Approximation and computation: In honor of Gradimir V. Milovanovic PDF
Preview Approximation and computation: In honor of Gradimir V. Milovanovic
APPROXIMATION AND COMPUTATION Formoretitlesinthisseries,goto http://www.springer.com/series/7393 Springer Optimization and Its Applications VOLUME42 ManagingEditor PanosM.Pardalos(UniversityofFlorida) Editor—CombinatorialOptimization Ding-ZhuDu(UniversityofTexasatDallas) AdvisoryBoard J.Birge(UniversityofChicago) C.A.Floudas(PrincetonUniversity) F.Giannessi(UniversityofPisa) H.D.Sherali(VirginiaPolytechnicandStateUniversity) T.Terlaky(McMasterUniversity) Y.Ye(StanfordUniversity) AimsandScope Optimizationhasbeenexpandinginalldirectionsatanastonishingratedur- ing the last few decades. New algorithmic and theoretical techniques have beendeveloped,thediffusionintootherdisciplineshasproceededatarapid pace, and our knowledge of all aspects of the field has grown even more profound.At thesametime, oneofthe moststriking trendsin optimization is the constantly increasing emphasis on the interdisciplinary nature of the field.Optimizationhasbeenabasictoolinallareasofappliedmathematics, engineering,medicine,economicsandothersciences. The series Springer Optimization and Its Applications publishes under- graduate and graduate textbooks, monographs and state-of-the-art exposi- tory works that focus on algorithms for solving optimization problems and alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered includenonlinearoptimization(convexandnonconvex),networkflowprob- lems, stochastic optimization, optimal control, discrete optimization, mul- tiobjective programming, description of software packages, approximation techniquesandheuristicapproaches. APPROXIMATION AND COMPUTATION In Honor of Gradimir V. Milovanovic´ EditedBy WALTERGAUTSCHI PurdueUniversity DepartmentofComputerScience WestLafayette,Indiana,USA GIUSEPPEMASTROIANNI UniversityofBasilicata DepartmentofMathematicsandComputerSciences Potenza,Italy THEMISTOCLESM.RASSIAS NationalTechnicalUniversityofAthens DepartmentofMathematics Athens,Greece ABC Editors WalterGautschi ThemistoclesM.Rassias PurdueUniversity NationalTechnicalUniversityofAthens DepartmentofComputerScience DepartmentofMathematics 305N.UniversityStreet ZografouCampus WestLafayette,Indiana,47907 15780Athens USA Greece [email protected] [email protected] GiuseppeMastroianni UniversityofBasilicata DepartmentofMathematicsand ComputerSciences Vialedell’AteneoLucano,10 85100Potenza Italy [email protected] ISSN1931-6828 ISBN978-1-4419-6593-6 e-ISBN978-1-4419-6594-3 DOI10.1007/978-1-4419-6594-3 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2010937561 MathematicsSubjectClassification(2010):37D40,65Kxx,68U35,68Txx (cid:2)c SpringerScience+BusinessMedia,LLC2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Coverillustration:“DynamicsofPebbleEnergy”–PhototakenbyEliasTyligadas Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The book “APPROXIMATION AND COMPUTATION” deals with recent results in approximationtheory, numericalanalysis, and various applications of an inter- disciplinary character. It contains papers written by experts from 15 countries in theirrespectivesubjects.Mostofthese paperswere presentedin personduringan InternationalConference,whichwasheldattheUniversityofNisˇ, Serbia(August 25–29,2008),tohonorthe60thanniversaryofthewell-knownmathematicianPro- fessor Gradimir V. Milovanovic´.Professor Milovanovic´,nextto other distinctions he has received, is a correspondingmember of the Serbian Academy of Sciences andArts. ThebookconsistsofthefollowingfiveParts: 1. Introduction 2. PolynomialsandOrthogonalSystems 3. QuadratureFormulae 4. DifferentialEquations 5. Applications Ineachpart,thepapersappearinalphabeticalorderwithrespecttothelastname of the first-named author, except in Part 1. The latter contains three contributions connectedwiththescientificworkofG.V.Milovanovic´.Besidesbiographicaldata, A.Ivic´presentscontributionsofMilovanovic´inthefieldofpolynomialsandquadra- tureprocesses.W.GautschidescribeshiscollaborationwithMilovanovic´overmany years,andTh.M.Rassiasgivesanaccountonsomemajortrendsinmathematics. Polynomials,algebraicandtrigonometric,andcorrespondingorthogonalsystems are treated as basic constructive elements in Part 2. P. Barry, P.M. Rajkovic´, and M.D.Petkovic´ giveanapplicationofSobolevorthogonalpolynomialstothecom- putation of a special Hankel determinant. Extremal problems for polynomials in the complex plane, including the well-known conjecture of Sendov about critical points of algebraic polynomials and the mean value conjecture of Smale, as well as certain relations between these two famous open problems, are considered by B. Bojanov. He also formulates a conjecture that seems to be a natural complex v vi Preface analog of Rolle’s theorem and contains as a particular case Smale’s conjecture. V. Bozˇin and M. Mateljevic´ characterize graphs of maximal energy by means of orthogonalmatrices.Theirresultmakesitpossibletoestimatetheenergyofgraphs without direct computationof eigenvalues. A.S. Cvetkovic´ further develops inter- lacingpropertiesofzerosofshiftedJacobipolynomials,theinvestigationofwhich hasrecentlybeeninitiatedbyK.DriverandK.Jordaan.He,infact,provescertain improvementsoftheirresults.A.S.Cvetkovic´ andM.P.Stanic´ investigatetrigono- metricpolynomialsofsemi-integerdegreewithrespecttosomeweightfunctionson [−π,π)andorthogonalsystemsconnectedwithinterpolatoryquadratureruleswith anevenmaximaltrigonometricdegreeofexactness.W. Gautschigivesanaccount of computational work in support of conjectured inequalities for zeros of Jacobi polynomials, the sharpness of Bernstein’s inequality for Jacobi polynomials, and thepositivityofcertainquadratureformulaeofNewton–Cotes,Gauss–Radau,and Gauss–Lobattotype.Theuseofsymboliccomputationisdescribedforgenerating Gaussquadratureruleswithexoticweightfunctions,specifically weightfunctions decayingsuper-exponentiallyatinfinity,andweightfunctionsdenselyoscillatoryat zero.J.GilewiczandR.Jedynakilluminatethecompatibilityofcontinuedfraction convergentswithPade´approximants.Orthogonaldecompositionsoffractalsetsare considered by Lj.M. Kocic´, S. Gegovska-Zajkova,and E. Babacˇe. The last paper inthispartbyS.Koumandosgivesasystematicaccountofnewresultsonpositive trigonometricsums and applicationsto geometricfunctiontheory.His work is re- latedtorecentinvestigationsconcerningsharpeningandgeneralizationsofthecele- bratedVietorisinequalities.Far-reachingextensionsandresultsonstarlikefunctions areobtainedandnewpositivesumsofGegenbauerpolynomialsandadiscussionof somechallengingconjecturesarepresented. Part3isdedicatedtonumericalintegration,includingquadratureformulaswith equidistant nodes and formulas of Gaussian type having multiple nodes. Several quadraturerulesforthenumericalintegrationofsmooth(nonoscillatory)functions, definedonthereal(positive)semiaxisorontherealaxisanddecayingalgebraically atinfinity,areexaminedbyG.MonegatoandL.Scuderi.Amongthoseconsidered on the real axis, there are four new alternative numericalapproaches. The advan- tages and the disadvantages of each of them are pointed out through several nu- mericaltests,involvingeitherthecomputationofasingleintegralorthenumerical solution of some integral equations. G. Nikolov and C. Simian construct Gauss, Lobatto, and Radau quadrature formulae associated with the spaces of parabolic splineswithequidistantknots.Thesequadratureformulaeareknowntobeasymp- totically optimal in Sobolev spaces W3. Sharp estimates for the error constant in p W∞3 are given.Somequadraturerulesbasedonthe zerosof Freudpolynomialsfor computingCauchyprincipalvalueintegralsonthereallineareproposedbyI.No- tarangelo. M.M. Spalevic´, and M.S. Pranic´ give a survey on contour integration methods for estimating the remainder of Gauss–Tura´n quadrature rules involving analyticfunctions.Finally,J.Waldvogelproposestousethetrapezoidalruleonthe entire real line R as the standard algorithm for numerical quadrature of analytic functions.Otherintervalsandslowlydecayingintegrandsmayelegantlybehandled bymeansofsimpleanalytictransformationsoftheintegrationvariable. Preface vii Methods for differential equations are considered in Part 4. D.R. Bojovic´ and B.S. Jovanovic´ investigate the convergence of difference schemes for the one- dimensionalheatequationwith time-dependentoperatorandthe coefficientofthe time derivativecontaining a Dirac delta distribution. An abstract operator method isdevelopedforanalyzingthisequation.InthepaperbyS.Burke,Ch.Ortner,and E. Su¨li, the energy of the Francfort–Marigo model of brittle fracture is approxi- mated,inthesenseofΓ-convergence,bytheAmbrosio–Tortorellifunctional.They formulateand analyze an adaptivefinite elementalgorithm,combiningan inexact Newton method with residual-driven adaptive mesh refinement, for the computa- tionofits(local)minimizers.Thesequencegeneratedbythisalgorithmisprovedto convergetoa criticalpoint.C. Frammartinoproposesa Nystro¨mmethodforsolv- ing integralequationsequivalentto second-orderboundaryvalue problemson the real semiaxis. She proves the stability and convergence of such a procedure and gives some interesting numerical examples. B.S. Jovanovic´ investigates an initial boundaryvalueproblemforaone-dimensionalhyperbolicequationintwodisjoint intervals.Afinitedifferenceschemeapproximatingthisinitialboundaryvalueprob- lemisproposedandanalyzed.P.S.Milojevic´ developsanonlinearFredholmalter- nativetheoryinvolvingk-ballandk-setperturbationsof generalhomeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. He gives several applications to the unique and finite solvability of potential and semilinearproblemswithstronglynonlinearboundaryconditionsandtoquasilinear elliptic equations. The work of S. Pilipovic´, N. Teofanov, and J. Toft shows pos- sibledirectionsfornumericallyinterestedmathematicianstoapproximatedifferent typesofsingularsupports,wavefrontsets, andpseudodifferentialoperatorsinthe frameworkofFourier-Lebesguespaces.Itcontainsnewresultsonsingularsupports in Fourier-Lebesguespaces and on the continuity propertiesof certain pseudodif- ferentialoperators.Finally,B.M.Piperevskiconsidersaclassofmatrixdifferential equationsandgivesconditionsunderwhichthisclasshasapolynomialsolution. Finally,contributionsdiscussingvariousapplicationsarepresentedinPart5.An improvedalgorithmforPetviashvili’s heuristic numericalmethodfor findingsoli- tonsinopticallyinducedphotoniclatticesispresentedbyR.Jovanovic´andM.Tuba. An explicit method for the numerical solution of the Fokker–Planck equation of filtered phase noise in modern telecommunication systems is given by D. Milic´. Z.S.Nikolic´investigatesnumericallythedensificationduetogravity-inducedskele- talsettlingduringliquidphasesintering.Anewmethodologyisappliedforsimula- tionofmicrostructuralevolutionofa regularmultidomainmodel.P.Stanimirovic´, M. Miladinovic´,and I.M. Jovanovic´ investigatesymbolic transformationson “un- evaluated” expressions representing objective functions to generate unevaluated compositeobjectivefunctionsrequiredduringtheimplementationofunconstrained nonlinearoptimizationmethodsbased onthe exactline search. N. Stevanovic´ and P.V.Protic´ dealwithAbel-Grassmann’sgroupoidsandintroducetheclass“rootof aband”,ageneralizationofAG-bandandAG-3-band.B.T.Todorovic´,S.R.Rancˇic´, andE.H. Mulalic´ consideraversionofprobabilisticsupervisedmachine,learning classifierinnamedentityrecognition.Z.Udovicˇic´dealswithinterpolatingquadratic viii Preface splines,andfinally,Lj.S. Velimirovic´,S.R. Rancˇic´,andM.Lj. Zlatanovic´consider infinitesimalbendingofcurvesinE3. Thisbookaddressesresearchersandstudentsinmathematics,physics,andother computationalandappliedsciences. AscoauthorsofProfessorGradimirV.Milovanovic´,wefeelverypleasedtohave undertakenthepreparationofthispublication. Finally, we expressour warmest thanksto all of the scientists who contributed to this volume, to the referees for their carefulreading of the manuscripts, and to collaboratorsofProfessorMilovanovic´:LjubisˇaKocic´,AleksandarCvetkovic´,and MarijaStanic´,whohelpedintheorganizationoftheConferenceandintheprepara- tionofthisvolume. WestLafayette/Potenza/Athens WalterGautschi August,2009 GiuseppeV.Mastroianni ThemistoclesM.Rassias Contents Preface............................................................ v PartI Introduction TheScientificWorkofGradimirV.Milovanovic´ ..................... 3 AleksandarIvic´ 1 Introduction.............................................. 3 2 TheBiographyofG.V.Milovanovic´.......................... 3 3 FieldsofScientificWorkofGVM ........................... 5 4 GVMandQuadratureProcesses............................. 6 4.1 ConstructionofGaussianQuadratures ................ 6 4.2 Moment-PreservingSpline Approximationand Quadratures ...................................... 8 4.3 QuadratureswithMultipleNodes .................... 9 4.4 OrthogonalitywithRespecttoaMomentFunctional andCorrespondingQuadratures ..................... 10 4.5 NonstandardQuadraturesofGaussianType............ 13 4.6 GaussianQuadratureforMu¨ntzSystems .............. 15 5 GVMandPolynomials..................................... 16 5.1 ClassicalOrthogonalPolynomials.................... 16 5.2 ExtremalProblemsofMarkov–BernsteinTypefor Polynomials...................................... 17 5.3 OrthogonalPolynomialsonRadialRays .............. 21 6 GVMandInterpolationProcesses ........................... 24 7 TheSelectedBibliographyofGVM.......................... 25 References..................................................... 25 MyCollaborationwithGradimirV.Milovanovic´ ..................... 33 WalterGautschi References..................................................... 42 ix
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