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Mathematical Analysis : Linear and Metric Structures and Continuity
The book is divided into three parts. The first part introduces the basic ideas of linear and metric spaces, including the Jordan canonical form of matrices and the spectral theorem for self-adjoint and normal operators. The second part examines the role of general topology in the context of metric spaces and includes the notions of homotopy and degree. The third and final part is a discussion on Banach spaces of continuous functions, Hilbert spaces and the spectral theory of compact operators.
Lagrangian and Hamiltonian Methods for Nonlinear Control 2006 ; Proceedings from the 3rd IFAC Workshop, Nagoya, Japan, July 2006
A Differential-Geometric Approach for Bernstein’s Degrees-of-Freedom Problem.- Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping.- Synchronization of Networked Lagrangian Systems.- An Algorithm to Discretize One-Dimensional Distributed Port Hamiltonian Systems.- Virtual Lagrangian Construction Method for Infinitedimensional Systems with Homotopy Operators.- Direct Discrete-Time Design for Sampled-Data Hamiltonian Control Systems.- Kinematic Compensation in Port-Hamiltonian Telemanipulation.- Interconnection and Damping Assignment Passivity-Based Control of a Four-Tank System.- Towards Power-based Control Strategies for a Class of Nonlinear Mechanical Systems.- Power Shaping Control of Nonlinear Systems: A Benchmark Example.- Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations via Coordinate Changes.- Simultaneous Interconnection and Damping Assignment Passivity–Based Control: Two Practical Examples.
K-Theory : An Introduction
From the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch considered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory.
Braid Groups
Braids and braid groups have been at the heart of mathematical development over the last two decades. Braids play an important role in diverse areas of mathematics and theoretical physics. The special beauty of the theory of braids stems from their attractive geometric nature and their close relations to other fundamental geometric objects, such as knots, links, mapping class groups of surfaces, and configuration spaces. In this presentation the authors thoroughly examine various aspects of the theory of braids, starting from basic definitions and then moving to more recent results. The advanced topics cover the Burau and the Lawrence--Krammer--Bigelow representations of the braid groups, the Alexander--Conway and Jones link polynomials, connections with the representation theory of the Iwahori--Hecke algebras, and the Garside structure and orderability of the braid groups.
An Introduction to Manifolds
Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology.
Algebraic Cobordism
Algebraic Cobordism: is a theory satisfies the analogues of Quillen's theorems: the cobordism of the base field is the Lazard ring and the cobordism of a smooth variety is generated over the Lazard ring by the elements of positive degrees.
