Series:De Gruyter Series in Nonlinear Analysis and Applications 18
24 x 17 cmxix, 453 pages3 Fig.
Language: English
Type of Publication: Monograph
Keywords: Solution Set; Fixed Point Sets; Differential Equation; Differential Inclusion; Functional Differential Inclusions; Impulsive Differential Equation; Impulsive Differential Inclusion; Semigroup; Mild Solution; Impulsive Semilinear Differential Equation; Impulsive Semilinear Differential Inclusion
This monograph gives a systematic presentation of classical and recent results obtained in the last couple of years. It comprehensively describes the methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Many of the basic techniques and results recently developed about this theory are presented, as well as the literature that is disseminated and scattered in several papers of pioneering researchers who developed the functional analytic framework of this field over the past few decades. Several examples of applications relating to initial and boundary value problems are discussed in detail.
The book is intended to advanced graduate researchers and instructors active in research areas with interests in topological properties of fixed point mappings and applications; it also aims to provide students with the necessary understanding of the subject with no deep background material needed. This monograph fills the vacuum in the literature regarding the topological structure of fixed point sets and its applications.
Series:De Gruyter Textbook
Lots of examples of real life problems
Problem sets with solution hints
Suitable for a two semester course or self-study
24 x 17 cmix, 407 pages77 Fig. 22 Tables Language: English
Type of Publication: TextbookKey
words: Stochastics; Statistics; Probability Theory; Normal Distribution;
Regression Analysis, Maximum-Likelihood; Law of Large Numbers; Central
Limit Theorem; Markov Chains; Confidence Intervals
This textbook, now in its second revised and extended edition, presents the fundamental ideas and results of both probability theory and statistics. It comprises the material of a one-year course, which is addressed to students of mathematics and to scientists with an interest in the mathematical side of stochastics.
The stochastic concepts, models and methods are motivated by examples and then developed and analysed systematically. Some measure theory is included, but this is done at an elementary level that is in accordance with the introductory character of the book. A large number of problems, now in part with solutions, offer applications and supplements to the text.
ISBN : 978-2-85629-322-5
Cours specialises 18 (2012), 191 pages
Keywords: Gauge theory, Seiberg-Witten theory, Donaldson theory, monopoles, Yang-Mills, moduli spaces, differentiable manifolds, complex surfaces
L'idee fondamentale de la theorie de jauge (en mathematiques) est d'etudier les espaces de modules des solutions de certains systemes d'equations a derivees partielles sur une variete differentiable et d'obtenir des informations sur la variete (par exemple des informations sur son type de diffeomorphisme) a partir de ces espaces de modules. En partant de cela on a obtenu les premiers resultats spectaculaires en topologie differentielle 4-dimensionnelle : itemize on a montre que la forme d'intersection d'une 4-variete differentiable orientee compacte est standard sur si cette forme est definie (positivement ou negativement) ce qui, d'apres les resultats de Freedman concernant la classification des varietes topologiques, est totalement faux dans le contexte topologique ; on a introduit et calcule explicitement les premiers invariants en dimension 4, a savoir les invariants de Donaldson, a l'aide desquels on a trouve les premieres paires exotiques (paires de 4-varietes differentiables orientees, homeomorphes mais non-diffeomorphes). itemize Le but de ce cours specialise est de donner une introduction solide a la theorie de jauge et d'en presenter en detail quelques applications importantes en topologie differentielle 4-dimensionnelle, notamment le theoreme de Donaldson sur la forme d'intersection d'une 4-variete differentiable et la conjecture de Van de Ven sur la classification topologique-differentiable des surfaces complexes. Ce cours est essentiellement dedie a la theorie de Seiberg-Witten, qui est accessible aux etudiants, mais il contient aussi des elements de la theorie de Donaldson : le groupe de jauge d'un fibre principal, les equations de Yang-Mills, les equations d'anti-dualite, des exemples d'espaces de modules de connexions de Yang-Mills. Il est accessible aux etudiants ayant suivi des cours de geometrie differentielle et de topologie algebrique, et qui ont des notions de base de l'analyse moderne (espaces de Sobolev, distributions, operateurs differentiels).
Mots-clefs : Theorie de jauge, theorie de Seiberg-Witten, theorie de Donaldson, monopoles, Yang-Mills, espaces de modules, varietes differentiables, surfaces complexes
Introduction to gauge theory
The fundamental idea of mathematical gauge theory is to study the moduli spaces of solutions of certain systems of partial differential equations on a differentiable manifold, and to obtain information about this manifold (for instance information on its diffeomorphism type) using them. This idea brought the first spectacular results in 4-dimensional differential topology: itemize One was able to show that the intersection form of a compact, oriented, differentiable 4-manifold is standard over whenever it is (positively or negatively) defined. By Freedman's results on the classification of topological 4-manifolds, the analogue statement is definitely false in the topological framework. One was able to introduce and compute explicitly the first -invariants in dimension 4, which, in turn, were used to discover the first exotic pairs (i.e. homeomorphic but not diffeomorphic pairs of differentiable 4-manifolds). itemize The goal of these lecture notes is to give a solid introduction to the mathematical gauge theory and to explain in detail some of its important applications in 4-dimensional differential topology, e.g. the Donaldson theorem concerning the intersection form of differentiable 4-manifolds, and the Van de Ven conjecture concerning the differential topological classification of complex surfaces. This book deals essentially with Seiberg-Witten theory, which is easily accessible to the students, but also contains elements of Donaldson theory: the gauge group of a principal fiber-bundle, Yang-Mills equations, ASD-equations, examples of moduli spaces of Yang-Mills equations. These lecture notes are fully accessible to the students who attended lectures on differentiable geometry and algebraic topology, and have a basic background in modern analysis (Sobolev spaces, distributions, differential operators).
Simply-laced isomonodromy systems
p. 1-68
On the Functions counting walks with small steps in the quarter plane
p. 69-114
The Structure of approximate groups
p. 115-221
On Stably free modules over affine algebras
p. 223-243
Perfectoid spaces
p. 245-313
Johns Hopkins Studies in the Mathematical Sciences
HARDBACK
9781421407944
December 2012 784 pp., 31 line drawings
The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this text useful and engaging.
This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on
* fast transforms
* parallel LU
* discrete Poisson solvers
* pseudospectra
* structured linear equation problems
* structured eigenvalue problems
* large-scale SVD methods
* polynomial eigenvalue problems
Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature?everything needed to become a matrix-savvy developer of numerical methods and software.
"A mine of insight and information and a provocation to thought; the annotated bibliographies are helpful to those wishing to explore further. One could not ask for more, and the book should be considered a resounding success."?Bulletin of the Institute of Mathematics and Its Applications, reviewing a previous edition