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Computability in analysis and physics / Marian B. Pour-El, University of Minnesota, J. Ian Richards, University of Minnesota.

By: Contributor(s): Material type: TextTextSeries: Perspectives in logic ; 1.Publisher: Cambridge ; New York : Cambridge University Press, [2016]Copyright date: ©2016Description: 1 online resource : illustrationsContent type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781316754917
  • 131675491X
Subject(s): Genre/Form: Additional physical formats: Print version:: COMPUTABILITY IN ANALYSIS AND PHYSICS.DDC classification:
  • 511 23
LOC classification:
  • QA9.59 .P68 2016
Online resources:
Contents:
Cover; Half-title; Series information; Title page; Copyright information; Preface to the Series: Perspectives in Mathematical Logic; Authors' Preface; Table of Contents; Major Interconnections; Introduction; Prerequisites from Logic and Analysis; Part I. Computability in Classical Analysis; Chapter 0. An Introduction to Computable Analysis; Introduction; 1. Computable Real Numbers; 2. Computable Sequences of Real Numbers; 3. Computable Functions of One or Several Real Variables; 4. Preliminary Constructs in Analysis; 5. Basic Constructs of Analysis
6. The Max-Min Theorem and the Intermediate Value Theorem7. Proof of the Effective Weierstrass Theorem; Chapter 1. Further Topics in Computable Analysis; Introduction; 1. Cn Functions, 1 ≤ n ≤ ∞; 2. Analytic Functions; 3. The Effective Modulus Lemma and Some of Its Consequences; 4. Translation Invariant Operators; Part II. The Computability Theory of Banach Spaces; Chapter 2. Computability Structures on a Banach Space; Introduction; 1. The Axioms for a Computability Structure; 2. The Classical Case: Computability in the Sense of Chapter 0; 3. Intrinsic Lp-computability
4. Intrinsic lp-computability5. The Effective Density Lemma and the Stability Lemma; 6. Two Counterexamples: Separability Versus Effective Separability and Computability on L[sup(∞)] [0, 1]; 7. Ad Hoc Computability Structures; Chapter 3. The First Main Theorem and Its Applications; Introduction; 1. Bounded Operators, Closed Unbounded Operators; 2. The First Main Theorem; 3. Simple Applications to Real Analysis; 4. Further Applications to Real Analysis; 5. Applications to Physical Theory; Part III. The Computability Theory of Eigenvalues and Eigenvectors
Chapter 4. The Second Main Theorem, the Eigenvector Theorem, and Related ResultsIntroduction; 1. Basic Notions for Unbounded Operators, Effectively Determined Operators; 2. The Second Main Theorem and Some of Its Corollaries; 3. Creation and Destruction of Eigenvalues; 4. A Non-normal Operator with a Noncomputable Eigenvalue; 5. The Eigenvector Theorem; 6. The Eigenvector Theorem, Completed; 7. Some Results for Banach Spaces; Chapter 5. Proof of the Second Main Theorem; Introduction; 1. Review of the Spectral Theorem; 2. Preliminaries; 3. Heuristics; 4. The Algorithm
5. Proof That the Algorithm Works6. Normal Operators; 7. Unbounded Self-Adjoint Operators; 8. Converses; Addendum: Open Problems; Bibliography; Subject Index
Summary: The first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning.
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eBook eBook e-Library EBSCO Mathematics Available
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Includes bibliographical references and index.

Description based on print version record.

Cover; Half-title; Series information; Title page; Copyright information; Preface to the Series: Perspectives in Mathematical Logic; Authors' Preface; Table of Contents; Major Interconnections; Introduction; Prerequisites from Logic and Analysis; Part I. Computability in Classical Analysis; Chapter 0. An Introduction to Computable Analysis; Introduction; 1. Computable Real Numbers; 2. Computable Sequences of Real Numbers; 3. Computable Functions of One or Several Real Variables; 4. Preliminary Constructs in Analysis; 5. Basic Constructs of Analysis

6. The Max-Min Theorem and the Intermediate Value Theorem7. Proof of the Effective Weierstrass Theorem; Chapter 1. Further Topics in Computable Analysis; Introduction; 1. Cn Functions, 1 ≤ n ≤ ∞; 2. Analytic Functions; 3. The Effective Modulus Lemma and Some of Its Consequences; 4. Translation Invariant Operators; Part II. The Computability Theory of Banach Spaces; Chapter 2. Computability Structures on a Banach Space; Introduction; 1. The Axioms for a Computability Structure; 2. The Classical Case: Computability in the Sense of Chapter 0; 3. Intrinsic Lp-computability

4. Intrinsic lp-computability5. The Effective Density Lemma and the Stability Lemma; 6. Two Counterexamples: Separability Versus Effective Separability and Computability on L[sup(∞)] [0, 1]; 7. Ad Hoc Computability Structures; Chapter 3. The First Main Theorem and Its Applications; Introduction; 1. Bounded Operators, Closed Unbounded Operators; 2. The First Main Theorem; 3. Simple Applications to Real Analysis; 4. Further Applications to Real Analysis; 5. Applications to Physical Theory; Part III. The Computability Theory of Eigenvalues and Eigenvectors

Chapter 4. The Second Main Theorem, the Eigenvector Theorem, and Related ResultsIntroduction; 1. Basic Notions for Unbounded Operators, Effectively Determined Operators; 2. The Second Main Theorem and Some of Its Corollaries; 3. Creation and Destruction of Eigenvalues; 4. A Non-normal Operator with a Noncomputable Eigenvalue; 5. The Eigenvector Theorem; 6. The Eigenvector Theorem, Completed; 7. Some Results for Banach Spaces; Chapter 5. Proof of the Second Main Theorem; Introduction; 1. Review of the Spectral Theorem; 2. Preliminaries; 3. Heuristics; 4. The Algorithm

5. Proof That the Algorithm Works6. Normal Operators; 7. Unbounded Self-Adjoint Operators; 8. Converses; Addendum: Open Problems; Bibliography; Subject Index

The first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning.

Master record variable field(s) change: 050, 072, 082, 650

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