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. |
Contents
1 | |
Basic Concepts 23 | 22 |
Media and Wellgraded Families | 49 |
Closed Media and Closed Families 73 | 72 |
WellGraded Families of Relations | 101 |
Mediatic Graphs 123 | 122 |
Media and Partial Cubes | 139 |
Media and Integer Lattices | 161 |
Algorithms | 199 |
Visualization of Media | 229 |
Random Walks on Media | 263 |
Applications | 285 |
A Catalog of Small Mediatic Graphs | 305 |
311 | |
321 | |
Hyperplane arrangements and their media 177 | 176 |
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
algorithm apply arrangement assigned assume Axiom base bounded called Chapter classes closed complete concept concise message connected consider consistent construction contains contents coordinates corresponding defined Definition denote described dimension distance distinct drawing edge effective elements embedding equal equivalence example exists face fact Falmagne Figure finite follows function given graph holds hyperplane implies induced integer isometric isomorphic labeled lattice learning space Lemma length linear orders medium Note obtain orientation otherwise pair partial cube partial order partition path positive probability Problem producing projection Proof Prove pseudoline random region graph relation Remark representation representing respectively result reverse root satisfying semicubes sequence shown single strict structure subsets Suppose Take Theorem tiling token token system transformations transition tree vertex vertices walk weak order well-graded wg-family