Series: Springer Monographs in Mathematics
1st Edition., 2011, XVIII, 474 p.
Hardcover, ISBN 978-1-4471-2172-5
Due: November 30, 2011
This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing.
The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research.
This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field.
Overture: Ramsey's Theorem.- The Axioms of Zermelo-Fraenkel Set Theory.- Cardinal Relations in ZF only.- The Axiom of Choice.- How to Make Two Balls from One.- Models of Set Theory with Atoms.- Twelve Cardinals and their Relations.- The Shattering Number Revisited.- Happy Families and their Relatives.- Coda: A Dual Form of Ramsey's Theorem.- The Idea of Forcing.- Martin's Axiom.- The Notion of Forcing.- Models of Finite Fragments of Set Theory.- Proving Unprovability.- Models in which AC Fails.- Combining Forcing Notions.- Models in which p = c.- Properties of Forcing Extensions.- Cogen Forcing Revisited.- Silver-Like Forcing Notions.- Miller Forcing.- Mathias Forcing.- On the Existence of Ramsey Ultrafilters.- Combinatorial Properties of Sets of Partitions.- Suite.
Series: Universitext
English Language edition of 'Musubime to Sosu' Copyright c Springer Japan 2009
1st Edition., 2012, XII, 212 p. 31 illus.
Softcover, ISBN 978-1-4471-2157-2
Due: November 30, 2011
This is a foundation for arithmetic topology - a new branch of mathematics which is focused upon the analogy between knot theory and number theory.
Starting with an informative introduction to its origins, namely Gauss, this text provides a background on knots, three manifolds and number fields. Common aspects of both knot theory and number theory, for instance knots in three manifolds versus primes in a number field, are compared throughout the book. These comparisons begin at an elementary level, slowly building up to advanced theories in later chapters. Definitions are carefully formulated and proofs are largely self-contained.
When necessary, background information is provided and theory is accompanied with a number of useful examples and illustrations, making this a useful text for both undergraduates and graduates in the field of knot theory, number theory and geometry.
Preliminaries - Fundamental Groups and Galois Groups.- Knots and Primes, 3-Manifolds and Number Rings.- Linking Numbers and Legendre Symbols.- Decompositions of Knots and Primes.- Homology Groups and Ideal Class Groups I - Genus? Theory.- Link Groups and Galois Groups with Restricted Ramification.- Milnor Invariants and Multiple Power Residue Symbols.- Alexander Modules and Iwasawa Modules.- Homology Groups and Ideal Class Groups II - Higher Order Genus Theory.- Homology Groups and Ideal Class Groups III - Asymptotic Formulas.- Torsions and the Iwasawa Main Conjecture.- Moduli Spaces of Representations of Knot and Prime Groups.- Deformations of Hyperbolic Structures and of p-adic Ordinary Modular Forms.
Series: Springer Proceedings in Mathematics, Vol. 7
1st Edition., 2011, X, 252 p. 2 illus., 1 in color.
Hardcover, ISBN 978-3-642-22367-9
Due: October 31, 2011
Selected papers submitted by participants of the international Conference “Stochastic Analysis and Applied Probability 2010” ( www.saap2010.org ) make up the basis of this volume.
The SAAP 2010 was held in Tunisia, from 7-9 October, 2010, and was organized by the “Applied Mathematics & Mathematical Physics” research unit of the preparatory institute to the military academies of Sousse (Tunisia), chaired by Mounir Zili.
The papers cover theoretical, numerical and applied aspects of stochastic processes and stochastic differential equations. The study of such topic is motivated in part by the need to model, understand, forecast and control the behavior of many natural phenomena that evolve in time in a random way. Such phenomena appear in the fields of finance, telecommunications, economics, biology, geology, demography, physics, chemistry, signal processing and modern control theory, to mention just a few.
As this book emphasizes the importance of numerical and theoretical studies of the stochastic differential equations and stochastic processes, it will be useful for a wide spectrum of researchers in applied probability, stochastic numerical and theoretical analysis and statistics, as well as for graduate students.
To make it more complete and accessible for graduate students, practitioners and researchers, the editors Mounir Zili and Daria Filatova have included a survey dedicated to the basic concepts of numerical analysis of the stochastic differential equations, written by Henri Schurz.
- 1.H. Schurz: Basic Concepts of Numerical Analysis of Stochastic Differential Equations Explained by Balanced Implicit Theta Methods .- 2.C.A. Tudor: Kernel Density Estimation, Local Time and Chaos Expansion.- 3.W. Jedidi, J. Almhana, V. Choulakian, R. McGorman: General Shot Noise Processes and Functional Convergence to Stable Processes.- 4.C. El-Nouty: The Lower Classes of the Sub-Fractional Brownian Motion.- 5.M. Erraoui and Y. Ouknine: On the Bounded Variation of the Flow of Stochastic Differential Equation.- 6.A. Ayache, Q. Peng: Stochastic Volatility and Multifractional Brownian Motion.- 7.A. Gulisashvili, J. Vives: Two-sided Estimates for Distribution Densities in Models with Jumps.- 8.M. Lefebvre: Maximizing a Function of the Survival Time of a Wiener Process in an Interval.
Series: Springer Proceedings in Mathematics, Vol. 9
1st Edition., 2012, XXII, 284 p.
Hardcover, ISBN 978-1-4614-1218-2
Due: December 28, 2011
Contributions in Analytic and Algebraic Number Theory: Festschrift for S. J. Patterson is a collection of surveys and original work from experts in the fields of algebraic number theory, analytic number theory, harmonic analysis, and hyperbolic geometry. A portion of the collected contributions were developed from lectures presented at the “Patterson 60++ International Conference on the Occasion of the 60th Birthday of Samuel J. Patterson,” held at the University of Gottingen, July 27-29, 2009. Many of the included chapters were contributed by invited participants.
This volume presents and investigates the most recent developments in various key topics in analytic number theory and several related areas of mathematics. It is intended for graduate students and researchers of number theory as well as applied mathematicians interested in this broad field.
Preface.- The Density of Rational Points on a Certain Threefold (V. Blomer, J. Brudern).- Affine Gindikin-Karpelevich Formula via Uhlenbech Spaces (A. Braverman, M. Finklberg, D. Kazhdan).- Daniel’s Twists of Hooley’s Delta Function (J. Brudern).- Coefficients of the n-fold Theta Function and Weyl Group Multiple (B. Brubaker, D. Bump, S. Friedberg, J. Hoffstein).- Towards the Trace Formula for Convex-cocompact Groups (U. Bunkhe, M. Olbrich).- Double Dirichlet Series and Theta Functions (G. Chinta, S. Friedberg, J. Hoffstein).- The Patterson Measure: Classics, Variations, and Applications (M. Denker, B. O. Stratmann).- Moments for L-functions for GLr x GLr?1 (A. Diaconu, P. Garrett, D. Goldfeld).- Further Remarks on the Exponent of Convergence and the Hausdorff Dimension of the Limit Set of Kleinian Groups (M. R. Hille).- A Note on the Algebraic Growth Rate of Poinecare Series for Kleinian Groups (M. Kesselbohmer, B. O. Stratmann).- References.
Series: Lecture Notes in Mathematics, Vol. 2036
1st Edition., 2012, X, 110 p. 3 illus. in color.
Softcover, ISBN 978-3-642-23649-5
Due: November 30, 2011
The theory of random dynamical systems originated from stochastic
differential equations. It is intended to provide a framework and
techniques to describe and analyze the evolution of dynamical
systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural classification of the systems into two classes: quasi-deterministic systems, which share many
properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets.
1 Introduction.- 2 Expanding Random Maps.- 3 The RPF?theorem.- 4 Measurability, Pressure and Gibbs Condition.- 5 Fractal Structure of Conformal Expanding Random Repellers.- 6 Multifractal Analysis.- 7 Expanding in the Mean.- 8 Classical Expanding Random Systems.- 9 Real Analyticity of Pressure.