The Nature of Mathematical Modeling

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Overview

This is a book about the nature of mathematical modeling, and about the kinds of techniques that are useful for modeling. The text is in four sections. The first covers exact and approximate analytical techniques; the second, numerical methods; the third, model inference based on observations; and the last, the special role of time in modeling. Each of the topics in the book would be the worthy subject of a dedicated text, but only by presenting the material in this way is it possible to make so much material accessible to so many people. Each chapter presents a concise summary of the core results in an area. The text is complemented by extensive worked problems.
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Editorial Reviews

From the Publisher
"In a compact but accessible manner, Gershenfeld offers a wide-ranging overview of mathematical ideas and techniques that provide a number of effective approaches to problem solving...a great compendium of techniques. It should be kept within easy reach of anyone who wants to build computer models to help understand the world
Science

"...masterfully written, fun to read, and brimming over with useful information....This text is a marvelous handbook of mathematical modeling."
UMAP Journal

"Gershenfeld's style of giving brief overviews together with a concise summary of key results in the subject area makes [the book] invaluable for mathematical and computational modelers...Highly recommended."
Choice

"...I do not know any other single source of the material presented here."
Michael Marder, Physics Today

"Neil manages to combine new and old flavors like analysis and stochastic modeling, finite element methods and cellular automata, nonlinear function minimization and information-theoretic system identification in a single integrated overview that exposes the purposes and capabilities of each. Any of these areas can be a black hole, capable of swallowing a student in detail, with no observable output. This book performs the valuable service of teaching novice or practitioner of what the methods can offer to the overall purpose of modeling the world around us or before us."
Scott Kirkpatrick, IBM Research, inventor of simulated annealing

"This is a book for anyone who wants to use a computer to build models. The book draws on an enormous variety of sources, but Gershenfeld has seen through to the core ideas and has brought out the key relationships between the various methods. The book is a pleasure to read."
Michael I. Jordan, University of California, Berkeley

"Simulation and mathematical modeling will power the 21st Century the way steam powered the 19th. Gershenfeld masterfully compresses two armloads of dense textbooks into a single clear volume, including both classic and avant garde methods, and with well-selected references for further study. Every student of computing needs this book as the entry ticket into a vital and rapidly changing field."
William H. Press, Harvard University, author of Numerical Recipes

"The sheer breadth of material that this book surveys and unifies in surprising ways is one strong recommendation for including The Nature of Mathematical Modeling in your library. Readers will profit from physicist Neil Gershenfeld's background as he draws connections among the many, often seemingly disparate, facets of mathematical modeling...And, of course, this is a book for practicing mathematicians, computer scientists, physicists, engineers, and any and all others interested in both standard and not so standard techniques viewed in the rich and stimulating context that this book provides."
The American Mathematical Monthly

"This is a well-written and interesting book. It would make an excellent text for a final-year undergraduate course in modeling and a good reference for research students in any situation where data are to be examined."
Computing Reviews

"The exposition of the text is fluent and engrossing...As a mathematician I found many of the topics discussed in the text interesting and worthy of further investigation. I would certainly recommend the text to a colleague interested in understanding the connections that exist among the seemingly unrelated techniques of applied mathematics."
MAA Online

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Product Details

  • ISBN-13: 9780521570954
  • Publisher: Cambridge University Press
  • Publication date: 11/28/1998
  • Edition description: New Edition
  • Pages: 344
  • Product dimensions: 6.85 (w) x 9.72 (h) x 1.18 (d)

Table of Contents

Preface
1 Introduction 1
Pt. 1 Analytical Models 5
2 Ordinary Differential and Difference Equations 9
3 Partial Differential Equations 24
4 Variational Principles 34
5 Random Systems 44
Pt. 2 Numerical Models 63
6 Finite Differences: Ordinary Differential Equations 67
7 Finite Differences: Partial Differential Equations 78
8 Finite Elements 93
9 Cellular Automata and Lattice Gases 102
Pt. 3 Observational Models 111
10 Function Fitting 115
11 Transforms 128
12 Architectures 139
13 Optimization and Search 156
14 Clustering and Density Estimation 169
15 Filtering and State Estimation 186
16 Linear and Nonlinear Time Series 204
App. 1 Graphical and Mathematical Software 225
App. 2 Network Programming 250
App. 3 Benchmarking 257
App. 4 Problem Solutions 259
Bibliography 330
Index 340
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