Orthogonal Polynomial Series Research Articles

Dispersion of Ion Cyclotron Waves in Magnetoplasmas

Dispersion of waves propagated at frequencies near the ion cyclotron frequency in collision-dominated plasmas subjected to axial magnetic fields is investigated. Radial and axial propagation constants, satisfying Maxwell's equations, and including the contributions of binary elastic ion-ion and ion-electron interparticle forces to an ion conductivity tensor, are calculated. Elements of the ion conductivity tensor are found analytically by expressing the ion distribution function in the Boltzmann equation as a series of orthogonal Laguerre polynomials in velocity space with time-dependent expansion coefficients. Right-handed and left-handed elliptically polarized waves are found from a numerical solution for a range of values of the ion cyclotron frequency about the fixed wave frequency of 7.62 Mc. The axial wavelengths λzL, λzR, and damping lengths λDL, λDR are calculated as functions of the magnetic field for a plasma density of 1021 m−3, and temperature of 104, 105 and 106 °K. Enhanced dispersion occurs with increasing temperature. Stronger dispersion is associated with the left-handed elliptically polarized wave; for 105 ° and 106 °K, above resonance λDL ≫ λzL, whereas below λDL ≈ λzL. Ion-ion collisions broaden the resonance width and decrease absorption at resonance.

Microwave Conductivity of a Plasma in a Magnetic Field

The Boltzmann equation for electrons in a uniform isothermal plasma is solved by expressing the distribution function as a series of orthogonal polynomials in velocity space with time dependent expansion coefficients. The microwave conductivity is simply related to certain coefficients. Particular attention is devoted to the case in which the plasma is subject to a constant magnetic field and a microwave electric field. By introducing an "effective" electron temperature, convergence is attained for strong as well as weak electric fields. The formulation is particularly suited for problems involving partially ionized gases which contain several species of ions and neutrals. The conductivity of a completely ionized gas is calculated with and without consideration of electron-electron collisions, and the ratio (${\ensuremath{\gamma}}_{E}$) of the two results is graphically illustrated as a function of microwave frequency.

Lower bounds for the degree of approximation

is the optimal degree of approximation of 2W. In this paper we shall give simple methods which permit to find the order of magnitude of Dn(W) for several important classes ?1: for some classes of analytic functions (?6); for the unit ball AP+a of the space CP+a of functions with continuous derivatives of order p, which satisfy a Lipschitz condition with exponent a, 0 <a< 1 (?5). We also consider approximation of continuous functions (??2, 3) and give results about condensation of singularities (Theorems 2, 7). The main content of this paper consists of results which show that standard means (trigonometric approximation, series of orthogonal polynomials) give the best possible approximation, at least up to a bounded factor. Since estimates of Dn(21) from above follow from classical results [1; 5], we are interested in estimating Dn(2I) from below. Clearly, results of this type are the better, the smaller the norm used in the definition (1). This is why most of our theorems are for the LI-norm. Kolmogorov [2] (see also [6]) discussed D-n(2) in the L2-norm, and gave an asymptotic formula for it when 2 is the class of functionsf on an interval with f(P) bounded in the L2-norm. Originally the present paper was written to improve (for linear approximation) the

Thermal Buckling of Solid Wings of Arbitrary Aspect Ratio

SUMMARY The thermal buckling of solid wings with all edges free is analyzed by the Rayleigh-Ritz method. First wings of large aspect ratios are treated and the effect is neglected. A double series of orthogonal polynomials is assumed for the de­ flection function, and by the use of successively increasing num­ bers of terms the predominance of the simple torsional distortion is established for a number of typical temperature distributions for wings of constant thickness and of parabolic cross section. The effect of variation of aspect ratio in this analysis is also ex­ amined. The validity of the conclusion about the predominance of the simple torsional distortion is then verified by a buckling analysis for a wing of constant thickness and small aspect ratio, using an approximate solution of the thermal-stress problem of a finite rectangular plate. Having been established as a good approximation, the tor­ sional analysis is now modified to include wings of small aspect ratio also. It is shown that the effects of chordwise normal stresses and shear stresses cancel out, and that the critical tem­ perature of a finite wing can be obtained from that of an infinite wing by multiplication by an end effect factor. This factor is calculated for various temperature distributions for wings of constant thickness, parabolic cross section and double wedge shape, and shown to be approximately the same for different typical temperature distributions and nearly unaffected by the shape of the cross section. The effect is found to be signifi­ cant (more than 10 per cent for aspect ratios below 6), but can be included with very little additional work.

An Equivalence Theorem for Series of Orthogonal Polynomials

An Equivalence Theorem for Series of Orthogonal Polynomials

Societies and Academies

EDINBURGH Royal Society, December 2. J. WEIR and D. LEITCH: Zonal distribution of the non-marine lamellibranchs in the coal measures of Scotland. All zones from ovalis to tennis are represented in the Scottish Coal Measures, although they cannot all be delimited. The Productive Coal Measures fall within the ovalis, modiolaris and lower similis-pulchra zones, and are therefore homotaxially equivalent to the lower part of the Middle Coal Measures of Lancashire. In the Central Coalfield the boundary between ovalis and modiolaris zones is taken at the Kiltongue Musselband, and between modiolaris and similis-pulchra zones at the Musselband Coal. Correlations between Central, Douglas and Ayrshire coalfields are deduced. S. CHAPMAN: The lunar atmospheric tide at Glasgow. Reasons are given for believing that the lunar hourly inequalities of barometric pressure at Glasgow recently determined by Robb and Tannahill are substantially not of lunar origin at all (see NATURE, Nov. 16, p. 801). An earlier determination of the lunar atmospheric tide at Glasgow made in 1926 by Robb is now published with his permission, and appears to be a true lunar effect. It indicates that at Glasgow the lunar atmospheric tide is abnormally small for the latitude; the phase seems normal, and corresponds to a lag of high tide in the atmosphere of about one hour after lunar transit. ENID CHARLES: The effect of present trends in fertility and mortality upon the future population of Scotland and upon its age composition. Excluding the effects of migration, estimates of the population of Scotland during the next hundred years have been made on two assumptions, (a) that fertility and mortality remain at their present level, and (6) that they continue to decline at the rate shown in the past decade. According to (a), the population will begin to decline slowly about 1970, being 94 per cent of its present size in a hundred years. According to (6) the population will begin to decline about 1950 and thereafter will diminish rapidly, being 19 per cent of its present size in a hundred years. The effects of past declining fertility and of present low fertility will be seen in a continually decreasing proportion of children and increasing proportion of persons aged sixty years and more. The excess of females due to war losses will be replaced by an excess of males in 1960 by both estimates. In fifty years the increasing proportion of the older age groups will result in a higher crude death rate, between 18 and 21 per thousand. A. C. AITKEN and H. T. GONIN: Fourfold sampling with and without replacement. The correlation is examined of characters in a sample drawn at random from a limited population, the individuals of which are classed in four categories, as possessing two characters or their alternatives. In the cases both of replacement and of non-replacement of sampled individuals, the probability is found to be expressible as a terminating series in orthogonal polynomials. These are briefly studied, and applications are made to problems of regression and moments.

On the development of functions in series of orthogonal polynomials

On the development of functions in series of orthogonal polynomials

The summation of series of orthogonal polynomials

The summation of series of orthogonal polynomials

Series of Orthogonal Polynomials

Series of Orthogonal Polynomials