bluuebunny/crossref_metadata_2025_split huggingface dataset:
An Analysis of Triode Valve Rectification. Part II
Authors: Landale S. E. A. 
Published: November 2008

Generally, when an alternating voltage is applied to a cumulative grid rectifier no grid current flows at mean grid potential. The behaviour of a rectifier working under these conditions is examined. An expression is derived for calculating rectified current for any applied voltage. As this equation is rather cumbersome to apply, a very simply empirical formula is given which is applicable for any value of applied potential whatever.An expression is derived for the power absorbed by the rectifier. It is shown that as the applied voltage increases, the apparent resistance of the rectifier decreases and approaches half the value of the grid leak resistance.Further, it is shown that rectified current depends on the peak value of the applied potential and that it is almost independent of ordinary wave-form variations, even when the applied voltage is small.By slightly modifying the expression for rectified current we find that, in an amplifier, the grid current is a measure of V and, in an oscillator, the grid current is a measure of the output. The only condition is that V > 2b in both cases.In conclusion I wish to express my indebtedness to Professor C. E. Inglis for placing at my disposal the facilities of the Cambridge University Engineering Laboratory, and to Mr E. B. Moullin for the interest he has taken in the work.
The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra
Authors: Hartree D. R. 
Published: November 2008

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the formand require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.
Note on the theory of series (XXIII): On the partial sums of Fourier series
Authors: Hardy G. H. , Littlewood J. E. 
Published: November 2008

1. In what follows f(θ) is a periodic function of L2,is the Fourier series of f(θ), andis the nth partial sum of T(θ). We denote by n(θ) any function of θ which is measurable, is finite p.p. †, and assumes non-negative integral values only, and by n(θ, H) an n(θ) none of whose values exceeds H. We shall sometimes write N for n(θ), and N(H) for n(θ, H), to simplify the set-up of formulae.
Scattering of charged mesons under the influence of radiation damping
Authors: Ma S. T. , Hsüeh C. F. 
Published: November 2008

SummaryAn alternative derivation of the integral equation of Waller, Heitler and Wilson is given on the basis of the stationary method of the perturbation theory. The previous theoretical treatment of the scattering of charged vector mesons by nuclear particles through the nuclear interaction is extended to pseudoscalar mesons. Numerical results calculated from the theory are given for the total and differential scattering cross-sections.
The Stability of Viscous Fluid Flow under Pressure between Parallel Planes
Authors: Goldstein S. 
Published: November 2008

If we consider, by the method of small oscillations, the stability of a viscous fluid flow in which the undisturbed velocity is parallel to the axis ofxand its magnitudeUis a function ofyonly (x, y, zbeing rectangular Cartesian co-ordinates), and if we assume that any possible disturbance may be analysed into a number (usually infinite) of principal disturbances, each of which involves the time only through a single exponential factor, then it has been proved by Squire, by supposing the disturbance analysed also into constituents which are simple harmonic functions ofxandz, and considering only a single constituent, that if instability occurs at all, it will occur for the lowest Reynolds number for a disturbance which is two-dimensional, in thex, yplane. Hence only two-dimensional disturbances need be considered. The velocity components in the disturbed motion will be denoted by (U+u, v). Since only infinitesimal disturbances are considered, all terms in the equations of motion which are quadratic inuandvare neglected. Whenuandvare taken to be functions ofymultiplied byei(αx−βi), the equation of continuity becomesand the result of eliminating the pressure in the equations of motion then gives the following equation forv, where ν is the kinematic viscosity of the fluid:
On the Stability Problem in Hydrodynamics
Authors: Pekeris Chaim L. 
Published: November 2008

AbstractThe paper deals with the problem of the effect of two-dimensional first order disturbances on the linear and parabolic flows of a viscous fluid. If the laminar flow is in the direction x and is bounded by the planes y = ±h, and if the stream function of the disturbance is assumed to be of the form , it is found that, for small Reynolds numbers R (= Uh/v), C can be developed in the power seriesThe first term in (1) corresponds to a disturbance in still water and is known from Rayleigh's investigation of that problem; the succeeding C's are real and can be obtained by an application of Schrödinger's perturbation theory of wave mechanics. In the case of the parabolic flow, expressions for C0 and C1 were thus derived for both the symmetrical and anti-symmetrical types of disturbance.The imaginary part of C, as represented by in this approximation, is found to be positive. When α is of the order of unity, the value of R which makes the second term of the first is about 50 for the symmetrical disturbance and 100 for the antisymmetrical disturbance. C0, the first term in the expression for the phase velocity, is less than 1 and increases with α, which implies a group velocity greater than the phase velocity.In the case of the linear flow it is found that the C's with even subscripts vanish, implying a vanishing phase velocity. This verifies the assumption to that effect made by Southwell and Chitty, and it also suggests that the region of convergence of (A) is limited to the range of values of R covered by these authors. Expressions are derived for C1 and C3 and are evaluated for α=1. The formulae thus obtained for the damping constant give an initial decrease with R, which is not shown in the corresponding curve of Southwell and Chitty.An alternative method of determining C is outlined in the last section. Use is made of the orthogonal system of the characteristic functions for the vorticity of a disturbance in still water. The equation determining C is reduced to an infinite determinant in which C occurs only in the diagonal elements. When R→0, the determinant can be evaluated and the roots C are found to coincide with the first two terms in (A).Finally it is shown that the two-dimensional laminar flow is stable for values of R less than the smaller of the quantities 2α and α3.
Note on a theorem enunciated by Todd
Authors: Du Val P. 
Published: November 2008

J. A. Todd has enunciated the following theorem:“If the canonical adjoints ∑ of an irreducible curve Γ contain fixed parts, then, if Γ is not elliptic, all the fixed curves are simple, rational, determined uniquely by the base points, which present independent conditions for them, and have no free intersections with each other or with the variable part of the system ∑.”
The disintegration of nitrogen by slow neutrons
Authors: Burcham W. E. , Goldhaber M. 
Published: November 2008

The disintegration of nitrogen by slow neutrons has been studied in photographic emulsions of different sensitivity, which enable an unambiguous distinction to be made between the emission of α-particles and protons. Evidence has been obtained that the disintegration takes place according to the reactionwith a cross-section of about 10−24 cm.2
Note on the derivation of fluctuation formulae for statistical assemblies
Authors: Bartlett M. S. 
Published: November 2008

1. A familiar device in the study of statistical distributions is to form the moment-generating functionwhere the bar denotes averaging over all values of the statistical variate x. The moments μr of x are the coefficients of αr/r!, and the derived coefficients in the expansion of K ≡ log M are termed the semi-invariants kr. In particular,and, for the normal (Gaussian) law,we have the simple formula
Note on a Divergent Series
Authors: Hardy G. H. 
Published: November 2008

1. The seriesis the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, whenbyForsay;andoraccording as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.
On Hausdorff's methods of summability. II
Authors: Rogosinski W. W. 
Published: November 2008

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transformsof a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).
On certain expansions of the solutions of Mathieu'S differential equation
Authors: Erdélyi A. 
Published: November 2008

1. There are several known types of expansions of Mathieu functions, i.e. mod 2π periodic solutions of Mathieu's equation ((9), chap. 19),The simplest expansion is the Fourier seriesAlmost equally well known are Heine's expansion ((4), p. 414; see also (5) and (6))and
On the stability of crystal lattices VIII. Stability of rhombohedral Bravais lattices
Authors: Peng H. W. , Power S. C. 
Published: November 2008

The main purpose of the paper is an investigation of the stability of a certain class of Bravais lattices, namely, those with a rhombohedral cell of arbitrary angle. The potential energy is assumed to consist of two terms, each proportional to a reciprocal power of the distance. In the continuous series of lattices obtained by changing the rhombohedral angle, there are included the three cubic Bravais lattices, the simple (s), the face-centred (f) and the body-centred (b) lattices. It is shown that (f) and (b) correspond to a minimum of the potential energy, and (s) to a maximum. A method for calculating the potential energy for the intermediate rhombohedral lattices is developed, and, with the help of a certain characteristic function, it is shown by numerical calculation that the (f) lattice corresponds to the absolute minimum of potential energy, and that no extrema, other than (f), (s) and (b), exist. In the last section, the case of a compound (non-Bravais lattice) is considered, and it is shown that the equilibrium and stability conditions for the law of force assumed can be divided into one set for change of volume, and an independent set for change of shape.We take this opportunity of expressing our sincere thanks to Prof. Born for his interest in our work, and for much valuable advice.
An explanation of anomalous thermionic emission current constants
Authors: Nien-T'ai Sun , Band William 
Published: November 2008

AbstractIt is shown that anomalously large and anomalously small values of the constant A in the thermionic current formula for metals can be explained by the usual statistical equilibrium theory provided proper account is taken of the fact that free electrons may be shared by two competing overlapping energy bands. When this theory is developed there arises a comparatively large temperature variation of the statistical parameter ζ which occurs in the Fermi function, and when the exact formula is forced into the empirical form with a constant work function, the temperature factor appears as a multiplying factor of the current constant, this factor being greater or less than unity depending upon the nature of the overlapping bands. In particular the observed values of the constant A for nickel and for hafnium are well explained.
Some inequalities in the theory of functions
Authors: Hayman W. K. 
Published: November 2008

In this paper we investigate the following problem.We suppose given a sequence of complex values wn, defined for n = 0, 1, 2, …, and for n = ∞, and such thatwhile at least one wn differs from zero and ∞. We consider functions f(z), which are regular in | z | < 1, and take none of the sequence of values wn, and we investigate the effect of this restriction on the rate of growth of the function, as given by the maximum modulus
The interaction between floating particles
Authors: Nicolson M. M. 
Published: November 2008

A quantitative study of the law of force between bubbles floating on a fluid has been made and the results exhibited in the form of a potential energy-distance curve. The main approximations made are that the inclination of the surface of the fluid is small (so that the equations for the surface become linear), and that the submerged portion of the bubble remains spherical. The latter assumption restricts the validity of the results to small bubbles of about 3·0 mm. diameter and less.One object of the work was to determine the size of bubble which simulated most nearly a typical atomic interaction curve. This application arises in connexion with Prof. Bragg's ‘bubble’ model of crystal lattices. Exact simulation cannot be expected, for the bubble model is two-dimensional, and also does not exert forces analogous either to Coulomb repulsion or to the metallic bond. It can be shown, however, that the law of force between bubbles of a particular size is a remarkably close fit to the law of force between inert gas atoms, that is, to an interaction consisting of a van der Waals attraction and a short-range or ‘overlap’ repulsion. Fig. 4 illustrates this point. The two full curves are the energy-distance curves for krypton and argon atoms (plotted from Fowler, Statistical Mechanics, 2nd ed., Chapter x, Table 29). The circles mark the corresponding curve for bubbles with α = 1·0 (i.e. of diameter 3 mm. in soap solution). The units of energy and distance are chosen so as to bring the minimum of each curve to the point (1, − 50). It will be seen that the curves are remarkably similar over the range shown.
On the theory of statistical procedures
Authors: Pitt H. R. 
Published: November 2008

The essential elements in any application of statistical theory are(1) the set of hypothetical physical states θ which may give rise to the observed phenomena,(2) the set of all possible observations or samples x,(3) the set of actions α which may be taken when x is known.The whole procedure in the common applications can be divided into two parts:(A) the estimation of θ from the known observation x,(B) action on the basis of this estimation.
Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problem of the plane and related problems
Authors: Elliott H. A. 
Published: November 2008

The method of transforms (7), (2) is shown to be directly applicable to the case of axial symmetric stress distributions in hexagonal crystals. It is shown that solutions for problems of indentation of the hexagonal plane by rigid punches can be found for punches of arbitrary axial symmetric shape. Solutions are given in full for the cases of spherical, conical and circularly cylindrical punches.The same method is used to find the solutions for a material containing disk-shaped cracks between hexagonal planes and the results for the isotropic case deduced from the general solution.
Markoff chains with an enumerable number of states and a class of cascade processes
Authors: Foster F. G. 
Published: November 2008

AbstractIn § 1 a Markoff chain is defined, and a theorem of Kolmogoroff relating to its asymptotic behaviour is stated. Its stable distributions are examined in § 2 and some further results are obtained. These are then applied in §§ 3, 4 to a certain generalization of the cascade process, regarded as a Markoff chain with a special kind of matrix.
On the statistical distribution of the widths and spacings of nuclear resonance levels
Authors: Wigner Eugene P. 
Published: November 2008

AbstractIf the average spacing of the resonance levels is very small as compared with the range of energy in which the spacing or width of the levels changes appreciably on the average, one can speak of a statistical distribution of the level spacings and widths. The question then comes up naturally, whether the ‘distribution law’ for width and spacing is different for protons, neutrons, etc., i.e. whether it depends on the long-range interaction of the scattered particle. It is pointed out that while the average width must depend on the long-range interaction (on account of the penetration factor) the distribution of the widths, if these are measured in terms of their average, can be expected to be independent therefrom. In the case of the level spacings, not only the distribution about the average, but the average itself is also independent of the long-range interaction. It is pointed out, incidentally, that the logarithmic derivative of the wave function at the surface of the nucleus will have, on the whole, a positive value between resonances because of the greater effect of the high-energy resonances than that of the bound states.
Factorization of the algebra of particles of half-odd spin
Authors: Le Couteur K. J. 
Published: November 2008

AbstractIt is proved that the matrix algebra for any relativistic wave equation of half-odd integral spin can be factorized as the direct product of a Dirac algebra and another, called a ξ-algebra. The structure and representation of ξ-algebras are studied in detail. The factorization simplifies calculations with particles of spin > ½, because the ξ-algebra contains only one-sixteenth as many elements as the original matrix algebra.
Steady motion of conducting fluids in pipes under transverse magnetic fields
Authors: Shercliff J. A. 
Published: November 2008

ABSTRACTThis paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.
The couple on a rotating spheroid in a slow stream
Authors: Sowerby L. , Rosenhead L. 
Published: November 2008

AbstractThe problem considered here is to determine the couple required to maintain the steady rotation about its axis of symmetry of an oblate spheroid which is placed in a uniform stream of viscous fluid moving parallel to this axis. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. An expression in terms of spheroidal wave functions is derived for the transverse component of velocity in the fluid, and from this the expression for the couple is deduced. The disk rotating about the normal through its centre is considered as a special case, and some numerical results are given for certain values of a Reynolds number. A comparison is made, and agreement found, with results obtained by Jeffery in a previous paper which treated the case in which the mainstream was at rest.
The asymptotic distribution of serial correlation coefficients for autoregressive processes with dependent residuals
Authors: Walker A. M. 
Published: November 2008

In a recent paper (3) Diananda has given an extension of the central limit theorem to stationary sequences of random variables {Yt} (t = 1,2,3,…), which arem–dependent, i.e. are such thatm+ 1 is the smallest integerrhaving the property that two sets of the Y's are independent whenever the suffix of any member of one set differs from that of any member of the other set by at leastr. As Diananda points out, this can be used to show that for linear autoregressive processes withm–dependent residuals, the joint distribution of any finite number of the forms which occur in Bartlett and Diananda's goodness of fit test (see (1)) is asymptotically multivariate normal, provided also that the fourth moment of the residuals is finite. These forms are linear functions of the sample serial correlations rt, and if the autoregressive process is defined bythey may be denoted by, whereEtbeing the usual shift operator such that for any function ftdefined for integral t,In particular, when, as for a moving average process, the joint distribution of any finite number of the rt, is asymptotically multivariate normal.
Covariant integral equations for Heisenberg operators
Authors: Eden R. J. 
Published: November 2008

ABSTRACTSets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews & Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.
The ehrenfests' wind-wood model in two dimensions
Authors: Haar D. Ter , Green C. D. 
Published: November 2008

ABSTRACTThe consideration of the statistical aspect of the second law of thermodynamics of a previous paper is extended. The influence of fluctuations in the wind-wood model of the Ehrenfests is investigated, and the relative occurrence, mean life and time of recurrence of such fluctuations are calculated. These are found to agree with the results obtained for the simpler models considered previously.