Kanamori, Akihiro; Foreman, Matthew (Eds.)

Handbook Set Theory into the 21st Century, 4 vols.

2006, Approx. 2150 p., Hardcover
ISBN: 1-4020-4843-2
Due: September 2006

Table of contents

Akihiro Kanamori, Introduction

Volume - I

A. ZFC
Thomas Jech, Stationary Sets
Andras Hajnal and Jean Larson, Ordinary Partition Relations
Stevo Todorcevic, Ramsey Theory for Banach Spaces
Stevo Todorcevic, Coherent Sequences
Patrick Dehornoy, Elementary Embeddings and Algebra
B. Continuum
Greg Hjorth Borel Equivalence Relations
Andreas Blass, Combinatorial Invariants of the Continuum
Tomek Bartoszy´nski, Invariants of Measure and Category

Volume - II

C. Forcing
Uri Abraham, Proper Forcing
Ulrich Fuchs and Hans-Dieter Donder, Revised Countable Support It.
Sy Friedman, Class Forcing
James Cummings, Easton Extensions
Matthew Foreman, Ideals
Moti Gitik, Generalized Prikry Forcings and Singular Cardinals
D. Singular Cardinals
Uri Abraham and Menachem Magidor, Cardinal Arithmetic
Todd Eisworth, Successors of Singular Cardinals

Volume - III

E. Determinacy and Applications
Steve Jackson, Structural Consequences of AD
Paul Larson, P
Itay Neeman, Large Cardinals implies Determinacy
Peter Koellner and Hugh Woodin Determinacy implies Large Cardinals

Volume - IV

F. Fine Structure and Inner Models
Ralf-Dieter Schindler and Martin Zeman, Fine Structure Theory I
Philip Welch, Fine Structure Theory II
William Mitchell, Beginning Inner Model Theory
William Mitchell, The Covering Lemma
Ernest Schimmerling Core Models
John Steel, Inner Model Theory

Vath, Martin

Nonstandard Analysis

2006, Approx. 260 p., Hardcover.
ISBN: 3-7643-7773-9
Due: September 2006

About this textbook

This book presents an introduction into Robinson's nonstandard analysis.

Nonstandard analysis is the application of model theory in analysis. However, the reader is not expected to have any background in model theory; instead, some background in analysis, topology, or functional analysis would be useful - although the book is as much self-contained as possible and can be understood after a basic calculus course. Unlike some other texts, it does not attempt to teach elementary calculus on the basis of nonstandard analysis, but it points to some applications in more advanced analysis. Such applications can hardly be obtained by standard methods such as a deeper investigation of Hahn-Banach limits or of finitely additive measures.

Table of contents

Preface.- 1. Preliminaries.- 2. Nonstandard Models.- 3. Nonstandard Real Analysis.- 4. Enlargements and Saturated Models.- 5. Functionals, Generalized Limits, and Additive Measures.- 6. Nonstandard Topology and Functional Analysis.- 7. Miscellaneous.- Solutions to Exercises.- Bibliography.- Index.

Van der Hoeven, Joris

Transseries and Real Differential Algebra

Series: Lecture Notes in Mathematics , Vol. 1888
2006, Approx. 265 p., Softcover.
ISBN: 3-540-35590-1
Due: September 5, 2006

About this book

Transseries are formal objects constructed from an infinitely large variable x and the reals using infinite summation, exponentiation and logarithm. They are suitable for modeling "strongly monotonic" or "tame" asymptotic solutions to differential equations and find their origin in at least three different areas of mathematics: analysis, model theory and computer algebra. They play a crucial role in Ecalle's proof of Dulac's conjecture, which is closely related to Hilbert's 16th problem. The aim of the present book is to give a detailed and self-contained exposition of the theory of transseries, in the hope of making it more accessible to non-specialists.

Table of contents

Introduction.- Orderings.- Grid-based Series.- The Newton Polygon Method.- Transseries.- Operations on Transseries.- Grid-based operators.- Linear differential equations.- Algebraic Differential Equations.- The Intermediate Value Theorem.- References.- Glossary.- Index.

Osipenko, George

Dynamical Systems, Graphs, and Algorithms

Series: Lecture Notes in Mathematics , Vol. 1889
2006, XIII, 291 p., Softcover.
ISBN: 3-540-35593-6
Due: September 5, 2006

About this book

The modern theory and practice of dynamical systems requires the study of structures that fall outside the scope of traditional subjects of mathematical analysis. An important tool to investigate such complicated phenomena as chaos and strange attractors is the method of symbolic dynamics. This book describes a family of the algorithms to study global structure of systems.

By a finite covering of the phase space we construct a directed graph (symbolic image) with vertices corresponding to cells of the covering and edges corresponding to admissible transitions.

The method is used to localize the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, Lyapunov exponents and the Morse spectrum, to verify the hyperbolicity and the structural stability.

Considerable information can be obtained thus, and more techniques may be discovered in future research.

Table of contents

Introduction.- Symbolic Image.- Periodic Trajectories.- Newton's Method.- Invariant Sets.- Chain Recurrent Set.- Attractors.- Filtration.- Structural Graph.- Entropy.- Projective Space and Lyapunov Exponents.- Morse Spectrum.- Hyperbolicity and Structural Stability.- Controllability.- Invariant Manifolds.- Ikeda Mapping Dynamics.- A Dynamical System of Mathematical Biology.- Double Logistic Map.- Symbolic Image Implementation.- Bibliography.

Bunge, Marta, Funk, Jonathon

Singular Coverings of Toposes

Series: Lecture Notes in Mathematics , Vol. 1890
2006, XII, 230 p., 3 illus., Softcover.
ISBN: 3-540-36359-9
Due: August 17, 2006

About this book

This volume presents a fairly self-contained theory of certain singular coverings of toposes, including branched coverings.

This is a field that should be of interest to topologists working in knot theory, as well as also to certain categorists. An unusual feature which distinguishes this book from classical treatments of the subject is an unexpected connection with a topic from functional analysis, namely, distributions. Although primarily aimed at topos theorists, this book may also be used as a textbook for advanced graduate courses introducing topos theory with an emphasis on geometric applications.

Table of contents

Introduction.- Part I: Distributions and Complete Spreads.- 1.Lawvere Districutions on Toposes.- 2.Complete Spread Maps of Toposes.- 3.The Spread and Completeness Conditions.- Part II: An Axiomatic Theory of Complete Spreads.- 4.Completion KZ-Monads.- 5.Complete Spreads as Discrete M-fibrations.- 6. Closed and Linear KZ-Monads.- Part III: Aspects of Distributions and Complete Spreads.- 7.Lattice-Theoretic Aspects.- 8.Localic and Algebraic Aspects.- 9.Topological Aspects.- Bibliography.- Index

Brasselet, Jean-Paul; Soares Ruas, Maria Aparecida (Eds.)

Real and Complex Singularities
Sao Carlos Workshop 2004

Series: Trends in Mathematics
2006, Approx. 350 p., Hardcover.
ISBN: 3-7643-7775-5
Due: October 2006

About this book

The Sao Carlos Workshop on Real and Complex Singularities is the longest running workshop in singularities. It is held every two years and is a key international event for people working in the field. This volume contains papers presented at the eighth workshop, held at the IML, Marseille, July 19-23, 2004.

The workshop offers the opportunity to establish the state of the art and to present new trends, new ideas and new results in all of the branches of singularities - reflected by the contributions in this book: equisingularity of sets and mappings, geometry of singular complex analytic sets, singularities of mappings, characteristic classes, classification of singularities, interaction of singularity theory with some of the new ideas in algebraic geometry imported from theoretical physics, applications of singularity theory to geometry of surfaces in low dimensional euclidean spaces, to differential equations and to bifurcation theory.