Computability in Analysis and Physics

Résumé du livre

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 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