bluuebunny/crossref_metadata_2025_split huggingface dataset:
The theory of the structure and formation of a certain type of immobile adsorbed film with an application to the adsorption of oxygen on tungsten
Authors: Roberts J. K. 
Published: November 2008

The theory is given of the structure and behaviour of immobile adsorbed films on a simple quadratic lattice in which each particle is so large that it occupies more than one site and makes it impossible for adsorption to occur on sites which are neighbours to that on which it is adsorbed. The results are obtained partly by statistical methods and partly by the use of a model of the film.The final or complete film of this type is not homogeneous and has gaps in it. The sites on the surface can be divided into two groups, the gaps occurring where regions of the surface occupied by particles adsorbed on one group and those occupied by particles on the other group meet. The number of these gaps or “boundary sites” is determined, and the possible configurations of particles in the boundary regions are discussed in detail.The results are considered in connexion with some experiments on the adsorption of oxygen on tungsten. In this case a second layer is formed over the boundary sites and the amount that can be so adsorbed or the number of “second layer sites” to which the boundary sites give rise is obtained. It is shown that numerically the experimental results for oxygen are in better agreement with the view that the first or very stable layer is a molecular layer of the above type than with the view that it is atomic.
The variational equation of relativistic dynamics
Authors: Mathisson Myron 
Published: November 2008

If Tαβ is the energy-tensor, then, for any vector field ξα, the equationfollows from the energy equationSuppose first that the physical system considered is complete, i.e. that the energytensor vanishes beyond some world tube Z. Let L be a time-like world line running inside the tube. We integrate both sides of (a) over a portion of Z, and we transform an integral over a four-dimensional region into a linear integral over L. We obtain the variational equationin which the m's are tensors characteristic of the physical system. An essential feature of the m's is that they are symmetrical in their two last superscripts. Equation (c) has to be satisfied by every field ξα, provided that the ξ's vanish, with all their derivatives, at the ends of the integration path.
The Measure of Plane Sets
Authors: Moran P. A. P. 
Published: November 2008

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.
The theory of the mean square variation of a function formed by adding known functions with random phases, and applications to the theories of the shot effect and of light
Authors: Rowland E. N. 
Published: November 2008

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given byWe may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), whereBoth the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.
Electronic energy bands in metallic copper and silver
Authors: Tibbs S. R. 
Published: November 2008

The present discussion of the conduction levels of copper and silver was undertaken in view of the experimental investigations of the absorption and emission bands of metals, in the soft X-ray region, which have been carried out in this laboratory and elsewhere*. It is a well-known result of the electron theory of metals that, for cubic crystals, of lattice constant a, if k is the wave vector of an electron moving in the crystal lattice, there are discontinuities in the energy E of the electron for values of k such that(n1, n2, n3) being the Miller indices of the planes for which Bragg reflexion of the electron waves takes place. If we draw the energy E as a function of k for a particular direction in the lattice, then we obtain, using the usual notation, curves such as those shown in fig. 1, d being the half-distance between the corresponding planes for which Bragg reflexion takes place.
The properties of a perfect Einstein-Bose gas at low temperatures
Authors: Fowler R. H. , Jones H. 
Published: November 2008

In a recent letter to Nature F. London(3) has called attention to a discontinuity in the derivative dCV/dT of the specific heat with respect to temperature which arises in a perfect Einstein-Bose gas at low temperatures. When discussing the properties of a perfect gas satisfying the Bose statistics, Einstein(1) remarked that at low temperatures something resembling a condensed phase should appear. The temperature at which this condensation should begin is given by the equationwhere n is the number of particles of mass m per unit volume and the other symbols have their usual significance.
Zur Summabilitätstheorie der Fourierschen Doppelreihe
Authors: Grünwald Géza 
Published: November 2008

Es sei f(ξ) eine L-integrierbare und nach 2π periodische Funktion der reellen Variabeln ξ und es sei {sn} die Folge der Partialsummen ihrer Fourierschen Reihe an einer Stelle ξ = x, wo die Funktion stetig ist. Dann ist nach einem klassischen Satze von Fejérund nach Hardy und LittlewoodVor kurzem hat L. Fejér gefunden, dass der Hardy-Littlewoodsche Satz (2) Spezialfall eines “gewöhnlichen” Summabilitätssatzes wie (1) ist, der sich aber auf die Fouriersche Reihe einer Funktion zweier Variabeln bezieht. Dieser Satz ist der folgende.
Quantum electrodynamics. II
Authors: Hoyle F. 
Published: November 2008

It is shown in this paper and the preceding one that two separate forms of theory can be developed in which a “finite size” is attributed to a charged particle by means of its interaction with the radiation field. The region attributed in this way to the particle is four dimensional and is determined in such a manner that the usual difficulties with relativistic invariance do not arise.The advantage of such a theory becomes clear when the theory is applied to those problems in which the usual calculations give infinite results. The problem of the method of successive approximations is considered and satisfactory results are obtained provided that the space dimensions of the finite region are of the order of the classical radius of the electron, when the electron is at rest.It may be noted explicitly that the difficulty that has been associated with the emission of low energy quanta by “Bremsstrahlung” will not arise in the present formulation of the electromagnetic interaction between field and particles. This case is interesting since an infinity arises here which is not analogous to the self energy infinities, but occurs in the direct calculation of a physical process and not in a virtual transition.The theory seems satisfactory so far as low energy processes (< 137 mc2) are concerned and the real test of its applicability may be expected to arise in discussing processes of high energy. It is hoped to treat these in a later paper.
Continuants and precontinuants
Authors: Bennett G. T. 
Published: November 2008

1. It is here expedient to ignore the continued fractions from which they are usually derived and to define the simple continuants determined by the parameters a1, a2, a3, …, an, … as given by the chain of difference equationstogether with the initial values u0 = 0, u1 = 1: and these data lead successively to the functionsand
On certain expansions of the solutions of the general Lamé equation
Authors: Erdélyi A. 
Published: November 2008

1. In this paper I shall deal with the solutions of the Lamé equationwhen n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the formwhere θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.It is easy to obtain the system of recurrence relationsfor the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for whichk′ being the principal value of (1−k2)½
On the product of three non-homogeneous linear forms
Authors: Davenport H. 
Published: November 2008

Let ξ, η, ζ be linear forms inu, v, wwith real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbersa, b, cthere exist integral values ofu, v, wfor whichand the constant ⅛ on the right is best possible.
A theorem on positive harmonic functions
Authors: Verblunsky S. 
Published: November 2008

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can writewhere α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to writewhere h* is a harmonic function conjugate to h. Then the functionhas the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.
On the recurrent solutions of a class of non-linear differential equations
Authors: Friedlander F. G. 
Published: November 2008

1. This paper is concerned with certain properties of the solutions of differential equations of the typewhere f, g have continuous partial derivatives up to the second order satisfying Lipschitz conditions in some bounded domain and are periodic in t with period 2π, ω is a positive constant and k is a small parameter. If f = 0, (1·1) is equivalent toan equation representing the forced vibrations of a quasi-linear system of one degree of freedom.
Non-homogeneous binary cubic forms
Authors: Bambah R. P. 
Published: November 2008

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that
Superconductivity of tin isotopes
Authors: Lock J. M. , Pippard A. B. , Shoenberg D. 
Published: November 2008

AbstractDetailed measurements have been made of the superconducting transition temperatures and critical magnetic fields of the tin isotopes of mass 116, 120 and 124. The transition temperature varies with isotopic mass according to the law Tc ∞ M−n, with n = 0·462 ± 0·014, a result very similar to that already found in mercury. The critical field curves of isotopes 116 and 124 are geometrically similar, in the sense that both many be represented by the same equation, Hc/Ho = f(T/Tc), with the same ratio Ho/Tc for both. It is deduced that the electronic specific heat in normal tin varies only slowly, if at all, with isotopic mass. The variation of Tc and Ho with M is very close to that predicted by Fröhlich and Bardeen.
Core-preserving transformations of a vector space
Authors: Bonsall F. F. 
Published: November 2008

In the classical theory (3) due to Knopp, Agnew and others, the core K(x) of a sequence x = {ξn} of complex numbers is defined by where En(x) is the smallest closed convex set containing all ξk with k ≥ n. A matrix transformation T is said to be a core-preserving transformation ifholds for all sequences x. T is core-preserving for bounded sequences if (1·1) holds for all bounded sequences x. It is readily proved that K(x) is the set of complex numbers ζ such thatfor all complex numbers α (). Now is a sub-additive, positive-homogeneous real-valued functional defined on the vector space of bounded complex sequences. This suggests the construction of an abstract theory on the following lines.
A problem concerning three-dimensional convex bodies
Authors: Rankin R. A. 
Published: November 2008

1. The problem considered in this paper arose during an investigation of what Chabauty has called the ‘anomaly’ of convex bodies.Throughout the paper denotes a closed bounded convex body in three-dimensional Euclidean space , which is symmetric in the origin O and which contains O as an interior point. Such a body determines uniquely a distance-function f(x, y, z) which is defined and finite for each point (x, y, z) of and possesses the following properties
A note on the unsteady motion of a compressible fluid
Authors: Pack D. C. 
Published: November 2008

ABSTRACTA solution is found in closed form for the one-dimensional motion of an ideal, inviscid and isentropic gas with ratio of specific heats (2m + 3)/(2m + 1), m being any positive integer, when the gas is supposed to start from rest with an inhomogeneous temperature distribution.An example is given which is of interest in the study of interstellar gas clouds. A cloud is assumed to be initially at rest in contact with a vacuum; the Riemann invariant r is assumed to increase linearly for a finite distance into the cloud (from zero at the surface), beyond which it remains constant. The r-characteristics from the inhomogeneous strip meet simultaneously in the distance-time plane at a point of the gas surface; the surface does not move until this happens, when it acquires instantaneously a finite non-zero velocity. A numerical estimate is made of the time during which the surface remains at rest for an actual gas cloud, under the above conditions. In the subsequent development the gas motion takes the form of a growing simple wave.
On the circle polynomials of Zernike and related orthogonal sets
Authors: Bhatia A. B. , Wolf E. 
Published: November 2008

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.
Laminar flow in an axially symmetrical jet of compressible fluid, far from the orifice
Authors: Pack D. C. 
Published: November 2008

ABSTRACTWhen the boundary-layer approximations are applied to the laminar flow in an axially symmetrical jet of gas with Prandtl number unity, a particular integral of the equations of motion satisfying the required boundary conditions is found to be , where i1, is the (constant) enthalpy on the jet boundary. By a suitable transformation an equation is obtained which, to a first approximation, may be replaced by the equation for flowin an axially symmetrical jet of incompressible fluid at a sufficiently large distance from the orifice. For an ideal gas, the known solution of this equ`ation may be used as the starting point of an iterative method to build up the stream function as a power series. The next approximation is found explicitly. It shows that for subsonic Mach numbers the density at a point on the axis is greater and the temperature less than on the boundary by amounts which may be very closely represented by terms inversely proportional to the square of the distance of the point from the orifice.
Eddington's theory in terms of the concept of order
Authors: Bastin E. W. , Kilmister C. W. , Sciama D. W. 
Published: November 2008

The theory of a previous paper is used to discuss Eddington's work in fundamental physics.1. Introduction. A previous paper (2), referred to here as I, described a theory depending on an unorthodox view of the relation of mathematics to physics. This theory was begun from the study of the work of Eddington which we take to be summarized in Fundamental Theory (5). In the present paper is shown the extent to which, and the way in which, the two theories are connected.
Some inequalities concerning positive-definite Hermitian matrices
Authors: Fan Ky 
Published: November 2008

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatMand more generally,
The interpolation of atomic wave functions
Authors: Hartree D. R. 
Published: November 2008

ABSTRACTIf r̄nl is the mean radius for the radial wave function of a complete (nl) group in an atom of atomic number N, the variation of 1/r̄nl with N is nearly linear. Further the variation of a given (nl) radial wave function with N is such that for a given value of (r/r̄nl), the variation of the quantity (r̄nl)½P(nl; r) with r̄nl is nearly linear. These relations between the radial wave functions for different atoms are examined from the point of view of using them as a means of interpolating, with respect to atomic number, between results for atoms for which solutions of Fock's equations have been carried out.
A measure of dispersion applied to cosmic-ray and other problems
Authors: Broadbent S. R. 
Published: November 2008

ABSTRACTThe statistic is applied in several different situations to detect departure from uniform dispersion. The given sample points may lie on a square lattice (without repetition), a rectangular lattice (repetition being allowed), or within rectangular or circular plane boundaries. The problems considered include an analysis of cosmic ray shower structure, and the counting of dust particles and bacterial colonies.
Generalized Riemann spaces
Authors: Moffat John 
Published: November 2008

ABSTRACTThe recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.
Quantum electrodynamics and Hilbert space theory
Authors: Taylor J. G. 
Published: November 2008

ABSTRACTThis paper describes an attempt to formulate quantum field theory, in particular quantum electrodynamics, in terms of Hilbert space theory. The work of Cook (1) is extended to give a precise description of non-interacting electrons and positrons. The hole interpretation is not required in this extension, and no subtraction formalism is required. It is shown that the formalism can never reduce to that of intuitive quantum field theory except by an abuse of language associated with the δ-function. Interaction cannot be introduced in a simple manner into the rigorous formalism, so it seems extremely difficult to develop the Hilbert space formalism for quantum field theory in any useful manner.These difficulties indicate that an investigation of the Hilbert space basis of simple quantum theory is necessary before a rigorous mathematical formalism for intuitive quantum field theory can be developed.
An extension of the method of steepest descents
Authors: Chester C. , Friedman B. , Ursell F. 
Published: November 2008

ABSTRACTIn the integralthe functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle points varies with α, and if for some a (say α = 0) two saddle points coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In the present paper we consider this case of two nearly coincident saddle points and construct uniform expansions as follows. A new complex variable u is introduced by the implicit relationwhere the parameters ζ(α), A(α) are determined explicitly from the condition that the (u, z) transformation is uniformly regular near z = 0, α = 0 (see § 2 below). We show that with these values of the parameters there is one branch of the transformation which is uniformly regular. By taking u on this branch as a new variable of integration we obtain for the integral uniformly asymptotic expansions of the formwhere Ai and Ai′ are the Airy function and its derivative respectively, and A(α), ζ(α) are the parameters in the transformation. The application to Bessel functions of large order is briefly described.
The charge density near a sharp point on a conductor
Authors: Cade R. 
Published: November 2008

ABSTRACTA brief discussion is given of attempts that have been made to justify, in terms of electrostatic principles, the conjecture that surface charge on a conductor tends to infinity towards a convex sharp point and to zero towards a concave point, and it is concluded that the problem has not hitherto been solved. A new attempt is then made using potential-theoretic methods, and the problem is solved with a fair degree of generality. The limitations are of a kind familiar in this type of analysis and, roughly speaking, concern the degree of convexity of the conducting surface.