bluuebunny/crossref_metadata_2025_split huggingface dataset:
On a problem of Erdős concerning decompositions of the plane
Authors: Davies Roy O. 
Published: November 2008

Sierpiński ((6), (7)) proved that the continuum hypothesis implies the following proposition:Euclidean space E3 can be decomposed into three sets Si (i = 1, 2, 3) such that, for some three straight lines Di in E3, the intersection of each line parallel to Di with the corresponding set Si is finite.
Dispersion relations for a two-channel scattering process: residues of the scattering amplitudes
Authors: Underhill J. 
Published: November 2008

1. This paper is a continuation of a previous one(1), in which dispersion relations satisfied by the scattering amplitudes for a non-relativistic two-channel scattering process were derived. It was shown that for the process described by the integral equationswiththe scattering amplitudes Mi(E, τ) (where E is the incident energy and τ is the momentum transfer), satisfy the relations
Numerical evaluation of the integral
Authors: Catton Diana , Millis B. G. 
Published: November 2008

ABSTRACTNumerical values of the above integral are given for 3/2λ√3 = 0(0·1)1, to six decimals for |ω| = 0(0·l)2·0(0·2)4·8 and to seven decimals for |ω| = 5(0·5)10(1)15.The integral satisfies a third-order differential equation; this equation is integrated numerically for ω ≥ 0 and ω ≤ 0, and in each case the method of isolating the solution which satisfies f(ω) → 0 as ω → ∞ is described.
Variational principles in high-frequency scattering
Authors: Kodis Ralph D. 
Published: November 2008

ABSTRACTA pair of variational principles are formulated for two-dimensional scattering by obstacles. The first of these is in terms of the obstacle boundary values, and it is shown that a simple ‘optical’ trial function leads to an incorrect frequency dependence for the scattering cross-section. In the second, the obstacle is viewed as the analogue of an aperture coupling two half spaces. The geometric optics part of the cross-section can then be made explicit and is split off to leave a stationary form for the frequency correction. The zero-order calculation for the cross-section of a circle, using corresponding ‘optical’ trial functions, is found to have the correct (Ka)−2/3 frequency dependence.
The structure of radiation
Authors: Stoner Edmund C. 
Published: November 2008

It is shown that, assuming conservation of energy and momentum in individuai transitions, radiation must be regarded as propagated by linearly directed quanta. Evidence is brought forward in favour of the view that these quanta are spatially localised longitudinally as well as laterally. As these conclusions seem unavoidable, an attempt is made to see if there is any possibility of accounting for interference effects in terms of quants possessing periodic properties.The idea that a single quantum can produce interference effects is untenable. The conception of the phase of a quant is introduced, and it is supposed that atoms may absorb a quant in a given phase provided they are in a “state” corresponding to that phase. Considering radiation emanating from a point, it is shown that a tentative explanation of interference may be obtained if the quants have a constant phase at emission, and if they give atoms at any point an impulse towards the state corresponding to their phase at the point. According as these impulses reinforce or oppose each other, bright or dark parts of a fringe System are obtained. Practically the whole nomenclature and method of the wave theory may be taken over and applied to the quant phase field surrounding an emitting source. Essentially, on this view, an explanation of interference is sought in terms of different quants, and the dark and bright parte of a fringe system are to be explained, not in terms of different numbers of quante arriving at them, but in different phase relations of quante which do arrive.Quante are characterised by a magnetic and electric vector, whose magnitudes and directions define the state of polarisation and the phase. Electron orbite may also be regarded as characterised by two vectors in terms of which it seems as if the “state” may be specified.The relation between transition character and type of radiation emitted is finally considered and it is shown that the evidence so far available renders plausible an explanation of interference of the kind put forward. Much further experimental work, particularly in connection with the Stark and Zeeman effects, is, however, necessary, before sufficient precision can be given to the views put forward to make them satisfying. On the other hand, it seems premature to suppose that only a formal treatment is possible of the physical phenomena involved.
Some expansions in products of hypergeometric functions
Authors: Dhawan G. K. 
Published: November 2008

1. Making use of the familiar abbreviationlet us adopt a contracted notation for the generalized hypergeometric function AFB(x) and writewhere (a) denotes the sequence of parameters a1, a2,…, aA. It will be assumed throughout the present paper that there are A of the ‘a’ parameters, A′ of the ‘a′’ parameters, and so on. Thus [(a)]m is to be interpreted aswith similar interpretations for [(a′)]m, etc.
On the Rayleigh assumption in scattering by a periodic surface
Authors: Millar R. F. 
Published: November 2008

AbstractIn treating plane wave scattering by a periodic surface, Lord Rayleigh (10) assumed that the discrete, outgoing and evanescent plane wave representation for the scattered field was valid on the surface itself. Recently, this Rayleigh assumption has been questioned and criticized. For the surface y = b cos kx on which the total field vanishes, Petit and Cadilhac(8) have demonstrated its invalidity when Kb > 0·448. The present paper discusses scattering of a wave, incident from y > 0, by an analytic periodic surface y = f(x) ( – ∞ < x < ∞), and shows that the Rayleigh assumption is valid if and only if the solution can be continued analytically across the boundary at least to the line y = minf(x). Conformal mapping and results relating to the analytic continuation of solutions to elliptic partial differential equations reduce the problem to one involving the location of singularities and critical points of a potential Green's function. Provided that the perturbation of the surface from a plane is sufficiently gentle, the validity of the Rayleigh assumption is established. For the surface y = b cos kx it is shown that the assumption is valid if Kb < γ, where γ is a positive number no greater than 0·448, the precise value of which is unknown. Possible extensions of the analysis to different or more general situations are suggested.
The electromagnetic fields of moving dipoles
Authors: Ward G. N. 
Published: November 2008

AbstractRecent papers by J. R. Ellis and the author contain formulae for the fields of moving dipoles which are expressed in rather different forms in the two papers, and it appears that the author's results are more general than Ellis's. Here, the results are re-derived from a consideration of the Hertzian six-vector potential for a distribution of moving dipoles, and a comparison is made with the previous forms. It is found that Ellis's results are essentially as general as the author's. The expressions for the potentials and the field tensors are treated as weak functions in Temple's sense, and very few restrictions have to be imposed on the dipole moments and velocities, which may be expressed either as functions of coordinate time or as functions of proper-time.
A note on the problem of the penny-shaped crack
Authors: Sneddon Ian N. 
Published: November 2008

1. The problem of determining the distribution of stress in the neighbourhood of a penny-shaped crack defined in terms of cylindrical coordinates (ρ, φ, z) by 0 ≤ ρ ≤ α, z = 0, has been considered by Sneddon ((2)) and Sack ((1)). In the latter paper the solution is derived only in the case in which the stress field is due to the application of constant pressure to the faces of the crack. In the former paper the analysis given applies to an axisymmetric distribution of pressure p(ρ) applied to both the upper and lower face of the penny-shaped cavity, but the calculation of the stress intensity factorand of the energy W required to open up the crack is a complicated matter even in the case in which p(ρ) is a constant.
Some Relations between the Optical Spectra of Different Atoms of the same Electronic Structure. II. Aluminium-like and Copper-like Atoms
Authors: Hartree D. R. 
Published: November 2008

The relevant results from the writer's previous paper on the relation between spectra of atoms of different atomic structure are summarised.For non-penetrating orbits no new theoretical results are obtained, and there are few known spectra (other than those of lithium-like or sodium-like atoms already treated in the first paper) on which to test the relations previously obtained. Values of the polarisability for the A1+ and Si++ ions are calculated from terms of Al I and Si II respectively corresponding to non-penetrating orbits, and are shown to be very much greater than the values of the polarisability of the neon-like ions Al+++ and Si++++.The main new results are those for penetrating orbits. Assuming a central field of force, it is shown that the quantum defect q for such an orbit can be expressed as the sum of contributions from the electrons in groups of core orbits of different principal quantum number n, and further that if for a given atom in different states of ionisation corresponding orbits of the series electron are compared, the contribution to q from a set of core orbits of given n is very nearly independent of the degree of ionisation so long as the number of electrons in core orbits in the group remains the same. It follows that, if q is the quantum defect for a term of the spectrum of an atom core charge C, the core of which contains SM orbits of principal quantum number M and none of higher quantum number, and q″ is the quantum defect for the corresponding term in the spectrum of the atom of the same element with core charge C + sM, which differs from the atom of core charge C only in lacking the core orbits of principal quantum number M, then q −; q″ is approximately the contribution from the core orbits of principal quantum number M to the quantum defect for the term of the atom core charge C.Further, it is shown that if corresponding terms of different atoms of the same electronic structure are compared, then for large values of C the contribution to q from any group of core orbits should tend asymptotically to be proportional to the average time mean radius of these orbits, and its reciprocal should tend asymptotically to be linear in C.Somewhat similar relations are obtained for the quantity Q = dq/d (ν/R) which measures the variation of quantum defect within a sequence.These theoretical results, and in particular the result that l/(q − q″) should tend asymptotically to be proportional to C, are compared with the values of q deduced from the terms of such observed spectra of aluminium-like and copper-like atoms as are available, and it is found that though the theoretical relations are only established as asymptotically true for large C, there is a considerable measure of agreement with spectra for small values of C, which are the only ones which can be observed.
An apparatus for the investigation of the effect of poisonous substances, and mixed vapours on catalytic activity
Authors: Constable F. H. 
Published: November 2008

(1) An apparatus has been devised in which the composition of the vapour phase passing over the catalyst can be varied at will, or two vapours can be used alternately as reactants.(2) By using ethyl alcohol, and amyl alcohol from fusel oil alternately as reactants, the poisoning of a very active copper gauze catalyst has been followed and the temperature coefficient of the dehydrogenation of ethyl alcohol by the copper has been measured during the poisoning.(3) The activity of the catalyst, poisoned by amyl alcohol, fell in accordance with the simple logarithmic law(4) Confirmation was obtained that the rates of dehydrogenation of ethyl and amyl alcohols were equal.(5) The temperature coefficient of dehydrogenation on the catalyst was unchanged throughout the poisoning, though the reaction velocity was reduced 30 times.(6) The temperature coefficient was also shown to be unchanged during sintering, the reaction-velocity having been reduced 10 times.(7) Equations are derived for the rate of poisoning of an exponential distribution of centres of activity when the poison attacks the more active centres selectively.(8) The relation deducedcannot usually be distinguished (except with special relations between the constants) from the simple logarithmic law of poisoning.(9) Thus the exponential distribution normally poisons like a surface composed of homogeneous centres, the temperature coefficient being unchanged by the poisoning.(10) The equationdescribes the reaction-velocity as a function of time, and temperature, and gives adequate explanation of previous experimental results on the forms of the heating and cooling curves with “poisonous” alcohols.(11) Frenkel has shown , where τ0 is the vibration frequency of the molecule perpendicular to the surface, and μ0 is the heat of desorption of one gram molecule. Thus if the heat of desorption is not small compared with the energy necessary to cause chemical disruption of the molecule, the previous equation becomesthe form of the equation being unaltered.
The Action of Positive Ions of Caesium on a Hot Nickel Surface
Authors: Moon P. B. 
Published: November 2008

Experiments on the emission of secondary electrons from hot nickel surfaces bombarded by Cs+ ions at normal incidence show that the emission is a function of bombarding voltage and target temperature. The emission is negligible below 300 volts and increases steadily with further increase of the bombarding voltage, rising to 4% at 4000 volts with the target at 950° C. Increase of temperature causes the secondary emission to diminish. If positive ions are allowed to evaporate from the hot target the number of Cs+ ions leaving the surface may, under certain conditions, be equal (with an accuracy greater than 1%) to the number arriving in the incident beam.I much appreciate the kindness shown me in many ways by Lord Rutherford. The experiments were commenced at the instigation of Dr M. L. E. Oliphant and I gratefully acknowledge that this paper represents to a great extent the results of his advice and assistance. The work was made possible by a grant from the Department of Scientific and Industrial Research.
The application of quadratic forms in an infinity of variables to boundary problems in partial differential equations
Authors: Steen S. W. P. 
Published: November 2008

The object of this paper is to find the characteristic functions and the characteristic numbers of the partial differential equationvalid in a domain G, and where on Γ, the boundary of G. The method employed is to transform two quadratic forms to their common self-conjugate “triangle“ of reference. The solution to the problem is given by this method in a simple manner, without the use of the integral equation theory, or the use of minimal sequences.
The tangent planes to a degenerate surface
Authors: Roth L. 
Published: November 2008

1. In the Lehrbuch der abzählenden Methoden Zeuthen considers the problem of determining the nature and multiplicity of the tangent lines and planes to a degenerate surface in ordinary space. The surface φ in question consists of n arbitrary planes; these meet in pairs, forming double lines, and in threes, forming triple points. The tangent cone to φ from an arbitrary point consists of plane pencils, each counted twice, as follows from the known theory of degenerate plane curves. According to Zeuthen the tangent planes are of two types:(i) those passing through one of the triple points; there are thus such systems, and each is to be counted six times;(ii) those passing through certain points S lying on the double lines; these depend upon the passage to the limit, it being assumed that φ is obtained as the limiting form of some nondegenerate surface. Two such points lie on each double line, so that there are n(n−1) such systems of planes.
An integral equation
Authors: Hardy G. H. , Titchmarsh E. C. 
Published: November 2008

1. The integral equationwhere x, f (x), and λ are real and α positive, may be regarded as a differential equation of order α. Suppose for example that α is a positive integer p, that f (x) tends to 0, when x → ∞, with sufficient rapidity, and thatThen, if we integrate repeatedly by parts, and write z for fp (x), (1·1) becomesThe only solutions are finite combinations of exponentials.
The sums of powers as simultaneous canonical expressions
Authors: Bronowski J. 
Published: November 2008

1. In this paper I discuss the expression of m + 1 general forms F, F′, …, F(m), each of order n and homogeneous in r + 1 variables z0, z1, …, zl, …, zr, as the sums each of the nth powers of (the same) h + H linear forms in these variables. I take h > 0 of these linear forms to be undetermined, namely the formswhose coefficients are undetermined; and I take the remaining H ≥ 0 linear forms to be assigned, namely the formswhose coefficients are assigned.
On the minimum modulus of integral functions
Authors: Cartwright M. L. 
Published: November 2008

1·1. Let f (z) be an integral function of order ρ, and letIt is well known that, if ρ < 1,This result was first conjectured by Littlewood, who proved it with cos 2πρ in place of cosπρ; it was afterwards proved independently by Wiman and Valiron about the same time. Many other authors have written on the subject, considering among other things the set of valves of r for which m (r) is comparatively large.
Notes on the Theory of Series (XVIII): on the Convergence of Fourier Series
Authors: Hardy G. H. , Littlewood J. E. 
Published: November 2008

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We writebeing the complex Fourier series of f (θ).
Photomagnetic disintegration and magnetic moment of the deuteron in the meson theory
Authors: Ma S. T. 
Published: November 2008

The interaction between an external electromagnetic field and a nuclear system can be expressed in terms of the multipole moments. The electric quadripole and the magnetic dipole moments of the deuteron have been calculated, taking into account the exchange forces as given by the meson theory. The cross-section of the photomagnetic effect of the deuteron has been calculated.This work was carried out under the guidance of Dr Heitler and Dr Fröhlich. The writer wishes to express his sincerest thanks to them for suggesting the problem and many valuable comments. The writer is also indebted to Dr Kahn for discussions during the early stages of this work.
On the quartic surface
Authors: Segre B. 
Published: November 2008

Summary1. The projective transformations of F into itself … 1212. The flecnodal curve, and the lines of F … 1223. A geometric characterization of F, and the six different types of F in the real domain … 1234. The τ-points and τ-planes, and a notation for the lines of F … 1245. The incidence conditions for the lines of F … 1256. The tetrads of the first kind … 1267. The tetrads of the second kind … 1268. The pairs of lines of F … 1279. The tetrads of the third kind … 12910. The 16-tangent quadrics of F … 13011. The conics of the first kind … 13112. The conics of the second kind … 13213. No other irreducible conics lie on F … 13314. The tangent planes of F of multiplicity greater than 3 … 13615. On twisted curves, especially cubics and quartics, lying on F … 13816. The T-transformations … 13917. Construction of an infinite discontinuous group of birational transformations of F into itself … 14218. Deduction of an infinity of unicursal curves lying on F … 143
Thermal Properties of an Incompletely Degenerate Fermi Gas
Authors: Mott N. F. 
Published: November 2008

The purpose of this note is to calculate the specific heat and paramagnetic susceptibility of an electron gas obeying the Fermi-Dirac statistics for all temperatures, including those temperatures for which the gas is partially degenerate. The results are applicable to the electrons in a metal, whether free or moving in a periodic field, provided only that the number of electronic states per gram atom with energy between E and E + dE can be expressed in the formas for free electrons.
A generalization of Abel summability
Authors: Kuttner B. 
Published: November 2008

The Abel sum of the series can be written in the formwhereThis suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the formwhereA1(u) may conveniently be described as the “Abel transform” of A(u).
The second order electrical effects in metals
Authors: Wilson A. H. 
Published: November 2008

The theory of transport phenomena in metals depends upon the solution of an integral equation for the velocity distribution function f of the conduction electrons. This integral equation is formed by equating the rate of change in f due to external fields and temperature gradients to the rate of change in f due to the mechanism which produces the resistance. If this latter rate of change is denoted by [∂f/∂t]coll it happens with some mechanisms thatwhere f0 is the equilibrium distribution, and ℸ is the time of relaxation which does not depend on the external fields. When equation (1) is true, the problem is comparatively simple, but in general [∂/∂t]coll is an integral operator and it is not possible to define a time of relaxation and a free path. It is known that at high temperatures, such that (Θ/T)2 can be neglected, where Θ is the Debye temperature, a free path exists; but, in general, special methods have to be used to solve the integral equation.
Lorentz-invariance in quantum theory
Authors: Eddington Arthur 
Published: November 2008

The formal conditions of Lorentz-invariance apply to coordinates x1, x2, x3, x4 which form a 4-vector, and to expressions associated with them. In this paper attention is called to the frequent disregard in quantum investigations of the proviso that the coordinates form a 4-vector.In most quantum investigations with a practical application the coordinates employed are relative space coordinates ξ, η, ζ coupled with a progressive time coordinate, so that (ξ, η, ζ, it) is not a 4-vector. Nevertheless, conditions of Lorentz-invariance are applied by many authors. Alternatively, they attempt to base the investigation on wave functions of non-relative coordinates x, y, z, t. It is here pointed out that such wave functions give no information about the eigenstates, and that there is no means of deriving wave functions of ξ, η, ζ from those of x, y, z.Special attention is directed to the misapplication of Lorentz-invariance in Dirac's theory of the hydrogen atom, and in the calculation of degeneracy energy of ionized material.