Discrete mathematical models with applications to social, biological, and environmental problems
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Englewood Cliffs, New Jersey
Prentice-Hall
1976
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 245 | 1 | 0 | |a Discrete mathematical models |b with applications to social, biological, and environmental problems |c Fred S. Roberts |
| 264 | 1 | |a Englewood Cliffs, New Jersey |b Prentice-Hall |c 1976 | |
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| adam_text | Contents
Preface xi
Notation xviii
Introduction: The Scope of the Book 1
1: Mathematical Models 7
1.1 The Cyclical Nature of Mathematical Modelling 7
1.2 An Example: One Way Streets 8
1.3 Steps in the Mathematical Modelling Cycle 14
1.4 Types of Models 16
2: Graphs 20
2.1 Some Examples 20
2.2 Connectedness 31
2.2.1 Reaching 31
2.2.2 Distance 33
2.2.3 Joining 33
2.2.4 Connectedness Categories 34
Hi
iv Contents
2.2.5 Criteria for Connectedness 35
2.2.6 The Case of Graphs 38
2.3 Strong Components and the Vertex Basis 42
2.3.1 Vertex Bases and Communication Networks 42
2.3.2 Theorems and Proofs 46
2.3.3 Some Graph Theoretical Terminology 48
2.4 Digraphs and Matrices 52
2.4.1 The Adjacency Matrix 53
2.4.2 The Reachability Matrix 56
2.4.3 The Distance Matrix 58
3: Applications of Graphs 64
3.1 Signed Graphs and the Theory of Structural Balance 64
3.1.1 Signed Graphs 64
3.1.2 Balance in Small Groups 65
3.1.3 Proof of the Structure Theorem 70
3.2 Tournaments 81
3.3 Orientability and Vulnerability 93
3.3.1 Traffic Flow 93
3.3.2 Vulnerability 98
3.3.3 Transitive Orientability 100
3.4 Intersection Graphs 111
3.4.1 Definition of an Intersection Graph 111
3.4.2 Interval Graphs and their Applications 113
3.4.3 Characterization of Interval Graphs 121
3.4.4 Proofs of the Lemmas of Sec. 3,4.3 127
3.4.5 Other Intersection Graphs 129
3.4.6 Phasing Traffic Signals 129
3.5 Food Webs 140
3.5.1 The Dimension of Ecological Phase Space 140
3.5.2 Trophic Status 145
3.6 Garbage Trucks and Colorability 156
3.6.1 Tours of Garbage Trucks 156
3.6.2 Theorems on Colorability 158
3.6.3 The Four Color Problem 162
4: Weighted Digraphs and Pulse Processes 176
4.1 Introduction: Energy and other Applications 176
4.2 Excursis: Eigenvalues 178
Contents v
4.3 The Signed or Weighted Digraph as a Tool for
Modelling Complex Systems 186
4.4 Pulse Processes 207
4.5 Stability in Pulse Processes 277
4.5.1 Definition of Stability 217
4.5.2 Eigenvalues and Stability 219
4.5.3 Structure and Stability: Rosettes 222
4.6 Applications of the Theory of Stability 230
4.7 Proofs of Theorems 4.6 through 4.8 243
5: Markov Chains 258
5.1 Stochastic Processes and Markov Chains 258
5.2 Transition Probabilities and Transition Digraphs 263
5.3 Classification of States and Chains 274
5.4 Absorbing Chains 275
5.5 Regular Chains 289
5.5.1 Definition of Regular Chain 289
5.5.2 Fixed Point Probability Vectors 291
5.5.3 Mean First Passage 294
5.5.4 Proof of Theorem 5.8 296
5.6 Ergodic Chains 302
5.6.1 The Periodic Behavior of Ergodic Chains 302
5.6.2 Generalization from Regular Chains 306
5.7 Applications to Genetics 310
5.7.1 The Mendelian Theory 310
5.7.2 Predictions from the Mendelian Model 312
5.8 Flow Models 320
5.8.1 An Air Pollution Flow Model 320
5.8.2 A Money Flow Model 324
5.9 Mathematical Models of Learning 331
5.9.1 The Linear Model 332
5.9.2 The 1 element All or none Model 332
5.9.3 The 2 element All or none Model 336
5.9.4 The Estes Model 338
5.10 Influence and Social Power 346
5.11 Diffusion and Brownian Motion 358
vi Contents
6: n Person Games 364
6.1 Games in Characteristic Function Form 364
6.2 The Core 371
6.2.1 Effective Preference 371
6.2.2 Computation of the Core 373
6.3 Stable Sets 579
6.4 The Existence of a Nonempty Core 386
6.5 Existence and Uniqueness of Stable Sets 391
6.6 S Equivalence and the (0, 1) Normalization 400
6.6.1 Isomorphism and S equivalence 400
6.6.2 The Geometric Point of View 404
6.7 The Shapley Value 410
6.7.1 Shapley s Axioms 410
6.7.2 Examples 411
6.7.3 Simple Games 412
6.7.4 The Probabilistic Interpretation and Applications to Legislatures 414
6.7.5 Proof of the Value Formula 417
7: Group Decisionmaking 425
7.1 Social Welfare Functions 425
7.2 Arrow s Impossibility Theorem 433
7.2.1 Arrow s Axioms 433
7.2.2 Discussion 437
7.2.3 Proof of Arrow s Theorem 440
7.3 Joint Scales and Single Peakedness 447
7.4 Distance between Rankings 455
7.4.1 The Kemeny Snell Axioms 455
7.4.2 Calculating Distance 459
7.4.3 Medians and Means 462
7.4.4 Remarks 464
8: Measurement and Utility 473
8.1 Introduction 473
8.2 Relations 474
8.2.1 Definition of Relation 474
8.2.2 Properties of Binary Relations 476
8.2.3 Operations 478
8.3 The Theory of Measurement 484
Contents vii
8.4 Scale Type and the Theory of Meaningfulness 489
8.4.1 Regular Scales 489
8.4.2 Scale Type 491
8.4.3 Examples of Meaningful and Meaningless Statements 493
8.5 Examples of Fundamental Measurement I:
Ordinal Utility Functions 503
8.5.1 The Representation Theorem 503
8.5.2 The Expected Utility Hypothesis 508
8.5.3 The Uniqueness Theorem 510
8.5.4 Remarks 511
8.6 Examples of Fundamental Measurement II:
Extensive Measurement 516
8.6.1 Holder s Theorem 516
8.6.2 Uniqueness 520
8.6.3 A Comment 521
8.7 Examples of Fundamental Measurement III:
Conjoint Measurement 525
8.8 Semiorders 534
8.8.1 The Scott Suppes Theorem 534
8.8.2 Uniqueness 538
8.8.3 Interval Orders and Measurement Without Numbers 538
Author Index 549
Subject Index 551
|
| any_adam_object | 1 |
| author | Roberts, Fred S. 1943- |
| author_GND | (DE-588)13624419X |
| author_facet | Roberts, Fred S. 1943- |
| author_role | aut |
| author_sort | Roberts, Fred S. 1943- |
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| building | Verbundindex |
| bvnumber | BV002233926 |
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| dewey-full | 511/.8 |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.8 |
| dewey-search | 511/.8 |
| dewey-sort | 3511 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Soziologie Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV002233926 |
| illustrated | Illustrated |
| indexdate | 2025-02-14T17:22:40Z |
| institution | BVB |
| isbn | 013214171X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001468143 |
| oclc_num | 264953832 |
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| physical | XVI, 560 S. graph. Darst. |
| publishDate | 1976 |
| publishDateSearch | 1976 |
| publishDateSort | 1976 |
| publisher | Prentice-Hall |
| record_format | marc |
| spellingShingle | Roberts, Fred S. 1943- Discrete mathematical models with applications to social, biological, and environmental problems Wiskundige modellen gtt Mathematisches Modell Sozialwissenschaften Biology Mathematical models Social sciences Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd Mathematik (DE-588)4037944-9 gnd Biologie (DE-588)4006851-1 gnd Graphentheorie (DE-588)4113782-6 gnd Sozialwissenschaften (DE-588)4055916-6 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4037944-9 (DE-588)4006851-1 (DE-588)4113782-6 (DE-588)4055916-6 |
| title | Discrete mathematical models with applications to social, biological, and environmental problems |
| title_auth | Discrete mathematical models with applications to social, biological, and environmental problems |
| title_exact_search | Discrete mathematical models with applications to social, biological, and environmental problems |
| title_full | Discrete mathematical models with applications to social, biological, and environmental problems Fred S. Roberts |
| title_fullStr | Discrete mathematical models with applications to social, biological, and environmental problems Fred S. Roberts |
| title_full_unstemmed | Discrete mathematical models with applications to social, biological, and environmental problems Fred S. Roberts |
| title_short | Discrete mathematical models |
| title_sort | discrete mathematical models with applications to social biological and environmental problems |
| title_sub | with applications to social, biological, and environmental problems |
| topic | Wiskundige modellen gtt Mathematisches Modell Sozialwissenschaften Biology Mathematical models Social sciences Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd Mathematik (DE-588)4037944-9 gnd Biologie (DE-588)4006851-1 gnd Graphentheorie (DE-588)4113782-6 gnd Sozialwissenschaften (DE-588)4055916-6 gnd |
| topic_facet | Wiskundige modellen Mathematisches Modell Sozialwissenschaften Biology Mathematical models Social sciences Mathematical models Mathematik Biologie Graphentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001468143&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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