Thomas J. Watson IBM Research Center

Gregory. J. Chaitin

Chaitin, the inventor of algorithmic information theory, presents in this book the strongest possible version of Gödel's incompleteness theorem, using an information theoretic approach based on the size of computer programs. One half of the book is concerned with studying the halting probability of a universal computer if its program is chosen by tossing a coin. The other half is concerned with encoding the halting probability as an algebraic equation in integers, a so-called exponential diophantine equation.

Algorithmic Information Theory

C. C. Douglas

An Almost Assumptionless Convergence Theory for the Full Approximation Scheme Or Multiple Coarse Space Multigrid Algorithms

Specification and Utilization of a Transformational Grammar

Hyman Bass

Maria V. Otero-Espinar

Daniel Rockmore

Charles Tresser

The theme of the monograph is an interplay between dynamical systems and group theory. The authors formalize and study "cyclic renormalization", a phenomenon which appears naturally for some interval dynamical systems. A possibly infinite hierarchy of such renormalizations is naturally represented by a rooted tree, together with a "spherically transitive" automorphism; the infinite case corresponds to maps with an invariant Cantor set, a class of particular interest for its relevance to the description of the transition to chaos and of the Mandelbrot set. The normal subgroup structure of the automorphism group of such "spherically homogeneous" rooted trees is investigated in some detail. This work will be of interest to researchers in both dynamical systems and group theory.

Cyclic Renormalization and Automorphism Groups of Rooted Trees

Thomas J. Watson IBM Research Center

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