Expected Logarithm and Negative Integer Moments of a Noncentral 2-Distributed Random Variable
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Integration in Terms of Exponential Integrals and Incomplete Gamma Functions
Integration in terms of exponential integrals and incomplete gamma functions Waldemar Hebisch Wrocªaw University [email protected] 13 July 2016 Introduction What is FriCAS? I FriCAS is an advanced computer algebra system I forked from Axiom in 2007 I about 30% of mathematical code is new compared to Axiom I about 200000 lines of mathematical code I math code written in Spad (high level strongly typed language very similar to FriCAS interactive language) I runtime system currently based on Lisp Some functionality added to FriCAS after fork: I several improvements to integrator I limits via Gruntz algorithm I knows about most classical special functions I guessing package I package for computations in quantum probability I noncommutative Groebner bases I asymptotically fast arbitrary precision computation of elliptic functions and elliptic integrals I new user interfaces (Emacs mode and Texmacs interface) FriCAS inherited from Axiom good (probably the best at that time) implementation of Risch algorithm (Bronstein, . ) I strong for elementary integration I when integral needed special functions used pattern matching Part of motivation for current work came from Rubi testsuite: Rubi showed that there is a lot of functions which are integrable in terms of relatively few special functions. I Adapting Rubi looked dicult I Previous work on adding special functions to Risch integrator was widely considered impractical My conclusion: Need new algorithmic approach. Ex post I claim that for considered here class of functions extension to Risch algorithm is much more powerful than pattern matching. Informally Ei(u)0 = u0 exp(u)=u Γ(a; u)0 = u0um=k exp(−u) for a = (m + k)=k and −k < m < 0. -
Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images Hasni Idayu Binti Saidi
University of Northern Colorado Scholarship & Creative Works @ Digital UNC Dissertations Student Research 8-2018 Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images Hasni Idayu Binti Saidi Follow this and additional works at: https://digscholarship.unco.edu/dissertations Recommended Citation Saidi, Hasni Idayu Binti, "Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images" (2018). Dissertations. 507. https://digscholarship.unco.edu/dissertations/507 This Text is brought to you for free and open access by the Student Research at Scholarship & Creative Works @ Digital UNC. It has been accepted for inclusion in Dissertations by an authorized administrator of Scholarship & Creative Works @ Digital UNC. For more information, please contact [email protected]. ©2018 HASNI IDAYU BINTI SAIDI ALL RIGHTS RESERVED UNIVERSITY OF NORTHERN COLORADO Greeley, Colorado The Graduate School POWER COMPARISONS OF THE RICIAN AND GAUSSIAN RANDOM FIELDS TESTS FOR DETECTING SIGNAL FROM FUNCTIONAL MAGNETIC RESONANCE IMAGES A Dissertation Submitted in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy Hasni Idayu Binti Saidi College of Education and Behavioral Sciences Department of Applied Statistics and Research Methods August 2018 This dissertation by: Hasni Idayu Binti Saidi Entitled: Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images has been approved as meeting the requirement for the Degree of Doctor of Philosophy in College of Education and Behavioral Sciences in Department of Applied Statistics and Research Methods Accepted by the Doctoral Committee Khalil Shafie Holighi, Ph.D., Research Advisor Trent Lalonde, Ph.D., Committee Member Jay Schaffer, Ph.D., Committee Member Heng-Yu Ku, Ph.D., Faculty Representative Date of Dissertation Defense Accepted by the Graduate School Linda L. -
Hydrology 510 Quantitative Methods in Hydrology
New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology Hydrology 510 Quantitative Methods in Hydrology Motive Preview: Example of a function and its limits Consider the solute concentration, C [ML-3], in a confined aquifer downstream of a continuous source of solute with fixed concentration, C0, starting at time t=0. Assume that the solute advects in the aquifer while diffusing into the bounding aquitards, above and below (but which themselves have no significant flow). (A homology to this problem would be solute movement in a fracture aquitard (diffusion controlled) Flow aquifer Solute (advection controlled) aquitard (diffusion controlled) x with diffusion into the porous-matrix walls bounding the fracture: the so-called “matrix diffusion” problem. The difference with the original problem is one of spatial scale.) An approximate solution for this conceptual model, describing the space-time variation of concentration in the aquifer, is the function: 1/ 2 x x D C x D C(x,t) = C0 erfc or = erfc B vt − x vB C B v(vt − x) 0 where t is time [T] since the solute was first emitted x is the longitudinal distance [L] downstream, v is the longitudinal groundwater (seepage) velocity [L/T] in the aquifer, D is the effective molecular diffusion coefficient in the aquitard [L2/T], B is the aquifer thickness [L] and erfc is the complementary error function. Describe the behavior of this function. Pick some typical numbers for D/B2 (e.g., D ~ 10-9 m2 s-1, typical for many solutes, and B = 2m) and v (e.g., 0.1 m d-1), and graph the function vs. -
Multiple-Precision Exponential Integral and Related Functions
Multiple-Precision Exponential Integral and Related Functions David M. Smith Loyola Marymount University This article describes a collection of Fortran-95 routines for evaluating the exponential integral function, error function, sine and cosine integrals, Fresnel integrals, Bessel functions and related mathematical special functions using the FM multiple-precision arithmetic package. Categories and Subject Descriptors: G.1.0 [Numerical Analysis]: General { computer arith- metic, multiple precision arithmetic; G.1.2 [Numerical Analysis]: Approximation { special function approximation; G.4 [Mathematical Software]: { Algorithm design and analysis, efficiency, portability General Terms: Algorithms, exponential integral function, multiple precision Additional Key Words and Phrases: Accuracy, function evaluation, floating point, Fortran, math- ematical library, portable software 1. INTRODUCTION The functions described here are high-precision versions of those in the chapters on the ex- ponential integral function, error function, sine and cosine integrals, Fresnel integrals, and Bessel functions in a reference such as Abramowitz and Stegun [1965]. They are Fortran subroutines that use the FM package [Smith 1991; 1998; 2001] for multiple-precision arithmetic, constants, and elementary functions. The FM package supports high precision real, complex, and integer arithmetic and func- tions. All the standard elementary functions are included, as well as about 25 of the mathematical special functions. An interface module provides easy use of the package through three multiple- precision derived types, and also extends Fortran's array operations involving vectors and matrices to multiple-precision. The routines in this module can detect an attempt to use a multiple precision variable that has not been defined, which helps in debugging. There is great flexibility in choosing the numerical environment of the package. -
Intensity Distribution of Interplanetary Scintillation at 408 Mhz
Aust. J. Phys., 1975,28,621-32 Intensity Distribution of Interplanetary Scintillation at 408 MHz R. G. Milne Chatterton Astrophysics Department, School of Physics, University of Sydney, Sydney, N.S.W. 2006. Abstract It is shown that interplanetary scintillation of small-diameter radio sources at 408 MHz produces intensity fluctuations which are well fitted by a Rice-squared. distribution, better so than is usually claimed. The observed distribution can be used to estimate the proportion of flux density in the core of 'core-halo' sources without the need for calibration against known point sources. 1. Introduction The observed intensity of a radio source of angular diameter ;s 1" arc shows rapid fluctuations when the line of sight passes close to the Sun. This interplanetary scintillation (IPS) is due to diffraction by electron density variations which move outwards from the Sun at high velocities (-350 kms-1) •. In a typical IPS observation the fluctuating intensity Set) from a radio source is reCorded for a few minutes. This procedure is repeated on several occasions during the couple of months that the line of sight is within - 20° (for observatio~s at 408 MHz) of the direction of the Sun. For each observation we derive a mean intensity 8; a scintillation index m defined by m2 = «(S-8)2)/82, where ( ) denote the expectation value; intensity moments of order q QI! = «(S-8)4); and the skewness parameter Y1 = Q3 Q;3/2, hereafter referred to as y. A histogram, or probability distribution, of the normalized intensity S/8 is also constructed. The present paper is concerned with estimating the form of this distribution. -
NM Temme 1. Introduction the Incomplete Gamma Functions Are Defined by the Integrals 7(A,*)
Methods and Applications of Analysis © 1996 International Press 3 (3) 1996, pp. 335-344 ISSN 1073-2772 UNIFORM ASYMPTOTICS FOR THE INCOMPLETE GAMMA FUNCTIONS STARTING FROM NEGATIVE VALUES OF THE PARAMETERS N. M. Temme ABSTRACT. We consider the asymptotic behavior of the incomplete gamma func- tions 7(—a, —z) and r(—a, —z) as a —► oo. Uniform expansions are needed to describe the transition area z ~ a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for 7(0, z) and V(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given. 1. Introduction The incomplete gamma functions are defined by the integrals 7(a,*)= / T-Vcft, r(a,s)= / t^e^dt, (1.1) where a and z are complex parameters and ta takes its principal value. For 7(0,, z), we need the condition ^Ra > 0; for r(a, z), we assume that |arg2:| < TT. Analytic continuation can be based on these integrals or on series representations of 7(0,2). We have 7(0, z) + r(a, z) = T(a). Another important function is defined by 7>>*) = S7(a,s) = =^ fu^e—du. (1.2) This function is a single-valued entire function of both a and z and is real for pos- itive and negative values of a and z. For r(a,z), we have the additional integral representation e-z poo -zt j.-a r(a, z) = — r / dt, ^a < 1, ^z > 0, (1.3) L [i — a) JQ t ■+■ 1 which can be verified by differentiating the right-hand side with respect to z. -
Idealized Models of the Joint Probability Distribution of Wind Speeds
Nonlin. Processes Geophys., 25, 335–353, 2018 https://doi.org/10.5194/npg-25-335-2018 © Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License. Idealized models of the joint probability distribution of wind speeds Adam H. Monahan School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065 STN CSC, Victoria, BC, Canada, V8W 3V6 Correspondence: Adam H. Monahan ([email protected]) Received: 24 October 2017 – Discussion started: 2 November 2017 Revised: 6 February 2018 – Accepted: 26 March 2018 – Published: 2 May 2018 Abstract. The joint probability distribution of wind speeds position in space, or point in time. The simplest measure of at two separate locations in space or points in time com- statistical dependence, the correlation coefficient, is a natu- pletely characterizes the statistical dependence of these two ral measure for Gaussian-distributed quantities but does not quantities, providing more information than linear measures fully characterize dependence for non-Gaussian variables. such as correlation. In this study, we consider two models The most general representation of dependence between two of the joint distribution of wind speeds obtained from ide- or more quantities is their joint probability distribution. The alized models of the dependence structure of the horizon- joint probability distribution for a multivariate Gaussian is tal wind velocity components. The bivariate Rice distribu- well known, and expressed in terms of the mean and co- tion follows from assuming that the wind components have variance matrix (e.g. Wilks, 2005; von Storch and Zwiers, Gaussian and isotropic fluctuations. The bivariate Weibull 1999). -
Exponential Asymptotics with Coalescing Singularities
Exponential asymptotics with coalescing singularities Philippe H. Trinh and S. Jonathan Chapman Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford E-mail: [email protected], [email protected] Abstract. Problems in exponential asymptotics are typically characterized by divergence of the associated asymptotic expansion in the form of a factorial divided by a power. In this paper, we demonstrate that in certain classes of problems that involve coalescing singularities, a more general type of exponential-over-power divergence must be used. As a model example, we study the water waves produced by flow past an obstruction such as a surface-piercing ship. In the low speed or low Froude limit, the resultant water waves are exponentially small, and their formation is attributed to the singularities in the geometry of the obstruction. However, in cases where the singularities are closely spaced, the usual asymptotic theory fails. We present both a general asymptotic framework for handling such problems of coalescing singularities, and provide numerical and asymptotic results for particular examples. 1. Introduction Many problems in exponential asymptotics involve the analysis of singularly perturbed differential equations where the associated solution is expressed as a divergent asymptotic expansion. It has been noted by authors such as Dingle [14] and Berry [3] that in many cases, the divergence of the sequence occurs in the form of a factorial divided by a arXiv:1403.7182v2 [math.CA] 4 Oct 2014 power. In this paper, we use a model problem from the theory of water waves and ship hydrodynamics to demonstrate how in certain classes of problems, a more general form of divergence must be used in order to perform the exponential asymptotic analysis. -
Field Guide to Continuous Probability Distributions
Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations. -
A Note on the Bivariate Distribution Representation of Two Perfectly
1 A note on the bivariate distribution representation of two perfectly correlated random variables by Dirac’s δ-function Andr´es Alay´on Glazunova∗ and Jie Zhangb aKTH Royal Institute of Technology, Electrical Engineering, Teknikringen 33, SE-100 44 Stockholm, Sweden; bUniversity of Sheffield, Electronic and Electrical Engineering, Mappin Street, Sheffield, S1 3JD, UK Abstract In this paper we discuss the representation of the joint probability density function of perfectly correlated continuous random variables, i.e., with correlation coefficients ρ=±1, by Dirac’s δ-function. We also show how this representation allows to define Dirac’s δ-function as the ratio between bivariate distributions and the marginal distribution in the limit ρ → ±1, whenever this limit exists. We illustrate this with the example of the bivariate Rice distribution. I. INTRODUCTION The performance evaluation of wireless communications systems relies on the analysis of the behavior of signals modeled as random variables or random processes that obey prescribed probability distribution laws. The Pearson product-moment correlation coefficient is commonly used to quantify the strength of the relationship between two random variables. For example, in wireless communications, two-antenna diversity systems take advantage of low signal correlation. The diversity gain obtained by combining the received signals at the two antenna branches depends on the amount of correlation between them, e.g., the lower the correlation the higher the diversity gain, [1]. Moreover, when the correlation coefficient is 1 or 1 it is said that the two signals are perfectly correlated, which implies that there is no diversity gain. In this case the joint probability− distribution of the signals is not well-defined and, to the best knowledge of the authors, a thorough discussion of these limit cases is not found in the literature. -
POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged Edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A
POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A. Stegun (eds.) Material selected by Michael Danos and Johann Rafelski 1984 Verlag Harri Deutsch - Thun - Frankfurt/Main CONTENTS Forewordtothe Original NBS Handbook 5 Pref ace 6 2. PHYSICAL CONSTANTS AND CONVERSION FACTORS 17 A.G. McNish, revised by the editors Table 2.1. Common Units and Conversion Factors 17 Table 2.2. Names and Conversion Factors for Electric and Magnetic Units 17 Table 2.3. AdjustedValuesof Constants 18 Table 2.4. Miscellaneous Conversion Factors 19 Table 2.5. FactorsforConvertingCustomaryU.S. UnitstoSIUnits 19 Table 2.6. Geodetic Constants 19 Table 2.7. Physical andNumericalConstants 20 Table 2.8. Periodic Table of the Elements 21 Table 2.9. Electromagnetic Relations 22 Table 2.10. Radioactivity and Radiation Protection 22 3. ELEMENTARY ANALYTICAL METHODS 23 Milton Abramowitz 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometrie Progressions; Arithmetic, Geometrie, Harmonie and Generalized Means 23 3.2. Inequalities 23 3.3. Rules for Differentiation and Integration 24 3.4. Limits, Maxima and Minima 26 3.5. Absolute and Relative Errors 27 3.6. Infinite Series 27 3.7. Complex Numbers and Functions 29 3.8. Algebraic Equations 30 3.9. Successive Approximation Methods 31 3.10. TheoremsonContinuedFractions 32 4. ELEMENTARY TRANSCENDENTAL FUNCTIONS 33 Logarithmic, Exponential, Circular and Hyperbolic Functions Ruth Zucker 4.1. Logarithmic Function 33 4.2. Exponential Function 35 4.3. Circular Functions 37 4.4. Inverse Circular Functions 45 4.5. Hyperbolic Functions 49 4.6. InverseHyberbolicFunctions 52 5. EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS .... 56 Walter Gautschi and William F. -
Table of Contents
Table of Contents Preface............................................. ix Chapter 1. Elements of Analysis of Reliability and Quality Control......... 1 1.1. Introduction ..................................... 1 1.1.1. The importance of true physical acceleration life models (accelerated tests = true acceleration or acceleration)................ 3 1.1.2.Expressionforlinearaccelerationrelationships............... 4 1.2. Fundamental expression of the calculation of reliability ............. 5 1.3. Continuous uniform distribution .......................... 9 1.3.1.Distributionfunctionofprobabilities(densityofprobability) ....... 10 1.3.2.Distributionfunction.............................. 10 1.4.Discreteuniformdistribution(discreteU)..................... 12 1.5. Triangular distribution................................ 13 1.5.1. Discrete triangular distribution version .................... 13 1.5.2. Continuous triangular law version....................... 14 1.5.3.Linkswithuniformdistribution........................ 14 1.6.Betadistribution................................... 15 1.6.1.Functionofprobabilitydensity ........................ 16 1.6.2. Distribution function of cumulative probability ............... 18 1.6.3.Estimationoftheparameters(p,q)ofthebetadistribution......... 19 1.6.4.Distributionassociatedwithbetadistribution ................ 20 1.7.Normaldistribution................................. 20 1.7.1.Arithmeticmean ................................ 20 1.7.2.Reliability.................................... 22 1.7.3.Stabilizationandnormalizationofvarianceerror.............