Media Theory: Interdisciplinary Applied Mathematics

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Springer Science & Business Media, Oct 25, 2007 - Mathematics - 328 pages
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The focus of this book is a mathematical structure modeling a physical or biological system that can be in any of a number of ‘states. ’ Each state is characterized by a set of binary features, and di?ers from some other nei- bor state or states by just one of those features. In some situations, what distinguishes a state S from a neighbor state T is that S has a particular f- ture that T does not have. A familiar example is a partial solution of a jigsaw puzzle, with adjoining pieces. Such a state can be transformed into another state, that is, another partial solution or the ?nal solution, just by adding a single adjoining piece. This is the ?rst example discussed in Chapter 1. In other situations, the di?erence between a state S and a neighbor state T may reside in their location in a space, as in our second example, in which in which S and T are regions located on di?erent sides of some common border. We formalize the mathematical structure as a semigroup of ‘messages’ transforming states into other states. Each of these messages is produced by the concatenation of elementary transformations called ‘tokens (of infor- tion). ’ The structure is speci?ed by two constraining axioms. One states that any state can be produced from any other state by an appropriate kind of message. The other axiom guarantees that such a production of states from other states satis?es a consistency requirement.
 

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Contents

73 Semicubes of Media
149
74 Projections of Partial Cubes
151
75 Uniqueness of Media Representations
154
76 The Isometric Dimension of a Partial Cube
158
Problems
159
Media and Integer Lattices
161
82 Defining Lattice Dimension
162
83 Lattice Dimension of Finite Partial Cubes
167

19 Historical Note and References
17
Problems
19
Basic Concepts
22
22 Axioms for a Medium
24
23 Preparatory Results
27
24 Content Families
29
25 The Effective Set and the Producing Set of a State
30
26 Orderly and Regular Returns
31
27 Embeddings Isomorphisms and Submedia
34
28 Oriented Media
36
29 The Root of an Oriented Medium
38
210 An Infinite Example
39
211 Projections
40
Problems
45
Media and Wellgraded Families
49
32 The Grading Collection
52
33 Wellgradedness and Media
54
34 Cluster Partitions and Media
57
35 An Application to Clustered Linear Orders
62
36 A General Procedure
68
Closed Media and Closed Families
72
42 Learning Spaces and Closed Media
78
43 Complete Media
80
44 Summarizing a Closed Medium
83
45 Closed Families and their Bases
86
46 Projection of a Closed Medium
94
Problems
98
WellGraded Families of Relations
101
51 Preparatory Material
102
52 Wellgradedness and the Fringes
103
53 Partial Orders
106
54 Biorders and Interval Orders
107
55 Semiorders
110
56 Almost Connected Orders
114
Problems
119
Mediatic Graphs
122
62 Media Inducing Graphs
125
63 Paired Isomorphisms of Media and Graphs
130
64 From Mediatic Graphs to Media
132
Problems
136
Media and Partial Cubes
139
72 Characterizing Partial Cubes
142
84 Lattice Dimension of Infinite Partial Cubes
171
85 Oriented Media
172
Problems
174
Hyperplane arrangements and their media
176
92 The Lattice Dimension of an Arrangement
184
93 Labeled Interval Orders
186
94 Weak Orders and Cubical Complexes
188
Problems
196
Algorithms
199
102 Input Representation
202
103 Finding Concise Messages
211
104 Recognizing Media and Partial Cubes
217
105 Recognizing Closed Media
218
106 Black Box Media
222
Problems
227
Visualization of Media
229
111 Lattice Dimension
230
112 Drawing HighDimensional Lattice Graphs
231
113 Region Graphs of Line Arrangements
234
114 Pseudoline Arrangements
238
115 Finding Zonotopal Tilings
246
116 Learning Spaces
252
Problems
260
Random Walks on Media
263
121 On Regular Markov Chains
265
122 Discrete and Continuous Stochastic Processes
271
123 Continuous Random Walks on a Medium
273
124 Asymptotic Probabilities
279
125 Random Walks and Hyperplane Arrangements
280
Problems
282
Applications
285
132 The Entailment Relation
291
133 Assessing Knowledge in a Learning Space
293
134 The Stochastic Analysis of Opinion Polls
297
135 Concluding Remarks
302
Problems
303
A Catalog of Small Mediatic Graphs
305
Glossary
309
Bibliography
311
Index
321
Copyright

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Page 311 - SN Bhatt and SS Cosmadakis. The Complexity of minimizing wire lengths in VLSI layouts. Information Processing Letters, 25:263-267, 1987.
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