Media Theory

1. Front Matter

   Pages I-X

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2. Examples and Preliminaries

   Pages 1-21

3. Basic Concepts

   Pages 23-47

4. Media and Well-graded Families

   Pages 49-71

5. Closed Media and ∪-Closed Families

   Pages 73-99

6. Well-Graded Families of Relations

   Pages 101-121

7. Mediatic Graphs

   Pages 123-137

8. Media and Partial Cubes

   Pages 139-160

9. Media and Integer Lattices

   Pages 161-175

10. Hyperplane arrangements and their media

    Pages 177-198

11. Algorithms

    Pages 199-228

12. Visualization of Media

    Pages 229-262

13. Random Walks on Media

    Pages 263-284

14. Applications

    Pages 285-303

15. Back Matter

    Pages 305-328

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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.

• Artificial Intelligence
• Cognitive Science
• Combinatorics
• Educational Technology
• Mathematical Psychology
• algorithms
• visualization

From the reviews:

"The book takes its readers a long way from motivational examples at the beginning to the formal definition, algebraic, combinatorial, and geometric representations, to algorithmic problems, to several intuitive ways of visualizing media, and finally to serious applications. … The mathematician will find a nicely expounded theory with many ramifications. The theorems and proofs are clear and exhaustive, well balanced between rigor and intuition." (Reinhard Suck, Journal of Mathematical Psychology, Vol. 52, 2008)

"The book introduces a new mathematical structure called ‘medium’, 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 differs from some other neighbor state(s) by just one of those features. … The book does not require much background knowledge and therefore is easily readable. Graduate students, researchers and university media may take advantage of the ideas therein." (George Stoica, Zentralblatt MATH, Vol. 1149, 2008)

• Department of Computer Science, University of California, Irvine, Irvine, USA

  David Eppstein

• School of Sciences, Department of Cognitive Sciences, University of California, Irvine, Irvine, USA

  Jean-Claude Falmagne

• Deptartment of Mathematics, San Francisco State University, San Francisco, USA

  Sergei Ovchinnikov

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