Showing papers on "Gaussian published in 1973"

Statistical Dynamics of Classical Systems

Abstract: The statistical dynamics of a classical random variable that satisfies a nonlinear equation of motion is recast in terms of closed self-consistent equations in which only the observable correlations at pairs of points and the exact response to infinitesimal disturbances appear. The self-consistent equations are developed by introducing a second field that does not commute with the random variable. Techniques used in the study of the interacting quantum fields can then be employed, and systematic approximations can be obtained. It is also possible to carry out a "charge normalization" eliminating the nonlinear coupling in favor of a dimensionless parameter which measures the deviation from Gaussian behavior. No assumptions of spatial or time homogeneity or of small deviation from equilibrium enter. It is shown that previously inferred renormalization schemes for homogeneous systems were incomplete or erroneous. The application of the method to classical microscopic systems, where it leads from first principles to a coupled-mode description is briefly indicated.

Maximum likelihood identification of Gaussian autoregressive moving average models

Abstract: SUMMARY Closed form representations of the gradients and an approximation to the Hessian are given for an asymptotic approximation to the log likelihood function of a multidimensional autoregressive moving average Gaussian process. Their use for the numerical maximization of the likelihood function is discussed. It is shown that the procedure described by Hannan (1969) for the estimation of the parameters of one-dimensional autoregressive moving average processes is equivalent to a three-stage realization of one step of the NewtonRaphson procedure for the numerical maximization of the likelihood function, using the gradient and the approximate Hessian. This makes it straightforward to extend the procedure to the multidimensional case. The use of the block Toeplitz type characteristic of the approximate Hessian is pointed out.

General contraction of Gaussian atomic orbitals: Core, valence, polarization, and diffuse basis sets; Molecular integral evaluation

Abstract: Generally contracted Gaussian basis functions are defined as those for which each contracted function may have a nonzero contribution from each primitive Gaussian. Alternatives for choice of such bases are tested and guidelines proposed. The basis is compared with standard methods in current use and is shown to be superior in terms of energy lowering obtained per additional basis function beyond a minimal number. A new program for computation of the required multicentered integrals is described.

Hermite–gaussian functions of complex argument as optical-beam eigenfunctions

Abstract: Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite–gaussian eigenfunctions ψn (x,z) consisting of a hermite polynomial of real argument Hn [√2x/w(z)] times the complex gaussian function exp [−jkx2/2q(z)], in which q(z) is a complex quantity. This note shows that an alternative and in some ways more-elegant set of eigensolutions to the same basic wave equation is a hermite-gaussian set ψˆn(x,z) of the form Hn[√cx]exp [−cx2], in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)1/2x. The conventional functions ψn are orthogonal in x in the usual fashion. The new eigenfunctions ψˆn, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions ϕˆn(√cx). The new eigenfunctions ψˆn are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψˆn functions are not spherical for n > 1. However, they may still be useful as a basis set for other optical resonator and beam-propagation problems.

Joint measures and cross-covariance operators

Abstract: Let H1 (resp., H2) be a real and separable Hilbert space with Borel o-field r1 (resp., r2), and let (H1 x H2, r, x r2) be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on (H1 x H2, rJ x r2), i.e., joint measures, and the projections of such measures on (H1, rl) and (H2, r2). In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are ob- tained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized. (*) IH ~~~~~~~~~llx 11 2 djLi(x) < oo

Polymer conformational statistics. III. Modified Gaussian models of stiff chains

Abstract: Several attempts have been made to account for polymer chain stiffness or excluded volume effects in non‐Newtonian viscosity theory with the use of a modified Gaussian probability for intrachain distances. This method is here tested for the freely rotating chain by an application to the equilibrium expectation of the end‐to‐end vector R in the presence of an external potential −R· f. The Gaussian parameters are varied as f increases to keep Gaussian estimates of the bond length and bond angle constant. Numerical tests are made for the limits of a freely jointed chain and the worm chain, and the modified Gaussian approach is found to work very well.

Gaussian light beams in inhomogeneous media.

Abstract: Vector wave solutions are obtained for the propagation of beams of light in media having slow spatial variations of the gain, loss, or index of refraction. The formalism developed here is applicable to a wide range of problems, and an example considered in detail is the propagation of off-axis beams in lenslike laser materials and optical waveguides. A procedure is also described for the diagnosis of localized dielectric inhomogeneities such as plasmas by means of Gaussian laser beams.

Uncertainty, calibration and probability: The statistics of scientific and industrial measurement

Abstract: Uncertainties and frequency distributions The Gaussian distribution General distributions Rectangular distributions Applications Distributions ancillary to the Gaussian A general theory of uncertainty The estimation of calibration uncertainties Consistency and significance tests Method of least squares Theorems of Bernoulli and Stirling and the binomial, Poisson and hypergeometric distributions Appendices Bibliography Index

Asymptotic inference in stationary Gaussian time-series

Abstract: Conditions are given for the family of distributions of a stationary, discrete-time, Gaussian, vector-valued time-series with covariance structure given up to a finite number of parameters to satisfy the asymptotic differentiability conditions introduced by Le Cam (1969).

Simulation of Multicorrelated Random Processes Using the FFT Algorithm

Abstract: A technique for the digital simulation of multicorrelated Gaussian random processes is described. This technique is based upon generating discrete frequency functions which correspond to the Fourier transform of the random processes and then using the fast Fourier transform (FFT) algorithm to obtain the actual random processes. The main advantage of this method over other methods is computation time; it appears to be more than an order of magnitude faster than present methods of simulation. One of the main uses of multicorrelated simulated random processes is in solving nonlinear random vibration problems by numerical integration of the governing differential equations. [This research is supported in part by NASA.]

Correlated Gaussian wavefunctions for H3

Abstract: Quantum chemical calculations have been performed on H3+ using correlated Gaussian wavefunctions of a type suggested by Singer. The resulting upper bound energy, E = −1.34335 a.u., is lower than all previous results except those of Conroy. Vibrational frequencies were calculated to be ω1 = 3272 cm−1 (symmetrical stretch) and ω2 = 2735 cm−1 (bending mode). The kinetic energy, nuclear repulsion energy, interelectronic repulsion energy, nuclear‐electronic attraction energy, virial, and squared electronic position expectation values were computed.

Computer fitting of Voigt profiles to Raman lines

Abstract: A FORTRAN program has been developed for the accurate computation of the convolution of a Lorentzian with a Gaussian line shape (the Voigt function). This program has been included in a larger program written by the author to resolve overlapping lines in complex Raman spectra. It has been found that the program is very useful to determine accurate line widths and peak positions from slit-distorted spectra, which is important in high-resolution Raman spectroscopy.

Diode‐pumped solid‐state lasers

Abstract: Diode pumping of solid‐state lasers is analyzed. The analysis differs from more conventional analysis by the inclusion of a detailed calculation of the pumping rate and by the use of a Gaussian transverse intensity distribution as well as a circular approximation. Explicit expressions are obtained for the slope efficiency and the threshold. The temperature dependence of both parameters is investigated. Experimental results are obtained with a Nd : YAG diode‐pumped laser. These results are compared with the predictions of the analysis. The experimental results yielded a laser power output of 52 mW in TEM00 modes at 269°K.

Response of the impact damper to stationary random excitation

Abstract: The response of an impact damper system to an excitation with approximately white‐power spectral density and Gaussian probability distribution is determined, using two independent methods: digital computer and electronic‐analog techniques. Results are given for mean‐squared level, power spectral density, probability density, probability distribution, and amplitude probability density of the response. The impact damper is found to be a practical and efficient device for reducing the response amplitude of systems subjected to random excitation.

Study of self‐focusing damage in a high‐power Nd: glass‐rod amplifier

Abstract: Self‐focusing damage in a pattern of Fresnel fringes is observed in the final element of a high‐power glass laser amplifier as the result of the truncation of a Gaussian input beam. Detailed calculations give good agreement with the location and pattern of damage.

On joint detection, estimation and system identification: discrete data case†

Abstract: The recent results of Lainiotis (1971 a, b, 1971) on single-shot, as well as multishot, joint detection, estimation and system identification for continuous data and dynamics are extended to multishot, discrete data and discrete dynamical systems. The results are given for the signals generated by the linear dynamical systems with unknown parameter vectors and driven by white gaussian sequences, where the observation contains additive white gaussian noise. Specifically, it is shown that the above problem constitutes a class of non-linear mean-square estimation problems. By utilizing the adaptive approach, closed-form integral expressions are obtained for simultaneously optimal detection, estimation and system identification. In addition, several approximate algorithms that utilize linear Kalman estimators are presented to limit the storage requirement to finite size and reduce computational requirements. The results presented in this paper are applicable to both independent and Markov signalling sources

Calculation of molecular one-electron properties using coupled Hartree-Fock methods: II. The water molecule

Abstract: Various second-order properties for H2O have been computed using a near Hartree-Fock gaussian wave function and a coupled Hartree-Fock scheme. Satisfactory agreement with experimental data is obtai...

Even‐tempered atomic orbitals. IV. Atomic orbital bases with pseudoscaling capability for molecular calculations

Abstract: Even‐tempered atomic orbital bases are formulated which have the property that atomic orbital scaling can be closely simulated through variation of linear expansion coefficients. This ``adaptation to pseudoscaling'' involves two types of adjustments: (1) Optimal orbital exponents and basis sizes are determined for the even‐tempered exponential or Gaussian primitives and (2) optimal linear combinations of even‐tempered Gaussian primitives are found to serve as a ``reduced basis of contracted functions.'' The efficiency of the method is discussed.

No N4-dependence in the calculation of large molecules

Abstract: It is shown that the time needed for computing large molecules is proportional to N2 · (lnN)2, when Gaussian functions are used and the molecule is larger than a well-definable limit. Some proposals to overcome this limit are made.

Modified Gaussian functions and their use in quantum chemistry. I. Integrals

Abstract: A modified Gaussian function g(u, v, w, a, R) = const s(a, R) is considered where l = u + v + w, s (a, R) is a 1s-type Gaussian function centered at R, a is the coefficient in the exponent of the 1 s Gaussian function and X, Y, Z are components of R. General formulae are derived for overlap integrals, kinetic energy integrals, nuclear attraction integrals, and electron repulsion integrals, valid for any l. The formulae are much simpler than those derived by Huzinaga for Cartesian Gaussian functions.

Unified Analysis of a Class of Digital Systems in Additive Noise and Interference

Abstract: A unified approach is presented for evaluation of the error probabilities of a class of digital communications systems in additive noise and interference. This class of systems includes coherent systems such as M -ary amplitude-shift keying (ASK), M -ary phase-shift keying (PSK), and M -ary amplitude-and-phase keying (APK); it also includes differential coherent systems such as binary differential PSK (DPSK). The noise is not necessarily Gaussian. The interference can be intersymbol interference, co-channel interference, adjacent-channel interference, any of their linear combinations, or intermodulation prodducts at the output of some nonlinear device. This approach essentially expands the characteristic function of the interferences into a power series so that the desired error probability can be evaluated as the sum of terms representing perturbations around the error probability due to additive noise alone. Bounds on three kinds of truncation errors, which are simple and applicable to all aforementioned digital systems, are obtained. As a result, any desired accuracy in the evaluation of the error probabilities can be achieved with this approach. In the special case in which the noise is Gaussian, explicit bounds on truncation errors are also obtained. Examples are given to illustrate how the unified analysis can be applied to evaluate the error probabilities of various digital systems. More specifically, the combined effects of Gaussian noise, intersymbol interference, and co-channel interference on the error performance of M -ary coherent PSK and APK (MCPSK and MCAPK) systems are computed. The probability of error of a binary DPSK (BDPSK) system in the presence of Gaussian noise and intersymbol interference is analyzed. The intermodulation products at the output of a hardlimiter are also determined.

Conformational statistics of short chains of poly(L‐alanine) and poly(glycine) generated by Monte Carlo method and the partition function of chains with constrained ends

Abstract: We have calculated properties of samples of short poly(L‐alanine) and poly(glycine) chains generated by Monte Carlo method. We analyzed various distributions and averages of these samples. The average square of the end‐to‐end distance and the average end‐to‐end vector were very close to the values calculated according to the method of Flory. The distribution of is very different from a Gaussian function for poly(L‐alanine) chains of even 20 units, already close to Gaussian for 10 unit poly(glycine) chains. The average fourth power of the end‐to‐end distance was compared with that calculated for a Kratky‐Porod type wormlike chain, and the agreement was found to be good for all chain lengths, i.e., including those for which the distribution of r 2 is decidedly non‐Gaussian. The distribution of end‐to‐end vectors r was found to be approximately cylindrically symmetrical about the average . This distribution shows a pronounced maximum on the axis parallel to . Poly‐L‐alanine chains with short end‐to‐end distances are quite rare. By analyzing the distribution of samples in which each chain contains at least one (two, three, four) residue(s) in the ``α‐helical'' conformation, it was determined that all chains with very short end‐to‐end distance have several residues in this conformation. This is confirmed by the observation that the calculated average energy of the chains is several kcal/mole higher at short end‐to‐end distance. The results are discussed in terms of the ``stiffness'' of these chains, and in terms of the possibility of loop formation. It is found that probabilities of loop formation of short chains calculated on the basis of a Gaussian distribution may be seriously in error. Finally, a new approach is proposed for using Monte Carlo calculations of this type in order to calculate the partition function for looped chains for any fixed relative position‐orientation of the first and last unit of the chain (including, of course, true cyclic structures).

On Absolute Continuity of Measures Corresponding to Homogeneous Gaussian Fields

Abstract: General questions of absolute continuity and singularity of Gaussian measures have been considered in works of Ya. Gaek [1], J. Feldman [2] and Yu. A. Rozanov 3]. However, in considering concrete Gaussian measures it is desirable to be able to answer these questions using only the defining characteristics of the corresponding processes. As is known, to solve the problem of absolute continuity and to find the density it is necessary to solve a certain operator equation, which for ordinary processes leads to a Fredholm integral equation of the first type. The existence of a solution of this equation ensures absolute continuity. But the question of the existence of solutions of such equations is very complex. Hence there arises the problem: to find conditions of absolute continuity of measures which do not involve the existence of a solution of the corresponding equations. For stationary processes, several conditions expressed in terms of correlation functions or spectral densities have been given by Rozanov [4], [3]. Other general conditions appear in the summary report of I. I. Gikhman and A. V. Skorokhod [6], as well as in the book by the same authors [7] (Chapter 7, 5). In the present paper, analogous conditions using only spectral functions and densities are found for homogeneous Gaussian fields. The authors have restricted themselves only to the case when the means of the Gaussian fields are equal to zero and the correlation functions differ. The case of identical correlation functions and distinct means is studied by M. I. Yadrenko in [8]. Combining the results of [8] with those of.this paper one can obtain conditions of absolute continuity of homogeneous fields for distinct means and correlation functions. To be especially noted is the case of isotropic Gaussian fields which are considered separately. Conditions of absolute continuity and singularity of measures corresponding to Gaussian random fields have not yet been studied sufficiently. In this connection note the works of Z. S. Zerakidze [9, [10], G. M. Molchan and Yu. I. Golosov [11].

SCLC theory for a Gaussian trap distribution

Abstract: A simplified one-carrier space charge limited conduction theory has been developed for a Gaussian trap distribution in. an insulator. It is shown that the distribution behaves much like a discrete trap level at low applied fields, although the current has a temperature dependent activation energy. At higher fields the current-voltage relation becomes superquadratic, but this does not account for the behaviour normally attributed to an exponential trap distribution.

Gauge invariant gaussian orbitals and the ab initio calculation of diamagnetic susceptibility for molecules

Abstract: It is shown that gauge terms can be introduced into the Gaussian functions used as the basis functions for an ab initio calculation of the energy of a molecule in the presence of a uniform magnetic field so that all the integrals become independent of the origin of the vector potential. The perturbation treatment of the diamagnetic susceptibility is considered in the molecular orbital approximation. The results show that the susceptibility can be calculated using only the unperturbed orbitals and their first-order corrections. All the integrals that arise can be expressed in terms of known functions.

Probability distributions for discrete Fourier spectra

Abstract: The paper analyses power-spectrum estimates obtained by fast-Fourier-transform techniques Distributions are obtained for data, augmented, where necessary, by sequences of zeros, and the effect of data smoothing on the reliability of the estimates is considered The effect of segment averaging is analysed and a joint probability distribution is derived for the resulting spectrum estimates The number of degrees of freedom per estimate can then be directly determined 1st- and 2nd-order moments of logarithmic spectra are derived which lead to confidence bands on the spectral estimates Frequency-domain smoothing is then considered, and it is shown, that, for specified lengths of Gaussian random data, this, unlike data smoothing, does not lead to a reduction in the number of degrees of freedom Finally, the general case of frequency smoothing followed by adjacent estimate averaging is analysed A factor is proposed for assessing loss of stability of such estimates Computer results are given which demonstrate the effects of several data windows and sets of frequency-smoothing coefficients Results in the appendixes show that loss in degrees of freedom is related to the eigenvalues of a specific covariance matrix