New Optimization Algorithms in Physics

298 Pages·2004·5.452 MB·English

by  Alexander K. Hartmann, Heiko Rieger

Titelei_Hartmann 18.03.2004 14:22 Uhr Seite 3 (Black/Process Black Bogen)
New Optimization
Algorithms in Physics
Edited by
Alexander K.Hartmann and Heiko Rieger
Titelei_Hartmann 18.03.2004 14:22 Uhr Seite 4 (Black/Process Black Bogen)
Editors This book was carefully produced.Nevertheless,
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Universität Göttingen,Germany Readers are advised to keep in mind that state-
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Contents
ListofContributors XI
1 Introduction
(A.K.HartmannandH.Rieger) 1
PartI ApplicationsinPhysics 5
2 ClusterMonteCarloAlgorithms
(W.Krauth) 7
2.1 DetailedBalanceandaprioriProbabilities . . . . . . . . . . . . . . . . . . . 7
2.2 TheWolffClusterAlgorithmfortheIsingModel . . . . . . . . . . . . . . . 10
2.3 ClusterAlgorithmforHardSpheresandRelatedSystems . . . . . . . . . . . 12
2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 PhaseSeparationinBinaryMixtures. . . . . . . . . . . . . . . . . . 16
2.4.2 PolydisperseMixtures . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.3 Monomer-DimerProblem . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 LimitationsandExtensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 ProbingSpinGlasseswithHeuristicOptimization Algorithms
(O.C.Martin) 23
3.1 SpinGlasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.2 TheIsingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.3 ModelsofSpinGlasses. . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.4 SomeChallenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 SomeHeuristicAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 GeneralIssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.2 VariableDepthSearch . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.3 GeneticRenormalizationAlgorithm . . . . . . . . . . . . . . . . . . 36
3.3 ASurveyofPhysicsResults . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 ConvergenceoftheGround-stateEnergyDensity . . . . . . . . . . . 41
3.3.2 DomainWalls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.3 ClusteringofGroundStates . . . . . . . . . . . . . . . . . . . . . . 42
3.3.4 Low-energyExcitations . . . . . . . . . . . . . . . . . . . . . . . . 42
VI Contents
3.3.5 PhaseDiagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 ComputingExactGroundStatesofHardIsingSpinGlassProblemsby
Branch-and-cut
(F.Liers,M.Jünger,G.Reinelt,andG.Rinaldi) 47
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 GroundStatesandMaximumCuts . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 AGeneralSchemeforSolvingHardMax-cutProblems . . . . . . . . . . . . 51
4.4 LinearProgrammingRelaxationsofMax-cut . . . . . . . . . . . . . . . . . 55
4.5 Branch-and-cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 ResultsofExactGround-stateComputations . . . . . . . . . . . . . . . . . . 62
4.7 AdvantagesofBranch-and-cut . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.8 ChallengesfortheYearstoCome . . . . . . . . . . . . . . . . . . . . . . . 66
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 CountingStatesandCountingOperations
(A.AlanMiddleton) 71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 PhysicalQuestionsaboutGroundStates . . . . . . . . . . . . . . . . . . . . 72
5.2.1 HomogeneousModels . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 MagnetswithFrozenDisorder . . . . . . . . . . . . . . . . . . . . . 73
5.3 FindingLow-energyConfigurations . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 PhysicallyMotivatedApproaches . . . . . . . . . . . . . . . . . . . 75
5.3.2 CombinatorialOptimization . . . . . . . . . . . . . . . . . . . . . . 76
5.3.3 Ground-stateAlgorithmfortheRFIM . . . . . . . . . . . . . . . . . 78
5.4 TheEnergyLandscape: DegeneracyandBarriers . . . . . . . . . . . . . . . 80
5.5 CountingStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5.1 Ground-stateConfigurationDegeneracy . . . . . . . . . . . . . . . . 83
5.5.2 ThermodynamicState . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5.3 NumericalStudiesofZero-temperatureStates . . . . . . . . . . . . . 86
5.6 RunningTimesforOptimizationAlgorithms. . . . . . . . . . . . . . . . . . 91
5.6.1 RunningTimesandEvolutionoftheHeights . . . . . . . . . . . . . 92
5.6.2 HeuristicDerivationofRunningTimes . . . . . . . . . . . . . . . . 94
5.7 FurtherDirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 ComputingthePottsFreeEnergyandSubmodularFunctions
(J.-C.Anglèsd’Auriac) 101
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 ThePottsModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.1 DefinitionofthePottsModel. . . . . . . . . . . . . . . . . . . . . . 102
6.2.2 SomeResultsforNon-randomModels. . . . . . . . . . . . . . . . . 103
6.2.3 TheFerromagneticRandomBondPottsModel . . . . . . . . . . . . 103
Contents VII
6.2.4 HighTemperatureDevelopment . . . . . . . . . . . . . . . . . . . . 103
6.2.5 LimitofanInfiniteNumberofStates . . . . . . . . . . . . . . . . . 104
6.3 BasicsontheMinimizationofSubmodularFunctions . . . . . . . . . . . . . 105
6.3.1 DefinitionofSubmodularFunctions . . . . . . . . . . . . . . . . . . 105
6.3.2 ASimpleCharacterization . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.4 MinimizationofSubmodularFunction. . . . . . . . . . . . . . . . . 106
6.4 FreeEnergyofthePottsModelintheInfiniteq-Limit . . . . . . . . . . . . . 107
6.4.1 TheMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.2 TheAuxiliaryProblem . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.3 TheMax-flowProblem: theGoldbergandTarjanAlgorithm . . . . . 111
6.4.4 AbouttheStructureoftheOptimalSets . . . . . . . . . . . . . . . . 111
6.5 ImplementationandEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5.2 ExampleofApplication . . . . . . . . . . . . . . . . . . . . . . . . 114
6.5.3 EvaluationoftheCPUTime . . . . . . . . . . . . . . . . . . . . . . 114
6.5.4 MemoryRequirement . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.5.5 VariousPossibleImprovements . . . . . . . . . . . . . . . . . . . . 115
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
PartII PhaseTransitionsinCombinatorialOptimizationProblems 119
7 TheRandom3-satisfiabilityProblem: FromthePhaseTransitiontotheEfficient
GenerationofHard,butSatisfiableProblemInstances
(M.Weigt) 121
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Random3-SATandtheSAT/UNSATTransition . . . . . . . . . . . . . . . . 122
7.2.1 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2.2 UsingStatisticalMechanics . . . . . . . . . . . . . . . . . . . . . . 124
7.3 SatisfiableRandom3-SATInstances . . . . . . . . . . . . . . . . . . . . . . 127
7.3.1 TheNaiveGenerator . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3.2 UnbiasedGenerators . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8 AnalysisofBacktrackingProceduresforRandomDecisionProblems
(S.Cocco,L.Ein-Dor,andR.Monasson) 139
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.2 PhaseDiagram,SearchTrajectoriesandtheEasySATPhase . . . . . . . . . 143
8.2.1 OverviewofConceptsUsefultoDPLLAnalysis . . . . . . . . . . . 144
8.2.2 ClausePopulations: Flows,AveragesandFluctuations . . . . . . . . 145
8.2.3 Average-caseAnalysisintheAbsenceofBacktracking . . . . . . . . 147
8.2.4 OccurrenceofContradictionsandPolynomialSATPhase. . . . . . . 150
VIII Contents
8.3 AnalysisoftheSearchTreeGrowthintheUNSATPhase . . . . . . . . . . . 153
8.3.1 NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . . 153
8.3.2 ParallelGrowthProcessandMarkovianEvolutionMatrix . . . . . . 155
8.3.3 GeneratingFunctionandLarge-sizeScaling . . . . . . . . . . . . . . 158
8.3.4 InterpretationinTermsofGrowthProcess . . . . . . . . . . . . . . . 161
8.4 HardSATPhase: AverageCaseandFluctuations . . . . . . . . . . . . . . . 164
8.4.1 MixedBranchandTreeTrajectories . . . . . . . . . . . . . . . . . . 164
8.4.2 DistributionofRunningTimes . . . . . . . . . . . . . . . . . . . . . 165
8.4.3 LargeDeviationAnalysisoftheFirstBranchintheTree . . . . . . . 167
8.5 TheRandomGraphColoringProblem . . . . . . . . . . . . . . . . . . . . . 171
8.5.1 DescriptionofDPLLAlgorithmforColoring . . . . . . . . . . . . . 171
8.5.2 ColoringintheAbsenceofBacktracking . . . . . . . . . . . . . . . 172
8.5.3 ColoringinthePresenceofMassiveBacktracking . . . . . . . . . . 173
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9 NewIterativeAlgorithmsforHardCombinatorialProblems
(R.Zecchina) 183
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 CombinatorialDecisionProblems,K-SATandtheFactorGraphRepresentation185
9.2.1 RandomK-SAT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.3 GrowthProcessAlgorithm: Probabilities,MessagesandTheirStatistics . . . 190
9.4 TraditionalMessage-passingAlgorithm: BeliefPropagationasSimpleCavity
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.5 SurveyPropagationEquations . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.6 DecimatingVariablesAccordingtoTheirStatisticalBias . . . . . . . . . . . 195
9.7 ConclusionsandPerspectives . . . . . . . . . . . . . . . . . . . . . . . . . . 197
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
PartIII NewHeuristicsandInterdisciplinaryApplications 203
10 HystereticOptimization
(K.F.Pál) 205
10.1 HystereticOptimizationforIsingSpinGlasses. . . . . . . . . . . . . . . . . 206
10.2 GeneralizationtoOtherOptimizationProblems . . . . . . . . . . . . . . . . 214
10.3 ApplicationtotheTravelingSalesmanProblem . . . . . . . . . . . . . . . . 221
10.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
11 ExtremalOptimization
(S.Boettcher) 227
11.1 EmergingOptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.2 ExtremalOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
11.2.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
11.2.2 EOAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Contents IX
11.2.3 ExtremalSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
11.2.4 RankOrdering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
11.2.5 DefiningFitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
11.2.6 DistinguishingEOfromotherHeuristics . . . . . . . . . . . . . . . 235
11.2.7 ImplementingEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
11.3 NumericalResultsforEO. . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
11.3.1 EarlyResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
11.3.2 ApplicationsofEObyOthers . . . . . . . . . . . . . . . . . . . . . 242
11.3.3 Large-scaleSimulationsofSpinGlasses . . . . . . . . . . . . . . . . 243
11.4 TheoreticalInvestigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
12 SequenceAlignments
(A.K.Hartmann) 253
12.1 MolecularBiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
12.2 AlignmentsandAlignmentAlgorithms. . . . . . . . . . . . . . . . . . . . . 259
12.3 Low-probabilityTailofAlignmentScores . . . . . . . . . . . . . . . . . . . 266
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
13 ProteinFoldinginSilico–theQuestforBetterAlgorithms
(U.H.E.Hansmann) 275
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
13.2 EnergyLandscapePaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
13.3 BeyondGlobalOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . 280
13.3.1 ParallelTempering . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
13.3.2 MulticanonicalSamplingandOtherGeneralized-ensembleTechniques 283
13.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
13.4.1 HelixFormationandFolding . . . . . . . . . . . . . . . . . . . . . . 288
13.4.2 StructurePredictionsofSmallProteins . . . . . . . . . . . . . . . . 290
13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Index 297
List of Contributors
•Jean-ChristianAnglèsd’Auriac, Ch.6 •UlrichH.E.Hansmann, Ch.13
CentredeRecherchessurlesTresBassesTem- DepartmentofPhysics
peratures MichiganTechnologicalUniversity
BP166,F-38042Grenoble,France Houghton
MI49931-1295,USA
e-mail:[email protected]
e-mail:[email protected]
•StefanBoettcher, Ch.11 •AlexanderK.Hartmann, Ch.1,12
InstitutfürTheoretischePhysik
StefanBoettcher
UniversitätGöttingen
PhysicsDepartment
Tammanstraße1
EmoryUniversity
D-37077Göttingen,Germany
Atlanta,GA30322;USA
e-mail:
e-mail:[email protected]
[email protected]
www.physics.emory.edu/faculty/boettcher
•MichaelJünger, Ch.4
•SimonaCocco, Ch.8 InstitutfürInformatik
UniversitätzuKöln
Laboratoire de Dynamique des Fluides Com-
Pohligstraße1
plexes
D-50969Köln,Germany
3ruedel’Université
e-mail:[email protected]
67000Strasbourg,France
e-mail:[email protected] •WernerKrauth, Ch.2
CNRS-LaboratoiredePhysiqueStatistique
•LiatEin-Dor, Ch.8 EcoleNormaleSupérieure
24,rueLhomond
DepartmentofPhysicsofComplexSystems F-75231ParisCedex05,France
WezmannInstitute
e-mail:[email protected]
Rehovot76100,Israel
•FraukeLiers, Ch.4
and InstitutfürInformatik
UniversitätzuKöln
LaboratoiredePhysiqueThéoriquedel’ENS Pohligstraße1
24rueLhomond D-50969Köln,Germany
75005Paris,France e-mail:[email protected]
NewOptimizationAlgorithmsinPhysics.EditedbyAlexanderK.Hartmann,HeikoRieger
Copyright(cid:1)c 2004Wiley-VCHVerlagGmbH&Co.KGaA
ISBN:3-527-40406-6
XII ListofContributors
•OlivierC.Martin, Ch.3 •GerhardReinelt, Ch.4
LPTMS,UniversitéParis-Sud InstitutfürInformatik
OrsayCedex91405,France Ruprecht-Karls-UniversitätHeidelberg
ImNeuenheimerFeld368
e-mail:[email protected]
D-69120Heidelberg,Germany
e-mail:
•A.AlanMiddleton, Ch.5 [email protected]
DepartmentofPhysics •HeikoRieger, Ch.1
SyracuseUniversity
TheoretischePhysik
Syracuse,NY,USA
UniversitätdesSaarlandes
e-mail:[email protected]
D-66041Saarbrücken,Germany
e-mail:[email protected]
•RemiMonasson, Ch.8
•GiovanniRinaldi, Ch.4
LaboratoiredePhysiqueThéoriquedel’ENS
IstitutodiAnalisideiSistemiedInformatica
24rueLhomond
‘AntonioRuberti’-CNR
75005Paris,France
VialeManzoni,30
e-mail:[email protected]
00185Roma,Italy
e-mail:[email protected]
and
•MartinWeigt, Ch.7
LaboratoiredePhysiqueThéorique InstitutfürTheoretischePhysik
3ruedel’Université UniversitätGöttingen
67000Strasbourg,France Tammanstraße1
D-37077Göttingen,Germany
e-mail:[email protected]
•KárolyF.Pál, Ch.10
•RiccardoZecchina, Ch.9
DepartmentofTheoreticalPhysics
InstituteofNuclearResearchofthe International Centre for Theoretical Physics
HungarianAcademyofSciences (ICTP)
Bemtér18/c StradaCostiera,11 P.O.Box586
H-4026Debrecen,Hungary I-34100 Trieste,Italy
e-mail:[email protected] e-mail:[email protected]
1 Introduction
AlexanderK.HartmannandHeikoRieger
Optimization problems occur very frequently in physics. Some of them are easy to handle
withconventionalmethodsalsousedinotherareassuchaseconomyoroperationsresearch.
Butassoonasahugenumberofdegreesoffreedomareinvolved,asistypicallythecasein
statisticalphysics,condensedmatter,astrophysicsandbiophysics,conventionalmethodsfail
to find the optimum in a reasonable time and new methods have to be invented. This book
containsarepresentativecollectionofnewoptimizationalgorithmsthathavebeendevisedby
physicistsfromvariousfields,sometimesbasedonmethodsdevelopedbycomputerscientists
and mathematicians. However, it is not a mere collection of algorithms but tries to demon-
strates their scope and efficiency by describing typical situations in physics where they are
useful.
Theindividualarticlesofthiscollectionsareself-containedandshouldbeunderstandable
forscientistsroutinelyusingnumericaltools. Amorebasicandpedagogicalintroductioninto
optimizationalgorithmsisourbookonOptimizationAlgorithmsinPhysics,whichcanserve
asanappendixforthenewcomertothisfieldofcomputationalphysicsorforundergraduate
students. The reason why we found it necessaryto compose another book in this field with
agreaterfocusisthefactthattheapplicationofoptimizationmethodsisoneofthestrongest
growingfieldsinphysics. Themainreasonsforthesecurrentdevelopmentsarethefollowing
keyfactors:
First of all great progress has been made in the developmentof new combinatorial opti-
mization methods in computer science. Using these sophisticated approaches, much larger
systemsizesofthecorrespondingphysicalsystemscanbetreated. Formanymodelsthesys-
temssizeswhichwereaccessiblebefore,weretoosmalltoobtainreliableandsignificantdata.
However,thisisnowpossible. Inthiswaycomputersciencehashelpedphysics.
Butknowledgetransferalsoworkstheotherwayround.Physicsprovidesstillnewinsights
andmethodsoftreatingoptimizationproblems,suchastheearlierinventionofthesimulated
annealingtechnique. Recentalgorithmicdevelopmentsinphysicsare,e.g.,theextremalopti-
mizationmethodorthehystericoptimizationapproach,bothcoveredinthisbook.
Moreover, phase transitions were recently found in “classical” optimization problems
within theoretical computer science, during the study of suitably parameterized ensembles.
Thesephasetransitionsveryoftencoincidewithpeaksoftherunningtimeorwithchangesof
thetypical-casecomplexityfrompolynomialtoexponential. Aswellasthegainfromtaking
the physical viewpoint, by mapping the optimization problems to physical systems and ap-
plyingmethodsfromstatisticalphysics,itispossibletoobtainmanyresults,whichhavenot
beenfoundwithtraditionalmathematicaltechniques. Thisistruealsofortheanalysisofthe
typical-casecomplexityof(random)algorithms.
NewOptimizationAlgorithmsinPhysics.EditedbyAlexanderK.Hartmann,HeikoRieger
Copyright(cid:1)c 2004Wiley-VCHVerlagGmbH&Co.KGaA
ISBN:3-527-40406-6