bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Distribution of Charge and Current in an Atom consisting of many Electrons obeying Dirac's equations
Authors: Hartree D. R. 
Published: November 2008

The approximation is made that in a many-electron atom each electron is in a stationary state in the field of the nucleus and the remaining electrons, and further that this field is central.To this approximation it is known that, if the electrons obey Schrödinger's equation, then a complete n, l group of electrons [i. e. 2 (2l + 1) electrons with the same n, l] can be divided into two half-groups for each of which the distribution of charge is spherically symmetrical. To the same approximation it is shown that, if the electrons obey Dirac's equation, a complete Stoner sub-group [i. e. 2j + 1 electrons with the same n, l, j (j = l ± £)] can be divided into two halves, for each of which the distribution of charge is spherically symmetrical.The distribution of current and the resultant magnetic moment for a solution of Dirac's equation in a central field are obtained, and it is shown that for a complete Stoner sub-group the net current and magnetic moment are zero. The Landé g-formula for one electron is derived from the formula for the magnetic moment by neglecting ‘relativity’ terms.The formula for the magnetic moment suggests the question whether it is ever possible to specify the direction of the spin axis of the electron (as distinct from the magnetic moment of the whole atom). This is investigated, and it seems justifiable to specify the direction of the spin axis for states for which the magnetic quantum number m has its extreme values ±j, but not otherwise.
Probability relations between separated systems
Authors: Schrödinger E. 
Published: November 2008

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.
Some applications of conformal transformation to airscrew theory
Authors: Westwater F. L. 
Published: November 2008

ABSTRACTRecent reports contain tables of a parameter K required in calculating the performance of an airscrew by a new method.The method of calculating K (due to Goldstein) is unsuitable for large values of the pitch especially near the tip of the airscrew.In the case of infinite pitch we fall back, for a two-bladed airscrew, on the problem of a rotating lamina in two dimensions.The solution for a cross lamina (corresponding to a four-bladed propeller) is given below and the tables of K for four blades are completed.A formula for the limit of K/Kp at the airscrew tip is given for a propeller with any number of blades, where Kp is an approximate value of K due to Prandtl.K for any number of blades is given in the form of an infinite series. The case of three blades is discussed in detail.
The number of configurations in an assembly and cooperative phenomena
Authors: Chang T. S. 
Published: November 2008

We consider an ideal problem of adsorption of single and double particles upon a solid surface which has its sites of accommodation regularly arranged, and by comparing the equilibrium properties obtained by Bethe's method with the ordinary statistical formulae, we obtain approximate expressions for:(1) g(N, n, X), the number of ways of arranging n particles upon N sites of a lattice so that the number of neighbouring sites occupied by the particles is X.(2) g2(N, n, X), the number of ways of arranging n double particles upon N sites so that each of the double particles takes up two adjacent sites and the number of neighbouring sites occupied by two different particles is X.Both these expressions are found to agree with the exact values when the N sites lie on a straight line. When we use the first expression to construct the configurational partition functions of certain physical assemblies and expand them in powers of 1/kT, they are found to agree with the corresponding rigorous expressions as far as (1/kT)3, which is the highest power which we can find rigorously at present. With the help of the first expression, formal equations for superlattice formation in an alloy with the composition 1: 1 and equations for the separation into phases of regular liquids are given. Lastly we show that atoms and molecules in a regular liquid may dissociate or recombine suddenly accompanied by a latent heat. This is a new cooperative phenomenon, which may bear some resemblance to the melting process between the solid and liquid states.
Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
Authors: Rankin R. A. 
Published: November 2008

Suppose thatis an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Thenwhere a, b, c, d are integers such that ad − bc = 1.
The number of configurations of a cooperative assembly
Authors: Miller A. R. 
Published: November 2008

Formulae are found for the number of configurations of particles on two-and three-dimensional lattices when each particle (a) occupies two closest neighbour sites, and (b) consists of three groups which occupy three sites on the lattice in such a way that adjacent groups in the molecule occupy closest neighbour sites on the lattice. Bethe's method is used to determine the equilibrium conditions of the corresponding order-disorder problem, and the number of configurations is determined from these equilibrium conditions. For the case in which the molecules occupy two closest neighbour sites on the surface the determination of the number of configurations from geometrical considerations is discussed.It is found that for molecules which occupy two closest neighbour sites the number of configurations of particles for a square lattice, a simple cubic lattice and a body-centred cubic lattice respectively are and For molecules which occupy three sites on the lattice the corresponding results are and when the molecules are perfectly flexible and and when the molecules are completely inflexible, Ns being the total number of sites in the lattice.The author wishes to thank Dr J. K. Roberts for suggesting this work; the geometrical treatment given in § 3 was developed from manuscript notes prepared by him. The problem arose during an investigation of the vapour pressure equations of solutions in which the solute molecules are chain molecules consisting of a large number of groups, undertaken as part of the programme of fundamental research of the British Rubber Producers' Research Association, whom the author wishes to thank for the grant of a Research Scholarship.
A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type
Authors: Crank J. , Nicolson P. 
Published: November 2008

This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equationwheresubject to the boundary conditionsA, k, q are known constants.Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).
The lattice properties of asymmetric hyperbolic regions
Authors: Cassels J. W. S. 
Published: November 2008

Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequalityfor any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.
Rectilinear plastic flow of a Bingham solid
Authors: Oldroyd J. G. 
Published: November 2008

It is apparent from the two previous papers of the same main title (1, 2) that velocity distributions in steady rectilinear plastic flow of a Bingham solid between moving boundaries are not easy to determine, even when the boundaries are of simple shape. In the familiar case of a Newtonian liquid (which can be regarded as a special case of a Bingham solid with zero yield value) the velocity ω is a harmonic function and can be obtained by a conformal transformation of the region of flow of the typeIn order to generalize and include materials of finite yield value, in which ω is not harmonic, one must first regard the Newtonian case from a slightly different point of view. The transformationmust be regarded as defining a change of coordinates, from Cartesian coordinates x, y to the natural curvilinear coordinates for a particular problem ω, W, one of which is the velocity. For a Bingham solid in general, it is shown in the present paper that natural coordinates ω, W exist, but are not obtainable by a conformal transformation from Cartesian coordinates, except in the limit when the yield value tends to zero. In practice it may be difficult to determine the natural coordinate system in a given problem, but when it is found the velocity distribution is automatically known.
The lattice properties of asymmetric hyperbolic regions
Authors: Cassels J. W. S. 
Published: November 2008

1. Letwhere a > 0, be an indefinite quadratic form, so that d = b2 − 4ac > 0. A classical theorem of Minkowski states that, if (x0, y0) is any pair of real numbers, there are numbers (x, y) congruent (mod 1) to (x0, y0), such thatand, more recently, Davenport has shown that this theorem can be sharpened for certain special f, for instance that it is always possible to satisfy
Experimental evidence for the theory of local isotropy
Authors: Townsend A. A. , Taylor Geoffrey 
Published: November 2008

Some new measurements of isotropic turbulence produced behind a biplane grid have been made at high Reynolds numbers, and these results are compared with the predictions of the theory of local isotropy developed by A. N. Kolmogoroff. The transverse double-velocity correlation has been measured at mesh Reynolds numbers up to 3·2 × 105, and the observed form agrees well with the predicted form. Measurements of the skewness factor of velocity differences over finite intervals have also been made, and the factor is nearly constant and equal to −0·38, if the interval is small compared with the integral scale. The invariance of dimensionless functions of the velocity derivatives has been confirmed for the flattening factor of ∂u/∂x, namely,which is nearly constant over a wide range of conditions. It is concluded that the theory of local isotropy is substantially correct for isotropic turbulence of high Reynolds number.
The definition of measure in function space
Authors: Pitt H. R. 
Published: November 2008

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).
Rayleigh's problem for a semi-infinite plate
Authors: Howarth L. 
Published: November 2008

The problem considered in this paper is the fluid motion arising from a thin semi-infinite plate started to move impulsively from rest (in viscous incompressible fluid at rest) with a velocity, subsequently maintained uniform, parallel to the edge. Two solutions are given, one obtained in polar coordinates is in the form of an infinite series, whilst the other, derived operationally in parabolic coordinates, leads to a single integral for the velocity distribution. The former is convenient for computation in the vicinity of the edge, the latter being more convenient elsewhere.A hypothesis originally introduced by Rayleigh after he had discussed the corresponding flow for an infinite plane has been used here to draw deductions about the steady flow past a quarter-plane whose leading edge is normal to the direction of flow and also to obtain approximate expressions for the effect of the edges on the skin friction of a sufficiently broad rectangle whose length is parallel to the incident stream.
The expected number of aggregates in a random distribution of n points
Authors: Mack C. 
Published: November 2008

ABSTRACTWhen n points are distributed at random (equally likely anywhere) over a large area or volume, they may form small clusters or aggregates. In this paper a k-aggregate is defined more precisely as k points which can be covered by a small figure of given size, shape and orientation, but whose position is not otherwise specified. Exact formulae for the expected number of such k-aggregates are given in some cases, while a good approximate formula (accurate in most practical cases) and limits are stated, covering all cases.
On a theorem of Zygmund
Authors: Kac M. 
Published: November 2008

Zygmund (1) proved in 1936 that if f(x) ∈Lp(−∞, ∞) (1 ≤ p ≤ 2), the Fourier transformexists for almost every λ, in the sense thatexists for almost every λ. The purpose of this note is to provide a short proof of this theorem.
On semi-groups in which xr = x
Authors: Green J. A. , Rees D. 
Published: November 2008

In this note we are concerned with the semi-group Snr, generated by n of its elements, in which each element x satisfies the equation xr = x, the semi-group being otherwise free. The main result proved is that the proposition(Ar) Snris finite for all finite n, is equivalent to the proposition(Br) the group Bn, r−1, generated by n of its elements, in which each element x satisfies the equation xr−1 = 1, the group being otherwise free, is finite for all finite n.
Algebraic integers with two conjugates outside the unit circle
Authors: Samet P. A. 
Published: November 2008

In this paper we consider algebraic integers, θ, such that θ and θ¯ (θ ╪ θ¯) are the only algebraic conjugates of θ that lie outside the unit circle.Let S1 be the set of algebraic integers, ρ, necessarily real, all of whose conjugates, except ρ, satisfy | z < 1.
Experiments with a rotating cylinder viscometer in liquid helium II
Authors: Hollis-Hallett A. C. 
Published: November 2008

ABSTRACTLiquid helium II was contained in the annular space between two co-axial cylinders, the inner of which was suspended by a torsion fibre while the outer was rotated at constant speeds. The torque upon the inner cylinder produced by the rotating fluid was measured for various steady velocities between 0·1 and 3 cm.sec.1, and was not found to be directly proportional to the velocity of rotation at any temperature between the lambda-point and 1·15° K. This result suggests that there must be some new type of non-linear frictional force acting in the liquid, possibly in addition to the Gorter-Mellink force of mutual friction.Extrapolation of the experimental results to zero velocity gives values of the coefficient of viscosity of the normal component which agree with the oscillating disk values between the lambda-point and about 1·6° K. At lower temperatures, the present results are significantly lower, suggesting, perhaps, that the values of the normal component density used in the analysis of the oscillating disk results were too low.
The calculation of transients in dynamical systems
Authors: Ward E. E. 
Published: November 2008

ABSTRACTThe paper shows that the calculation of transients by Tricomi's method using Laguerre functions is a practical alternative to the use of partial fractions. By taking numerical examples of Laplace transforms ranging from quadratic to sixth-power polynomials it shows that the approximation offered by this method would be acceptable for many practical uses. It analyses the composition of the coefficients of the Laguerre series, thus disclosing the conditions for quick convergence; for some transforms these conditions are not attainable.
The dimension of Cartesian product sets
Authors: Marstrand J. M. 
Published: November 2008

Given a plane set E, we denote by Ex the set of its points whose abscissae are equal to x.Throughout this paper we use the letters A and B to denote subsets of the x-axis and y-axis respectively, and we denote by A × B their Cartesian product set. We use the letters s and t to denote positive numbers; we denote by ΛsE the outer Hausdorff s-dimensional measure of the set E.
A note on commutators in free products. II
Authors: Griffiths H. B. 
Published: November 2008

The purpose of this note is to improve Theorem 1·1 of (1) to the following result:Theorem 1. Let H be the ordinary free product of a sequence of groups Gn, and let m be any integer > 0. Ifwhere each gi є Gi and is not the identity, then h is not equal to any product of fewer than m commutators from H.
On the Taylor coefficients of rational functions
Authors: Mahler K. , Cassels J. W. S. 
Published: November 2008

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power seriesThe problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.
The thermodynamics of bodies in static electromagnetic fields
Authors: Heine V. 
Published: November 2008

ABSTRACTThe thermodynamic behaviour of a body in a static electromagnetic field depends only on the field over the region of the body itself, whence its own thermodynamic behaviour is determined even in the presence of other polarizable media. Its behaviour depends only on the entropy density, for in fact no real energy density need exist. Also new formulae are developed for the energy and for the entropy of a body in electric and magnetic fields, including those due to permanent magnets.
The rheology of some two-dimensional disperse systems
Authors: Oldroyd J. G. 
Published: November 2008

ABSTRACTAn investigation is made of the deformation and flow properties of two-dimensional disperse systems consisting of small circular patches of one component widely dispersed in a continuous component with different rheological properties. Attention is restricted to small deformations (or small rates of deformation in the case of a fluid system) so that the equations of state are linear and the properties of each component can be characterized by two elastic moduli, or by two operators involving d/dt which take the place of moduli. Surface tension in each component and boundary tension between the components are taken into account, so that the theory can be applied to interfacial films at liquid-liquid or liquid-gas interfaces as well as to thicker sheets and films. General formulae are derived, expressing the two modulus operators of a disperse system in terms of those of the components, and their use is illustrated by means of examples.
Polar modules
Authors: Rees D. 
Published: November 2008

This paper is concerned with what, for lack of a better name, the author proposes to call geometric modules over a fieldk. These modules are defined as follows. A geometric moduleMconsists of a vector space, also denoted byM, overk, together with a ring of linear transformationsAofMinto itself subject to the following conditions:(i)Ais a homomorphic image of the ringSn=k[X1, …,Xn] of polynomials innindeterminatesX1, …,Xnfor some value ofn;(ii)M, considered as anA-module, is finitely generated.
Precompact linear operators in locally convex spaces
Authors: Ringrose J. R. 
Published: November 2008

This paper is concerned with the properties of precompact and compact linear operators from a locally convex Hausdorff space into itself, the field of scalars being the complex number field.The Riesz–Schauder theory for the equationswhere T is a compact linear operator from the Bacach space into itself, is well known (see, for example, Banach(1), Chapter 10, §2) and has been extended to the more general setting of locally convx Hausdorff spaces by Leray (4). In particular, the ‘Fredholm alternative theorem’ remains valid.
Double refraction phenomena in quantum field theory
Authors: Stephen M. J. 
Published: November 2008

ABSTRACTIt is shown that the phenomena of double refraction—optical activity, Faraday rotation, Kerr effect, etc.—may be treated as scattering problems in quantum field theory. An expression for the scattering cross-section of an atom or molecule for photons may be easily obtained and from this expression using the idea of Stokes operators, equating their expectation values to the classical Stokes parameters, the usual quantum mechanical expressions for optical activity, etc. may be obtained. This approach does not require the ideas of refractive index or polarization of the medium, and emphasizes what are the actual observables in these experiments.
The analysis of singularly loaded and rigidly clamped thin elastic slabs with curvilinear boundaries. I
Authors: Bassali W. A. 
Published: November 2008

ABSTRACTIn this paper complex variable methods are used to derive exact solutions in closed forms for the small deflexions of certain thin elastic plates due to transverse concentrated forces or couples applied at arbitrary or specified points. The isotropic plates considered are bounded by curvilinear edges of certain types along which the plates are rigidly clamped. Plates bounded by quartic curves having the forms of the inverses of an ellipse with respect to its centre or its focus are included as special cases.
On 3N points in a plane
Authors: Birch B. J. 
Published: November 2008

In this note I prove the following theorem:Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles will all have a common point.The proof depends on three lemmas; the first two are well known, and are quoted without proof; the third I believe to be new, and leads directly to the proof of the theorem.
Mutual Magnetic Interactions and Oscillator Strengths in the First Spectrum of Oxygen
Authors: Garstang R. H. 
Published: November 2008

ABSTRACTThe effects of the spin-other-orbit and spin-spin interactions on the energy levels of the 2p3 3s configuration of oxygen are investigated, and they are found to make an appreciable contribution to the fine structure. The interactions are shown to have a negligible effect on the strengths of all the stronger lines of the 2p4-2p3 3s array, and the more important of the weaker transitions are likewise little affected. The value to be adopted for the radial integral involved in the absolute oscillator strengths is discussed, and numerical values of the oscillator strengths are given.