Transversal theory: an account of some aspects of combinatorial mathematics
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; London
Academic Press
1971
|
| Series: | Mathematics in science and engineering
75 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 255 Seiten |
| ISBN: | 0124985505 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
1 Sets, Topological Spaces, Graphs
1.1 Sets and mappings 1
1.2 Families 5
1.3 Mapping theorems and cardinal numbers 9
1.4 Boolean atoms 14
1.5 The lemmas of Zorn and Tukey 16
1.6 Tychonoff s theorem 20
1.7 Graphs 21
Notes on Chapter 1 23
2 Hall s Theorem and the Notion of Duality
2.1 Transversals, representatives, and representing sets 24
2.2 Proofs of the fundamental theorem for finite families 27
2.3 Duality 32
Notes on Chapter 2 38
3 The Method of Elementary Constructions
3.1 Elementary constructions 39
3.2 Transversal index 40
3.3 Further extensions of Hall s theorem 44
3.4 A self dual variant of Hall s theorem 48
Notes on Chapter 3 50
4 Rado s Selection Principle
4.1 Proofs of the selection principle 52
4.2 Transfinite form of Hall s theorem 55
4.3 A theorem of Rado and Jung 59
4.4 Dilworth s decomposition theorem 61
4.5 Miscellaneous applications of the selection principle 64
Notes on Chapter 4 71
vii
viii CONTENTS
5 Variants, Refinements, and Applications of Hall s Theorem
5.1 Disjoint partial transversals 74
5.2 Strict systems of distinct representatives 78
5.3 Latin rectangles 81
5.4 Subsets with a prescribed pattern of overlaps 84
Notes on Chapter 5 88
6 Independent Transversals
6.1 Pre independence and independence 90
6.2 Rado s theorem on independent transversals 93
6.3 A characteristic property of independence structures 99
6.4 Finite independent partial transversals 100
6.5 Transversal structures and independence structures 101
6.6 Marginal elements 105
6.7 Axiomatic treatment of the rank function 107
Notes on Chapter 6 110
7 Independence Structures and Linear Structures
7.1 A hierarchy of structures 112
7.2 Bases of independence spaces 119
7.3 Totally admissible sets 124
7.4 Set theoretic models of independence structures 125
Notes on Chapter 7 128
8 The Rank Formula of Nash Williams
8.1 Sums of independence structures 130
8.2 Disjoint independent sets 134
8.3 A characterization of transversal structures 138
8.4 Symmetrized form of Rado s theorem on independent transversals 140
Notes on Chapter 8 145
9 Links of Two Finite Families
9.1 The notion of a link 147
9.2 Common representatives 148
9.3 The criterion of Ford and Fulkerson 150
9.4 Common representatives with restricted frequencies 154
9.5 An insertion theorem for common transversals 158
9.6 Harder results for a single family 161
Notes on Chapter 9 167
CONTENTS ix
10 Links of Two Arbitrary Families
10.1 The theorem of Mendelsohn and Dulmage and its interpretations 169
10.2 Systems of representatives with repetition 173
10.3 Common systems of representatives with defect 175
10.4 Common transversals of two families 176
10.5 Common transversals of maximal subfamilies 181
Notes on Chapter 10 182
11 Combinatorial Properties of Matrices
11.1 The language of matrix theory 183
11.2 Theorems of Konig, Frobenius, and Rado 187
11.3 Diagonals of doubly stochastic matrices 192
11.4 Doubly stochastic patterns 199
11.5 Existence theorems for integral matrices 204
Notes on Chapter 11 211
12 Conclusion
12.1 Current trends in transversal theory 214
12.2 Future research and open questions 218
Miscellaneous Exercises 229
Bibliography 236
Index of Symbols 247
Index of Authors 249
General Index 252
|
| any_adam_object | 1 |
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| dewey-full | 511/.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.6 |
| dewey-search | 511/.6 |
| dewey-sort | 3511 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:04:15Z |
| institution | BVB |
| isbn | 0124985505 |
| language | English |
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| physical | IX, 255 Seiten |
| publishDate | 1971 |
| publishDateSearch | 1971 |
| publishDateSort | 1971 |
| publisher | Academic Press |
| record_format | marc |
| series | Mathematics in science and engineering |
| series2 | Mathematics in science and engineering |
| spelling | Mirsky, Leonid 1918-1983 (DE-588)1104525941 aut Transversal theory an account of some aspects of combinatorial mathematics L. Mirsky New York ; London Academic Press 1971 IX, 255 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematics in science and engineering 75 Analyse combinatoire Analyse combinatoire ram combinatoire inriac Combinatorial analysis Kombinatorik (DE-588)4031824-2 gnd rswk-swf Transversalentheorie (DE-588)4294375-9 gnd rswk-swf Transversalentheorie (DE-588)4294375-9 s DE-604 Kombinatorik (DE-588)4031824-2 s Erscheint auch als Online-Ausgabe 978-1-282-29043-3 Erscheint auch als Online-Ausgabe 978-0-12-498550-6 Mathematics in science and engineering 75 (DE-604)BV000001196 75 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002363480&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mirsky, Leonid 1918-1983 Transversal theory an account of some aspects of combinatorial mathematics Mathematics in science and engineering Analyse combinatoire Analyse combinatoire ram combinatoire inriac Combinatorial analysis Kombinatorik (DE-588)4031824-2 gnd Transversalentheorie (DE-588)4294375-9 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4294375-9 |
| title | Transversal theory an account of some aspects of combinatorial mathematics |
| title_auth | Transversal theory an account of some aspects of combinatorial mathematics |
| title_exact_search | Transversal theory an account of some aspects of combinatorial mathematics |
| title_full | Transversal theory an account of some aspects of combinatorial mathematics L. Mirsky |
| title_fullStr | Transversal theory an account of some aspects of combinatorial mathematics L. Mirsky |
| title_full_unstemmed | Transversal theory an account of some aspects of combinatorial mathematics L. Mirsky |
| title_short | Transversal theory |
| title_sort | transversal theory an account of some aspects of combinatorial mathematics |
| title_sub | an account of some aspects of combinatorial mathematics |
| topic | Analyse combinatoire Analyse combinatoire ram combinatoire inriac Combinatorial analysis Kombinatorik (DE-588)4031824-2 gnd Transversalentheorie (DE-588)4294375-9 gnd |
| topic_facet | Analyse combinatoire combinatoire Combinatorial analysis Kombinatorik Transversalentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002363480&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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