## Media Theory: Interdisciplinary Applied MathematicsThe 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

1 | |

4 | |

6 | |

7 | |

8 | |

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 |

311 | |

321 | |

### Other editions - View all

Media Theory: Interdisciplinary Applied Mathematics David Eppstein,Jean-Claude Falmagne,Sergei Ovchinnikov No preview available - 2009 |

Media Theory: Interdisciplinary Applied Mathematics David Eppstein,Jean-Claude Falmagne,Sergei Ovchinnikov No preview available - 2010 |

Media Theory: Interdisciplinary Applied Mathematics David Eppstein,Jean-Claude Falmagne,Sergei Ovchinnikov No preview available - 2007 |

### Common terms and phrases

ac-order adjacency list algorithm Axiom biorder bipartite Chapter closed medium concise message producing convex coordinates corresponding defined Definition denote digraph dimZ(G distinct Doignon and Falmagne drawing Eppstein equivalence classes example family F finite set follows function graph G Hasse diagram hyperplane arrangement implies induced integer lattice interval orders isometric embedding isometric subgraph isomorphic labeled lattice dimension Lemma Let G Markov chain mediatic graph medium S,T mutually reverse oriented medium pair partial cube partial order partition permutohedron planar positive contents positive tokens Problem Proof Prove random walk region graph representation representing result satisfying semicube graph semiorder sequence shortest path st-planar learning space stepwise effective strict linear order strict partial order strict weak order submedium subsets Suppose symmetric Theorem token system transition table unbounded face vacuous vector vertex vertices virtual edge weak order weak pseudoline arrangement well-graded wellgradedness wg-family zonotopal tiling