Computable number

Abstract: 1. Computing machines. 2. Definitions. Automatic machines. Computing machines. Circle and circle-free numbers. Computable sequences and numbers. 3. Examples of computing machines. 4. Abbreviated tables Further examples. 5. Enumeration of computable sequences. 6. The universal computing machine. 7. Detailed description of the universal machine. 8. Application of the diagonal process. Pagina 1 di 38 On computable numbers, with an application to the Entscheidungsproblem A. M. ...

Abstract: Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

Abstract: This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning. Among the topics dealt with are: classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The book is self-contained, and yet sufficiently detailed to provide an introduction to research in this area.

Abstract: Computable Analysis studies those functions on the real numbers and related sets which can be computed by machines such as digital computers. It connects two traditionally disjoint fields, namely Analysis/Numerical Analysis on the one hand and Computability/Computational Complexity on the other hand, combining concepts of approximation and of computation. In particular, Computable Analysis supplies an algorithmic foundation of numerical computation. While essentially all computablity models for number functions are equivalent (Church's Thesis) several non-equivalent mathematical models for real number computation are still being discussed.