Computable Analysis
An Introduction
Published on 20. November 2013
Book
Paperback/Softback
X, 288 pages
978-3-642-63102-3 (ISBN)
Description
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.
More details
Series
Edition
Softcover reprint of the original 1st ed. 2000
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Primary & secondary/elementary & high school
Graduate
Illustrations
44 s/w Abbildungen, 1 farbige Abbildung
X, 288 p. 45 illus., 1 illus. in color.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 17 mm
Weight
464 gr
ISBN-13
978-3-642-63102-3 (9783642631023)
DOI
10.1007/978-3-642-56999-9
Schweitzer Classification
Other editions
Additional editions
Book
09/2000
Springer
€53.49
Shipment within 10-15 days
Content
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 Grzegorczyk's Characterizations.- 9.3 The Pour-El/Richards Approach.- 9.4 Ko's Approach.- 9.5 Domain Theory.- 9.6 Markov's Approach.- 9.7 The real-RAM and Related Models.- 9.8 Comparison.- References.