ISBN: 978-0-470-52833-4
Hardcover
672 pages
June 2010
Fundamentals of Matrix Computations, Third Edition thoroughly details matrix computations and the accompanying theory alongside the author’s useful insights. Featuring many new and updated examples and exercises that use the MATLABR language, this revision presents the most important algorithms of numerical linear algebra and helps readers to understand how the algorithms are developed and why they work. It also includes modern coverage of Singular Value Decomposition, a streamlined discussion of the Gram-Schmidt process, and a discussion on balancing the eigenvalue problem. Practicing scientists and graduate and advanced undergraduate students will find this popular book more than meets their needs.
Series: Lecture Notes in Mathematics, Vol. 1997
2010, XVIII, 213 p., Softcover
ISBN: 978-3-642-12588-1
Due: May 2010
Intersection cohomology assigns groups which satisfy a generalized form of Poincare duality over the rationals to a stratified singular space. The present monograph introduces a method that assigns to certain classes of stratified spaces cell complexes, called intersection spaces, whose ordinary rational homology satisfies generalized Poincare duality. The cornerstone of the method is a process of spatial homology truncation, whose functoriality properties are analyzed in detail. The material on truncation is autonomous and may be of independent interest to homotopy theorists. The cohomology of intersection spaces is not isomorphic to intersection cohomology and possesses algebraic features such as perversity-internal cup-products and cohomology operations that are not generally available for intersection cohomology. A mirror-symmetric interpretation, as well as applications to string theory concerning massless D-branes arising in type IIB theory during a Calabi-Yau conifold transition, are discussed.
Iterated Truncation ; 1.7 Localization at Odd Primes; 1.8 Summary; 1.9 The Interleaf Category; 1.10 Continuity; Properties of Homology Truncation; 1.11 Fiberwise Homology Truncation; 1.12 Remarks on Perverse Links and Basic Sets Spaces; 2.1 Reflective Algebra; 2.2 The Intersection Space in the Isolated Singularities Case; 2.3 Independence of Choices of the Intersection Space Homology; 2.4 The Homotopy Type of Intersection Spaces for Interleaf Links ; 2.5 The Middle Dimension; 2.6 Cap products for Middle Perversities; 2.7 L-Theory; 2.8 Intersection Vector Bundles and K-Theory; 2.9 Beyond Isolated Singularities; 3 String Theory; 3.1 Introduction3.2 The Topology of 3-Cycles in 6-Manifolds; 3.3 The Conifold Transition; 3.4 Breakdown of the Low Energy Effective Field Theory Near a Singularity; 3.5 Massless D-Branes; 3.6 Cohomology and Massless States; 3.7 The Homology of Intersection Spaces and Massless D-Branes; 3.8 Mirror Symmetry; 3.9 An Example; References; Index
Series: Selected Works in Probability and Statistics
2010, XXVI, 496 p., Hardcover
ISBN: 978-1-4419-5820-4
Due: June 29, 2010
For almost fifty years, Richard M. Dudley has been extremely influential in the development of several areas of Probability. His work on Gaussian processes led to the understanding of the basic fact that their sample boundedness and continuity should be characterized in terms of proper measures of complexity of their parameter spaces equipped with the intrinsic covariance metric. His sufficient condition for sample continuity in terms of metric entropy is widely used and was proved by X. Fernique to be necessary for stationary Gaussian processes, whereas its more subtle versions (majorizing measures) were proved by M. Talagrand to be necessary in general.
Together with V. N. Vapnik and A. Y. Cervonenkis, R. M. Dudley is a founder of the modern theory of empirical processes in general spaces. His work on uniform central limit theorems (under bracketing entropy conditions and for Vapnik-Cervonenkis classes), greatly extends classical results that go back to A. N. Kolmogorov and M. D. Donsker, and became the starting point of a new line of research, continued in the work of Dudley and others, that developed empirical processes into one of the major tools in mathematical statistics and statistical learning theory.
As a consequence of Dudley's early work on weak convergence of probability measures on non-separable metric spaces, the Skorohod topology on the space of regulated right-continuous functions can be replaced, in the study of weak convergence of the empirical distribution function, by the supremum norm. In a further recent step Dudley replaces this norm by the stronger p-variation norms, which then allows replacing compact differentiability of many statistical functionals by Frechet differentiability in the delta method.
Richard M. Dudley has also made important contributions to mathematical statistics, the theory of weak convergence, relativistic Markov processes, differentiability of nonlinear operators and several other areas of mathematics.
Chapter 1: Convergence in Law: Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces.- Measures on non-separable metric spaces.- Distances of probability measures and random variables.- An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions.- Chapter 2: Markov Processes: Lorentz-invariant Markov processes in relativistic phase space.- A note on Lorentz-invariant Markov processes.- Asymptotics of some relativistic Markov processes.- Chapter 3: Gaussian Processes: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes.- On seminorms and probabilities, and abstract Wiener spaces.- Sample functions of the Gaussian process.- On the lower tail of Gaussian seminorms.- Chapter 4: Empirical Processes: Central limit theorems for empirical measures.- Empirical and Poisson processes on classes of sets or functions too large for central limit theorems.- Invariance principles for sums of Banach space valued random variables and empirical processes.- Universal Donsker classes and metric entropy.- Chapter 5: Nonlinear functionals and p-variation: Frechet differentiability, p-variation and uniform Donsker classes.- The order of the remainder in derivatives of composition and inverse operators for p-variation norms.- Empirical processes and p-variation.- Chapter 6: Miscellanea: Pathological topologies and random walks on Abelian groups.- Metric entropy of some classes of sets with differentiable boundaries.- In Wiener functionals as Ito integrals.- A metric entropy bound is not sufficient for learnability.- Asymptotic normality with small relative errors of posterior probabilities of half-spaces.
Series: Lecture Notes in Mathematics, Vol. 1991
2010, 400 p., Softcover
ISBN: 978-3-642-12244-6
Due: June 2010
This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on “near positivity.” The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the ?rst part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenberg-type symmetry result is included. Finally, some recent progress on the Dirichlet problem for Willmore surfaces under symmetry assumptions is discussed.
Preface.- Acknowledgements.- 1. Models of higher order.- 2. Linear Problems.- 3. Eigenvalue Problems.- 4. Kernel estimates.- 5. Positivity and lower order perturbations.- 6. Dominance of positivity in linear equations.- 7. Semilinear problems.- 8. Willmore surfaces of revolution Notations, citations and indexes
Series: Lecture Notes in Mathematics, Vol. 1996
2010, XIV, 508 p., Softcover
ISBN: 978-3-642-12470-9
Due: June 2010
Ordinary differential equations play a central role in science and have been extended to evolution equations in Banach spaces. For many applications, however, it is difficult to specify a suitable normed vector space. Shapes without a priori restrictions, for example, do not have an obvious linear structure. This book generalizes ordinary differential equations beyond the borders of vector spaces with a focus on the well-posed Cauchy problem in finite time intervals. Here are some of the examples: - Feedback evolutions of compact subsets of the Euclidean space - Birth-and-growth processes of random sets (not necessarily convex) - Semilinear evolution equations - Nonlocal parabolic differential equations - Nonlinear transport equations for Radon measures - A structured population model - Stochastic differential equations with nonlocal sample dependence and how they can be coupled in systems immediately - due to the joint framework of Mutational Analysis. Finally, the book offers new tools for modelling.
Preface Acknowledgments 0 Introduction 1 Extending ordinary differential equations to metric spaces 2 Adapting mutational equations to examples in vector space 3 Continuity of distances replaces the triangle inequality 4 Introducing distribution-like solutions to mutational equations 5 Mutational inclusions in metric spaces Tools Bibliographical Notes References Index of Notation Index