Series: Lecture Notes of the Unione Matematica Italiana, Vol. 9
1st Edition., 2010, X, 106 p., Softcover
ISBN: 978-3-642-14239-0
Due: August 2010
Hilbert functions play major parts in Algebraic Geometry and Commutative Algebra, and are also becoming increasingly important in Computational Algebra. They capture many useful numerical characters associated to a projective variety or to a filtered module over a local ring. Starting from the pioneering work of D.G. Northcott and J. Sally, we aim to gather together in one book a broad range of new developments in this theory by using a unifying approach which yields self-contained and easier proofs. The extension of the theory to the case of general filtrations on a module, and its application to the study of certain graded algebras which are not associated to a filtration are two of the main features of this work. The material is intended for graduate students and researchers who are interested in Commutative Algebra, particularly in the theory of the Hilbert functions and related topics.
Keywords ā Associated graded module - Filtration - Hilbert coefficients - Hilbert function - Superficial element
Related subjects ā Algebra
Series: Stochastic Modelling and Applied Probability, Vol. 64
1st Edition., 2010, XXVI, 856 p.
ISBN: 978-3-642-12057-2
Due: August 2010
In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.
Preface.- Suggestions for the Reader.- Basic Notation.- Motivation and Brief Survey.- 1. SDEs with Jumps.- 2. Exact Simulation of Solutions of SDEs.- 3. Benchmark Approach to Finance.- 4. Stochastic Expansions.- 5. Introduction to Scenario Simulation.- 6. Regular Strong Taylor Approximations.- 7. Regular Strong Ito Approximations.- 8. Jump-Adapted Strong Approximations.- 9. Estimating Discretely Observed Diffusions.- 10. Filtering.- 11. Monte Carlo Simulation of SDEs.- 12. Regular Weak Taylor Approximations.- 3. Jump-Adapted Weak Approximations.- 14. Numerical Stability.- 15. Martingale Representations and Hedge Ratios.- 16. Variance Reduction Techniques.- 17. Trees and Markov Chain Approximations.- 18. Solutions for Exercises.- Acknowledgements.- Bibliographical Notes.- References.- Author Index.- Index.
Series: Lecture Notes in Mathematics, Vol. 1999
1st Edition., 2010, X, 200 p., Softcover
ISBN: 978-3-642-13367-1
Due: August 2010
In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.
Content Level ā Research
Keywords ā - homological conjectures - tight closure - ultraproduct - uniform bounds; flatness
Related subjects ā Algebra
Series: Lecture Notes in Mathematics, Vol. 2001
1st Edition., 2010, XVI, 195 p., Softcover
ISBN: 978-3-642-14006-8
Due: August 2010
This is the first volume of a subseries of the Lecture Notes in Mathematics called Levy Matters, which will appear randomly over the next years. Each volume will describe some important topic in the theory or applications of Levy processes and pay tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world. The three expository articles of this first volume have been chosen to reflect the breadth of the area of Levy processes. The first article by Ken-iti Sato characterizes extensions of the class of selfdecomposable distributions on R^d. The second article by Thomas Duquesne discusses Hausdorff and packing measures of stable trees. The third article by Oleg Reichmann and Christoph Schwab presents numerical solutions to Kolmogoroff equations, which arise for instance in financial engineering, when Levy or additive processes model the dynamics of the risky assets.
Content Level ā Research
Keywords ā Feller processes - distribution - stochastic analysis - trees
Related subjects ā Probability Theory
Series: Universitext
2010, X, 543 p., Softcover
ISBN: 978-88-470-1783-2
Due: September 17, 2010
The purpose of this textbook is to present an array of topics in Calculus, and conceptually follow our previous effort Mathematical Analysis I.The present material is partly found, in fact, in the syllabus of the typical second lecture course in Calculus as offered in most Italian universities. While the subject matter known as `Calculus 1' is more or less standard, and concerns real functions of real variables, the topics of a course on `Calculus 2'can vary a lot, resulting in a bigger flexibility. For these reasons the Authors tried to cover a wide range of subjects, not forgetting that the number of credits the current programme specifications confers to a second Calculus course is not comparable to the amount of content gathered here. The reminders disseminated in the text make the chapters more independent from one another, allowing the reader to jump back and forth, and thus enhancing the versatility of the book. On the website: http://calvino.polito.it/canuto-tabacco/analisi 2, the interested reader may find the rigorous explanation of the results that are merely stated without proof in the book, together with useful additional material. The Authors have completely omitted the proofs whose technical aspects prevail over the fundamental notions and ideas. The large number of exercises gathered according to the main topics at the end of each chapter should help the student put his improvements to the test. The solution to all exercises is provided, and very often the procedure for solving is outlined.
Content Level ā Lower undergraduate
Keywords ā Differential Equations - Functional Analysis
Related subjects ā Analysis - Dynamical Systems & Diff. Equations - Mathematics
Series: Springer Monographs in Mathematics
1st Edition., 2010, 620 p., Hardcover
ISBN: 978-1-4419-7116-6
Due: September 27, 2010
Makes the theory of QRT maps accessible to non-specialists in algebraic geometry
May be used as an introduction to the theory of general elliptic surfaces
Applies theory to Poncelet mappings, the elliptic billiard, and difference equations from mathematical physics
The rich subject matter in this book brings in mathematics from different domains, especially from the theory of elliptic surfaces and dynamics.The material comes from the authorfs insights and understanding of a birational transformation of the plane derived from a discrete sine-Gordon equation, posing the question of determining the behavior of the discrete dynamical system defined by the iterates of the map. The aim of this book is to give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based. Readers with a good knowledge of algebraic geometry will be interested in Kodairafs theory of elliptic surfaces and deal with the collection of nontrivial applications presented here. While prerequisites for some readers will demand their knowledge of quite a bit of algebraic- and complex analytic geometry, different categories of readers will be able to become familiar with any selected interest in the book without having to make an extensive journey through the literature.
Content Level ā Graduate
Related subjects ā Algebra - Analysis - Number Theory & Combinatorics - Physics
Introduction.- The QRT Map.- Complex Projective Curves.- The QRT Surface.- Pencils of Cubic Curves in the Projective Plane.- The Action of the QRT Map on Homology Classes.- Elliptic Surfaces.- Rational Elliptic Surfaces.- Examples from the Literature.- Appendices.- Index.