bluuebunny/crossref_metadata_2025_split huggingface dataset:
On iterative solutions of algebraic equations
Authors: Domb C. 
Published: November 2008

1. In a recent paper (1) Hartree has given a systematic classification of iterative processes, and has shown how to construct processes of any order for the solution of equations of the formwhere p is a positive or negative integer. It is the purpose of the present note to describe general methods of generating iterative processes, and in particular to show how to construct processes of any order for the solution of algebraic equations.
The cardinal series in Hilbert space
Authors: Weston J. D. 
Published: November 2008

The name ‘cardinal function’ was given toby E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined byform a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.
Sets of uniqueness for trigonometric series and integrals
Authors: Henstock R. 
Published: November 2008

It is well known that, if a trigonometric seriesconverges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.
The extremal values of Legendre polynomials and of certain related functions
Authors: Cooper R. 
Published: November 2008

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show thatwhere jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.
Unsteady laminar flow of gas near an infinite flat plate
Authors: Illingworth C. R. 
Published: November 2008

AbstractBoundary-layer equations for the unsteady flow near an effectively infinite flat plate set into motion in its own plane are subjected to von Mises's transformation. Solutions are obtained for the flows in which gravity is neglected, the Prandtl number σ is arbitrary, and the plate has a constant temperature and a velocity that is either uniform or, with dissipation neglected, non-uniform. Explicit solutions are obtained for the case in which the viscosity μr varies directly as the absolute temperature Tr. Solutions are also obtained for the diffusion of a plane vortex sheet in a gas, and for the boundary layer near a uniformly accelerated plate of constant temperature when gravity is not neglected. For the non-uniform motion of a heat-insulated plate, dissipation not being negligible, a solution is obtained when σ is 1 and μr ∝ Tr. The relative importance of free convection due to gravity and forced convection due to viscosity is discussed, and a solution is obtained, with μr ∝ Tr, for the free convection current set up near a plate that is at rest in a gas at a temperature different from that of the plate, dissipation being neglected.
A theorem on multiple integrals
Authors: Hammersley J. M. , Moran P. A. P. 
Published: November 2008

Suppose thatdυ anddυ′ are two volume elements situated at pointsPand P′ respectively in a three-dimensional right circular cylinder, thatyis the distancePP′, thatz(y) is a given function ofy, and that we wish to evaluate the sixfold integraltaken over all pairs of pointsP, P′within the cylinder. We observe thatz(y) is a function ofyonly; so that the sixfold integral can be expressed as a single integralthat is to say a weighted mean ofz(y) over the relevant values ofy,where the weight function is evidently given by
The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms
Authors: Cassels J. W. S. 
Published: November 2008

In this paper I prove the following results.Theorem 1. Let f(x, y) = ax2 + bxy + cy2be an indefinite form. Then there exist real (x0, y0) such thatfor all (x, y) ≡ (x0, y0) (mod 1). If f represents 0 there are indenumerably many such incongruent (x0, y0).
A superconducting galvanometer
Authors: Pippard A. B. , Pullan G. T. 
Published: November 2008

AbstractA superconducting tangent galvanometer suitable for the measurement of thermoelectric voltages in circuits of resistance about 10−7 ohm is described. The inductance has been kept low by the use of only a single turn in each of the deflecting coils, but a high sensitivity has been attained by reducing the controlling field to about 0·01 gauss. The theory of operation of such a galvanometer in circuits of low resistance is discussed; the actual behaviour of the instrument is in satisfactory agreement with the theory. The galvanometer will detect a current of 10−5 A. corresponding to an e.m.f. of 10−12 V.
The stability of hexagonal lattices with a simple law of force
Authors: Nabarro F. R. N. , Varley J. H. O. 
Published: November 2008

AbstractA hexagonal lattice bound by central forces between nearest neighbours is always close-packed. If there is in addition an energy which, like the Fermi energy of free electrons, depends only on the volume, a close-packed equilibrium structure is still possible, but there may be another hexagonal structure of lower energy. A numerical example is given in which this occurs, the lattice spacing, binding energy and elastic constants being comparable with those of Zn, and the law of force showing no obvious peculiarities.
The classification of multipole radiation
Authors: Jeffreys Bertha 
Published: November 2008

AbstractA detailed account is given of the classification of the electromagnetic fields arising from a vibrating charge-current distribution. The work owes much to that of Berestetzky and Fierz, but differs from their treatments in that the exact expansions are taken as a starting point. Hence the general forms of the potentials and fields and their approximate forms are obtained directly. Formulae appropriate to large distances are given in §§2·1 and 3·1. The relation to the approach of Condon and Shortley (4) is discussed.
Gravitation and cosmic expansion in conformal space-time
Authors: Gürsey Feza , Bondi H. 
Published: November 2008

AbstractA simple theory of gravitation is formulated in conformal Riemannian space-time. The metric is determined by a scalar function which satisfies a linear equation. A conclusion in favour of Einstein's general tensor theory is drawn from a discussion of the corrections to the Newtonian theory for purely gravitational phenomena. Finally the theory is applied to the cosmological problem and especially to the possibility of a steady-state universe. The velocity-distance law is shown to be compatible with a constant uniform distribution of matter without the need of artificial assumptions.
On the evaluation of the confluent hypergeometric function
Authors: Slater L. J. 
Published: November 2008

ABSTRACTThis paper contains a table of the confluent hypergeometric function over the range a = − 1·0(0·1) + 1·0, b = 0·1(0·1)1·0, x= 1·0(1·0)10·0, and the expansions in converging factors by means of which the accuracy of the asymptotic expansions for higher values of x can be improved.
The evaluation of a diffraction integral
Authors: Hartree D. R. 
Published: November 2008

The calculation of the diffraction pattern for an out-of-focus line source viewed through an axially symmetrical lens involves the evaluation of the integralfor a constant value of β (related to the numerical aperture of the lens) and a range of values ofxandy. For small values ofxandythis can best be done by quadrature; for larger values ofxoryor both it is more practicable to use numerical evaluation of an appropriate ordinary differential equation. It is shown that the variation ofwithxfory= 0, and withyforx= const., can both be expressed by second-order ordinary differential equations of a convenient form for numerical integration; the evaluation ofI1(x,y) from solutions of these equations is straightforward. For comparison with experiment, results are required in the form ofI1(x,y) and a derived quantity as functions ofy, at relatively widely spaced values ofx; numerical work for β = 0.72 has been carried out on the EDSAC and has been organized to produce the results in the required form directly, without interpolation.Comparison of results obtained by using different Gauss quadrature formulae and different interval lengths show some interesting features of the errors of high-order integration formulae, and show that care is necessary in estimating these errors from the differences between quadratures carried out with intervals of different lengths.
Temporally ordered graphs in quantum field theory
Authors: Polkinghorne J. C. 
Published: November 2008

ABSTRACTThe S matrix is expressed in terms of momentum-space operators and its Cayley transform exhibited. A new graphical formalism called O-graphs is developed in which the vertices are temporally ordered and the lines represent real particles. This is used to prove a conjecture of Gupta concerning the relation of real and virtual processes to the Cayley transform. The transformation operator for a semi-infinite time interval is also considered and some of its properties demonstrated. In conclusion, another graphical formalism with temporally ordered vertices is developed and renormalized.
A note on the P-homomorphism in homotopy groups of spheres
Authors: Hilton P. J. 
Published: November 2008

The homomorphism of the title is a homomorphism of πr(Sn) into πr+n−1(Sn) given bywhere ι generates πn(Sn). Various results on the P-homomorphism were proved in (4); in particular it was shown that, if ηn generates πn+1(Sn), n ≥ 2, then
Thermal conduction in the intermediate state of superconductors
Authors: Laredo S. J. , Pippard A. B. 
Published: November 2008

ABSTRACTThe maximum in thermal resistivity exhibited by superconductors in the intermediate state is found to be particularly marked in tin at 0-5° K., at which temperature heat transport in the pure superconducting state is almost entirely by phonons. A simple theory of the effect is developed, based on the idea that phonon free paths are considerably shorter in the intermediate than in the superconducting state, owing to the high scattering caused by normal regions. Theory and experiment are in satisfactory agreement.
Note on Whitham's ‘The propagation of weak spherical shocks in stars’
Authors: Jones D. S. 
Published: November 2008

In his paper on ‘The propagation of weak spherical shocks in stars’ Whitham(1) uses as the basis of his non-linear theory the solution of the linear equationwhere the suffixes denote partial derivatives and A, B, C are functions of r only. In obtaining his solution he assumes (i) thatwhere ξ = t – ∫dr/A and fn(ξ) is the nth integral of f with respect to ξ, and (ii) that the function f may be identified with the corresponding function which occurs when gravity is neglected, i.e. when A becomes the constant A0, B = −2/r and C = 0. Having assumed that (2) exists, Whitham is able to show that it will hold asymptotically for Aξ/r ≪ 1 and A′ = O (A/r).
The effect of configuration interaction on forbidden line strengths
Authors: Garstang R. H. 
Published: November 2008

ABSTRACTThe calculation of the strengths of forbidden lines in the 1s22s22p2 configuration is extended to include the effect of interaction with the 1s22p4 configuration. For this case it is shown that if observed energy levels are used instead of theoretical energies in line-strength formulae which neglect configuration interaction the major part of the effect of configuration interaction is nevertheless taken into account. The results are illustrated by detailed computations on the Ca xv spectrum. The 1s22s22p4 configuration is also discussed.
Logic on electronic computers: a practical method for reducing expressions to conjunctive normal form
Authors: Routledge N. A. 
Published: November 2008

ABSTRACTIn § 1 we introduce our system and prove a theorem about its syntax. In § 2 we recall some stock results about the propositional calculus. In § 3 we consider a method of deriving an expression from a given expression and a real number. In § 4 we use this to derive a sequence of expressions from a given expression. In § 5 this sequence is shown to be just all the terms of a conjunctive normal form of the given expression. In § 6 we note that we may not need to produce all of these terms. In § 7 we describe a practical method (suitable for a binary digital electronic computer) which results from all this, and in § 8 we attempt to explain just why this is so.
Kinematic wave theory of bottlenecks of varying capacity
Authors: De S. C. 
Published: November 2008

ABSTRACTThis paper discusses a problem in traffic flow by the method of kinematic waves developed by Lighthill and Whitham(1,2). The theory of kinematic waves introduced by Lighthill and Whitham has been extended here to include the case when the flow q varies with the time, and it is seen that the expression for the wave velocity is of the same form as before, namely, ∂q/∂k, where q is the flow (quantity passing a given point in unit time) and k is the concentration (quantity per unit distance). The theory is applied to the problem of estimating how a uniform oncoming flow behaves on entering a bottleneck, the capacity of which varies with time. This capacity has initially a higher value than the oncoming flow but falls at a uniform rate to a lower value, where it remains constant for a time, and again rises at a uniform rate to the original value. A shock wave is found to move back from the bottleneck, and later forward again and through it, much as in the case of a bottleneck of constant capacity with varying oncoming flow studied by Lighthill and Whitham.
Entry of conducting and non-conducting fluids in pipes
Authors: Shercliff J. A. 
Published: November 2008

ABSTRACTThe problem of estimating the entry length over which the laminar motion of an electrically conducting, viscous fluid in a pipe under a transverse field adjusts to a steady state from some initial state is linearized by a Rayleigh approximation adapted for cases where there is a pressure gradient. Solutions are found which express the difference between the initial and ultimate velocity profiles as a series of orthogonal, exponentially decaying terms. Approximate entry lengths for the most persistent terms are calculated for flow between two infinite parallel planes and in rectangular tubes when magnetic effects are present. Entry lengths in circular and rectangular pipes and between infinite planes are calculated in the case where magnetic effects are negligible to permit a comparison of the results with those derived by other methods. Pressure losses at entry are considered in the simpler cases.
Applications of the theory of cluster sets to a class of meromorphic functions
Authors: Collingwood E. F. , Lohwater A. J. 
Published: November 2008

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined bywhere m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned byfrom which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.
Eigenvalue problems treated by finite-difference methods. II. Two-dimensional Schrödinger equations
Authors: Bolton H. C. , Scoins H. I. 
Published: November 2008

ABSTRACTA discussion is given of the convergence of the eigenvalues Λ (N) of two-dimensional finite-difference equations towards the eigenvalues Λ of the corresponding second-order differential equation, and it is shown thatwhere h = N−1 and ν2 is a constant. As in our previous paper (4), this can be used to make an accurate estimate of λ by extrapolating to h = 0. After an account of the relaxation method used for computing Λ(N) and a discussion of the residual vector, results are presented for an approximation to the lowest spatially symmetric and antisymmetric states of two electrons in a sphere, interacting through their Coulomb potential.
On the continuous solution of integral equations by an electronic analogue. I
Authors: Fisher Michael E. 
Published: November 2008

ABSTRACTA scheme is proposed for solving a class of integral equations by electronic analogue computing techniques in times as short as one-tenth of a second. The scheme utilizes a recently developed high-speed analogue function store for carrying out a special iterative procedure which is shown to be more efficient than the classical Neumann process. The problem of the kernel generation at high repetition rates is considered and a novel method based on pivotal function generators is described. Likely errors are analysed and an overall accuracy of the order of 1% is shown to be attainable with known techniques.
An application of Sturm-Liouville theory to a class of two-part boundary-value problems
Authors: Karp Samuel N. 
Published: November 2008

ABSTRACTA simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.
On the Crank-Nicolson procedure for solving parabolic partial differential equations
Authors: Juncosa M. L. , Young David 
Published: November 2008

ABSTRACTProof of convergence of the Crank-Nicolson procedure, an ‘implicit’ numerical method for solving parabolic partial differential equations, is given for the case of the classical ‘problem of limits’ for one-dimensional diffusion with zero boundary conditions. Orders of convergence are also given for different classes of initial functions. Results do not support the validity of so-called h2-extrapolation in some cases.
Formation of an alpha-group in the shell-model of a heavy nucleus
Authors: Strachan Charles 
Published: November 2008

ABSTRACTBy means of a variation principle applied to a nuclear wave function for four nucleons in the field of a closed core, equations are derived giving the relative proportions of a component describing an alpha-particle group and components describing the four nucleons in shell-model states with spin-orbit coupling wave functions, the two neutron states and the two proton states each paired to give zero resultant angular momentum. Numerical calculations, taking into account only 1h2/3 states for protons and the 2g3/2 states for neutrons, give the alpha-group wave function a relative weight (in the linear combination) of about 1%.
The second-order effect of free electrons on lattice conduction
Authors: Pyle I. C. 
Published: November 2008

ABSTRACTSecond-order perturbation theory, in conjunction with the usual treatment of electron-phonon interaction, allows us to calculate the correction to the first-order result for the scattering of phonons by electrons. It is shown that the second-order term is much smaller, and therefore negligible. This justifies the use of the first-order theory in the treatment of the interaction in metals and semi-conductors.
A simplified method for integrating over Feynman histories
Authors: Brush S. G. 
Published: November 2008

ABSTRACTIn Feynman's ‘space-time’ formulation of quantum mechanics the Green's function for the Schrödinger equation is defined by an integral over all histories of the system. By integrating over one-parameter sets of functions, one gets the same Green's function as by integrating over a Fourier series, in simple cases. The method may be useful for estimating the result in cases when the integration over all histories cannot be performed exactly.
Abelian groups, graphs and generalized knights
Authors: Nash-Williams C. St J. A. 
Published: November 2008

ABSTRACTIf g is a set of generatore of an enumerably infinite Abelian group A, it is proved that the elements of A can be arranged in both a one-ended and an endless infinite sequence in which successive terms differ by ± an element of g, except that the one-ended arrangement is impossible if g is finite and the rank of A is 1. Let ν be a cardinal number. Consider an infinite ‘chessboard’ whose positions are those lattice points of ν-dimensional space which have only finitely many non-zero coordinates. A piece associated with this chessboard is a generalized knight if every vector obtainable from a move of the piece by permuting its components and changing the signs of a subset of them is itself a permitted move. It is ascertained which positions a given generalized knight can reach in a finite sequence of moves starting at the origin, and whether or not, if it can trace out the whole chessboard in (i) a one-ended, (ii) an endless infinite sequence of moves visiting each position exactly once.
Hilbert's function in a semi-lattice
Authors: Learner A. 
Published: November 2008

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).
Extensions of ideals in associative algebras
Authors: Smithies F. 
Published: November 2008

1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way thatfor x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.
Quadratic Lagrangians and gauge-invariance in covariant field theories
Authors: Stephenson G. 
Published: November 2008

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformationwhere is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed assuch that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).