Ken Palmer
Div. of Mathematical & Statistical Sciences, La Trobe University, Bundoora, Australia

Shadowing in Dynamical Systems
Theory and Applications

MATHEMATICS AND ITS APPLICATIONS Volume 501

In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.

It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that
they possess periodic orbits of long periods and that it is really true that they are chaotic.

Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

Contents
Preface. 1. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds. 2. Hyperbolic Sets of Diffeomorphisms. 3. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets. 4. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms. 5. Symbolic Dynamics Near
a Transversal Homoclinic Point of a Diffeomorphism. 6. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and asymptotic Phase. 7. Hyperbolic Sets of Ordinary Differential Equations. 8. Transversal Homoclinic Points and Hyperbolic Sets in Differential Equations. 9. Shadowing Theorems for Hyperbolic Sets of Differential Equations. 10. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations. 11. Numerical Shadowing. References.

Hardbound, ISBN 0-7923-6179-2
February 2000, 316 pp.

edited by : B. Brent Gordon / James D. Lewis /Stefan Muller-Stach
Shuji Saito / Tokyo Unstitute of Technology, Japan
Noriko Yui / Queen's University, Kingston, ON, Canada

The Arithmetic and Geometry of AlgebraicCycles
Proceedings of the NATO Advanced Study Institute held in Banff,
Alberta, Canada, June 7・9, 1998

NATO SCIENCE SERIES: C Mathematical and Physical Sciences Volume 548

The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel
褒acobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson.

The immense recent progress in algebraic cycles, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well.

This book offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods. Topics include a discussion of the arithmetic Abel-Jacobi mapping, higher Abel-Jacobi regulator maps, polylogarithms and L-series, candidate Bloch-Beilinson filtrations, applications of Chern亡imons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection, motivic
cohomology, Chow groups of singular varieties, and recent progress on the Hodge and Tate conjectures for Abelian varieties.

Contents and Contributors
Preface. Conference Programme. Conference Picture. List of participants. Authors' addresses. Cohomological Methods. Lectures on algebro-geometric Chern-Weil and Cheeger-Chern-Simons theory for vector brundles; Bloch, et al. Deligne cohomology and the geometric (co-)bar constructions; P. Gajer.
Kuga-Satake varieties and the Hodge conjecture; B. van Geemen.Hodge and Weil classes on abelian varieties; K.V. Murty. Bloch-Kato conjecture and motivic cohomology with finite coefficients; A. Suslin, V. Voevodsky. Chow Groups and Motives. Indecomposable higher Chow cycles; B.B. Gordon, J.D.
Lewis. Equivalence relations on algebraic cycles; U. Jannsen. Letter to Dick Gross on higher Abel-Jacobi maps; U. Jannsen. Finiteness of torsion in thecodimension-two Chow group: An Axiomatic Approach; A. Langer. Algebraic cycle complexes Basic Properties; S. Muller-Stach. Algebraic cycles on abelian varieties Application of abstract Fourier theory; J.P. Murre. Motives and filtrations on Chow groups, II; S. Saito. Zero cycles on singular varietis; V. Srinivas. Arithmetic Methods. Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry; M. Saito. Weight-monodromy conjecture for l-adic representations associated to modular forms: A supplement to the paper `IO'; T. Saito. Cohomology computations related to the l-adic Abel-Jacobi map module l; C. Schoen. Integral elements in
K-theory and products of modular curves; A.J. Scholl. Appendix to Scholl's article: A counterexample to a conjecture of Beilinson; R. de Jeu. Reduction of abelian varieties; A. Silverberg, Y. Zahrin. The arithmetic of certain Calabi-You varieties over number fields; N. Yui. Classical and elliptic polylogarithms and special values of L-series; D. Zagier, H. Gangl.

Hardbound, ISBN 0-7923-6193-8
March 2000, 648 pp.

Paperback, ISBN 0-7923-6194-6
March 2000, 648 pp.

Michael B. Mensky
P.N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow

Quantum Measurements and Decoherence
Models and Phenomenology

FUNDAMENTAL THEORIES OF PHYSICS Volume 110

This book is devoted to the theory of quantum measurements, an area that recently has attracted much attention because of its new applications for quantum information technology. The phenomenon of decoherence of a measured system is investigated and simple techniques for the description of
a wide class of measurements are developed. An individual continuously measured (decohering) system is presented by an effective complex Hamiltonian which supplies a phenomenological theory of gradual
decoherence.

The work, which features a clear presentation of physical processes leading to quantum measurement (decoherence) and simple mathematical formalisms, concentrates on the physical nature of quantum measurements and the behaviour of measured (open) quantum systems, but conceptual problems are
also treated. The analysis of interrelations between different approaches to quantum measurement is given. The methods developed in this volume are applicable for the description of individual continuously measured (decohering) systems, not only to a whole set of such systems.

Audience: This work will be of interest to both researchers and graduate students in the fields of quantum mechanics, metaphysics, probability theory, stochastic processes, the mathematics of physics and computational physics.

Contents
List of Figures. Preface. 1. Introduction. I: The Nature of Quantum Measurements. 2. The Physics Behind Quantum Measurements. 3. Measurement induced uncertainties. II: Continuous Quantum
Measurements: Theory. 4. Continuous Quantum Measurements: The General Notion. 5. Fuzzy Continuous Measurements: Phenomenology. 6. Restricted Path Integrals: General Features. III: Continuous Quantum
Measurements: Applications. 7. The Monitoring of Energy. 8. Monitoring by a series of observations. 9. The Quantum Zeno effect in a passive setup. IV: Conceptual Problems and Prospects. 10. RPI and the Philosophy of Quantum Mechanics. 11. Conclusion: New Prospects. A. The Formalism of Quantum Mechanics. References. Index.

Hardbound, ISBN 0-7923-6227-6
March 2000, 244 pp.

edited by : Emily Grosholz / Dept. of Philosophy, The Pennsylvania State University, USA
Herbert Breger / Leibniz-Archiv, Hannover, Germany

The Growth of Mathematical Knowledge

SYNTHESE LIBRARY Volume 289

This book draws its inspiration from Hilbert, Wittgenstein, Cavaills and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support,
complicate or question it.

The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how
progress may be understood.

Readership:
Students and scholars concerned with the history and philosophy of mathematics and the formal sciences.

Contents and Contributors
Acknowledgments. Introduction. Notes on Contributors. Part I: The Question of Empiricism. The Role of Scientific Theory and Empirical Fact in the Growth of Mathematical Knowledge. 1. Knowledge of Functions in the Growth of Mathematical Knowledge; J. Hintikka. Huygens and the Pendulum: From Device to Mathematical Relation; M.S. Mahoney. 2. An Empiricist Philosophy of Mathematics and Its Implications for the History of Mathematics; D. Gillies. The Mathematization of Chance in the Middle of the 17th Century;
I. Schneider. Mathematical Empiricism and the Mathematization of Chance: Comment on Gillies and Schneider; M. Liston. 3. The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge; E. Grosholz. Hamilton-Jacobi Methods and Weierstrassian Field Theory in the Calculus of Variations; C. Fraser. 4. On Mathematical Explanation; P. Mancosu. Mathematics and the Reelaboration of Truths; F. de Gandt. 5. Penrose and Platonism; M. Steiner. On the Mathematics of Spilt Milk; M.
Wilson. Part II: The Question of Formalism. The Role of Abstraction, Analysis, and Axiomatization in the Growth of Mathematical Knowledge. 1. The Growth of Mathematical Knowledge: An Open World View; C. Cellucci. Controversies about Numbers and Functions; D. Laugwitz. Epistemology, Ontology, and the Continuum; C. Posy. 2. Tacit Knowledge and Mathematical Progress; H. Breger. The Quadrature of Parabolic Segments 1635-1658: A Response to Herbert Breger; M.M. Muntersbjorn. Mathematical Progress:
Ariadne's Thread; M. Liston. Voir-Dire in the Case of Mathematical Progress; C. Mclarty. 3. The Nature of Progress in Mathematics: The Significance of Analogy; H. Sinaceur. Analogy and the Growth of Mathematical Knowledge; E. Knobloch. 4. Evolution of the Modes of Systematization of Mathematical
Knowledge; A. Barabashev. Geometry, the First Universal Language of Mathematics; I. Bashmakova, G.S. Smirnova. Part II: The Question of Progress. Criteria for and Characterizations of Progress in Mathematical Knowledge. 1. Mathematical Progress; P. Maddy. Some Remarks on Mathematical Progress from a Structuralist's Perspective; M.D. Resnik. 2. Scientific Progress and Changes in Hierarchies of Scientific Disciplines; V. Peckhaus. On the Progress of Mathematics; S. Demidov. Attractors of
Mathematical Progress: The Complex Dynamics of Mathematical Research; K. Mainzer. On Some Determinants of Mathematical Progress; C. Thiel.

Hardbound, ISBN 0-7923-6151-2
March 2000, 460 pp.

edited by : David Basin /Albert-Ludwigs-Universitat Freiburg, Germany
Fred D'Agostino /University of New England, Australia
Dov M. Gabbay /King's College, Dept. of Computer Science, London, UK
Sen Matthews / Logica GmbH, Frankfurt am Main, Germany
Luca Vigan / Institut fur Informatik, Albert-Ludwigs-Universitat, Freiburg, Germany

Labelled Deduction

APPLIED LOGIC SERIES Volume 17

Labelled deduction is an approach to providing frameworks for presenting and using different logics in a uniform and natural way by enriching the language of a logic with additional information of a semantic proof-theoretical nature.

Labelled deduction systems often possess attractive properties, such as modularity in the way that families of related logics are presented, parameterised proofs of metatheoretic properties, and ease of
mechanisability. It is thus not surprising that labelled deduction has been applied to problems in computer science, AI, mathematical logic, cognitive science, philosophy and computational linguistics ・for example, formalizing and reasoning about dynamic `state oriented' properties such as knowledge, belief, time, space, and resources.

Contents and Contributors
Preface. Contributing Authors. Labelled Proof Systems for Intuitionistic Provability; V. Balat, D. Galmiche. Normal Multimodal Logics with Interaction Axioms; M. Baldoni. The SAT Problem of Signed CNF Formulas; B. Beckers, et al. Discipline as Logic: Treating Labels as First Class Citizens; P. Blackburn. Labelled Abduction (I); K. Broda, D.M. Gabbay. Labelled Tableaux for Propositional Linear Time Logic over Finite Frames; S. Cerrito, M.C. Mayer. Fibred Modal Tableaux; D.M. Gabby, G. Governatori. Labelled Deduction for
the Guarded Fragment; M. Marx, et al. Semantics for Temporal Annotated Constraint Logic Programming; A. Raffaet?, T. Fr?hwirth. The Logic of Reusable Propositional Output with the Fulfilment Constraint; L. van der Torre.

Hardbound, ISBN 0-7923-6237-3