Power Series Representation Research Articles

Correction of instrumental distortion by analytical deconvolution of data

A general, analytical theorem developed by van de Hulst for inverting the convolution integral is reviewed and illustrated both with synthetic data and with experimental data from time-of-flight measurements. If the undesired influence of an instrument used in an experimental measurement can be represented by the convolution integral, the original undistorted or ’’true’’ distribution may sometimes be recovered in post-processing the data by means of deconvolution. Analytic deconvolution is achieved by using the coefficients from a power series representation of the distorted output distribution and a set of ’’solving polynomials’’ which may be readily derived from the response function of the instrument. The distortion due to the excitation pulse duration in time-of-flight molecular beam data has been removed by application of the theorem. Some possible advantages of this analytical method over conventional deconvolution techniques are discussed.

Extensions of von Neumann’s method for generating random variables

Von Neumann’s method of generating random variables with the exponential distribution and Forsythe’s method for obtaining distributions with densities of the form e − G ( x ) {e^{ - G(x)}} are generalized to apply to certain power series representations. The flexibility of the power series methods is illustrated by algorithms for the Cauchy and geometric distributions.

Variational calculation of dynamic polarizabilities

The finite field method of obtaining static multipole polarizabilities is extended to the calculation of dynamic polarizabilities. The first-order frequency dependent wavefunction is obtained variationally using a trial function, λ(ω)ψ 1(0). ψ 1(0) is the first-order wavefunction obtained in the static calculation. The results are applied to the approximation of the dynamic polarizability at imaginary frequencies. These are then used to calculate the coefficients for the inverse power series representation of the interatomic potential energy. The method has been applied to two simple cases, hydrogen and helium. For hydrogen the present method yields results close to the exact values for the static dipole polarizability and the dipole-dipole van der Waals coefficient; C 6. In the case of helium a simple Hartree-Fock function was perturbed but the results are still encouraging with the dipole-dipole van der Waals coefficient, C 6, calculated to within 7% of the accurate value.

On the theory for closed shell intermolecular forces

The electron gas model of Gordon and Kim is applied to the interaction of inert gases with a hydrogen molecule. Comparison of the isotropic potential wells with available experimental data shows clearly that modification of the exchange potential along the lines suggested by Rae produces greatly improved agreement when simple power series representations of the dispersion energy are included.

An analytical solution for the laminar flow over a flat plate with similarity preserving suction

An exact analytical solution is presented for the laminar boundary-layer flow over a semi-infinite flat plate subjected to a type of similarity preserving suction. The solution is developed for the case of a plate immersed in either a uniform compressible stream with viscosity proportional to temperature or a uniform incompressible stream with constant viscosity. The problem is formulated in Crocco's variables. It is described by a second-order, non-linear, ordinary differential (and singular) boundary-value problem for the shear stress as a function of the velocity in the boundary layer. A unique solution is shown to exist and to possess a power series representation for all magnitudes of suction. The series is constructed explicitly and provides a transcendental equation for the shear stress at the plate (the important skin friction) which can be solved to any desired accuracy. Examples of upper and lower bounds for the wall shear are presented for several magnitudes of suction and confirm the reasonable accuracy of results obtained heretofore only by numerical solutions of the problem. In addition to the intrinsic value of the technique developed, it can be the basis of accurate checks for the numerical solution of more complex problems.

On the moments of the distribution of widths corresponding to an ensemble of random matrices

In this work, we discuss the moments of the width distribution corresponding to an ensemble of random matrices. The ensemble considered is one where the real and imaginary parts of the Hamiltonian matrices have Gaussian distributions with different widths. In particular, we give exact expressions for the moments when N = 2 (N = dimension of the Hamiltonian submatrix with a fixed set of quantum numbers) and give a method for obtaining a power series representation when N is arbitrary.

A theory of high frequency vibrations of piezoelectric cyrstal bars

This paper presents a higher order, linear theory of piezoelectric crystal bars as in the same spirit as that of Mindlin. First, a power series representation in aerial coordinates is employed for both the mechanical displacement and electric potential fields. Next, with the help of a variational theorem deduced from Hamilton's principle, together with these series, the theory is established consistently. A hierarchy of 1-dimensional approximate equations of motion, charge equations of electrostatics, initial and boundary conditions, strain-displacement and electric field-electric potential relations, and macroscopic constitutive equations constitutes the theory, and it governs all the types of motions of piezoelectric crystal bars of uniform cross-section. Further, special cases of interest are pointed out. The solutions of the initial mixed boundary-value problems defined by this theory are proven to be unique.

Fixed point of polynomial type nonlinear operators

A concept of generalized polynomial operators is considered, and a fixed-point problem with these operators is posed. The existence of a fixed point, with a minimum norm property, is stated and a power series representation is obtained. Problems of this kind appear, for instance, in some two-point boundary-value problems in optimal control.

Generalized Euler-Pochhammer integral representation for single-loop Feynman amplitudes

Kershaw's power-series representation for the single-loop Feynman amplitudes is cast into a new integral representation which, as a generalization of the Euler-Pochhammer type, is closer in spirit to the Veneziano representation than the conventional Feynman-parameter representation. Landau singularities, which are obscure in the power-series expansion, are recovered in the new integral representation. Explicit calculation is carried out for the cases $N=3$ and $N=4$.

Inversion of generalized power series representation

The aim of this concise paper is to present an extension of the local inverse theorem of polynomial type operators, recently considered by the same authors in [l-4] for nonlinear systems analysis. This extension gives a local inversion of certain operators which are represented by generalized power series. The homogeneous operator terms (i.e., powers) of the series are defined algebraically via multilinear mappings. In this generality, theory of polynomial operators and power series is previously introduced by, e.g., Michal [5] and Cartan [6], by using a somewhat different presentation. However, the results concerning the inversion are not presented anywhere, as far as the authors know. Among others, a general recursive representation is obtained which gives the homogeneous operators of the inverted power series successively with increasing orders. The main theorem is presented here without proof because it can be performed by a pedantic modification of the corresponding one in [l, 31.

Low temperature thermocouples: KP, "normal" silver, and copper versus Au-0.02 at% Fe and Au-0.07 at% Fe

The Seebeck thermoelectric voltages of two dilute alloys of iron in gold, Au 0.02 at% Fe and Au-0.07 at% Fe, have been determined with respect to KP (a particular Ni-Cr alloy), "normal" silver, and copper in the temperature range from 4 to 280 K. The power series representation of these data, along with the calculated Seebeck coefficients and derivatives of the Seebeck coefficients, have been extrapolated to 0 K and are presented as a function of temperature. In addition to these reference data, seven different Au-0.07 at% Fe alloys were thermoelectrically intercompared in order to determine the variability in wires from different melts and from different manufacturers. The largest deviation found amounted to about 9 percent of the output of a KP versus Au-0.07 at% Fe thermocouple pair between 4 and 20 K. A more typical variation for this temperature range was 2 to 4 percent. Initial indications are that the reference data can be adjusted satisfactorily with data from spot calibrations on particular wires. The effect of heat treatment is illustrated by comparing our results to Rosenbaum's data for annealed and unannealed specimens of both Au-Fe alloys.

Neutron Diffusion from a Fast Point Source in Aqueous Absorbing Solutions

Transport parameters (migration area, age to indium resonance) of fast neutrons from a plutonium -beryllium source have been measured in aqueous absorbing solutions at several temperatures (35, 40, 55, and 75°C), using boric acid as the 1/ absorber. For the measurements at 35 and 40°C, the saturation concentrations of boric acid were attained at 70 and 80 g/liter, respectively.For a 1/ absorber, a temperature-dependent power series representation of k2 in terms of absorption cross section ∑ao was proposed, based on the concept of neutron temperature. The temperature range wherein such an expansion remains valid was experimentally determined.It was found that strong concentrations of a 1/ absorber caused much difficulty in experimentally resolving the thermal neutron spatial distributions, an observation which might have a direct relation to the (∑t)min limit of Corngold.

Power series representation of partial derivatives required in orbit determination

A program to integrate the equations of motion by series in powers of the time step can be easily modified to furnish the elements of the matrizant in power series of the time step. In particular, if the series representing the motion are obtained recursively, differentiation of the recurrence relations will provide immediately a recursive scheme for computing the coefficient in the power series for the elements of the matrizant.

Asymptotics for a Class of Nonanalytic Second Order Differential Equations

New formulas giving asymptotic behavior (at $\infty $) of the solutions to $y'' + py' + qy = 0$ on $(a,\infty ]$ are developed. The coefficient functions p and q must satisfy certain regularity and relative rate of growth conditions, but it is not required that they be analytic nor that they possess asymptotic power series representations.

An Iterated Logarithm Theorem for Some Weighted Averages of Independent Random Variables

for all m < n, provided I t t Cn < 1 . Let f be a real-valued function which is continuous on [0, 1], and define Sn = k lf(k/n)Xk. Then (2) lim supn,,00,(2n log log n) <Sn _ f II a.e. where ||fl| = (flf22(x) dx)-. Furthermore, any of the following conditions is sufficient to ensure that equality holds in (2): (i) f is a polynomial: (ii) f is an absolutely continuous or monotone function which can be written as a power series f(x) = Zo akXk on [0, 1] where lim sup la,'ln1, |a < 0o, ja. = f(1) = 0. (iii)f has a power series representation with radius of convergence greater than 1.

Ideal fluid flow in an enclosure of toroidal geometry

Ideal fluid flow in a space of toroidal geometry is studied with the goal of obtaining steady state solutions for flow patterns useful in the design of turbomachinery. Potential flow solutions of Laplace's equation referred to a bi-polar toroidal coordinate system are shown to be inadmissible for application in the turbomachinery field since such solutions do not include the possibility of momentum transfer. Attention is, therefore, focused on rotational flow. The governing equations are referred to a “polar toroidal” coordinate system. The problem is reduced to the solution of a homogeneous, non-linear, third order partial differential equation for a stream function with an associated set of boundary conditions. An inner boundary (the core) is determined by prescribing a suitable fluid velocity distribution at the outer boundary (the shell). Solutions of the stream function equation are obtained by a power series representation valid in some neighborhood of the outer boundary. This results in a set of linear first order ordinary differential equations which can be solved consecutively, and exhibits explicitly the connection between the fluid velocity and the body forces. Some numerical examples of particular flows are presented. When the velocity distribution at the outer boundary (the shell) is uniform a singular flow region appears around the center of the meriodional section, bounded by a critical streamline; only a small portion of the flow circulates completely around the meridional section.

Multiple fourier series analysis for mixers and modulators

Well-known results of double-Fourier-series analysis for bilinear or linear/ square-law mixers operating in the switching mode are generalized to permit computation of output product levels in devices subjected to an arbitrary number of input signals and possessing conducting-region nonlinearities requiring power series representation.

Distribution and Frequency Dependence of Incidental Man-Made HF/VHF Noise in Metropolitan Areas

Evaluation and consolidation have been performed on HF and VHF man-made incidental radio noise data obtained from journal publications and corporate reports released during the past seventeen years by several investigators. This summary represents an extention to 20 MHz of a compilation and analysis of metropolitan area incidental radio noise data initially begun for the 200- to 500-MHz range. The data sources included in the present summary represent the noise environments of metropolitan areas located in Australia, Asia, Europe, and North, Central, and South America. The noise regions, Urban, Suburban I, and Suburban II, previously identified for the 200- to 500-MHz frequency range and representing concentric zones in urbanized areas have been found appropriate for classifying the HF/VHF noise data. Prior to graphically consolidating the information, all data were converted into units of noise power in dB relative to one mW/kHz of receiver bandwidth. To the consolidated data of each noise region has been fitted a power series representation. It has been found necessary, because of differences in magnitude, to segregate the noise region data into two group: one set recorded with quasi-peak field intensity meters, and the other obtained with detectors calibrated to measure available power.

Drift elimination in system identification using binary m sequences

It is shown that, by crosscorrelating over two periods only of the output when using msequences to identify linear time-invariant systems, the elimination of the effect of drift up to the (N − 2)th power in the power-series representation of the distrubrance is possible, where Nδ is the period of the sequence. The method is more complex than that proposed by Brown.

Method of minimising the effect of disturbances on the estimate of the impulse response of a linear system

A method is proposed whereby the errors in the estimate of the impulse response, obtained from pseudorandom-binary-signal correlation experiments, due to low-frequency drift at the output of the system may be reduced. In particular, errors due to a constant, a linear, a quadratic and a cubic term in the power-series representation of the noise are eliminated. Moreover, the bias on the estimate of the impulse response, due to the nonzero value of the autocorrelation of the binary sequence for nonzero delays, is eliminated.