3)) which is of importance in genetics. If aij, pi, qj (i, j = 1, 2, 3,…) are non-negative numbers withand ifthenand if then This result is shown in (2) to be a special case of a large class of inequalities involving sets of non-negative numbers depending on several indices, and ‘partial averages’ over subsets of those indices.
Investigations into a new stratified hyperbolic profile
Authors: Heading John 
Published: November 2008

AbstractThe symmetrical Epstein refractive index profile (with free space conditions on each side of the transition) is generalized to a non-symmetrical form including both a barrier and a well by the discovery of a profile that cannot be obtained from the standard Epstein profile merely by adjusting the parameters. The analytical evaluation of the reflexion and transmission properties of the model is more complicated than that which occurs in the study of the Epstein profile.