Dictionary of Applied Math for Engineers and Scientists (Comprehensive Dictionary of Mathematics)

165 Pages·2002·1.25 MB·English

by  Emma Previato

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 PR ESS
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

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PREFACE
To describe the scope of this work, I must go back to when Stan Gibilisco, editorial advisor of the
dictionaryseries,askedmetobeinchargeofthisvolume. Iappreciatedtheideaofacompendium
of mathematical terms used in the sciences and engineering for two reasons. Firstly, mathematical
definitionsarenoteasilylocated;whenIneedinsightonatechnicalterm,Iturntotheanalyticindex
ofamonographthatseemsrelated;recentlyIwasatalosswhentryingtofind“Vie`te’sformulas∗,”
atermusedbyanEastern-Europeanstudentinhishomework. IfinallylocateditintheEncyclopaedic
Dictionary of Mathematics, and that brought home the value of a collection of esoteric terms, put
togetherbymanypeopleacquaintedwithdifferentsectorsoftheliterature. Secondly,atthistimewe
donotyethaveatraditionofcross-disciplinaryterms;infact,muchinteractionbetweenmathematics
andotherscientificareasisinthemaking, andtimes(andtiming)couldnotbemoreexciting. 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
suspicionasbeinginterestedinproblemsfarremovedfromtherealworld. But...thingsarechanging.”
Rapidly,though,myenthusiasmturnedtodismayuponrealizingthatanystrategyIcoulddevise
wasdoomedtofailthetestof“completeness.” Whatisadictionary? Atbest, arapidlysuperseded
record of word/symbol usage by some groups of people; the only really complete achievement in
thatrespectis,inmyview,theOED.Notonlywassuchanundertakingbeyondme,theveryattempt
at bridging disciplines and importing words from one to another is still an ill-defined endeavor —
scientiststhemselvesareunsurehowtotranslateatermintootherdisciplines.
AsaconsequencewhatserviceIcanhopethisbooktoprovide,atbest,isthatofapocketmanual
withwhichavoyagercanatleastgetbyinabasicfashioninaforeign-speakingcountry. Ialsohope
thatitwillhavethesmallvirtuetobeafirstofitskind,apath-breakerthatwillpromptotherstofollow.
Notbeinganappliedmathematicianmyself,Ireliedonthegenerosityofthefollowingteamofauthors:
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-
ationsresearch,GiovanniAndreatta(UniversityofPadua,Italy),directedmetoH.J.Greenberg’sweb
glossary, andToni Kazic referred me to the most extensive web glossary in chemistry, authored by
A.D.McNaughtandA.Wilkinson. ToallthesepeopleIowemuchmorethanthanksfortheirwork. I
knowtherewardthatwouldmostpleasethemisforthisbooktohaveserveditsreaderswell: please
writemeanycommentsorsuggestions,andIwillgratefullytrytoputthemtofutureuse.
EmmaPreviato,DepartmentofMathematicsandStatistics
BostonUniversity,Boston,MA02215-2411–USA
e-mail: [email protected]
∗Theyarejusttheelementarysymmetricpolynomials,incaseanyonebesidemedidn’tknow
∗∗EngineeringandPhysicalSciencesResearchCouncil,UK.
v

CONTRIBUTORS
LorenzoFatibene ToProfessorGreenbergandDr. McNaught,a
IstitutodiFisicaMatematica greatmanythanksaredueforamostcourteous,
Universita` diTorino promptandgenerouspermissiontouseoftheir
Torino,Italy glossaries.
HarveyJ.Greenberg
MauroFrancaviglia
MathematicsDepartment
IstitutodiFisicaMatematica
UniversityofColoradoatDenver
Universita` diTorino
Denver,Colorado,U.S.
Torino,Italy
A.D.McNaught
RalfHiptmair
GeneralManager,ProductionDivision
MathematischesInstitut RSCPublishing,RoyalSocietyofChemistry
Universita¨tTu¨bingen Cambridge,U.K.
Tu¨bingen,Germany
ToniKazic
DepartmentofComputerEngineeringand
ComputerScience
UniversityofMissouri—Columbia
Columbia,Missouri,U.S.
HongQian
DepartmentofAppliedMathematics
UniversityofWashington
Seattle,Washington,U.S.
RudolfSchmid
DepartmentofMathematicsand
ComputerScience
EmoryUniversity
Atlanta,Georgia,U.S.
Inaddition,thetwofollowingdatabaseshave
beenusedwithpermission:
IUPACCompendiumofChemicalTerminology,
2nded.(1997),compiledbyAlanD.McNaught
andAndrewWilkinson,RoyalSocietyofChem-
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 Thedeviationofasphericalmir-
rorfromperfectfocusing.
A
abscissa Inarectangularcoordinatesystem
(Cartesiancoordinates)(x,y)oftheplaneR2,x
iscalledtheabscissa,y theordinate.
(cid:1)
absoluteconvergence Aseries x issaid
n
aposteriorierrorestimator Analgorithm
tobeab(cid:1)soluteconvergentiftheseriesofabsolute
forobtaininginformationaboutadiscretization values |x |converges.
n
errorforaconcretediscreteapproximationuhof (cid:1)
thecontinuoussolutionu. Twoprincipalfeatures absolute con(cid:1)vergence test If |xn| con-
areexpectedfromsuchdevice: verges,then xnconverges.
(i.) Itshouldbereliable: theestimatederror absolute error The difference between the
(norm)mustbeproportionaltoanupperbound exact value of a number x and an approximate
forthetrueerror(norm). Thus,discretesolutions value a is called the absolute error (cid:21) of the
a
that do not meet a prescribed accuracy can be approximatevalue,i.e.,(cid:21) =|x−a|. Thequo-
a
detected. tientδ = (cid:21)a iscalledtherelativeerror.
a a
(cid:1)
(ii.) Itshouldbeefficient: theerrorestimator absoluteratiotest Let x beaseriesof
n
sehrroourl(dnoprrmov)i.dTehissohmeelpsloawvoeirdbroejuencdtinfogratdhiesctrreutee ρno.nnegativetermsandsupposelimn→∞ |x|xn+n|1| =
solutionneedlessly.
(i.) Ifρ <1,theseriesconvergesabsolutely
In the case of a finite element discretization an (henceconverges);
additional requirement is the locality of the a
(ii.) Ifρ >1,theseriesdiverges;
posteriorierrorestimator. Itmustbepossibleto
(iii.) Ifρ =1,thetestisinconclusive.
extractinformationaboutthecontributionsfrom
individualcellsofthemeshtothetotalerror. This absolutetemperature −273.15◦C.
isessentialfortheuseofanaposteriorierroresti-
matorintheframeworkofadaptiverefinement. absolutevalue Theabsolutevalueofareal
numberx,denotedby|x|,isdefinedby|x|=x
abacus Oldest known “computer” circa ifx ≥0 and|x|=−x ifx <0.
1100BCfromChina,aframewithslidingbeads
absolute value of an operator Let A be a
fordoingarithmetic.
boundedlinearoperatoronaHilbertspace, H.
Abbe’s sine condition (Ernst Abbe 1840– T√hentheabsolutevalueofAisgivenby|A| =
1905) n(cid:1)l(cid:1)sinβ(cid:1) = nlsinβ where n,n(cid:1),β,β(cid:1) A∗A,whereA∗istheadjointofA.
are the refraction indices and refraction angles, absolutely continuous A function x(t)
respectively. definedon[a,b]iscalledabsolutelycontinuous
on[a,b]iftheree(cid:2)xistsafunctiony ∈ L1[a,b]
Abelian group (Niels Henrik Abel 1802– such that x(t) = ty(s)ds +C, where C is a
1829)Agroup(G,·)iscalledAbelianor com- a
constant.
mutativeifa·b=b·aforalla,b∈G.
(cid:1) absorbance Alogarithmoftheratioofinci-
Abeliantheorems (1)Suppose ∞n=0 anxn dent to transmitted radiant power through a
convergesfor|x| < R andforx = R. Thenthe sample (excluding the effects on cell walls).
seriesconvergesuniformlyon0≤x ≤R. Dependingonthebaseofthelogarithm,decadic
(2)Forn≥5thegeneralequationofnthorder or Napierian absorbance are used. Symbols:
cannotbesolvedbyradicals. A,A ,A . This quantity is sometimes called
10 e
(cid:2) extinction, although the term extinction, better
Abel’sintegralequation f(x)= x √φ(ξ)
0 x−ξ called attenuance, is reserved for the quantity
dξ, where f(x) is C1 with f(0) = 0, is called whichtakesintoaccounttheeffectsoflumines-
Abel’sintegralequation. cenceandscatteringaswell.
1-58488-050-3/01/$0.00+$.50
(cid:1)c 2003byCRCPressLLC 1