Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 001–0
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0373-3114应用数学ANN MAT PUR APPLAPPL MATH COMPUTAPPLIED MATHEMATICS AND COMPUTATION0096-3003应用数学Boundary Value Problems 1687-2762应用数学BOUND VALUE PROBCALCULUS OF VARIATIONS AND PARTIAL DIFFE0944-2669应用数学CALC VAR PARTIALCarpathian Journal of Mathematics1584-2851应用数学CARPATHIAN J MAT0219-1997应用数学COMMUN CONTEMP MCOMMUNICATIONS IN CONTEMPORARY MATHEMATI0360-5302应用数学COMMUN PART DIFFCOMMUNICATIONS IN PARTIAL DIFFERENTIAL E0925-7721应用数学COMPUTATIONAL GEOMETRY-THEORY AND APPLICCOMP GEOM-THEOR1078-0947应用数学DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMDISCRETE CONT DY0764-583X应用数学ESAIM-MATHEMATICAL MODELLING AND NUMERICESAIM-MATH MODELFixed Point Theory 1583-5022应用数学FIXED POINT THEOFixed Point Theory and Applications 1687-1820应用数学FIXED POINT THEO1615-3375应用数学FOUNDATIONS OF COMPUTATIONAL MATHEMATICSFOUND COMPUT MATFUZZY SET SYSTFUZZY SETS AND SYSTEMS0165-0114应用数学IMA JOURNAL OF NUMERICAL ANALYSIS0272-4979应用数学IMA J NUMER ANALInternational Journal of Numerical Analy1705-5105应用数学INT J NUMER ANAL1040-7294应用数学Journal of Dynamics and Differential EquJ DYN DIFFER EQU1435-9855应用数学J EUR MATH SOCJOURNAL OF THE EUROPEAN MATHEMATICAL SOCJ FOURIER ANAL A1069-5869应用数学JOURNAL OF FOURIER ANALYSIS AND APPLICATJ GLOBAL OPTIMJOURNAL OF GLOBAL OPTIMIZATION0925-5001应用数学JOURNAL OF MATHEMATICAL ANALYSIS AND APP0022-247X应用数学J MATH ANAL APPLJ MOD DYNAM Journal of Modern Dynamics1930-5311应用数学Journal of Noncommutative Geometry1661-6952应用数学J NONCOMMUT GEOMJ NONLINEAR SCI J OURNAL OF NONLINEAR SCIENCE0938-8974应用数学J SCI COMPUT JOURNAL OF SCIENTIFIC COMPUTING0885-7474应用数学KINET RELAT MOD K inetic and Related Models1937-5093应用数学。
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Optimization349 177 Mathematical Control and Related Fields104 178 Journal of Pseudo-Differential Operators and Applications84 179 Journal of Systems Engineering and Electronics825 180 Journal of Electrical Engineering & Technology768 181 IEEJ Transactions on Electrical and Electronic Engineering489 182 DESIGN AUTOMATION FOR EMBEDDED SYSTEMS139 183 ELECTROMAGNETICS410 184 IET Computers and Digital Techniques231 185 Journal of Semiconductor Technology and Science275 186 MINDS AND MACHINES358 187 CHINESE JOURNAL OF ELECTRONICS529 188 Econometrics Journal791 189 Numerical Mathematics-Theory Methods and Applications190 190 IEICE TRANSACTIONS ON ELECTRONICS1075 191 ACM Journal on Computing and Cultural Heritage90192 Journal of Grey System356 193 Revista Iberoamericana de Automatica e Informatica Industrial144 194 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS598 195 Journal of Nanoelectronics and Optoelectronics420 196 JOURNAL OF ALGEBRA AND ITS APPLICATIONS602 197 COMPUTING AND INFORMATICS338198 COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING711199 Homology Homotopy and Applications247 200 JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS317 201 JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL362 202 Social Semiotics344 203 Journal of Electrical Engineering-Elektrotechnicky Casopis417 204 JOURNAL OF CIRCUITS SYSTEMS AND COMPUTERS588205 INFORMACIJE MIDEM-JOURNAL OF MICROELECTRONICS ELECTRONIC COMPONENTS AND MATERIALS95206 Annals of Functional Analysis137 207 Information Technology and Control163 208 Discrete Optimization458 209 Continuum-Journal of Media & Cultural Studies496 210 JOURNAL OF INFORMATION SCIENCE AND ENGINEERING634 211 Journal of Transportation Safety & Security93 212 Revista de la Union Matematica Argentina87213 International Journal of Wavelets Multiresolution and InformationProcessing327214 FREQUENZ207 215 Fixed Point Theory299 216 JOURNAL OF DATABASE MANAGEMENT182 217 QUARTERLY JOURNAL OF SPEECH737 218 INTEGRATED FERROELECTRICS913 219 Milan Journal of Mathematics267 220 IEICE Electronics Express979 221 Computational Methods and Function Theory250 222 KSII Transactions on Internet and Information Systems569 223 Journal of Function Spaces150 224 FUNCTIONAL ANALYSIS AND ITS APPLICATIONS1841 225 Communication Culture & Critique209 226 Text & Talk297 227 ACTA MATHEMATICA SINICA-ENGLISH SERIES1443 228 JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS935 229 APPLIED COMPUTATIONAL ELECTROMAGNETICS SOCIETY JOURNAL481 230 Statistics and Its Interface421 231 COMPUTATIONAL COMPLEXITY636 232 JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY578 233 MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS345 234 Proceedings of the Steklov Institute of Mathematics860 235 Journal of Mathematics and Music53236 Revista Internacional de Metodos Numericos para Calculo y Diseno enIngenieria84237 East Asian Journal on Applied Mathematics60238 NATURAL RESOURCE MODELING353 239 COMPUTER ANIMATION AND VIRTUAL WORLDS367 240 MATHEMATICAL SOCIAL SCIENCES751 241 Analele Stiintifice ale Universitatii Ovidius Constanta-Seria144 242 Journal of Mass Media Ethics243 243 Theory and Applications of Categories303 244 Mathematics and Financial Economics163 245 Periodica Mathematica Hungarica344 246 JAVNOST-THE PUBLIC191 247 IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS1448 248 COMPUTER MUSIC JOURNAL402 249 Journal of Numerical Mathematics276 250 Funkcialaj Ekvacioj-Serio Internacia504 251 Neural Network World251 252 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE616 253 Automatika159 254 KYBERNETIKA808 255 TOPOLOGY AND ITS APPLICATIONS1952 256 INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING EDUCATION152 257 JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING332 258 Romanian Journal of Information Science and Technology108 259 PMM JOURNAL OF APPLIED MATHEMATICS AND MECHANICS1864 260 COMPUTER SYSTEMS SCIENCE AND ENGINEERING155 261 Media International Australia264 262 ALGEBRA COLLOQUIUM375 263 CMC-Computers Materials & Continua315 264 ACM SIGPLAN NOTICES2541 265 Advances in Difference Equations849266 Iranian Journal of Science and Technology-Transactions of ElectricalEngineering22267 Rhetoric Society Quarterly204 268 Glasnik Matematicki256 269 NARRATIVE INQUIRY379 270 Mathematical Communications204 271 ARCHIVE FOR HISTORY OF EXACT SCIENCES285 272 JOURNAL OF APPLIED COMMUNICATION RESEARCH692 273 Bollettino di Storia delle Scienze Matematiche15 274 Economic Computation and Economic Cybernetics Studies and Research103 275 INTERNATIONAL JOURNAL OF SOFTWARE ENGINEERING AND KNOWLEDGE ENGINEERING345 276 DYNAMIC SYSTEMS AND APPLICATIONS264 277 Mathematical Population Studies170278 University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics184279 IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES1381280 UTILITAS MATHEMATICA482 281 HISTORIA MATHEMATICA221 282 MICROWAVE JOURNAL396 283 CRYPTOLOGIA160 284 Applied Mathematics-A Journal of Chinese Universities Series B174285 Acta Mathematicae Applicatae Sinica-English Series538 286 PROGRAMMING AND COMPUTER SOFTWARE114 287 Ukrainian Mathematical Journal699 288 International Journal of Transport Economics189 289 JOURNAL OF MEDIA ECONOMICS187 290 Electronics and Communications in Japan167 291 FUJITSU SCIENTIFIC & TECHNICAL JOURNAL173 292 INFOR380 293 ELECTRICAL ENGINEERING IN JAPAN398 294 African Journalism Studies12 295 Tijdschrift voor Communicatiewetenschap42 296 Journal of African Media Studies66 297 ICGA JOURNAL63 298 Pure and Applied Mathematics Quarterly241 299 Light & Engineering41 300 EPE Journal101 301 SOLID STATE TECHNOLOGY162 302 Journal of the Institute of Telecommunications Professionals7 303 Traitement du Signal72 304 ELECTRONICS WORLD45 305 Road & Transport Research115 306 IEEE Transactions on Cognitive and Developmental Systems7影响因子eigenFac torScore1.1470.00259 1.130.00268 1.1180.00026 1.1160.00199 1.0790.00047 1.0750.00154 1.0750.00256 1.0670.00255 1.0660.00263 1.060.00221 1.0560.00071 1.0390.00113 1.0370.00082 1.0340.00068 1.0320.00826 1.0310.00048 1.0290.00039 1.0250.0011 1.010.0019310.0010210.0003710.000310.0007710.00015 0.9940.00151 0.9760.00109 0.9640.00094 0.9560.00213 0.9510.00704 0.950.00089 0.9450.00191 0.9430.00039 0.920.00244 0.9090.00033 0.9090.00028 0.9030.00178 0.9020.000520.90.00064 0.8970.00021 0.8930.00155 0.8850.00058 0.8780.00115 0.8770.0015 0.8750.00035 0.8750.001410.8670.00027 0.8630.00089 0.8590.00126 0.850.00116 0.850.00275 0.8490.00036 0.8380.00073 0.8370.00093 0.8270.00279 0.8270.00196 0.8130.00085 0.8120.00113 0.8110.00071 0.8090.00033 0.8070.001130.80.00168 0.7910.00103 0.7890.00048 0.7810.00166 0.7780.00063 0.7780.00084 0.7750.00031 0.7720.00113 0.7720.00063 0.7710.00099 0.770.00081 0.7690.00031 0.7690.00137 0.7660.00162 0.7650.00137 0.7595.00E-05 0.7580.00018 0.7570.00654 0.7540.00261 0.7530.00029 0.750.00054 0.7480.00625 0.7460.00106 0.740.00067 0.7380.00104 0.7370.00029 0.7350.00516 0.7330.0004 0.7320.00052 0.7310.00754 0.730.00145 0.7290.00109 0.7270.00024 0.7270.00019 0.7240.006450.7240.0003 0.7230.001760.720.00137 0.7130.00235 0.7110.00301 0.7110.00349 0.710.000630.7060.00048 0.7050.00064 0.6980.01629 0.6960.00112 0.6960.00024 0.6940.00057 0.6920.00017 0.6880.00125 0.6870.0028 0.6860.00028 0.6770.00081 0.6750.00073 0.6740.00046 0.670.0028 0.6670.00099 0.6670.00024 0.6610.00337 0.6570.0003 0.6540.00098 0.6520.00049 0.6470.0037 0.6470.00055 0.6460.00131 0.6450.0029 0.6440.00041 0.6430.00078 0.6430.00081 0.6420.00151 0.6410.00128 0.6380.00057 0.6350.00038 0.6320.00082 0.6310.00102 0.6280.00031 0.6270.00047 0.6250.00235 0.6230.00216 0.6220.00016 0.6220.00065 0.6220.001030.6190.00061 0.6180.00058 0.6090.00048 0.6090.00097 0.6050.002090.60.000420.68.00E-05 0.5970.00124 0.5970.00354 0.5970.00047 0.5950.00038 0.5810.00081 0.580.00019 0.5790.00023 0.5780.00107 0.5750.00025 0.5750.0015 0.5730.00199 0.5690.00028 0.5670.0006 0.5650.00321 0.5620 0.5620.00043 0.5620.00024 0.560.00063 0.5580.00067 0.5560.00106 0.5560.00037 0.5480.00163 0.5480.00142 0.5480.00036 0.5440.0022 0.5430.01147 0.5430.00127 0.5420.00125 0.5290.00056 0.5290.00152 0.5250.00183 0.5170.00076 0.5160.00035 0.5160.00058 0.5150.00049 0.5150.00073 0.5140.00045 0.5130.00075 0.5130.00287 0.5090.00091 0.5030.001560.50.000190.50.00020.50.00028 0.4970.00346 0.4970.00073 0.4890.00428 0.4880.000530.4870.00121 0.4860.00149 0.4860.00078 0.4840.00051 0.4840.00072 0.4830.00043 0.4810.000750.4780.00016 0.4770.00098 0.4750.00031 0.4690.00161 0.4680.00091 0.4680.00089 0.4650.00034 0.4640.00046 0.4630.00053 0.4620.00025 0.4620.00044 0.4628.00E-05 0.460.00027 0.4570.00109 0.4570.0012 0.4560.00189 0.4550.00111 0.4520.00119 0.4510.00068 0.450.00144 0.4480.00071 0.4480.00083 0.4460.0042 0.4460.00076 0.4440.00099 0.4440.00201 0.4410.00124 0.4410.00194 0.4390.00074 0.4360.00318 0.4350.000110.4310.00017 0.4260.000370.4260.0006 0.4240.00099 0.4230.00131 0.4220.00052 0.4190.00037 0.4190.00145 0.4150.00129 0.4150.00079 0.4130.00033 0.4110.00196 0.4050.00018 0.4050.00163 0.3940.00125 0.3940.00023 0.390.00185 0.380.00041 0.3790.00125 0.3770.00551 0.3750.0001 0.3650.00087 0.3650.0002 0.3570.00117 0.3480.00015 0.3460.00068 0.3430.00142 0.3390.00093 0.3350.0032 0.3350.002330.3335.00E-05 0.3330.0003 0.3280.00049 0.3170.00043 0.3160.00059 0.3080.00042 0.3080.000720.30 0.2990.00019 0.2990.00054 0.2980.00052 0.2860.000210.2790.000340.2740.00196 0.2610.00082 0.2580.00028 0.2580.00063 0.2563.00E-05 0.2470.000490.2420.001060.230.000170.2280.000790.220.000210.2170.000170.2020.000360.1910.000220.1890.000250.1880.000330.1712.00E-050.1717.00E-050.1540.000180.1520.000280.1490.001410.1183.00E-050.0910.000110.0820.00010.07800.0281.00E-050.0262.00E-050.0210.00014 -999.9991.00E-05。
运筹与管理与本领域相关SCI期刊4ORACM Journal of Experimental Algorithms AlgorithmicaAnnals of Operations ResearchAsia-Pacific Journal of Operational ResearchCentral European Journal of Operations Research Computational Geometry: Theory and Applications Computational Optimization and Applications Computers & Industrial EngineeringComputers & Operations ResearchDiscrete Applied MathematicsDiscrete OptimizationDiscrete Mathematics & Theoretical Computer Science European Journal of Operational ResearchIIE TransactionsINFORMS Journal on ComputingInternational Journal of Computational Geometry and Applications International Transactions in Operational ResearchJournal of Discrete AlgorithmsJournal of HeuristicsJournal of the Operational Research SocietyJournal of Mathematical Modelling and AlgorithmsJournal of SchedulingLinear Algebra and its ApplicationsManagement ScienceMathematical and Computer ModellingMathematical Methods of Operations ResearchMathematical ProgrammingMathematics of Operations ResearchNaval Research LogisticsNetworksOmegaOperations ResearchOperations Research LettersOptimizationOptimization Methods and SoftwareSIAM Journal on OptimizationTransportation ScienceEI收录Journal of Algorithms and Computational TechnologyJournal of Applied Mathematics and Computing(EI 符合范围)Journal of Computational and Applied Mathematics(SCI难度不大)Journal of Systems Science and Systems Engineering(中国EI)Journal of the Operations Research Society of China(不被收录,中国刚办)Journal of Mathematical Modelling and Algorithms(不被收录,运筹优化的)The International Journal of Advanced Manufacturing Technology (SCI期刊,容易中,3个月审稿) International Journal of Innovative Computing Information and Control (SCI)International Journal of Systems Science (SCI)International Journal of Mathematics in Operational research (EI)International Journal of Mathematical Modelling and Numerical OptimisationPacific Journal of Optimization (华人多)Optimization Methods and Software (SCI期刊,据说容易中,2个月审稿,方向对口(小木虫说慢))优化相关资料与网址收集Optimization and Operations Research:Operations Research: the science of better :/Science_of_Better/htdocs/prospect/index.aspOR directory (a bit out of date): http://www.math.tu-bs.de/mo/orlistings.htmlLP FAQs: /otc/Guide/faq/linear-programming-faq.htmlsome free software by Prof. John O. McClain:/faculty/mcclain/Software/Software.htmIFORS tutorial modules:/tutorial/benchmark for optimization software: /bench.htmlThe French OR Society /The European OR Society /The international federation of OR societies /The Canadian OR Society http://www.cors.ca/OR Society of the US /THE OR web page of Michael Trick /e-optimization /OR library of optimization instances OR-library/~mastjjb/jeb/info.htmlOperations Research Journals RSS ListOperations Research:Annals of Operations ResearchComputational Management ScienceComputers & Operations ResearchEuropean Journal of Operational ResearchINFORMS Journal on ComputingInterfacesInternational Transactions in Operational Research Journal of Optimization Theory and Applications Journal of the Operational Research Society Management ScienceMathematical ProgrammingMathematical Programming Computation Mathematics of Operations ResearchNetworksOmegaOperations ResearchOR SpectrumIndustrial Engineeing and Operations Management: Computers & Industrial Engineering International Journal of Production Economics International Journal of Production Research Manufacturing & Service Operations Management Naval Research LogisticsTransportation:Transportation Research Part B: MethodologicalTransportation Research Part E: Logistics and Transportation Review Transportation ScienceArtificial Intelligence:Advances in Engineering SoftwareApplied Soft ComputingDecision Support SystemsEngineering Applications of Artificial IntelligenceExpert Systems with Applications管理科学与运筹学(MS/OR)国际期刊最新权威排名首先声明:这份MS/OR国际期刊排名是完全根据2011年JCR(Journal Citation Reports)的Article Influence Score(AIS)而给出的。
数据挖掘18种候选算法和十大经典算法国际权威的学术组织the IEEE International Conference on Data Mining (ICDM) 2006(香港召开)年12月评选出了数据挖掘领域的十大经典算法。
不仅仅是选中的十大算法,其实参加评选的18种算法,实际上随便拿出一种来都可以称得上是经典算法,它们在数据挖掘领域都产生了极为深远的影响。
一、数据挖掘18种候选算法Classification(分类)==============#1. C4.5Quinlan, J. R. 1993. C4.5: Programs for Machine Learning.Morgan Kaufmann Publishers Inc.Google Scholar Count in October 2006: 6907#2. CARTL. Breiman, J. Friedman, R. Olshen, and C. Stone. Classification andRegression Trees. Wadsworth, Belmont, CA, 1984.Google Scholar Count in October 2006: 6078#3. K Nearest Neighbours (kNN)Hastie, T. and Tibshirani, R. 1996. Discriminant Adaptive NearestNeighbor Classification. IEEE Trans. PatternAnal. Mach. Intell. (TPAMI). 18, 6 (Jun. 1996), 607-616.DOI= /10.1109/34.506411Google SCholar Count: 183#4. Naive BayesHand, D.J., Yu, K., 2001. Idiot's Bayes: Not So Stupid After All?Internat. Statist. Rev. 69, 385-398.Google Scholar Count in October 2006: 51Statistical Learning(统计学习)====================#5. SVMVapnik, V. N. 1995. The Nature of Statistical LearningTheory. Springer-Verlag New York, Inc.Google Scholar Count in October 2006: 6441#6. EMMcLachlan, G. and Peel, D. (2000). Finite Mixture Models.J. Wiley, New York.Google Scholar Count in October 2006: 848Association Analysis(关联分析)====================#7. AprioriRakesh Agrawal and Ramakrishnan Srikant. Fast Algorithms for MiningAssociation Rules. In Proc. of the 20th Int'l Conference on Very LargeDatabases (VLDB '94), Santiago, Chile, September 1994./agrawal94fast.htmlGoogle Scholar Count in October 2006: 3639#8. FP-TreeHan, J., Pei, J., and Yin, Y. 2000. Mining frequent patternswithoutcandidate generation. In Proceedings of the 2000 ACM SIGMODinternational Conference on Management of Data (Dallas, Texas, UnitedStates, May 15 - 18, 2000). SIGMOD '00. ACM Press, New York, NY, 1-12.DOI= /10.1145/342009.335372Google Scholar Count in October 2006: 1258Link Mining(链接挖掘)===========#9. PageRankBrin, S. and Page, L. 1998. The anatomy of a large-scale hypertextualWeb search engine. In Proceedings of the Seventh internationalConference on World Wide Web (WWW-7) (Brisbane,Australia). P. H. Enslow and A. Ellis, Eds. Elsevier SciencePublishers B. V., Amsterdam, The Netherlands, 107-117.DOI= /10.1016/S0169-7552(98)00110-XGoogle Shcolar Count: 2558#10. HITSKleinberg, J. M. 1998. Authoritative sources in a hyperlinked environment. In Proceedings of the Ninth Annual ACM-SIAM Symposium onDiscrete Algorithms (San Francisco, California, United States, January25 - 27, 1998). Symposium on Discrete Algorithms. SocietyforIndustrial and Applied Mathematics, Philadelphia, PA, 668-677.Google Shcolar Count: 2240Clustering(聚类)==========#11. K-MeansMacQueen, J. B., Some methods for classification and analysis ofmultivariate observations, in Proc. 5th Berkeley Symp. MathematicalStatistics and Probability, 1967, pp. 281-297.Google Scholar Count in October 2006: 1579#12. BIRCHZhang, T., Ramakrishnan, R., and Livny, M. 1996. BIRCH: an efficientdata clustering method for very large databases. In Proceedings of the1996 ACM SIGMOD international Conference on Management of Data(Montreal, Quebec, Canada, June 04 - 06, 1996). J. Widom, Ed.SIGMOD '96. ACM Press, New York, NY, 103-114.DOI= /10.1145/233269.233324Google Scholar Count in October 2006: 853Bagging and Boosting(袋装与推进)====================#13. AdaBoostFreund, Y. and Schapire, R. E. 1997. A decision-theoreticgeneralization of on-line learning and an application toboosting. J. Comput. Syst. Sci. 55, 1 (Aug. 1997), 119-139.DOI= /10.1006/jcss.1997.1504Google Scholar Count in October 2006: 1576Sequential Patterns(序列模式)===================#14. GSPSrikant, R. and Agrawal, R. 1996. Mining Sequential Patterns: Generalizations and Performance Improvements. In Proceedings of the5th international Conference on Extending Database Technology:Advances in Database Technology (March 25 - 29, 1996). P. M. Apers,M. Bouzeghoub, and G. Gardarin, Eds. Lecture Notes In ComputerScience, vol. 1057. Springer-Verlag, London, 3-17.Google Scholar Count in October 2006: 596#15. PrefixSpanJ. Pei, J. Han, B. Mortazavi-Asl, H. Pinto, Q. Chen, U. Dayal and M-C. Hsu. PrefixSpan: Mining Sequential Patterns Efficiently byPrefix-Projected Pattern Growth. In Proceedings of the 17th international Conference on Data Engineering (April 02 - 06, 2001). ICDE '01. IEEE Computer Society, Washington,DC.Google Scholar Count in October 2006: 248Integrated Mining(集成挖掘)=================#16. CBALiu, B., Hsu, W. and Ma, Y. M. Integrating classification and association rule mining. KDD-98, 1998, pp. 80-86./liu98integrating.htmlGoogle Scholar Count in October 2006: 436Rough Sets(粗糙集)==========#17. Finding reductZdzislaw Pawlak, Rough Sets: Theoretical Aspects of Reasoning aboutData, Kluwer Academic Publishers, Norwell, MA, 1992Google Scholar Count in October 2006: 329Graph Mining(图挖掘)============#18. gSpanYan, X. and Han, J. 2002. gSpan: Graph-Based Substructure PatternMining. In Proceedings of the 2002 IEEE International Conference onData Mining (ICDM '02) (December 09 - 12, 2002). IEEE ComputerSociety, Washington, DC.Google Scholar Count in October 2006: 155二、数据挖掘十大经典算法十大经典算法是:C4.5、k-Means、SVM、Apriori、EM、PageRank、AdaBoost、kNN、Naive Bayes、and CART。
计算机科学国际刊物排名Computer Science Journal Rankings计算机刊物分为以下几大领域:人工智能;算法、理论和相关领域;程序设计语言和软件工程;软件技术;系统技术;普通计算机科学刊物及相关刊物。
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Electronic Networking Applications and Policy Computer CommunicationsJnl of Computer CommunicationsJnl of Network and Computer ApplicationsParallel ComputingMobile Networks and ApplicationsJnl of High Speed NetworksWireless NetworksQueueing SystemsThe Intl Jnl of Supercomputer ApplicationsConcurrency - Practice and ExperienceIEEE Journal of Internet ComputingACM Transactions on Information and System SecurityReputable:Computers & SecurityIEE Proc. - CommunicationsIEICE Trans on Fundamentals of Elec., Comm., and Comp.IEICE Trans on CommunicationsMobile Computing and Communicatons ReviewDistributed Systems Engineering JnlIntl Jnl of High Speed ComputingWorld Wide Web JournalElectronic NetworkingIntl Jnl of Communication SystemsSignal Processing: Image CommunicationTelecommunication SystemsInternational Journal of Systems ScienceJournal of CryptographyInternet Security JournalJournal of Networks and Systems ManagementInternational Journal of Wireless Information NetworksAn Interactive Journal of Mobile ComputingOthers:Information & Systems TechnologyJnl of Computer SecurityParallel Processing LettersIBM Systems JournalThe International Journal of Computer and Telecommunications NetworkingUSENIX Computing Systems JournalA T&T Technical JournalJournal of Internetworking: Research and ExperienceFujitsu Scientific and Technical JournalComputer Music JournalJournal of Computer-based InstructionICL Technical JournalInternational Journal of Imaging Systems and TechnologyInternational Journal of Digital and Analog Communication SystemJournal of Supercomputer Application of High Performance ComputingJournal of Decision Support SystemsJournal of Computer Research and DevelopmentBritish Telecommunications Technology JournalThe Journal of Interactive Media and EducationInternational Journal of Time-Critical Computing SystemsInternational. 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国外数学期刊英汉对照作者:admin 日期:2009-06-17字体大小: 小中大Advances in Applied Mathematics 应用数学Advances in Applied Probability 应用概率论进展Advances in Computational Mathematics 计算数学进展Advances in Mathematics 数学进展Algebra Colloquium 代数学讨论会Algebras and Representation Theory 代数和表示理论American Mathem atical Monthly 美国数学月刊American Statistician 美国统计员Annals of Applied Probability 应用概率论年报Annals of Global Analysis and Geometry 整体分析与几何学年报Annals of Mathematics and Artificial Intelligence 人工智能论题年报Annals of Operations Research 运筹学研究年报Annals of Probability 概率论年报Annals of Pure and Applied Logic 抽象和应用逻辑年报Annals of Statistics 统计学年报Annals of The Institute of Statistical Mathematics 统计数学学会年报Applicable Algebra in Engineering Communication and Computing 代数在工程通信与计算中的应用Applied and Computational Harmonic Analysis 调和分析应用和计算Applied Categorical Structures 应用范畴结构Applied Mathematics and Computation 应用数学与计算Applied Mathematics and Optimization 应用数学与最优化Applied Mathematics Letters 应用数学快报Archive for History of Exact Sciences 科学史档案Archive for Mathem atical Logic 数理逻辑档案Archive for Rational Mechanics and Analysis 理性力学和分析档案Archives of Computational Methods in Engineering 工程计算方法档案Asymptotic Analysis 渐近线分析Autonomous Robots 机器人British Journal of Mathematical & Statistical Psychology 英国数学与统计心理学杂志Bulletin of The American Mathematical Society 美国数学会快报Bulletin of the London Mathematical Society 伦敦数学会快报Calculus of Variations and Partial Differential Equations 变分法与偏微分方程Combinatorics Probability & Computing 组合概率与计算Combustion Theory and Modeling 燃烧理论建模Communications in Algebra 代数通讯Communications in Contemporary Mathematics 当代数学通讯Communications in Mathematical Physics 数学物理学通讯Communications in Partial Differential Equations 偏微分方程通讯Communications in Statistics-Simulation and Computation 统计通讯–模拟与计算Communications on Pure and Applied Mathematics 纯数学与应用数学通讯Computational Geometry-Theory and Applications 计算几何- 理论与应用Computational Optimization and Applications 优化计算与应用Computational Statistics & Data Analysis 统计计算与数据分析Computer Aided Geometric Design 计算机辅助几何设计Computer Physics Communications 计算机物理通讯Computers & Mathematics with Applications 计算机与数学应用Computers & Operations Research 计算机与运筹学研究Concurrent Engineering-Research and Applications 共点工程- 研究与应用Conformal Geometry and Dynamics 投影几何与力学Decision Support Systems 决策支持系统Designs Codes and Cryptography 编码设计与密码系统Differential Geom etry and Its Applications 微分几何及其应用Discrete and Continuous Dynamical Systems 离散与连续动力系统Discrete Applied Mathematics 应用离散数学Discrete Computational Geometry 离散计算几何学Discrete Event Dynamic Systems-Theory and Applications 离散事件动态系统- 理论和应用Discrete Mathematics 离散数学Educational and Psychological Measurement 教育与心理测量方法Engineering Analysis with Boundary Elem ents 工程边界元素分析Ergodic Theory and Dynamical System s 遍历理论和动力系统European Journal of Applied Mathematics 欧洲应用数学杂志European Journal of Combinatorics 欧洲组合数学杂志European Journal of Operational Research 欧洲运筹学杂志Experimental Mathematics 实验数学Expert Systems with Applications 专家系统应用Finite Fields and Their Applications 有限域及其应用Foundations of Computational Mathematics 计算数学基础Fuzzy Sets and Systems 模糊集与模糊系统Glasgow Mathematical Journal 英国格拉斯哥数学杂志Graphs and Combinatorics 图论与组合数学IEEE Robotics & Automation Magazine IEEE机器人与自动化杂志IEEE Transactions on Robotics and Automation IEEE机器人与自动化学报IMA Journal of Applied Mathem atics IMA应用数学学报IMA Journal of Mathem atics Applied in Medicine and Biology IMA数学在医药与生物中的应用杂志IMA Journal of Numerical Analysis IMA数值分析学报Information and Computation 信息与计算Insurance Mathematics & Economics 保险数学和经济学International Journal for Numerical Methods in Engineering 国际工程数值方法杂志International Journal of Algebra and Computation 国际代数和计算杂志International Journal of Computational Geometry & Applications 国际计算几何应用杂志International Journal of Computer Integrated Manufacturing 国际计算机集成制造业国际杂志International Journal of Game Theory 国际对策论国际杂志International Journal of Mathematics 国际数学杂志International Journal of Production Research 国际研究成果杂志International Journal of Robotics Research 国际机器人研究杂志International Journal of Systems Science 国际系统科学杂志International Statistical Review 国际统计评论Inverse Problems 反比问题Journal of Algebra 代数学报Journal of Algebraic Combinatorics 代数组合数学学报Journal of Algebraic Geometry 代数几何学报Journal of Algorithm s 算法学报Journal of Applied Mathematics and Mechanics 数学应用与力学学报Journal of Applied Probability 应用概率杂志Journal of Approximation Theory 近似值理论杂志Journal of Combinatorial Optimization 组合最优化学报Journal of Combinatorial Theory Series B 组合理论学报B辑Journal of Combinatorial Theory Series A 组合理论学报A辑Journal of Computational Acoustics 声学计算杂志Journal of Computational Analysis and Applications 计算分析与应用学报Journal of Computational and Applied Mathematics 计算与应用数学杂志Journal of Computational Biology 生物计算杂志Journal of Computational Mathematics 计算数学学报Journal of Computational Neuroscience 神经系统计算杂志Journal of Computational Neuroscience 神经系统计算杂志Journal of Differential Equations 微分方程组学报Journal of Econom etrics 计量经济学会会刊Journal of Engineering Mathematics 工程数学学报Journal of Functional Analysis 泛函分析学报Journal of Geometry and Physics 几何学和物理学学报Journal of Global Optimization 整体优化学报Journal of Graph Theory 图论理论学报Journal of Group Theory 群论理论学报Journal of Lie Theory 展开理论杂志Journal of Mathematical Analysis and Applications 数学分析和应用学报Journal of Mathematical Biology 数学生物学杂志Journal of Mathematical Economics 数学经济学学报Journal of Mathematical Imaging and Vision 成像和视觉数学学报Journal of Mathematical Physics 数学物理学的杂志Journal of Mathematical Psychology 数学心理学杂志Journal of Multivariate Analysis 多变量分析杂志Journal of Nonlinear Science 非线性数学学报Journal of Number Theory 数论学报Journal of Operations Management 操作管理杂志Journal of Pure and Applied Algebra 纯代数与应用代数学学报Journal of Statistical Planning and Inference 统计规划和推论杂志Journal of Symbolic Computation 符号计算学报Journal of the American Mathematical Society 美国数学会杂志Journal of the London Mathematical Society-Second Series 伦敦数学会杂志- 第二辑Journal of the Royal Statistical Society Series B-Statistical Methodology 皇家学会系列杂志B 辑-统计方法学Journal of The Royal Statistical Society Series C-Applied Statistics 皇家学会系列杂志C - 应用统计Journal of Theoretical Probability 概率理论杂志K-Theory K-理论Lecture Notes in Economics and Mathematical Systems 经济学和数学体系讲座Lecture Notes on Mathematics 数学讲座Letters in Mathematical Physics 数学物理学通讯Linear Algebra and Its Applications 线性代数及其应用Mathematical Biosciences 数理生物学Mathematical Finance 数理财金学Mathematical Geology 数理地质学Mathematical Intelligencer 数学情报Mathematical Logic Quarterly 数理逻辑学报Mathematical Methods in the Applied Sciences 数学应用学科Mathematical Methods of Operations Research 运筹学研究Mathematical Models & Methods in Applied Sciences 数学模拟与应用方法Mathematical Physics Electronic J 数学物理电子杂志Mathematical Proceedings of the Society 数学学会学报Mathematical Programming 数学规划Mathematical Social Sciences 数学社会科学Mathematics and Computers in Simulation 数学与电脑模拟Mathematics and Mechanics of Solids 数学与固体力学Mathematics Archives 数学档案库Mathematics Magazine 数学杂志Mathematics of Computation 计算数学Mathematics of Control Signals and Systems 控制信号系统数学Memoirs of the American Mathematical Society 美国数学会备忘录Modem Logic 现代逻辑学Nonlinear Analysis-Theory Methods & Applications 非线性分析- 理论与应用Nonlinearity 非线性特性Notices 短评Numerical Algorithms 数字算法Numerical Methods for Partial Differential Equations 偏微分方程式数值方法Oxford Bulletin of Economics and Statistics 牛津大学经济与统计快报Potential Analysis 位势分析Proceedings of the American Mathem atical Society 美国数学会会议录Proceedings of the London Mathematical Society 伦敦数学会会议录Proceedings of the Royal Society of Edinburgh Section A-Mathem atics 英国爱丁堡皇家学会会议录A分册数学Quarterly Journal of Mathematics 数学季刊Quarterly Journal of Mechanics and Applied Mathematics 数学与应用数学季刊Quarterly of Applied Mathematics 应用数学季刊Queueing Systems 排列系统Random Structures and Algorithms 随机结构与算法Reports on Mathematical Physics 数学物理学报告Representation Theory 表示法理论Robotics and Autonomous Systems 机器人技术和自动系统Rocky Mountain Journal of Mathematics 数学难题学报Set-Valued Analysis 精点分析Statistical Papers 统计学论文Statistics & Probability Letters 统计和概率通讯Statistics and Computing 统计和计算Stochastic Analysis and Applications 随机分析和应用Stochastic Environm ental Research and Risk Assessm ent 随机环境论研究和风险评估Stochastic Processes and Their Applications 随机过程及其应用Studies in Applied Mathematics 应用数学研究The College Mathematics Journal 大学数学杂志The Electronic Journal of Combinatorics 组合数学电子期刊Theory of Computing Systems 计算方法理论Theory of Probability and Its Applications 概率理论及其应用程序理论Topology 拓扑学Topology and Its Applications 拓扑学及其应用Transactions of the American Mathematical Society 美国数学会学报Applied Numerical Mathematics《应用数值数学》荷兰Annales Scientifiques de l'École Normale Supérieure《高等师范学校科学纪事》法国Applied and Computational Harmonic Analysis《应用和计算谐波分析》美国Applied Stochastic Models in Business and Industry《商业与工业应用随机模型》英国Acta Applicandae Mathematicae 《应用数学学报》荷兰Advances in Computational Mathematics《计算数学进展》荷兰Annals of Mathematics and Artificial Intelligence《数学与人工智能纪事》荷兰Annals of Operations Research《运筹学纪事》荷兰Annals of the Institute of Statistical Mathematics《统计数理研究所纪事》日本。
江西科技师范大学计算机网络信息检索检索课题:(中文):数学在计算机中的应用(外文):The application of mathematics incomputer院(系):数学与计算机科学学院专业:计算机科学与技术学生姓名:学号:2017 年 12 月 13 日《数学在计算机中的应用》检索综述一、分析研究课题随着计算机技术的快速发展,数学在计算机技术的发展中,尤其是在计算机应用编程设计中处于非常重要的地位。
程序员要能用数学思维解决各种编程方面的问题,将实际问题转换为程序,建立起完善的数学模型,才能设计出好的软件。
另外,软件编程的思维定式决定了一个人编程的水平,如果在编程的过程中,只是根据语法而编写程序,完全脱离软件编程的思维,这种思维定式编写出来的程序非常糟糕,没有一点逻辑。
相反,在编程过程中,数学思维清晰,编写出来的程序让人耳目一新。
所以数学思想在计算机技术中最直接的体现便是编程。
数学对于计算机科学的发展以及应用作用是不容小觑的。
那么计算机科学作为数学的一个分支,其内容实际上是很多数学知识的融合。
在学习数学的过程中,要多思考,建立起数学的思维模式。
在计算机科学的应用中,使用好这种思维模式,才能将数学知识游刃有余地应用到计算机科学中来。
二、制定检索策略1、选择检索手段因为我主要研究的是数学在计算机中的应用,而编程是其主要的应用方面,所以我在选关键词的时候用编程和数学。
而计算机这个关键词简单而明了,所以我选择编程,数学,计算机,应用为我的关键字。
又因为编程,计算机是近义词,我用(编程or计算机)and数学and应用为我的计算机检索式。
2、选择检索工具(1)中国知网(2)万方数据库(3)SPRINGER数据库(4)超星数学图书馆(5)维普中文科技期刊3、选择检索方法关键词:编程数学计算机应用Programming Mathematics Computer application4、选择检索途径进入江西科技师范大学首页,选择图书馆藏:/tsg/myindex.asp。
数学与应用数学专业人才培养方案学科门类理学专业代码 070101 授予学位理学学士(2016级本科生开始执行)一、培养目标本专业培养具备系统扎实的专业知识和良好的数学素养,适应社会发展需要,能够在科技、教育、经济和管理等机构、行业中从事数学理论及应用研究、教学、软件研制、决策管理等工作的创新型复合型人才。
具体目标如下:(1)掌握数学与应用数学的基本理论与方法;(2)具备运用数学理论和方法解决理论问题的能力;(3)具备借助计算机进行数学建模和定量分析解决实际问题的能力;(4)受到一定科学研究训练;(5)有合作意识和创新精神;(6)具有诚信意识和社会责任感。
二、毕业生能力要求1.素质结构要求1.1思想道德素质:热爱祖国,有科学的世界观、人生观和价值观,有责任心和社会责任感,自觉遵纪守法,注重职业道德,具有诚信意识和团队精神;1.2文化素质:有较高的文化素养,有一定的文学艺术修养、人际交往能力和现代意识;1.3专业素质:掌握较多的数学知识,学会“数学方式”的理性思维和科学的研究方法,能够对实际问题建立数学模型,能够用规范的数学语言表达自己的思想,具备求实创新意识;1.4身心素质:身体健康,心理健康。
2.能力结构要求2.1获取知识的能力:具有较强的分析能力、归纳能力、抽象能力、空间想象能力、演绎推理能力以及具有进行准确计算、使用运用数学软件和学习新数学知识的能力;2.2应用知识的能力:具有较高的理论联系实际的能力、较强的解决实际问题的能力;2.3创新的能力:有创造性思维,有一定的科学研究能力,对新知识和新技术有敏锐的洞察能力。
3.知识结构要求3.1工具性知识:熟练掌握一门外国语,熟练使用计算机,会进行文献检索,懂科技写作;3.2人文社会科学知识:有一定的文学、哲学、历史、经济等社会科学知识;3.3自然科学知识:具有较好的物理基础和实验技能,对化学、生命科学和地球科学有较好的了解;3.4专业知识:掌握较多的分析、代数、几何和随机数学方面的知识,达到本专业规范规定的总学分要求和分类学分要求。
Abbreviations of Names of SerialsThis list gives the form of references used in Mathematical Reviews(MR).The abbreviation is followed by the complete title,the place of publication and other pertinent information.∗not previously listed E available electronically §journal reviewed cover-to-cover V videocassette series †monographic series¶bibliographic journal∗Abh.Braunschw.Wiss.Ges.Abhandlungen derBraunschweigischen Wissenschaftlichen Gesellschaft.J.Cramer Verlag,Braunschweig.(Formerly Abh.Braunschweig.Wiss.Ges.)Abh.Braunschweig.Wiss.Ges.Abhandlungen derBraunschweigischen Wissenschaftlichen Gesellschaft.Goltze,G¨o ttingen.(Continued as Abh.Braunschw.Wiss.Ges.)§Abh.Math.Sem.Univ.Hamburg Abhandlungen aus dem Mathematischen Seminar der Universit¨a t Hamburg.Vandenhoeck&Ruprecht,G¨o ttingen.ISSN0025-5858.†Abh.Math.-Naturwiss.Kl.Akad.Wiss.Lit.Mainz Abhandlungen der Mathematisch-NaturwissenschaftlichenKlasse.Akademie der Wissenschaften und der Literaturin Mainz.[Transactions of the Mathematical-ScientificSection.Academy of Sciences and Literature in Mainz]Steiner,Stuttgart.ISSN0002-2993.§Abstr.Appl.Anal.Abstract and Applied Analysis.Mancorp,Tampa,FL.ISSN1085-3375.¶Abstracts Amer.Math.Soc.Abstracts of Papers Presented to the American Mathematical Society.Amer.Math.Soc.,Providence,RI.ISSN0192-5857.Acad.Roy.Belg.Bull.Cl.Sci.(6)Acad´e mie Royale deBelgique.Bulletin de la Classe des Sciences.6e S´e rie.Acad.Roy.Belgique,Brussels.ISSN0001-4141.Acad.Roy.Belg.Cl.Sci.M´e m.Collect.8o(3)Acad´e mieRoyale de Belgique.Classe des Sciences.M´e moires.Collection in-8o.3e S´e rie.Acad.Roy.Belgique,Brussels.ISSN0365-0936.Acad.Serbe Sci.Arts Glas Acad´e mie Serbe des Scienceset des Arts.Glas.Classe des Sciences Naturelles etMath´e matiques.Srpska Akad.Nauk.i Umetnost.,Belgrade.ISSN0374-7956.†Acc`e s Sci.Acc`e s Sciences.[Access to Sciences]De Boeck Univ.,Brussels.§E ACM J.Exp.Algorithmics The ACM Journal ofExperimental Algorithmics.ACM,New York.ISSN1084-6654.E ACM Trans.Math.Software Association for ComputingMachinery.Transactions on Mathematical Software.ACM,New York.ISSN0098-3500.∗§Acta Acad.Paedagog.Agriensis Sect.Mat.(N.S.)Acta Academiae Paedagogicae Agriensis.Nova Series.SectioMatematicae.Eszterh´a zy K´a roly Coll.,Eger.∗§Acta Anal.Funct.Appl.Acta Analysis Functionalis Applicata.AAFA.Yingyong Fanhanfenxi Xuebao.SciencePress,Beijing.ISSN1009-1327.§E Acta Appl.Math.Acta Applicandae Mathematicae.An International Survey Journal on Applying Mathematics andMathematical Applications.Kluwer Acad.Publ.,Dordrecht.ISSN0167-8019.§Acta Arith.Acta Arithmetica.Polish Acad.Sci.,Warsaw.ISSN0065-1036.Acta Astronom.Sinica Acta Astronomica Sinica.TianwenXuebao.Kexue Chubanshe(Science Press),Beijing.(Translated in Chinese Astronom.Astrophys.)ISSN0001-5245.Acta Astrophys.Sinica Acta Astrophysica Sinica.TiantiWuli Xuebao.Kexue Chubanshe(Science Press),Beijing.(Translated in Chinese Astronom.Astrophys.)ISSN0253-2379.Acta Automat.Sinica Acta Automatica Sinica.ZidonghuaXuebao.Kexue Chubanshe(Science Press),Beijing.ISSN0254-4156.Acta Cienc.Indica Math.Acta Ciencia Indica.Mathematics.Pragati Prakashan,Meerut.ISSN0970-0455.Acta Cient.Venezolana Acta Cient´ıfica Venezolana.Asociaci´o n Venezolana para el Avance de la Ciencia.Asoc.Venezolana Avance Cien.,Caracas.ISSN0001-5504.Acta Comment.Univ.Tartu.Math.Acta etCommentationes Universitatis Tartuensis de Mathematica.Univ.Tartu,Fac.Math.,Tartu.ISSN1406-2283.E Acta Cryst.Sect.A Acta Crystallographica.Section A:Foundations of Crystallography.Munksgaard,Copenhagen.ISSN0108-7673.§Acta Cybernet.Acta Cybernetica.J´o zsef Attila Univ.Szeged,Szeged.ISSN0324-721X.Acta Hist.Leopold.Acta Historica Leopoldina.DeutscheAkad.Naturforscher Leopoldina,Halle an der Saale.ISSN0001-5857.§E Acta Inform.Acta Informatica.Springer,Heidelberg.ISSN0001-5903.§Acta Math.Acta Mathematica.Inst.Mittag-Leffler, Djursholm.ISSN0001-5962.§E Acta Math.Acad.Paedagog.Nyh´a zi.(N.S.)Acta Mathematica.Academiae Paedagogicae Ny´ıregyh´a ziensis.New Series.Bessenyei Gy¨o rgy Coll.,Ny´ıregyh´a za.ISSN0866-0182.§Acta Math.Appl.Sinica Acta Mathematicae Applicatae Sinica.Yingyong Shuxue Xuebao.Kexue Chubanshe(Science Press),Beijing.ISSN0254-3079.§Acta Math.Appl.Sinica(English Ser.)Acta Mathematicae Applicatae Sinica.English Series.Yingyong ShuxueXuebao.Science Press,Beijing.ISSN0168-9673.§E Acta Math.Hungar.Acta Mathematica Hungarica.Akad.Kiad´o,Budapest.ISSN0236-5294.§Acta rm.Univ.Ostraviensis Acta Mathematica et Informatica Universitatis Ostraviensis.Univ.Ostrava,Ostrava.ISSN1211-4774.§Acta Math.Sci.(Chinese)Acta Mathematica Scientia.Series A.Shuxue Wuli Xuebao.Chinese Edition.KexueChubanshe(Science Press),Beijing.(See also Acta Math.Sci.(English Ed.))ISSN1003-3998.§Acta Math.Sci.(English Ed.)Acta Mathematica Scientia.Series B.English Edition.Shuxue Wuli Xuebao.SciencePress,Beijing.(See also Acta Math.Sci.(Chinese))ISSN0252-9602.§E Acta Math.Sin.(Engl.Ser.)Acta Mathematica Sinica.English Series.Springer,Heidelberg.ISSN1000-9574.§Acta Math.Sinica Acta Mathematica Sinica.Chinese Math.Soc.,Acta Math.Sinica m.,Beijing.ISSN0583-1431.§E Acta enian.(N.S.)Acta Mathematica Universitatis Comenianae.New enius Univ.Press,Bratislava.ISSN0862-9544.§Acta Math.Vietnam.Acta Mathematica Vietnamica.Nat.Center Natur.Sci.Tech.,Hanoi.ISSN0251-4184.∗Acta Mech.Sin.Engl.Ser.Acta Mechanica Sinica.English Series.The Chinese Society of Theoretical and AppliedMechanics.Chinese J.Mech.Press,Beijing.(FormerlyActa Mech.Sinica(English Ed.))ISSN0567-7718.Acta Mech.Sinica(Beijing)Acta Mechanica Sinica.LixueXuebao.Chinese J.Mech.Press,Beijing.(See also ActaMech.Sinica(English Ed.))ISSN0459-1879.Acta Mech.Sinica(English Ed.)Acta Mechanica Sinica.English Edition.Lixue Xuebao.Kexue Chubanshe(SciencePress),Beijing.(Continued as Acta Mech.Sin.Engl.Ser.)(See also Acta Mech.Sinica(Beijing))ISSN0567-7718.∗Acta Mech.Solida Sin.Acta Mechanica Solida Sinica.Chinese Journal of Solid Mechanics.Huazhong Univ.Sci.Tech.,Wuhan.ISSN0894-9166.†Acta Numer.Acta Numerica.Cambridge Univ.Press, Cambridge.ISSN0962-4929.Acta Phys.Polon.B Jagellonian University.Institute ofPhysics and Polish Physical Society.Acta Physica PolonicaB.Jagellonian Univ.,Krak´o w.ISSN0587-4254.Acta Phys.Sinica Acta Physica Sinica.Wuli Xuebao.Chinese Phys.Soc.,Beijing.ISSN1000-3290.Acta put.Manage.Eng.Ser.Acta Polytechnica Scandinavica.Mathematics,Computingand Management in Engineering Series.Finn.Acad.Tech.,Espoo.ISSN1238-9803.§Acta Sci.Math.(Szeged)Acta Universitatis Szegediensis.Acta Scientiarum Mathematicarum.Univ.Szeged,Szeged.ISSN0001-6969.Acta Sci.Natur.Univ.Jilin.Acta Scientiarum NaturaliumUniversitatis Jilinensis.Jilin Daxue.Ziran Kexue Xuebao.Jilin University.Natural Sciences Journal.Jilin Univ.Nat.Sci.J.,Editor.Dept.,Changchun.ISSN0529-0279.Acta Sci.Natur.Univ.Norm.Hunan.Acta ScientiarumNaturalium Universitatis Normalis Hunanensis.HunanShifan Daxue Ziran Kexue Xuebao.J.Hunan Norm.Univ.,Editor.Dept.,Changsha.ISSN1000-2537.Acta Sci.Natur.Univ.Pekinensis See Beijing DaxueXuebao Ziran Kexue BanActa Sci.Natur.Univ.Sunyatseni Acta ScientiarumNaturalium Universitatis Sunyatseni.Zhongshan DaxueXuebao.Ziran Kexue Ban.Journal of Sun Yatsen University.Natural Sciences.J.Zhongshan Univ.,Editor.Dept.,Guangzhou.ISSN0529-6579.Acta Tech.CSA V Acta Technica CSA V.Acad.Sci.CzechRepub.,Prague.ISSN0001-7043.§Acta Univ.Carolin.Math.Phys.Acta Universitatis Carolinae.Mathematica et Physica.Karolinum,Prague.ISSN0001-7140.§Acta Univ.Lodz.Folia Math.Acta UniversitatisLodziensis.Folia Mathematica.Wydawn.Uniw.Ł´o dzkiego,Ł´o d´z.ISSN0208-6204.Acta Univ.Lodz.Folia Philos.Acta UniversitatisLodziensis.Folia Philosophica.Wydawn.Uniw.Ł´o dzkiego,Ł´o d´z.ISSN0208-6107.∗§Acta Univ.M.Belii Ser.Math.Acta Universitatis Matthiae Belii.Natural Science Series.Series Mathematics.MatejBel Univ.,Bansk´a Bystrica.(Formerly Acta Univ.MathaeiBelii Nat.Sci.Ser.Ser.Math.)§Acta Univ.Mathaei Belii Nat.Sci.Ser.Ser.Math.Matej Bel University.Acta.Natural Science Series.Series Mathematics.Matej Bel Univ.,Bansk´a Bystrica.(Continued as Acta Univ.M.Belii Ser.Math.)Acta Univ.Oulu.Ser.A Sci.Rerum Natur.ActaUniversitatis Ouluensis.Series A.Scientiae RerumNaturalium.Univ.Oulu,Oulu.ISSN0355-3191.§Acta Univ.Palack.Olomuc.Fac.Rerum Natur.Math.Acta Universitatis Palackianae Olomucensis.Facultas Rerum Naturalium.Mathematica.ISSN0231-9721.†Acta Univ.Ups.Stud.Philos.Ups.Acta Universitatis Upsaliensis.Studia Philosophica Upsaliensia.Uppsala Univ., Uppsala.ISSN0585-5497.†Acta Univ.Upsaliensis Skr.Uppsala Univ.C Organ.Hist.Acta Universitatis Upsaliensis.Skrifter r¨o randeUppsala anisation och Historia.[ActaUniversitatis Upsaliensis.Publications concerning Uppsalaanization and History]Uppsala Univ.,Uppsala.ISSN0502-7454.†Actualit´e s Math.Actualit´e s Math´e matiques.[Current Mathematical Topics]Hermann,Paris.†Actualit´e s Sci.Indust.Actualit´e s Scientifiques etIndustrielles.[Current Scientific and Industrial Topics]Hermann,Paris.†Adapt.Learn.Syst.Signal mun.Control Adaptive and Learning Systems for Signal Processing,Communications,and Control.Wiley,New York.Adv.Appl.Clifford Algebras Advances in Applied Clifford Algebras.Univ.Nac.Aut´o noma M´e xico,M´e xico.ISSN0188-7009.†Adv.Appl.Mech.Advances in Applied Mechanics.Academic Press,Boston,MA.ISSN0065-2165.∗†Adv.Astron.Astrophys.Advances in Astronomy andAstrophysics.Gordon and Breach,Amsterdam.ISSN1025-8206.†Adv.Book Class.Advanced Book Classics.Perseus, Reading,MA.†Adv.Bound.Elem.Ser.Advances in Boundary Elements put.Mech.,Southampton.(Continued as Int.Ser.Adv.Bound.Elem.)ISSN1368-258X.†Adv.Chem.Phys.Advances in Chemical Physics.Wiley, New York.†put.Econom.Advances in Computational Economics.Kluwer Acad.Publ.,Dordrecht.§E put.Math.Advances in ComputationalMathematics.Baltzer,Bussum.ISSN1019-7168.†put.Sci.Advances in Computing Science.Springer,Vienna.ISSN1433-0113.∗†Adv.Des.Control Advances in Design and Control.SIAM, Philadelphia,PA.§Adv.Differential Equations Advances in Differential Equations.Khayyam,Athens,OH.ISSN1079-9389.†Adv.Discrete Math.Appl.Advances in DiscreteMathematics and Applications.Gordon and Breach,Amsterdam.ISSN1028-3129.†Adv.Fluid Mech.Advances in Fluid put.Mech.,Southampton.ISSN1353-808X.†Adv.Fuzzy Systems Appl.Theory Advances in Fuzzy Systems—Applications and Theory.World Sci.Publishing,River Edge,NJ.§E Adv.in Appl.Math.Advances in Applied Mathematics.Academic Press,Orlando,FL.ISSN0196-8858.§Adv.in Appl.Probab.Advances in Applied Probability.Appl.Probab.Trust,Sheffield.ISSN0001-8678.∗†Adv.Ind.Control Advances in Industrial Control.Springer, London.†Adv.Lectures Math.Advanced Lectures in Mathematics.Vieweg,Braunschweig.ISSN0932-7134.§E Adv.Math.Advances in Mathematics.Academic Press, Orlando,FL.ISSN0001-8708.§Adv.Math.(China)Advances in Mathematics(China).Shuxue Jinzhan.Peking Univ.Press,Beijing.ISSN1000-0917.†Adv.Math.Econ.Advances in Mathematical Economics.Springer,Tokyo.†Adv.Math.Sci.Advances in the Mathematical Sciences.Amer.Math.Soc.,Providence,RI.§Adv.Math.Sci.Appl.Advances in Mathematical Sciences and Applications.An International Journal.Gakk¯o tosho,Tokyo.ISSN1343-4373.§Adv.Nonlinear Var.Inequal.Advances in Nonlinear Variational Inequalities.An International Journal.Internat.Publ.,Orlando,FL.ISSN1092-910X.†Adv.Numer.Math.Advances in Numerical Mathematics.Teubner,Stuttgart.†Adv.Partial Differ.Equ.Advances in Partial Differential Equations.Wiley-VCH,Berlin.†Adv.Partial Differential Equations Advances in Partial Differential Equations.Akademie Verlag,Berlin.†Adv.Ser.Dynam.Systems Advanced Series in Dynamical Systems.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Math.Phys.Advanced Series in Mathematical Physics.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Math.Sci.Eng.Advanced Series in Mathematical Science and Engineering.World Fed.Publ.,Atlanta,GA.†Adv.Ser.Neurosci.Advanced Series in Neuroscience.World Sci.Publishing,River 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Nou˘a.Ed.Univ.“Al.I.Cuza”,Ias¸i.ISSN1224-2268.§An.S¸tiint¸.Univ.Al.I.Cuza Ias¸i.Mat.(N.S.)Analele S¸tiint¸ifice ale Universit˘a tii“Al.I.Cuza”din Ias¸i.SerieNou˘a.Matematic˘a.Univ.Al.I.Cuza,Ias¸i.ISSN1221-8421.§An.S¸tiint¸.Univ.Ovidius Constant¸a Ser.Mat.Universit˘a t¸ii “Ovidius”Constant¸a.Analele S¸tiint¸ifice.Seria Matematic˘a.“Ovidius”Univ.Press,Constant¸a.ISSN1223-723X.§An.Univ.Bucures¸ti Mat.Analele Universit˘a t¸ii Bucures¸ti.Matematic˘a.Univ.Bucharest,Bucharest.ISSN1013-4123.An.Univ.Craiova rm.Analele Universitˇa t¸iidin Craiova.Seria Matematic˘a-Informatic˘a.Univ.Craiova,Craiova.ISSN1223-6934.∗§An.Univ.Oradea Fasc.Mat.Analele Universit˘a t¸ii din Oradea.Fascicola Matematica.Univ.Oradea,Oradea.ISSN1221-1265.§An.Univ.Timis¸oara Ser.Mat.-Inform.Universit˘a t¸ii din Timis¸oara.Analele.Seria Matematic˘a-Informatic˘a.Univ.Timis¸oara,Timis¸oara.ISSN1224-970X.An.Univ.Timis¸oara Ser.S¸tiint¸.Fiz.Analele Universit˘a t¸iidin Timis¸oara.Seria S¸tiint¸e Fizice.Univ.Vest Timis¸oara,Timis¸oara.†Anal.Appl.Analysis and its Applications.IOS,Amsterdam.ISSN1345-4240.§E Anal.Math.Analysis Mathematica.Akad.Kiad´o,Budapest.ISSN0133-3852.†Anal.Methods Spec.Funct.Analytical Methods and Special Functions.Gordon and Breach,Amsterdam.ISSN1027-0264.†Anal.Modern.Apl.Analiz˘a Modern˘a s¸i Aplicat¸ii.[Modern Analysis and Applications]Ed.Acad.Romˆa ne,Bucharest.§Analysis(Munich)Analysis.International Mathematical Journal of Analysis and its Applications.Oldenbourg,Munich.ISSN0174-4747.E Analysis(Oxford)Analysis.Blackwell,Oxford.ISSN0003-2638.†Angew.Statist.¨Okonom.Angewandte Statistik und¨Okonometrie.[Applied Statistics and Econometrics]Vandenhoeck&Ruprecht,G¨o ttingen.§E Ann.Acad.Sci.Fenn.Math.Annales AcademiæScientiarium Fennicæ.Mathematica.Acad.Sci.Fennica,Helsinki.ISSN1239-629X.§Ann.Acad.Sci.Fenn.Math.Diss.AcademiæScientiarum Fennicæ.Annales.Mathematica.Dissertationes.Acad.Sci.Fennica,Helsinki.ISSN1239-6303.§E Ann.Appl.Probab.The Annals of Applied 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Geometry.Kluwer Acad.Publ.,Dordrecht.ISSN0232-704X.∗E Ann.Henri Poincar´e Annales Henri Poincar´e.A Journal of Theoretical and Mathematical Physics.Birkh¨a user,Basel.(Merged from Ann.Inst.H.Poincar´e Phys.Th´e or.and Helv.Phys.Acta)ISSN1424-0637.∗Ann.I.S.U.P.Annales de l’I.S.U.P..Univ.Paris,Inst.Stat., Paris.§E Ann.Inst.Fourier(Grenoble)Universit´e de Grenoble.Annales de l’Institut Fourier.Univ.Grenoble I,Saint-Martin-d’H`e res.ISSN0373-0956.§E Ann.Inst.H.Poincar´e Anal.Non Lin´e aire Annales de l’Institut Henri Poincar´e.Analyse Non Lin´e aire.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.ISSN0294-1449.Ann.Inst.H.Poincar´e Phys.Th´e or.Annales de l’InstitutHenri Poincar´e.Physique Th´e orique.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.(Merged into Ann.HenriPoincar´e)ISSN0246-0211.§E Ann.Inst.H.Poincar´e Probab.Statist.Annales de l’Institut Henri Poincar´e.Probabilit´e s et Statistiques.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.ISSN0246-0203.E Ann.Inst.Statist.Math.Annals of 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ndUniv.,Budapest.ISSN0138-9491.†Annu.Rev.Fluid Mech.Annual Review of FluidMechanics.Annual Reviews,Palo Alto,CA.ISSN0066-4189.Annuaire Univ.Sofia rm.Godishnik naSofi˘ıskiya Universitet“Sv.Kliment Okhridski”.Fakultetpo Matematika i Informatika.Annuaire de l’Universit´e deSofia“St.Kliment Ohridski”.Facult´e de Math´e matiques etInformatique.Presses Univ.“St.Kliment Ohridski”,Sofia.ISSN0205-0808.Annuaire Univ.Sofia Fac.Phys.Godishnik na Sofi˘ıskiyaUniversitet“Sv.Kliment Okhridski”.Fizicheski Fakultet.Annuaire de l’Universit´e de Sofia“St.Kliment Ohridski”.Facult´e de Physique.Presses Univ.“St.Kliment Ohridski”,Sofia.ISSN0584-0279.†Anu.Filol.Univ.Barc.Anuari de Filologia(Universitat de Barcelona).[Philology Yearbook(University ofBarcelona)]Univ.Barcelona,Barcelona.†Anwend.orientier.Stat.Anwendungsorientierte Statistik.[Applications-Oriented Statistics]Lang,Frankfurt am Main.ISSN1431-7982.Anz.¨Osterreich.Akad.Wiss.Math.-Natur.Kl.¨Osterreichische Akademie der Wissenschaften.Mathematisch-Naturwissenschaftliche Klasse.Anzeiger.¨Osterreich.Akad.Wissensch.,Vienna.∗§E ANZIAM J.The ANZIAM Journal.The Australian& New Zealand Industrial and Applied Mathematics Journal.Austral.Math.Soc.,Canberra.(Formerly J.Austral.Math.Soc.Ser.B)ISSN0334-2700.†Aportaciones Mat.Aportaciones Matem´a ticas.[Mathematical Contributions]Soc.Mat.Mexicana,M´e xico.†Aportaciones un.Aportaciones Matem´a ticas: Comunicaciones.[Mathematical Contributions:Communications]Soc.Mat.Mexicana,M´e xico.∗†Aportaciones Mat.Invest.Aportaciones Matem´a ticas: Investigaci´o n.[Mathematical Contributions:Research]Soc.Mat.Mexicana,M´e xico.(Formerly Aportaciones Mat.Notas Investigaci´o n)†Aportaciones Mat.Notas Investigaci´o n Aportaciones Matem´a ticas:Notas de Investigaci´o n.[MathematicalContributions:Research Notes]Soc.Mat.Mexicana,M´e xico.(Continued as Aportaciones Mat.Invest.)†Aportaciones Mat.Textos Aportaciones Matem´a ticas: Textos.[Mathematical Contributions:Texts]Soc.Mat.Mexicana,M´e xico.§E Appl.Algebra put.Applicable Algebra in Engineering,Communication and Computing.Springer,Heidelberg.ISSN0938-1279.§E Appl.Anal.Applicable Analysis.An International Journal.Gordon and Breach,Yverdon.ISSN0003-6811.§E Appl.Categ.Structures Applied Categorical Structures.A Journal Devoted to Applications of Categorical Methods inAlgebra,Analysis,Order,Topology and Computer Science.Kluwer Acad.Publ.,Dordrecht.ISSN0927-2852.†put.Control Signals Circuits Applied and Computational Control,Signals,and Circuits.Birkh¨a userBoston,Boston,MA.ISSN1522-8363.§E put.Harmon.Anal.Applied and Computational Harmonic Analysis.Time-Frequency and Time-Scale Analysis,Wavelets,Numerical Algorithms,andApplications.Academic Press,Orlando,FL.ISSN1063-5203.†Appl.Log.Ser.Applied Logic Series.Kluwer Acad.Publ., Dordrecht.†Appl.Math.Applications of Mathematics.Springer,New York.ISSN0172-4568.§Appl.Math.Applications of Mathematics.Acad.Sci.Czech 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Discrete Mathematics and Theoretical Computer Science Proceedings AA(DM-CCG),2001,001–022 Mixing Times of Plane Random Rhombus TilingsNicolas DestainvilleLaboratoire de Physique Quantique,UMR-CNRS5626,Universit´e Paul Sabatier,118route de Narbonne,31062Cedex5,Toulouse,France1IntroductionAfter the discovery of quasicrystals about15years ago[1],Penrose tilings[2]and more generally quasiperiodic tilings as well as their randomized counterparts–namely random rhombus tilings–rapidly became a popular topic for a large community of researchers.Indeed,they appeared to be suitable paradig-matic models for quasicrystalline alloys[3,4].However,it rapidly became clear that characterizing and establishing the properties of this new class of models in solid state and statistical physics would be a formidable and long-term task.Simultaneously,these systems also became an activefield of investiga-tion for computer scientists and mathematicians due to the richness of their properties(see[5]or[6]for examples).An example of random tiling is displayed infigure1.It belongs to the class of two-dimensional tilings with octagonal(or8-fold)symmetry.Beyond this example,plane tilings with larger symmetries including Penrose tilings[2]and space tilings with icosahedral symmetries were proposed to model every kind of real quasicrystal.Indeed,the complex microscopic structures of the highest quality alloys turn out to be faithfully described by these tilings.Note however that the origin of the stability of quasicrystals is still debated.Physical explanations ranging from an electronic stabilization in perfect quasiperiodic structures to an entropic stabilization allowed by the large amount of microscopic configurations in a random tiling ensemble have been put forward.Our purpose is not to discuss the relative merits of these approaches but to analyze the combinatorial problematic associated with the random tiling model[7].1365–8050c2001Maison de l’Informatique et des Math´e matiques Discr`e tes(MIMD),Paris,France2Nicolas DestainvilleFig.1:Example of randomly sampledfixed boundary tiling.Itfills a centrally symmetric octagonal region of sides 565and7.The present paper is devoted to dynamical properties of random rhombus tilings in terms of local dy-namical rules,the so-called phason-flips,which consists of local rearrangement of tiles.It is the activation of these degrees of freedom which gives access to a large number of configurations in the random tiling model.These dynamics are of interest for several reasons.On the one hand,it is more and more clear that phason-flips exist in real quasi-crystals(see[8])and can be modeled in afirst approximation by tile-flips;they are local rearrangements of atoms and are therefore a new source of atomic mobility,as compared to usual crystalline materials.In particular,these atomic moves could carry their own contribution to self-diffusion in quasicrystalline materials,as was first pointed out in reference[9],even if the efficiency of such processes in real alloys remains controver-sial[10].Moreover,they are certainly involved in some specific mechanical properties of quasicrystals, such as plasticity via dislocation mobility[11].The two processes are related since long-range diffusion is necessary to allow dislocation formation[11].Therefore a complete understanding offlip dynamics is essential in quasicrystal physics.On the other hand,as it became clear that the characterization of statistical properties of random tiling sets would be a long process,a lot of numerical work has been carried out,a part of which was based on Monte Carlo techniques which rely on a faithful sampling of the tiling sets(see[12,13,14]for examples). So far,no systematic study of the relaxation times between two independent numerical measures has been accomplished,whereas it is an essential ingredient for a suitable control of error bars.However,there exist exact results in the simplest case of random rhombus tilings with hexagonal symmetry[15,16,17]and several estimates of relaxation times in larger symmetries,either numerical or in the approximate frame of Langevin dynamics[12,13,18].Note that these approximate results miss logarithmic corrections provided by an exact treatment.In this paper we examine,by probability theory techniques,mixing times offlip dynamics,also called correlation or ergodic times,which characterize how manyflips the system needs to(nearly)forget its initial configuration and to reach any configuration with uniform probability,in other words at which rate the system converges towards its stationary distribution.Indeed,we suppose here that all tilings have theMixing Times of Plane Random Rhombus Tilings3 same probability;in physical terms we work at infinite temperature.As compared to the hexagonal symmetry case,the main complication in the analysis of mixing times in sets of rhombus tilings with8-fold or larger symmetries seems to come from the fact that there is no convenient way to define underlying lattices in these tilings.Our method relies on the general idea that these sets can be disconnected(or decomposed)in smaller subsets,in each of which all tilings can be mapped onto families of paths(“routings”)running on an underlying rhombus tiling with smaller symmetry[19].On each subset,which is called a“fiber”,we will establish that theflip dynamics is rapidly mixing.Then the difficulty will be to reconnect the differentfibers and to reconstruct the initial dynamics,in order to show that it is in turn rapidly mixing.Before going on,we mention that this paper is restricted to tilings withfixed polygonal boundaries,for the sake of technical simplicity[20].Thefirst section is devoted to preliminaries(state of the art,definitions,known techniques and results). In particular,we present in this section the coupling method to estimate mixing times and the“disconnec-tion”process of the tiling set.The second section contains the main results of the paper,in the specialized case of octagonal tilings:mixing times onfibers and the“reconnection”process.Beyond the octagonal case,the last section is devoted to the generalization of our results to the larger class of plane rhombus tilings with any2D-fold symmetry.Appendices are devoted to technical proofs or developments,which need not be exposed in the body of the paper.This paper follows and completes a shorter version which was dedicated to octagonal tilings only[21].2Preliminaries2.1Definitions and state of the artRandom rhombus tilings of plane(and of space)are made of rhombi of unitary side length.They are classified according to their global symmetry groups[7].The simplest class of hexagonal tilings–made of60o rhombi with3possible orientations–has been widely explored.Indeed,these tilings are commonly mapped onto assemblies of random walkers on square lattices,or“routings”(see[22]or[16]for instance), which highly facilitates their study.Tilings with octagonal symmetry are made of6different tiles:two squares differently oriented and four45o rhombi(seefigures1and2).Beyond these two cases,one can define tilings with higher symmetries(e.g.Penrose tilings[2])or of higher dimensions[7].For sake of technical simplicity,we focus on tilingsfilling a centrally symmetric polygonal region with integral side lengths(figure1;see also reference[20]).For a2D-fold symmetry,this polygon is a2D-gon.As it is usual in thefield of random tilings,we are interested in the large size limit where the polygon becomes infinitely large,while keeping afixed shape:its i-th side is equal to x i k,where x i remainsfixed while k goes to infinity(see also reference[20]).However,most of our statements will concern the diagonal case x1x2x D1for the sake of simplicity but can easily be extended to non-diagonal cases.As far as dynamics is concerned,these tiling sets are endowed with rearrangements of tiles.In this paper,we will focus on local3-tileflips,or elementaryflips:asfigure2illustrates it,whenever3rhombic tilesfill an hexagon in a tiling,they can be“rotated”in that hexagon,thus generating a new tiling.The set of all the tilings of such a region together with the above discrete dynamical rule define a discrete time Markov chain[23,24]:at each step,a vertex of the tiling is uniformly chosen at random and if this vertex is surrounded by3tiles inflippable configuration,then weflip it.Starting from an initial tiling, this process can reach certain configurations with certain probabilities.Since the sets of plane tilings are4Nicolas DestainvilleFig.2:An example of elementaryflip,involving3(white)tiles,in a patch of octagonal random tiling. connected via elementaryflips[25,19],this process can reach any tiling.Moreover,it converges toward the uniform stationary(or equilibrium)distribution,since it satisfies the detailed balance condition(in probability theory,one says equivalently that the Markov chain is reversible[24]).All the difficulty is to characterize the convergence rates:how manyflips are necessary to get close to this stationary distribution?So far they have only been calculated for hexagonal tilings[15,17]:the characteristic times behave like the square of the system size,up to logarithmic corrections.Note that in this specific case,this result still holds for more general,non-local tileflips,the so-called tower moves[15,17].Before going on,we need to define precisely such characteristic times,and thereforefirstly how to measure the distance from the probability distribution at any time to equilibrium.Generally speaking,let us consider a Markov chain on afinite configuration space L,which converges toward a stationary distributionπon L.Let x0denote an initial configuration and P x t x00be the probability that the process has reached the configuration x after t steps.Then the variation distance1∆t x0Mixing Times of Plane Random Rhombus Tilings5 A coupling is a Markov chain on L L;couples of configurations are updated simultaneously and are strongly correlated,but each configuration,viewed in isolation,performs transitions of the original Markov chain under interest.Moreover,the coupled process is constructed so that when the two config-urations happen to be identical,then they follow the same evolution and remain identical forever(coa-lescence property).Then the central idea of the technique is that the average time the two configurations need to couple(or to coalesce)provides a good upper bound on the original mixing timeτε.More precisely,given an initial couple x0y0at time t0,define the coalescence time asT x0y0inf t x t y t x0x0y0y0(4)and the coupling time asT supT x0y0(5)x0y0L Lwhere the last mean is taken over realizations of the coupled Markovian process.The following result links the mixing time of the original Markov chain to this coupling time[26,16]:τεTe ln1ε1Te ln1ε(6)All the difficulty consists in exhibiting Markovian processes where this method can be technically imple-mented.In particular,the coalescence property is an essential ingredient of the coupling technique.If the configuration set L can be endowed with a partial order relation with unique minimum and max-imum elements(denoted byˆ0andˆ1),the implementation of the technique is highly facilitated,provided the coupled dynamics is monotonous.By monotonous,we mean that if x t y t,then x t1y t1. In this case,let x0y0be any initial configuration:ˆ0x0y0ˆ1,and after any number of steps,the four configurations remain in the same order.Therefore when the iterates ofˆ0andˆ1have coalesced,one is sure that the iterates of x0and y0also have.As a consequence,T x0y0Tˆ0ˆ1and the coupling time satisfies T Tˆ0ˆ1.2.3De Bruijnfibration of the configuration spaceA convenient representation of random rhombus tilings was introduced by de Bruijn[27,28].It consists of following in a random tiling lines of tiles made of adjacent tiles sharing and edge with a given orientation. Figure3(left)displays such de Bruijn lines in an octagonal tiling.The set of lines associated with an orientation is called a de Bruijn family.In a2D-fold tiling,there are D orientations of edges and therefore D de Bruijn families.In this representation,a tile corresponds to the intersection of two de Bruijn lines. When removing a de Bruijn family from a2D-fold tiling,one gets a2D1-fold tiling.Conversely, this remark enables one to propose a convenient iterative algorithm for generating random tilings(see[14, 19]):paths are chosen on a2D1-fold tiling.They are represented by dark lines infigure3(right). They go from left to right and do not intersect(but they can have contacts;see alsofigure4).When they are“opened”following a new edge orientation,they generate de Bruijn lines of a new D-th family.There is a one-to-one correspondence between D-tilings and families of non-intersecting paths on a D1-tiling.The latter is called the base tiling.In the following,this idea will be adapted tofixed-boundary tilings in order to study their dynamical properties.At this point,note that a tilingflip involving tiles of the D-th family become a lineflip in this representation,as illustrated in thefigure.The combinatorial structure of plane rhombic tiling sets has been studied in detail[19,6].Unfortu-nately,even if these sets have unique maximum and minimum elements,which could simplify greatly the6Nicolas DestainvilleFig.3:Left:a de Bruijn family(among4)in a patch of octagonal tiling;Right:iterative process for the construction of octagonal tiling,the left-hand side tiling can be seen as a family of non-intersecting oriented paths on a hexagonal base tiling.The tilingflip inside the black hexagon(left)would be translated into a lineflip(right):the line jumps from one side of a tile to the opposite side(broken line).The same viewpoint can be applied to the polygonal boundary case.implementation of a coupling technique,defining couplings on the whole configuration spaces seems to be an infeasible task.Consequently we shall instead use the paths-on-tiling point of view defined above to decompose the configuration space into smaller subsets on which couplings can easily be defined.The main difficulty will be then to connect those subsets in order to re-construct the complete dynamics.We consider each set of tilings which have the same base hexagonal tiling as a subset J a of the whole configuration space L;L is the disjoint union of the J a’s.In reference[19],these subsets are improperly called“fibers”;nevertheless,we shall go on using this terminology which will prove to be illuminating in the following.In this point of view,the only possibleflips in J a are those which involve a given family of de Bruijn lines,which is arbitrarily chosen.Since there are four families of lines,there are four ways to construct such afibration.Now,let M denote the transition matrix associated with the complete Markov chain on the whole set on octagonal tilings:given two configurations x and y,the matrix coefficient M x y is equal to the transition probability P x t1y t.In the same way,we define the transition matrices M i associated with the Markov chains where onlyflips involving the i-th de Bruijn family are allowed.Notefirst that if M a i is the transition matrix on thefiber J a of thefibration i–also denoted by J i a–,then M i is block-diagonal, where each block corresponds with afiber:M i M1iM N i i(7) where N i is the number offibers J i a in thefibration i.As compared to M,M i have some entries M i x y replaced by0when x and y are no longer connected by aflip,and possibly larger diagonal coefficients in order to insure that sums on columns are still equal to1.The following result will be of great use in the following,since it interconnects the fourfibrations:M M1M2M3M43(8)where Id is the identity matrix.Indeed,each coefficient M x y appears in all four matrices M i but one, since the correspondingflip involves3de Bruijn lines.The last term of the sum insures that M is indeed a transition matrix.Mixing Times of Plane Random Rhombus Tilings7 Note that M as well as M i are symmetric matrices.Therefore their spectrum is real and they have an orthonormal eigenbasis.These two points will turn out to be quite useful in the following.3Mixing times of octagonal tilingsOur strategy in this central section is as follows:(i)we implement the above coupling technique in the case of octagonal tilings.More precisely,we construct a coupling technique on eachfiber,then breaking the symmetry between the four de Bruijn families,since one family is arbitrarily chosen among4in the fibration viewpoint;(ii)couplings are then studied numerically;(iii)at the end of the section,we restore the lost symmetry using equation(8)to relate the dynamics on the whole lattice L with dynamics on individualfibers.We also discuss the possible existence and the effect of“slower”fibers.3.1Mixing times on afiberThe average coupling time on afiber is estimated numerically.Note that,anticipating an extension of previous results on hexagonal tilings[17,16],we expect the coupling time to behave like a power law of the system size,up to some possible logarithmic corrections.Note also that numerical techniques usually have some difficulties in exhibiting such logarithmic corrections:this point will deserve a careful analysis.3.1.1Couplings on afiberOur point of view is quite similar to the“routing”method developed in references[15,16].The bijection between octagonal tilings and families of paths on hexagonal tilings has been previously introduced in section2.On afiber,the base tiling isfixed whereas the de Bruijn lines of the remaining family canflip (seefigure3).To begin with,let us suppose that there is only one line in the latter family,denoted by.As in references[15,16],wefirst slightly modify the Markov chain in order to define a suitable coupling:at each step,we choose uniformly an internal point of,the i-th vertex(starting from the left),as well as a number r01.If the i-th vertex isflippable upward(resp.downward)and r12(resp.r12),then weflip it.Note that,as compared to the original dynamics M(or M i),this dynamics has a different time scale:on the one hand,it is faster since the vertex to beflipped is chosen a priori on.On the other hand, it is slowed down by a factor2since a feasibleflip is actually realized with probability1/2,depending on r.However this modification does not change the nature of thefixed point or of the dynamics since it only implies a rescaling of the time unit.As far as couplings are concerned,the order relation between lines is the following:a line1is greater than another one2(12)if1is entirely(but not strictly)above2.Figure4(Left)provides an exam-ple.Note that,as compared to the“routing”point of view of[16],we allow lines to cover(overlap)locally, since we have contracted de Bruijn lines of the corresponding family.The maximum(resp.minimum) configuration clearly lies on the top(resp.bottom)boundary of the hexagonal domain.If the twoflips on 1and2occur with same i and r and if12then their images satisfy the same relation.Indeed,as in reference[15],by construction–namely the introduction of r–if aflip could bring1below2,then the sameflip would also apply to2,thus preserving the order between lines.As a consequence the coupling is monotonous.In the general case with p non-intersecting lines in each configuration(p2infigure4,right),let us denote by j i,j1p,the p lines of each configurationγi.Thenγ1γ2if for any j,j1j2.The configurationγis maximum(resp.minimum)when each of its lines is maximum(resp.minimum).As8Nicolas DestainvilleFig.4:Left:single-line coupling on afiber.Right:multi-line coupling.compared to the p1case,at each step,the index j of the line to beflipped is chosen randomly between 1and p,the same j forγ1andγ2.This also leads to a valid monotonous coupling.3.1.2Heuristic argumentWith the underlying idea that couplings in paths-on-tilings problems do not differ qualitatively from cou-plings in a problem(of similar size)of paths on a square grid,we expect the following average coupling times:for a single-line coupling on a base tiling of sides of order k,T k3ln k;for a p-line coupling, T k4ln k.Indeed,D.B.Wilson predicts a similar result for hexagonal tilings[17]and therefore in thecase of paths on a square grid(even in the case of singleflips dynamics).Note however that we can already predict that the numerical prefactors in this asymptotic law will differ between hexagonal and octagonal cases;even if we ignore details of the local dynamics,for a given n,the distance(in number of flips)between extremal configurations is bigger in the octagonal case than in the hexagonal one,leading to a bigger prefactor in the former case.The two followings paragraphs are devoted to the quantitative numerical study of this point.3.1.3Mixing times of single-line couplingsTo begin with,we study the case p1,when there is only one de Bruijn line in each component of the coupling.We also suppose in afirst time that the hexagonal base tilings are diagonal:all their sides are equal to k.Following reference[17],we expect T to be asymptotic to k3ln k,up to a numerical prefactor. For a given base hexagonal tiling T a associated with thefiber J a,we run a number m of couplings until they coalesce,and then estimate the coupling time T T a.We then make a second average on M different tilings,in order to get the timeT seems to decrease toward a constant(007)as k∞,which means that the average cou-pling time T T a goes on depending on the base tiling T a at the large size limit.However,most T T a are of order T.Nevertheless, the effect of few“slower”fibers will deserve a detailed discussion below(section3.3.2).The measures of T T a are compatible with a k3ln k behavior,asfigure5illustrates:the graphMixing Times of Plane Random Rhombus Tilings 901000020000k 3 ln(k)02e+054e+056e+05a v e r a g e c o u p l i n g t i m e Fig.5:In the case of 1-line systems,numerical estimates ofT is also about 0.07.Moreover,the slope is equal to 4279011.We also have explored non-diagonal cases where all the sides of the hexagonal boundary are not neces-sarily equal.For example,single-line couplings on hexagonal base tilings of sides k 2k 3k also display a k 3ln k asymptotic behavior with a slope 346117.However in this last case,∆T10Nicolas Destainville 0500001e+05 1.5e+05k 4 ln(k)02e+064e+066e+06a v e r a g e c o u p l i n g t i m e Fig.6:In the case of p -line systems,numerical estimates of2g M τεln 1ε2g M τεln 1εln NMixing Times of Plane Random Rhombus Tilings11Fig.7:The lattice of tilingsfilling an octagon of sides1,1,1and2.The complete lattice can be found in reference[6]. Twofibrations among four are represented.They are equivalent because in both cases de Bruijn lines of the chosen family play a symmetric role.whereµ1is the second largest eigenvalue and N denotes the number of configurations[29,15].Since N grows exponentially with the number of tiles N T,one getsln1εSN Tτε(14)g Mwhereτth is the above threshold.The upper bound in(12)matches this form,whereas the upper bound provided by the coupling technique does not,and might certainly be refined.In particular,the coupling time T provides an upper bound forτth[17].As a consequence,the knowledge of the gap g M is not sufficient to derive a tight upper bound for τεat short times.However,as detailed below,relation(13)is sufficient to derive a polynomial upper bound forτεand to prove thatflip dynamics are rapidly mixing.12Nicolas Destainville3.3.2Statistics and effect of rare “slow”fibersIn the previous sections,we have proved that the coalescence timesT κ4N 2T ln N T (15)where κ411890003in the original Markov chain time unit (see discussion about time units in paragraph 3.1.1).However,the coupling time T depends on the associate fiber and the distribution of times T in a given fibration have a certain width,as it is illustrated in figure 8.Now our proof is based 15002000250030003500T 02004006008001000N u m b e r o f b a s e t i l i n g s 4500055000650007500085000T050100150200250N u m b e r o f b a s e t i l i n g s (a)(b)Fig.8:Numerical distributions of coupling times T around the average value T asdiscussed above.Therefore the typical values of coupling times should remain of the order of magnitude of the average value (15).Nevertheless,the above arguments do not exclude definitively the existence of rare slow fibers in the upper tails of these distributions.We propose two arguments to treat this question of slow fibers.The first one is the less rigorous but provides a better upper bound.It is developed in appendix C.The second argument is based upon a careful analysis of numerical distributions of times T T a .We recall that coupling times T are maxima of coalescence time distributions (see relation (5)).Therefore the expected shape of these numerical distributions ought to be sought in the specific class of distributions of extrema of families of random variables,namely Gumbel distributions [30,31]:we first recall some standard results of extreme-value statistics.Consider N independent and identically distributed random variables T s ,s 1N .We suppose thatMixing Times of Plane Random Rhombus Tilings13 the probability densities decay rapidly at large T:C1p T14Nicolas Destainville where C is a constant of orderκ4.We should also mention that coupling techniques do not necessary provide the best upper bounds[26]and that mixing times might even be shorter than(21)(see in particular the discussion of section3.3.1).Moreover,the argument at the beginning of section3.2suggests that mixing times in the whole set L are of order of magnitude of mixing times infibrations.Now the N2T ln N T ln1εmixing times infibers are valid even at short times.As a consequence,we believe that the above upper bound(21)should also be valid at short times.4Higher symmetry plane tilingsIn this section,we discuss what the above result becomes in the case of two-dimensional tilings of larger symmetriesfilling2D-gons.These tilings are made of rhombi with angles multiple ofπD(see[7,14]) and they are also endowed with3-tile elementaryflips which still connect the set of configurations.All the above approach can be extended to higher-symmetry tilings.Some points need to be clarified further.Thefibration concept is general[19]:there are Dfibrations associated with D de Bruijn families. Moreover,as far as coupling times onfibers are concerned,our phenomenological arguments remains valid in this case.We also have tested coupling times onfibers numerically.For example,for D5 coupling times onfibers still behave as in the octagonal case:in the diagonal cases of side k with k-line couplings,we stillfind that coupling times are asymptotic to N2T ln N T,up to a different numerical prefactorsκ5280010in the original time unit.We also note that Gumbel distributions remain the good description of coupling time distributions but that∆T3D3Dinfi1Dg M i(23)andg M D1Mixing Times of Plane Random Rhombus Tilings15 Tab.1:Numerical constants3κD D1.For diagonal tilings of symmetry2D,the constant C in relation(21)is of order3κD D1.D5210008D1T also increases with D should be related to this point as well,since dynamics on rare base tilings with a“large”amount of large coordination vertices must be particularly slow.The question of slow dynamics in rhombus random tilings in the presence of energetic interactions between tiles should also be addressed in the frame of this analysis:do they remain rapidly mixing,or can sufficiently many configurations appear which significantly slow down the dynamics[32,33]?Using the same construction as in the paper,if the dynamics happen to be rapid infibers atfinite temperature, the system will be rapidly mixing.Conversely,the existence of slow dynamics should be detectable in fiber dynamics,even though,as it was discussed in this paper,the existence of slow dynamics in some fibers does not involve their existence in the whole system.Moreover,note that relation(37)has been established in the case where all transition matrices are symmetric,which is false in the presence of energetic interactions.Therefore the methods of the present paper cannot be generalized without care at finite temperature.At last,what does the present analysis become in the case of larger dimension tilings,related for example to quasicrystals with icosahedral symmetry?Even if thefibration process remains valid,it should be considered with care since the connectivity offibers has not even been establish with certainty[19].。