Contributions in Mathematics and Engineering: In Honor of Constantin Carathéodory PDF
Preview Contributions in Mathematics and Engineering: In Honor of Constantin Carathéodory
Panos M. Pardalos Themistocles M. Rassias Editors Contributions in Mathematics and Engineering In Honor of Constantin Carathéodory Foreword by Terry Rockafellar Contributions in Mathematics and Engineering Panos M. Pardalos • Themistocles M. Rassias Editors Contributions in Mathematics and Engineering In Honor of Constantin Carathéodory Foreword by R. Tyrrell Rockafellar 123 Editors PanosM.Pardalos ThemistoclesM.Rassias DepartmentofIndustrial DepartmentofMathematics andSystemsEngineering NationalTechnicalUniversity UniversityofFlorida ofAthens Gainesville,Florida,USA Athens,Greece ISBN978-3-319-31315-3 ISBN 978-3-319-31317-7 (eBook) DOI10.1007/978-3-319-31317-7 LibraryofCongressControlNumber:2016941334 MathematicsSubjectClassification(2010):33–XX,34–XX,35–XX,39–XX,41–XX,46–XX,47–XX, 49–XX,51–XX,52–XX,53–XX,58–XX,65–XX,90–XX,91–XX,93–XX ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Foreword ConstantinCarathéodory,1873–1950,wasoneofthemostimportantandinfluential mathematicians of his era and certainly the preeminent Greek on the scene. His workcombinedtheory-buildingwithpracticalapplicationsinawaythatisnolonger so common. His career spanned a remarkable range of locations and historical upheavals,evenbytoday’sstandards. Carathéodory’s accomplishments are too voluminous to describe in balanced detail, but from a personal perspective, my connection began with his famous theoremonconvexhulls.Thatresult,withmajorconsequencesinconvexanalysis and optimization technology, confirms that in n-dimensional space, the hull can be formed by taking convex combinations of just n C 1 elements at a time. His innovative developments in the calculus of variations, beginning with his doctoral dissertation and carried on in retirement with his editing of Euler’s works in that subject,alsoattractedmeearlyonbecauseoftheirmoderntiestooptimizationand control. Theory-building was certainly the theme in those efforts in the calculus of variations,influenced byconsiderations inoptics,buteven moresoinhisground- breaking axiomatic formulation of thermodynamics. These achievements under- scoretheback-and-forthbetweenmathematicsandphysicsthatwasthecenterpiece of science a century ago but now is just one of many lines of inspiration and progress.TheyalsoreflectCarathéory’spracticalbent,havingonceworkedforthe BritishColonialServiceasanengineerindamconstructioninEgypt. We think it normal now that faculties of universities include multilingual professors of many nationalities who have lived, studied, and taught in many places, but that was true in Carathéodory’s time as well. His own life provides an instructive example which is full of reminders of how world events can intervene. Born in Berlin of Greek parents in the diplomatic circuit, not for Greece but the Ottoman Empire (with Constantinople as their home), he grew up in Brussels, summeringontheFrenchRiviera.Hewent ontoGermany asauniversitystudent and eventually to a prominent academic career with posts shifting from Göttingen v vi Foreword toBonntoHannovertoBreslautoGöttingenagainandfinallytoBerlin.WorldWar Ibroughthimhardshipsandemptiedtheuniversitiesofcolleaguesandstudentsof militaryage. In1920,afteraformalinvitationfromtheprimeministerofGreeceEleftherios Venizelos,hemovedtoGreeceinordertooverseethefoundingofanewuniversity inSmyrna.However,thatcamequicklytoanabruptendwhentheTurksoverranthe regionandforcedtheGreeksout.In1922hemovedtoAthensobtainingprofessorial positions at the University of Athens and subsequently at the National Technical University of Athens. In 1924, on the recommendation of A. Sommerfeld, he was appointed professor at the University of Munich succeeding F. Lindemann. A few yearslater,in1930,CarathéodorywasinvitedyetagainbytheGreekprimeminister E.VenizelosinordertoundertakeadministrativedutiesattheUniversityofAthens as well as at the Aristotle University of Thessaloniki, which he accepted and then offeredhisvaluableservicesforthenext2years. Carathéodory visited the United States twice and could have accepted an offer fromStanford,butdidnot.ThencameWorldWarII,whichherodeoutinMunich. That marked the demise of Germany’s longtime dominance in mathematics and othersciences. ThededicationofthisvolumetoConstantinCarathéodoryisafittingtributetoa greatdiscovereroffactsandideaswhichcontinuetoenrichusall. Seattle,WA,USA R.TyrrellRockafellar November2015 Preface This volume consists of scientific articles dedicated to the work of Constantin Carathéodory. These articles deepen our understanding of some of the current researchproblemsandtheorieswhichhavetheiroriginorhavebeeninfluencedby Carathéodory. The presentation of concepts and methods featured in this volume make it an invaluable reference for teachers and other professionals in Mathematics, who are interestedinpureandappliedresearch. It is our pleasure to express our warmest thanks to all of the scientists who contributed to this volume and to Professor R. Tyrrell Rockafellar for writing the Forewordtothisvolume.Wewouldalsoliketoacknowledgethesuperbassistance thatthestaffofSpringerhasprovidedforthispublication. Gainesville,FL,USA PanosM.Pardalos Athens,Greece ThemistoclesM.Rassias vii Contents ApplicationsofQuasiconvexity................................................. 1 S.Abramovich Taylor’sFormulaandIntegralInequalitiesforConformable FractionalDerivatives ........................................................... 25 DouglasR.Anderson Sobolev-TypeInequalitiesonManifoldsinthePresenceof SymmetriesandApplications................................................... 45 AthanaseCotsiolisandNikosLabropoulos GeneralizedMinkowskiFunctionals........................................... 69 StefanCzerwikandKrzysztofKról ANetworkDesignModelUnderUncertainty................................. 81 E.D’Amato,E.Daniele,andL.Mallozzi OptimalRationalApproximationNumberSets:Applicationto NonlinearDynamicsinParticleAccelerators................................. 95 NicholasJ.DarasandMichaelN.Vrahatis A Characterization Theorem for the Best L1 Piecewise MonotonicDataApproximationProblem..................................... 117 IoannisC.Demetriou SupermeasuresAssociatedtoSomeClassicalInequalities .................. 127 SilvestruSeverDragomir RemarksonSolutionsofaFunctionalEquationArisingfrom anAsymmetricSwitch........................................................... 153 El-Sayed El-Hady, Janusz Brzde˛k, Wolfgang Förg-Rob, and HamedNassar Hyers–UlamStabilityofWilson’sFunctionalEquation..................... 165 ElhoucienElqorachi,YoussefManar,andThemistoclesM.Rassias ix
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