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Stability Analysis of Fitzhughâ•Finagumo With
Proceedings of GREAT Day Volume 2011 Article 4 2012 Stability Analysis of FitzHugh–Nagumo with Smooth Periodic Forcing Tyler Massaro SUNY Geneseo Follow this and additional works at: https://knightscholar.geneseo.edu/proceedings-of-great-day Creative Commons Attribution 4.0 License This work is licensed under a Creative Commons Attribution 4.0 License. Recommended Citation Massaro, Tyler (2012) "Stability Analysis of FitzHugh–Nagumo with Smooth Periodic Forcing," Proceedings of GREAT Day: Vol. 2011 , Article 4. Available at: https://knightscholar.geneseo.edu/proceedings-of-great-day/vol2011/iss1/4 This Article is brought to you for free and open access by the GREAT Day at KnightScholar. It has been accepted for inclusion in Proceedings of GREAT Day by an authorized editor of KnightScholar. For more information, please contact [email protected]. Massaro: Stability Analysis of FitzHugh–Nagumo with Smooth Periodic Forcin Stability Analysis of FitzHugh – Nagumo with Smooth Periodic Forcing Tyler Massaro 1 Background Since the concentration of K+ ions is so much higher inside the cell than outside, there is a As Izhikevich so aptly put it, tendency for K+ to flow out of these leak channels “If somebody were to put a gun to the head of along its concentration gradient. When this the author of this book and ask him to name the happens, there is a negative charge left behind by single most important concept in brain science, he the K+ ions immediately leaving the cell. This would say it is the concept of a neuron[16].” build-up of negative charge is actually enough to, in a sense, catch the K+ ions in the act of leaving By no means are the concepts forwarded in his and momentarily halt the flow of charge across the book restricted to brain science. -
3 Autocorrelation
3 Autocorrelation Autocorrelation refers to the correlation of a time series with its own past and future values. Autocorrelation is also sometimes called “lagged correlation” or “serial correlation”, which refers to the correlation between members of a series of numbers arranged in time. Positive autocorrelation might be considered a specific form of “persistence”, a tendency for a system to remain in the same state from one observation to the next. For example, the likelihood of tomorrow being rainy is greater if today is rainy than if today is dry. Geophysical time series are frequently autocorrelated because of inertia or carryover processes in the physical system. For example, the slowly evolving and moving low pressure systems in the atmosphere might impart persistence to daily rainfall. Or the slow drainage of groundwater reserves might impart correlation to successive annual flows of a river. Or stored photosynthates might impart correlation to successive annual values of tree-ring indices. Autocorrelation complicates the application of statistical tests by reducing the effective sample size. Autocorrelation can also complicate the identification of significant covariance or correlation between time series (e.g., precipitation with a tree-ring series). Autocorrelation implies that a time series is predictable, probabilistically, as future values are correlated with current and past values. Three tools for assessing the autocorrelation of a time series are (1) the time series plot, (2) the lagged scatterplot, and (3) the autocorrelation function. 3.1 Time series plot Positively autocorrelated series are sometimes referred to as persistent because positive departures from the mean tend to be followed by positive depatures from the mean, and negative departures from the mean tend to be followed by negative departures (Figure 3.1). -
The Probability Lifesaver: Order Statistics and the Median Theorem
The Probability Lifesaver: Order Statistics and the Median Theorem Steven J. Miller December 30, 2015 Contents 1 Order Statistics and the Median Theorem 3 1.1 Definition of the Median 5 1.2 Order Statistics 10 1.3 Examples of Order Statistics 15 1.4 TheSampleDistributionoftheMedian 17 1.5 TechnicalboundsforproofofMedianTheorem 20 1.6 TheMedianofNormalRandomVariables 22 2 • Greetings again! In this supplemental chapter we develop the theory of order statistics in order to prove The Median Theorem. This is a beautiful result in its own, but also extremely important as a substitute for the Central Limit Theorem, and allows us to say non- trivial things when the CLT is unavailable. Chapter 1 Order Statistics and the Median Theorem The Central Limit Theorem is one of the gems of probability. It’s easy to use and its hypotheses are satisfied in a wealth of problems. Many courses build towards a proof of this beautiful and powerful result, as it truly is ‘central’ to the entire subject. Not to detract from the majesty of this wonderful result, however, what happens in those instances where it’s unavailable? For example, one of the key assumptions that must be met is that our random variables need to have finite higher moments, or at the very least a finite variance. What if we were to consider sums of Cauchy random variables? Is there anything we can say? This is not just a question of theoretical interest, of mathematicians generalizing for the sake of generalization. The following example from economics highlights why this chapter is more than just of theoretical interest. -
At Work in Qualitative Analysis: a Case Study of the Opposite
HISTORIA MATHEMATICA 25 (1998), 379±411 ARTICLE NO. HM982211 The ``Essential Tension'' at Work in Qualitative Analysis: A Case Study of the Opposite Points of View of Poincare and Enriques on the Relationships between Analysis and Geometry* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Giorgio Israel and Marta Menghini Dipartimento di Matematica, UniversitaÁ di Roma ``La Sapienza,'' P. le A. Moro, 2, I00185 Rome, Italy An analysis of the different philosophic and scienti®c visions of Henri Poincare and Federigo Enriques relative to qualitative analysis provides us with a complex and interesting image of the ``essential tension'' between ``tradition'' and ``innovation'' within the history of science. In accordance with his scienti®c paradigm, Poincare viewed qualitative analysis as a means for preserving the nucleus of the classical reductionist program, even though it meant ``bending the rules'' somewhat. To Enriques's mind, qualitative analysis represented the af®rmation of a synthetic, geometrical vision that would supplant the analytical/quantitative conception characteristic of 19th-century mathematics and mathematical physics. Here, we examine the two different answers given at the turn of the century to the question of the relationship between geometry and analysis and between mathematics, on the one hand, and mechanics and physics, on the other. 1998 Academic Press Un'analisi delle diverse posizioni ®loso®che e scienti®che di Henri Poincare e Federigo Enriques nei riguardi dell'analisi qualitativa fornisce un'immagine complessa e interessante della ``tensione essenziale'' tra ``tradizione'' e ``innovazione'' nell'ambito della storia della scienza. -
The Bayesian Approach to Statistics
THE BAYESIAN APPROACH TO STATISTICS ANTHONY O’HAGAN INTRODUCTION the true nature of scientific reasoning. The fi- nal section addresses various features of modern By far the most widely taught and used statisti- Bayesian methods that provide some explanation for the rapid increase in their adoption since the cal methods in practice are those of the frequen- 1980s. tist school. The ideas of frequentist inference, as set out in Chapter 5 of this book, rest on the frequency definition of probability (Chapter 2), BAYESIAN INFERENCE and were developed in the first half of the 20th century. This chapter concerns a radically differ- We first present the basic procedures of Bayesian ent approach to statistics, the Bayesian approach, inference. which depends instead on the subjective defini- tion of probability (Chapter 3). In some respects, Bayesian methods are older than frequentist ones, Bayes’s Theorem and the Nature of Learning having been the basis of very early statistical rea- Bayesian inference is a process of learning soning as far back as the 18th century. Bayesian from data. To give substance to this statement, statistics as it is now understood, however, dates we need to identify who is doing the learning and back to the 1950s, with subsequent development what they are learning about. in the second half of the 20th century. Over that time, the Bayesian approach has steadily gained Terms and Notation ground, and is now recognized as a legitimate al- ternative to the frequentist approach. The person doing the learning is an individual This chapter is organized into three sections. -
Central Limit Theorem and Its Applications to Baseball
Central Limit Theorem and Its Applications to Baseball by Nicole Anderson A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honours Seminar) Lakehead University Thunder Bay, Ontario, Canada copyright c (2014) Nicole Anderson Abstract This honours project is on the Central Limit Theorem (CLT). The CLT is considered to be one of the most powerful theorems in all of statistics and probability. In probability theory, the CLT states that, given certain conditions, the sample mean of a sufficiently large number or iterates of independent random variables, each with a well-defined ex- pected value and well-defined variance, will be approximately normally distributed. In this project, a brief historical review of the CLT is provided, some basic concepts, two proofs of the CLT and several properties are discussed. As an application, we discuss how to use the CLT to study the sampling distribution of the sample mean and hypothesis testing using baseball statistics. i Acknowledgements I would like to thank my supervisor, Dr. Li, who helped me by sharing his knowledge and many resources to help make this paper come to life. I would also like to thank Dr. Adam Van Tuyl for all of his help with Latex, and support throughout this project. Thank you very much! ii Contents Abstract i Acknowledgements ii Chapter 1. Introduction 1 1. Historical Review of Central Limit Theorem 1 2. Central Limit Theorem in Practice 1 Chapter 2. Preliminaries 3 1. Definitions 3 2. Central Limit Theorem 7 Chapter 3. Proofs of Central Limit Theorem 8 1. -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Table of Contents More Information
Cambridge University Press 978-1-107-03042-8 - Lyapunov Exponents: A Tool to Explore Complex Dynamics Arkady Pikovsky and Antonio Politi Table of Contents More information Contents Preface page xi 1Introduction 1 1.1 Historical considerations 1 1.1.1 Early results 1 1.1.2 Biography of Aleksandr Lyapunov 3 1.1.3 Lyapunov’s contribution 4 1.1.4 The recent past 5 1.2 Outline of the book 6 1.3 Notations 8 2Thebasics 10 2.1 The mathematical setup 10 2.2 One-dimensional maps 11 2.3 Oseledets theorem 12 2.3.1 Remarks 13 2.3.2 Oseledets splitting 15 2.3.3 “Typical perturbations” and time inversion 16 2.4 Simple examples 17 2.4.1 Stability of fixed points and periodic orbits 17 2.4.2 Stability of independent and driven systems 18 2.5 General properties 18 2.5.1 Deterministic vs. stochastic systems 18 2.5.2 Relationship with instabilities and chaos 19 2.5.3 Invariance 20 2.5.4 Volume contraction 21 2.5.5 Time parametrisation 22 2.5.6 Symmetries and zero Lyapunov exponents 24 2.5.7 Symplectic systems 26 3 Numerical methods 28 3.1 The largest Lyapunov exponent 28 3.2 Full spectrum: QR decomposition 29 3.2.1 Gram-Schmidt orthogonalisation 31 3.2.2 Householder reflections 31 v © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03042-8 - Lyapunov Exponents: A Tool to Explore Complex Dynamics Arkady Pikovsky and Antonio Politi Table of Contents More information vi Contents 3.3 Continuous methods 33 3.4 Ensemble averages 35 3.5 Numerical errors 36 3.5.1 Orthogonalisation 37 3.5.2 Statistical error 38 3.5.3 -
Lecture 4 Multivariate Normal Distribution and Multivariate CLT
Lecture 4 Multivariate normal distribution and multivariate CLT. T We start with several simple observations. If X = (x1; : : : ; xk) is a k 1 random vector then its expectation is × T EX = (Ex1; : : : ; Exk) and its covariance matrix is Cov(X) = E(X EX)(X EX)T : − − Notice that a covariance matrix is always symmetric Cov(X)T = Cov(X) and nonnegative definite, i.e. for any k 1 vector a, × a T Cov(X)a = Ea T (X EX)(X EX)T a T = E a T (X EX) 2 0: − − j − j � We will often use that for any vector X its squared length can be written as X 2 = XT X: If we multiply a random k 1 vector X by a n k matrix A then the covariancej j of Y = AX is a n n matrix × × × Cov(Y ) = EA(X EX)(X EX)T AT = ACov(X)AT : − − T Multivariate normal distribution. Let us consider a k 1 vector g = (g1; : : : ; gk) of i.i.d. standard normal random variables. The covariance of g is,× obviously, a k k identity × matrix, Cov(g) = I: Given a n k matrix A, the covariance of Ag is a n n matrix × × � := Cov(Ag) = AIAT = AAT : Definition. The distribution of a vector Ag is called a (multivariate) normal distribution with covariance � and is denoted N(0; �): One can also shift this disrtibution, the distribution of Ag + a is called a normal distri bution with mean a and covariance � and is denoted N(a; �): There is one potential problem 23 with the above definition - we assume that the distribution depends only on covariance ma trix � and does not depend on the construction, i.e. -
The Development of Hierarchical Knowledge in Robot Systems
THE DEVELOPMENT OF HIERARCHICAL KNOWLEDGE IN ROBOT SYSTEMS A Dissertation Presented by STEPHEN W. HART Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2009 Computer Science c Copyright by Stephen W. Hart 2009 All Rights Reserved THE DEVELOPMENT OF HIERARCHICAL KNOWLEDGE IN ROBOT SYSTEMS A Dissertation Presented by STEPHEN W. HART Approved as to style and content by: Roderic Grupen, Chair Andrew Barto, Member David Jensen, Member Rachel Keen, Member Andrew Barto, Department Chair Computer Science To R. Daneel Olivaw. ACKNOWLEDGMENTS This dissertation would not have been possible without the help and support of many people. Most of all, I would like to extend my gratitude to Rod Grupen for many years of inspiring work, our discussions, and his guidance. Without his sup- port and vision, I cannot imagine that the journey would have been as enormously enjoyable and rewarding as it turned out to be. I am very excited about what we discovered during my time at UMass, but there is much more to be done. I look forward to what comes next! In addition to providing professional inspiration, Rod was a great person to work with and for|creating a warm and encouraging labora- tory atmosphere, motivating us to stay in shape for his annual half-marathons, and ensuring a sufficient amount of cake at the weekly lab meetings. Thanks for all your support, Rod! I am very grateful to my thesis committee|Andy Barto, David Jensen, and Rachel Keen|for many encouraging and inspirational discussions. -
A Review of Graph and Network Complexity from an Algorithmic Information Perspective
entropy Review A Review of Graph and Network Complexity from an Algorithmic Information Perspective Hector Zenil 1,2,3,4,5,*, Narsis A. Kiani 1,2,3,4 and Jesper Tegnér 2,3,4,5 1 Algorithmic Dynamics Lab, Centre for Molecular Medicine, Karolinska Institute, 171 77 Stockholm, Sweden; [email protected] 2 Unit of Computational Medicine, Department of Medicine, Karolinska Institute, 171 77 Stockholm, Sweden; [email protected] 3 Science for Life Laboratory (SciLifeLab), 171 77 Stockholm, Sweden 4 Algorithmic Nature Group, Laboratoire de Recherche Scientifique (LABORES) for the Natural and Digital Sciences, 75005 Paris, France 5 Biological and Environmental Sciences and Engineering Division (BESE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia * Correspondence: [email protected] or [email protected] Received: 21 June 2018; Accepted: 20 July 2018; Published: 25 July 2018 Abstract: Information-theoretic-based measures have been useful in quantifying network complexity. Here we briefly survey and contrast (algorithmic) information-theoretic methods which have been used to characterize graphs and networks. We illustrate the strengths and limitations of Shannon’s entropy, lossless compressibility and algorithmic complexity when used to identify aspects and properties of complex networks. We review the fragility of computable measures on the one hand and the invariant properties of algorithmic measures on the other demonstrating how current approaches to algorithmic complexity are misguided and suffer of similar limitations than traditional statistical approaches such as Shannon entropy. Finally, we review some current definitions of algorithmic complexity which are used in analyzing labelled and unlabelled graphs. This analysis opens up several new opportunities to advance beyond traditional measures. -
Introduction to Bayesian Inference and Modeling Edps 590BAY
Introduction to Bayesian Inference and Modeling Edps 590BAY Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2019 Introduction What Why Probability Steps Example History Practice Overview ◮ What is Bayes theorem ◮ Why Bayesian analysis ◮ What is probability? ◮ Basic Steps ◮ An little example ◮ History (not all of the 705+ people that influenced development of Bayesian approach) ◮ In class work with probabilities Depending on the book that you select for this course, read either Gelman et al. Chapter 1 or Kruschke Chapters 1 & 2. C.J. Anderson (Illinois) Introduction Fall 2019 2.2/ 29 Introduction What Why Probability Steps Example History Practice Main References for Course Throughout the coures, I will take material from ◮ Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., & Rubin, D.B. (20114). Bayesian Data Analysis, 3rd Edition. Boco Raton, FL, CRC/Taylor & Francis.** ◮ Hoff, P.D., (2009). A First Course in Bayesian Statistical Methods. NY: Sringer.** ◮ McElreath, R.M. (2016). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Boco Raton, FL, CRC/Taylor & Francis. ◮ Kruschke, J.K. (2015). Doing Bayesian Data Analysis: A Tutorial with JAGS and Stan. NY: Academic Press.** ** There are e-versions these of from the UofI library. There is a verson of McElreath, but I couldn’t get if from UofI e-collection. C.J. Anderson (Illinois) Introduction Fall 2019 3.3/ 29 Introduction What Why Probability Steps Example History Practice Bayes Theorem A whole semester on this? p(y|θ)p(θ) p(θ|y)= p(y) where ◮ y is data, sample from some population.