Showing papers in "The Mathematical Gazette in 1995"
2,345 citations
TL;DR: The second edition of Stroock's text as discussed by the authors is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis and includes more than 750 exercises.
Abstract: This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, where they are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motion to the analysis of partial differential equations and potential theory.
269 citations
223 citations
196 citations
TL;DR: This book introduces chaos and economics in a continuous-time model of inventory business cycles and applications to Economics, and discusses the role of software in this transformation.
Abstract: Preface Part I. Theory: 1. General introduction: chaos and economics 2. Basic mathematical concepts 3. A user's guide 4. Surfaces of sections and Poincare maps 5. Spectral analysis 6. Lyapunov characteristic exponents 7. Dimensions 8. Symbolic dynamics 9. Transition to chaos: theoretical predictive criteria 10. Analysis of experimental signals: some theoretical problems Part II. Applications to Economics: 11. Discrete and continuous chaos 12. Cycles and chaos in overlapping generations models with production 13. Chaos in a continuous-time model of inventory business cycles 14. Analysis of experimental signals Part III. Software: 15. DMC manual 16. MODEL 17. EVAL 18. PLOT 19. STAT 20. FILES 21. UTIL 22. OPTS 23. QUIT 24. Internal menu commands 25. DMC internal compiler.
152 citations
74 citations
TL;DR: This chapter discusses Fourier's series in detail, focusing on the summations of the Fourier series, which are concerned with the convergence of infinite series.
Abstract: Preface 1. Crisis in mathematics: Fourier's series 2. Infinite summations 3. Differentiability and continuity 4. The convergence of infinite series 5. Understanding infinite series 6. Return to Fourier series 7. Epilogue A. Explorations of the infinite B. Bibliography C. Hints to selected exercises.
70 citations
63 citations
TL;DR: The collection and representation of data estimation of error sampling frequency distributions measure of central tendency measure of dispersion regression and correlation applications of the binomial distribution the normal distribution sample measures significance tests demographic statistics answers as discussed by the authors.
Abstract: The collection and representation of data estimation of error sampling frequency distributions measure of central tendency measure of dispersion regression and correlation applications of the mean probability the binomial distribution the normal distribution sample measures significance tests demographic statistics answers.
55 citations
41 citations
TL;DR: In this article, the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle, is considered, and the techniques are applied to produce new results in the geometry of the triangle.
Abstract: This article considers the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle. The techniques are applied to produce new results in the geometry of the triangle. This is an extended version of part of the contribution: How do computers change the way we do mathematics? given at the Association’s 1994 Easter Conference.
TL;DR: A talk given by the first author to students and staff of the Department of Geometria e Topologia at the University of Seville in November 1993 is described in this article, where the issues presented there have been part of a continued debate and discussion at Bangor over many years, and this explains why this is a joint paper.
Abstract: This essay is based on a talk given by the first author to students and staff of the Departmento de Geometria e Topologia at the University of Seville in November 1993. The issues presented there have been part of a continued debate and discussion at Bangor over many years, and this explains why this is a joint paper. The aim of the talk, and the reason for discussing these topics, was to give students an understanding and a sense of pride in the aims and achievements of their subject, and so help them explain these aims and achievements to their friends and relatives. This pride in itself would be expected to contribute to their enjoyment of the subject, whatever their own level of achievement. Because of this, and because of its origin, the tone of the article is principally that of an address to students.
TL;DR: Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader.
Abstract: Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.
TL;DR: In this paper, a variation of this introductory material which involves the Fibonacci numbers is considered, where the student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions.
Abstract: A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.