bluuebunny/crossref_metadata_2025_split huggingface dataset:
Uniqueness theorems and the maximum-minimum principle for a type of non-linear partial differential equations
Authors: Fuchs W. H. J. , Weiss P. 
Published: November 2008

It is well known that solutions of partial linear differential equations of the second order and of elliptic type are uniquely determined by their boundary data, and that they assume their maximum and minimum values on the boundary. The usual proofs make use of the principle of superposition and are therefore not applicable to non-linear problems. But recently Pryce has proved the uniqueness theorem for the non-linear equations of minimal surfaces and of Born's electrostatics. These equations are the Euler equations of the variational problemk = + 1 corresponds to the case of minimal surfaces in n + 1 dimensions; k = − b−2, n = 3 corresponds to Born's electrostatics. Pryce's procedure depends essentially on the notion of conjugate variables in the calculus of variations for multiple integrals and can therefore be extended to a wide class of differential equations arising from variational problems (for several functions of several variables) as we show in § 3.
On the easier Waring problem for powers of primes. II
Authors: Erdös P. 
Published: November 2008

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:The density of each of the sets of integersis positive.
On a Hardy-Littlewood problem of diophantine approximation
Authors: Spencer D. C. 
Published: November 2008

1. Let .When r is a positive integer, various writers have considered sums of the formwhere ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which
On the determination of intermolecular energies by inductive analysis
Authors: Orr William J. C. 
Published: November 2008

For a direct comparison of the individual attractive and repulsive terms of an intermolecular potential determined by the inductive analysis of themodynamic data with the same terms calculated by quantal methods it is desirable to carry out the analyses, in the first approximation, with an intermolecular potential of the form ø(R) = Pe−aR − A1/R6 − A2/R8. For mathematical convenience, in place of the above expression, two potential functions,andare considered, the first being taken to be adequate in the range of values of R between 0 and R0 (the minimum of the potential function) and the second, in the range from R0 to ∞. By dividing the problem in this way it is possible to find substitutions which permit the integration of the classical expression for the second virial coefficients (and other appropriate thermodynamic data) directly in terms of fairly simple series in | ψ0 |, R0, a and r. Finally it is pointed out that for such simple atoms or molecules as the rare gases, oxygen, nitrogen and methane r may be taken as 0·15 throughout, which considerably simplifies the application of the method to the experimental data.
Momentum distribution in molecular systems. Part VI. Shape of the Compton line for methane CH4
Authors: Coulson C. A. , Duncanson W. E. 
Published: November 2008

Calculations have been made of the momentum distribution and the shape of the Compton profile for gaseous methane, using the Dirac transformation theory and three distinct approximations (molecular orbital, electron-pair and self-consistent-field) to the true molecular wave functions. The bearing of the results upon the validity of the original wave functions is discussed; it is concluded that the exponents in molecular wave functions should be approximately 1·1 times greater than in corresponding atomic wave functions. Comparison is made with experimental electron scattering results.Our thanks are due to Prof. Massey for providing us with the numerical tables of the self-consistent-field for methane in advance of publication, and to Mr E. G: Phillips of the Mathematics Department of Univeristy College of North Wales, for the loan of a calculating machine.
Minimum variance and the estimation of several parameters
Authors: Rao C. Radhakrishna 
Published: November 2008

With the help of certain inequalities concerning the elements of the dispersion matrix of a set of statistics, and of the information matrix, the following results have been proved. Some of these inequalities are extensions of results given by Fisher (1) in the case of a single parameter.(i) Efficient statistics are explicit functions of the minimal set of sufficient statistics.(ii) Functions of the minimal set of sufficient statistics, satisfying the property of uniqueness defined in the text, are best unbiased estimates. Under certain conditions estimates possessing exactly the minimum possible variance can be obtained by the method of maximum likelihood.(iii) In large samples maximum likelihood estimates supply efficient statistics in the case of several parameters.(iv) The importance of replacing the sample by an exhaustive set of sufficient statistics (referred to in this paper as the minimal set) as a first step in any methodological problem has been stressed by R. A. Fisher in various articles and lectures. The above discussion supplies a formal demonstration of this view so far as the problem of estimation is concerned.
An equation of quantum mechanics
Authors: Littlewood D. E. 
Published: November 2008

The complete set of matrix representations of a set of n quantities β1, β2,…,βn satisfyingis obtained for all values of n. It is found that if n = 2v or n = 2v + 1, there are v + 1 irreducible representations with the respective degrees 1, but if n = 2v + 1 there are in addition two conjugate representations of order , the symbol denoting the binomial coefficient. The explicit representations are given for n = 2, 3, 4, 5.
On the matrix equation (AX)n = ∥A∥I
Authors: Grimshaw M. E. 
Published: November 2008

It was proved by J. Williamson (1) that when C is any circulant matrix of order n and Ω is a certain diagonal matrix thenwhere ∥ C ∥ is the determinant of C and I is the unit matrix of order n. A simpler proof was given by U. Wegner (2), and later the characteristic equation of ΩC was discussed independently by L. Toscano (3). In the present note we prove the more general theorem that there corresponds to any matrix A of order n with simple elementary divisors a solution X of the equation (AX)n = ∥ A ∥ I. The theorem has a simple geometrical interpretation; the proof is almost immediate.
The problem of a rotating incompressible disk
Authors: Clark G. L. 
Published: November 2008

In this paper an investigation is made of the properties of material which is under isotropic stress p when in a strained state, and is such that the relation between the stress and the dilatation Δ iswhere α is the density, assumed constant in the unstrained state, and c is the velocity of light. It is shown that in this material the waves of dilatation travel with the velocity of light and that a disk or cylinder composed of this kind of matter suffers no change in radius when it is made to rotate.It is suggested that it is not unreasonable to attach the label incompressible to matter having these properties.
On non-linear differential equations of the second order
Authors: Cartwright M. L. 
Published: November 2008

1. The problem of automatic synchronization of triode oscillators was studied by Appleton† and van der Pol‡; it gives rise to the differential equationwhere α, γ, ω, E, ω1 are positive constants such that α/ω, γ/ω, (ω − ω1)/ω are small and dots denote differentiations with respect to t. When these conditions are satisfied, it is easy to see thatis an approximate solution over a limited time for any b1, b2 chosen to fit initial conditions on v and ṿ provided that v and ṿ are not too large. If it is assumed that b1 and b2 vary slowly compared with ω1t, so that can be neglected, and ḃ1, ḃ2 are comparatively small, the equationswhereare obtained. These are sufficiently accurate for the discussion of most of the physical phenomena, and have been used in this form (or in the polar-coordinate form obtained by putting b1 = b cos ø, b2 = b sin ø) by various authors §. Solutions of (1) with period 2π/ω1 are obtained approximately by putting ḃ1 = ḃ2 = 0 in (3). The steady-state solutions of (1) other than those with period 2π/ω1 correspond to periodic solutions of (3). All other solutions converge to one or other of these types of solution.
An approximate relation between the energy levels of a particle in a field of given potential energy, calculated on the relativistic and non-relativistic theories
Authors: Sewell Geoffrey L. 
Published: November 2008

The relativistic Hamiltonian H for a particle in a field of potential energy V is given, in atomic units, by the formulawhere any ρ commutes with any σ, any σ anticommutes with any other σ, ρ1 anticommutes with ρ3 and
A modification of the molecular orbital method: the LCMO method
Authors: Dewar M. J. S. 
Published: November 2008

A modification of the MO method is described which leads to the same polynomial secular equation as does the LCAO method, but via a simpler secular determinant. The basic idea is to divide the whole molecule into parts, calculate the MO's for these by the usual method, and then use linear combinations of these MO's as trial eigen-functions in a variational treatment of the whole molecule.Equations are developed for a number of cases, and it is shown that the computations of energy levels and MO's from them is often much simpler than from the corresponding LCAO equations.
The forces between floating bubbles and a quantitative study of the Bragg ‘Bubble Model’ of a crystal
Authors: Lomer W. M. 
Published: November 2008

The object of the paper was to find the forces acting in bubble rafts and to use them for a description of the mechanical properties of these rafts. The first section is based entirely on Nicolson's work and gives expressions for the cohesive forces in the lattice. The second and third sections deal with the repulsive forces arising when a bubble is compressed uniaxially, and are really the basis of the paper. The fourth section introduces the idea of ‘interaction’ between compressions in the three close-packed directions—a conception which explains the breakdown of Cauchy's relations. The conditions of equilibrium in the lattice are investigated in § 5, and in § 6 all these forces are combined to give a complete account of the elastic properties of the rafts.The remaining section gives a brief account of the experimental checks of the theory; it appears to be a reasonable approximation for small bubbles, and, indeed, we could certainly not expect it to apply to large ones, since for them not only are all the calculations of the attractive potentials invalid because of the large local slopes of the surface, but also the ‘flats’ and ‘caps’ become so large as to interfere with one another.The forces are considered well enough supported to justify their further use in an investigation of the plastic behaviour of the bubble rafts—an investigation which has explained several interesting experimental facts but brought us little nearer an understanding of the low yield strains of metallic single crystals (see (2)).
On two theorems of S. Verblunsky
Authors: Delange Hubert 
Published: November 2008

In connexion with moment problems, S. Verblunsky proved the following two theorems:Theorem I. (a) If f(x) is integrable in (−∞, +∞) and satisfies 0 ≤f(x) ≤ 1, then there exists a function σ(x), bounded and non-decreasing in (−∞, +∞), such that, for ζ = ξ + iη and η ≠ 0,
Some metrical theorems in diophantine approximation
Authors: Cassels J. W. S. 
Published: November 2008

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequalityhas infinitely many integer solutions q > 0. In particular, if α is any real number, the inequalityhas infinitely many solutions.
Pressure fluctuations in isotropic turbulence
Authors: Batchelor G. K. 
Published: November 2008

ABSTRACTThis paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.
A generalization of the Hurwitz composition theorem to irregular power series
Authors: Eggleston H. G. 
Published: November 2008

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the functionin terms of the positions of the singularities of f(z) and g(z).
The surface energies of superconductors
Authors: Pippard A. B. 
Published: November 2008

AbstractMeasurements of the areas of magnetization curves of superconducting colloids and films indicate that the difference between the surface energies of the normal and super-conducting states at the same temperature is much smaller than has been suggested hitherto. An attempt is made to explain the high critical fields of thin films, which led to the introduction of the surface energy, by the hypothesis that a negative interphase surface energy in films favours the persistence of superconducting threads in fields higher than those normally regarded as sufficient to destroy superconductivity. A model is proposed of the interface between normal and superconducting phases which allows the interphase surface energy to be positive in pure macroscopic specimens but negative in alloys and thin films and at lattice defects. This enables a qualitative explanation to be given of some of the details, such as supercooling, of the phase transition in pure metals and alloys.
On the order of magnitude of Ramanujan's arithmetical function τ(n)
Authors: Pennington W. B. 
Published: November 2008

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansionThis function appears in the discussion of an asymptotic formula for the functionand also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.
The energy loss of free magnetic poles in passing through matter
Authors: Bauer E. 
Published: November 2008

AbstractDirac has shown that in quantum theory free magnetic poles are possible only if their pole strength g is an integral multiple of ℏc/(2e) = 137e/2. The ionization produced by such particles is calculated by the classical impact parameter method and also by quantal perturbation theory; at sufficiently large velocities the rate of loss of energy of poles of unit strength is g2/e2 ≑ 5000 times as large as that of an electron of the same velocity; at low velocities the energy loss of a monopole decreases steadily without rising to a maximum as in the case of electric particles.
Congruence properties of the partition function and associated functions
Authors: Rushforth J. M. 
Published: November 2008

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled‘Properties ofp(n) and τ(n) defined by the relationsand was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties ofp(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems
The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains
Authors: Jones D. S. 
Published: November 2008

ABSTRACTThe spectrum of −∇2 (and of −∇2 + b) is investigated when the boundary conditions are given on surfaces which extend to infinity. Simple criteria are obtained for determining whether point-eigenvalues are present in the lower part of the spectrum.Semi-infinite domains which are conical at infinity are found to possess purely continuous spectra when the boundary condition is u = 0 or ∂u/∂v = 0; the radiation condition ensures a unique solution. A counter-example shows that this is not true in general for the boundary condition ∂u/∂v + σu = 0.Semi-infinite domains which are cylindrical at infinity have a continuous spectrum with a discrete spectrum embedded in it. An example is given.The results are applied to the theory of surface waves. It is shown that Ursell's ‘trapping modes’ can occur in a canal of finite width when the bed has a protrusion over a finite longth but is otherwise of uniform depth. Trapping modes can also occur when the canal contains a submerged cylinder (not necessarily small in cross-section).
On linearly polarized electromagnetic waves of arbitrary form
Authors: Nisbet A. , Wolf E. 
Published: November 2008

ABSTRACTTwo simple laws connecting the amplitude and phase functions of a monochromatic electromagnetic wave of arbitrary form are derived, holding in the case when one of the field vectors is linearly polarized. The first is a generalized Fermat's principle which enables determination of the phase when the amplitude is known; the second expresses the propagation of the (vector) amplitude along the curves orthogonal to the co-phasal surfaces. Some other general properties of linearly polarized fields are also discussed, and illustrative examples are given.
The character of the equilibrium of an incompressible heavy viscous fluid of variable density
Authors: Chandrasekhar S. 
Published: November 2008

ABSTRACTThis paper is devoted to a consideration of the following problem: Given a static state in which an incompressible viscous fluid is arranged in horizontal strata and the density is a function of the vertical coordinate only, to determine the initial manner of development of an infinitesimal disturbance. The mathematical problem is reduced to one in characteristic values in a fourth-order differential equation and a variational principle characterizing the solution is enunciated. The particular case of two uniform fluids of different densities (but the same kinematic viscosity) separated by a horizontal boundary is considered in some detail. The mode of maximum instability in case the upper fluid is more dense and the manner of decay in case the lower fluid is more dense are determined; and the results of the calculations are illustrated graphically. Gravity waves (obtained in the limit when the density of the upper fluid is zero) are also treated.
The character of the equilibrium of an incompressible heavy viscous fluid of variable density: an approximate theory
Authors: Hide Raymond 
Published: November 2008

ABSTRACTThe equilibrium of a layer of a viscous incompressible fluid of density ρ variable in the vertical (z) direction, the fluid extending indefinitely in the horizontal directions and being confined between rigid horizontal boundaries, is examined by the usual method of studying the initial behaviour of a small disturbance. Diffusion effects are ignored. Chandra-sekhar has shown quite generally that a variational procedure for solving the resulting characteristic value problem is admissible, suggesting a method whereby an approximate solution can be obtained analytically. In this paper, by making use of appropriate approximate methods of solution, three types of density configuration, ρ(z), are studied. The first corresponds to two superposed fluids of great depth, the upper and lower fluids haying densities and coefficients of viscosity ρ2 and μ2, and ρ1 and μ1, respectively. Chandrasekhar has given an exact numerical solution to this problem for the case where μ2/ρ2 = μ1/ρ1 The results of the approximate theory, by which not all the boundary conditions can be satisfied, and the exact theory are in good agreement. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (ρ2 > ρ1) are found. The stable case (ρ2 > ρ1) is also discussed in detail; the formulae obtained for the phase and group velocities of gravity waves in deep viscous fluids are especially interesting. There is an upper limit to the natural frequency of oscillation of the system which would be well within the range of experimental investigation.The second problem considered is that of two superposed fluids of small depth. Formulae for the phase and group velocities of ‘long gravity waves’ (which are not equal when viscous effects are included) are obtained.Finally, a continuously stratified fluid of finite depth d in which ρ(z) = ρ0eβz is investigated. The character of the equilibrium depends on a Grashof number G = (gβd4)/(π4ν2) where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (G > 0) are obtained. In the stable case (G < 0) all modes are aperiodically damped if – G < . Otherwise, the possibility of a disturbance being propagated as a horizontal wave does arise, but only within a limited wave-length range.
The irrotational motion outside a free turbulent boundary
Authors: Phillips O. M. 
Published: November 2008

ABSTRACTIn this paper there is considered the irrotational motion of an infinite fluid when the normal velocity across a plane is specified as a stationary random function of position in the plane, and a solution is obtained in terms of the specified boundary conditions. It is shown that the mean square velocity normal to the plane is equal to the sum of the mean squares of the velocities in the other two orthogonal directions. The asymptotic variations with distance normal to the plane are found for functions representing the important properties of the motion, and, in particular, the energy of the fluctuations is shown to be inversely proportional to the fourth power of the distance from the plane. The conditions postulated are shown to correspond closely to the motion outside a free turbulent boundary, and good agreement is found between the predictions of the theory and the available experimental results.