bluuebunny/crossref_metadata_2025_split huggingface dataset:
On an urn problem of Paul and Tatiana Ehrenfest
Authors: Takács Lajos 
Published: November 2008

A short solution is given for the urn problem proposed by Paul and Tatiana Ehrenfest in 1907.In 1907 P. and T. Ehrenfest(3) proposed an urn model for the resolution of the apparent discrepancy between irreversibility and recurrence in Boltzmann's theory of gases (2). In this model it is assumed that m balls numbered 1, 2, …, m are distributed in two boxes. We perform a series of trials. In each trial we choose a number at random among 1, 2, …, m in such a way that each number has probability 1/m. If we choose j, then we transfer the ball numbered j from one box to the other. Denote by ξn the number of balls in the first box at the end of the nth trial. Initially there are ξ0 balls in the first box. If the trials are independent, then the sequence {ξn;n = 0, 1, 2,…} forms a homogeneous Markov chain with state space I = {0, 1, 2,…, m} and transition probabilities pi,i+1 = (m − i)/m for i = 0, l,…, m − 1, Pi,i−1 = i/m for i = 1, 2,…, m, and pi,k = 0 otherwise. The problem is to determine the transition probabilitiesfor i∈I, k∈I and n = 0,1, 2,….
Commuting flows and conservation laws for Lax equations
Authors: Wilson George 
Published: November 2008

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the formwhere each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equationwhich has a Lax representation with(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.
Selection Rules in Nuclear Radiation
Authors: Taylor H. M. 
Published: November 2008

AbstractIn the first part of this paper we discuss the radiation from a single charged particle moving in an arbitrary central field of force and obeying Dirac's equation. We consider the electric quadripole and magnetic dipole radiation as well as the electric dipole. We derive the selection rules for the magnetic dipole radiation and collect together for reference the corresponding selection rules for the electric dipole and quadripole radiations. In the second part we discuss the relative intensities of the various types of radiation, treating in detail the cases where the selection rules for magnetic dipole and electric quadripole are simultaneously satisfied. Finally we show that these results have an important bearing on the theory of internal conversion of γ-rays. The internal conversion of soft γ-rays occurs with such high probability that the theory is unable to account for the experimental results unless it is assumed that the radiation is largely magnetic dipole in character. On the other hand, Fisk and Taylor (loc. cit.) were unable to account for the presence of magnetic dipole radiation in appreciable amounts. We show that this is due to the fact that, of the two possible transitions (a and e of § 2) in which both magnetic dipole and electric quadripole radiation can be emitted, Fisk and Taylor considered only the second. In the case of the second, corresponding to a transition between two distinct terms, we show that Fisk and Taylor were correct in predicting a negligible amount of magnetic dipole radiation, but in the case of the first, corresponding to a transition between two levels of one multiplet term, we find that there is indeed a high percentage of magnetic dipole radiation.
A second note on the simple group of order 6048
Authors: Edge W. L. 
Published: November 2008

1. The simple group of order 6048 has a permutation representation of degree 28 and two of degree 63. These are conspicuous in SU(3,32), the representation as a group of 3-rowed matrices, unitary and of determinant + 1, whose elements all belong to GF(32), or to J as we name this field. When these matrices as linear transformations in the projective plane ϖ over J they leave invariant the unit Hermitian formand the single capital U will henceforth signify the group so acting. There are 28 isotropic points n in ϖ satisfying H3 = 0; they furnish the representation of degree 28. The other 63 points p in ϖ are non-isotropic and furnish one of the representations of degree 63. The other is furnished by 63 triangles T; the vertices of a T are all p and every pair of them is linked in the antipolarity Π set up by H3; if their coordinate vectors are y and z then y′z = 0. Each p is a vertex of 3 T. These matters are fully explained in (7). Those points linked by II to a given point A are collinear, on the polar of A; no p lies on its polar, every n does. The polars t of points n are called tangents of H3 = 0, and the only isotropic point on t is its pole. The polars s of points p are secants of H3 = 0: each consists of 4 n with 3 pairs of linked p and, given s, it is these 3 pairs that complete those T having a vertex at the pole of s.
On the one centre expansion of scattered distorted spherical and hyperspherical waves
Authors: Edelstein L. A. 
Published: November 2008

1. In many problems in theoretical physics one encounters situations in which it is desirable to be able to expand orbitals centred at one particle about another centre, frequently, but not always, another particle.There have long been available methods for permitting this to be effected for bound states but no such procedures are available for the continuum.Typical examples of physical situations in which such procedures are desirable are positron scattering from atomic and molecular systems, electron scattering from molecules, and nucleon scattering from light nuclei, to mention only a few.The author's interest arose from a desire to establish a general theory of slow electron scattering from molecules, and an account of such a theory will appear shortly (1).
The region of convergence of simple Taylor–Dirichlet series
Authors: Read G. A. 
Published: November 2008

In several recent papers G. L. Lunc (7, 8, 9, 10, 11) has examined the serieswhere {mn} is a sequence of positive integers or zeros, and μn > 0, μn ↑ ∞. The discussion has been aimed towards extending results which are well-known for Dirichlet series to series (1) which Lunc has very reasonably called a Taylor–Dirichlet series.
Concepts of differentiability and analyticity on certain classes of topological groups
Authors: Reid G. A. 
Published: November 2008

AbstractWe introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.
On Hausdorff's proof of the extended Riesz-Fischer theorem
Authors: Pollard S. 
Published: November 2008

1. The extended Riesz-Fischer theorem, in the special case of the trigonometrical orthogonal system, states that, if a and β are real numbers such thatthenimplies, whenever α ≤ 2, the existence of a solution of the system of integral equations,such thatand furtherWhen a = 2, this reduces to the original Riesz-Fischer theorem.
Some factors in the design of hot cathode X-ray tubes for steady running
Authors: Harper W. R. 
Published: November 2008

Some general considerations regarding the design of hot cathode X-ray tubes for producing an accurately constant and repeatable beam of X-rays are first given. The construction of a tube in accordance with these considerations is then described. This tube has been used to show that except with small tube-currents the quality of the vacuum in such a tube has little influence on the steadiness of the beam if the electrical input to the tube can be satisfactorily controlled.It is shown that the X-radiation associated with the fluorescence often observed in X-ray tubes has usually but not always a negligible effect on the steadiness of the beam.Some factors limiting tube current when the filament is enclosed in a box containing a focussing slit are mentioned, and the mechanism of formation of the focus is discussed with reference to its size. The use of an auxiliary electrode to increase the tube current and improve the focussing is also discussed.
The number of α-particles emitted by thorium C + C′
Authors: Watson S. W. , Henderson M. C. 
Published: November 2008

The number of α-particles emitted per second by thorium (C + C′) has been determined by an ionisation method. The number obtained is 4·26 ± ·08 × 1010 α-particles per second per curie equivalent γ-ray activity when in equilibrium with radio-thorium and when measured by the γ-rays of thorium C″ through 18 mm of lead. The result agrees within 1 per cent, with that extrapolated from Shenstone and Schlundt's values.The same apparatus was used to determine the slope of the Bragg curve over the first three centimetres of the range. The data fit the curve as given by I. Curie and Behounek within the accuracy of experiment. The curve given by G. H. Henderson falls too rapidly as it approaches the axis of ordinates.
The theory of ionisation measurements in gases at high pressures
Authors: Lea D. E. 
Published: November 2008

The columnar theory developed by Jaffé to account for the recombination of ions in alpha particle tracks is extended to beta rays by taking account of the clusters of secondary ionisation. Reasonable agreement is obtained with experiment. Recombination in proton tracks produced in hydrogen by neutrons is shown to be in agreement with the columnar theory, but in the case of nitrogen nuclear tracks in nitrogen the recombination is only a hundredth of that predicted by the theory. An explanation of this effect is advanced, and it is suggested that recombination is likely to be abnormally small for all heavy nuclei of velocities not exceeding 5 × 108 cm. per sec.An experimental determination of the coefficient of recombination of ions in nitrogen and hydrogen at pressures of 20, 40 and 90 atmospheres is reported.My thanks are due to Dr Chadwick for interest in this work, and to Dr Gray and Dr Tarrant for advice on the experimental technique of high pressure ionisation measurements. I am indebted also to the Department of Scientific and Industrial Research for a maintenance grant.
A Helical Method of focussing β-Rays
Authors: Tricker R. A. R. 
Published: November 2008

Experiments with a helical method of focussing β-rays are described. Although the method does not appear to be so generally useful for obtaining ordinary β-ray spectra, it possesses certain advantages, of which may be mentioned:(1) Large dispersion;(2) Greater intensity;(3) Good screening.The method produces some evidence on the question of the continuous background, and experiments with it in this connection are described.
Note on Pauli's Exclusion Principle
Authors: Ursell H. D. 
Published: November 2008

A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the formin nature, but only of the formHence σ, τ,…ω must be all distinct.
The Real Representation of the Commutator S−1 T−1 ST in Four Dimensions
Authors: Robinson G. de B. 
Published: November 2008

It has been remarked in a former paper that perhaps a fuller understanding of the theory of linear groups may be arrived at by considering the real representations. It is sufficient that these be irreducible in the real field. In the present paper we continue this investigation and deal with the angles of rotation; in particular we find the form of the commutatorwhere S and T are substitutions of order p1 and p2 and have angles of rotation θ, θ′, θ″, … and φ, φ′, φ″, …. If certain conclusions regarding the orders of S, T, and C may be drawn we shall then be able to attack a very important problem—;that of Primitivity. This is in essence very similar to the method of Blichfeldt, since the commutator is the product ofwhich are both of order p2.
The structure of a β-ray line by semicircular magnetic focusing
Authors: Li K. T. 
Published: November 2008

The actual distribution of a homogeneous group of electrons by semicircular magnetic focusing is calculated for different types of source.The dependence of maximum intensity on the radius of curvature of the path of electrons is investigated and the approximate power laws assigned to different cases. The results can be used in the measurement of the intensity of β-ray line spectra.Tables of numerical values of effective solid angle subtended by the photographic plate at the source are given for some special cases.The results of the above calculations have been verified experimentally.
The azimuthal dependence of processes involving mesons
Authors: Chang T. S. 
Published: November 2008

The elastic scattering of Yukawa's particles, or mesons, by a static electric field is recalculated by considering the scattering as a process in which the incident meson is absorbed and another meson, i.e. the scattered meson, is created, and Laporte's result is confirmed. This method promises an easier handling of the second approximation.A similar dependence of the angular distribution of the emitted quanta upon the spin of the incident mesons in a process like Y− + P → N + hν is found. Thus, the angular distribution contains a dependence upon the azimuthal angle in general. For the reverse process, the angular distribution of the emitted mesons, both longitudinal and transverse, contains also a dependence upon the azimuthal angle, which disappears only when the incident light is circularly polarized.The writer wishes to thank Prof. W. Pauli for calling his attention to the azimuthal dependence in the problem of elastic scattering and Dr C. Møller for help given during the preparation of the paper.
On the subadditivity of the transfinite diameter
Authors: Schiffer Menahem 
Published: November 2008

Let M be a bounded and closed set of points in the complex z plane; d(M), a set-function which is of great importance in potential and function theory, may then be defined as follows. n points z1, z2, …, zn in M are so chosen that the product of the mutual distanceshas the greatest possible value Then it can be proved thatexists. Thus the set-function d(M), named by Fekete the transfinite diameter of M, is defined.
On surfaces of sectional genus six
Authors: Roth Leonard 
Published: November 2008

In two previous notes I have given the various types of non-singular surfaces with sectional genera 4 and 5. In this paper the methods previously adopted are applied to the non-singular surfaces of sectional genus 6. For purposes of classification these may be divided into the four classes:(1) Regular, non-rational.(2) Rational.(3) Irregular, non-referable.(4) Referable.
A note on molecular orbitals
Authors: Devonshire A. F. 
Published: November 2008

A method of obtaining an approximate solution of Schrödinger's equation with a non-central field of force is given for the case when a solution of a similar equation involving the angular coordinates only is known.In conclusion I should like to thank Prof. J. E. Lennard-Jones, F.R.S., for suggesting the problem and for his continued interest in it, and also the Department of Scientific and Industrial Research for a grant.
The generalized Fermi interaction
Authors: Hoyle F. 
Published: November 2008

Konopinski and Uhlenbeck have shown that the empirical β-ray spectra seem to require a modification of Fermi's original expression for the interaction leading to β decay‡. Their expression includes a derivative of the neutrino wave function. The purpose of the present paper is to investigate the most general type of such an interaction term and the possibility of introducing second-order derivatives. We have to deal with the transition§neutron + neutrino → proton + electronor proton + anti-neutrino → neutron + positron.The whole system has the equations of motionwhere ψ1 and ψ2 are the wave functions of the initial and final states in configuration space, H1 and H2 are the Hamiltonians of the particles in the initial and final states, and A is a small interaction operator.
The effect of temperature on the reflexion of X-rays
Authors: Blackman M. , Fowler R. H. 
Published: November 2008

1. The reflexion of X-rays from an ideal crystal would be independent of temperature if the particles forming the crystal were at rest. Because of the motion of the particles a temperature effect does exist. It can easily be seen that only the component of the displacement at right angles to the reflecting plane is of importance, since it is the path difference between such planes which matters. The complete theory has been worked out by Debye and by Waller. The result is that, owing to the influence of the temperature, the intensity is multiplied by a factor e−2m, whereHere and μs is the mean square of the component of the displacement (in a direction at right angles to the reflecting planes) due to a normal mode π is the glancing angle of incidence of the X-rays on the reflecting planes, and λ is the wave-length.
The penetrating power of cosmic-ray shower particles
Authors: Jánossy L. 
Published: November 2008

The absorption curve for cosmic-ray shower particles has been observed in lead and iron with a counter arrangement. Since the observed curves are in good agreement with the absorption curve of cosmic ray electrons calculated by Heitler, it has been concluded that nearly all the showers are of the cascade type.Observations of Schmeisser and Bothe which have previously been interpreted as giving experimental evidence for the occurrence of penetrating particles in showers are not in contradiction with the observations given here, so that they also must be interpreted in terms of energetic electron cascades.
On the generalized helices of Hayden and Sypták in an N-space
Authors: Wong Yung-Chow , Hodge W. V. D. 
Published: November 2008

In this paper we are concerned with curves (C) of the following types:where k1, k2, …, kh−1, kh = 0 (h ≤ n) are the curvatures of (C) relative to the space Vn in which (C) lie. Hayden proved that a curve in a Vn is an (A)2m, h = 2m + 1, if and only if it admits an auto-parallel vector along it which lies in the osculating space of the curve and makes constant angles with the tangent and the principal normals. Independently, Sypták∥ stated without proof that a curve in an Rn is a (B)n if and only if it admits a certain number of fixed R2's having the same angle properties; he also gave to such a curve a set of canonical equations from which many interesting properties follow as immediate consequences.
The influence of radiation damping on the scattering of light and mesons by free particles. I
Authors: Heitler W. 
Published: November 2008

1. In the classical theory there is no difficulty in treating the effect of radiation damping on the scattering of light by a free electron in so far as it is a result of the conservation of energy. In the non-relativistic approximation the equation of motion of a free electron under the influence of a light wave iswith the periodic solution The total energy radiated per second is thenand the total cross-section(1) isFormula (1) differs from the Thomson formula by the factor 1/(1 +κ2). This factor becomes appreciable for energies ħν ≥ 137mc2.
On Hausdorff's methods of summability
Authors: Rogosinski W. W. 
Published: November 2008

The aim of this paper is to give a general theory of the ‘strength’ of Hausdorff methods of summability. These methods are defined by the linear transformsof a sequence (sk). Here the μk form a given sequence of real or complex numbers and the Δp(μk) denote their differences of order p; i.e. Δ0(μk) = μk and
On a class of linear integro-differential equations
Authors: Pitt H. R. 
Published: November 2008

This paper is a sequel to a previous one(1) with the same title which dealt with the general solution of equations of the typeWe consider here the more general equationwhere g(x) is a given function. We are interested particularly in the existence and uniqueness of solutions of the latter equation and show how these are related to the closure and completeness properties of sets of functions {xneωnx} derived from the kernels kr(y).
Inequalities for the derivatives of a bounded harmonic function
Authors: Verblunsky S. 
Published: November 2008

If h(r, θ) is harmonic in the unit circle | r | < 1 and satisfies the condition | h | ≤ 1, then there is a function u(ø) which satisfies | u | ≤ 1 such thatand conversely. Hence, any properties of such harmonic functions should be deducible from equation (1). A number of such properties have been proved by Koebe (Math. Z. 6 (1920), 52–84, 69), using Schwarz's lemma and the geometry of simple conformal transformations. They can be deduced from (1) together with an elementary lemma on the rearrangement of a function (Lemma 1 below). As, however, students of this subject will regard Koebe's method as the one best adapted to establish his theorems, we shall illustrate the alternative method by considering two new problems, namely to find max ∂h/∂r, max ∂h/∂θ, where the maximum in each case is taken for all harmonic functions h which satisfy
The disintegration of boron by slow neutrons. II
Authors: Gilbert C. W. 
Published: November 2008

A further study has been made of the disintegration of boron by slow neutrons, B10 (n, α) Li, in the cloud chamber. The total ranges of the α-particles and the lithium particles were measured and two groups found. 91·4 % of the disintegrations form the main group, which was previously measured, and 8·6 % a long-range group corresponding to the lithium being left in the ground state. The results of the previous measurements are corrected for the variation of stopping power with velocity and the ranges in standard air of the particles of the two groups are now given as 7·7 mm. α-particle and 4·8 mm. lithium particle, total 12·5 mm. for the main group, and 9·3 mm. α-particle and 5·6 mm. lithium particle, total 14·9 mm. for the long-range group.These results, taken together with those of other workers, make it certain that in about 92 % of disintegrations the lithium is left in an excited state of about 0·47 MeV. energy and in the remaining 8 % in the ground state. The energies calculated from this scheme by using the known masses do not agree with those deduced from the measured ranges and the range-energy relation for a-particles. It is suggested that the rangeenergy relation is at fault and that these measurements provide two experimental points to which a range-energy relation should be fitted.