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Two computer programs developed according to two general types of exponential models for conducting nonlinear exponential regression analysis are described. Least squares procedure is used in which the nonlinear problem is linearized by expanding in a Taylor series. Program is written in FORTRAN 5 for the Univac 1108 computer.
A computer-oriented technique is presented for performing a nonlinear exponential regression analysis on decay-type experimental data. The technique involves the least squares procedure wherein the nonlinear problem is linearized by expansion in a Taylor series. A linear curve fitting procedure for determining the initial nominal estimates for the unknown exponential model parameters is included as an integral part of the technique. A correction matrix was derived and then applied to the nominal estimate to produce an improved set of model parameters. The solution cycle is repeated until some predetermined criterion is satisfied.
When incorporating formulations of instantaneous solar heating or photolytic rates as functions of altitude and sun angle into long range forecasting models, it may be desirable to replace the time integrals by daily average rates that are simple functions of latitude and season. This replacement is accomplished by approximating the integral over the solar day by a pure exponential. This gives a daily average rate as a multiplication factor times the instantaneous rate evaluated at an appropriate sun angle. The accuracy of the exponential approximation is investigated by a sample calculation using an instantaneous ozone heating formulation available in the literature.
A theoretical analysis is presented of certain peculiarities of the current-voltage characteristics of silicon solar cells, involving high values of the empirical constant A in the diode equation for a p-n junction. An attempt was made in a lab experiment to demonstrate that the saturation current which is associated with the exponential term qV/A2kT of the I-V characteristic, with A2 roughly equal to 2, originates in the space charge region and that it can be increased, as observed on ATS-1 cells, by the introduction of additional defects through low energy proton irradiation. It was shown that the proton irradiation introduces defects into the space charge region which give rise to a recombination current from this region, although the I-V characteristic is, in this case, dominated by an exponential term which has A = 1.
Failure times of a software reliabilty growth process are modeled as order statistics of independent, nonidentically distributed exponential random variables. The Jelinsky-Moranda, Goel-Okumoto, Littlewood, Musa-Okumoto Logarithmic, and Power Law models are all special cases of Exponential Order Statistic Models, but there are many additional examples also. Various characterizations, properties and examples of this class of models are developed and presented.
Failure times of a software reliability growth process are modeled as order statistics of independent, nonidentically distributed exponential random variables. The Jelinsky-Moranda, Goel-Okumoto, Littlewood, Musa-Okumoto Logarithmic, and Power Law models are all special cases of Exponential Order Statistic Models, but there are many additional examples also. Various characterizations, properties and examples of this class of models are developed and presented.
Adopting the formalism of Parsons and Daly (1983), analytical integral equations (Green's function integrals) are derived which relate gravity anomalies and dynamic boundary topography with temperature as a function of wavenumber for a fluid layer whose viscosity varies exponentially with depth. In the earth, such a viscosity profile may be found in the asthenosphere, where the large thermal gradient leads to exponential decrease of viscosity with depth, the effects of a pressure increase being small in comparison. It is shown that, when viscosity varies rapidly, topography kernels for both the surface and bottom boundaries (and hence the gravity kernel) are strongly affected at all wavelengths.
A mechanism for the formation of the exponential stellar disk distribution in spiral galaxies is proposed. It is assumed that the gas which falls in to form the disk behaves like an accretion disk which evolves on a viscous time scale t(v) and forms stars on a time scale t(star). It is demonstrated that if t(v) roughly equals t(star) the resulting stellar disk has an exponential distribution independent of the disk rotation law and of the assumed viscosity prescription. Why the viscous and star formation time scales might be related is briefly commented on.
Creep tests conducted on NaCl single crystals in the temperature range from 373 to 1023 K show that true steady state creep is obtained only above 873 K when the ratio of the applied stress to the shear modulus is less than or equal to 0.0001. Under other stress and temperature conditions, corresponding to both power law and exponential creep, the creep rate decreases monotonically with increasing strain. The transition from power law to exponential creep is shown to be associated with increases in the dislocation density, the cell boundary width, and the aspect ratio of the subgrains along the primary slip planes. The relation between dislocation structure and creep behavior is also assessed.
One-point and two-point exponential functions have been developed and proved to be very effective approximations of structural response. The exponential has been compared to the linear, reciprocal and quadratic fit methods. Four test problems in structural analysis have been selected. The use of such approximations is attractive in structural optimization to reduce the numbers of exact analyses which involve computationally expensive finite element analysis.
The point exponential approximation method was introduced by Fadel et al. (Fadel, 1990), and tested on structural optimization problems with stress and displacement constraints. The reports in earlier papers were promising, and the method, which consists of correcting Taylor series approximations using previous design history, is tested in this paper on optimization problems with frequency constraints. The aim of the research is to verify the robustness and speed of convergence of the two point exponential approximation method when highly non-linear constraints are used.
This paper presents an application of matrix exponentiation via Krylov subspace projection, to the solution of fluid dynamics problems. The main idea is to approximate the operation exp(A)v by means of a projection-like process onto a Krylov subspace. This results in a computation of an exponential matrix vector product similar to the one above but of a much smaller size. Time integration schemes can then be devised to exploit this basic computational kernel. The motivation of this approach is to provide time-integration schemes that are essentially of an explicit nature but which have good stability properties.
An application of matrix exponentiation via Krylov subspace projection to the solution of fluid dynamics problems is presented. The main idea is to approximate the operation exp(A)v by means of a projection-like process onto a krylov subspace. This results in a computation of an exponential matrix vector product similar to the one above but of a much smaller size. Time integration schemes can then be devised to exploit this basic computational kernel. The motivation of this approach is to provide time-integration schemes that are essentially of an explicit nature but which have good stability properties.
It is well known that the Fourier series of an analytic or periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. Here, we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed.
A systematic examination is presented of the observational effects that can provide information on the departure of the cosmic-ray path length distribution (PLD) for propagation in the Galaxy from a simple exponential distribution. The possible physical causes for this departure are examined in order to set realistic limits on how well a simple exponential PLD actually describes the distribution of sources and propagation of cosmic rays in the galaxy. From the findings, it is argued that the cosmic ray source distribution must be quite uniform, probably on a scale of about 50 pc or less near the sun. Extending this uniformity to the Galaxy as a whole would imply more than 100,000 active sources. The cosmic-ray propagation can be described quite well by a leaky box model which is equivalent to observing the particles through a dense fog.
A medium-sized band model for water vapor and CO2 absorption is developed using the one-parameter scaling approximation. The infrared spectrum is divided into 10 bands. The Planck-weighted diffuse transmittance is reduced to a function dependent only upon the scaled absorber amount and fit by an exponential sum. By selecting specific sets of absorption coefficients for exponential-sum fitting, computations of fluxes and cooling rate are made very fast. Compared to a broadband model, the accuracy, speed, and versatility are all enhanced. With absorption due to water vapor line, continuum, CO2 as well as O3 included, the parameterization introduces an error of less than 1.5 W/sq m in fluxes and less than 0.15 C/day in the tropospheric and lower stratospheric cooling rates.
An airplane in steady rectilinear flight was assumed to experience an initial disturbance in rolling or yawing velocity. The equations of motion were solved to see if it was possible to hasten recovery of a stable airplane or to secure recovery of an unstable airplane by the application of a single lateral control following an exponential law. The sample computations indicate that, for initial disturbances complex in character, it would be difficult to secure correlation with any type of exponential control. The possibility is visualized that the two-control operation may seriously impair the ability to hasten recovery or counteract instability.
The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.