Discrete mathematics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1989
|
| Ausgabe: | Revised edition |
| Schriftenreihe: | Oxford science publications
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xiv, 480 Seiten Illustrationen, Diagramme |
| ISBN: | 0198534272 0198534264 |
Internformat
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| 100 | 1 | |a Biggs, Norman |d 1941- |e Verfasser |0 (DE-588)136105580 |4 aut | |
| 245 | 1 | 0 | |a Discrete mathematics |c Norman L. Biggs, Professor of Mathematics, London School of Economics, University of London |
| 250 | |a Revised edition | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 1989 | |
| 264 | 4 | |c © 1985 | |
| 300 | |a xiv, 480 Seiten |b Illustrationen, Diagramme | ||
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| 650 | 4 | |a Informatik | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Computer science |x Mathematics | |
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Datensatz im Suchindex
| _version_ | 1804117957199003648 |
|---|---|
| adam_text | t
Contents
PART I NUMBERS AND COUNTING
1. Integers
1.1 Arithmetic 3
1.2 Ordering the integers 5
1.3 Recursive definitions 8
1.4 The principle of induction 11
1.5 Quotient and remainder 14
1.6 Divisibility 17
1.7 The greatest common divisor 18
1.8 Factorization into primes 21
1.9 Miscellaneous exercises 25
2. Functions and counting
2.1 Functions 27
2.2 Surjections, injections, bijections 29
2.3 Counting 33
2.4 The pigeonhole principle 36
2.5 Finite or infinite? 38
2.6 Miscellaneous exercises 42
3. Principles of counting
3.1 The addition principle 44
3.2 Counting sets of pairs 46
3.3 Euler s function 49
3.4 Functions, words, and selections 52
3.5 Injections as ordered selections without repetition 54
3.6 Permutations 55
3.7 Miscellaneous exercises 59
x Contents
4. Subsets and designs
4.1 Binomial numbers 62
4.2 Unordered selections with repetition 66
4.3 The binomial theorem 68
4.4 The sieve principle 71
4.5 Some arithmetical applications 74
4.6 Designs 79
4.7 t-designs 83
4.8 Miscellaneous exercises 86
5. Partition, classification, and distribution
5.1 Partitions of a set 89
5.2 Classification and equivalence relations 92 .
5.3 Distributions and the multinomial numbers 95
5.4 Partitions of a positive integer 100
5.5 Classification of permutations 101
5.6 Odd and even permutations 105
5.7 Miscellaneous exercises 110
6. Modular arithmetic
6.1 Congruences 113
6.2 lm and its arithmetic 115
6.3 Invertible elements of Zm 119
6.4 Cyclic constructions for designs 122
6.5 Latin squares 126
6.6 Miscellaneous exercises 129
PART II GRAPHS AND ALGORITHMS
7. Algorithms and their efficiency
7.1 What is an algorithm? 133
7.2 The language of programs 135
7.3 Algorithms and programs 139
7.4 Proving that an algorithm is correct 142
7.5 Efficiency of algorithms 145
7.6 Growth rates: the O notation 148
7.7 Comparison of algorithms 150
7.8 Introduction to sorting algorithms 153
7.9 Miscellaneous exercises 156
Contents xi
8. Graphs
8.1 Graphs and their representation 158
8.2 Isomorphism of graphs 161
8.3 Valency 163
8.4 Paths and cycles 165
8.5 Trees 169
8.6 Colouring the vertices of a graph 171
8.7 The greedy algorithm for vertex-colouring 173
8.8 Miscellaneous exercises 177
9. Trees, sorting, and searching
9.1 Counting the leaves on a rooted tree 180
9.2 Trees and sorting algorithms 184
9.3 Spanning trees and the MST problem 189
9.4 Depth-first search 193
9.5 Breadth-first search 197
9.6 The shortest-path problem 201
9.7 Miscellaneous exercises 203
10. Bipartite graphs and matching problems
10.1 Relations and bipartite graphs 206
10.2 Edge-colourings of graphs 208
10.3 Application of edge-colouring to latin squares 212
10.4 Matchings 215
10.5 Maximum matchings 219
10.6 Transversals for families of finite sets 222
10.7 Miscellaneous exercises 225
11. Digraphs, networks, and flows
11.1 Digraphs 228
11.2 Networks and critical paths 231
11.3 Flows and cuts 234
11.4 The max-flow min-cut theorem 237
11.5 The labelling algorithm for network flows 241
11.6 Miscellaneous exercises 245
12. Recursive techniques
12.1 Generalities about recursion 248
12.2 Linear recursion 250
xii Contents
12.3 Recursive bisection 253
12.4 Recursive optimization 255
12.5 The framework of dynamic programming 258
12.6 Examples of the dynamic programming method 261
12.7 Miscellaneous exercises 265
PART III ALGEBRAIC METHODS
13. Groups
13.1 The axioms for a group 271
13.2 Examples of groups 272
13.3 Basic algebra in groups 276
13.4 The order of a group element 278
13.5 Isomorphism of groups 280
13.6 Cyclic groups 282
13.7 Subgroups 285
13.8 Cosets and Lagrange s theorem 288
13.9 Characterization of cyclic groups 293
13.10 Miscellaneous exercises 296
14. Groups of permutations
14.1 Definitions and examples 300
14.2 Orbits and stabilizers 303
14.3 The size of an orbit 306
14.4 The number of orbits 309
14.5 Representation of groups by permutations 313
14.6 Applications to group theory 316
14.7 Miscellaneous exercises 318
15. Rings, fields, and polynomials
15.1 Rings 321
15.2 Invertible elements of a ring 323
15.3 Fields 324
15.4 Polynomials 327
15.5 The division algorithm for polynomials 330
15.6 The Euclidean algorithm for polynomials 333
15.7 Factorization of polynomials in theory 336
15.8 Factorization of polynomials in practice 338
15.9 Miscellaneous exercises 341
Contents xiii
16. Finite fields and some applications
16.1 A field with nine elements 344
16.2 The order of a finite field 346
16.3 Construction of finite fields 34g
16.4 The primitive element theorem 350
16.5 Finite fields and latin squares 354
16.6 Finite geometry and designs 357
16.7 Projective planes 361
16.8 Squares in finite fields 364
16.9 Existence of finite fields 368
16.10 Miscellaneous exercises 372
17. Error-correcting codes
17.1 Words, codes, and errors 375
17.2 Linear codes 379
17.3 Construction of linear codes 382
17.4 Correcting errors in linear codes 384
17.5 Cyclic codes 389
17.6 Classification and properties of cyclic codes 392
17.7 Miscellaneous exercises 397
18. Generating functions
18.1 Power series and their algebraic properties 399
18.2 Partial fractions 403
18.3 The binomial theorem for negative exponents 408
18.4 Generating functions 411
18.5 The homogeneous linear recursion 414
18.6 Nonhomogeneous linear recursions 418
18.7 Miscellaneous exercises 421
19. Partitions of a positive integer
19.1 Partitions and diagrams 423
19.2 Conjugate partitions 425
19.3 Partitions and generating functions 427
19.4 Generating functions for restricted partitions 431
19.5 A mysterious identity 433
19.6 The calculation of p(n) 437
19.7 Miscellaneous exercises 439
xiv Contents
20. Symmetry and counting
20.1 The cycle index of a group of permutations 441
20.2 Cyclic and dihedral symmetry 444
20.3 Symmetry in three dimensions 448
20.4 The number of inequivalent colourings 452
20.5 Sets of colourings and their generating functions 456
20.6 Polya s theorem 458
20.7 Miscellaneous exercises 462
Answers to selected exercises 465
Index 477
t
Contents
PART I NUMBERS AND COUNTING
1. Integers
1.1 Arithmetic 3
1.2 Ordering the integers 5
1.3 Recursive definitions 8
1.4 The principle of induction 11
1.5 Quotient and remainder 14
1.6 Divisibility 17
1.7 The greatest common divisor 18
1.8 Factorization into primes 21
1.9 Miscellaneous exercises 25
2. Functions and counting
2.1 Functions 27
2.2 Surjections, injections, bijections 29
2.3 Counting 33
2.4 The pigeonhole principle 36
2.5 Finite or infinite? 38
2.6 Miscellaneous exercises 42
3. Principles of counting
3.1 The addition principle 44
3.2 Counting sets of pairs 46
3.3 Euler s function 49
3.4 Functions, words, and selections 52
3.5 Injections as ordered selections without repetition 54
3.6 Permutations 55
3.7 Miscellaneous exercises 59
x Contents
4. Subsets and designs
4.1 Binomial numbers 62
4.2 Unordered selections with repetition 66
4.3 The binomial theorem 68
4.4 The sieve principle 71
4.5 Some arithmetical applications 74
4.6 Designs 79
4.7 t designs 83
4.8 Miscellaneous exercises 86
5. Partition, classification, and distribution
5.1 Partitions of a set 89
5.2 Classification and equivalence relations 92 .
5.3 Distributions and the multinomial numbers 95
5.4 Partitions of a positive integer 100
5.5 Classification of permutations 101
5.6 Odd and even permutations 105
5.7 Miscellaneous exercises 110
6. Modular arithmetic
6.1 Congruences 113
6.2 lm and its arithmetic 115
6.3 Invertible elements of Zm 119
6.4 Cyclic constructions for designs 122
6.5 Latin squares 126
6.6 Miscellaneous exercises 129
PART II GRAPHS AND ALGORITHMS
7. Algorithms and their efficiency
7.1 What is an algorithm? 133
7.2 The language of programs 135
7.3 Algorithms and programs 139
7.4 Proving that an algorithm is correct 142
7.5 Efficiency of algorithms 145
7.6 Growth rates: the O notation 148
7.7 Comparison of algorithms 150
7.8 Introduction to sorting algorithms 153
7.9 Miscellaneous exercises 156
Contents xi
8. Graphs
8.1 Graphs and their representation 158
8.2 Isomorphism of graphs 161
8.3 Valency 163
8.4 Paths and cycles 165
8.5 Trees 169
8.6 Colouring the vertices of a graph 171
8.7 The greedy algorithm for vertex colouring 173
8.8 Miscellaneous exercises 177
9. Trees, sorting, and searching
9.1 Counting the leaves on a rooted tree 180
9.2 Trees and sorting algorithms 184
9.3 Spanning trees and the MST problem 189
9.4 Depth first search 193
9.5 Breadth first search 197
9.6 The shortest path problem 201
9.7 Miscellaneous exercises 203
10. Bipartite graphs and matching problems
10.1 Relations and bipartite graphs 206
10.2 Edge colourings of graphs 208
10.3 Application of edge colouring to latin squares 212
10.4 Matchings 215
10.5 Maximum matchings 219
10.6 Transversals for families of finite sets 222
10.7 Miscellaneous exercises 225
11. Digraphs, networks, and flows
11.1 Digraphs 228
11.2 Networks and critical paths 231
11.3 Flows and cuts 234
11.4 The max flow min cut theorem 237
11.5 The labelling algorithm for network flows 241
11.6 Miscellaneous exercises 245
12. Recursive techniques
12.1 Generalities about recursion 248
12.2 Linear recursion 250
xii Contents
12.3 Recursive bisection 253
12.4 Recursive optimization 255
12.5 The framework of dynamic programming 258
12.6 Examples of the dynamic programming method 261
12.7 Miscellaneous exercises 265
PART III ALGEBRAIC METHODS
13. Groups
13.1 The axioms for a group 271
13.2 Examples of groups 272
13.3 Basic algebra in groups 276
13.4 The order of a group element 278
13.5 Isomorphism of groups 280
13.6 Cyclic groups 282
13.7 Subgroups 285
13.8 Cosets and Lagrange s theorem 288
13.9 Characterization of cyclic groups 293
13.10 Miscellaneous exercises 296
14. Groups of permutations
14.1 Definitions and examples 300
14.2 Orbits and stabilizers 303
14.3 The size of an orbit 306
14.4 The number of orbits 309
14.5 Representation of groups by permutations 313
14.6 Applications to group theory 316
14.7 Miscellaneous exercises 318
15. Rings, fields, and polynomials
15.1 Rings 321
15.2 Invertible elements of a ring 323
15.3 Fields 324
15.4 Polynomials 327
15.5 The division algorithm for polynomials 330
15.6 The Euclidean algorithm for polynomials 333
15.7 Factorization of polynomials in theory 336
15.8 Factorization of polynomials in practice 338
15.9 Miscellaneous exercises 341
Contents xiii
16. Finite fields and some applications
16.1 A field with nine elements 344
16.2 The order of a finite field 346
16.3 Construction of finite fields 34g
16.4 The primitive element theorem 350
16.5 Finite fields and latin squares 354
16.6 Finite geometry and designs 357
16.7 Projective planes 361
16.8 Squares in finite fields 364
16.9 Existence of finite fields 368
16.10 Miscellaneous exercises 372
17. Error correcting codes
17.1 Words, codes, and errors 375
17.2 Linear codes 379
17.3 Construction of linear codes 382
17.4 Correcting errors in linear codes 384
17.5 Cyclic codes 389
17.6 Classification and properties of cyclic codes 392
17.7 Miscellaneous exercises 397
18. Generating functions
18.1 Power series and their algebraic properties 399
18.2 Partial fractions 403
18.3 The binomial theorem for negative exponents 408
18.4 Generating functions 411
18.5 The homogeneous linear recursion 414
18.6 Nonhomogeneous linear recursions 418
18.7 Miscellaneous exercises 421
19. Partitions of a positive integer
19.1 Partitions and diagrams 423
19.2 Conjugate partitions 425
19.3 Partitions and generating functions 427
19.4 Generating functions for restricted partitions 431
19.5 A mysterious identity 433
19.6 The calculation of p(n) 437
19.7 Miscellaneous exercises 439
xiv Contents
20. Symmetry and counting
20.1 The cycle index of a group of permutations 441
20.2 Cyclic and dihedral symmetry 444
20.3 Symmetry in three dimensions 448
20.4 The number of inequivalent colourings 452
20.5 Sets of colourings and their generating functions 456
20.6 Polya s theorem 458
20.7 Miscellaneous exercises 462
Answers to selected exercises 465
Index 477
|
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| discipline | Informatik Mathematik |
| edition | Revised edition |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:02:39Z |
| institution | BVB |
| isbn | 0198534272 0198534264 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002297064 |
| oclc_num | 19921617 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-706 DE-83 DE-384 DE-29T DE-703 |
| owner_facet | DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-706 DE-83 DE-384 DE-29T DE-703 |
| physical | xiv, 480 Seiten Illustrationen, Diagramme |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford science publications |
| spelling | Biggs, Norman 1941- Verfasser (DE-588)136105580 aut Discrete mathematics Norman L. Biggs, Professor of Mathematics, London School of Economics, University of London Revised edition Oxford Clarendon Press 1989 © 1985 xiv, 480 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Oxford science publications Hier auch später erschienene, unveränderte Nachdrucke Combinatieleer gtt Informatik Mathematik Computer science Mathematics Diskrete Mathematik (DE-588)4129143-8 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content 2\p (DE-588)4143389-0 Aufgabensammlung gnd-content Diskrete Mathematik (DE-588)4129143-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002297064&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002297064&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Biggs, Norman 1941- Discrete mathematics Combinatieleer gtt Informatik Mathematik Computer science Mathematics Diskrete Mathematik (DE-588)4129143-8 gnd |
| subject_GND | (DE-588)4129143-8 (DE-588)4151278-9 (DE-588)4143389-0 |
| title | Discrete mathematics |
| title_auth | Discrete mathematics |
| title_exact_search | Discrete mathematics |
| title_full | Discrete mathematics Norman L. Biggs, Professor of Mathematics, London School of Economics, University of London |
| title_fullStr | Discrete mathematics Norman L. Biggs, Professor of Mathematics, London School of Economics, University of London |
| title_full_unstemmed | Discrete mathematics Norman L. Biggs, Professor of Mathematics, London School of Economics, University of London |
| title_short | Discrete mathematics |
| title_sort | discrete mathematics |
| topic | Combinatieleer gtt Informatik Mathematik Computer science Mathematics Diskrete Mathematik (DE-588)4129143-8 gnd |
| topic_facet | Combinatieleer Informatik Mathematik Computer science Mathematics Diskrete Mathematik Einführung Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002297064&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002297064&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT biggsnorman discretemathematics |
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