قابلیت محاسبه در تحلیل و فیزیک (چشم‌اندازهای منطق)

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. 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 This is the first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning. The authors are concerned with the computability or noncomputability of standard processes in analysis and physics. The book is self-contained and serves well as an introduction to research in this area.