Table Of ContentPanos 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
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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
