The complexity of partition functions:
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| Main Author: | |
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| Format: | Thesis Book |
| Language: | English |
| Published: |
2009
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII S., S. 9 - 236 graph. Darst. |
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MARC
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| adam_text | THE COMPLEXITY OF PARTITION FUNCTIONS DISSERTATION ZUR ERLANGUNG DES
AKADEMISCHEN GRADES DOCTOR RERUM NATURALIUM (DR. RER. NAT.) IM FACH
INFORMATIK EINGEREICHT AN DER MATHEMATISCH-NATURWISSENSCHAFTLICHEN
FAKULTAET II HUMBOLDT-UNIVERSITAET ZU BERLIN VON HERR DIPL.-INF. MARC
THURLEY GEBOREN AM 28.03.1981 IN STAAKEN PRAESIDENT DER
HUMBOLDT-UNIVERSITAET ZU BERLIN: PROF. DR. DR. H.C. CHRISTOPH MARKSCHIES
DEKAN DER MATHEMATISCH-NATURWISSENSCHAFTLICHEN FAKULTAET II: PROF. DR.
PETER FRENSCH GUTACHTER: 1. PROF. DR. MARTIN GROHE 2. PROF. DR. HERIBERT
VOLLMER 3. PROF. ANDREI BULATOV EINGEREICHT AM: 07. JULI 2009 TAG DER
MUENDLICHEN PRUEFUNG: 18. SEPTEMBER 2009 CONTENTS 1 INTRODUCTION 9 1.1
CONTRIBUTIONS OF THIS THESIS 14 1.2 BASIC NOTATION 16 2 ORIGINS OF
PARTITION FUNCTIONS 19 2.1 STATISTICAL PHYSICS 19 2.1.1 STATISTICAL
MECHANICS MODELS ON GRAPHS 21 2.1.2 SOLVING MODELS AND COMPUTING
PARTITION FUNCTIONS 23 2.2 COMBINATORICS 27 2.2.1 THE TUTTE POLYNOMIAL
27 2.2.2 PARTITION FUNCTIONS 29 3 A COMBINATORIAL PROPERTY OF PARTITION
FUNCTIONS 35 3.1 DETERMINING MATRICES BY PARTITION FUNCTIONS 36 3.1.1
THE TWIN REDUCTION LEMMA 37 3.1.2 THE RECONSTRUCTION LEMMA 39 3.1.3 THE
PROOF OF THEOREM 3.1.1 45 4 PARTITION FUNCTIONS AND THEIR COMPLEXITY 47
4.1 COUNTING COMPLEXITY 48 4.2 COUNTING COMPLEXITY AND PARTITION
FUNCTIONS 49 4.3 THE ARITHMETICAL STRUCTURE OF PARTITION FUNCTIONS 52
4.3.1 FUNDAMENTAL TECHNICAL TOOLS 52 4.3.2 THE EQUIVALENCE OF EVAL PIN
(A, ) AND COUNT PIN (A, ) 53 4.3.3 LOOKING AT THE STRUCTURE: THE
PROOF OF LEMMA 4.3.1 55 4.4 FURTHER PREPARATORY CONSIDERATIONS 58 4.4.1
GENERAL PRINCIPLES 58 4.4.2 EDGE PRODUCTS 60 4.4.3 RANK 1 MATRICES AND
TRACTABILITY 61 4.5 PINNING VERTICES 63 4.6 BOUNDED DEGREES AND
ELIMINATING VERTEX WEIGHTS 67 4.6.1 SOME TECHNICAL TOOLS 68 4.6.2 THE
PROOF OF LEMMA 4.6.2 70 V * 4.6.3 THE PROOF OF LEMMA 4.6.1 73 5
NON-NEGATIVE MATRICES 79 5.1 PRELIMINARIES 80 5.2 AN ITINERARY 81 5.3
THE GENERAL CONDITIONING LEMMA 82 5.3.1 FROM GENERAL MATRICES TO
POSITIVE MATRICES . 82 5.3.2 FROM POSITIVE MATRICES TO X-MATRICES 82
5.3.3 FROM X-MATRICES TO THE GENERAL CONDITIONING LEMMA 84 5.4 THE TWO
1-CELL LEMMA 85 5.5 THE SINGLE 1-CELL LEMMA 89 5.5.1 PROOF OF THE
SINGLE-1-CELL TECHNICAL CORE LEMMA 5.5.4 94 6 PARTITION FUNCTIONS ON
HERMITIAN MATRICES 99 6.1 CONGRUENTIAL PARTITION FUNCTIONS 100 6.2 AN
ITINERARY 100 6.2.1 THE GENERAL CASE 101 6.2.2 HADAMARD COMPONENTS 102
6.2.3 THE POLYNOMIAL TIME CASE 107 6.2.4 THE PROOF OF THEOREM 6.1 107
6.3 TECHNICAL PRELIMINARIES 109 6.3.1 GENERAL PRINCIPLES FOR
CONGRUENTIAL PARTITION FUNCTIONS 109 6.3.2 EDGE PRODUCTS 110 6.3.3 BASIC
COMPLEXITY RESULTS FOR CONGRUENTIAL PARTITION FUNCTIONS . . . . 110 7
CONNECTED HERMITIAN MATRICES 111 7.1 SOME TECHNICAL PRELIMINARIES 112
7.2 THE NON-BIPARTITE CASE . 114 7.2.1 SATISFYING SHAPE CONDITIONS (CI)
AND (DL) 118 7.2.2 THE REMAINING CONDITIONS (C2), (D2) AND (D3) 124
7.2.3 FINISHING THE PROOF OF LEMMA 7.1 131 7.3 THE PROOF OF THE
BIPARTITE CASE . 132 7.3.1 SATISFYING THE BIPARTITE SHAPE CONDITIONS
(B-CL) AND (B-DL) ... 139 7.3.2 THE REMAINING CONDITIONS (B-C2), (B-D2)
AND (B-D3) 146 7.3.3 FINISHING THE PROOF OF LEMMA 7.2 153 8 HADAMARD
COMPONENTS 155 8.1 BOUNDING THE MAXIMUM DEGREE 156 8.2 NON-BIPARTITE
HADAMARD COMPONENTS 158 8.2.1 THE GROUP CONDITION (GC) 159 8.2.2 THE
REPRESENTATION CONDITIONS (RL) THROUGH (R5) 161 8.2.3 THE AFFINITY
CONDITION (AF) 164 8.2.4 THE PROOF OF THE NON-BIPARTITE CASE LEMMA 8.1
173 VI 8.3 BIPARTITE HADAMARD COMPONENTS 173 8.3.1 THE GROUP CONDITION
(GC) 173 8.3.2 THE REPRESENTATION ACCORDING TO (B-RL) THROUGH (B-R5) 176
8.3.3 THE AFFINITY CONDITION (B-AF) 180 8.3.4 THE PROOF OF THE BIPARTITE
CASE LEMMA 8.2 189 9 POLYNOMIAL TIME COMPUTABLE PARTITION FUNCTIONS 191
9.1 A POLYNOMIAL TIME COMPUTABLE PROBLEM 192 9.1.1 SOLVING EVAL(G) FOR Q
A POWER OF AN ODD PRIME 193 9.1.2 SOLVING EVAL(Q) FOR Q A POWER OF 2 196
9.2 COMPUTING PARTITION FUNCTIONS * THE NON-BIPARTITE CASE 200 9.2.1 THE
STRUCTURE OF THE MAPPINGS P C 200 9.2.2 THE FINAL REDUCTION 203 9.3
COMPUTING PARTITION FUNCTIONS * THE BIPARTITE CASE 210 9.3.1 THE
STRUCTURE OF THE MAPPINGS P CJROW AND P CICO I 210 9.3.2 THE PROOF OF
LEMMA 9.2 211 10 EPILOGUE 221 10.1 IMPLICATIONS ON POLYNOMIAL TIME
COMPUTABILITY 221 10.2 OPEN QUESTIONS 223 A SOME MATHEMATICAL BACKGROUND
225 A.L VANDERMONDE DETERMINANTS 225 A.2 IMPORTANT FACTS FROM GROUP
THEORY 225 A.2.1 FOURIER ANALYSIS OF ABELIAN GROUPS 226 BIBLIOGRAPHY 231
VLL
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| any_adam_object | 1 |
| author | Thurley, Marc 1981- |
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| dewey-full | 511.352 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.352 |
| dewey-search | 511.352 |
| dewey-sort | 3511.352 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| format | Thesis Book |
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| indexdate | 2024-07-09T22:10:30Z |
| institution | BVB |
| language | English |
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| physical | VII S., S. 9 - 236 graph. Darst. |
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| spelling | Thurley, Marc 1981- Verfasser (DE-588)140655131 aut The complexity of partition functions Marc Thurley 2009 VII S., S. 9 - 236 graph. Darst. txt rdacontent n rdamedia nc rdacarrier Berlin, Humboldt-Univ., Diss., 2009 Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Partitionsfunktion (DE-588)4319801-6 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Partitionsfunktion (DE-588)4319801-6 s Berechnungskomplexität (DE-588)4134751-1 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018949401&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Thurley, Marc 1981- The complexity of partition functions Berechnungskomplexität (DE-588)4134751-1 gnd Partitionsfunktion (DE-588)4319801-6 gnd |
| subject_GND | (DE-588)4134751-1 (DE-588)4319801-6 (DE-588)4113937-9 |
| title | The complexity of partition functions |
| title_auth | The complexity of partition functions |
| title_exact_search | The complexity of partition functions |
| title_full | The complexity of partition functions Marc Thurley |
| title_fullStr | The complexity of partition functions Marc Thurley |
| title_full_unstemmed | The complexity of partition functions Marc Thurley |
| title_short | The complexity of partition functions |
| title_sort | the complexity of partition functions |
| topic | Berechnungskomplexität (DE-588)4134751-1 gnd Partitionsfunktion (DE-588)4319801-6 gnd |
| topic_facet | Berechnungskomplexität Partitionsfunktion Hochschulschrift |
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