DICTIONARY OF Applied math for engineers and scientists Comprehensive Dictionary of Mathematics Douglas N. Clark Editor-in-Chief Stan Gibilisco Editorial Advisor PUBLISHED VOLUMES Analysis, Calculus, and Differential Equations Douglas N. Clark Algebra, Arithmetic, and Trigonometry Steven G. Krantz Classical and Theoretical Mathematics Catherine Cavagnaro and William T. Haight, II Applied Mathematics for Engineers and Scientists Emma Previato FORTHCOMING VOLUMES The Comprehensive Dictionary of Mathematics Douglas N. Clark a Volume in the Comprehensive Dictionary of Mathematics DICTIONARY OF applied math for engineers and scientists Edited by Emma Previato CRC PRESS Boca Raton London New York Washington, D.C. 3122 disclaimer Page 1 Friday, September 27, 2002 9:47 AM Library of Congress Cataloging-in-Publication Data Dictionary of applied math for engineers and scientists/ edited by Emma Previato. p. cm. ISBN 1-58488-053-8 1. Mathematics—Dictionaries. I. Previato, Emma. QA5 .D49835 2002 510¢.3—dc21 2002074025 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58488-053-8 Library of Congress Card Number 2002074025 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper PREFACE To describe the scope of this work, I must go back to when Stan Gibilisco, editorial advisor of the dictionary series, asked me to be in charge of this volume. I appreciated the idea of a compendium of mathematical terms used in the sciences and engineering for two reasons. Firstly, mathematical definitions are not easily located; when I need insight on a technical term, I turn to the analytic index of a monograph that seems related; recently I was at a loss when trying to find “Vi`ete’s formulas ∗,” a term used by an Eastern-European student in his homework. I finally located it in the Encyclopaedic Dictionary of Mathematics, and that brought home the value of a collection of esoteric terms, put together by many people acquainted with different sectors of the literature. Secondly, at this time we do not yet have a tradition of cross-disciplinary terms; in fact, much interaction between mathematics and other scientific areas is in the making, and times (and timing) could not be more exciting. The EPSRC∗∗ newsletter Newsline (available on the web at www.epsrc.ac.uk), devoted to mathematics, in July 2001 rightly states “Even amongst fellow scientists, mathematicians are often viewed with suspicion as being interested in problems far removed from the real world. But ...things are changing.” Rapidly, though, my enthusiasm turned to dismay upon realizing that any strategy I could devise was doomed to fail the test of “completeness.” What is a dictionary? At best, a rapidly superseded record of word/symbol usage by some groups of people; the only really complete achievement in that respect is, in my view, the OED. Not only was such an undertaking beyond me, the very attempt at bridging disciplines and importing words from one to another is still an ill-defined endeavor — scientists themselves are unsure how to translate a term into other disciplines. As a consequence what service I can hope this book to provide, at best, is that of a pocket manual with which a voyager can at least get by in a basic fashion in a foreign-speaking country. I also hope that it will have the small virtue to be a first of its kind, a path-breaker that will prompt others to follow. Not being an applied mathematician myself, I relied on the generosity of the following team of authors: Lorenzo Fatibene, Mauro Francaviglia, and Rudolf Schmid, experts of mathematical physics; Toni Kazic, a biologist with broad and daring interdisciplinary experience; Hong Qian, a mathematical biologist; and Ralf Hiptmair, who works on numerical solution of differential equations. For oper- ations research, Giovanni Andreatta (University of Padua, Italy), directed me to H.J. Greenberg’s web glossary, and Toni Kazic referred me to the most extensive web glossary in chemistry, authored by A.D. McNaught and A. Wilkinson. To all these people I owe much more than thanks for their work. I know the reward that would most please them is for this book to have served its readers well: please write me any comments or suggestions, and I will gratefully try to put them to future use. Emma Previato, Department of Mathematics and Statistics Boston University, Boston, MA 02215-2411 – USA e-mail: [email protected] ∗They are just the elementary symmetric polynomials, in case anyone beside me didn’t know ∗∗Engineering and Physical Sciences Research Council, UK. v CONTRIBUTORS Lorenzo Fatibene To Professor Greenberg and Dr. McNaught, a Istituto di Fisica Matematica great many thanks are due for a most courteous, Universit`a di Torino prompt and generous permission to use of their Torino, Italy glossaries. Harvey J. Greenberg Mauro Francaviglia Mathematics Department Istituto di Fisica Matematica University of Colorado at Denver Universit`a di Torino Denver, Colorado, U.S. Torino, Italy A.D. McNaught Ralf Hiptmair General Manager, Production Division Mathematisches Institut RSC Publishing, Royal Society of Chemistry Universit¨at T¨ubingen Cambridge, U.K. Tubingen,¨ Germany Toni Kazic Department of Computer Engineering and Computer Science University of Missouri — Columbia Columbia, Missouri, U.S. Hong Qian Department of Applied Mathematics University of Washington Seattle, Washington, U.S. Rudolf Schmid Department of Mathematics and Computer Science Emory University Atlanta, Georgia, U.S. In addition, the two following databases have been used with permission: IUPAC Compendium of Chemical Terminology, 2nd ed. (1997), compiled by Alan D. McNaught and Andrew Wilkinson, Royal Society of Chem- istry, Cambridge, U.K. http://www.iupac.org/ publications/compendium/index.html H. J. Greenberg. Mathematical Programming Glossary http://carbon.cudenver.edu/˜hgreenbe/ glossary/ glossary.html ,1996-2000. vii absorbance aberration The deviation of a spherical mir- ror from perfect focusing. abscissa In a rectangular coordinate system (Cartesian coordinates) (x, y) of the plane R2, x A is called the abscissa, y the ordinate. absolute convergence A series xn is said a posteriori error estimator An algorithm to be absolute convergent if the series of absolute for obtaining information about a discretization values |xn| converges. error for a concrete discrete approximation uh of the continuous solution u. Two principal features absolute convergence test If |xn| con- are expected from such device: verges, then xn converges. (i.) It should be reliable: the estimated error absolute error The difference between the (norm) must be proportional to an upper bound exact value of a number x and an approximate for the true error (norm). Thus, discrete solutions value a is called the absolute error a of the that do not meet a prescribed accuracy can be approximate value, i.e., a =|x − a|. The quo- δ = a detected. tient a a is called the relative error. (ii.) It should be efficient: the error estimator absolute ratio test Let xn be a series of |x | should provide some lower bound for the true n+1 = nonnegative terms and suppose limn→∞ |x | error (norm). This helps avoid rejecting a discrete n ρ. solution needlessly. (i.) If ρ<1, the series converges absolutely In the case of a finite element discretization an (hence converges); additional requirement is the locality of the a (ii.) If ρ>1, the series diverges; posteriori error estimator. It must be possible to ρ = extract information about the contributions from (iii.) If 1, the test is inconclusive. individual cells of the mesh to the total error. This absolute temperature −273.15◦C. is essential for the use of an a posteriori error esti- mator in the framework of adaptive refinement. absolute value The absolute value of a real number x, denoted by |x|, is defined by |x|=x abacus Oldest known “computer” circa if x ≥ 0 and |x|=−x if x<0. 1100 BC from China, a frame with sliding beads absolute value of an operator Let A be a for doing arithmetic.