Approximate calculation of cumulative probability from a moment-generating function.
Cumulative probability distribution of positive random variable from moment generating function, exemplifying exponential and Poisson functions
Helstrom, C. W.↗
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Indefinite integrals of products of some exponential and trigonometric functions
Two integrals have been developed which are needed to solve certain problems of mathematical physics related to the Bessel and Euler equations. Integrals of this type have known application in problems using cylindrical coordinates.
Rostafinski, W. A.↗
Exponential estimates and the saddle point property for neutral functional differential equations
Exponential estimates and saddle point property for neutral functional differential equations
Cruz, M. A.↗
Atmospheric attenuation for correlated satellite communication ground sites
Link margin estimates are calculated for 30/20 GHz satellite communication systems employing closely-spaced (4 to 100 km) 'dual diversity' switched ground sites. The link margin estimates are based on a new analysis in which the bivariate rain attenuation density function for two correlated ground sites is modelled by an exponential density function. The results of the exponential density function analysis enable derivation of a direct relation between rain margin estimates and probability of exceedance (link availability). Margins typically in the range 2 to 12 dB are calculated for various ground site separations and summarized for seven city locations and five satellite orbit positions over the U.S. The results can be easily extended to other EHF satellite frequencies and other locations.
Christopher, P.↗
A table of integrals of the error function. II - Additions and corrections.
Integrals of products of error functions with other functions are presented, taking into account a combination of the error function with powers, a combination of the error function with exponentials and powers, a combination of the error function with exponentials of more complicated arguments, definite integrals from Laplace transforms, and a combination of the error function with trigonometric functions. Other integrals considered include a combination of the error function with logarithms and powers, a combination of two error functions, and a combination of the error function with other special functions.
Geller, M.↗
A Parametric Study of Fine-scale Turbulence Mixing Noise
The present paper is a study of aerodynamic noise spectra from model functions that describe the source. The study is motivated by the need to improve the spectral shape of the MGBK jet noise prediction methodology at high frequency. The predicted spectral shape usually appears less broadband than measurements and faster decaying at high frequency. Theoretical representation of the source is based on Lilley's equation. Numerical simulations of high-speed subsonic jets as well as some recent turbulence measurements reveal a number of interesting statistical properties of turbulence correlation functions that may have a bearing on radiated noise. These studies indicate that an exponential spatial function may be a more appropriate representation of a two-point correlation compared to its Gaussian counterpart. The effect of source non-compactness on spectral shape is discussed. It is shown that source non-compactness could well be the differentiating factor between the Gaussian and exponential model functions. In particular, the fall-off of the noise spectra at high frequency is studied and it is shown that a non-compact source with an exponential model function results in a broader spectrum and better agreement with data. An alternate source model that represents the source as a covariance of the convective derivative of fine-scale turbulence kinetic energy is also examined.
Khavaran, Abbas↗
Spatial modelling of total storm rainfall
The spatial structure of the depth of rainfall from a stationary storm event is investigated by using point process techniques. Cells are assumed to be stationary and to be distributed in space either independently according to a Poisson process, or with clustering according to a Neyman-Scott scheme. Total storm rainfall at the centre of each cell is a random variable and rainfall is distributed around the centre in a way specified by a spread function that may incorporate random parameters. The mean, variance and covariance structure of the precipitation depth at a point are obtained for different spread functions. For exponentially distributed centre depth and a spread function having quadratically exponential decay, the total storm depth at any point in the field is shown to have a gamma distribution. The probability of zero rainfall at a point is investigated, as is the stochastic variability of model parameters from storm to storm. Data from the Upper Rio Guaire basin in Venezuela are used in illustration.
Rodriguez-Iturbe, I.↗
A property of the order statistics of an exponential variate
Order statistics of exponential variate used in estimating scale of exponential distribution of know location
DISTRIBUTION FUNCTION↗
Analytical solution for boundary heat fluxes from a radiating rectangular medium
Reference is made to the work of Shah (1979) which demonstrated the possibility of partially integrating the radiative equations analytically to obtain an 'exact' solution. Shah's solution was given as a double integration of the modified Bessel function of order zero. Here, it is shown that the 'exact' solution for a rectangular region radiating to cold black walls can be conveniently derived, and expressed in simple form, by using an integral function, Sn, analogous to the exponential integral function appearing in plane-layer solutions.
Siegel, R.↗
Deep Learning without Global Optimization by Random Fourier Neural Networks
Here we introduce a new training algorithm for deep neural networks that utilize random complex exponential activation functions. Our approach employs a Markov chain Monte Carlo sampling procedure to iteratively train network layers, avoiding global and gradient-based optimization while maintaining error control. It consistently attains the theoretical approximation rate for residual networks with complex exponential activation functions, determined by network complexity. Additionally, it enables efficient learning of multiscale and high-frequency features, producing interpretable parameter distributions. Despite using sinusoidal basis functions, we do not observe Gibbs phenomena in approximating discontinuous target functions.
97 MATHEMATICS AND COMPUTING↗
Correlation function studies for snow and ice
The random medium model is used to characterize snow and ice fields in the interpretation of active and passive microwave remote sensing data. A correlation function is used to describe the random permittivity fluctuations with the associated mean and variance and correlation lengths; and several samples are investigated to determine typical correlation functions for snow and ice. It is shown that correlation functions are extracted directly from appropriate ground truth data, and an exponential correlation function is observed for snow and ice with lengths corresponding to the actual size of ice particles or air bubbles. Thus, given that a medium has spatially stationary statistics and a small medium, the random medium model can interpret remote sensing data where theoretical parameters correspond to actual physical parameters of the terrain.
Vallese, F.↗
Evidence for biasing in the CfA survey
Intrinsically bright galaxies appear systematically more correlated than faint galaxies in the Center for Astrophysics redshift survey. The amplification of the two-point correlation function behaves exponentially with luminosity, being essentially flat up to the knee of the luminosity function, then increasing markedly. The amplification reaches a factor of 3.5e + or - 0.4 in the very brightest galaxies. The effect is dominated by spirals rather than ellipticals, so that the correlation function of bright spirals becomes comparable to that of normal ellipticals. Similar results are obtained whether the correlation function is measured in two or three dimensions. The effect persists to separations of a correlation length or more, and is not confined to the cores of the Virgo, Coma, and Abell 1367 clusters, suggesting that the effect is caused by biasing, that is, galaxies kindle preferentially in more clustered regions, rather than by gravitational relaxation.
Hamilton, A. J. S.↗
Spectral Modeling of the EGRET 3EG Gamma Ray Sources Near the Galactic Plane
The third EGRET catalog lists 84 sources within 10 deg of the Galactic Plane. Five of these are well-known spin-powered pulsars, 2 and possibly 3 others are blazars, and the remaining 74 are classified as unidentified, although 6 of these are likely to be artifacts of nearby strong sources. Several of the remaining 68 unidentified sources have been noted as having positional agreement with supernovae remnants and OB associations. Others may be radio-quiet pulsars like Geminga, and still others may belong to a totally new class of sources. The question of the energy spectral distributions of these sources is an important clue to their identification. In this paper, the spectra of the sources within 10 deg of Galactic Plane are fit with three different functional forms; a single power law, two power laws, and a power law with an exponential cutoff. Where possible, the best fit is selected with statistical tests. Twelve, and possibly an additional 5 sources, are found to have spectra that are fit by a breaking power law or by the power law with exponential cutoff function.
Bertsch, D. L.↗
Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
Pratt, D. T.↗
Interactive grid generation for fighter aircraft geometries
An interactive procedure to construct grids about fighter aircraft configurations using a 'state-of-the-art' workstation is described. A multiple-block grid topology is used to minimize grid skewness. Grid points on the surface of a configuration are precomputed and stored on the faces of appropriate blocks. Grid points at intermediate and far field boundary surfaces are computed with analytical functions and stored on appropriate faces. Grid points in the interior of blocks are computed using transfinite interpolation with Lagrangian and exponential blending functions. Interactive software has been written for the IRIS 3030 graphics workstation to control the shape and spacing of grid points on intermediate and far field boundaries, and to control the interior interpolation.
Smith, Robert E.↗
Triangle Method for Dense ReLU Layers [SWR-25-72]
This software is an implementation of the methods for initializing and training neural networks to be more efficient per parameter, described more fully below and in the related publication: In theory, depth should make a ReLU network EXPONENTIALLY more efficient by enabling it to produce an exponential number of piecewise linear sections in its output. This reasoning is largely based on the work of mathematicians that have hand-constructed networks that make good use of depth. In practice however, even very deep ReLU networks that have been randomly initialized will behave identically to their shallow counterparts - missing an entire exponential dimension of efficiency. The triangle method is a first attempt at realizing the exponential potential of deep networks. Instead of randomly setting weights, we force pairs of neurons in each layer learn to build triangles (i.e. functions from [0,1] -> [0,1] that look like triangles). This is a very efficient pattern for generating lots of linear pieces because composing two triangular functions doubles the number of pieces with each composition. The triangle method is more than just a different initialization, it is a new paradigm of training. Instead of making direct updates to the matrix weights, we do an extra step of backpropagation to collect the derivatives of the loss function with respect to the shapes of the triangles, training them to tilt left or right. This process essentially holds the networks hand throughout the loss landscape and forces it to always use depth effectively by producing triangular shapes internally. This can produce several orders of magnitude of improvement on convex one-dimensional regression problems. Much more theoretical work is needed to realize its full potential beyond this context, but the implementation in this repository will still work in arbitrary numbers of dimensions. The file Triangle_Method.py is a generalized form of the method that will build each neuron its own custom 1-d convex activation function (with exponential efficiency). Example usage on one dimensional problems can be found in Example_Usage.ipynb and an example of using this in a real neural network can be found in Example_VGG16_CIFAR10.ipynb.
Milkert, Max [National Renewable Energy Laboratory↗
Performance and state-space analyses of systems using Petri nets
The goal of any modeling methodology is to develop a mathematical description of a system that is accurate in its representation and also permits analysis of structural and/or performance properties. Inherently, trade-offs exist between the level detail in the model and the ease with which analysis can be performed. Petri nets (PN's), a highly graphical modeling methodology for Discrete Event Dynamic Systems, permit representation of shared resources, finite capacities, conflict, synchronization, concurrency, and timing between state changes. By restricting the state transition time delays to the family of exponential density functions, Markov chain analysis of performance problems is possible. One major drawback of PN's is the tendency for the state-space to grow rapidly (exponential complexity) compared to increases in the PN constructs. It is the state space, or the Markov chain obtained from it, that is needed in the solution of many problems. The theory of state-space size estimation for PN's is introduced. The problem of state-space size estimation is defined, its complexities are examined, and estimation algorithms are developed. Both top-down and bottom-up approaches are pursued, and the advantages and disadvantages of each are described. Additionally, the author's research in non-exponential transition modeling for PN's is discussed. An algorithm for approximating non-exponential transitions is developed. Since only basic PN constructs are used in the approximation, theory already developed for PN's remains applicable. Comparison to results from entropy theory show the transition performance is close to the theoretic optimum. Inclusion of non-exponential transition approximations improves performance results at the expense of increased state-space size. The state-space size estimation theory provides insight and algorithms for evaluating this trade-off.
Watson, James Francis, III↗
Further Improvement in 3DGRAPE
3DGRAPE/AL:V2 denotes version 2 of the Three-Dimensional Grids About Anything by Poisson's Equation with Upgrades from Ames and Langley computer program. The preceding version, 3DGRAPE/AL, was described in Improved 3DGRAPE (ARC-14069) NASA Tech Briefs, Vol. 21, No. 5 (May 1997), page 66. These programs are so named because they generate volume grids by iteratively solving Poisson's Equation in three dimensions. The grids generated by the various versions of 3DGRAPE have been used in computational fluid dynamics (CFD). The main novel feature of 3DGRAPE/AL:V2 is the incorporation of an optional scheme in which anisotropic Lagrange-based trans-finite interpolation (ALBTFI) is coupled with exponential decay functions to compute and blend interior source terms. In the input to 3DGRAPE/AL:V2 the user can specify whether or not to invoke ALBTFI in combination with exponential-decay controls, angles, and cell size for controlling the character of grid lines. Of the known programs that solve elliptic partial differential equations for generating grids, 3DGRAPE/AL:V2 is the only code that offers a combination of speed and versatility with most options for controlling the densities and other characteristics of grids for CFD.
Alter, Stephen↗