View of Book Reviews

REVUE

D'ANAI,YSE

NUN{ÉRIQUF] I.]'I' DE TITÉORITJ

DE

I,'APPROXII\IA1'ION T'rimr 28, N" 2, 1999, pp.

l')l-200

BOOK REVIEWS

JAMES

w.

DEMMEL, Appried Nntneritttl Linerir Argebn¡.,S|AM, phirattelphia, r997, ISBN 0-89871-3tì9-7, xi + 4t9 pp.

This book is irttended to provide an introduciion to rhe lìunlerical liriear algebra.

ln

chapter.

l -

Intn¡du<:¡ir¡rr _ _{there are} inttcjducéd and motivated somelbasic notions and concepts.

Equution Soh,in¡¡

-

^deals with direct nlethóûs for linear systems,,Thei

G

various aspects involvecl are described for general linear systems: per.-

tu

alysis, improving the accuracy

oi

a solution, blocking.algorithms for

hi

^:

192 Book reviews ₂

'Ihere are presented classical results but also current research trends. rvhich makes this book both an introductory text and a leference describing the current state of the art in the domain of numerical linear algebra Hence, the broad audience of this book includes not only students but also scientists interested in this tield.

Enil Cãrinas

A.

GREENBAI)M; Iteratit'e Methods

for

^{Solving l-inear Srsle,m, SIAM,} Philadelphia, 1997 ISBN 0-89871-396-X. xiii +220pp.

The numerical solving of linear systerns is a very important fiekì of fhe numerical analysis.

since many of the techniques uscd, in practice finaìly lead to such problems. The present book of- fers an overlook on the modern topics on solving lineal systems: Krylov methods and precondi- tioning techniques. These domains were subject to active research in thc last deca<ies and they still are continuously growing. In some.cases the existing methods,are well understood (e.g. hermitian systerns) white in othcr cases few things are known.

This book is published inlhe Fronriers in Applietl Malhenntics series. Ir contains l2,chap- ters, a bibliography and an index, Its struoturc is the following:

l. Introduc'tit¡n :r

^,,

ParÍ one: Krtlov Subspace Approxinntíotts

, :

:

2. Sone heratiott Methods, _S. Error Bounds,for CG, MTNRES and CMRES, 4, Effects

of

Fi_

niîe Prec.ission.Arilhmetic, 5. B|CG.and Related Methods, 6. l,t Tltere a.short Rscurrent'e.for a N ea r- Op t i nnl Altp,roximat ion ?,'1 . M i sce Il ane ou s I ss ue s.

Part hro : P reconditione rs

,,

.8. oveniev' an¿l Preconditione¡t Algorithnts, g,Tv'tt E:ample prol:lens, _10. Cotnparison

rif

Prg lit¡tt.s,

t L)e.contpo:;itìott Me¡hr¡els

, tound

mathods. pointing our the

esse

d and

oretic

the balance inclining not in favo¡ of the algorithmicalìy tietr¡ils but in the f'avor ot the underlying marhenìätics of thc mcthods.

We warnrly recommend this hook both to specialists and nonspecialists as well in numerical linear

algebra.,

ì ^,

Emil Cãtittuç

M. MATZEU,

A.

VIGNOLI, ktpologit'ul Nonlincur Analvsis

ll

Degree, Singularirv antl Vuriu- rioris, Birkhäuser, Boston-Basel;Bcrlin. 1997,601 pp,, ISBN 0-8176-3886-5.

The present volume is intended, at least partly, to be a contìnuation of the successtul book Ttipo'logical Nonlínear Ar¡al'r.risr Degree, Sitrgùluritt'ttnd Variarions, published by Birkhäuser in 199-5.

It

contains survey articles concerning three'nrain streams ofresearch: topological degree, singularity theory and variational methods. Each article starts with an historical introduction, con- cludes with thè discussion of signifìcant achievemcnts obtained duling thc last decades an{ fin- ishes with a rich bibliography. The result is a dynamic ovcrview on rhe field ro which the author, a

distinguished specialist, has himself brought major contributions. Thc most oi the marerials in this book werc presented by the authors atthe "second Topological Analysis Workshop'on Degree, Singularity and Variations: Developnrents of the Last 25 Years" hekl in June 1995 at Villa Tus- colana. Frascati, near Ronrc.

Book rcviews _r93

The contents are:

C.

Dell'Antoni<i. Classical solutit-urs for.

a

per.tur.berj N-body sysrenr (86pp,.with proofs.23 references); H. Brczis, Dcgrcc thcory: old and niw (22pp. withour proofs, 23 referçnces); P. T. Church _{and J.} G. Timourian, clobal siructur. ro, nonìi*u,.operators in dif.- fþrential and inregrar equations r. Fbrcrs (52 pp. wìth some proofs, g5 references); p. T. church and

spectn¡m (48 pp. with proofs. 84 references); p. p. Zabrqko, Rotàtion of vector fields: definition, basic proprieties. and carcuration

(t57 pp.

without prcofs. many comments and exampres.

726 rcfërences).

Such survey papers are vcry wclcomc: thcy arc to systematizc and make clear ncw matcrial.

point our the main problems and results, _ancl guide the future

r

_{I be of} par_

ticular interest to postgraduate students and young

mathematic

_undersrand

basic problems and methods in nonlinear anatysis ana make

ra

_{c more and}

more branchy literature in the fields. We also recommend rhis book ro all specialist,

t, ;;;;';;r:

R tulu

pre,cup

:

3

GHEORGHE MICULA and

sANDA

MICULA, _Handbook

,f

sptine, Kruwer Acadernic ^pub_

lisher, DordrechlBoston/London.

The purpose of this book is to give a cornprehcnsivc inrrorluction ro the theory of'spline tunctions, together with some applications to various fields, enrphasizing the signifìcance of the relationship between the general theory and its applications.

At

the same time, the goal of the book is also to provide new material on spline function the-ory, as well as a fresh look at old results, being written for people interested in research, as well as for those ùho are interésted in appìications.

The theory of spline functions and their applications is a relatively reccur field olappliett mathematics.

' ^ln

the last 50 years, spline function thcory has undergone a wonde¡ful ttevelopment with ma¡ly lìew directions appearing during this time. This book has its origins

in

the wish to adc- quately dcscribc this dcvclopment from the notion of ."spline" inrroduied by L

J,

Schoenberg (1901-1990)

in

1946, to the newest recent theories of "spline wavelers" o,

"rjlin.

fractals,'. Iso- lated fãcts about the functions now called "splines,' can be found

in

the japers

of l-.

^Euler,

A. Lebesgue,

C. Birklrofi

J. Favard.

L.

Tschakaloff,

L.

Collatz,

T.

popovíciu, D.

V.

Ionescu.

However, the theory of spline functions has been developecl in the last 40 years through the effort of many mathematicians. As latc as 1960, there were no more than a hantlful of papers men(ioning spline functions by name. Today, less than 40 years later. there are, to our knowledge, nore thair 357 books, monographies and conf'erence reports, many thousands original papers, and more than 3l

I

dissertations for a doctoral degree or habilitation, on various aspccts

oi

spline functions an¿

their,applications, and it is still an active research area.

.

The rapid devcloprnent of spline functions is due piimarily _{to their} gr.eat usefulness in ap- plications. Classes of spline functions possess many nice *tru.trrãl ptop.rrË. as well as excellenr approximation powcrs. Since they are easy to evaluâtc and manipulate on a computer, a myriad of applications to the numerical solution of a varicty of problcms in applicd nrathcmatics _{has bcen} fbund.'Ihe enorntous lileruturc publishcd tluring the lasr rlecaclcs, shows rhat thc actual devclop-

194 Book rcviews

While this book attçmpts to give a comprehensive treatment of the basic methods in spline and their applications,

it

is not meant to provide solutions to all problems that have arisen in this field in the last ycars.

However, it is our hope that enough information has been included which may be of interest to the reader.

Ion Pãvãloiu

M'

G'

NADKARNI, _Basi¿

E odic

Theon,. Birkhäuser Advanccd Texts, Birkhäuser Verlag, Bascl-Bosron-Berlin, t998, vi + 149 pp., ISBN 0-8t76-5816-5 un¿

¡-iä¿¡-s¡lo-s.

Book reviews t95

Thc book was first published

in

1995 and in ihis sccond cdition. besides.the corrcciion of

some etrors ânt1

i thc

intner and

a section on

rank

_n

C

tivcly.

Thc book

short time and

on lng

relativelY

c

re

_c theorv.

S. Cobatl

M. C. NADKARNI, ,Spec'rrzrl T'heor.t,t¡f D.vnanúc.ol Sy.rten,¡, Birkhäuser Advanced Texts, Birkhäu- ser Verlag, Basel-Bosron-Berlin. I 99g. vii + I tì2 pp., ISBN 0-g l 7ó-5g I 7-3 _and 3-7643-5g I 7-1.

The book is concerned with the spectral theorl,of dynamical systems; r,r,hcre by a dynamical systcn onc unclcrstands a lneasure spacc on which a gloup of automorphisfirs acts preserying _sets of lucasurc zcro.

Bringin

_{in a unitìed and ac_}

ccssible

way,

_{esearch, the book is}

dirccted

borh

_{romising area of in_}

vcstigations, a

S. Cob:u.t

SERGEY BAGDASARoY,: Chebt'thev S¡tlitrcs urtd Kolgotnoroy Ittequulirit.r, operator The-

'

ot'y-Aclvances iind Applications,

vol.

10.5, Bilkhäuser Vellag. Basel-lJoston-Bcrlin. 199g,

'

ISBN 3-7643-5981-6 and 0-8t76-5984_6. xiii + 205 pp.

)

4

The book is an introductory text in ergodic theory, requiring from the reader only a knowl- edge of the basic measure theory and metric topology. The exposition focusses more on interac- tions with clas-sical descriptive set theory than àtheitexts on

tle

same topic. For instancé, some b¿sic topics of ergodic theory such as Poincaré recurrence lemma, induced automorphisms and Kakutani towers, comPressibility and E. Hopf's theorem, the theorem of Ambrose on the repre- sentation

of

flows. are treated first

(in

Chaprers

I

^and 2) using only descriptive set-theoretical tools, before presenting theirmeasure theoretìc or topological u"Ãionr. These iirst two chapters óf the book, Ch. I

' ^{The Poincaré} reculrence lemma. and Ch. 2, Ergodic rheorems of Birkhoff and von Neumann, can serve as a base f'or a course of four tp.six leciures at the advanced or beginning graduate leve.l (as it has been done by the author at some universities,in lndia). The other c-hapteri of the book are headcd _as follows:

l.

nrgoaicity, 4. M.ixing.conditions, ,5. Bernoulli shifrs and re- ìated concepts, 6. Discrete spectrum theorenr, 7. Induced autonlor:phisnrs and related concepts.

8. Borel automorphisms are polish ho.nreomorphisnls, _9. The clinm--Etfros rheorem. r0. E. Hopf's thcorem, I

l.

H. Dye's theorem. 12. Flows andtheir rcprescntations

The bibliography at the entl of'the book is arranged by chapters. Some chaprer.s en¿s with,a section entit.led Asides, conraining vqr), intcrcsting historicat remarks and comments on rhe rele_

vance of crgodic theory for othcr disciplines, mainly nlechanics (cclestial) _and physics.

Thc perlìct pol¡'nonrial spline functions ol'rJcgree

r(i.c.

tunctions whose dcrivafives o,f or- rler

r

_{takc Ìhc} values

+l

¿rnrl

-l

^on ad.jactrnt inlcr.vals) plirr an csrt,nti¿rl rolc in thc study of some

trl|emal

prohlems

in l4ti.(t)

sttch its: !lìt' hcst approri4r,rìion ol.a tìrnction hy elerncnts

in

li¡ite

)96 Book reviews

dintensiorr¿l subspaces. the problenr ol sharp Kol¡nogo¡'ov inequalirie.s f<¡r inter.metliate derir,,aiiycs of lunctiolrs, etc.

Considçr the classes of functions:

'

ry.'

Ht"(a,bl):--{.re C'[u,bl:o(.rt')lr)<o¡(l). re[0,b-n]l

Wt Htrt ₌

{re

l{,H(,)(Ift ) :.r(¡ + 2nl) =.r(t), ^t e

l*n,nl,

^t e

Zl

( l,l/t' ¡1u = ^Hu, ), wherc cu : lR*

-r

^lR

*

^{is a} concave nrodulus oi continuity.

The declared aims

of

the book are the lbllowing:

(l)

ro inrroduce rhe notion

of

^perfecr rusplines

in

W'Ht't: (2) to describe various cxtremal properties of these functions; and (3) to ap-

ply thô gcncral thcory oi pcrt'côt splihcs to the calculation of N-width of classès

W,Ha(t).

The author applies also this theory ro give a solution to the famous Kolmogorov problenr on sharp ine- qualities between intermediate dcrivalives

in

the Hölder classes

w'Ho(x), for _{x =lR*}

and

X=R. :

^.

The book contains l7 chapters, two Appendices, _a Bibliography of 93 rirles and an Index.

Chapters

0-3

contain some auxiliary results as Borsuk ìheorem, Chebyshev theorenl. the notion of simple kernel and that of a rearengenrent of it: The enrphasis is on results needed for the study of extremal problems

in

W' Ho.

Chapter 4 is concerned'with the defìnition of pert'ect Chebyshev splines and their properties, while in Chapter 5 one obtains a formuìa of numerical differentialion anä a sufficienr condiiion for

a function _.f e W' H,ù[0,1] ro be extl'enaj for the Kolmogorov_Landau (K_L) problem.

Chapter 6 cont¿ìins the main result of the book: the description of the family of Chebyshev trrsplines lz,,; ⁿ 2

r)

of the Kol'rog.rov-La^dau _¡ roûlenr on a linite interval.

In chaptcrs 7 ro l6 ihcrc arc studicd sonìc cxrreniar probrcms ot (K-L)-typc (fï,,,',-+ sup) or concerning N-width, in various classes bt:functions.

The Appcndices are concerned with Kolmogolov problenr t'or I'uncrions .f e lV,H,')(lR-):

llfll,,

,u.,

(Appendix A) respectivety

'n W-rÉlr(R*)

and WrHu'(lRn) (Appendix B).

The book is clearly written, contains a lot of results (including author's original results) _and

will

be.of.interest for researchers in approximation theory, applied finctional anaiysis and nunreri- cal ànalysis,

C. Mustrita

C' H'

GOLUB and C. F.

VAN

LOAN, Matrix contpuîorior¡.r, third edition, The Johns Hopkins univcrsity Press. Balrimore ancl London. 1996, ISBN 0-g0lg-5413-x, xxvii + 694 pp,

f t is harcl to say in f'ew words all the things that should be said about rhc imprcssive work of Colub and Van Loan. wc begin by noting that this book ref'ers to thc nrajor aspecrs occurring in

main problems concerning nlatr.ices: linear. systenrs, eigenvalue

/

squares problems. The mentioncd aspects arc: pcrtut.balion theorv.

for inrplementing the methods, round_off error analisis and cor,.e_

t

softwaie. ^.

ⁱ

The implessive bibliography constilutes another distinct characteristic. Every section frorn each chapter contains notes and refþrences to relevant books and articles. Moreover, in tbe end of the book there is given a comprehensive list of bibliography items.

All

these t'easons make u.s believe that this b¡Jok is addrcssed, as any such eniyclopedic work, to all persons intercsted in numerical mathetnatics.

Book reviews

Emil Cãrinas

ln our world' dominated by more and more sophisticated and complex plants, r.eliability is one of thc móst important task of technological iesearch, dcsign anrl production. Reliability theory is based on statistical and prohabilistic nlodels. Modelling and modËls appcar in all stages of rhe reliabiliiy analysis, a model being a representation i'n matn'ematicaiterÀ,

ãi r¿¡i,v' ¡0."ã.,n .-r.

hypotheses representing particular feátures of technical systems. These hypotheses are given by the techrolog¡sts, whilc the mathematicians have thc task to build up the models starting fr.om these hypotheses and io give numerical evaluations. It is obvious that the elaboration of a working lnodcl needs cooperation of'both catcgorics ol'analysts, technologisls and nraihernaticians. A tcchl nologist

will

hardly be able to identify the right hypotheses without knowing rhe mode¡ing rech- niques and

a

tnathematician

will

never be able to build purely thcoretical models capable to answer all technical ^problems. In fact, due to the complexity of the problems, the reliability analy- sis is dec'omposed and modulated in specific submodels.

The.present volume conlains 24 papers seleçted from the

ót

^presented at the

lst

lnterna- tional Conference on Mathqmatical Methods in Reliability, held from l6 to

l9

September 1997 ar the Polytechnic University of Bucharest, Romania, and organized in cooperation

with

thc Tech- nological University of Compiègne, France. The main target

of

the Conference was to bring te- gether mathematicians and technologists interested

in

reliability theory,

in

order

to

exchange information, to discuss open problems, and to

tìll

in the gap between theory and real-lifê problems.

The papers are grouped in rhree parts:

I

Statistical methods.

II.

Probabilistic methods, and

lll.

Special techniques and applications. This volume contains botb surve¡¿ and contributed papers dealing with topics as: estimation of accelerating of Iil'e date. modclling of the components subject to random diffuse stress cnvironmcnt, semi-Markov rcliability nrodels, statistical methods lbr re- pairable systems. accclcrating life testing. asymptotic methods in

rcliability

analysis of stochastic systems. clc.

(r 7

and

Statistical untl Probabilistic MorJel.s

in

Retiabilitt', D. C. lonescu and N. Limnios (Editors) Statis- tics For Indu.stry and Technology., Birkhäuser Verlag, Basel-Bosron-Berlin, 1999,

xxxri

+ 352 pp.. ISBN 0-8176-4068-t and 3-7ó43-4068.1.

tt 9 Book rivicu's ¹⁹⁹

'l'hc,book contains 74 casc studics and tXr end-ot'-châpter problcnrs supplicd with hints:

Concerhing the computational aspects. manl' ol thc progranìs included in the book are avàilable iri the public domain and the includetl algorithnrs are detailed enough to generate cotllputer progralnS with case.

'Ihe book is clearly written, the topics ^are carefully selected and nreticulously prcsented.

The bibliography is extensivc, including relclcrlces to computer plogralns.

I¡r conclusion, rhe book is highly recomnìcrrded as a lelèrence lext for applied nrathenrati- cians, computer scientists, physicists and cngineers. interoslcd in numerical aspccts of contplcx analysis with applications to intcgrùl cquations. It can be used alstl as a textbook fbr teaching or l'or sclt'-study.

S. Crrl¡:¿.r'

wERNER

C

RHEINBOLD'I, M¿thxl't

lttr

^Solvirtg SJ'rr¿rrt"'

ol

Ntuilinear Equarion't' SlÃM' Philadelphia, l99tì, CtsMS-NSf.' Regional Clonf'erencc Series

in

Applied Mathenratics.

ISBN 0-89871-41-5-X. ix + l4tÌ pp

'fhis _{is thc second edition}

ol

this nronograph. the lìrst cdition being published in'1974. lt comprises l0 chapters. a bibliogLaphy and an index.

Chapter

I

contains an overview ol the problcnr of solving nonlinear systçlns o[ equations.

Ioilt¡wc'd b,v somc notaf ions antl backgrtlurtd rcstrlts.

ln

the second chaptcr there arc descríbcd sor¡c model pi'oblents which lcad to nonlinear svstenrs

in

iR" .

Chapter 3 deals with general iterativê ^{proccsscs ¿rncl} their rates ol convergencc. The claisi- cal rigorou.s definitions and -properties

of the q- and r-conveigence orders are reúiewed, For fixed point iterations the Ostrowski theorenì and ^{sorne àspects} concerning superlineal rates

ol

^conver-

gence are treated.

Chapter 4

-

^{Methods oJ Nev'ton rrpc}

-

^{is the lirst chapter} dévoted to such methods. There is prescnted thc linearization conccpt, together with somc classical local conVergence resûlts

fol

the

Newton method. The discretized Ncwton nethods and attraction basins are then discussed.

Chaptcr 5 tlcals with the ntcthods of secant type.

ll

describes the general secant nlethods and presents rcsults.based on consistent approximation. The uprlatc lnethods âre lhen thortlughly ^ana- lysed.

Chapter ó is cntitlcd C'ombinution qf ¡troce.s.tc,t.lt begins with the discussitln of using the cl¿rssical irerative nlethods in solving thc linear systems at each Newton stepr lt continues with the analysis oI the nonlinear SOR mcthods. The inexact Newton ^nìethod.s are then'Presented. together with resulis clcalìng with thcir locat and global convergence, and lesidual controls. Thc GMRES rncthod is briefly cliscussccl, but the tinite-diltcrence Newton-Krylov methods ate not mcntionçd,

The following chapter is concerncd with paranretr:ized systenls of equatitlns. Some back- grounrJ results arc presentcd. tlre fbllorving topics being then analysed: continuätion using ODEs.

continuation u,ith local paranrctrizatiou. and sintplicial approxirrration on nlanilblds.

Chapter

8 is

devoted

to

urlconstrainetl rninimization lìlethotls and discusses admissiblc step-lt'ngth algorithnrs. gradient lelated nicthotls. collcctivcly gradicnt rclatcd dircctions ^and

tr ust-rcgiorr nlethods.

The ninth chaptcl is conccrned *,ith nonlinear getterulizatirtrs ol scvcral trutrix ^cllsses.

The lasr chapter'

-

^{Otttlo¡tk ur .fitrther method.s}

-

discussetl sotttc higher-ordcr methorls.

¡ricct'-wise li¡tear nrcthods ¿ìntl solllc lclditional nlinintization rncthods.

'f'he iormer book of Rheinbolclt

(J

l\4 ()rtegn and W

C

Rhcinboldt. ltcnilive St¡lutions of Nottlt¡¿,rtr lx¡uutions in Sever¿tl l'ttriul.tle.;, Acadcmic Plcss. Nerv Yulk. 1970) is onc

ol

the l'un-

r9E Book reviews

rnaking it accessible to a broad audience'

The volume is reconrmended to mathematicians and engineers interested in statistical and probabilistic rnodels. in reliability ^theory, reliability enginccring ^and risk analysis.

C. Mustãta ^.

pRENi K. KYTHE, Cont¡tuÍutiotrct! Confornul Mappittg. Birkhãuse¡ Vcrlag, Basel-Boston-Berlirr,

I 998, xvi + 462 pp.. ISBN 3-7643-3996-9 ^and 0-8 ^I 76-3996-9.

As it is well known. Green's functions for simple ^{regions. as} circles, squares, or annuli, can be calculated effectively, allowing to find ^exact solutions ofboundary ^problems for these regions.

even for rather complicated boundary conditions. But this ^method doesn't work for more compli- cated regions. even for simplc boundary problenrs such as the Dirichlet problem.

A

possible ap- proach to overpass this difficulty is to map coflfbrmally multiple ^connected legions onto simpler connected regions.'yielding in'change ^not only the regions an<J the boundary conditions. but also ir¡

the governing difTerential equations. Conformal mappings

of

multiple connected regions ^are harder to handle than'itt{ose ofì,simp}e connÞcted''ones; most'of the conrputationak'details ^being canied out only numerically.

The main purpbse

of

the present book is to provide a self-contained and systematic intro- ducrion ro the theory and computation of conformal mappings

of

simply or multiply connected regions onro the unit diòc or onto other canonical ^regions, with applications to boundary ^problems fbr integral equations. It is based on a graduate course for students in applied mathematics or engi- neering, taught by the author at the University of New O¡leans

in

1991 . The prerequisites for its reading are a first course iú compìex analysis and familiarity with basic iesults in numerical analy- sis and integral equations (of Fredholm ^and Stieltjes type). A good working knowledge of Mathe- matica as well as a knowlcdgc of á programming language (Fortran or C++) are also requir-éd'

]'he book is divided into l-5 chapters and _^1 Appendices. The first one. Chapter 0, contains a

hystorical overview

of

the problem: backgrounds and modern developments. The basic concepts of complex analysis including Scwartz-Christófi'el trârisformatións of polygonal domains, ^{are pre-} sented (some without proofs)

in

Chapters

I

and 2. Chapter

3

is conccrned with cornputational methods (e.g. Ncwtoir method) for Schwarz-Christoffel improper integrals. This chapter contaìtls also the rcccnt complete solution, given by A. L. Elcrat and L. N. Trelihen'irl 198ó, to the ol<J fluw problem stated by Kirchoff

in

1868. Polynomial approxination, mininíum area problenl and Ritz and Bergmann kernel álgorithms'are ^treated in Ch. 4. Chapter 5 is conccfned rvith nearly circular regions and Ch, 5 with numefical evaluations of Creen functions foi valioirs kinds of regions.

Numerìcal nìethods for various integral equation formulations of'the conforlnal ^mapping ploblenr are discussed in Chaptcrs 7. ^8, 9, l

l

an<l ,l 3. Ch.

l0

is concérned with Jukowski lunction and airfoils. Ch. l2,presents an important ^aspect of the conlbrmal mapping problem, namely fbr regiòns with boundaries with corners. The behavior öf univalent mapping for doubly connected ' domains are srúdied near the boundary in Ch. 12. this study being continued ih Ch.

l3

fbr multiple corìnected <Jontains. The last chapter of the bookr Ch. ^14. gives application.s of coirformal map- pings in adaptive grid generation

200 Beok reviÊws

l0

damental books in'the fi.elcl of solving

nonline

systems of equarions. Horye'er. a lot oll progl.ess has'becn made since this book has,aipeare¿, t,.inúor¿t gathers in thc prcsent monograph sorre important results achieled since thci _incl presents them on the rigorous bases previously _setdecl

with Onega. '-' -"- "êv¡ver _"qrLr I

I

The monograph _'is intended to pre.senr,the thcoretical f'oundations of the,fncthods in fãvor of thc conrputational _aspects: Many ol'ihe results arc accompanied by proof.s. which niake ihis,sec- ond edition ¡nore self'cbrltained tharr the first one: it i,s worth noting that the author _h¿s contributed (o this brx¡k with many person:il _results,

'

^Wo believe that for its'conciscncss and clcàrncss, this nionograph can be a valuable tool not only for the scientist concerned with the theoretical aspects in solvin! _{nonlinear sysrems of óqua.}

tions' but _musr fi¡'st k¡row'how also for those intereste<j. _the methods _behave in the practical ìn the ideal setting.usp..t.. Thc reason is obvious: the practitioners

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E.

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p.

_stanley (Editors)' _Progress

in

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I 998.,483 p, Ir.n IS BN : 0; ^g l 7ó_3 872--5 ⁿ n,| 3. 7643_38 72_-s.

j, i

O. M. D'An¡ona. p. .Diaconis, ,A.

Di

Bucchianic

A.M.

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Tiu.

D. E. l-oeb, M.

A.

Mcndcz.

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R R. P. Stanley, D. Sraton. B, Sturrnf.cls, B. D. T :.The papers inclu.ded

in

thc volunre _are combinatorics (lecture hall panitiôn, _zonotopcs, of trivectofs. axioniati2ation _<rl cubic ofg.Ur;,

;,

c.ial f'unctiohs, comnlutârive algebra ûnd ltatisrics.

L A. Rus