Number Systems

Bayesian Computation with R (Use R)

There has been a dramatic growth in the development and application of Bayesian inferential methods. Some of this growth is due to the availability of powerful simulation-based algorithms to summarize posterior distributions. There has been also a growing interest in the use of the system R for statistical analyses. R's open source nature, free availability, and large number of contributor packages have made R the software of choice for many statisticians in education and industry.

Bayesian Computation with R introduces Bayesian modeling by the use of computation using the R language. The early chapters present the basic tenets of Bayesian thinking by use of familiar one and two-parameter inferential problems. Bayesian computational methods such as Laplace's method, rejection sampling, and the SIR algorithm are illustrated in the context of a random effects model. The construction and implementation of Markov Chain Monte Carlo (MCMC) methods is introduced. These simulation-based algorithms are implemented for a variety of Bayesian applications such as normal and binary response regression, hierarchical modeling, order-restricted inference, and robust modeling. Algorithms written in R are used to develop Bayesian tests and assess Bayesian models by use of the posterior predictive distribution. The use of R to interface with WinBUGS, a popular MCMC computing language, is described with several illustrative examples.

This book is a suitable companion book for an introductory course on Bayesian methods. Also the book is valuable to the statistical practitioner who wishes to learn more about the R language and Bayesian methodology. The LearnBayes package, written by the author and available from the CRAN website, contains all of the R functions described in the book.

An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.

The purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. Over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.

Monte Carlo Strategies in Scientific Computing (Springer Series in Statistics)

This paperback edition is a reprint of the 2001 Springer edition.

This book provides a self-contained and up-to-date treatment of the Monte Carlo method and develops a common framework under which various Monte Carlo techniques can be "standardized" and compared. Given the interdisciplinary nature of the topics and a moderate prerequisite for the reader, this book should be of interest to a broad audience of quantitative researchers such as computational biologists, computer scientists, econometricians, engineers, probabilists, and statisticians. It can also be used as the textbook for a graduate-level course on Monte Carlo methods. Many problems discussed in the alter chapters can be potential thesis topics for masters’ or Ph.D. students in statistics or computer science departments.

Jun Liu is Professor of Statistics at Harvard University, with a courtesy Professor appointment at Harvard Biostatistics Department. Professor Liu was the recipient of the 2002 COPSS Presidents' Award, the most prestigious one for statisticians and given annually by five leading statistical associations to one individual under age 40. He was selected as a Terman Fellow by Stanford University in 1995, as a Medallion Lecturer by the Institute of Mathematical Statistics (IMS) in 2002, and as a Bernoulli Lecturer by the International Bernoulli Society in 2004. He was elected to the IMS Fellow in 2004 and Fellow of the American Statistical Association in 2005. He and co-workers have published more than 130 research articles and book chapters on Bayesian modeling and computation, bioinformatics, genetics, signal processing, stochastic dynamic systems, Monte Carlo methods, and theoretical statistics.

"An excellent survey of current Monte Carlo methods. The applications amply demonstrate the relevance of this approach to modern computing. The book is highly recommended." (Mathematical Reviews)

"This book provides comprehensive coverage of Monte Carlo methods, and in the process uncovers and discusses commonalities among seemingly disparate techniques that arose in various areas of application. … The book is well organized; the flow of topics follows a logical development. … The coverage is up-to-date and comprehensive, and so the book is a good resource for people conducting research on Monte Carlo methods. … The book would be an excellent supplementary text for a course in scientific computing … ." (SIAM Review)

"The strength of this book is in bringing together advanced Monte Carlo (MC) methods developed in many disciplines. … Throughout the book are examples of techniques invented, or reinvented, in different fields that may be applied elsewhere. … Those interested in using MC to solve difficult problems will find many ideas, collected from a variety of disciplines, and references for further study." (Technometrics)

Implementing Models in Quantitative Finance: Methods and Cases (Springer Finance)

This book puts numerical methods into action for the purpose of solving concrete problems arising in quantitative finance. Part one develops a comprehensive toolkit including Monte Carlo simulation, numerical schemes for partial differential equations, stochastic optimization in discrete time, copula functions, transform-based methods and quadrature techniques. The content originates from class notes written for courses on numerical methods for finance and exotic derivative pricing held by the authors at Bocconi University since the year 2000. Part two proposes eighteen self-contained cases covering model simulation, derivative valuation, dynamic hedging, portfolio selection, risk management, statistical estimation and model calibration. It encompasses a wide variety of problems arising in markets for equity, interest rates, credit risk, energy and exotic derivatives. Each case introduces a problem, develops a detailed solution and illustrates empirical results. Proposed algorithms are implemented using either Matlab^® or Visual Basic for Applications^® in collaboration with contributors.

Functional Equations and How to Solve Them (Problem Books in Mathematics)

Over the years, a number of books have been written on the theory of functional equations. However, very little has been published which helps readers to solve functional equations in mathematics competitions and mathematical problem solving. This book fills that gap. The student who encounters a functional equation on a mathematics contest will need to investigate solutions to the equation by finding all solutions, or by showing that all solutions have a particular property. The emphasis here will be on the development of those tools which are most useful in assigning a family of solutions to each functional equation in explicit form.

At the end of each chapter, readers will find a list of problems associated with the material in that chapter. The problems vary greatly, with the easiest problems being accessible to any high school student who has read the chapter carefully. The most difficult problems will be a reasonable challenge to advanced students studying for the International Mathematical Olympiad at the high school level or the William Lowell Putnam Competition for university undergraduates. The book ends with an appendix containing topics that provide a springboard for further investigation of the concepts of limits, infinite series and continuity.

Random Number Generation and Monte Carlo Methods (Statistics and Computing)

Monte Carlo simulation has become one of the most important tools in all fields of science. Simulation methodology relies on a good source of numbers that appear to be random. These "pseudorandom" numbers must pass statistical tests just as random samples would. Methods for producing pseudorandom numbers and transforming those numbers to simulate samples from various distributions are among the most important topics in statistical computing. This book surveys techniques of random number generation and the use of random numbers in Monte Carlo simulation. The book covers basic principles, as well as newer methods such as parallel random number generation, nonlinear congruential generators, quasi Monte Carlo methods, and Markov chain Monte Carlo. The best methods for generating random variates from the standard distributions are presented, but also general techniques useful in more complicated models and in novel settings are described. The emphasis throughout the book is on practical methods that work well in current computing environments. The book includes exercises and can be used as a test or supplementary text for various courses in modern statistics. It could serve as the primary test for a specialized course in statistical computing, or as a supplementary text for a course in computational statistics and other areas of modern statistics that rely on simulation. The book, which covers recent developments in the field, could also serve as a useful reference for practitioners. Although some familiarity with probability and statistics is assumed, the book is accessible to a broad audience. The second edition is approximately 50% longer than the first edition. It includes advances in methods for parallel random number generation, universal methods for generation of nonuniform variates, perfect sampling, and software for random number generation. The material on testing of random number generators has been expanded to include a discussion of newer software for testing, as well as more discussion about the tests themselves. The second edition has more discussion of applications of Monte Carlo methods in various fields, including physics and computational finance. James Gentle is University Professor of Computational Statistics at George Mason University. During a thirteen-year hiatus from academic work before joining George Mason, he was director of research and design at the world's largest independent producer of Fortran and C general-purpose scientific software libraries. These libraries implement several random number generators, and are widely used in Monte Carlo studies. He is a Fellow of the American Statistical Association and a member of the International Statistical Institute. He has held several national offices in the American Statistical Association and has served as an associate editor for journals of the ASA as well as for other journals in statistics and computing. Recent activities include serving as program director of statistics at the National Science Foundation and as research fellow at the Bureau of Labor Statistics.

Modeling and Simulation in Scilab/Scicos

Scilab is a free open-source software package for scientific computation. It includes hundreds of general purpose and specialized functions for numerical computation, organized in libraries called toolboxes, which cover such areas as simulation, optimization, systems and control, and signal processing. One important Scilab toolbox is Scicos. Scicos provides a block diagram graphical editor for the construction and simulation of dynamical systems. The objective of this book is to provide a tutorial for the use of Scilab/Scicos with a special emphasis on modeling and simulation tools. While it will provide useful information to experienced users it is designed to be accessible to beginning users from a variety of disciplines. Students and academic and industrial scientists and engineers should find it useful.

The book is divided into two parts. The first part concerns Scilab and includes a tutorial covering the language features, the data structures and specialized functions for doing graphics, importing, exporting data and interfacing external routines. It also covers in detail Scilab numerical solvers for ordinary differential equations and differential-algebraic equations. Even though the emphasis is placed on modeling and simulation applications, this part provides a global view of Scilab. The second part is dedicated to modeling and simulation of dynamical systems in Scicos. This type of modeling tool is widely used in industry because it provides a means for constructing modular and reusable models. This part contains a detailed description of the editor and its usage, which is illustrated through numerous examples. All codes used in the book is made available to the reader.

The Universal History of Numbers: From Prehistory to the Invention of the Computer

The title doesn't lie. Mathematician Georges Ifrah's masterpiece, The Universal History of Numbers, is a wonderfully comprehensive overview of numbers and counting spanning all the inhabited continents as far back in time as records will allow us to look. Beyond the ancient Babylonians, Sumerians, and Indians, Ifrah takes us farther south into Africa to examine an early decimal counting system and into ancient Mexico to reconstruct what we can of the Mayan calendar and numerical system. The 27 chapters are chiefly organized by culture, though there are some cross-cultural overviews of topics like letters and numbers.

The author's aim was grand: "to provide in simple and accessible terms the full and complete answer to all and any questions ... about the history of numbers and counting, from prehistory to the age of computers." This led him to wander the world for 10 years, studying and learning; this scholastic pilgrim has returned with amazing stories to tell. Toward the end of the book, Ifrah makes the book truly universal by refuting alien-intervention theories of cultural origins--surely our benefactors would have given us an efficient decimal counting system, zero and all, before helping us build pyramids and such. Such charming ideas, combined with such rigorously researched facts, make The Universal History of Numbers a uniquely important and fascinating volume. --Rob Lightner

"Georges Ifrah is the man. This book, quite simply, rules. . . . It is outstanding . . . a mind-boggling and enriching experience." -The Guardian (London) "Monumental. . . . a fascinating journey taking us through many different cultures."-The Times (London)"Ifrah's book amazes and fascinates by the scope of its scholarship. It is nothing less than the history of the human race told through figures." -International Herald Tribune Now in paperback, here is Georges Ifrah's landmark international bestseller-the first complete, universal study of the invention and evolution of numbers the world over. A riveting history of counting and calculating, from the time of the cave dwellers to the twentieth century, this fascinating volume brings numbers to thrilling life, explaining their development in human terms, the intriguing situations that made them necessary, and the brilliant achievements in human thought that they made possible. It takes us through the numbers story from Europe to China, via ancient Greece and Rome, Mesopotamia, Latin America, India, and the Arabic countries. Exploring the many ways civilizations developed and changed their mathematical systems, Ifrah imparts a unique insight into the nature of human thought-and into how our understanding of numbers and the ways they shape our lives have changed and grown over thousands of years. "Dazzling."-Kirkus Reviews "Sure to transfix readers."-PublishersWeekly

Complex Numbers from A to ...Z

It is impossible to imagine modern mathematics without complex numbers. Complex Numbers from A to . . . Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics.

The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.

The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.

The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics.

Level Set Methods and Dynamic Implicit Surfaces

This book is an introduction to level set methods and dynamic implicit surfaces. These are powerful techniques for analyzing and computing moving fronts in a variety of different settings. While it gives many examples of the utility of the methods to a diverse set of applications, it also gives complete numerical analysis and recipes, which will enable users to quickly apply the techniques to real problems. The book begins with a description of implicit surfaces and their basic properties, then devises the level set geometry and calculus toolbox, including the construction of signed distance functions. Part II adds dynamics to this static calculus. Topics include the level set equation itself, Hamilton-Jacobi equations, motion of a surface normal to itself, re-initialization to a signed distance function, extrapolation in the normal direction, the particle level set method and the motion of co-dimension two (and higher) objects. Part III is concerned with topics taken from the fields of Image Processing and Computer Vision. These include the restoration of images degraded by noise and blur, image segmentation with active contours (snakes), and reconstruction of surfaces from unorganized data points. Part IV is dedicated to Computational Physics. It begins with one phase compressible fluid dynamics, then two-phase compressible flow involving possibly different equations of state, detonation and deflagration waves, and solid/fluid structure interaction. Next it discusses incompressible fluid dynamics, including a computer graphics simulation of smoke, free surface flows, including a computer graphics simulation of water, and fully two-phase incompressible flow. Additional related topics include incompressible flames with applications to computer graphics and coupling a compressible and incompressible fluid. Finally, heat flow and Stefan problems are discussed. A student or researcher working in mathematics, computer graphics, science, or engineering interested in any dynamic moving front, which might change its topology or develop singularities, will find this book interesting and useful.