Series: Stochastic Modelling and Applied Probability, Vol. 68
2013, XVI, 260 p. 4 illus.
Hardcover
ISBN 978-3-642-39362-4
.Combines advanced mathematical tools and theoretical analysis of stochastic numerical methods at a high level
Provides methods to reach optimal results on the accuracy of Monte Carlo simulations of stochastic processes
Contains exercises in the text and problem sets of increasing demand at the end of each chapter ?
In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Ito integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view.
The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.
Series: Universitext
Original French edition published by EDP Sciences, Les Ulis Cedex A, 2010
2013, XVI, 618 p. 110 illus.
Softcover
ISBN 978-1-4471-5495-2
Due: September 30, 2013
Translation of the popular French textbook
Provides a unified presentation of Morse theory and Floer homology that is unique in the English language
Explains all the required background on symplectic geometry, differential geometry, algebraic topology and analysis
This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold.
The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications.
Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part.
The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis.
The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.
Introduction to Part I.- Morse Functions.- Pseudo-Gradients.- The Morse Complex.- Morse Homology, Applications.- Introduction to Part II.- What You Need To Know About Symplectic Geometry.- The Arnold Conjecture and the Floer Equation.- The Maslov Index.- Linearization and Transversality.- Spaces of Trajectories.- From Floer To Morse.- Floer Homology: Invariance.- Elliptic Regularity.- Technical Lemmas.- Exercises for the Second Part.- Appendices: What You Need to Know to Read This Book.
Series: SpringerBriefs in Mathematics
2013, VIII, 112 p. 41 illus.
Softcover
ISBN 978-1-4614-8698-5
Due: September 30, 2013
Geodesic Convexity in Graphs ?is a self-contained monograph which is devoted to geodesic convexity on finite, simply connected graphs
Includes specific definitions, discussion and examples, results, proofs, exercises and open problems ?on geodesic convexity
Examines results obtained during the past 15 years, relating cardinality of minimum geodetic and hull sets
Geodesic Convexity in Graphs is devoted to the study of the geodesic convexity
on finite, simple, connected graphs. The first chapter includes the main
definitions and results on graph theory, metric graph theory and graph
path convexities. The following chapters focus exclusively on the geodesic
convexity, including motivation and background, specific definitions, discussion
and examples, results, proofs, exercises and open problems. The main and
most st?udied parameters involving geodesic convexity in graphs are both
the geodetic and the hull number which are defined as the cardinality of
minimum geodetic and hull set, respectively. This text reviews various
results, obtained during the last one and a half decade, relating these
two invariants and some others such as convexity number, Steiner number,
geodetic iteration number, Helly number, and Caratheodory number to a wide
range a contexts, including products, boundary-type vertex sets, and perfect
graph families. This monograph can serve as a supplement to a half-semester
graduate course in geodesic convexity but is primarily a guide for postgraduates
and researchers interested in topics related to metric graph theory and
graph convexity theory. ?
Content Level ≫ Research
Keywords ≫ Convex hull - Geodesic convexity - Geodetic closure - Graph convexity - Hull set - Metric graph theory
Related subjects ≫ Dynamical Systems & Differential Equations - Geometry & Topology
*
Series: Developments in Mathematics, Vol. 33
2013, XII, 124 p.
Hardcover
ISBN 978-3-319-01332-9
Due: October 31, 2013
Presents a comprehensive treatment of material previously available in journals only
Contains a number of new results and extensions of known results
States a number of open and accessible problems
Unified notation is used for a cohesive presentation
Two prisoners are told that they will be brought to a room and seated so that each can see the other. Hats will be placed on their heads; each hat is either red or green. The two prisoners must simultaneously submit a guess of their own hat color, and they both go free if at least one of them guesses correctly. While no communication is allowed once the hats have been placed, they will, however, be allowed to have a strategy session before being brought to the room. Is there a strategy ensuring their release? The answer turns out to be yes, and this is the simplest non-trivial example of a “hat problem.”
This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domain based on some knowledge of its values at other points. Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics. For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a ? 1, a), and for every such function the prediction is incorrect only on a countable set that is nowhere dense.
The monograph progresses from topics requiring fewer prerequisites to those requiring more, with most of the text being accessible to any graduate student in mathematics. The broad range of readership includes researchers, postdocs, and graduate students in the fields of set theory, mathematical logic, and combinatorics, The hope is that this book will bring together mathematicians from different areas to think about set theory via a very broad array of coordinated inference problems.
1. Introduction.- 2. The Finite Setting.- 3. The Denumerable Setting: Full Visibility.- 4. The Denumerable Setting: One-Way Visibility.- 5. Dual Hat Problems and the Uncountable.- 6. Galvin's Setting: Neutral and Anonymous Predictors.- 7. The Topological Setting.- 8. Universality of the μ-Predictor.- 9. Generalizations and Galois-Tukey Connections.- Bibliography.- Index.
Series: Applied Mathematical Sciences, Vol. 187
2013, X, 372 p. 30 illus., 29 illus. in color.
Hardcover
ISBN 978-1-4614-8258-1
Due: September 30, 2013
Aimed at a broad spectrum of scientists and engineers
Contains detailed description of practical numerical algorithms
Written by top international researchers in field
Prolate Spheroidal Wave Functions (PSWFs) are the eigenfunctions of the bandlimited operator in one dimension. As such, they play an important role in signal processing, Fourier analysis, and approximation theory. While historically the numerical evaluation of PSWFs presented serious difficulties, the developments of the last fifteen years or so made them as computationally tractable as any other class of special functions. As a result, PSWFs have been becoming a popular computational tool.
The present book serves as a complete, self-contained resource for both theory and computation. It will be of interest to a wide range of scientists and engineers, from mathematicians interested in PSWF as an analytical tool to electrical engineers designing filters and antennas.
Introduction.- Mathematical and Numerical Preliminaries.- Overview.- Analysis
of the Differential Operator.- Analysis of the Integral Operator.- Rational
Approximations of PSWFs.-Miscellaneous Properties of PSWFs.- Asymptotic
Analysis of PSWFs.- Quadrature Rules and Interpolation via PSWFs.- Numerical
Algorithms-