bluuebunny/crossref_metadata_2025_split huggingface dataset:
The roots of Yn(λη)Jn(λ) – Jn(λη) Yn(λ) = 0
Authors: Chandrasekhar S. , Elbert Donna 
Published: November 2008

Cylinder functions, (λx), of integral orders which vanish at x = 1 and x = η (where η is an assigned positive constant less than 1) occur in the solution of many problems in applied mathematics. Such functions can be expressed in terms of the Bessel functions Jn(x) and Yn(x) of the two kinds in the formwhere λ is a root of the equation
Physical properties of particles obeying generalized statistics
Authors: McCarthy I. E. 
Published: November 2008

ABSTRACTThe Feynman probability amplitudes for interactions between spin ½ fields and fields of integral spin quantized according to the generalized method of Green are investigated. It is shown that the scattering amplitudes are independent of the method of quantization though the amplitudes of the vacuum fluctuations are not. The question of distinguishing particles satisfying the generalized statistics, if they exist in nature, from Fermi or Bose particles is considered. It is shown that the statistical thermodynamics of the particles depends on the statistics, but as this requires a large number of particles in thermodynamical equilibrium, it offers no means of distinction for particles with short lifetimes. In practice, an examination of particles in bound states would identify particles obeying generalized statistics.
A differential equation for undamped forced non-linear oscillations. I
Authors: Morris G. R. 
Published: November 2008

In this paper and its sequels I consider the differential equationin which e(t) is an even periodic function of t and dots denote differentiation with respect to t. We can regard (1·1) as a special case of the equation(the ‘differential equation of non-linear oscillations’), which describes the motion on the x–axis of a particle of unit mass subject to a restoring force g(x), a variable damping force f(x, ẋ)ẋ and an external force p(t); the dynamical description used in the title appeals to this interpretation. The most interesting problems on (1·2) and its specializations concern the behaviour of solutions as t → ∞ for example, we may seek conditions on f, g and p which allow us to assert that some or all solutions are bounded or, having assumed that p(t) is periodic, seek conditions on f and g which allow us to assert the existence of periodic solutions.
The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables
Authors: Cox D. R. 
Published: November 2008

ABSTRACTCertain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.
Molecules with tetrahedral and octahedral symmetry
Authors: Ballinger R. A. , March N. H. 
Published: November 2008

ABSTRACTDetailed energy calculations for CH4, SiH4 and GeH4 have been carried out using the Thomas-Fermi approximation together with a nuclear field which is averaged over all orientations about the central nucleus. The energy, including exchange, has been calculated using the unmodified Thomas-Fermi electron densities as variation functions, and in this way bond lengths and force constants for the three molecules have been obtained. The effect of exchange is very marked, agreement with experiment being quite reasonable when it is included in this way.In order to test the accuracy of this procedure, a scale factor has been introduced and used as a variational parameter. In this refined approximation, calculations have been made for SiH4, thus enabling improved charge distributions to be obtained. Energy curves are also presented for this molecule, showing separately the variations of kinetic, potential and exchange energies with internuclear distance. It is shown that whilst the scale factor has an appreciable effect on the separate energy curves, the total energy is virtually unchanged.Finally, the reason why the exchange energy is of considerable importance in the theory is briefly discussed, and some observations are made concerning the rather abnormal bonding in GeH4.
The stability of a homopolar dynamo
Authors: Bullard Edward 
Published: November 2008

ABSTRACTThe stability of the self-exciting disk dynamo is considered. If there is no electrical load in parallel with the field coil and no friction at the axle, the dynamo can perform oscillations of constant amplitude about its state of steady motion. Viscous forces on the axle cause the dynamo to settle to a steady motion. A parallel load may have an effect similar to that of viscous forces or may cause the oscillations to grow without limit. With both a parallel load and viscous forces the amplitude of the oscillations is bounded.Possible analogies between these results and the motion of a body of electrically conducting fluid in a magnetic field are discussed. The main applications are astronomical and geophysical.
A property of Hausdorff measure
Authors: Davies Roy O. 
Published: November 2008

From the fact that Hausdorffs-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result:TheoremA.If {En} is any increasing sequence of sets, then∧sEn→ ∧s(ΣEn)as n→ ∞.Since ∧sXis denned (for every setX) as, the problem arises whether for every positive δ and every increasing sequence of sets one can proveNow there are four possible ways of defining; it is the lower bound oftaken over coverings ΣUrofXby convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be (i) open and of diameter less than δ, (ii) closed and of diameter less than δ, (iii) open and of diameter not exceeding δ, or (iv) closed and of diameter not exceeding δ.
Molecules with tetrahedral and octahedral symmetry
Authors: Banyard K. E. , March N. H. , Coulson C. A. 
Published: November 2008

ABSTRACTThe modified Thomas-Fermi method reported in (2) has been applied numerically to calculate the distribution of valence electrons in CC14. Slater wave functions have been used to describe the Cl inner-shell electrons, whilst a description of the C ls electrons based on the Thomas-Fermi model for atoms has been adopted. The electron distribution thus found is compared with the two approximations discussed in part I of this series (3). The first of these, in which valence and inner-shell electrons are treated in the same way, is shown to give a distribution of valence electrons in good agreement with the present results. The second approximation, that of effective nuclear charges, is less satisfactory and could only lead to a distribution of valence electrons in reasonable agreement with the present work if the effective nuclear charges were considerably reduced. This seems to throw some light on the results of Bowers for the force constants of tetrachlorides and tetrabromides.The overall distribution of electrons obtained here seems to be the inevitable result of assuming the valence electrons to move in any plausible central field. It is, of course, by no means certain that such an approximation is near to the truth. In fact, the results of (3) and the present paper leave little doubt that the electron cloud obtained by such an approximate method will be too diffuse. This seems at least partly responsible for our finding that rather more than 20 of the 28 Cl valence electrons lie outside the sphere drawn through the Cl nuclei.
Statistical dynamics of multiply-periodic systems
Authors: Born M. , Hooton D. J. , Mott N. F. 
Published: November 2008

ABSTRACTClassical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.
Approximation to transients by means of Laguerre series
Authors: Head J. W. 
Published: November 2008

ABSTRACTWard(1) has discussed a method, introduced by Tricomi (6), of calculating transients by means of series involving Laguerre functions which in some cases makes it unnecessary to determine poles of the relevant characteristic function. This method is here investigated with special reference to conditions for convergence and adjustments for improving convergence; some of the examples discussed by Ward are reconsidered.Both in Ward's paper and here the location of poles of the characteristic function is assumed to be approximately known. In some cases the determination of poles of outstandingly small or large modulus and their separation from the remainder may be the most satisfactory procedure. Lin's method (2) for determining a quadratic factor of a polynomial is more widely applicable than has previously been supposed, and this is discussed in bare outline, without proof, here, but in detail, with adequate numerical examples, elsewhere (3).
Some arithmetic properties of the Legendre polynomials
Authors: Carlitz L. 
Published: November 2008

1. Good (4) has proved the formulawhere Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put
Homology and ringoids. II
Authors: Hilton P. J. , Ledermann W. 
Published: November 2008

1. Introduction and recapitulation. In a previous paper (4) we have defined a ringoid (with identities) as a set of elements α, β,… in which addition and multiplication are defined for certain pairs of elements in such a way that, whenever operations are defined, the usual ring axioms are satisfied. Precisely, is an abstract category ((3), p. 108) with an addition operation such that the following axioms hold:(R 1) For any two identity elements ι, ι′ of, let ℋ(ι, ι′) be the set of elements ofwhose left identity is ι and right identity is ι′; then ℋ(ι, ι′) is empty or an Abelian group under addition.
Equivalence of two absorption problems with Markovian transitions and continuous or discrete time parameters
Authors: Sack R. A. 
Published: November 2008

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given bywhere the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form aswhere P(t) is the vector with components pi(t) and the constant matrix A has elements
Diffusion in branching regions
Authors: Jaeger J. C. 
Published: November 2008

ABSTRACTThe problem of diffusion from an instantaneous point source in a thin plane sheet which is intersected by one or more other thin plane sheets of the same material is considered. This problem arises in the study of the diffusion of packets of transmitter substance in synaptic clefts, and some numerical results for the amphibian neuro-muscular junction are presented. A similar discussion applies to branching rods and to cases where there is loss of diffusing substance at the surfaces of the sheets.
Flexural problems of circular ring plates and sectorial plates. I
Authors: Bassali W. A. , Gorgui M. A. 
Published: November 2008

ABSTRACTIn this paper explicit expressions in closed forms are first obtained for the complex potentials and deflexion at any point of a circular annular plate under various edge conditions when the plate is acted upon by general line loadings distributed along the circumference of a concentric circle. These solutions are then used to discuss the bending of a circular plate with a central hole under a concentrated load or a concentrated couple acting at any point of the plate. Solutions for singularly loaded sectorial plates bounded by two arcs of concentric circles and two radii are also derived when the plate is simply supported along the straight edges. The boundary conditions along the circular edges include the cases of a free boundary as well as the elastically restrained boundary which covers the usual rigidly clamped and simply supported boundaries as special cases. The usual restrictions relating to the small deflexion theory of thin plates of constant thickness are assumed. Limiting forms of the resulting solutions are investigated.
Expansions of certain Whittaker functions
Authors: Meligy A. S. 
Published: November 2008

ABSTRACTAn expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.
Permutation endomorphisms and refinement of a theorem of Birkhoff
Authors: Farahat H. K. , Mirsky L. 
Published: November 2008

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .
Degree Functions in Local Rings
Authors: Rees D. 
Published: November 2008

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identityThe main purpose of this paper is to obtain a formulawhere vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).
The number of polygons on a lattice
Authors: Hammersley J. M. 
Published: November 2008

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such thatHere I shall prove the truth of the long-standing conjecture thatI shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.
Transport properties of thin metallic films
Authors: Baines M. J. 
Published: November 2008

ABSTRACTIn using the Boltzmann equation to calculate the transport properties of thin metallic films, it is usually assumed that a time of relaxation exists for the scattering in the film which is the same as for the bulk metal, although this is only strictly justified in the ideal case of isotropic scattering. The correct Boltzmann equation in the case of elastic scattering, when the cross-section depends on the angle of scatter, is set up and possible methods of solution are discussed.The equation is solved for a simple anisotropic scattering law and the electrical resistivity and thermo-electric power are found to depend upon two parameters which are measures respectively of the thickness of the film and the anisotropy of the scattering mechanism. The numerical results for the electrical resistivity, however, differ only slightly from those given by the one-parameter formula obtained when a time of relaxation is assumed to exist.
Road Traffic Flow Considered as a Stochastic Process
Authors: Miller A. J. 
Published: November 2008

AbstractEquations are derived for the process of catching-up and overtaking which occurs in one lane of traffic for a model in which vehicles travel in random bunches and overtake at random times. The model allows vehicle velocities to be distributed and allows for several vehicles at a time to overtake. At first only stationary states are considered, that is states in which the rate of catching-up equals, on average, the rate of overtaking.
Gradient configurations and quadratic functions
Authors: Afriat S. N. 
Published: November 2008

A normal preference system for a combination of goods is represented by an increasing function φ with convex levels. From Gossen's law, that preference and price directions coincide in equilibrium (a special consequence of his Second Law), it follows that, on the data that xr is the vector of quantities purchased at prices given by a vector pr (r = 1,…, k), the gradient gr = g(xr) of the function φ at the point xr is given byfor some positive multiplier μr;. There may be considered the class of preference functions thus satisfying Gossen's law in respect to the data, and thus with gradients taking prescribed directions at k prescribed points. In particular, there may be considered the subclass of these which are quadratic in some convex region containing the points xr. By choosing any multipliers μr, there is obtained a set of gradients gr associated with the points xr. It is asked if there exists a quadratic function which is increasing and has convex levels in a convex neighbourhood of the points xr, and whose gradient at xr is gr; also it is required to characterize the class of such functions, which, if any exist, form an infinite variety. This is the background of the questions which are going to be investigated, and which are of importance in a general method of empirical preference analysis in economics.
The solution of Bessel function dual integral equations by a multiplying-factor method
Authors: Noble B. 
Published: November 2008

In this paper we first of all consider the dual integral equationswhere f(ρ), g(ρ) are given, A(t) is unknown, and α is a given constant. This system, with g(ρ) = 0, was originally considered by Titchmarsh ((13), p. 337), and Busbridge (1), who obtained a solution by the use of Mellin transforms and analytic continuation in the complex plane. The method described in this paper involves the application of certain multiplying factors to the equations. In the present case it is relatively easy to guess the multiplying factors and then the method is essentially a real-variable technique. It is presented in this way in § 2 below.
The rotating body problem
Authors: Levy H. , Robinson W. J. 
Published: November 2008

AbstractIt is shown that for an axisymmetric stationary system there exists, in vacuo, a canonical coordinate system, analogous to Weyl's canonical coordinates for the static case. Some approximate solutions of the vacuum field equations are then obtained in these canonical coordinates.In the second part of this paper, the vacuum field equations for an isolated axisymmetric stationary system are set up and approximate solutions obtained by means of a multipole expansion. The physical components of the Riemann tensor are then examined and it is found that the ‘mass term’ predominates, the ‘rotation term’ being of the next order of magnitude.
A finite set of generators for the homeotopy group of a 2-manifold
Authors: Lickorish W. B. R. 
Published: November 2008

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A byH is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.
On complex Stiefel manifolds
Authors: Adams J. F. , Walker G. 
Published: November 2008

We shall study the following complex Stiefel fibring:.In particular we shall study the problem: for what values of n and k does the fibring(1.1) admit a cross-section? A necessary condition for the existence of a cross-section has been found by Atiyah and Todd (8). We shall show (Theorem 1.1 below) that the condition of Atiyah and Todd is sufficient (as well as necessary) for the existence of a cross-section. The problem stated above is therefore completely solved.