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Space, Time, and Mechanics
In connection with the "Philosophy of Science" research program conducted by the Deutsche Forschungsgemeinschaft a colloquium was held in Munich from 18th to 20th May 1919. This covered basic structures of physical theories, the main emphasis being on the interrelation of space, time and mechanics. The present volume contains contributions and the results of the discussions. The papers are given here in the same order of presentation as at the meeting. The development of these "basic structures of physical theories" involved diverging trends arising from different starting points in philosophy and physics. In order to obtain a clear comparison between these schools of thought, it was appropr...
Toni Sussmann
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The Mining Engineer
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Mathematics of Complexity and Dynamical Systems
Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
Applied Differential Geometry
Introduction -- Technical preliminaries: tensors, actions and functors -- Applied manifold geometry -- Applied bundle geometry -- Applied jet geometry -- Geometrical path integrals and their applications
Nonlinear Optics, Quantum Optics, and Ultrafast Phenomena with X-Rays
Nonlinear Optics, Quantum Optics, and Ultrafast Phenomena with X-Rays is an introduction to cutting-edge science that is beginning to emerge on state-of-the-art synchrotron radiation facilities and will come to flourish with the x-ray free-electron lasers currently being planned. It is intended for the use by scientists at synchrotron radiation facilities working with the combination of x-rays and lasers and those preparing for the science at x-ray free-electron lasers. In the past decade synchrotron radiation sources have experienced a tremendous increase in their brilliance and other figures of merit. This progress, driven strongly by the scientific applications, is still going on and may actually be accelerating with the advent of x-ray free-electron lasers. As a result, a confluence of x-ray and laser physics is taking place, due to the increasing importance of laser concepts, such as coherence and nonlinear optics to the x-ray community and the importance of x-ray optics to the laser-generation of ultrashort pulses of x-rays.
Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning
Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.
Nonlinear Control Systems Design 1992
This volume represents most aspects of the rich and growing field of nonlinear control. These proceedings contain 78 papers, including six plenary lectures, striking a balance between theory and applications. Subjects covered include feedback stabilization, nonlinear and adaptive control of electromechanical systems, nonholonomic systems. Generalized state space systems, algebraic computing in nonlinear systems theory, decoupling, linearization and model-matching and robust control are also covered.
Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions
A sub-Riemannian manifold ([italic capitals]M, E, G) consists of a finite-dimensional manifold [italic capital]M, a rank-two bracket generating distribution [italic capital]E on [italic capital]M, and a Riemannian metric [italic capital]G on [italic capital]E. All length-minimizing arcs on ([italic capitals]M, E, G) are either normal extremals or abnormal extremals. Normal extremals are locally optimal, i.e., every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal was recently settled by R. G. Montgomery, who exhibited a counterexample. The present work proves that regular abnormal extremals are locally optimal, and, in the case that [italic capital]E satisfies a mild additional restriction, the abnormal minimizers are ubiquitous rather than exceptional. All the topics of this research report (historical notes, examples, abnormal extremals, Hamiltonians, nonholonomic distributions, sub-Riemannian distance, the relations between minimality and extremality, regular abnormal extremals, local optimality of regular abnormal extremals, etc.) are presented in a very clear and effective way.
