# Media Theory: Interdisciplinary Applied Mathematics

Springer Science & Business Media, Oct 25, 2007 - Mathematics - 328 pages
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

 Examples and Preliminaries 1 12 A Geometrical Example 4 13 The Set of Linear Orders 6 14 The Set of Partial Orders 7 15 An Isometric Subgraph of ℤƞ 8 16 Learning Spaces 10 17 A Genetic Mutations Scheme 11 18 Notation and Conventions 12
 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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