bluuebunny/crossref_metadata_2025_split huggingface dataset:
The production of 8Li by bombardment of boron with neutrons
Authors: Lawrance A. M. 
Published: November 2008

The decay period of the α-activity produced when boron is bombarded with neutrons from a (Li + D) source has been measured, and found to agree with the known decay period of 8Li. It has been shown that no activity is produced when (Be + D) neutrons (maximum energy 4·5 × 106 e.V.) are used. The results are explained by postulating the formation of radioactive 8Li according to the processA short investigation of the number-range distribution for the α-particles emitted by 8Li indicated that a maximum occurred in the distribution at a range of about 6 mm.
On sets of fractional dimensions (II)
Authors: Best E. 
Published: November 2008

1. Any number x between 0 and 1 may be expressed uniquely in the formwhere xr is a non-negative integer less than r (r = 2,3,…). We consider the set E of numbers x for whichWe establish an inequality connecting the dimensional number of the set E with certain constants of the seriesin particular we show that, when ξτ = rθ, the dimensional number of E is θ. We are concerned with the measure of Hausdorff.
The action in a uniform field
Authors: Pars L. A. 
Published: November 2008

We consider the motion of a particle in a plane field of force. We take rectangular cartesian coordinates (x, y) in the plane, and denote by V(x, y) the potential of the field, and by h the (constant) energy of the motion. If A and B, whose coordinates are (x0, y0) and (x1, y1), are any two points on an orbit, the orbit is characterized by the property thattaken along the orbit is stationary as compared with the integral taken along a neighbouring curve joining the same points. Here s denotes the length of the are measured from A to B. This is one form, sometimes called Jacobi's form, of the principle of least action. The value of the integral, taken along the orbit, and expressed in terms of the coordinates of the termini and the constant of energy,is the action function for the field V.
A note on k-connexes
Authors: Hodge W. V. D. 
Published: November 2008

The expressionfor the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisam's result by determining the number of linearly independent algebraic forms of a special type to which we give the name k-connex.
On the representations of a number as a sum of squares and certain related identities
Authors: Rankin R. A. 
Published: November 2008

LetRc(n) be the number of representations of any non-negative integernas the sum ofcsquares; i.e.Rc(n) is the number of different solutions (x1,x2, …,xc) of the equationwhere thexμare integers, and may be positive, negative, or zero. Two such solutions, (x1,x2, …,xc) andare only considered to be the same solution if
On the summation of certain trigonometric series
Authors: Goddard L. S. 
Published: November 2008

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the serieswhere α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), whereWhen κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.
On the classical theory of radiating electrons
Authors: Eliezer C. Jayaratnam , Mailvaganam A. W. 
Published: November 2008

1. In Dirac's classical theory of radiating electrons, the relativistic equations of motion of a point-electron in an electromagnetic field arewhere (x0, x1, x2, x3) denote the electron's coordinates in flat space-time, dots denote differentiation with respect to the proper time , and the external electromagnetic field is described by the usual field quantities Fμν. The units are chosen so that the velocity of light is unity. These equations, which are derived from the principles of conservation of energy and of momentum, are the same as those obtained by Lorentz, when he used the spherical model of the electron and included radiation damping in an approximate way. But Dirac's method of derivation suggests that this treatment of radiation damping, and therefore the resulting scheme of equations, is exact within the limits of the classical theory.
Field theories with high derivatives
Authors: Chang T. S. 
Published: November 2008

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative ofwith respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.
The statistical distribution of the length of a rubber molecule
Authors: Moran P. A. P. 
Published: November 2008

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the formwhere σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.
Notes on Fourier series (V): Summability (R1)
Authors: Hardy G. H. , Rogosinski W. W. 
Published: November 2008

1·1. There is a familiar method of summation of series usually called the method (R, 1), or sometimes Lebesgue's method. If sn = u0 + u1 + … + un and, as it will be convenient to suppose throughout, u0 = 0, thenwhen h → + 0§ the convergence of the series for small positive h is presupposed. The method is not 'regular'; there are convergent series not summable (R, 1).
Inverse probability and sufficient statistics
Authors: Huzurbazar V. S. 
Published: November 2008

1. It is an interesting fact that in many problems of statistical estimation the results given by the theory of inverse probability (as modified by Jeffreys) are indistinguishable from those given by the methods of ‘fiducial probability’ or ‘confidence intervals’. The derivation of some of the important inverse distributions by Jeffreys(3) arouses one's curiosity. It seems that when this agreement is noticed there are usually sufficient statistics for parameters in the distribution. The object of this note is to throw some light, in general terms, on the similarity in form between the posterior probability-density function of the parameters and the probability-density function of the distribution when it admits sufficient statistics. For convenience the following notation in Jeffreys's probability logic is used below:P(q | p) is the probability of a proposition q on data p.
Diffraction by a wave-guide of finite length
Authors: Jones D. S. 
Published: November 2008

AbstractWhen the electric intensities on two parallel planes, of which the two perfectly conducting sides of a wave-guide of finite length and infinite width are portions, are taken as unknowns, the problem of the diffraction of a plane harmonic electromagnetic wave polarized parallel to the edges of the guide leads to two integral equations. By means of the Laplace transform these equations are converted into others suitable for solution by successive substitutions. The series thus obtained is too complex for practical purposes, and so an approximate solution is found for the case when the length of the guide is large compared with the wavelength. Finally, there is a brief discussion of the difference between the distant fields when l is large and when l is infinite.
Electronic bound states at the surface of a metal
Authors: Baldock G. R. 
Published: November 2008

AbstractThe conditions under which bound states associated with atoms in the surface of a metal may exist are investigated, using the tight-binding approximation. These states arise as a result of modifications in the parameters of certain atoms. The modifications required to produce (a) bound states associated with all the atoms in the surface (surface states) and (b) bound states associated with particular small groups of atoms are found for the simple cubic lattice. It is also shown that most of the simpler crystal structures do not exhibit surface states without such modifications; in the graphite and diamond lattices, however, surface states exist solely by virtue of the lattice geometry.
The stability of the surface of a cavitation bubble
Authors: Binnie A. M. 
Published: November 2008

ABSTRACTA theoretical examination is made of the stability of a spherical interface between two inviscid liquids accelerated radially from rest. If surface tension be neglected, it is found that the interface is unstable or stable according as the acceleration is directed towards or away from the liquid of greater density. Surface tension has a marked stabilizing effect, and numerical results are given for the minimum outward acceleration required to destroy the equilibrium of the surface of a cavity formed in water. These results are suggested as a possible explanation of the behaviour of cavitation bubbles seen to form near a fixed object under test in a water tunnel.
The roots of J−(l + ½) (λη)Jl + ½ (λ) − Jl + ½ (λη) J−(l + ½)(λ) = 0
Authors: Chandrasekhar S. , Elbert Donna 
Published: November 2008

Spherical Bessel functions which vanish at x = 1 and x = η (where η is an assigned positive constant less than 1) occur in the solution of many problems in applied mathematics. Such functions can be expressed in terms of Bessel's functions of the half odd integral orders in the formwhere λ is a root of the equation
Long waves in running water
Authors: Burns J. C. 
Published: November 2008

ABSTRACTClassical shallow-water theory for the propagation of long waves in running water is modified by the inclusion of the effects of the vorticity present in the main stream as the result of the action of viscosity. When this vorticity is assumed constant, a non-linear theory can be used, but for more general velocity distributions in the main stream it is necessary to linearize the problem.In the linearized theory, a general equation is obtained connecting the wave velocity with the velocity in the undisturbed stream and this is solved in several special cases. It is shown generally that the wave velocity relative to the mean flow is always greater than the value given by the classical theory. The wave velocity relative to the bottom of the stream has two values, one less than the minimum stream velocity and the other greater than the maximum stream velocity.
A singular integral equation in the theory of meson-nucleon scattering
Authors: Dalitz R. H. , Sundaresen M. K. , Bethe H. A. , Mott N. F. 
Published: November 2008

ABSTRACTThe Tamm-Dancoff theory of meson-nucleon scattering (pseudoscalar coupling) leads to a series of singular integral equations. The asymptotic behaviour of the solutions of these equations is obtained here for all scattering states. For attractive states there is a critical coupling constant beyond which no normalizable solutions exist for these equations; for repulsive states another critical coupling constant appears, beyond which the solutions oscillate infinitely often but are still normalizable. It is concluded that the renormalization procedures proposed previously (3) are consistent and successfully define finite vertex-functions, and that the equations obtained for these vertex-functions have satisfactory properties for numerical work.
Motion of a sphere through a conducting fluid in the presence of a strong magnetic field
Authors: Stewartson K. 
Published: November 2008

ABSTRACTThe steady motion of a perfectly conducting sphere in an inviscid conducting fluid in the presence of a strong magnetic field is discussed. It is shown that if the fluid velocity is ultimately steady then it is two-dimensional, and a cylinder of fluid whose generators are parallel to the direction of the field moves with the sphere as if solid. The streamlines outside are straight lines if the sphere moves in the direction of the field but have to execute sharp turns if it moves at right angles to the field. The motion to be expected in practice is discussed using an analogy.
Elements of an improved linear perturbation theory of steady supersonic flow with axial symmetry
Authors: Jones C. W. 
Published: November 2008

ABSTRACTThe usual linearized theory neglects small terms, of second and higher orders of the velocity perturbation, in the coefficients of the equation for the potential. In this paper these terms are approximated by constant mean values and the familiar formulae of the first approximation are then generalized. The characteristic lines of this improved linear theory thereby differ from the Mach lines of the undisturbed flow, though still being straight and parallel. The new formulae thus give a better calculation of the velocity field near bodies not slender enough for the ordinary theory to be accurate. A numerical example, for a 10° cone in a stream of Mach number about 3, illustrates this fact, and shows that a shock of constant strength may be fitted to the improved solution.Although the improved theory as it stands must fail at large distances from the axis of symmetry, just as the first approximation fails, the tentative suggestion is made that a technique due to G. B. Whitham may be applied to the former, as to the latter, perhaps leading to an improved description of the whole field of flow.
The penetration depth in impure superconducting tin
Authors: Chambers R. G. , Pippard A. B. 
Published: November 2008

ABSTRACTA new method is described for measuring the surface impedance of metals at low temperatures and at radio-frequencies. Using this method, the surface impedance of normal tin and the penetration depth λ in superconducting tin have been studied at about 9 Mc/s frequency as a function of impurity content. The measured surface impedances agree well with the values expected, and the penetration depths increase with impurity content, in confirmation of Pippard's observations (14) at 9400 Mc/s. The variation of λ with reduced temperature t is well represented by the equation λ/λ0 = (1 − t4)−½, in impure as well as in pure tin. The absolute value of λ found for pure tin is about 4 × 10−6 cm., significantly lower than the values previously found.
Finite-by-nilpotent groups
Authors: Hall P. 
Published: November 2008

1. Introduction. 1·1. Notation. Letandbe, respectively, the upper and lower central series of a group G. By definition, Zi+1/Zi is the centre of G/Zi and Γj+1 = [Γj, G] is the commutator subgroup of Γj with G. When necessary for clearness, we write ZiG) for Zi and Γj(G) for Γj.
Some formulae for the product of hypergeometric functions
Authors: Ragab F. M. 
Published: November 2008

The formula to be established iswhere α > ½, β > ½and b is real and positive. In § 3 we shall generalize this by provingwhere α > (p−l)/p, β > (p−l)/p, b > 0 and p = 2,3,….
The existence of subgroups of given order in finite groups
Authors: McLain D. H. 
Published: November 2008

1. Discussion of results1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.
Some properties of skew-symmetric elements of a ring
Authors: Barnett I. A. , Abian S. 
Published: November 2008

Let R* be a ring with unit element I. For every A ∈ R*, let the mapping A → A′ be an involutory anti-automorphism, i.e. a one-to-one mapping of R* onto itself with the following properties:Let C* denote the set of all symmetric elements of R*, i.e. C′ = C for all C ∈ C*.An element A ∈ R* is said to be skew-symmetric if A′ + A = 0 (the zero element of the ring).
Causal amplitudes and the Yang-Feldman formalism
Authors: Polkinghorne J. C. 
Published: November 2008

ABSTRACTThe Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.
Norms of summation methods
Authors: Petersen G. M. 
Published: November 2008

We shall understand the norm, h(A) of a regular matrix A = (amn) to beThe method has a norm, ‖ ‖, ‖ ‖ ≥ 1, given by ‖ ‖ = inf h(A) where the inf is taken over all the matrix methods equivalent to for bounded sequences (b-equivalent). These definitions are due essentially to Brudno(1) though his definition of
Some theorems on fractional integrals
Authors: Flett T. M. 
Published: November 2008

1. Let φ(z) be regular in │z│ < 1, letand for any real α letwhere (in)−α=n−αexp(— ½απi). The function φαis an adaptation to power series of the Weyl fractional integral of order a of α function of a real variable.† It exists and is regular in │z│ < 1 for all α. Let also
Diffraction by an imperfectly conducting right-angled wedge
Authors: Williams W. E. 
Published: November 2008

ABSTRACTA new and exact solution is given for the diffraction of the field of an electric line current by an imperfectly conducting wedge of exterior angle . Previous attempts at solving this problem assumed that the solution could be expressed as a series in inverse powers of the conductivity of the wedge, but this approach can only be applied to wedges of exterior angle less than π as the solution obtained is singular at the edge of the wedge for angles exceeding π. It is shown that in the present case no such expansion is uniformly valid; at large distances from the wedge, however, the present solution agrees with that obtained by assuming an expansion in inverse powers of the conductivity, even though this latter solution is singular at the edge, to order (conductivity of the wedge)−1; the next term in the expansion of the present exact solution is O(conductivity). The case of an incident plane wave may be obtained by letting the source tend to infinity.
The numerical integration of ordinary differential equations possessing exponential type solutions
Authors: Dennis S. C. R. 
Published: November 2008

Although there are many variations of finite-difference methods of obtaining approximate numerical solutions to ordinary differential equations they share the common feature that they tend to treat an equation of a given type as a standard problem and take no account of any special characteristics the wanted solution may have. We here suggest an alternative procedure when the wanted solution exhibits exponential characteristics. In essence the idea is that if a differential equation has an exponential type solution y(x) it is useful to solve numerically, instead of the equation for y, the equation for u = logey. The error-building and stability characteristics are then those of u rather than y and consequently the accuracy of the solutions may be improved. Although there is nothing basically new in this, of course, the point that we demonstrate is that the differential equation in y can be solved numerically in such a manner that the transformation from y to u is not actually carried out, i.e. we retain the original dependent variable but take account of the exponential variation by modifying the integration formula. Consider for example, in the usual notation, the first-order equationwith a given initial condition y(x0) = y0. If x0, x1, …, xr, xn is a set of pivotal values of x;, usually assumed equally spaced so that xr+1 − xr = h, the usual approach replaces (1) by the formulawhich, once the integral is expressed in terms of pivotal values of f using a difference series, represents a step-by-step formula for constructing successive values of y.
An extension of the Pitman–Koopman Theorem
Authors: Jeffreys Harold 
Published: November 2008

ABSTRACTThe theorems of Pitman and Koopman on laws of chance that yield sufficient statistics are extended to laws where the parameter or the observable, or both, may admit only a discrete set of values. The method depends on making changes, not necessarily small, within the values permitted for the variables, but it is still necessary to assume a solubility condition.
Waves on a Sloping Beach
Authors: Williams W. E. 
Published: November 2008

ABSTRACTA method employed by the author for a class of boundary-value problems arising in diffraction theory is applied to the problem of the oscillations of water waves on a sloping beach. The two cases of long and short waves are considered. The approach is valid for all values of the parameters and particular cases do not have to be treated separately. It is shown that the present approach enables certain solutions to be obtained which cannot arise in earlier solutions of the problem.
Two-dimensional elastic inclusion problems
Authors: Jaswon M. A. , Bhargava R. D. 
Published: November 2008

ABSTRACTAn account is given of Eshelby's point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.
The use of algebraic structures in physics
Authors: Kilmister C. W. , Newman D. J. 
Published: November 2008

ABSTRACTThe hypothesis of the existence of abstract structures in physics, which has been made in recent papers, is made precise. Difficulties in previous formulations are removed and insight into Maxwell electrodynamics is obtained. General relativity is related to other field theories which deal with flux from sources, and new field equations for general relativity are proposed.