Section 5: Laplace Transforms
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Discontinuous Forcing Functions
Math 135A, Winter 2012 Discontinuous forcing functions 1 Preliminaries If f(t) is defined on the interval [0; 1), then its Laplace transform is defined to be Z 1 F (s) = L (f(t)) = e−stf(t) dt; 0 as long as this integral is defined and converges. In particular, if f is of exponential order and is piecewise continuous, the Laplace transform of f(t) will be defined. • f is of exponential order if there are constants M and c so that jf(t)j ≤ Mect: Z 1 Since the integral e−stMect dt converges if s > c, then by a comparison test (like (11.7.2) 0 in Salas-Hille-Etgen), the integral defining the Laplace transform of f(t) will converge. • f is piecewise continuous if over each interval [0; b], f(t) has only finitely many discontinuities, and at each point a in [0; b], both of the limits lim f(t) and lim f(t) t!a− t!a+ exist { they need not be equal, but they must exist. (At the endpoints 0 and b, the appropriate one-sided limits must exist.) 2 Step functions Define u(t) to be the function ( 0 if t < 0; u(t) = 1 if t ≥ 0: Then u(t) is called the step function, or sometimes the Heaviside step function: it jumps from 0 to 1 at t = 0. Note that for any number a > 0, the graph of the function u(t − a) is the same as the graph of u(t), but translated right by a: u(t − a) jumps from 0 to 1 at t = a. -
Laplace Transforms: Theory, Problems, and Solutions
Laplace Transforms: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 43 The Laplace Transform: Basic Definitions and Results 3 44 Further Studies of Laplace Transform 15 45 The Laplace Transform and the Method of Partial Fractions 28 46 Laplace Transforms of Periodic Functions 35 47 Convolution Integrals 45 48 The Dirac Delta Function and Impulse Response 53 49 Solving Systems of Differential Equations Using Laplace Trans- form 61 50 Solutions to Problems 68 2 43 The Laplace Transform: Basic Definitions and Results Laplace transform is yet another operational tool for solving constant coeffi- cients linear differential equations. The process of solution consists of three main steps: • The given \hard" problem is transformed into a \simple" equation. • This simple equation is solved by purely algebraic manipulations. • The solution of the simple equation is transformed back to obtain the so- lution of the given problem. In this way the Laplace transformation reduces the problem of solving a dif- ferential equation to an algebraic problem. The third step is made easier by tables, whose role is similar to that of integral tables in integration. The above procedure can be summarized by Figure 43.1 Figure 43.1 In this section we introduce the concept of Laplace transform and discuss some of its properties. The Laplace transform is defined in the following way. Let f(t) be defined for t ≥ 0: Then the Laplace transform of f; which is denoted by L[f(t)] or by F (s), is defined by the following equation Z T Z 1 L[f(t)] = F (s) = lim f(t)e−stdt = f(t)e−stdt T !1 0 0 The integral which defined a Laplace transform is an improper integral. -
10 Heat Equation: Interpretation of the Solution
Math 124A { October 26, 2011 «Viktor Grigoryan 10 Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line u − ku = 0 (−∞ < x < 1; 0 < t < 1); t xx (1) u(x; 0) = φ(x); and derived the solution formula Z 1 u(x; t) = S(x − y; t)φ(y) dy; for t > 0; (2) −∞ where S(x; t) is the heat kernel, 1 2 S(x; t) = p e−x =4kt: (3) 4πkt Substituting this expression into (2), we can rewrite the solution as 1 1 Z 2 u(x; t) = p e−(x−y) =4ktφ(y) dy; for t > 0: (4) 4πkt −∞ Recall that to derive the solution formula we first considered the heat IVP with the following particular initial data 1; x > 0; Q(x; 0) = H(x) = (5) 0; x < 0: Then using dilation invariance of the Heaviside step function H(x), and the uniquenessp of solutions to the heat IVP on the whole line, we deduced that Q depends only on the ratio x= t, which lead to a reduction of the heat equation to an ODE. Solving the ODE and checking the initial condition (5), we arrived at the following explicit solution p x= 4kt 1 1 Z 2 Q(x; t) = + p e−p dp; for t > 0: (6) 2 π 0 The heat kernel S(x; t) was then defined as the spatial derivative of this particular solution Q(x; t), i.e. @Q S(x; t) = (x; t); (7) @x and hence it also solves the heat equation by the differentiation property. -
Delta Functions and Distributions
When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on. -
An Analytic Exact Form of the Unit Step Function
Mathematics and Statistics 2(7): 235-237, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020702 An Analytic Exact Form of the Unit Step Function J. Venetis Section of Mechanics, Faculty of Applied Mathematics and Physical Sciences, National Technical University of Athens *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, the author obtains an analytic Meanwhile, there are many smooth analytic exact form of the unit step function, which is also known as approximations to the unit step function as it can be seen in Heaviside function and constitutes a fundamental concept of the literature [4,5,6]. Besides, Sullivan et al [7] obtained a the Operational Calculus. Particularly, this function is linear algebraic approximation to this function by means of a equivalently expressed in a closed form as the summation of linear combination of exponential functions. two inverse trigonometric functions. The novelty of this However, the majority of all these approaches lead to work is that the exact representation which is proposed here closed – form representations consisting of non - elementary is not performed in terms of non – elementary special special functions, e.g. Logistic function, Hyperfunction, or functions, e.g. Dirac delta function or Error function and Error function and also most of its algebraic exact forms are also is neither the limit of a function, nor the limit of a expressed in terms generalized integrals or infinitesimal sequence of functions with point wise or uniform terms, something that complicates the related computational convergence. Therefore it may be much more appropriate in procedures. -
Learning Activation Functions in Deep Neural Networks
UNIVERSITÉ DE MONTRÉAL LEARNING ACTIVATION FUNCTIONS IN DEEP NEURAL NETWORKS FARNOUSH FARHADI DÉPARTEMENT DE MATHÉMATIQUES ET DE GÉNIE INDUSTRIEL ÉCOLE POLYTECHNIQUE DE MONTRÉAL MÉMOIRE PRÉSENTÉ EN VUE DE L’OBTENTION DU DIPLÔME DE MAÎTRISE ÈS SCIENCES APPLIQUÉES (GÉNIE INDUSTRIEL) DÉCEMBRE 2017 c Farnoush Farhadi, 2017. UNIVERSITÉ DE MONTRÉAL ÉCOLE POLYTECHNIQUE DE MONTRÉAL Ce mémoire intitulé : LEARNING ACTIVATION FUNCTIONS IN DEEP NEURAL NETWORKS présenté par : FARHADI Farnoush en vue de l’obtention du diplôme de : Maîtrise ès sciences appliquées a été dûment accepté par le jury d’examen constitué de : M. ADJENGUE Luc-Désiré, Ph. D., président M. LODI Andrea, Ph. D., membre et directeur de recherche M. PARTOVI NIA Vahid, Doctorat, membre et codirecteur de recherche M. CHARLIN Laurent, Ph. D., membre iii DEDICATION This thesis is dedicated to my beloved parents, Ahmadreza and Sholeh, who are my first teachers and always love me unconditionally. This work is also dedicated to my love, Arash, who has been a great source of motivation and encouragement during the challenges of graduate studies and life. iv ACKNOWLEDGEMENTS I would like to express my gratitude to Prof. Andrea Lodi for his permission to be my supervisor at Ecole Polytechnique de Montreal and more importantly for his enthusiastic encouragements and invaluable continuous support during my research and education. I am very appreciated to him for introducing me to a MITACS internship in which I have developed myself both academically and professionally. I would express my deepest thanks to Dr. Vahid Partovi Nia, my co-supervisor at Ecole Poly- technique de Montreal, for his supportive and careful guidance and taking part in important decisions which were extremely precious for my research both theoretically and practically. -
2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing – Continuous and Discrete 1 The Laplace Transform 1 Introduction In the class handout Introduction to Frequency Domain Processing we introduced the Fourier trans form as an important theoretical and practical tool for the analysis of waveforms, and the design of linear filters. We noted that there are classes of waveforms for which the classical Fourier integral does not converge. An important function that does not have a classical Fourier transform are is the unit step (Heaviside) function ( 0 t · 0; us(t) = 1 t > 0; Clearly Z 1 jus(t)j dt = 1; ¡1 and the forward Fourier integral Z 1 Z 1 ¡jt ¡jt Us(jΩ) = us(t)e dt = e dt (1) ¡1 0 does not converge. Aside: We also saw in the handout that many such functions whose Fourier integrals do not converge do in fact have Fourier transforms that can be defined using distributions (with the use of the Dirac delta function ±(t)), for example we found that 1 F fu (t)g = ¼±(Ω) + : s jΩ Clearly any function f(t) in which limj tj!1 f(t) 6= 0 will have a Fourier integral that does not converge. Similarly, the ramp function ( 0 t · 0; r(t) = t t > 0: is not integrable in the absolute sense, and does not allow direct computation of the Fourier trans form. -
On the Approximation of the Step Function by Raised-Cosine and Laplace Cumulative Distribution Functions
European International Journal of Science and Technology Vol. 4 No. 9 December, 2015 On the Approximation of the Step function by Raised-Cosine and Laplace Cumulative Distribution Functions Vesselin Kyurkchiev 1 Faculty of Mathematics and Informatics Institute of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, Bulgarian Academy of Sciences, 24 Tsar Assen Str., 4000 Plovdiv, Bulgaria Email: [email protected] Nikolay Kyurkchiev Faculty of Mathematics and Informatics Institute of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, Bulgarian Academy of Sciences, Acad.G.Bonchev Str.,Bl.8 1113Sofia, Bulgaria Email: [email protected] 1 Corresponding Author Abstract In this note the Hausdorff approximation of the Heaviside step function by Raised-Cosine and Laplace Cumulative Distribution Functions arising from lifetime analysis, financial mathematics, signal theory and communications systems is considered and precise upper and lower bounds for the Hausdorff distance are obtained. Numerical examples, that illustrate our results are given, too. Key words: Heaviside step function, Raised-Cosine Cumulative Distribution Function, Laplace Cumulative Distribution Function, Alpha–Skew–Laplace cumulative distribution function, Hausdorff distance, upper and lower bounds 2010 Mathematics Subject Classification: 41A46 75 European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk 1. Introduction The Cosine Distribution is sometimes used as a simple, and more computationally tractable, approximation to the Normal Distribution. The Raised-Cosine Distribution Function (RC.pdf) and Raised-Cosine Cumulative Distribution Function (RC.cdf) are functions commonly used to avoid inter symbol interference in communications systems [1], [2], [3]. The Laplace distribution function (L.pdf) and Laplace Cumulative Distribution function (L.cdf) is used for modeling in signal processing, various biological processes, finance, and economics. -
On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions
www.biomathforum.org/biomath/index.php/biomath ORIGINAL ARTICLE On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions Anton Iliev∗y, Nikolay Kyurkchievy, Svetoslav Markovy ∗Faculty of Mathematics and Informatics Paisii Hilendarski University of Plovdiv, Plovdiv, Bulgaria Email: [email protected] yInstitute of Mathematics and Informatics Bulgarian Academy of Sciences, Sofia, Bulgaria Emails: [email protected], [email protected] Received: , accepted: , published: will be added later Abstract—We study the uniform approximation practically important classes of sigmoid functions. of the sigmoid cut function by smooth sigmoid Two of them are the cut (or ramp) functions and functions such as the logistic and the Gompertz the step functions. Cut functions are continuous functions. The limiting case of the interval-valued but they are not smooth (differentiable) at the two step function is discussed using Hausdorff metric. endpoints of the interval where they increase. Step Various expressions for the error estimates of the corresponding uniform and Hausdorff approxima- functions can be viewed as limiting case of cut tions are obtained. Numerical examples are pre- functions; they are not continuous but they are sented using CAS MATHEMATICA. Hausdorff continuous (H-continuous) [4], [43]. In some applications smooth sigmoid functions are Keywords-cut function; step function; sigmoid preferred, some authors even require smoothness function; logistic function; Gompertz function; squashing function; Hausdorff approximation. in the definition of sigmoid functions. Two famil- iar classes of smooth sigmoid functions are the logistic and the Gompertz functions. There are I. INTRODUCTION situations when one needs to pass from nonsmooth In this paper we discuss some computational, sigmoid functions (e. -
Multilayer Perceptron
Advanced Introduction to Machine Learning, CMU-10715 Perceptron, Multilayer Perceptron Barnabás Póczos, Sept 17 Contents History of Artificial Neural Networks Definitions: Perceptron, MLP Representation questions Perceptron algorithm Backpropagation algorithm 2 Short History Progression (1943-1960) • First mathematical model of neurons . Pitts & McCulloch (1943) • Beginning of artificial neural networks • Perceptron, Rosenblatt (1958) . A single layer neuron for classification . Perceptron learning rule . Perceptron convergence theorem Degression (1960-1980) • Perceptron can’t even learn the XOR function • We don’t know how to train MLP • 1969 Backpropagation… but not much attention… 3 Short History Progression (1980-) • 1986 Backpropagation reinvented: . Rumelhart, Hinton, Williams: Learning representations by back-propagating errors. Nature, 323, 533—536, 1986 • Successful applications: . Character recognition, autonomous cars,… • Open questions: Overfitting? Network structure? Neuron number? Layer number? Bad local minimum points? When to stop training? • Hopfield nets (1982), Boltzmann machines,… 4 Short History Degression (1993-) • SVM: Vapnik and his co-workers developed the Support Vector Machine (1993). It is a shallow architecture. • SVM almost kills the ANN research. • Training deeper networks consistently yields poor results. • Exception: deep convolutional neural networks, Yann LeCun 1998. (discriminative model) 5 Short History Progression (2006-) Deep Belief Networks (DBN) • Hinton, G. E, Osindero, S., and Teh, Y. -
Maximizing Acquisition Functions for Bayesian Optimization
Maximizing acquisition functions for Bayesian optimization James T. Wilson∗ Frank Hutter Marc Peter Deisenroth Imperial College London University of Freiburg Imperial College London PROWLER.io Abstract Bayesian optimization is a sample-efficient approach to global optimization that relies on theoretically motivated value heuristics (acquisition functions) to guide its search process. Fully maximizing acquisition functions produces the Bayes’ decision rule, but this ideal is difficult to achieve since these functions are fre- quently non-trivial to optimize. This statement is especially true when evaluating queries in parallel, where acquisition functions are routinely non-convex, high- dimensional, and intractable. We first show that acquisition functions estimated via Monte Carlo integration are consistently amenable to gradient-based optimiza- tion. Subsequently, we identify a common family of acquisition functions, includ- ing EI and UCB, whose properties not only facilitate but justify use of greedy approaches for their maximization. 1 Introduction Bayesian optimization (BO) is a powerful framework for tackling complicated global optimization problems [32, 40, 44]. Given a black-box function f : , BO seeks to identify a maximizer X!Y x∗ arg maxx f(x) while simultaneously minimizing incurred costs. Recently, these strategies have2 demonstrated2X state-of-the-art results on many important, real-world problems ranging from material sciences [17, 57], to robotics [3, 7], to algorithm tuning and configuration [16, 29, 53, 56]. From a high-level perspective, BO can be understood as the application of Bayesian decision theory to optimization problems [11, 14, 45]. One first specifies a belief over possible explanations for f using a probabilistic surrogate model and then combines this belief with an acquisition function to convey the expected utility for evaluating a set of queries X. -
Expanded Use of Discontinuity and Singularity Functions in Mechanics
AC 2010-1666: EXPANDED USE OF DISCONTINUITY AND SINGULARITY FUNCTIONS IN MECHANICS Robert Soutas-Little, Michigan State University Professor Soutas-Little received his B.S. in Mechanical Engineering 1955 from Duke University and his MS 1959 and PhD 1962 in Mechanics from University of Wisconsin. His name until 1982 when he married Patricia Soutas was Robert William Little. Under this name, he wrote an Elasticity book and published over 20 articles. Since 1982, he has written over 100 papers for journals and conference proceedings and written books in Statics and Dynamics. He has taught courses in Statics, Dynamics, Mechanics of Materials, Elasticity, Continuum Mechanics, Viscoelasticity, Advanced Dynamics, Advanced Elasticity, Tissue Biomechanics and Biodynamics. He has won teaching excellence awards and the Distinguished Faculty Award. During his tenure at Michigan State University, he chaired the Department of Mechanical Engineering for 5 years and the Department of Biomechanics for 13 years. He directed the Biomechanics Evaluation Laboratory from 1990 until he retired in 2002. He served as Major Professor for 22 PhD students and over 100 MS students. He has received numerous research grants and consulted with engineering companies. He now is Professor Emeritus of Mechanical Engineering at Michigan State University. Page 15.549.1 Page © American Society for Engineering Education, 2010 Expanded Use of Discontinuity and Singularity Functions in Mechanics Abstract W. H. Macauley published Notes on the Deflection of Beams 1 in 1919 introducing the use of discontinuity functions into the calculation of the deflection of beams. In particular, he introduced the singularity functions, the unit doublet to model a concentrated moment, the Dirac delta function to model a concentrated load and the Heaviside step function to start a uniform load at any point on the beam.