Selected Topics in the Geometrical Study of Differential Equations
Author | : Niky Kamran |
Publisher | : American Mathematical Soc. |
Total Pages | : 138 |
Release | : 2002-01-01 |
Genre | : Mathematics |
ISBN | : 9780821889404 |
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Author | : Niky Kamran |
Publisher | : American Mathematical Soc. |
Total Pages | : 138 |
Release | : 2002-01-01 |
Genre | : Mathematics |
ISBN | : 9780821889404 |
Author | : V.I. Arnold |
Publisher | : Springer Science & Business Media |
Total Pages | : 366 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461210372 |
Since the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has been made partly with the help of computers. Much of this progress is represented in this revised, expanded edition, including such topics as the Feigenbaum universality of period doubling, the Zoladec solution, the Iljashenko proof, the Ecalle and Voronin theory, the Varchenko and Hovanski theorems, and the Neistadt theory. In the selection of material for this book, the author explains basic ideas and methods applicable to the study of differential equations. Special efforts were made to keep the basic ideas free from excessive technicalities. Thus the most fundamental questions are considered in great detail, while of the more special and difficult parts of the theory have the character of a survey. Consequently, the reader needs only a general mathematical knowledge to easily follow this text. It is directed to mathematicians, as well as all users of the theory of differential equations.
Author | : |
Publisher | : American Mathematical Soc. |
Total Pages | : 135 |
Release | : |
Genre | : |
ISBN | : 0821826395 |
Author | : Zhenghan Wang |
Publisher | : American Mathematical Soc. |
Total Pages | : 134 |
Release | : 2010 |
Genre | : Computers |
ISBN | : 0821849301 |
Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. In this approach, information is stored in the lowest energy states of many-anyon systems and processed by braiding non-abelian anyons. The computational answer is accessed by bringing anyons together and observing the result. Besides its theoretical esthetic appeal, the practical merit of the topological approach lies in its error-minimizing hypothetical hardware: topological phases of matter are fault-avoiding or deaf to most local noises, and unitary gates are implemented with exponential accuracy. Experimental realizations are pursued in systems such as fractional quantum Hall liquids and topological insulators. This book expands on the author's CBMS lectures on knots and topological quantum computing and is intended as a primer for mathematically inclined graduate students. With an emphasis on introducing basic notions and current research, this book gives the first coherent account of the field, covering a wide range of topics: Temperley-Lieb-Jones theory, the quantum circuit model, ribbon fusion category theory, topological quantum field theory, anyon theory, additive approximation of the Jones polynomial, anyonic quantum computing models, and mathematical models of topological phases of matter.
Author | : George Finlay Simmons |
Publisher | : |
Total Pages | : 465 |
Release | : 1972 |
Genre | : Differential equations |
ISBN | : |
Author | : Martin Markl |
Publisher | : American Mathematical Soc. |
Total Pages | : 143 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 0821889796 |
This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.
Author | : Richard A. Brualdi |
Publisher | : American Mathematical Soc. |
Total Pages | : 110 |
Release | : 2011-07-06 |
Genre | : Mathematics |
ISBN | : 0821853155 |
Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterizes certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.
Author | : Giuseppe Gaeta |
Publisher | : World Scientific |
Total Pages | : 347 |
Release | : 2005-01-25 |
Genre | : Science |
ISBN | : 9814481114 |
This proceedings volume is a collection of papers presented at the International Conference on SPT2004 focusing on symmetry, perturbation theory, and integrability.The book provides an updated overview of the recent developments in the various different fields of nonlinear dynamics, covering both theory and applications. Special emphasis is given to algebraic and geometric integrability, solutions to the N-body problem of the “choreography” type, geometry and symmetry of dynamical systems, integrable evolution equations, various different perturbation theories, and bifurcation analysis.The contributors to this volume include some of the leading scientists in the field, among them: I Anderson, D Bambusi, S Benenti, S Bolotin, M Fels, W Y Hsiang, V Matveev, A V Mikhailov, P J Olver, G Pucacco, G Sartori, M A Teixeira, S Terracini, F Verhulst and I Yehorchenko.
Author | : Mohamed Ben Ayed |
Publisher | : Cambridge University Press |
Total Pages | : 471 |
Release | : 2019-05-02 |
Genre | : Mathematics |
ISBN | : 1108431631 |
Presents the state of the art in PDEs, including the latest research and short courses accessible to graduate students.
Author | : Maria Ulan |
Publisher | : Springer Nature |
Total Pages | : 231 |
Release | : 2021-02-12 |
Genre | : Mathematics |
ISBN | : 3030632539 |
This volume presents lectures given at the Wisła 19 Summer School: Differential Geometry, Differential Equations, and Mathematical Physics, which took place from August 19 - 29th, 2019 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures were dedicated to symplectic and Poisson geometry, tractor calculus, and the integration of ordinary differential equations, and are included here as lecture notes comprising the first three chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and mathematical physics. Specific topics covered include: Parabolic geometry Geometric methods for solving PDEs in physics, mathematical biology, and mathematical finance Darcy and Euler flows of real gases Differential invariants for fluid and gas flow Differential Geometry, Differential Equations, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry is assumed.