Showing papers on "Gaussian published in 1984"

Compact effective potentials and efficient shared‐exponent basis sets for the first‐ and second‐row atoms

Abstract: Compact effective potentials, which replace the atomic core electrons in molecular calculations, are presented for atoms in the first and second rows of the periodic table. The angular‐dependent components of these potentials are represented by compact one‐ and two‐term Gaussian expansions obtained directly from the appropriate eigenvalue equation. Energy‐optimized Gaussian basis set expansions of the atomic pseudo‐orbitals, which have a common set of exponents (shared exponents) for the s and p orbitals, are also presented. The potentials and basis sets have been used to calculate the equilibrium structures and spectroscopic properties of several molecules. The results compare extremely favorably with corresponding all‐electron calculations.

Universal coding, information, prediction, and estimation

Abstract: A connection between universal codes and the problems of prediction and statistical estimation is established. A known lower bound for the mean length of universal codes is sharpened and generalized, and optimum universal codes constructed. The bound is defined to give the information in strings relative to the considered class of processes. The earlier derived minimum description length criterion for estimation of parameters, including their number, is given a fundamental information, theoretic justification by showing that its estimators achieve the information in the strings. It is also shown that one cannot do prediction in Gaussian autoregressive moving average (ARMA) processes below a bound, which is determined by the information in the data.

Extended Gaussian images

Abstract: This is a primer on extended Gaussian images Extended Gaussian images are useful for representing the shapes of surfaces They can be computed easily from: 1 needle maps obtained using photometric stereo; or 2 depth maps generated by ranging devices or binocular stereo Importantly, they can also be determined simply from geometric models of the objects Extended Gaussian images can be of use in at least two of the tasks facing a machine vision system: 1 recognition, and 2 determining the attitude in space of an object Here, the extended Gaussian image is defined and some of its properties discussed An elaboration for nonconvex objects is presented and several examples are shown

A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations

Abstract: A justification is given for the use of non-spreading or frozen gaussian packets in dynamics calculations In this work an initial wavefunction or quantum density operator is expanded in a complete set of grussian wavepackets It is demonstrated that the time evolution of this wavepacket expansion for the quantum wavefunction or density is correctly given within the approximations employed by the classical propagation of the avarage position and momentum of each gaussian packet, holding the shape of these individual gaussians fixed The semiclassical approximation is employed for the quantum propagator and the stationary phase approximation for certain integrals is utilized in this derivation This analysis demonstrates that the divergence of the classical trajectories associated with the individual gaussian packets accounts for the changes in shape of the quantum wavefunction or density, as has been suggested on intuitive grounds by Heller The method should be exact for quadratic potentials and this is verified by explicitly applying it for the harmonic oscillator example

Asymptotics of Graphical Projection Pursuit

Abstract: Mathematical tools are developed for describing low-dimensional projections of high-dimensional data. Theorems are given to show that under suitable conditions, most projections are approximately Gaussian.

Optimum quantizer performance for a class of non-Gaussian memoryless sources

Abstract: The performance of optimum quantizers subject to an entropy constraint is studied for a wide class of memoryless sources. For a general distortion criterion, necessary conditions are developed for optimality and a recursive algorithm is described for obtaining the optimum quantizer. Under a mean-square error criterion, the performance of entropy encoded uniform quantization of memoryless Gaussian sources is well-known to be within 0.255 bits/sample of the rate-distortion bound at relatively high rates. Despite claims to the contrary, it is demonstrated that similar performance can be expected for a wide range of memoryless sources. Indeed, for the cases considered, the worst case performance is observed to be less than 0.3 bits/sample from the rate-distortion bound, and in most cases this disparity is less at Iow rates.

The capacity of the white Gaussian multiple access channel with feedback

Abstract: Since the appearance of [10] by Gaarder and Wolf, it has been well known that feedback can enlarge the capacity region of the multiple access channel. In this paper a deterministic feedback code is presented for the two-user Gaussian multiple access channel, which is shown to allow reliable communication at all points inside a region larger than any previously obtained. An outer bound is given which is shown to coincide with the achievable region, thus yielding the capacity region of this channel exactly.

Crossings of non-gaussian translation processes

Abstract: Mean upcrossing rates are determined for translation processes obtained from normal processes by univariate, nonlinear transformations. Monotonic and more general transformations are studied. It is shown that translation processes can have any marginal distribution and autocorrelation function and that approximations proposed previously for the mean upcrossing rate of non‐Gaussian processes can be unsatisfactory. These approximations assume that the process and its time‐derivative, considered to follow a Gaussian distribution, are independent. Theoretical findings are applied to determine crossing characteristics of wind speeds, river flows, and other non‐Guassian processes.

Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations

Abstract: The statistical moments of a non-linear system responding to random excitations are governed by an infinite hierarchy of equations; therefore, suitable closure schemes are needed to compute the more important lower order moments approximately. One easily implemented and versatile scheme is to set the cumulants of response variables higher than a given order to zero. This is applied to three non-linear oscillators with very different dynamic properties, and with Gaussian white noises acting as external and/or parametric excitations. It is found that the accuracy of computed second moments can be improved greatly by extending from the second order closure (Gaussian closure) to the fourth order closure, and that further refinement is unnecessary for practical purposes. Treatment of nonstationary transient response is also illustrated.

Time delay estimation by generalized cross correlation methods

Abstract: The problem of estimating time delay by cross correlation methods is reexamined for the whole class of stationary signals. Expressions are derived for the estimation mean square error (MSE) by the cross correlation method, and are shown to be identical to previously published results for Gaussian signals. The generalized cross correlation method is also analyzed, and the optimal weight function for this method is derived. It is shown to be identical to that derived for Gaussian signals by the maximum likelihood method. For the cross correlation method a simplified MSE expression is derived, which is to be used instead of a previously published result.

Differential detection of Gaussian MSK in a mobile radio environment

Abstract: Minimum shift keying with Gaussian shaped transmit pulses is a strong candidate for a modulation technique that satisfies the stringent out-of-band radiated power requirements of the mobil radio application. Numerous studies and field experiments have been conducted by the Japanese on urban and suburban mobile radio channels with systems employing Gaussian minimum-shift keying (GMSK) transmission and differentially coherent reception. A comprehensive analytical treatment is presented of the performance of such systems emphasizing the important trade-offs among the various system design parameters such as transmit and receiver filter bandwidths and detection threshold level. It is shown that two-bit differential detection of GMSK is capable of offering far superior performance to the more conventional one-bit detection method both in the presence of an additive Gaussian noise background and Rician fading.

Random walk modelling of diffusion in inhomogeneous turbulence

Abstract: The use of random walk techniques in diffusion modelling has been very successful when the turbulence is not too inhomogeneous. In this paper the difficulties associated with inhomogeneous turbulence are investigated, and a random walk model is developed which ensures the correct steady state distribution of particles in phase space. The model requires the generation of random numbers from non-Gaussian distributions, and this makes the model difficult to apply in cases of severe inhomogeneity. An alternative model is presented which overcomes this problem in cases where the velocity distribution is Gaussian.

Signal/noise separation and velocity estimation

Abstract: A signal/noise separation must recognize the lateral coherence of geologic events and their statistical predictability before extracting those components most useful for a particular process, such as velocity analysis Events with recognizable coherence we call signal; the rest we term noise Let us define “focusing” as increasing the statistical independence of samples with some invertible, linear transform L By the central limit theorem, focused signal must become more non‐Gaussian; the same L must defocus noise and make it more Gaussian A measure F defined from cross entropy measures non‐Gaussianity from local histograms of an array, and thereby measures focusing Local histograms of the transformed data and of transformed, artificially incoherent data provide enough information to estimate the amplitude distributions of transformed signal and noise; errors only increase the estimate of noise These distributions allow the recognition and extraction of samples containing the highest percentage of sig

An Optimal Global Nearest Neighbor Metric

Abstract: A quadratic metric dAO (X, Y) =[(X - Y)T AO(X - Y)]? is proposed which minimizes the mean-squared error between the nearest neighbor asymptotic risk and the finite sample risk. Under linearity assumptions, a heuristic argument is given which indicates that this metric produces lower mean-squared error than the Euclidean metric. A nonparametric estimate of Ao is developed. If samples appear to come from a Gaussian mixture, an alternative, parametrically directed distance measure is suggested for nearness decisions within a limited region of space. Examples of some two-class Gaussian mixture distributions are included.

Evaluation of the use of Gaussian phi(roz) curves in quantitative electron probe microanalysis: a new optimization

Abstract: A computer program based on the use of Gaussian expressions for the x-ray distribution with depth [+(pz) curves] was tested on its usefulnw for quantitative electron probe microanalysis. As the good results originally claimed for a similar program could not be reproduced initiaUy, it was subjected to a detailed analysis. As a result, some modi6cations in the approach are proposed. Apart from increasing the speed of calculation considerably, the modified expressions provide a better insight into the delicate balance which has to exist between the relevant quantities involved. After a new optimization process the modified program was tested on about 450 published microanalyses. The results show that the +(pz) approach is indeed very promising as a narrow histogram with an r.m.s. value of 5.4% could be produced. Finally, some suggestions are made for future improvements.

First and second derivatives of two electron integrals over Cartesian Gaussians using Rys polynomials

Abstract: Formulas are developed for the first and second derivatives of two electron integrals over Cartesian Gaussians Integrals and integral derivatives are evaluated by the Rys polynomial method Higher angular momentum functions are not used to calculate the integral derivatives; instead the integral formulas are differentiated directly to produce compact and efficient expressions for the integral derivatives The use of this algorithm in the ab initio molecular orbital programs gaussian 80 and gaussian 82 is discussed Representative timings for some small molecules with several basis sets are presented This method is compared with previously published algorithms and its computational merits are discussed

Generalized rays in first-order optics: Transformation properties of Gaussian Schell-model fields

Abstract: Propagation characteristics of Gaussian Schell-model fields through first-order optical systems and in free space are analyzed by the method of generalized rays. This allows the development of a simple geometrical description of these processes. The invariance of the degree of global coherence is established in full generality. Asymptotic behavior under free propagation and the emergence of a far-zone universal structure are analyzed. New invariants associated with incoherent superpositions of such fields are found.

Characterization and estimation of two-dimensional ARMA models

Abstract: A class of finite-order two-dimensional autoregressive moving average (ARMA) is introduced that can represent any process with rational spectral density. In this model the driving noise is correlated and need not be Gaussian. Currently known classes of ARMA models or AR models are shown to be subsets of the above class. The three definitions of Markov property are discussed, and the class of ARMA models are precisely stated which have the noncausal and semicausal Markov property without imposing any specific boundary conditions. Next two approaches are considered to estimate the parameters of a model to fit a given image. The first method uses only the empirical correlations and involves the solution of linear equations. The second method is the likelihood approach. Since the exact likelihood function is difficult to compute, we resort to approximations suggested by the toroidal models. Numerical experiments compare the quality of the two estimation schemes. Finally the problem of synthesizing a texture obeying an ARMA model is considered.

The two-dimensional Gaussian beam synthetic method: Testing and application

Abstract: The Gaussian beam method of Cervený et al. (1982) is an asymptotic method for the computation of wave fields in inhomogeneous media. The method consists of tracing rays and then solving the wave equation in “ray-centered coordinates.” The parabolic approximation is applied to find the asymptotic local solution in the neighborhod of each ray. The approximate global solution for a given source is then constructed by a superposition of Gaussian beams along nearby rays. The Gaussian beam method is tested in a two-dimensional inhomogeneous medium using two approaches. One is the application of the reciprocal theorem for Green's functions in an arbitrarily heterogeneous medium. The discrepancy between synthetic seismograms for reciprocal cases is considered as a measure of the error. The other approach is to apply Gaussian beam synthesis to cases for which solutions are known by other approximate methods. This includes the soft basin problem that has been studied by finite difference, finite element, discrete wavenumber, and glorified optics. We found that the results of these tests were in general satisfactory. We have used the Gaussian beam method for two applications. First, the method is used to study volcanic earthquakes at Mount Saint Helens. The observed large differences in amplitude and arrival time between a station inside the crater and stations on the flanks can be explained by the combined effects of an anomalous velocity structure and a shallow focal depth. The method is also applied to scattering of teleseismic P waves by a lithosphere with randomly fluctuating velocities.

Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams

Abstract: Summary. The Gaussian beam method has recently been introduced into synthetic seismology to overcome shortcomings of the ray method, especially in transition regions due to focusing or diffraction where ray theory fails. One proceeds by discretizing the initial data as a superposition of paraxial Gaussian beams, each of which is then traced through the seismic environment. Since Gaussian beam fields do not diverge in ray transition regions, they are ‘uniformly regular’ although the quality of this regularity depends on the beam parameters and on the ‘numerical distance’ which defines the extent of the transitional domain. However, when Gaussian beam patches are used to simulate non-Gaussian initial data, there arise ambiguities due to choice of patch size and location, beam width, etc., which are at the user's disposal. The effects of this arbitrariness have customarily been explored by trial and error numerical experiment but no quantitative recommendations have emerged as yet. As a step toward a priori predictive capability, it is proposed here to perform a systematic study on analytically tractable prototype models of how the parameters and location of a single beam affect the quality of the observed seismic field, especially in ray transition regions. The conversion of ordinary ray fields into beam fields in canonical configurations can be accomplished conveniently by displacing a real source point into a complex coordinate space. Thus, the desired beam solutions can be obtained directly from available ray, and even paraxial ray, fields. Complex ray theory and its implications are reviewed here, with an emphasis on improvements of beam tracking schemes employed at present.

The Radon transform and its properties

Abstract: This paper presents a summary of the fundamental properties of the Radon transform, including delay effects, data shifting, rotation, scaling, windowing, bowtie events, energy conservation, etc., and includes representative examples on standard data sets such as 2-D delta functions, boxcar events, Gaussian bell, and conic sections which reinforce the basic concepts of the Radon transform.

The effects of focusing and refraction on Gaussian ultrasonic beams

Abstract: A scalar theory of the propagation of Gaussian ultrasonic beams through lenses and interfaces is presented. For radiation into a fluid, the Fresnel approximation is employed to derive the laws of propagation of Guassian beams (previously employed in the analysis of coherent optical systems). These are then generalized to situations commonly found in nondestructive evaluation by treating the effects of propagation through lenses and through curved interfaces at oblique incidence. A numerical example illustrates the ease with which insight into diffraction phenomena for complex geometries can be gained by this approach. The limitations imposed on the theory by aberrations and the scalar assumption are discussed, and the relationship of the Gaussian theory to the radiation of piston transducers is explored.

The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions

Abstract: We produce a self contained account of the relationship between the Gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for x is one of the by-products.

Packetlike solutions of the homogeneous-wave equation

Abstract: We have obtained packetlike solutions to the free-space homogeneous-wave equation. These solutions are Gaussian Laguerre (or Hermite) beams that propagate in a straight line at a light velocity remaining focused for all time.

Compact contracted Gaussian‐type basis sets for halogen atoms. Basis‐set superposition effects on molecular properties

Abstract: Compact, contracted Gaussian basis sets for halogen atoms are generated and tested in ab initio molecular calculations. These basis sets have similar structure to that of Huzinaga and co-workers' (HTS) sets; however, they give both better atomic total energies and better properties of atomic valence orbitals. These sets, after splitting of valence orbitals and augmenting with polarization functions, provide molecular results that agree well with those given by extended calculations. Basis set superposition error (BSSE) is calculated using the counterpoise method. BSSE has only slight influence on calculated equilibrium geometry, shape of potential curve, and electric properties (dipole and quadrupole moments) of molecules. However, atomization energies may be significantly changed by the BSSE.

Techniques for measuring 1-μm diam Gaussian beams

Abstract: This paper reviews various techniques for measuring the diameter of Gaussian beams and, in particular, those of ~1-μm diam. A description of measurement techniques for nonideal conditions is also included. A novel ruling used for beam-size measurement is discussed.