Media Theory: Interdisciplinary Applied Mathematics

ISBN-13: 9783540716969 / Angielski / Twarda / 2007 / 328 str.

David Eppstein; Jean-Claude Falmagne

Opis

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."

Recenzje

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)

Spis treści

Examples and Preliminaries.- Basic Concepts.- Media and Well-graded Families.- Closed Media and ?-Closed Families.- Well-Graded Families of Relations.- Mediatic Graphs.- Media and Partial Cubes.- Media and Integer Lattices.- Hyperplane arrangements and their media.- Algorithms.- Visualization of Media.- Random Walks on Media.- Applications.

Blurb

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 differs from some other neighbor state or states by just one of those feature. A simple example of a `state’ is a partial solution of a jigsaw puzzle, which can be transformed into another partial solution or into the final solution just by adding or removing a single adjoining piece. The evolution of such a system over time is considered. Such a structure is analyzed from algebraic and probabilistic (stochastic) standpoints.