1st Edition., 2011, XVI, 237 p. 41 illus., Hardcover
ISBN: 978-1-4419-7235-4
.Simplicial Structures in Topology provides a clear and comprehensive introduction
to the subject. Ideas are developed in the first four chapters. The fifth
chapter studies closed surfaces and gives their classification. The last
chapter of the book is devoted to homotopy groups, which are used in a
short introduction on obstruction theory. The text is more in tune with
the original development of algebraic topology as given by Henri Poincare
(singular homology is not discussed). Illustrative examples throughout
and extensive exercises at the end of each chapter for practice enhance
the text. Advanced undergraduate and beginning graduate students will benefit
from this book. Researchers and professionals interested in topology and
applications of mathematics will also find this book useful.
Content Level ≫ Graduate
Related subjects ≫ Geometry & Topology
Lecture Notes in Mathematics, Vol. 2020
1st Edition., 2011, 230 p. 1 illus., Softcover
ISBN: 978-3-642-19782-6
Due: April 2011
.We examine Levi non-degenerate tube hypersurfaces in complex linear space which are "spherical," that is, locally CR-equivalent to the real hyperquadric. Spherical hypersurfaces are characterized by the condition of the vanishing of the CR-curvature form, so such hypersurfaces are flat from the CR-geometric viewpoint. On the other hand, such hypersurfaces are also of interest from the point of view of affine geometry. Thus our treatment of spherical tube hypersurfaces in this book is two-fold: CR-geometric and affine-geometric. As the book shows, spherical tube hypersurfaces possess remarkable properties. For example, every such hypersurface is real-analytic and extends to a closed real-analytic spherical tube hypersurface in complex space. One of our main goals is to provide an explicit affine classification of closed spherical tube hypersurfaces whenever possible. In this book we offer a comprehensive exposition of the theory of spherical tube hypersurfaces, starting with the idea proposed in the pioneering work by P. Yang (1982) and ending with the new approach put forward by G. Fels and W. Kaup (2009).
Content Level ≫ Research
Keywords ≫ 32-XX - CR-geometry - affine homogeneity - spherical hypersurfaces - tube hypersurfaces
Related subjects ≫ Analysis
Series: Lecture Notes in Mathematics, Vol. 2019
Subseries: Ecole d'Ete de Probabilites de Saint-Flour
1st Edition., 2011, 126 p. 27 illus., 10 in color., Softcover
ISBN: 978-3-642-19579-2
Due: May 2011
These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors’ 2007 Springer monograph “Random Fields and Geometry.” While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.
Content Level ≫ Research
Keywords ≫ - Differential topology - Gaussian extrema - Gaussian processes - Random fields - Stochastic geometry
Related subjects ≫ Geometry & Topology - Statistical Theory and Methods
1 Introduction.- 2 Gaussian Processes.- 3 Some Geometry and Some Topology.- 4 The Gaussian Kinematic Formula.- 5 On Applications: Topological Inference.- 6 Algebraic Topology of Excursion Sets: A New Challenge
Series: Frontiers in Mathematics, 9
1st Edition., 2011, 150 p., Softcover
ISBN: 978-3-0348-0118-8
This book presents new accurate and efficient exponentially convergent methods for abstract differential equations with unbounded operator coefficients in Banach space. These methods are highly relevant for the practical scientific computing since the equations under consideration can be seen as the meta-models of systems of ordinary differential equations (ODE) as well as the partial differential equations (PDEs) describing various applied problems. The framework of functional analysis allows one to obtain very general but at the same time transparent algorithms and mathematical results which can be then applied to the mathematical models of the real world. The problem class includes initial value problems (IVP) for the first order differential equations with constant and variable unbounded operator coefficient in a Banach space (the heat equation is a simple example), boundary value problem for the second order elliptic differential equation with an operator coefficient (e.g. the Laplace equation), IVPs for the second order strongly damped differential equation as well as exponentially convergent methods to IVPs for the first order nonlinear differential equation with unbounded operator coefficients.
Researchers and students from numerical functional analysis, engineering and other sciences will find this book provides highly efficient algorithms for numerical solution of differential equations and applied problems.
Content Level ≫ Research
Related subjects ≫ Mathematics