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Monte Carloand Quasi-Monte Carlo Methods2002
Springer-Verlag Berlin Heidelberg GmbH
Harald Niederreiter Editor
Monte Carlo and Quasi-Monte Carlo Methods 2002 Proceedings of a Conference held at the National University of Singapore, Republic of Singapore, November 25-28, 2002
Springer
Editor
Harald Niederreiter
Department of Mathematics N ationa1 University of Singapore 2 Science Drive 2
Singapore 117543 Republic of Singapore e-mail: [email protected]
Cataloging-in-Publication Data applied for
A catalog record for this book is available from the Library of Congress.
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
Mathematics Subject Classification (2000): Primary: llK45, 65-06, 65C05, 65ClO Secondary: nK38, 65D30, 65D32, 65R20, 91B28
ISBN 978-3-540-20466-4 ISBN 978-3-642-18743-8 (eBook) DOI 10.1007/978-3-642-18743-8
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations. recitation. broadcasting. reproduction on microfilm or in any other way. and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965. in its current version. and permis sion for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
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© Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg in 2004
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Preface
This volume contains the refereed proceedings of the Fifth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2002) which was held at the National University of Singapore from 25-28 November 2002. The programme of this conference wasarranged by a committee consisting of Kai-Tai Fang (Hong Kong BaptistUniversity), Paul Glasserman (Columbia University), Stefan Heinrich (Universitat Kaiserslautern), Fred J . Hickernell (Hong Kong Baptist University),Pierre L'Ecuyer (Universite de Montreal), Harald Niederreiter (National University of Singapore, chair), Erich Novak (Universitat Jena), Art B. Owen(Stanford University), Ian H. Sloan (University of New South Wales), JeromeSpanier (Claremont Graduate University), Denis Talay (INRIA Sophia Antipolis), Simon Tavare (University of Southern California), Jian-Sheng Wang(National University of Singapore) and Henryk Wozniakowski (Columbia University and University of Warsaw).
MCQMC 2002 continued the tradition of biennial MCQMC conferenceswhich was begun at the University of Nevada in Las Vegas, Nevada, USA, inJune 1994 and followed by conferences at the University of Salzburg, Austria,in July 1996, the Claremont Colleges in Claremont, California, USA, in June1998 and Hong Kong Baptist University in Hong Kong, China, in November2000. The proceedings of these previous conferences were all published bySpringer-Verlag, under the titles Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P.J.-S. Shiue, eds.), MonteCarlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter, P. Hellekalek,G. Larcher and P. Zinterhof, eds.), Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.) and Monte Carlo and QuasiMonte Carlo Methods 2000 (K.-T. Fang, F.J. Hickernell and H. Niederreiter,eds.). The next MCQMC conference will be held at Juan-les-Pins, France, inJune 2004.
These proceedings contain papers based on all eight invited plenary talks atthe conference as well as 20 contributed papers that passed rigorous screeningand refereeing procedures. Many of these contributed papers were presented in
VI Preface
special sessions at the conference that were devoted to topics of great currentinterest. The papers in this volume cover both the theory and the applicationsof Monte Carlo and quasi-Monte Carlo methods, as has been the custom forprevious proceedings.
I gratefully acknowledge generous financial support of the conference bythe Department of Mathematics and the Centre for Financial Engineering(CFE) at the National University of Singapore, the Lee Foundation and DBSBank in Singapore. A special feature of MCQMC 2002 was a one-day seminaron Computational Finance in Banking which was held on 26 November 2002.I want to thank Kah Hwa Ng, the Director of the CFE, for helping to organise this event and raising sponsorship funds from CFE and DBS Bank. Myspecial appreciation and thanks go to Seng Luan Lee, the Head of the Department of Mathematics at the National University of Singapore, for granting thenecessary administrative and technical support of the conference and to mycolleague Chaoping Xing for helping with various tasks. Among the administrative staff I want to single out the indomitable Stella Pang who put in atremendous effort to make the conference a success. The high scientific level ofthe conference was due in great measure to the Programme Committee whichscreened the submitted abstracts and provided valuable advice on the selection of invited speakers. I want to extend my cordial thanks to the refereesof the proceedings manuscripts who worked with care and professionalism tomaintain the customary standards of the MCQMC proceedings.
Finally, I want to express my gratitude to Springer-Verlag, and especiallyto Dr. Martin Peters, for continuing to publish the MCQMC proceedings andfor the very helpful support and kind advice I have received from his staff.
September 2003 Harald Niederreiter
Table of Contents
Preface V
Conference Participants XI
Invited Papers
Finance: A Fertile Field for Applications of MC and QMCPhelim P. B oyle 1
How Many Random Bits Do We Need for Monte Carlo Integration?Stefan Heinrich, Erich Novak, and Harald Pfeiffer 27
On Tractability of Weighted Integration for Certain BanachSpaces of FunctionsFred J. Hickern ell, Ian H. Sloan, and Grzegorz W. Wasi lkowski 51
Polynomial Integration LatticesPierre L'Ecuyer 73
Approximate Bayesian Computation and MCMCVincent Plagnol and Sim on Tauare 99
New Challenges for the Simulation of Stochastic ProcessesDenis Talay 115
Stochastic Models and Monte Carlo Algorithms for BoltzmannType Equat ionsWolfgang Wagner 129
Digital Nets , Duality, and Algebraic CurvesChaoping X ing and Harald Niederreiier 155
VIII Table of Contents
Contributed Papers
Generalized Mersenne Prime Number and Its Application toRandom Number GenerationLih- Yuan Deng 167
Constructing Good Lattice Rules with Millions of PointsJosef Dick and Frances Y. Kuo 181
Lattice Structure of Nonlinear Pseudorandom Number Generatorsin Parts of the PeriodGerhard Dorfer and Arne Winterhof 199
Simulation for American Options: Regression Now orRegression Later?Paul Glasserman and Bin Yu 213
Perturbation Monte Carlo Methods for the Solution ofInverse ProblemsCarole K. Hayakawa and Jerome Spanier 227
Quantum Boolean Summation with Repetitions in theWorst-Average SettingStefan Heinrich, Marek Kwas, and Henryk Woiniakowski 243
The Strong Tractability of Multivariate Integration UsingLattice RulesFred J. Hickernell, Ian H. Sloan, and Grzegorz W. Wasilkowski 259
Minimizing Effective Dimension Using Linear TransformationJunichi Imai and Ken Seng Tan 275
Component by Component Construction of Rank-1 LatticeRules Having O(n-1(1n(n))d) Star DiscrepancyStephen Joe 293
Stratification by Rank-1 LatticesAlexander Keller 299
Walsh Series Analysis of the Star Discrepancy of Digital Netsand SequencesGerhard Larcher and Friedrich Pillichshammer 315
Quasi-Monte Carlo Methods for Estimating Transient Measuresof Discrete Time Markov ChainsChristian Lecot and Bruno Tuffin 329
Quasi-Monte Carlo Methods for Elliptic BVPsMichael Mascagni, Aneta Karaivanova, and Chi-Ok Hwang 345
Table of Contents IX
Stable Connectivity of Networks and Its Monte Carlo EstimationHozumi Morohosi and Tatsuo Oyama 357
Random Numb er Generators Based on Linear Recurrences in 1F2 w
Francois Panneton and Pierre L 'Ecuyer 367
Using Quasi-Monte Carlo Scenarios in Risk ManagementFilip Pistooaik and Thoma s Breuer 379
Adaptive Quasi-Monte Carlo Integration Based on MISERand VEGASRudolf Schiirer 393
Wh en Does Monte Carlo Depend Polynomially on theNumber of Vari ables ?Ian H. Sloan and Hen ryk Woiniakowski 407
A New Adaptive Method for Geometric ConvergenceJerome Spanier and Rang Kong 439
Polynomial Arithmetic Analogue of Hickernell SequencesShu Tezuka 451
Conference Participants
Martin Ambrose5864 Los Ranchos DriveLos Angeles, C A 90620US Amart in @md s.com
Michael BoothSu it e 50125 Lime StreetSydney, NSW 2000Au [email protected]
Phelim BoyleCentre for Advances Studi es in FinanceSchool of Accou nt ancyUni versity of Wa t erlooWaterloo , Ont ario N2 L 3G lCa nadapboyle@uwater loo.ca
Chin Hiang ChanUOB Limited Head Office80 Raffles Place#25-00, UO B Plaza 2Singapore 04 [email protected]
Scott Chastain3900 W isconsin Ave NWWashington. DC 2001 6-2 892USAscott_chastain@fanniemae .com
Wai-Yan ChengDepartmen t of Ec onomics a nd Fi nanceCity Un ivers ity of Hong KongKowloon TongHong Kong SA RChinayan.cheng@ci tyu .edu.hk
Philippe DamayDBS Bank8 Cross Street#22-01/07, PWC BuildingSingapore 048424philippedamay @dbs.com
Paolo BaldiDipartimento di MatematicaUnlversi ta di Roma Tor VergataVia della Ri cerce Scientifica1-00133 Ro maIt alybaldi@ma t .uniroma2.it
Artan BoridJ CMBUni versi ty of EdinburghMayfield RoadEdi nb ur gh E H9 3JZScotland , UKbo rici @ph .ed .ac.uk
Chee Hian ChaiBlk 133 Bedok Reservoir Ro ad# 10-1205Singapore 470133br [email protected] .sg
Ling Yau ChanDep artmen t of Indust rial and
Manufact uri ng Systems E ng ineeringThe Uni vers ity of Hong KongP okfula m RoadHong Kong SARChinap lychan@h ku .hk
Anthony Dien Yiu ChengDBS Bank8 Cross St ree t# 22-01/07, PWC BuildingSingapore 048424
Ronald CoolsDepartment of Computer ScienceKatholieke Universite it LeuvenCe lest ij nenlaan 200AB- 3001 HeverleeBelg iumRonald.Cools@cs. ku leuven .ac.be
Lih-Yuan DengDep ar tment of Mathematical SciencesUniversity of Memp hisMemphis , TN 381 52USAlih [email protected]
XII Conference Participants
Josef DickSchool of MathematicsUniversity of New South WalesKensingtonSydney, NSW [email protected]
Gerhard DorferDepartment of Algebra and Compo Math .Vienna University of TechnologyWiedner Hauptstr. 8-10/118A-1040 ViennaAustriag.dorfer@tuwien .ac .at
Gangzhi FanDepartment of Real EstateSchool of Design and EnvironmentNational University of Singapore4 Architecture DriveSingapore [email protected] .sg
Henri FaureInstitut de Mathemat.iques de Luminy163 Avenue de LuminyCase 907F-13288 Marseille Cedex [email protected] .fr
Yuan GaoNUS Centre for F inancial EngineeringKent Ridge Post OfficeP .O. Box 1130Singapore 911105
Emmanuel GobetCMAPEcole PolytechniqueRoute de SaclayF-91128 Palaiseau [email protected]
Carole HayakawaUniversity of California. Irvine916 Engineering TowerIrvine, CA [email protected]
Zhi DongDepartment of Real EstateSchool of Design and /';nvironmentNational University of Singapore4 Architecture DriveSingapore [email protected]
Sivasankar Elambooranan10 Shenton WayMAS Building, 23rd FloorSingapore [email protected] .sg
Kai-Tai FangDepartment of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong SARC [email protected]
Masanori FushimiDepartment of MS & IENanzan University217 SeireiSeto 489-0863Japanfushimi@nanzan-u .ac.jp
Paul Glasserman403 Uris HallColumbia UniversityNew York , NY [email protected]
Hilde GovaertDepartment of Computer ScienceKatholieke Universiteit LeuvenCelestijnenlaan 200AB-3001 HeverleeBelgiumhilde [email protected]
Stefan HeinrichFachbereich InformatikUnlversitat KaiserslauternPostfach 3049D-67653 [email protected]
Fred J. HickernellDepartment of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong [email protected] .hk
Peter HoyerDepartment of Computer ScienceUniversity of Calgary2500 University Drive NWCalgary, Alberta T 2N IN4Canadahoyer@ cpsc.ucalgary.ca
Junichi IrnatIwate Prefectural University152-52, Takizawa-aza-sugoTakizawaIwate 020-0193Japanim ai @iwat e-pu .ac. j p
Stephen JoeDepartment of MathematicsUniversity of WaikatoPrivate Bag 3105Hamilton 2001New Zealandstephenj@math .waikato.ac .nz
Aneta KaraivanovaDepartment of Com put er ScienceFlorida State Universi ty253 Love BuildingTallahassee, F L 32306-4530USAaneta@csit .fsu .ed u
Alexander KellerDepartment of Computer Scien ceUniversity of UlmD-89069 UlmGermanykeller@informat ik.uni-ulm.de
Thomas KolIigFachbereich InformatikUnlversitat KaiserslauternPostfach 3049D-67653 K aiserslauternGermanykollig@informatik .uni-kl .de
Conference Particip ants XIII
Hee Sun Regina HongDepartment of Economics and FinanceCity University of Hong KongKowloon TongHong Kong [email protected]
Fanglun HuangDepartment of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong [email protected]
Hongmei JinInstitute of High Performance Computing1 Sci en ce Park Ro ad #01-01T he CapricornSingapore Science Park IISinga pore 117528jinhm@ihp c.a-star.edu.sg
Reinhold KainhoferDepartment of Mathemat icsGraz University of TechnologySteyrerg. 30/IIA-8010 GrazAustriare inhold@ka inhofer .com
1wan KawrakowNational Research Council1500 Montreal Ro ad, Bldg M35Ottawa KIA OR6Canadaiwan@irs .phy.nrc.ca
Jai Sam KimDep artment of PhysicsPohang University of Science and TechnologyP ohang 790-784South KoreajSk@po st ech .ac.kr
Wai Ming KooDB S Bank8 Cross Street#22-01/07, PWC BuildingSingapore [email protected]
XIV Conference Participants
Frances Y. KuoSchool of MathematicsUniversity of New South WalesKensingtonSydney, NSW [email protected]
Kevin Lai31 Ruden Cr.Toronto, Ontario M3A 3H4Canadakc3Iai@student .math.uwaterloo.ca
Steven LaValleDepartment of Computer ScienceUniversity of Illinois at Urbana-Champaign3270 Digital Computer Lab1304 W . Springfield AveUrbana, IL [email protected]
Hwee Kuan LeeDepartment of Physics and AstronomyUniversity of GeorgiaAthens, GA [email protected]
William LeesonRuwenzori, Ballyman LaneBray, Co . [email protected]
Josef LeydoldDepartment of StatisticsVienna University of EconomicsAugasse 2-6A-1090 [email protected]
Meng Chiew LimBlk 321C, Anchorvale Drive#09-32Singapore [email protected]
Sabrina Lee Lian KwekDBS Bank8 Cross Street#22-01/07, PWC BuildingSingapore 048424
David P. LandauCenter for Simulational PhysicsUniversity of GeorgiaAthens, GA [email protected]
Pierre L'EcuyerDeparternent d 'informatique et de
recherche operationnelleUniverslte de MontrealC.P. 6128, Succ. Centre-VilleMontreal, Quebec H3C [email protected]
Nai Jia LeeDepartment of Real EstateSchool of Design and EnvironmentNational University of Singapore4 Architecture DriveSingapore [email protected]
Christiane LemieuxDepartment of Mathematics and StatisticsUniversity of Calgary2500 University Drive NWCalgary, Alberta T2N [email protected]
Hua LiNorthwest Institute of Nuclear TechnologyXi 'an , ShaanxiChina
Xiaoai LinDBS Bank8 Cross Street#22-01/07, PWC BuildingSingapore [email protected]
Vivien LingDBS Bank8 Cross Street#22-01/07, PWC BuildingSingapore 0484 24
Xiaoqlng LiuDepartment of MathematicsNational University of Singa pore2 Science Drive 2Singapore [email protected]
Xuan LuDepartment of Mathem atical ScienceTsinghua UniversityBeijing 100084Chinaxlu @math.tsinghua.edu .cn
Gianvittorio MauriVIR Foppa 311-20144 MilanoItalygianvittorio.mauri@bancaimi. it
John Mll1erDepartment of Computationa l ScienceNational Un iver sity of Singapore3 Science Drive 2Singapore [email protected] .sg
Hozumi Morohosi2-2 Wakamatsu-choShinjuku-kuTokyo 162-8677Japanmorohosi@grips .ac .jp
Poh Har NeoBlk 5 Marsiling Drive#13-53Singapore [email protected] .sg
Sok Eng Ng490B, Tampines Stree t 45#05-193Singapore 521490sokeng@berryh appy.com
Conference Participant s XV
X ia LluDBS Bank8 C ross Street# 22-01 /07, PWC BuildingSingapore 0484 24liu xia@dbs .com
Teck N gee LowBlk 336 Hougang Ave 7#04-377Singapore 530336t [email protected] .sg
Tilo LumppSwiss ReMyt henquai 40/50C H-8022 ZurichSwitz erlandT [email protected]
Hrushikesh N. MhaskarDepartment of MathematicsCalifornia State Un iversityLos Angeles, CA [email protected]
Sanjay MirpuriBlk 111 , Bed ok Nor th Road#06-333Singapore 460111sjay@pa cific. net .sg
Vittal Naidu30 Cecil Street#22-01 Prudential TowerSingapore 049712vit t a l.naid [email protected]
Kah Hwa NgNUS Cent re for Financial EngineeringKent Ridge Post OfficeP .O . Box 1130Singa pore 911105
Harald NiederreiterDepartment of MathematicsNational University of Singapore2 Science Dri ve 2Singapore 117543nied @math.nus.edu .sg
XVI Conference Participants
Syoiti NinomiyaCenter for Research in Advanced
Financial TechnologyTokyo Institute of Technology2-12-1 Ookayama, Meguro-kuTokyo 152-8550Japanninomiya@craft .titech.ac.jp
Yutaka OkabeDepartment of PhysicsTokyo Metropolitan UniversityHachiojiTokyo [email protected] .ac .jp
Clement Chuan Hock Ooi30 Cecil Street$22-01 Prudential TowerSingapore [email protected]
Ronald E. Pevey11905 W . Fox Chase CircleKnoxville, TN [email protected]
Tim PiIlardsDepartment of Computer ScienceKatholieke Universiteit LeuvenCelestijnenlaan 200AB-3001 [email protected]
Gottlieb PirsicAustrian Academy of SciencesSonnenfelsgasse 19A-lOW [email protected] .at
Leszek PlaskotaInstitute of Applied MathematicsWarsaw UniversityBanacha 202-097 [email protected] .pl
Erich NovakMathematisches InstitutUniversitat JenaErnst-Abbe-Platz 1-4D-07743 [email protected]
Giray OktenDepartment of MathematicsBall State UniversityMuncie, IN [email protected]
Franc;ois PannetonDeparternent d'informatique et de
recherche operatlonnelleUniversite de MontrealC .P. 6128, Succ. Centre-VilleMontreal, Quebec H3C [email protected]
Georg PflugInstitut fiir Statistik und
Decision Support SystemsUniversitat WienUnlversitatsstralle 5/3/9A-lOW [email protected]
Friedrich PiIlichshammerInstitut filr AnalysisUnlverslt.at LinzAltenbergerstr. 69A-4040 LinzAustriafriedrich [email protected]
Filip PtstovcakFH VorarlbergAchstrafle 1A-6850 DornbirnAustriafilip [email protected]
Hong QinDepartment of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong SARChinahqin@math .hkbu.edu.hk
Klaus ScheicherInstitut fiir AnalysisUniversitiit LinzAltenbergerstr. 69A-4040 [email protected]
Rudolf SchiirerDepartment of MathematicsUniversity of SalzburgHellbrunnerstr. 34A-5020 [email protected] .at
Ian H . SloanSchool of MathematicsUniversity of New South WalesSydney, NSW 2052AustraliaLsloan@unsw .edu.au
Yang SunNUS Centre for Financial EngineeringKent Ridge Post OfficeP.O. Box 1130Singapore 911105
Norman TangStandard Chartered6 Battery Road#03-00Singapore [email protected] .com
Teck How TayDBS Bank8 Cross Street#22-01/07, PWC BuildingSingapore 048424
Michal TomekCentre for Quantitative FinanceImperial College LondonExhibition RoadLondon SW7 2BXEnglandmichaI. [email protected] .uk
Bruno TufflnIRISACampus Universitaire de BeaulieuF-35042 Rennes [email protected]
Conference Participants XVII
Wolfgang Ch. SchrnldDepartment of MathematicsUniversity of SalzburgHellbrunnerstr. 34A-5020 [email protected]
Meng Chan ShuMalaysia Derivatives Exchange10th Floor, Exchange SquareBukit KewanganKuala LumpurMalaysia
Jerome SpanierBeckman Laser Institute1002 Health Sciences RoadUniversity of California, IrvineIrvine, C A [email protected]
Denis TalayINRIA Sophia Antlpot- s2004 Route des LuciolesB.P.93F-06902 Sophia [email protected]
Simon TavareDepartment of Mathematics, DRB 1551042 West 36th PlaceUniversity of Southern CaliforniaLos Angeles, CA [email protected]
Shu Tezuka1149-13, ChigasakiKanagawa [email protected]
Yiu Kuen TseSchool of Economics and Social SciencesSingapore Management UniversitySingapore [email protected]
Wolfgang WagnerWeierstrass Institute for Applied Analysis
and StochasticsMohrenstralle 39D-10117 [email protected]
XVIII Conference Participants
Vasant WaikarDepartment of Mathematics and Statist icsMiami UniversityOxford, OH [email protected]
Jian-Sheng WangDepartment of Computational ScienceNational University of Singapore3 Science Drive 2Singapore [email protected] .sg
Sick Leng Wang6th Floor, Mayban Lif TowerDataran MaybankNo.1, Jalan Maarof59000 Kuala [email protected] y
Grzegorz WasilkowskiDepartment of Computer ScienceUniversity of KentuckyLexington, KY [email protected]
Carsten WiegandFachbereich InformatikUniversitat KaiserslauternPostfach 3049D-67653 [email protected]\.de
Michael C. WongFlat C, 2/FWah Pont House69-75 Fuk Wa StreetShamshuipoHong Kong SARChinacywong@math .cuhk.edu .hk
Tien Chi WongNUS Centre for Financial EngineeringKent Ridge Post OfficeP .O. Box 1130Singapore [email protected]
Chaoping XingDepartment of MathematicsNational University of Singapore2 Science Drive 2Singapore [email protected] .sg
Justin WanSchool of Computer ScienceUniversity of Waterloo200 University Avenue W.Waterloo, Ontario N2L 3G1Can ad ajwlwan @uwaterloo.ca
Qiang WangDBS Bank8 Cross Street#22-01/07, PWC BuildingSingapore 048424wangqiang@dbs. com
Xiaoqun WangSchool of MathematicsUniversity of New South WalesSydney, NSW 2052Australiaxwang111 @maths.unsw.edu .au
Woon Gek WeeDBS Bank8 Cross Street#22-01 /07 , PWC BuildingSingapore 048424
Arne WinterhofRICAMUnlversitat LinzAl tenbergerstr . 69A-4040 LinzAustriaa rn e [email protected]
Thian Boon WongStandard Chartered6 Battery Road#03-00Singapore 049909Wong. [email protected]
Henryk WozniakowskiDepartment of Computer ScienceColumbia UniversityNew York , NY [email protected]
Jianxing YinDepartment of MathematicsSu zhou UniversitySu zhou 215006C [email protected]
Shoji YurnaeNLI Research Institute1-1- 1, YurakuchoChiyoda-kuTokyo 100 -0006Japanyumae@nli-research .co.jp
Bo ZhengPhysics DepartmentZhejiang Universit yHangzhou [email protected] .edu .cn
Conference Participants XIX
Yubei ZhangDep artment of MathematicsNational Uni vers ity of Singapore2 Scien ce D ri ve 2Si nga p ore 117543scipI 370 @nus.edu.sg
