bluuebunny/crossref_metadata_2025_split huggingface dataset:
An extension of a theorem of Kulakoff
Authors: Easterfield T. E. 
Published: November 2008

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:(1) The dihedral group, of order 2k, generated by A and B, where(2) The quaternion group, of order 2k, generated by A and B, where(3) The "mixed" group, of order 2k, generated by A and B, where
The dynamics of a simple system. I. Observables
Authors: Powell F. C. 
Published: November 2008

Any particular form of mechanics (e.g. classical or quantum) makes use of a particular ‘formalism’ or set of rules governing the use of the symbols representing dynamical variables and the symbols (such as +, =) representing relations between dynamical variables. In the present paper an attempt is made to examine the physical content of this formalism, particularly that of quantum mechanics. This is done by building up a formalism as the direct expression of a number of physical postulates having direct operational meaning. It is shown that, in a simple case, with any two observables A and B can be associated a unique sum observable A + B and a unique (real) symmetric product observable A.B (= B.A).It is next shown that, if, like a classical Hamiltonian system, the system has the property that the time rate of change of an observable depends linearly on some observable H when the environment in which the system moves is varied, then a (real) skew product observable A × B (= − B × A) can be defined.Finally, if the equations of motion are ‘of second order', it is possible to express the second rate of change of any observable as an algebraic function of that observable and H. This leads to algebraic identities from which it follows that ‘complex multiplication’, defined by AB = A.B + iA × B, is associative (but not commutative). Observables are thus shown to possess the properties usually ascribed to them in quantum mechanics. These properties make possible a representation by Hermitian matrices.
Notes on the theory of series (XXIV): a curious power-series
Authors: Hardy G. H. , Littlewood J. E. 
Published: November 2008

1. This note originates from a question put to us by Mr W. R. Dean concerning series of the typewhere θ is irrational. Series of the typeare familiar: it is well known, for example, thatmay have any radius of convergence from 0 to 1 inclusive, according to the arithmetic nature of θ. It is natural to ask how this radius is connected with that ofand, more generally, how those of (1·1) and (1·2) are connected.
A note on the Petzval field curvature in electron-optical systems
Authors: Goddard L. S. 
Published: November 2008

1. The Petzval field curvature produced in a compound glass optical system of axial symmetry is given by a well-known formula. If the system consists of a number of media, of refractive indices n0, n1, n2, …, having spherical faces whose radii of curvature are r0, r1, r2, …, the formula is [see for example, (1) or (2)]and this is in common use amongst those concerned with the design of optical equipment. The analogous integral expression in the electron optical case has not, however, received the attention it deserves, in spite of the developments in electron optics during the past decade. In 1935 Glaser(3) presented the third-order error theory of an axially symmetric electron optical system.
Collective electron assemblies in a metal with overlapping energy bands
Authors: Band William 
Published: November 2008

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equationswhere kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.
Notes on Fourier series (IV): Summability (R2)
Authors: Hardy G. H. , Rogosinski W. W. 
Published: November 2008

1.1. There are two familiar methods of summation of divergent series usually called the methods (R, 1) and (R, 2). 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.
Some remarks on Schottky's theorem
Authors: Hayman W. K. 
Published: November 2008

1. We denote by the class of all functions regular in |z| < 1, and never taking the values 0 or 1 in |z| < 1. Then Schottky's(1) theorem states:Theorem A. Let f(z) = a0 + a1z + … belong to . Thenand hencewhereand the constantsK(a0), K(a0,ρ) depend only on the quantities indicated.
Notes on the molecular orbital treatment of linearly coupled systems
Authors: Coulson C. A. , Rushbrooke G. S. 
Published: November 2008

It is well known (1) that for an unsaturated hydrocarbon of the type CnHn+2 (where n is the number of carbon atoms in the linear chain), the theory of molecular orbitals applied to calculate the binding energy of the mobile π-electrons leads to the secular determinant, of continuant type,where β is the resonance integral between adjacent carbon centres and x stands for the quantity generally written E0 − E, E being the energy of a mobile electron, which we seek to calculate. The equation for the allowed energies E is, of course, simply Pn = 0.
The relation between different types of Abel summability
Authors: Kuttner B. 
Published: November 2008

1. Let {μn} be a sequence of positive numbers increasing to infinity. If the seriesconverges for s > 0, and if f(s) → l as s → 0, then ∑an is said to be summable (A, μn) to l; this generalization of Abel summability, first introduced by Hardy, has long been familiar. Suppose that, for x ≥ μ0, ø(x)is a positive increasing function of x, which tends to infinity as x → ∞. We write throughout λn = ø(μn), andif this exists.
On a class of probability distributions
Authors: Chandrasekhar S. 
Published: November 2008

1. The statement of the problem. A problem which arises in certain physical (5) and astronomical (1, 3) contexts relates to the probability distribution of the infinite sumwhere the rj's are the position vectors from a fixed origin of an infinite uniform Poisson distribution of points with a given constant mean space density N. While certain special cases of the sum (1) have been considered in the literature (cf. (2)) the study of the general case discloses properties of a class of probability distributions which appear to have some interest. For example, we shall show that the probability distribution of F exists only for that the moments of F (= |F|) exist only for orders 0 ≤ p < 3/n, and finally that an explicit formula can be given for all the finite moments.
Numerical integration by systematic sampling
Authors: Moran P. A. P. 
Published: November 2008

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sumfor some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.
Space-time on the rotating disk
Authors: Kurşunogğlu Behram 
Published: November 2008

1.Introduction.In a recent paper Clark (1) has dealt with the problem of the rotating disk, the material of which is such that the waves of dilatation in this particular material travel with the velocity of light. The material of the disk is supposed to be under an isotropic stresspwhen in a strained state, and the relation between stresspand the dilatation Δ is found to be connected by an expressionwherea= density in the unstrained state, and Δ is given byUi(i= 1, 2, 3) are the components of the strain.
Tables for numerical integration at non-equidistant argument steps
Authors: Hartley H. O. 
Published: November 2008

1. Published tables facilitating the numerical evaluation of the definite integralfrom given numerical valuesf(xi) of the integrand in the form of a weighted sumessentially deal with the following situations:(a) The argumentsxiare equidistant, i.e.xi=x0+iw;i=integer and the weight coefficientsciare tabulated for all argumentsxioccurring in the summation. Among these tables the most important are:(a1) panels in which the end-points of the integration a andbare two of thexi, and in particular(a11) panels confined to the range of the integration in which(a12) panels extending beyond the range of integration which usually have the same number of arguments,k, below a and beyondb, so that(a2) panels in which a and b are at distance ½wfrom one of thexi, and in particular(a21) panels confined to the range of integration in which(a22) panels extending beyond the range of integration which usually have the same number of (k) points belowaand beyondb, so that(b) Thenargumentsxiare not equidistant but are specially chosen so that for a givennthe sumSapproximates toIwith highest precision. Gauss's formula (see, for example, Milne-Thomson (4)) and various formulae of approximate product integration (see, for example, Beard (l)) are examples of these. In Chebyshev's integration formula (see, for example, Runge and König (5)) thexirequired for equal weights ctare used, resulting in an important formula when the ordinatesf(xi) are affeected by observational errors of equal variance.
Overlapping energy bands and the theory of collective electron ferromagnetism
Authors: Lidiard A. B. 
Published: November 2008

ABSTRACTThe theory of collective electron ferromagnetism given by Stoner applies to a system of electrons in a single energy band; for iron, cobalt and nickel this is identified with the band of states derived from atomic 3d functions. To bring in the generally assumed overlapping of the 3d band by the wide 4s band in these metals, the theory has been extended to take account of the transfer of electrons from 3d to 4s states with change of temperature. A previous calculation of this transfer effect must be regarded as inadequate, for the part played by the exchange energy in determining the distribution of electrons between the two sets of states was omitted. The general equations are derived in § 2 and used as a basis for discussion of the properties of nickel-copper alloys at absolute zero in § 3. In §§4 and 5 numerical results are presented which show the effect of the overlapping 4s band on the magnetic properties of a system such as nickel both above and below its Curie point. Comparison with the measured paramagnetic susceptibility of pure nickel above the Curie point strongly suggests that for this metal the overlapping 4s band has only a minor influence, although in principle the effect could be very large (cf. Fig. 4). This result is not unambiguous, however, because values thus inferred for the two unknown parameters lead to inaccurate predictions below the Curie point. First, the predicted values for the spontaneous magnetization are too small. Secondly, the theory demands that the nickel-copper alloys should only be ferromagnetic below a copper content of about 20 %, whereas experimentally the limit is known to be about 60 %. In conclusion, it is suggested that the implicit assumption of Stoner's theory that the exchange integrals between all pairs of 3d states are equal to one another is a more serious restriction on the theory than the consideration of only a single energy band.
The isotherms of an imperfect gas
Authors: ter Haar D. 
Published: November 2008

ABSTRACTThe liquid drop model of an imperfect gas in the form introduced by Wergeland is discussed by using the method of the grand ensembles and the equation of state of the system is derived. This equation of state is of the same general type as the one derived by Mayer for a more general model. It is shown that in both cases the isotherms consist of two analytically different parts in the limit where the number of particles in the system, N, goes to infinity.
A case of gaseous accretion
Authors: Dodd K. N. 
Published: November 2008

ABSTRACTThe isothermal flow of gas past a gravitating point-sink has been studied for four values of the undisturbed gas velocity, all of which are supersonic. From these studies, estimates have been made of the rates of accretion of an isothermal gas by a star moving with a supersonic velocity. These estimates are of the same order as those obtained by earlier workers who used simpler mathematical models.
A free-electron theory of conjugated molecules
Authors: Griffith J. Stanley 
Published: November 2008

ABSTRACTThe values of a free-electron eigenfunotion at the carbon nuclei of a conjugated hydrocarbon are found to satisfy a system of algebraic equations. These equations are similar in form to those obtained in the method known as the linear combination of atomic orbitale but only coincide with them for linear polyenes and benzene. The symmetry, degeneracy and energy of the eigenvectors of these free-electron equations correspond exactly to those of the free-electron wave functions found by the usual methods. From this correspondence, a theorem is deduced about the free-electron charge density in alternant hydrocarbons.
Self-consistent electrodynamics
Authors: Buneman O. 
Published: November 2008

ABSTRACTThe idea of direct action between streams is applied to a continuous charged fluid and combined with the new formulation of the electrodynamical laws of motion in terms of conservation of circulation. A simple and rigorous integrated formulation is thus obtained from the Maxwell-Lorentz differential equations, applicable to co-existing positive and negative fluids, as well as vacuum. Exact solutions are obtained, among them one which represents self-consistent, self-maintained flow in a hollow tubular region of infinite axial extent. It is hoped this tube might be bent into a torus and that an electron model will result from merely quantizing the one or two vortices around which this flow-pattern circulates.
The concept of order I. The space-time structure
Authors: Bastin E. W. , Kilmister C. W. 
Published: November 2008

ABSTRACTEven if ideas of measurement did not exist, a great deal of the simple mechanical and electromagnetic structure of the physical world could be understood. To see how this can be so, the orthodox ideas of the relationship of mathematics to physics are inadequate. In this paper a development of this fundamental non-metrical physics is made to depend on a different view of this relationship.
Bounds for the integral of a non-negative function in terms of its Fourier coefficients
Authors: Longuet-Higgins M. S. 
Published: November 2008

ABSTRACTThe first 2N + 1 Fourier coefficients of an unknown, non-negative function f(θ) are given, and it is required to find bounds for ∫Ef(θ) dθ, where E is some given region of integration. We also wish to find the interval E for which the bounds are most strict, when the width of E is specified. f(θ) may represent a distribution of energy in the interval 0 ≤ θ ≤ 2π; the object is to determine where the energy is chiefly located.In the present paper we show that if the energy is located mainly in the neighbourhood of not more than M distinct points, significant lower bounds for ∫Ef(θ) dθ can be found in terms of the first 2M + 1 Fourier coefficients. The effectiveness of the method is illustrated by applying the inequalities to some known functions.The results have application in determining the direction of propagation of ocean waves and other forms of energy.
Functions regular in the unit circle
Authors: Linden C. N. , Cartwright M. L. 
Published: November 2008

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.
The real zeros of the confluent hypergeometric function
Authors: Slater L. J. 
Published: November 2008

This paper contains a discussion of various points which arise in the numerical evaluation of the small real zeros of the confluent hypergeometric functionwhereThere are two distinct problems, first the determination of those values of x for which M(a, b; x) = 0, given a and b, and secondly the study of the curves represented by M (a, b; x) = 0, for fixed values of x. These curves all lie on the surface M(a, b; x) = 0, of course.
Transverse bending of a thin circular plate loaded normally over an eccentric circle
Authors: Bassali W. A. 
Published: November 2008

ABSTRACTComplex variable methods have been applied to isotropic and aelotropic plate problems by several authors. The notation used here is that of Stevenson(14). Dawoud(5) has expressed the continuity conditions between two differently loaded regions in terms of the complex potentials and the particular integrals for the two regions.The problem of a transverse load at any point of a clamped circular plate was solved by Clebsch(4), Michell(11), Melan(10) and Flügge(6). A series solution for the simply supported circular plate under the same load was given by Foeppel(8). Using Stevenson's tentative method Dawoud(5) applied complex potentials to solve the problem of an eccentric isolated load under certain boundary conditions. Applying Muskhelishvili's method, Washizu(15) obtained the same results for clamped and simply supported boundaries.It is easy to get solutions for a circular plate concentrically and uniformly loaded. For non-uniform loadings there are the solutions found by Sen (13) for certain distributions of normal thrust over the complete plate or over a concentric circle and the solution of Flügge (7) for a linearly varying load over the simply supported circular plate. The present author and Dawoud(3) obtained the solutions for a circular plate with the load over the complete plate or over a concentric circle, under a general boundary constraint including as special cases the usual clamped and hinged boundaries. Ghose (9) worked out the problem of a clamped circular plate when the load is uniformly distributed between two concentric circles and two radii. Schmidt (12) found the solution for a clamped circular plate uniformly loaded over an eccentric circle. The complex variable method was applied by the author and Dawoud(2) to obtain the solutions for a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under the general boundary condition mentioned before. The author (1) also found the solution for a linearly varying load over an eccentric circle under the same boundary condition. In this paper the power of the complex variable method is exhibited by rinding the appropriate complex potentials corresponding to the loadover an eccentric circular patch, where R, θ are measured from the centre of the patch and the common diameter of the plate and the patch. Since the two cases n = 0, 1 require special consideration and were dealt with separately (in (2) and (1) respectively), we see that this paper completes the solution of the problem of a circular plate with an eccentric circular patch symmetrically loaded with respect to the common diameter of the plate and patch, the load being in this case expressible in the form .For a clamped boundary the solution is obtained in finite terms.
On a scale of Abel-type summability methods
Authors: Borwein D. 
Published: November 2008

1. Introduction. In this note Abel-type summability methods (Aλ) are defined and some of their properties investigated.Letand let {sn} be any sequence of numbers. Ifis convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we shall say that the sequence is Aλ-convergent to s and write sn → s (Aλ;). The A0 method is the ordinary Abel method.
The static spherically symmetric solutions in a unified field theory
Authors: Moffat John 
Published: November 2008

ABSTRACTA brief account is given of the fundamental properties of a new generalization ((1), (2)) of Einstein's gravitational theory. The field equations are then solved exactly for the case of a static spherically symmetric gravitational and electric field due to a charged particle at rest at the origin of the space-time coordinates. This solution provides information about the gravitational field produced by the electric energy surrounding a charged particle and yields the Coulomb potential field. The solution satisfies the required boundary conditions at infinity, and it reduces to the Schwarzschild solution of general relativity when the charge is zero.
Some self-reciprocal functions and kernels
Authors: Rao V. V. L. N. 
Published: November 2008

1. The object of this note is to deduce some self-reciprocal functions and kernels for Hankel transforms. We denote by Rμ the class of functions which are self-reciprocal for Hankel transforms of order μ; that is, those that satisfywhere Jμ(x) is a Bessel function of order μ. In particular we write R8 for R½ and Rc for R−½.
Two bodies at rest in general relativity
Authors: Marder L. 
Published: November 2008

ABSTRACTIt is well known that there is no static axisymmetric two-body solution of Einstein's gravitational field equations, if it is assumed that the bodies are separated in a certain definite sense. In this paper it is shown, by the construction of a complete physically sensible model, that static two-body solutions do exist for systems in which one body is hollow and contains the other. The stability of the particular system described is briefly discussed.
The distribution of the sizes of images reflected in a random surface
Authors: Longuet-Higgins M. S. 
Published: November 2008

It has been shown recently that a fruitful method of studying the formation of waves on the sea surface is through the average intensity of reflected sunlight, as seen from different angles ((3), (4)). This gives, in effect, the statistical distribution of the components of surface slope.Following the same line of thought, we may inquire what information could be derived from the distribution of the sizes of the reflected images of the sun. The size of an image depends essentially on the total curvature of the surface at the point of reflexion (see § 2).