Algebraic and Algorithmic Aspects of Differential and Integral Operators: 5th International Meeting, AADIOS 2012, Held at the Applications of Computer Algebra Conference, ACA 2012, Sofia, Bulgaria, June 25-28, 2012, Selected and Invited Papers PDF
Preview Algebraic and Algorithmic Aspects of Differential and Integral Operators: 5th International Meeting, AADIOS 2012, Held at the Applications of Computer Algebra Conference, ACA 2012, Sofia, Bulgaria, June 25-28, 2012, Selected and Invited Papers
Moulay Barkatou Thomas Cluzeau Georg Regensburger Markus Rosenkranz (Eds.) Algebraic and Algorithmic Aspects 2 7 3 of Differential 8 S C and Integral Operators N L 5th International Meeting, AADIOS 2012 Held at the Applications of Computer Algebra Conference, ACA 2012 Sofia, Bulgaria, June 25-28, 2012 Selected and Invited Papers 123 Lecture Notes in Computer Science 8372 CommencedPublicationin1973 FoundingandFormerSeriesEditors: GerhardGoos,JurisHartmanis,andJanvanLeeuwen EditorialBoard DavidHutchison LancasterUniversity,UK TakeoKanade CarnegieMellonUniversity,Pittsburgh,PA,USA JosefKittler UniversityofSurrey,Guildford,UK JonM.Kleinberg CornellUniversity,Ithaca,NY,USA AlfredKobsa UniversityofCalifornia,Irvine,CA,USA FriedemannMattern ETHZurich,Switzerland JohnC.Mitchell StanfordUniversity,CA,USA MoniNaor WeizmannInstituteofScience,Rehovot,Israel OscarNierstrasz UniversityofBern,Switzerland C.PanduRangan IndianInstituteofTechnology,Madras,India BernhardSteffen TUDortmundUniversity,Germany DemetriTerzopoulos UniversityofCalifornia,LosAngeles,CA,USA DougTygar UniversityofCalifornia,Berkeley,CA,USA GerhardWeikum MaxPlanckInstituteforInformatics,Saarbruecken,Germany Moulay Barkatou Thomas Cluzeau GeorgRegensburger MarkusRosenkranz(Eds.) Algebraic and Algorithmic Aspects of Differential and Integral Operators 5th International Meeting,AADIOS 2012 Held at theApplications of ComputerAlgebra Conference,ACA 2012 Sofia, Bulgaria, June 25-28, 2012 Selected and Invited Papers 1 3 VolumeEditors MoulayBarkatou ThomasCluzeau UniversityofLimoges XLIM,DMI 123,avenueAlbertThomas,87060Limogescedex,France E-mail:[email protected] E-mail:[email protected] GeorgRegensburger JohannRadonInstituteforComputationalandAppliedMathematics(RICAM) AustrianAcademyofSciences AltenbergerStr.69,4040Linz,Austria E-mail:[email protected] MarkusRosenkranz UniversityofKent SMAS Canterbury,CT27NF,UK E-mail:[email protected] ISSN0302-9743 e-ISSN1611-3349 ISBN978-3-642-54478-1 e-ISBN978-3-642-54479-8 DOI10.1007/978-3-642-54479-8 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014931855 CRSubjectClassification(1998):F.4,G.1.7-9,G.1-2,I.1,D.3,F.3 LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues ©Springer-VerlagBerlinHeidelberg2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’slocation, inistcurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissionsforuse maybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliabletoprosecution undertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication, neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrorsor omissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothe materialcontainedherein. Typesetting:Camera-readybyauthor,dataconversionbyScientificPublishingServices,Chennai,India Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Symbolic analysis may be defined as the study of differential equations in sym- bolic computation; this implies a certain preference of algebraic over geometric concepts anda strong bias towardconstructive methods. Overthe lastdecades, this field has seen an exceptionally fruitful period of growth in many different directions.Itisthereforeimportanttorefocusthesubjectareabyofferingasuit- able vantage point for workshopsand papers. Choosing the operator perspective assuchavantagepoint,twoofus(GeorgRegensburger,MarkusRosenkranz)set upthefirstAlgebraicandAlgorithmicAspectsofDifferentialandIntegralOper- atorSession(AADIOS)attheConferenceonApplicationsofComputerAlgebra (ACA), held in July 2008 at the Research Institute for Symbolic Computation (RISC) at Hagenberg, Austria. The perspective of algebraic operator rings was chosen not only to provide a common leading theme for much interesting work in symbolic analysis. It is also instrumental in bringing together the two main streams of analysis— differentiation and integration. There is no question that between the two, the algebraic theory of differential operators is by far the more developed one. On the other hand, the theory of Rota–Baxter algebras is also rapidly expanding with new applications in fields as diverse as perturbative quantum field theory and multiple zeta values. Therefore, we have always given special attention to approaches that provide a link between these theories. The venues of previous AADIOS were Hagenberg in 2008, Montr´ealin 2009, Vlora in 2010, Houston in 2011, Sofia in 2012, Malaga in 2013, and the next AADIOS is already scheduled for 2014 in New York. Since 2012, the AADIOS has been co-organized with Moulay Barkatou and Thomas Cluzeau. Based on the AADIOS 2008–2010 events, we have published an MCS Special Issue (Vol. 4, Nos. 2–3, 2010). For the current proceedings volume of AADIOS 2012, the Call for Papers listed the following (non-exclusive) topics of interest: – Symbolic Computation for Operator Algebras – Factorization of Differential/Integral Operators – Linear Boundary Problems and Green’s Operators – Initial Value Problems for Differential Equations – Symbolic Integration and Differential Galois Theory – Symbolic Operator Calculi – Algorithmic D-Module Theory – Rota–Baxter Algebra – Differential Algebra – Discrete Analogs of the above – Software Aspects of the above VI Preface Wereceived13submissionsfromAustria,Bulgaria,France,TheNetherlands, Russian Federation, Spain, USA, and Vietnam; of these, seven research papers containing new results were finally accepted (acceptance rate 0.54). To ensure a high-quality review process, we asked six associate editors to join our team. All reviewswerediscussedviaEasyChairbyalleditors,andjointdecisionswere reached. A total of 33 reviews were carried out (30 of these from a pool of 27 external reviewers). In addition to the contributed papers (which were not lim- ited to AADIOS participants), we have also included two invited survey papers on crucial AADIOS topics. For these papers we drew on informative feedback from selected external reviewers and some associate editors. The survey paper by Xing Gao and Li Guo features integro-differential algebras—one of those algebraic structures that bring together differential and integral operators—and reports on new constructions of the free object in this category. The survey by Jos´e G´omez-Torrecillas takes a more abstract view- point, studying operatoralgebrasfromthe unified perspective of computational module theory. It ends with an appendix on computer algebra systems written by Viktor Levandovskyy. The paper by Ivan Dimovski and Margarita Spiridonova sets up new opera- tional calculi in the spirit ofMikusin´ski (introducing the differential operatoras a suitable convolution inverse) for nonlocal Cauchy problems with resonances. The intertwining Laplace transformations studied by Elena I. Ganzha are an interesting class of operators generalizing Laplace’s classic cascade method for solvinghyperboliclinearpartialdifferentialequations.Integraloperatorsarein- vestigatedin the paper by Anja Korporal and Georg Regensburger, inparticular the compositionandfactorizationofgeneralizedGreen’s operators(solutionop- erators for singular boundary problems). In their paper on relativistic Coulomb integrals,Christoph Koutschan, Peter Paule, and Sergei K. Suslov employ Zeil- berger’s machinery of holonomic functions (those defined by holonomic ideals of differential operators with polynomial coefficients) in a novel and sophisti- cated way. The paper by Alexander Levin takes up the classic topic of mea- suring the dimension of differential field extensions (correspondingto Einstein’s notion of strength in relativistic equations) and applies it to a new class of in- versivedifference fieldextensions.The paperbyLaurent Poinsot isdevotedtoa functor-based investigation and generalization of differential monoid and semi- groupalgebras(their main flavor—theringof differentialpolynomials—forming thealgebraicstandardmodelofnonlineardifferentialoperators).Inthepaperby Yulian Tsankov, asuitableoperationalcalculus(asabove)isappliedtoobtaina detailed solutionto a class of local as well as nonlocal boundary value problems for the Laplace equation in a rectangle. Itisourpleasuretothanktheauthorsofsubmittedpapersforalltheirefforts. Wearealsoverygratefultothe authorsofthe invitedpapersforprovidingtheir topical surveys. We would also like to thank our associate editors (see below) and all external reviewers; without their diligent work a proceedings volume suchas this would not be possible. In the same vein, we would like to creditthe great work of the ACA organizers and the ACA working group for providing Preface VII a constant and reliable platform to present and discuss hot topics in computer algebra (those of the AADIOS as well as many other interesting themes)! Last but not least we thank Springer for all their professional support in putting together this LNCS volume. January 2014 Moulay Barkatou Thomas Cluzeau Georg Regensburger Markus Rosenkranz Organization Session Organizers and Volume Editors Moulay Barkatou University of Limoges, XLIM, DMI, France Thomas Cluzeau University of Limoges, XLIM, DMI, France Georg Regensburger RICAM, Austrian Academy of Sciences, Austria Markus Rosenkranz University of Kent, SMSAS, UK Associate Editors Fran¸cois Boulier Universit´e Lille 1, LIFL, France Ryong Feng Chinese Academy of Science, Key Laboratory of Mathematics Mechanization, China George Labahn University of Waterloo, Cheriton School of Computer Science, Canada Viktor Levandovskyy RWTH Aachen University, Lehrstuhl D fu¨r Mathematik, Germany Margarita Spiridonova Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Bulgaria Franz Winkler Johannes Kepler University Linz, RISC, Austria Table of Contents Constructions of Free Commutative Integro-Differential Algebras....... 1 Xing Gao and Li Guo BasicModuleTheoryoverNon-commutativeRingswithComputational Aspects of Operator Algebras...................................... 23 Jos´e G´omez-Torrecillas Operational Calculi for Nonlocal Cauchy Problems in Resonance Cases........................................................... 83 Ivan Dimovski and Margarita Spiridonova Intertwining Laplace Transformations of Linear Partial Differential Equations....................................................... 96 Elena I. Ganzha Composing and Factoring GeneralizedGreen’s Operators and Ordinary Boundary Problems .............................................. 116 Anja Korporal and Georg Regensburger Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach II..................................................... 135 Christoph Koutschan, Peter Paule, and Sergei K. Suslov Multivariate Dimension Polynomials of Inversive Difference Field Extensions ...................................................... 146 Alexander Levin Differential (Monoid) Algebra and More ............................ 164 Laurent Poinsot Exact Solution of Local and Nonlocal BVPs for the Laplace Equation in a Rectangle ................................................... 190 Yulian Tsankov Author Index.................................................. 201 Constructions of Free Commutative Integro-Differential Algebras Xing Gao1 and Li Guo1,2 1 Department of Mathematics, Lanzhou University,Lanzhou, Gansu 730000, China [email protected] 2 Department of Mathematics and Computer Science, RutgersUniversity,Newark, NJ 07102, USA [email protected] Abstract. In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the con- struction offree commutativeRota-Baxteralgebras bymixableshuffles. The first is by evaluations. The second is by the method of Gro¨bner- Shirshov bases. Keywords: Differential algebra, Rota-Baxter algebra, integro- differential algebra, Gro¨bner-Shirshov basis, free algebra, shuffle product, mixable shuffleproduct. Table of Contents Constructions of Free Commutative Integro-Differential Algebras Xing Gao and Li Guo .......................................... 1 1 Introduction................................................... 2 2 Definitions and Preliminary Constructions ........................ 3 2.1 The Definitions ........................................... 3 2.2 Free Differential Rota-Baxter Algebras ....................... 5 2.3 Free Commutative Operated Algebras........................ 8 3 Free Commutative Integro-Differential Algebras by Initialization ..... 10 3.1 Regular Differential Algebras ............................... 10 Quasi-Antiderivatives and Regularity. ........................ 10 Differential Polynomial Algebras............................. 11 Rational Functions......................................... 12 3.2 Construction of ID(A)∗ .................................... 13 4 Free Commutative Integro-Differential Algebras by Gro¨bner-Shirshov Bases...................................... 14 4.1 Weakly Monomial Order ................................... 14 4.2 Composition-Diamond Lemma .............................. 17 4.3 Free Commutative Integro-Differential Algebras by Gr¨obner-Shirshov Bases ................................. 19 M.Barkatouetal.(Eds.):AADIOS2012,LNCS8372,pp.1–22,2014. (cid:3)c Springer-VerlagBerlinHeidelberg2014
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