Download or read book Geometric and Algebraic Structures in Differential Equations written by P.H. Kersten and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Bäcklund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg-de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability. Nowadays the geometrical and algebraic approach to partial differential equations constitutes a special branch of modern mathematics. In 1993, a workshop on algebra and geometry of differential equations took place at the University of Twente (The Netherlands), where the state-of-the-art of the main problems was fixed. This book contains a collection of invited lectures presented at this workshop. The material presented is of interest to those who work in pure and applied mathematics and especially in mathematical physics.
Download or read book Geometric and Algebraic Structures in Differential Equations written by I. S Krasil'shchik and published by . This book was released on 1995 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Analytic Algebraic and Geometric Aspects of Differential Equations written by Galina Filipuk and published by Birkhäuser. This book was released on 2017-06-23 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Będlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics. The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers. It will be of interest to graduate students, early-career researchers and specialists in analysis, geometry, algebra and related areas, as well as anyone interested in learning new methods and techniques.
Download or read book Power Geometry in Algebraic and Differential Equations written by A.D. Bruno and published by Elsevier. This book was released on 2000-08-03 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed. The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems. The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.
Download or read book Applications of Analytic and Geometric Methods to Nonlinear Differential Equations written by P.A. Clarkson and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years. (1) The inverse scattering transform (IST), using complex function theory, which has been employed to solve many physically significant equations, the `soliton' equations. (2) Twistor theory, using differential geometry, which has been used to solve the self-dual Yang--Mills (SDYM) equations, a four-dimensional system having important applications in mathematical physics. Both soliton and the SDYM equations have rich algebraic structures which have been extensively studied. Recently, it has been conjectured that, in some sense, all soliton equations arise as special cases of the SDYM equations; subsequently many have been discovered as either exact or asymptotic reductions of the SDYM equations. Consequently what seems to be emerging is that a natural, physically significant system such as the SDYM equations provides the basis for a unifying framework underlying this class of integrable systems, i.e. `soliton' systems. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. The majority of nonlinear evolution equations are nonintegrable, and so asymptotic, numerical perturbation and reduction techniques are often used to study such equations. This book also contains articles on perturbed soliton equations. Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations. (ABSTRACT) In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years; the inverse scattering transform (IST), for `soliton' equations and twistor theory, for the self-dual Yang--Mills (SDYM) equations. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. Additionally, it contains articles on perturbed soliton equations, Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations.
Download or read book Space Time Matter written by Jochen Brüning and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-04-09 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph describes some of the most interesting results obtained by the mathematicians and physicists collaborating in the CRC 647 "Space – Time – Matter", in the years 2005 - 2016. The work presented concerns the mathematical and physical foundations of string and quantum field theory as well as cosmology. Important topics are the spaces and metrics modelling the geometry of matter, and the evolution of these geometries. The partial differential equations governing such structures and their singularities, special solutions and stability properties are discussed in detail. Contents Introduction Algebraic K-theory, assembly maps, controlled algebra, and trace methods Lorentzian manifolds with special holonomy – Constructions and global properties Contributions to the spectral geometry of locally homogeneous spaces On conformally covariant differential operators and spectral theory of the holographic Laplacian Moduli and deformations Vector bundles in algebraic geometry and mathematical physics Dyson–Schwinger equations: Fix-point equations for quantum fields Hidden structure in the form factors ofN = 4 SYM On regulating the AdS superstring Constraints on CFT observables from the bootstrap program Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities Yangian symmetry in maximally supersymmetric Yang-Mills theory Wave and Dirac equations on manifolds Geometric analysis on singular spaces Singularities and long-time behavior in nonlinear evolution equations and general relativity
Download or read book Difference Algebra written by Alexander Levin and published by Springer Science & Business Media. This book was released on 2008-04-19 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings. The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. The book is self-contained; it requires no prerequisites other than the knowledge of basic algebraic concepts and a mathematical maturity of an advanced undergraduate.
Download or read book Fundamentals of Advanced Mathematics 1 written by Henri Bourles and published by Elsevier. This book was released on 2017-07-10 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This precis, comprised of three volumes, of which this book is the first, exposes the mathematical elements which make up the foundations of a number of contemporary scientific methods: modern theory on systems, physics and engineering. This first volume focuses primarily on algebraic questions: categories and functors, groups, rings, modules and algebra. Notions are introduced in a general framework and then studied in the context of commutative and homological algebra; their application in algebraic topology and geometry is therefore developed. These notions play an essential role in algebraic analysis (analytico-algebraic systems theory of ordinary or partial linear differential equations). The book concludes with a study of modules over the main types of rings, the rational canonical form of matrices, the (commutative) theory of elemental divisors and their application in systems of linear differential equations with constant coefficients. Part of the New Mathematical Methods, Systems, and Applications series Presents the notions, results, and proofs necessary to understand and master the various topics Provides a unified notation, making the task easier for the reader. Includes several summaries of mathematics for engineers
Download or read book Combinatorial Structures in Algebra and Geometry written by Dumitru I. Stamate and published by Springer Nature. This book was released on 2020-09-01 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: This proceedings volume presents selected, peer-reviewed contributions from the 26th National School on Algebra, which was held in Constanța, Romania, on August 26-September 1, 2018. The works cover three fields of mathematics: algebra, geometry and discrete mathematics, discussing the latest developments in the theory of monomial ideals, algebras of graphs and local positivity of line bundles. Whereas interactions between algebra and geometry go back at least to Hilbert, the ties to combinatorics are much more recent and are subject of immense interest at the forefront of contemporary mathematics research. Transplanting methods between different branches of mathematics has proved very fruitful in the past – for example, the application of fixed point theorems in topology to solving nonlinear differential equations in analysis. Similarly, combinatorial structures, e.g., Newton-Okounkov bodies, have led to significant advances in our understanding of the asymptotic properties of line bundles in geometry and multiplier ideals in algebra. This book is intended for advanced graduate students, young scientists and established researchers with an interest in the overlaps between different fields of mathematics. A volume for the 24th edition of this conference was previously published with Springer under the title "Multigraded Algebra and Applications" (ISBN 978-3-319-90493-1).
Download or read book Geometry of PDEs and Mechanics written by Agostino Prastaro and published by World Scientific. This book was released on 1996 with total page 764 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents the theory of partial differential equations (PDEs) from a modern geometric point of view so that PDEs can be characterized by using either technique of differential geometry or algebraic geometry. This allows us to recognize the richness of the structure of PDEs. It presents, for the first time, a geometric theory of non-commutative (quantum) PDEs and gives a general application of this theory to quantum field theory and quantum supergravity.
Download or read book Glimpses of Soliton Theory written by Alex Kasman and published by American Mathematical Society. This book was released on 2023-03-30 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar. —William J. Satzer, MAA Reviews on Glimpses of Soliton Theory (First Edition) Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity. Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous. Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass $wp$-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians. Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica® to facilitate computation and animate solutions. The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
Download or read book Modern Geometric Structures and Fields written by Сергей Петрович Новиков and published by American Mathematical Soc.. This book was released on 2006 with total page 658 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the important structures on them. This book shows that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications.
Download or read book A Guide To Lie Systems With Compatible Geometric Structures written by Javier De Lucas Araujo and published by World Scientific. This book was released on 2020-01-22 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents a comprehensive guide to the study of Lie systems from the fundamentals of differential geometry to the development of contemporary research topics. It embraces several basic topics on differential geometry and the study of geometric structures while developing known applications in the theory of Lie systems. The book also includes a brief exploration of the applications of Lie systems to superequations, discrete systems, and partial differential equations.Offering a complete overview from the topic's foundations to the present, this book is an ideal resource for Physics and Mathematics students, doctoral students and researchers.
Download or read book The Diverse World of PDEs written by I. S. Krasil′shchik and published by American Mathematical Society. This book was released on 2023-08-23 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13–17, 2021, at Independent University of Moscow and Moscow State University, Moscow, Russia. The papers reflect the modern interplay between partial differential equations and various aspects of algebra and computer science. The topics discussed are: relations between integrability and differential rings, supermanifolds, differential calculus over graded algebras, noncommutative generalizations of PDEs, quantum vector fields, generalized Nijenhuis torsion, cohomological approach to the geometry of differential equations, the argument shift method, Frölicher structures in the formal Kadomtsev–Petviashvili hierarchy, and computer-based determination of optimal systems of Lie subalgebras. The companion volume (Contemporary Mathematics, Volume 788) is devoted to Geometry and Mathematical Physics.
Download or read book Metric Structures in Differential Geometry written by Gerard Walschap and published by Springer Science & Business Media. This book was released on 2012-08-23 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.
Download or read book Geometric Numerical Integration written by Ernst Hairer and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.
Download or read book Lie Groups Geometric Structures and Differential Equations written by Tohru Morimoto and published by . This book was released on 2002 with total page 514 pages. Available in PDF, EPUB and Kindle. Book excerpt: The blending of algebra, geometry, and differential equations has a long and distinguished history, dating back to the work of Sophus Lie and Elie Cartan. Overviewing the depth of their influence over the past 100 years presents a formidable challenge. A conference was held on the centennial of Lie's death to reflect upon and celebrate his pursuits, later developments, and what the future may hold. This volume showcases the contents, atmosphere, and results of that conference. Ofparticular importance are two survey articles: Morimoto develops a synthetic study of Lie groups, geometric structures, and differential equations from a unified viewpoint of nilpotent geometry. Yamaguchi and Yatsui discuss the geometry of higher order differential equations of finite type. Contributedresearch articles cover a wide range of disciplines, from geometry of differential equations, CR-geometry, and differential geometry to topics in mathematical physics. This volume is intended for graduate students studying differential geometry and analyis and advanced graduate students and researchers interested in an overview of the most recent progress in these fields. Information for our distributors: Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributedworldwide, except in Japan, by the AMS. All commercial channel discounts apply.
