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E-raamat: Computable Analysis: An Introduction

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Computable Analysis: An IntroductionSuurem pilt
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Is the exponential function computable? Are union and intersection of closed subsets of the real plane computable? Are differentiation and integration computable operators? Is zero finding for complex polynomials computable? Is the Mandelbrot set decidable? And in case of computability, what is the computational complexity? Computable analysis supplies exact definitions for these and many other similar questions and tries to solve them. - Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid basis 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.

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Springer Book Archives
1. Introduction.- 1.1 The Aim of Computable Analysis.- 1.2 Why a New
Introduction?.- 1.3 A Sketch of TTE.- 1.4 Prerequisites aud Notation.-
2.
Computability on the Cantor Space.- 2.1 Type-2 Machines and Computable String
Functions.- 2.2 Computable String Functions are Continuous.- 2.3 Standard
Representations of Sets of Continuous String Functions.- 2.4 Effective
Subsets.-
3. Naming Systems.- 3.1 Continuity and Computability Induced by
Naming Systems.- 3.2 Admissible Naming Systems.- 3.3 Constructions of New
Naming Systems.-
4. Computability on the Real Numbers.- 4.1 Various
Representations of the Real Numbers.- 4.2 Computable Real Numbers.- 4.3
Computable Real Functions.-
5. Computability on Closed, Open and Compact
Sets.- 5.1 Closed Sets and Open Sets.- 5.2 Compact Sets.-
6. Spaces of
Continuous Functions.- 6.1 Various representations.- 6.2 Computable Operators
on Functions. Sets and Numbers.- 6.3 Zero-Finding.- 6.4 Differentiation and
Integration.- 6.5 Analytic Functions.-
7. Computational Complexity.- 7.1
Complexity of Type-2 Machine Computations.- 7.2 Complexity Induced by the
Signed Digit Representation.- 7.3 The Complexity of Some Real Functions.- 7.4
Complexity on Compact Sets.-
8. Some Extensions.- 8.1 Computable Metric
Spaces.- 8.2 Degrees of Discontinuity.-
9. Other Approaches to Computable
Analysis.- 9.1 Banach/Mazur Computability.- 9.2 Grzegorczyks
Characterizations.- 9.3 The Pour-El/Richards Approach.- 9.4 Kos Approach.-
9.5 Domain Theory.- 9.6 Markovs Approach.- 9.7 The real-RAM and Related
Models.- 9.8 Comparison.- References.
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