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Author: J. F. Pommaret
Publisher: CRC Press
ISBN: 9780677002705
Category : Mathematics
Languages : en
Pages : 428
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Book Description
Author: J. F. Pommaret
Publisher: CRC Press
ISBN: 9780677002705
Category : Mathematics
Languages : en
Pages : 428
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Book Description
Author: Antonio Kumpera
Publisher: Presses de l'Université de Montréal
ISBN:
Category : Differential equations, Linear
Languages : en
Pages : 108
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Book Description
The main goal of these notes is the description of a non-linear complex into which the integrability (or compatibility) condition is inserted as a non-linear operator in such a way that exactness implies the integrability of the almost-structure (existence of local coordinates for the structure) or, by the introduction of parameters, the existence of a (germ of) deformation of the structure. To the non-linear complex are attached some fundamental identities and a structure equation. The non-linear complex is a finite form of the initial portion of a linear complex which is a differential graded Lie algebra. The operators in the non-linear and linear complexes are of first order.
Author: J. F. Pommaret
Publisher:
ISBN:
Category :
Languages : en
Pages : 412
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Book Description
Author: J.F. Pommaret
Publisher: Springer Science & Business Media
ISBN: 940172539X
Category : Mathematics
Languages : en
Pages : 481
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Book Description
Ordinary differential control thPory (the classical theory) studies input/output re lations defined by systems of ordinary differential equations (ODE). The various con cepts that can be introduced (controllability, observability, invertibility, etc. ) must be tested on formal objects (matrices, vector fields, etc. ) by means of formal operations (multiplication, bracket, rank, etc. ), but without appealing to the explicit integration (search for trajectories, etc. ) of the given ODE. Many partial results have been re cently unified by means of new formal methods coming from differential geometry and differential algebra. However, certain problems (invariance, equivalence, linearization, etc. ) naturally lead to systems of partial differential equations (PDE). More generally, partial differential control theory studies input/output relations defined by systems of PDE (mechanics, thermodynamics, hydrodynamics, plasma physics, robotics, etc. ). One of the aims of this book is to extend the preceding con cepts to this new situation, where, of course, functional analysis and/or a dynamical system approach cannot be used. A link will be exhibited between this domain of applied mathematics and the famous 'Backlund problem', existing in the study of solitary waves or solitons. In particular, we shall show how the methods of differ ential elimination presented here will allow us to determine compatibility conditions on input and/or output as a better understanding of the foundations of control the ory. At the same time we shall unify differential geometry and differential algebra in a new framework, called differential algebraic geometry.
Author: Olle Stormark
Publisher: Cambridge University Press
ISBN: 9780521780889
Category : Mathematics
Languages : en
Pages : 604
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Book Description
Here is a lucid and comprehensive introduction to the differential geometric study of partial differential equations (PDE). The first book to present substantial results on local solvability of general and nonlinear PDE systems without using power series techniques, it describes a general approach to PDE systems based on ideas developed by Lie, Cartan and Vessiot. The central theme is the exploitation of singular vector field systems and their first integrals. These considerations naturally lead to local Lie groups, Lie pseudogroups and the equivalence problem, all of which are covered in detail. This book will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.
Author: J. F. Pommaret
Publisher: CRC Press
ISBN: 9782881242137
Category : Mathematics
Languages : en
Pages : 612
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Book Description
Author: Peter J. Vassiliou
Publisher: Cambridge University Press
ISBN: 9780521775984
Category : Mathematics
Languages : en
Pages : 242
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Book Description
A concise and accessible introduction to the wide range of topics in geometric approaches to differential equations.
Author: Victor G. Kac
Publisher: Springer
ISBN: 3030013766
Category : Mathematics
Languages : en
Pages : 199
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Book Description
Based on the third International Conference on Symmetries, Differential Equations and Applications (SDEA-III), this proceedings volume highlights recent important advances and trends in the applications of Lie groups, including a broad area of topics in interdisciplinary studies, ranging from mathematical physics to financial mathematics. The selected and peer-reviewed contributions gathered here cover Lie theory and symmetry methods in differential equations, Lie algebras and Lie pseudogroups, super-symmetry and super-integrability, representation theory of Lie algebras, classification problems, conservation laws, and geometrical methods. The SDEA III, held in honour of the Centenary of Noether’s Theorem, proven by the prominent German mathematician Emmy Noether, at Istanbul Technical University in August 2017 provided a productive forum for academic researchers, both junior and senior, and students to discuss and share the latest developments in the theory and applications of Lie symmetry groups. This work has an interdisciplinary appeal and will be a valuable read for researchers in mathematics, mechanics, physics, engineering, medicine and finance.
Author: Michael Eastwood
Publisher: Springer Science & Business Media
ISBN: 0387738312
Category : Mathematics
Languages : en
Pages : 565
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Book Description
This three-week summer program considered the symmetries preserving various natural geometric structures. There are two parts to the proceedings. The articles in the first part are expository but all contain significant new material. The articles in the second part are concerned with original research. All articles were thoroughly refereed and the range of interrelated work ensures that this will be an extremely useful collection.
Author: Jean-Francois Pommaret
Publisher:
ISBN: 9781536134100
Category : Mathematical physics
Languages : en
Pages : 146
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Book Description
The concept of "group" has been introduced in mathematics for the first time by E. Galois (1830) and slowly passed from algebra to geometry with the work of S. Lie on Lie groups (1880) and Lie pseudogroups (1890) of transformations. The concept of a finite length differential sequence, now called the Janet sequence, had been described for the first time by M. Janet (1920). Then, the work of D. C. Spencer (1970) has been the first attempt to use the formal theory of systems of partial differential equations (PDE) in order to study the formal theory of Lie pseudogroups. However, the linear and nonlinear Spencer sequences for Lie pseudogroups, though never used in physics, largely supersede the "Cartan structure equations " (1905) and are quite different from the "Vessiot structure equations " (1903), introduced for the same purpose but never acknowledged by E. Cartan or successors. Meanwhile, mixing differential geometry with homological algebra, M. Kashiwara (1970) created "algebraic analysis" in order to study differential modules and double duality. By chance, unexpected arguments have been introduced by the brothers E. and F. Cosserat (1909) in order to revisit elasticity and by H. Weyl (1918) in order to revisit electromagnetism through a unique differential sequence only depending on the structure of the conformal group of space-time.The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. It has been the dream of many mathematicians at the end of the nineteenth century to generalize these results to systems of linear or algebraic PDE and the corresponding finitely generated differential extensions, in order to be able to add the word differential in front of any classical statement. The achievement of the Picard-Vessiot theory by E. Kolchin and coworkers between 1950 and 1970 is now well-known. However, the work of Vessiot on the differential Galois theory (1904), that is on the possibility to extend the classical Galois theory to systems of algebraic PDE and algebraic Lie pseudogroups, namely groups of transformations solutions for systems of algebraic PDE, has also never been acknowledged. His main idea has been to notice that the Galois theory (old and new) is a study of principal homogeneous spaces (PHS) for algebraic groups or pseudogroups described by what he called "automorphic systems" of PDE.The purpose of this book is first to revisit Gauge Theory and General Relativity in light of the latest developments just described and then to apply the differential Galois theory in order to revisit various domains of mechanics (Shell theory, Chain theory, Frenet-Serret formulas, Hamilton-Jacobi equations). All the results presented are new. (Nova)
