Computability in Analysis and Physics ( Perspectives in Logic )

Publication series ：Perspectives in Logic

Author： Marian B. Pour-El; J. Ian Richards

Publisher： Cambridge University Press‎

Publication year： 2017

E-ISBN: 9781316731758

P-ISBN(Paperback): 9781107168442

Subject： O141.3 recursive function (recursion theory, the effectiveness theory)

Keyword：一般性问题

Language： ENG

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Computability in Analysis and Physics

Description

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the first publication in the Perspectives in Logic series, Pour-El and Richards present the first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning. The book focuses on the computability or noncomputability of standard processes in analysis and physics. Topics include classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The work is self-contained, and although it is intended primarily for logicians and analysts, it should also be of interest to researchers and graduate students in physics and computer science.

Chapter

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 Theorem

7. 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-computability

5. 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 Results

Introduction

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 Works

6. Normal Operators

7. Unbounded Self-Adjoint Operators

8. Converses

Addendum: Open Problems

Bibliography

Subject Index

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