Series: Algebra and Applications
2010, 188 p., Hardcover
ISBN: 978-1-84882-241-2
Due: September 29, 2010
The specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid-1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed mainly for fields of characteristic different from 2, are explored here without this restriction. In this book, a quadratic form Ķ over a field of characteristic 2 is allowed to have a big quasilinear part QL(Ķ) (defined as the restriction of Ķ to the radical of the bilinear form associated to Ķ), while in most of the literature QL(Ķ) is assumed to have dimension at most 1. Of course, in nature, quadratic forms with a big quasilinear part abound. In addition to chapters on specialization theory, generic splitting theory and their applications, the book's final chapter contains research never before published on specialization with respect to quadratic places and will provide the reader with a glimpse towards the future.
Fundamentals of Specialization Theory.- Generic Splitting Theory.- Some Applications.- Specialization with Respect to Quadratic Places.- Forms.- References.- Index
Series: Applied Mathematical Sciences, Vol. 149
2010, X, 523 p., Hardcover
ISBN: 978-1-4419-7075-6
Due: September 28, 2010
Exercises at the ends of chapters or sections
Solutions to selected exercises
Detailed illustrations
This book provides a modern investigation into the bifurcation phenomena of physical and engineering problems. Systematic methods - based on asymptotic, probabilistic, and group-theoretic standpoints - are used to examine experimental and computational data from numerous examples (soil, sand, kaolin, concrete, domes).
For mathematicians, static bifurcation theory for finite-dimensional systems, as well as its implications for practical problems, is illuminated by the numerous examples. Engineers may find this book, with its minimized mathematical formalism, to be a useful introduction to modern bifurcation theory.
This second edition strengthens the theoretical backgrounds of group representation theory and its application, uses of block-diagonalization in bifurcation analysis, and includes up-to-date topics of the bifurcation analysis of diverse materials from rectangular parallelepiped sand specimens to honeycomb cellular solids.
Introduction to Bifurcation Behavior * Critical Point and Local Behavior * Imperfection Sensitivity Laws * Critical Initial Imperfections (I) * Stochasticity of Initial Imperfections (I) * Experimentally-observed Bifurcation Diagrams * Group-theoretic Bifurcation Theory * Bifurcation Behavior of Dn-equivariant Systems * Critical Initial Imperfections (II) * Stochasticity of Initial Imperfections (II) * Description of Bifurcation Behaviors * Bifurcation of Cylindrical Sand Specimens * Echelon-mode Formation * Bifurcation of Steel Specimens * Miscellaneous Aspects of Bifurcation Phenomena * References * Index
Series: Lecture Notes in Mathematics, Vol. 2003
2010, X, 278 p., Softcover
ISBN: 978-3-642-14659-6
Due: October 2010
The Paris-Princeton Lectures in Financial Mathematics, of which this is the fourth volume, publish cutting-edge research in self-contained, expository articles from outstanding specialists - established or on the rise! The aim is to produce a series of articles that can serve as an introductory reference source for research in the field. The articles are the result of frequent exchanges between the finance and financial mathematics groups in Paris and Princeton. The present volume sets standards with articles by Areski Cousin, Monique Jeanblanc and Jean-Paul Laurent, Stephane Crepey, Olivier Gueant, Jean-Michel Lasry and Pierre-Louis Lions, David Hobson, and Peter Tankov.
Hedging CDO Tranches in a Markovian Environment.- About the Pricing Equations in Finance.- Mean Field Games and Applications.- The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices.- Pricing and Hedging in Exponential Levy Models: Review of Recent Results
Series: Lecture Notes in Mathematics, Vol. 2004
2010, X, 246 p. 20 illus., 10 in color., Softcover
ISBN: 978-3-642-14573-5
Due: September 2010
Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.
1 Introduction.- 2 Riemann-Liouville Differential and Integral Operators.- 3 Caputo's Approach.- 4 Mittag-Leffler Functions.- 5 Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations.- 6 Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results.- 7 Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases.- 8 Multi-Term Caputo Fractional Differential Equations.
Series: Undergraduate Texts in Mathematics
2010, 580 p. 278 illus., Hardcover
ISBN: 978-1-4419-7331-3
Due: October 13, 2010
Offers important geometric approach to advanced calculus Integrates text fully with 250+ illustrations Treats classical advanced calculus topics Uses 2D and 3D graphics to study maps Magnifies images to carry out local analysis Gives visual insight into the derivative Gives geometric interpretation of implicit function theorems Analyzes physical meaning of divergence and curl Presents Morse's lemma and Poincare lemma
With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse's lemma and the Poincare' lemma. The ideas behind most topics can be understood with just two or three variables. This invites geometric visualization; the book incorporates modern computational tools to give visualization real power.
Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps, such as the role of the derivative as the local linear approximation to a map and its role in the change of variables formula for multiple integrals. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books.
Advanced Calculus: A Geometric View is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry. It avoids duplicating the material of real analysis. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.
1 Starting Points.-1.1 Substitution.- Exercises.- 1.2 Work and path integrals.- Exercises.- 1.3 Polar coordinates.- Exercises.- 2 Geometry of Linear Maps.- 2.1 Maps from R2 to R2.- Exercises.- 2.2 Maps from Rn to Rn.- Exercises.- 2.3 Maps from Rn to Rp, n 6= p.- Exercises.- 3 Approximations.- 3.1 Mean-value theorems.- Exercises.- 3.2 Taylor polynomials in one variable.- Exercises.- 3.3 Taylor polynomials in several variables.- Exercises.- 4 The Derivative.- 4.1 Differentiability.- Exercises.- 4.2 Maps of the plane.- Exercises.- 4.3 Parametrized surfaces.- Exercises.- 4.4 The chain rule.- Exercises.- 5 Inverses.- 5.1 Solving equations.- Exercises.- 5.2 Coordinate Changes.- Exercises.- 5.3 The Inverse Function Theorem.- Exercises.- 6 Implicit Functions.- 6.1 A single equation.- Exercises.- 6.2 A pair of equations.- Exercises.- 6.3 The general case.- Exercises.- 7 Critical Points.- 7.1 Functions of one variable.- Exercises.- 7.2 Functions of two variables.- Exercises.- 7.3 Morsefs lemma.- Exercises.- 8 Double Integrals.- 8.1 Example: gravitational attraction.- Exercises.- 8.2 Area and Jordan content.- Exercises.- 8.3 Riemann and Darboux integrals.- Exercises.- 9 Evaluating Double Integrals.- 9.1 Iterated integrals.- Exercises.- 9.2 Improper integrals.- Exercises.- 9.3 The change of variables formula.- 9.4 Orientation.- Exercises.- 9.5 Greenfs Theorem.- Exercises.- 10 Surface Integrals.- 10.1 Measuring flux.- Exercises.- 10.2 Surface area and scalar integrals.- Exercises.- 10.3 Differential forms.- Exercises.- 11 Stokesf Theorem.- 11.1 Divergence.- Exercises.- 11.2 Circulation and Vorticity.- Exercises.- 11.3 Stokesf Theorem.- 11.4 Closed and Exact Forms.- Exercises