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E-raamat: Discrete Structures and Their Interactions

(Dalhousie University, Halifax, Nova Scotia, Canada)
  • Formaat: 224 pages
  • Ilmumisaeg: 19-Apr-2016
  • Kirjastus: CRC Press Inc
  • ISBN-13: 9781482203790
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Discrete Structures and Their InteractionsSuurem pilt
  • Formaat: 224 pages
  • Ilmumisaeg: 19-Apr-2016
  • Kirjastus: CRC Press Inc
  • ISBN-13: 9781482203790
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"Discover the Connections between Different Structures and Fields Discrete Structures and Their Interactions highlights the connections among various discrete structures, including graphs, directed graphs, hypergraphs, partial orders, finite topologies, and simplicial complexes. It also explores their relationships to classical areas of mathematics, such as linear and multilinear algebra, analysis, probability, logic, and topology.The text introduces a number of discrete structures, such as hypergraphs, finite topologies, preorders, simplicial complexes, and order ideals of monomials, that most graduate students in combinatorics, and even some researchers in the field, seldom experience. The author explains how these structures have important applications in many areas inside and outside of combinatorics. He also discusses how to recognize valuable research connections through the structures.Intended for graduate and upper-level undergraduate students in mathematics who have taken an initial course in discrete mathematics or graph theory, this book shows how discrete structures offer new insights into the classical fields of mathematics. It illustrates how to use discrete structures to represent the salient features and discover the underlying combinatorial principles of seemingly unrelated areas of mathematics"--

Discover the Connections between Different Structures and Fields

Discrete Structures and Their Interactions highlights the connections among various discrete structures, including graphs, directed graphs, hypergraphs, partial orders, finite topologies, and simplicial complexes. It also explores their relationships to classical areas of mathematics, such as linear and multilinear algebra, analysis, probability, logic, and topology.

The text introduces a number of discrete structures, such as hypergraphs, finite topologies, preorders, simplicial complexes, and order ideals of monomials, that most graduate students in combinatorics, and even some researchers in the field, seldom experience. The author explains how these structures have important applications in many areas inside and outside of combinatorics. He also discusses how to recognize valuable research connections through the structures.

Intended for graduate and upper-level undergraduate students in mathematics who have taken an initial course in discrete mathematics or graph theory, this book shows how discrete structures offer new insights into the classical fields of mathematics. It illustrates how to use discrete structures to represent the salient features and discover the underlying combinatorial principles of seemingly unrelated areas of mathematics.

Arvustused

"The book is a collection of examples, each of which shows either how discrete structures interact with each other or how discrete structures interact with other parts of mathematics. I am certain I will use some examples I found in this book when I teach combinatorics in the upcoming semester." Miklós Bóna, MAA Reviews, December 2013

List of Figures
xi
Preface xv
About the Author xvii
1 Introduction
1(4)
1.1 Sets
1(1)
1.2 Sequences
2(1)
1.3 Asymptotics
3(1)
1.4 Computational Complexity
3(2)
2 Discrete Structures - A Common Framework
5(18)
2.1 Isomorphism
9(1)
2.2 Substructures
10(1)
2.3 Properties, Parameters and Operations
11(1)
2.4 Representations and Models
12(11)
2.4.1 Geometric Models
13(1)
2.4.2 Algebraic Models
13(3)
2.4.3 Logical Models
16(1)
2.4.4 Probabilistic Models
17(6)
3 Graphs and Directed Graphs
23(42)
3.1 Graphs and Directed Graphs as Models
29(9)
3.1.1 Graph Colourings
29(2)
3.1.2 Reliability
31(3)
3.1.3 Proofs in Matrix Theory
34(4)
3.2 Graphs and Other Branches of Mathematics
38(27)
3.2.1 Graphs and Topology
38(2)
3.2.2 Graphs and Algebra
40(5)
3.2.3 Graphs and Analysis
45(8)
3.2.4 Graphs and Logic
53(2)
3.2.5 Graphs and Probability
55(10)
4 Preorders and Partial Orders
65(24)
4.1 Finite Topologies and Preorders
69(8)
4.1.1 The Correspondence
69(2)
4.1.2 Open Sets
71(1)
4.1.3 The Lattice of All Topologies
72(1)
4.1.4 Algorithmic Considerations
72(5)
4.2 Representing Preorders and Partial Orders
77(12)
4.2.1 Random Preorders and Partial Orders
77(3)
4.2.2 Graphs for Preorders
80(9)
5 Hypergraphs
89(28)
5.1 Applying Hypergraphs
89(13)
5.1.1 Hypergraphs and Graph Colourings
89(4)
5.1.2 Hypergraphs and Generalized Ramsey Theory
93(1)
5.1.3 Designs and Graphs
94(5)
5.1.4 Hypergraphs and Dimension of Partial Orders
99(3)
5.2 Modeling Hypergraphs
102(15)
5.2.1 Criticality and Matrix Rank
102(2)
5.2.2 Criticality and Multilinear Algebra
104(2)
5.2.3 Finite Geometries and Orthogonality
106(3)
5.2.4 Designs from Codes
109(8)
6 Complexes and Multicomplexes
117(46)
6.1 Representations of Complexes and Multicomplexes
126(11)
6.1.1 Topological Realizations of Complexes
126(4)
6.1.2 Connections to Commutative Algebra
130(7)
6.2 Applications of Complexes and Multicomplexes
137(26)
6.2.1 A "Complex" View of Partial Orders
137(2)
6.2.2 Order Ideals of Monomials and Graph Colourings
139(24)
7 Research Problems
163(4)
Selected Solutions 167(4)
Appendix A Set Theory 171(2)
Appendix B Matrix Theory and Linear Algebra 173(2)
Appendix C Abstract Algebra 175(2)
Appendix D Probability 177(2)
Appendix E Topology 179(2)
Appendix F Logic 181(4)
Bibliography 185(10)
Index 195
Jason I. Brown is a professor of mathematics at Dalhousie University. He received a Ph.D. from the University of Toronto and has written over 70 refereed articles. His research interests include graphs, hypergraphs, partial order, finite topologies, and simplicial complexes, with a focus on the applications of other fields of mathematics to discrete problems. His mathematical research that uncovered how the Beatles played the opening chord of "A Hard Days Night" was featured in various media, including NPR and BBC radio, Guitar Player Magazine, and the Wall Street Journal website.
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