Series: Lecture Notes in Mathematics, Vol. 2009
Subseries: Fondazione C.I.M.E., Firenze
1st Edition., 2011, X, 238 p., Softcover
ISBN: 978-3-642-15944-2
Due: November 2010
Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties over arbitrary rings, in particular over non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory.
A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry.
This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties.
The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene Peter Swinnerton Dyer and Paul Vojta.
Varietes presque rationnelles, leurs points rationnels et leurs degenerescences.-
Topics in Diophantine Equations.- Diophantine Approximation and Nevanlinna
Theory.
Series: Studies in Universal Logic
1st Edition., 2011, Approx. 500 p., Softcover
ISBN: 978-3-0346-0144-3
Due: December 2010
A collection of papers from Paul Hertz to Dov Gabbay - through Tarski, Godel, Kripke - giving a general perspective about logical systems. These papers discuss questions such as the relativity and nature of logic, present tools such as consequence operators and combinations of logics, prove theorems such as translations between logics, investigate the domain of validity and application of fundamental results such as compactness and completeness. Each of these papers is presented by a specialist explaining its context, import and influence.
1. Paul Hertz (1922), ≪Uber Axiomensysteme fur beliebige Satzsysteme, Teil. 1≫ presented by Javier Legris // 2. Paul Bernays (1926), ≪Axiomatische Untersuchung des Aussagenkalkuls des Principia Mathematica≫ presented by Walter Carnielli // 3. Alfred Tarski (1928), ≪Remarques sur les notions fondamentales de la methodologie des mathematiques≫, presented by Stan Surma and Jan Zygmunt // 4. Kurt Godel (1933), ≪Eine Interpretation des intuitionistischen Aussagenkalkuls≫ presented by presented by Itala D'Ottavianao and Hercules Feitosa// 5. Louis Rougier (1941), ≪The relativity of logic≫ presented by Mathieu Marion // 6. Haskell B. Curry (1952), Lecons de logique algebrique (Translation of extracts) presented by Johnatan Seldin // 7. Jerzy Los and Ronam Suszko (1958), ≪Remarks on sentential logics≫ presented by Jan Zygmunt // 8. Saul Kripke (1963), ≪Semantical Considerations on Modal Logic≫ presented by Johan van Benthem // 9. Jean Porte (1965), Recherches sur la theorie generale des systemes formels et sur les systemes connectifs (Translation of extracts) presented by Marcel Guillaume // 10. Per Lindstrom (1969), ≪ On extensions of elementary logic ≫ presented by Jouko Vaananen // 11. Stephen L.Bloom, Donald J.Brown et Roman Suszko (1970), ≪Some theorems on abstract logics≫ presented by Ramon Jansana // 12. Dana Scott (1974), ≪Completeness and axiomatizability in many-valued logic≫ presented by Lloyd Humberstone // 13. Joseph Goguen and Rod Burstall (1984), ≪Introducing institutions≫ presented by Razvan Diaconescu // 14. Andrea Loparic and Newton da Costa (1984), ≪Paraconsistency, paracompleteness and valuations≫ presented by Jean-Yves Beziau // 15. Dov Gabbay, ≪Fibred Semantics and the Weaving of Logics, Part 1: Modal and Intuitionistic logic≫ (extracts) presented by Amilcar Sernadas and Carlos Caleiro
Series: Selected Works in Probability and Statistics
1st Edition., 2011, XXX, 570 p., Hardcover
ISBN: 978-1-4419-7244-6
Due: December 29, 2010
This book chronicles Donald Burkholder's thirty five year study of martingales, and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well known open problems, one involving the two dimensional Hilbert transform and its connection to quasiconformal mappings and the other a conjecture in the calculus of variations concerning rank-one convex and quasiconvex functions. This book includes reprints of many of Burkholder's papers, together with two commentaries on his work and its continuing impact.
On a class of stochastic approximation processes.- Sufficiency in the undominated case.- Iterates of conditional expectation operators.- On the order structure of the set of sufficient subfields.- Semi-Gaussian subspaces.- Successive conditional expectations of an integrable function.- Maximal inequalities as necessary conditions for almost everywhere convergence.- Martingale transforms. Ann. Math. Statist.- Extrapolation and interpolation of quasilinear operators on martingales.- A maximal function characterization of the class Hp.- Distribution function inequalities for the area integral.- Distribution function inequalities for martingales.- Boundary behaviour of harmonic functions in a half-space and Brownian motion.- One-sided maximal functions and Hp. J. Functional Analysis.- Boundary value estimation of the range of an analytic function.- A sharp inequality for martingale transforms.- Weak inequalities for exit times and analytic functions.- A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional.- Martingale transforms and the geometry of Banach spaces.- A nonlinear partial differential equation and the unconditional constant of the Haar system in Lp.- Boundary value problems and sharp inequalities for martingale transforms.- An elementary proof of an inequality of R. E. A. C. Paley.- Martingales and Fourier analysis in Banach spaces.- A sharp and strict Lp-inequality for stochastic integrals.- A proof of Pe?czyn´ski’s conjecture for the Haar system.- Differential subordination of harmonic functions and martingales.- Explorations in martingale theory and its applications.- Strong differential subordination and stochastic integration.- The best constant in the Davis inequality for the expectation of the martingale square function.- Joseph L. Doob.
Series: Probability and Its Applications
1st Edition., 2011, XVI, 274 p., Hardcover
ISBN: 978-3-642-16193-3
Due: December 2010
The systematic study of existence, uniqueness, and properties of solutions to stochastic differential equations in infinite dimensions arising from practical problems characterizes this volume that is intended for graduate students and for pure and applied mathematicians, physicists, engineers, professionals working with mathematical models of finance. Major methods include compactness, coercivity, monotonicity, in a variety of set-ups. The authors emphasize the fundamental work of Gikhman and Skorokhod on the existence and uniqueness of solutions to stochastic differential equations and present its extension to infinite dimension. They also generalize the work of Khasminskii on stability and stationary distributions of solutions. New results, applications, and examples of stochastic partial differential equations are included. This clear and detailed presentation gives the basics of the infinite dimensional version of the classic books of Gikhman and Skorokhod and of Khasminskii in one concise volume that covers the main topics in infinite dimensional stochastic PDE’s. By appropriate selection of material, the volume can be adapted for a 1- or 2-semester course, and can prepare the reader for research in this rapidly expanding area.
Preface.- Part I: Stochastic Differential Equations in Infinite Dimensions.- 1.Partial Differential Equations as Equations in Infinite.- 2.Stochastic Calculus.- 3.Stochastic Differential Equations.- 4.Solutions by Variational Method.- 5.Stochastic Differential Equations with Discontinuous Drift.- Part II: Stability, Boundedness, and Invariant Measures.- 6.Stability Theory for Strong and Mild Solutions.- 7.Ultimate Boundedness and Invariant Measure.- References.- Index.
Series: Universitext
1st Edition., 2011, XIV, 325 p. 34 illus., Softcover
ISBN: 978-0-85729-111-0
Due: October 2010
An extension of different lectures given by the authors, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems provides the reader with a comprehensive overview of these topics.
Starting with the simplest bifurcation problems arising for ordinary differential equations in one- and two-dimensions, this book describes several tools from the theory of infinite dimensional dynamical systems, allowing the reader to treat more complicated bifurcation problems, such as bifurcations arising in partial differential equations. Attention is restricted to the study of local bifurcations with a focus upon the center manifold reduction and the normal form theory; two methods that have been widely used during the last decades.
Through use of step-by-step examples and exercises, a number of possible applications are illustrated, and allow the less familiar reader to use this reduction method by checking some clear assumptions. Written by recognised experts in the field of center manifold and normal form theory this book provides a much-needed graduate level text on bifurcation theory, center manifolds and normal form theory. It will appeal to graduate students and researchers working in dynamical system theory.
Elementary Bifurcations.- Center Manifolds.- Normal Forms.- Reversible Bifurcations.- Applications.- Appendix.
Series: Applied and Numerical Harmonic Analysis
1st Edition., 2011, XXVI, 536 p. 42 illus., Hardcover
ISBN: 978-0-8176-4686-8
Due: December 17, 2010
Unique book in the literature
A good text to learn from, with detailed explanations of abstract material that are usually explained with the remark "clearly..." in other texts
Covers abstract material with a high degree of relevance to a wide variety of modern topics
Written for a broad audience of graduate students, pure and applied mathematicians as well as engineers
Includes exercises at the end of each chapter
The classical subject of bases in Banach spaces has taken on a new life in the modern development of applied harmonic analysis. This textbook is a self-contained introduction to the abstract theory of bases and redundant frame expansions and its use in both applied and classical harmonic analysis.
The four parts of the text take the reader from classical functional analysis and basis theory to modern time-frequency and wavelet theory.
* Part I develops the functional analysis that underlies most of the concepts presented in the later parts of the text.
* Part II presents the abstract theory of bases and frames in Banach and Hilbert spaces, including the classical topics of convergence, Schauder bases, biorthogonal systems, and unconditional bases, followed by the more recent topics of Riesz bases and frames in Hilbert spaces.
* Part III relates bases and frames to applied harmonic analysis, including sampling theory, Gabor analysis, and wavelet theory.
* Part IV deals with classical harmonic analysis and Fourier series, emphasizing the role played by bases, which is a different viewpoint from that taken in most discussions of Fourier series.
* Self-contained presentation with clear proofs is accessible to graduate students, pure and applied mathematicians, and engineers interested in the mathematical underpinnings of applications.
* Extensive exercises complement the text and provide opportunities for learning-by-doing, making the text suitable for graduate-level courses; hints for selected exercises are included at the end of the book.
* A separate solutions manual is available for instructors upon request at www.birkhauser-science.com/978-0-8176-4686-8/.
* No other text develops the ties between classical basis theory and its modern uses in applied harmonic analysis.
A Basis Theory Primer is suitable for independent study or as the basis for a graduate-level course. Instructors have several options for building a course around the text depending on the level and background of their students.
ANHA Series Preface.- Preface.- General Notation.- Part I. A Primer on
Functional Analysis .- Banach Spaces and Operator Theory.- Functional Analysis.-
Part II. Bases and Frames.- Unconditional Convergence of Series in Banach
and Hilbert Spaces.- Bases in Banach Spaces.- Biorthogonality, Minimality,
and More About Bases.- Unconditional Bases in Banach Spaces.- Bessel Sequences
and Bases in Hilbert Spaces.- Frames in Hilbert Spaces.- Part III. Bases
and Frames in Applied Harmonic Analysis.- The Fourier Transform on the
Real Line.- Sampling, Weighted Exponentials, and Translations.- Gabor Bases
and Frames.- Wavelet Bases and Frames.- Part IV. Fourier Series.- Fourier
Series.- Basic Properties of Fourier Series.- Part V. Appendices.- Lebesgue
Measure and Integration.- Compact and Hilbert-Schmidt Operators.- Hints
for Exercises.- Index of Symbols.- References.- Index.