bluuebunny/crossref_metadata_2025_split huggingface dataset:
Diffusion in a field of homogeneous turbulence
Authors: Batchelor G. K. 
Published: November 2008

AbstractIn an earlier paper the author considered the problem of the turbulent diffusion, relative to a fixed origin, of a cloud of marked fluid whose initial position is given. This was found to be determined by the initial shape of the cloud and the statistical properties of the displacement of a single fluid particle. The present paper is concerned with the relative diffusion of the cloud, i.e. with the tendency to change its shape, or, more precisely, with that part of the relative diffusion which is described by the probability that a given vector y can lie with both its ends in marked fluid at time t. This aspect of the relative diffusion is found to be determined by the initial shape of the cloud and the statistical properties of the separation, at time t, of two fluid particles of given initial separation. The statistical functions introduced to describe the relative diffusion are found to be related to Richardson's distance-neighbour function.The relative diffusion of two particles is a more complex problem than diffusion of a single particle about a fixed origin because the relative diffusion depends on the initial separation. The closer the particles are together, the smaller is the range of eddy sizes that contributes to their relative velocity; for the same reason, relative diffusion is an accelerating process, until the particles are very far apart and wander independently. The hypothesis is made that if the initial separation is small enough, the probability distribution of the separation will tend asymptotically to a form independent of the initial separation, before the particles move independently. This hypothesis permits various simple deductions, some of which make use of Kolmogoroff's similarity theory. The important question of the description of the relative diffusion by a differential equation is examined; Richardson has put forward one suggestion, and another, based on a normal distribution of the separation, is made herein.
Some properties of Legendre functions
Authors: Cooke J. C. 
Published: November 2008

Nicholson (3) stated that the Legendre functions of the second kind, Qn(x), are orthogonal between − 1 and 1. This is incorrect in general, and one purpose of this paper is to give the correct value ofwhen m ╪ n. Nicholson's value of the integral when m = n, namely,is correct.
On the product of n linear forms
Authors: Davenport H. 
Published: November 2008

Let L1, …, Ln be n homogeneous linear forms in n variables u1, …, un, with non-zero determinant Δ. Suppose that L1, …, Lr have real coefficients, that Lr+1, …, Lr+s have complex coefficients, and that the form Lr+s+j is the complex conjugate of the form Lr+j for j = 1, …, s, where r + 2s = n. Letfor integral u1, …, un, not all zero. For any n numbers α1, …, αn of the same ‘type’ as the forms L1, …, Ln (that is, α1, …, αr real, αr+1, …, αr+s complex, αr+s+j = ᾱr+j), let
A correction to a paper on the dimension of Cartesian product sets
Authors: Eggleston H. G. 
Published: November 2008

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.
One-dimensional unsteady motion of a gas initially at rest and the dam-break problem
Authors: Mackie A. G. 
Published: November 2008

ABSTRACTThe object of this paper is to discuss the one-dimensional unsteady adiabatic motion of a gas which is initially at rest with a prescribed density distribution such that the specific entropy is uniform. The contour integral methods which Copson developed recently for even analytic functions are extended to apply to general analytic initial conditions. The solution is valid in the range 1 < Υ > 3, where y is the adiabatic index of the gas. Of particular interest, in view of the hydraulic analogy, is the case Υ = 2 for which real variable methods cannot readily be adapted. The motion of the front of a water column Sowing into a dry, horizontal stream bed is discussed. A curious type of solution, corresponding to a particular choice of initial distribution, which was established by Pack for a, countable sequence of values of Υ, is verified to hold over the whole range and is interpreted in terms of the dam-break problem.
The calculation of the latent roots and vectors of matrices on the pilot model of the A.C.E.
Authors: Wilkinson J. H. 
Published: November 2008

ABSTRACTThe problem of finding the latent roots and vectors of matrices has been treated in a number of papers ((2)–(7)) mainly from the point of view of desk computers. In this paper the problem is treated from the standpoint of users of high-speed automatic computers. In the first section a number of iterative processes are described and, in the second, the techniques developed for using these processes on the Pilot Model of the Automatic Computing Engine. It is shown that the methods give very high accuracy and can be used to deal with matrices of high orders even on a machine of very limited storage capacity. They have been used on numerous matrices of orders up to 60 mainly on problems arising in the aircraft industry and on eigenvalue problems for systems of ordinary differential equations.
A modified form of Siegel's mean-value theorem
Authors: Macbeath A. M. , Rogers C. A. 
Published: November 2008

The Minkowski–Hlawka theorem† asserts that, if S is any n-dimensional star body, with the origin o as centre, and with volume less than 2ζ(n), then there is a lattice of determinant 1 which has no point other than o in S. One of the methods used to prove this theorem splits up into three stages, (a) A function ρ(x) is considered, and it is shown that some suitably defined mean value of the sumtaken over a suitable set of lattices Λ of determinant 1, is equal, or approximately equal, to the integralover the whole space. (b) By taking ρ(x) to be equal, or approximately equal, towhere σ(x) is the characteristic function of S, and μ(r) is the Möbius function, it is shown that a corresponding mean value of the sumwhere Λ* is the set of primitive points of the lattice Λ, is equal, or approximately equal, to
The approach to scalar growth of a vector transformed by an increasing power of a matrix
Authors: Afriat S. N. , Eggleston H. G. 
Published: November 2008

The solution of a system of simultaneous homogeneous linear differential equations with constant coefficients has been obtained (1) in the formwhere the elements of the vector Z are the k functions determined by the system and their derivatives of order up to m – 1, where m is the order of the system. If Zn is the value of Z at t = n, then
Eigenvalues of differential equations by finite-difference methods
Authors: Bolton H. C. , Scoins H. I. , Rushbrooke G. S. 
Published: November 2008

ABSTRACTThe paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue Λ(h) of the corresponding finite difference equations towards the eigenvalue λ of the differential equation itself and it is shown thatwhere h is the size of the interval of the grid covering the range of the independent variable; the constant ν is usually a negative number and consequently Λ(h) may well be a lower bound to λ. This convergence property is used in the numerical calculation of λ by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.
A numerical method of solving second-order linear differential equations with two-point boundary conditions
Authors: Ridley E. Cicely 
Published: November 2008

ABSTRACTA direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.
The partial wave theory of electron-hydrogen atom collisions
Authors: Percival I. C. , Seaton M. J. 
Published: November 2008

ABSTRACTThe paper is concerned with the solution of the algebraic problems arising in the partial wave treatment of electron-hydrogen atom collisions. Explicitly antisymmetrized wave functions are used throughout. The boundary conditions are written in S-matrix notation and expressions for total and differential cross-sections obtained. The algebraic coefficients fλ and gλ occurring in the continuous state Hartree-Fock equations are expressed in terms of Racah coefficients, and tabulated as functions of the total angular momentum for atomic s, p and d electrons and all angular momenta of the scattered electron. Expressions are given for the calculation of first-order corrections to the results obtained using approximate wave functions.
The calculation of the electromagnetic field in an Alvarez type linear accelerator
Authors: Ford Fred 
Published: November 2008

ABSTRACTA recently developed theory of the electromagnetic field in nearly periodic structures is used to calculate the field in an Alvarez type linear accelerator. The problem is reduced to the solution of the transmission line equations for the ‘voltage’ and ‘current’ of the dominant mode. Formulae are given for the coefficients in these equations, and also for the ratio of mode current to accelerating field, as functions of the cavity dimensions of the accelerator; numerical values are given for the Berkeley accelerator.
The use of associated Legendre polynomials for interpolation
Authors: Albasiny E. L. 
Published: November 2008

ABSTRACTSeveral writers have dealt with the use of Chebyshev polynomials as aids to interpolation. The present paper considers the theory with associated Legendre polynomials replacing those of Chebyshev. Interpolation formulae of both Bessel and Everett type are derived, and formulae and expansions are given for evaluating the coefficients required in them either directly or in terms of derivatives or central differences. The introduction summarizes a number of formulae for polynomial interpolation and indicates the relationship between the new formulae and the old.
Natural maps of extension functors and a theorem of R. G. Swan
Authors: Hilton P. J. , Rees D. 
Published: November 2008

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.
On Self-reciprocal functions for Fourier-Bessel integral transforms
Authors: Ahmad Afzal , Lakshmikanth V. 
Published: November 2008

Following Hardy and Titchmarsh(1) a function f(x) is said to be self-reciprocal if it satisfies the Fourier-Bessel integral transformwhere Jp(x) is a Bessel function of order P ≥ –½. This integral is denoted by Rp. The special cases P ½ and P ½, we denote by Rs and Rc, respectively.
On the direct calculations of the representations of the three-dimensional pure rotation group
Authors: Lehrer-Ilamed Yehiel 
Published: November 2008

AbstractExplicit formulae are given for calculating the matrix elements of the irreducible representations of the three-dimensional pure rotation group by the direct method. In addition explicit formulae are derived to calculate the representations of the finite elements of any group when the eigenvalues of the matrix representing the corresponding infinitesimal elements are given.
On the differentials in the Adams spectral sequence
Authors: Maunder C. R. F. 
Published: November 2008

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements ofcorresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.
Two-dimensional oscillations in a canal of arbitrary cross-section
Authors: Davis A. M. J. 
Published: November 2008

AbstractAn infinitely long canal with uniform cross-section is filled with inviscid fluid. It is required first to show that any small two-dimensional motion of the fluid can be represented as the superposition of normal mode disturbances. A suitable generalized Green's function G(x, y; ξ) is constructed and is used to set up an integral equation (2·9) for the velocity potential on the free surface. It is shown that the eigenfunctions are complete and so are their (possibly time-dependent) extensions to the whole canal, in the sense that an arbitrary disturbance possesses a unique representation. In section 5, it is required to find asymptotic approximations to the large eigenvalues of (2·9). For this purpose a different integral equation (5·5) is set up on the canal, the kernel of which is the sum of a degenerate kernel and a small kernel. The solutions of this equation can therefore be obtained by iteration. The form of the mth eigenvalue is shown to befor sufficiently large m.
A campanological problem in group theory. II
Authors: Rankin R. A. 
Published: November 2008

1. Let E be a finite non-null set and write (E) for the family of all permutations of E. Let be a non-null subset of (E) and write () for the subgroup of (E) generated by the members of . For any α ∈ we putso that () is a subgroup of () and is independent of the choice of α in . We suppose that E splits into k disjoint transitivity sets (orbits) Ei(1 ≤ i ≤ k) with respect to (); thus σEi = Ei for all σ ∈ ().
Bending of an eccentrically loaded and concentrically supported thin circular plate. I
Authors: Bassali W. A. , Barsoum F. R. 
Published: November 2008

AbstractWithin the limitations of the classical small deflexion theory of thin plates and using complex variable methods, exact expressions are obtained in series form for the deflexion at any point of a thin isotropic circular plate simply supported along a concentric circle and subject to loading symmetrically distributed over an eccentric circular patch which lies inside the circle of support. In special and limiting cases the solutions reduce to those obtained before.
An approximate method in high-frequency scattering
Authors: Sharples A. 
Published: November 2008

ABSTRACTThe diffraction of a high-frequency plane sound wave by a circular cylinder is investigated when the boundary condition on the cylinder is expressed by means of an equation of the form The special feature of this investigation is that an extended form of the Kirchhoff-Fresnel theory of diffraction is used to find an integral representation for the scattering coefficient. In order to avoid the complicated analysis which would be necessary to evaluate the integrals concerned, the more natural geometrical acoustics approach is used to find the first correction term in the scattering coefficient. Numerical results are given for large and small values of the impedance Z.
On a class of integral operators
Authors: Verblunsky S. 
Published: November 2008

Introduction. The integral operators we shall be concerned with can be described with the help of the following:Definition. A function F(u, v) will be called a primitive, if it is defined on the square 0 ≥ u, v ≥ 1, and satisfies the conditions (i) F(u, v) = 0 if u ≥ v; (ii) F(u, v) is a.c. (absolutely continuous) in each variable; (iii) there is a function p ∈ L(0, 1) such that |F1(u, v)| ≤ p(u), |F2(u, v)| ≤ p(v)a.e. (almost everywhere) on the square, where F1(u, v), F2(u, v) denote the partial derivatives of F(u, v) with respect to u, v respectively. It is not difficult to see that conditions (ii), (iii) imply that F(u, v) is a.c. in each variable uniformly with respect to the other variable or, as we say, is equi-a.c. in each variable. Thus, in the above definition, condition (iii) can be replaced by (iii), there is a p ∈ L(0, 1) such that for each v ∈ [0, 1], |F1(u, v)| ≤ p(u) a.e. in [0, 1] and for each v ∈ [0, 1], |F2(u, v)| ≤ p(u)a.e. in [0, 1].
Energy of a charged particle in Møller's tetrad theory
Authors: Shah K. B. 
Published: November 2008

AbstractEnergy of the exterior and the interior fields of a particle, charged or otherwise, embedded in an empty space is calculated, using Møller's tetrad formulation of the energy-momentum complex in general relativity. It is found that the exterior field contains positive gravitational energy, and in addition, the electromagnetic energy if the particle is a charged one and that the sum total of the energies of the interior and the exterior fields is equivalent to the Newtonian mass of the particle.
On a generalized L–H transform
Authors: Kapoor V. K. , Masood S. 
Published: November 2008

AbstractThe authors, while attempting to give a new generalization of the Laplace transform, have come across a particular function in the form of Meijer's G-functionwhich, when taken as a nucleus of the transformation K(x) indefines a new transform. This transform besides serving as a generalization of the Laplace transform and most of its generalizations existing in the literature bears the characteristic property of generalizing the Hankel transform (and hence some of its generalizations) as well. In this paper the authors after having defined the transform have given an inversion theorem which is supported by means of two examples.
The scale effect in compressible turbulence
Authors: Crighton D. G. 
Published: November 2008

AbstractThe pressure fluctuations in a compressible fluid in statistically steady isotropic motion are studied. These depend crucially on the scale of the turbulent region. If the region is small enough, the mean square acoustic pressure varies as the linear scale of the region, and as the fourth power of the turbulence Mach number. In the limit of infinite scale, however, diffusive effects limit the otherwise infinite pressure fluctuations as first shown by Lighthill(6). The mean square acoustic pressure then varies as the Mach number and as the turbulence Reynolds number. The pressure fluctuations above a sheet composed of statistically uniformly fluctuating sources are also examined. Here the mean square pressure diverges, initially, in proportion to the logarithm of the scale of the sheet, until viscous effects again become significant in limiting the pressure.
The flow around a semi-infinite oscillating plate and the skin friction on arbitrarily cross-sectioned infinite cylinders oscillating parallel to their length
Authors: Wilks Graham 
Published: November 2008

AbstractIn Part I of the work that follows solutions for the flow around a semi-infinite oscifiating plate are obtained, one by an operational method using parabolic coordinates and one in the form of a series solution. The latter is used to obtain an estimate of the skin friction distribution about the edge of the plate and hence the excess skin friction over the infinite oscilliating plate value. A valuable check on this result is provided by an equivalent consideration of the operational solution. Having demonstrated the usefulness of the series solution we proceed to extend its application to a wedge of arbitrary angle in Part II and eventually use the results to derive a general expression for the skin friction on an infinite cylinder of arbitrary cross-section oscifiating rapidly parallel to its length.
Exact solutions for vertically polarized electromaǵnetic waves in a horizontally stratified maǵneto-plasma
Authors: Westcott B. S. 
Published: November 2008

AbstractIn a previous paper (11) refractive index profiles capable of yielding exact solutions for vertically polarized electromagnetic waves propagating in horizontally stratified isotropic media were derived systematically. The present work extends the method to deal with anisotropic media in which propagation is transverse to a horizontally applied static magnetic field. The relevance to ELF radio wave propagation in the terrestrial ionosphere is noted.
The propagation of waves in an elastic half-space containing a cylindrical cavity
Authors: Gregory R. D. 
Published: November 2008

AbstractThe problem of the propagation of time harmonic waves in an isotropic elastic half-space containing a submerged cylindrical cavity is solved analytically. Linear plane strain conditions are assumed. Using an expansion theorem proved in a previous paper (Gregory (3)), the elastic potentials are expanded in a series form which automatically satisfies the governing equations, the conditions of zero stress on the flat surface, and the radiation conditions at infinity. The conditions of prescribed normal and tangential stresses on the cavity walls are shown to lead to an infinite system of equations for the expansion coefficients. This system of equations is shown to be a regular L2-system of the second kind and from its unique l2-solution, the solution to the problem is constructed. The fundamental questions of existence and uniqueness are fully treated and methods are described for constructing the solution.Three applications of the general theory are presented dealing respectively with the production, amplification and reflexion of Rayleigh waves.
Montel's theorem for subharmonic functions and solutions of partial differential equations
Authors: Beardon A. F. 
Published: November 2008

1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-stripand if there exists an x0 in (a, b) such thatas y → ∞, thenas y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and ifas y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such thatas y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)asy → ∞, thenas y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker conditionas n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.