bluuebunny/crossref_metadata_2025_split huggingface dataset:
Power-bounded elements and tauberian theorems in locally convex algebras
Authors: Hennings M. A. 
Published: November 2008

AbstractThe tauberian theorems concerning power-bounded elements of Banach algebras studied by Katznelson and Tzafriri, Allan, O'Farrell and Ransford and Allan are considered, and it is shown that (almost) exactly the same results are true for power-bounded elements in a very large class of locally convex topological algebras, the pseudo-complete algebras. The submultiplicativity of the Banach algebra norm is, for once, inessential to the proof of these theorems.
On the Differential Invariants of the Laguerre Group
Authors: Kubota Tadahiko 
Published: November 2008

1. In a paper “Beiträge zur Inversionsgeometric,” which will appear in the forthcoming volume of theScience Reports of the Tôhoku Imperial University, I have treated the problem of determining the necessary and sufficient condition in order that two plane curves which are given by natural equations should be transformable into each other by inversion. In connection with that paper I propose here to treat the analogous problem for Laguerre transformations:Having given two plane curves, by natural equations, in which the functional relations are all supposed to be analytic, it is required to determine the necessary and sufficient condition in order that the two plane curves should be transformable into each other by a Laguerre transformation.
The Problems of Random Flight and Conduction of Heat
Authors: Burnside W. 
Published: November 2008

In a paper on random flight Lord Rayleigh proved the following result: A number is formed by adding together n numbers each of which is equally likely to have any value from − a to + a. Then, if f (n, s) ds is the probability that the number so formed lies between s and s + ds, and if n is sufficiently great,This result may be stated as follows: A point moves discontinuously in a straight line. For a time τ it has a constant velocity. During the next time-interval τ it again has a constant velocity, and so on. Then if each of these velocities is equally likely to have any value from − v to + v, the probability that in the time nτ, the point moves a distance lying between s and s + ds is f (n, s) ds, with vτ written for a.
Diffraction by a wide slit and complementary strip. II
Authors: Millar R. F. 
Published: November 2008

ABSTRACTThe notions developed in Part I in connexion with the diffraction of an E-polar-ized wave are here applied to the case of H-polarization. Induced current densities, aperture and far fields, and the transmission coefficient are again found in the form of infinite series in inverse powers of the slit-width to wavelength ratio. By means of Babinet's principle, the solution for diffraction by a strip is obtained.A detailed comparison is made with values of the transmission coefficients for both polarizations found previously and by the present method. It appears that this theory provides accurate information when the slit-width is greater than about a wavelength.
A property of the zeros of a cross-product of Bessel functions
Authors: Willis D. M. 
Published: November 2008

In this note it is shown that any positive root of the transcendental equationis definable as a continuous increasing function of the real variable ν, provided ν is positive. Here Jν and Yν denote respectively the Bessel functions of the first and second kind of order ν, and k is a positive constant. Watson ((4)) has established the corresponding result for the simpler equations Jν(z) = 0 and Jν(z) cosα − Yν(z) sinα = 0, where α is a constant. The extension of the result to the positive roots of equation (1) is important because this equation occurs quite frequently in physical problems.
An inversion formula for the kernel Kv(x)
Authors: Fox Charles 
Published: November 2008

AbstractThe problem discussed is that of solving the integral equationwhere g(x) is given, Kv(z) is associated with Bessel functions of purely imaginary argument and f(x) is to be determined.I prove that, by means of fractional integration, it is possible to reduce this equation to the form of a Laplace transform which can be solved by known methods.
On the Cesàro summability of Fourier series and Allied series
Authors: Paley R. E. A. C. 
Published: November 2008

For r>-1, let denoteIfwe say that the series a0 + a1 + a2 +…+an+… is summable by Cesàro mean of order r, or more shortly summable (C, r) to sum s. If r >−1, andwe say that the series is summable by Rieszian mean of order r to the sum s. It has been shown that these two methods of summation are equivalent. Throughout this paper I shall deal with the Rieszian mean, but I shall retain the symbol (C, r). It is known† that if a series is summable (C, r), it is also summable (C, r′) to the same sum for all numbers r′ greater than r.
On the demonstration of electric lines of force, and a new method of measuring the electric moment of tourmaline
Authors: Maurice M. E. 
Published: November 2008

1. A description of a simple method of demonstrating electric lines of force around a charged body is given.2. The instability of liquid surfaces in an electric field is used to obtain an idea of the order of magnitude of the charge developed on tourmaline on cooling through about 140° C.3. A new method of measuring the electric moment of tourmaline is described.
A constant deviation X-ray vacuum monochromator
Authors: Harper G. I. 
Published: November 2008

An instrument is described which provides an accurately monochromatised beam of X-rays by reflection at a crystal and permits the wave-length to be changed without moving either the X-ray tube or the measuring apparatus. This is effected by an automatic adjustment which moves a second crystal into the correct position and orientation to reflect the monochromatised beam along a fixed emergent direction. Twenty wave-lengths well distributed in the range 0·8Å.—4Å. are available, the time required to select any one being about one minute. The whole instrument is contained in a small metal box which is evacuated.I am indebted to Professor A. M. Tyndall, F.R.S., for extending to me the facilities of his laboratory, and for his interest in this work.
The calculations of processes involving mesons by matrix methods
Authors: Wilson A. H. 
Published: November 2008

In a previous paper a new method, based on Kemmer's β-formalism, of calculating meson processes was given for the case in which the meson interacts with an electromagnetic field. This method is now extended to the nuclear interaction, so that the whole of the meson theory can be given either in tensor or in matrix form, the former being preferable when the wave aspect of the meson is important and the latter when the particle aspect is dominant.As examples of the matrix method, derivations are given of the cross-sections for the nuclear scattering of mesons and for the production of mesons from nuclei by photons. It is pointed out that the usual non-relativistic theory of the nuclear interaction is inadequate even for very small velocities.
Superconducting cylinders
Authors: Shoenberg D. 
Published: November 2008

The magnetization curves of short tin, lead and tantalum cylinders were measured by the force method, and it was shown that although for tin there was no appreciable hysteresis in the transverse position, there was a very marked hysteresis in the longitudinal position, of the same form at different temperatures but depending on the sharpness of the cylinder rims. Lead and tantalum showed additional temperature dependent features which could be attributed to impurities.The hysteresis due to shape in the longitudinal cylinder is analogous to the hysteresis of alloy ellipsoids, in that the magnetization for a given field always lies within a certain boundary curve and varies “classically” within this boundary. This analogy suggests that the mechanism of the shape hysteresis for a pure superconductor may be similar to the sponge mechanism proposed by Mendelssohn for alloys, but it is also possible that the hysteresis is due to the formation of macroscopic superconducting rings. The considerations which fix the boundary curve are probably similar to those in the superconducting ring, and in this connection some new measurements on a lead ring are reported. The paper concludes with a discussion of the time lag between magnetization and field, and it is suggested that this is not a primary phenomenon.
The kinetics of adsorption with long-range interaction between adsorbed particles
Authors: Wang J. S. 
Published: November 2008

The rates of condensation and evaporation of adsorbed particles when they interact with long-range forces are deduced from kinetic considerations, including also the case of adsorption of diatomic molecules with dissociation. The rate of evaporation is compared with experiment, and the variation of the dipole moment with θ required to make the theory agree with experiment is discussed.Finally the formulae for the rates of condensation and evaporation are applied to the problem of the diffusion of gases through metals, and it is found that for the processes considered the effect of interaction does not alter the conclusion arrived at in an earlier paper concerning the diffusion equation at small p and at large p.
Remarks on the Dirac δ-operator
Authors: Hamburger Hans Ludwig 
Published: November 2008

1. In this note we shall prove theTheorem 1. Letbe a linear space of (real or complex) functions f(s) defined in the interval 0 ≤ s ≤ 1 subject to the following two conditions:(i) every function of the infinite sequence 1, s, s2, …, sn, … is an element of;(ii) two elements, f(s) and g(s), ofare to be considered as distinct if, and only if, they differ on a set of positive measure in the interval 0 ≤ s ≤ 1.
Some metrical theorems in Diophantine approximation. III
Authors: Cassels J. W. S. 
Published: November 2008

Letbe w sequences of differentiable functions defined in the rangeswhere aj, bj may be − ∞, + ∞ respectively. For each j and each m ≠ n letbe monotonic and let there be a constant K independent of m, n, j, θj such that
The oscillatory behaviour of moving averages
Authors: Moran P. A. P. 
Published: November 2008

The theory of graduation discusses methods of obtaining a smoothed series of values of a function from a given empirical set of values. This is usually done by replacing each observation by a weighted average of it and neighbouring observations. Thus if {xi} (i = 0, ± 1, …) is a sequence of values, we can replace them by the serieswhere the A's are constants which are chosen in some suitable manner. If we regard the xi as the sum of a functional part fi and an error term εi we may attempt to choose the A's in such a way that the functional part is reproduced as well as possible, e.g. thatfor some desired type of function. If this is so and the error terms εi are independently distributed with zero mean and finite standard deviation σ, we find thatwhere the ηi are a series of random variables which are no longer independent of each other but which have zero mean and a standard deviation σ1 such that
The relativistic mass of a rotating cylinder
Authors: Gilloch Josephine M. , McCrea W. H. 
Published: November 2008

AbstractAn investigation by G. L. Clark is found to provide exceptionally instructive illustrations of the calculation of the variously defined ‘densities’ and ‘masses’ occurring in general relativity theory and of their physical significance. In particular, the effects of motion and stress upon gravitational mass are elucidated. An explicit example shows how kinetic energy is included in the gravitational mass.
On the interaction of colloidal particles IV. general mathematical theory for two identical particles
Authors: Levine S. 
Published: November 2008

AbstractA general theory of the interaction of two charged identical colloidal particles of arbitrary shape is developed. An expression for the Helmholtz free energy of the electric double layers is obtained by the methods of statistical mechanics. The condition that there is equilibrium between the ions adsorbed on the surfaces of the colloidal particles and those dissolved in the dispersion medium is accounted for by requiring that the free energy of the whole system be a minimum with respect to variation of the ionic density on the surfaces.The theory presented here is a further development of the work of Verwey and Overbeek. The conclusions of this paper are that in dilute sols, to which the present investigations are restricted, the results of these authors require extension in two directions. First, there is a correction to the mutual energy of two particles, due to the Coulomb interaction of the ions in the bulk of the solution. Secondly, no special assumption concerning the relation between the surface potential (or charge) and interparticle separation need be introduced. The equations set up to determine the free energy of interaction at the same time yield the ‘adsorption isotherm’ for the ion type which is common to the solution and the surface of the particle.
On the interaction between shock waves and boundary layers
Authors: Stewartson K. 
Published: November 2008

AbstractThe effect on the boundary-layer equations of a weak shock wave of strength ∈ has been investigated, and it is shown that ifRis the Reynolds number of the boundary layer, separation occurs when ∈ =o(R−i). The boundary-layer assumptions are then investigated and shown to be consistent. It is inferred that separation will occur if a shock wave meets a boundary and the above condition is satisfied.
On the law of decay of homogeneous isotropic turbulence and the theories of the equilibrium and similarity spectra
Authors: Goldstein S. 
Published: November 2008

AbstractThe requirements of Kolmogoroff's theory of the equilibrium spectrum are satisfied only at very high Reynolds numbers, higher than any at which experiments have yet been done. In particular, when the theory holds, the rate of decay of the mean-square vorticity ω must be negligible compared with either its rate of increase due to the stretching of the vortex filaments or the rate of dissipation due to viscosity.An extended version of Kolmogoroff's hypothesis may be proposed, in which the statistical properties of the turbulence in a range of wave-numbers (range of eddy sizes) depend not only on the rate of dissipation ∈ per unit volume and the viscosity ν, but also on the time rate of change d∈/dt of ∈. The result is to introduce a dependence on the Reynolds number R of the turbulence into quantities and constants which, on Kolmogoroff's original hypothesis, were independent of R. The Reynolds number R is defined from the decay law; u2t, with an origin of time suitably chosen, is a function of t, finite when t = 0, and R is defined as (u2t)t=0/ν. Lin's decay law follows logically from the extended hypothesis, according to which the rates of change of ω−1, due to the causes mentioned above, are constant during the decay; Lin's decay law would also follow if this less general part only of the extended hypothesis be assumed. The same decay law is also obtained if the similarity spectrum of Heisenberg is taken to apply not to the whole of the energy-bearing eddies, but only to the energy-dissipating eddies. But it is suggested that further generalization of the theory of the similarity spectrum, and of the decay law, is necessary; that the similarity spectrum is probably only asymptotically correct for a range of large wave-numbers, the range depending on the initial conditions and decreasing as the decay proceeds; that the general decay law is u2t = μRd(t), where d(t) is an integral function of t, such that d(0) = 1, and with an asymptotic value for large t to give correctly the law of decay in the final period; and that d(t), and the number of constants needed to specify it approximately, depend on the initial conditions. An experiment is suggested to test the dependence of the law of decay on the initial conditions. It is also suggested that the recently observed constancy of u2t in the initial period in the turbulence behind a single grid is only approximate, and this approximate constancy still needs explanation. Remarks are also included on the range of application of the equilibrium spectrum, for which some formulae are given when there is a definite cut-off in the spectrum.
A method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder functions
Authors: Miller J. C. P. 
Published: November 2008

1. The method of converging factors, for hastening the convergence of slowly convergentseries and improving the accuracy of asymptotic expansions, was introduced by J. R. Airey and is well known to computers (see Airey(1) and Rosser(2)). The principle is as follows. It is required to compute a quantity which is expressed as an infinite seriesThe series may be either convergent or asymptotic and divergent.
Fredholm solutions of partial integral equations
Authors: Salam Abdus , Smithies F. 
Published: November 2008

The following ‘partial’ integral equation arises in the quantum theory of fields:where f(x1x2) = f(x2x1), K(x1x2, y) = K(x2x1, y), and a symmetric solution of the equation is desired. The solution of (1) will be obtained as a special case of the solution of the following equation:
The diamagnetism of quasi-bound conduction electrons
Authors: Wilson A. H. 
Published: November 2008

The diamagnetism of the conduction electrons gives rise to some of the most difficult problems in the theory of metals, the complete solution of which has not yet been found. Formally, the problem is equivalent to determining the density matrixand the exact expression for ψ(r′, r, γ) for perfectly free electrons in a constant magnetic field H has recently been found by Sondheimer and Wilson(2). The extension of the theory to deal with quasi-bound electrons for all values of H seems to be out of the question, but an approximate partition function was given by Peierls (1), excluding terms of higher order than H2. In The theory of metals ((3), referred to as T.M.) I gave a more powerful and simpler method of dealing with the problem, based upon the properties of ψ(r′, r, γ), but since the solution was obtained as a power series in μ0Hγ, where it could at best determine only the normal diamagnetism.
A certain triple Whitehead product
Authors: Hilton P. J. 
Published: November 2008

The triple Whitehead product we consider in this note is [[ɩn, ɩn], ɩn] ∈ π3n−2(Sn), where ɩn generates πn(Sn). It follows from the Jacobi identity for Whitehead productsα ∈ πp(X), β ∈ πq(X), γ ∈ πr(X), that 3[[ɩn, ɩn], ɩn] = 0. Now, if n is odd, 2[ɩn, ɩn] = 0, so that 2[[ɩn, ɩn], ɩn] = 0, whence
The coefficients of certain integral modular forms
Authors: Rankin R. A. , Rushforth J. M. 
Published: November 2008

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of formswhere k is defined by (2·10) of R and the coefficients possess the following properties:for a prime p, where p is a positive integer.
Some new results on equivalent products
Authors: Slater L. J. 
Published: November 2008

ABSTRACTIn this paper, I prove some new results which are identities connecting groups of general infinite products, and I give two tables, calculated using EDSAC, of the function 1/II(1 − aqn−1) over the range a = − 0·90(0·05) + 0·95, q = 0·0(0·05) + 1·00, and of the function 1/n(1 − qn) over the range q = 0(0·005)+ 0·890.
On Kepler's equation
Authors: Cherry T. M. 
Published: November 2008

1. Kepler's equationhas, when θ, x are real with 0 < x < 1, just one real root ξ = ξ*(θ, x). For this root there are the well-known formulae, dating from Lagrange and Bessel,The objects of this paper are (i) to obtain analogous formulae for the unreal roots ξ of the equation, and (ii) to sum the conjugates (as Fourier series in θ) of the series on the right of (2) and (3).
On the infinity in the second Born approximation for the Coulomb field
Authors: Elton L. R. B. 
Published: November 2008

ABSTRACTIt is shown that the infinity arising in the second Born approximation to the scattering amplitude of a particle scattered by a Coulomb field is due to the distortion at infinity of Coulomb waves. Ways of subtracting out this infinity, which does not lead to any observable effects, are indicated.
On the stretching of material lines and surfaces in isotropic turbulence with zero fourth cumulants
Authors: Reid W. H. 
Published: November 2008

ABSTRACTThe convective effects of isotropic turbulence on arbitrary vector and scalar fields satisfying certain conservation conditions are studied under the assumption that all fourth-order cumulant tensors are zero. In the absence of molecular conduction, the behaviours of such vector and scalar fields are related to those of material lines and surfaces respectively, and the diffusive action of the turbulence results in a stretching of these lines and surfaces. The present analysis leads to a definite prediction for the corresponding strain rates which suggests that an initially spherical material volume element is drawn out into a long thin ribbon of constant width.
The interpolation of atomic fields
Authors: Ridley E. Cicely 
Published: November 2008

ABSTRACTZ(nl; r) is the contribution to Z(r) from an electron in the (nl) wave function. The Z(nl; r) vary systematically with atomic number and, as N becomes large, tend to the corresponding hydrogen-like functions, ZH(nl; r). A two-parameter method of fitting the Z(nl; r) to the ZH(nl; r) is described. This involves a ‘screening constant’ and a ‘slope constant’, both of which are defined. From published data, the two parameters have been obtained as functions of atomic number. The parameters for an unsolved atom can then be found by interpolation and approximate Z(nl; r) derived by appropriate adjustment of the functions for the nearest atom in the periodic table for which they are known. The method has been tested by interpolating for the (3d) function between Cu+ and Rb+ and by preparing estimates of the Z(nl; r) for the unknown structure Mo+. The results were good for all but Z(4d; r) for Mo+, where the number of values of the screening and slope constants already known was insufficient for reliable interpolation.
The Stokes phenomenon and certain nth-order differential equations II. The Stokes phenomenon
Authors: Heading J. 
Published: November 2008

1. Introduction. In paper I(2), solutions have been found for the nth-order differential equations (I, 3), (I, 9), (I, 18), namely,where ϑ = wd/dw, m is any rational fraction and a any constant. Power-series solutions have been found for these equations, together with integral representations and their asymptotic expressions valid for restricted ranges of arg z. The object of this second paper is to consider these asymptotic solutions in more detail, and to extend these expressions to all values of arg z. The Stokes phenomenon will be manifest throughout, and this will be treated in a manner suitable for further application.
The concept of order III. General relativity as a technique for extrapolating over great distances
Authors: Bastin E. W. , Kilmister C. W. 
Published: November 2008

ABSTRACTIt is shown that a combination of the methods developed in the two previous papers in the series allows the reformulation of general relativity as a technique for extrapolating physical theory to distant regions, whose accessibility to experiment is limited. General relativity is thus shown to be one of a number of extrapolation theories allowed by the spread theory of Paper II.