Topics In Critical Point Theory
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Topics In Critical Point Theory
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Author : Kanishka Perera
language : en
Publisher: Cambridge University Press
Release Date : 2013
Topics In Critical Point Theory written by Kanishka Perera and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Mathematics categories.
Provides an introduction to critical point theory and shows how it solves many difficult problems.
Topics In Critical Point Theory
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Author : Kanishka Perera
language : en
Publisher:
Release Date : 2014-05-14
Topics In Critical Point Theory written by Kanishka Perera and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-14 with MATHEMATICS categories.
Provides an introduction to critical point theory and shows how it solves many difficult problems.
Critical Point Theory And Its Applications
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Author : Wenming Zou
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-09-10
Critical Point Theory And Its Applications written by Wenming Zou and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-09-10 with Mathematics categories.
This book presents some of the latest research in critical point theory, describing methods and presenting the newest applications. Coverage includes extrema, even valued functionals, weak and double linking, sign changing solutions, Morse inequalities, and cohomology groups. Applications described include Hamiltonian systems, Schrödinger equations and systems, jumping nonlinearities, elliptic equations and systems, superlinear problems and beam equations.
Critical Point Theory
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Author : Martin Schechter
language : en
Publisher: Springer Nature
Release Date : 2020-05-30
Critical Point Theory written by Martin Schechter and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-30 with Mathematics categories.
This monograph collects cutting-edge results and techniques for solving nonlinear partial differential equations using critical points. Including many of the author’s own contributions, a range of proofs are conveniently collected here, Because the material is approached with rigor, this book will serve as an invaluable resource for exploring recent developments in this active area of research, as well as the numerous ways in which critical point theory can be applied. Different methods for finding critical points are presented in the first six chapters. The specific situations in which these methods are applicable is explained in detail. Focus then shifts toward the book’s main subject: applications to problems in mathematics and physics. These include topics such as Schrödinger equations, Hamiltonian systems, elliptic systems, nonlinear wave equations, nonlinear optics, semilinear PDEs, boundary value problems, and equations with multiple solutions. Readers will find this collection of applications convenient and thorough, with detailed proofs appearing throughout. Critical Point Theory will be ideal for graduate students and researchers interested in solving differential equations, and for those studying variational methods. An understanding of fundamental mathematical analysis is assumed. In particular, the basic properties of Hilbert and Banach spaces are used.
Duality And Perturbation Methods In Critical Point Theory
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Author : Nassif Ghoussoub
language : en
Publisher: Cambridge University Press
Release Date : 1993-08-19
Duality And Perturbation Methods In Critical Point Theory written by Nassif Ghoussoub and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-08-19 with Mathematics categories.
The calculus of variations has been an active area of mathematics for over 300 years. Its main use is to find stable critical points of functions for the solution of problems. To find unstable values, new approaches (Morse theory and min-max methods) were developed, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a point of view that may help when dealing with such problems. Building upon min-max methods, he systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book reasonably self-contained. Consequently, it should be accessible to all mathematicians, pure or applied, economists and engineers working in nonlinear analysis or optimization.
Minimax Methods In Critical Point Theory With Applications To Differential Equations
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Author : Paul H. Rabinowitz
language : en
Publisher: American Mathematical Soc.
Release Date : 1986-07-01
Minimax Methods In Critical Point Theory With Applications To Differential Equations written by Paul H. Rabinowitz and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986-07-01 with Mathematics categories.
The book provides an introduction to minimax methods in critical point theory and shows their use in existence questions for nonlinear differential equations. An expanded version of the author's 1984 CBMS lectures, this volume is the first monograph devoted solely to these topics. Among the abstract questions considered are the following: the mountain pass and saddle point theorems, multiple critical points for functionals invariant under a group of symmetries, perturbations from symmetry, and variational methods in bifurcation theory. The book requires some background in functional analysis and differential equations, especially elliptic partial differential equations. It is addressed to mathematicians interested in differential equations and/or nonlinear functional analysis, particularly critical point theory.
Linking Methods In Critical Point Theory
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Author : Martin Schechter
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Linking Methods In Critical Point Theory written by Martin Schechter and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif ferential equations? To find out, read on ...
The Cortex And The Critical Point
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Author : John M. Beggs
language : en
Publisher: MIT Press
Release Date : 2022-08-30
The Cortex And The Critical Point written by John M. Beggs and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-08-30 with Science categories.
How the cerebral cortex operates near a critical phase transition point for optimum performance. Individual neurons have limited computational powers, but when they work together, it is almost like magic. Firing synchronously and then breaking off to improvise by themselves, they can be paradoxically both independent and interdependent. This happens near the critical point: when neurons are poised between a phase where activity is damped and a phase where it is amplified, where information processing is optimized, and complex emergent activity patterns arise. The claim that neurons in the cortex work best when they operate near the critical point is known as the criticality hypothesis. In this book John Beggs—one of the pioneers of this hypothesis—offers an introduction to the critical point and its relevance to the brain. Drawing on recent experimental evidence, Beggs first explains the main ideas underlying the criticality hypotheses and emergent phenomena. He then discusses the critical point and its two main consequences—first, scale-free properties that confer optimum information processing; and second, universality, or the idea that complex emergent phenomena, like that seen near the critical point, can be explained by relatively simple models that are applicable across species and scale. Finally, Beggs considers future directions for the field, including research on homeostatic regulation, quasicriticality, and the expansion of the cortex and intelligence. An appendix provides technical material; many chapters include exercises that use freely available code and data sets.
Perturbation Methods And Semilinear Elliptic Problems On R N
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Author : Antonio Ambrosetti
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-03-21
Perturbation Methods And Semilinear Elliptic Problems On R N written by Antonio Ambrosetti and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-03-21 with Mathematics categories.
Several important problems arising in Physics, Di?erential Geometry and other n topics lead to consider semilinear variational elliptic equations on R and a great deal of work has been devoted to their study. From the mathematical point of view, the main interest relies on the fact that the tools of Nonlinear Functional Analysis, based on compactness arguments, in general cannot be used, at least in a straightforward way, and some new techniques have to be developed. n On the other hand, there are several elliptic problems on R which are p- turbative in nature. In some cases there is a natural perturbation parameter, like inthe bifurcationfromthe essentialspectrum orinsingularlyperturbed equations or in the study of semiclassical standing waves for NLS. In some other circ- stances, one studies perturbations either because this is the ?rst step to obtain global results or else because it often provides a correct perspective for further global studies. For these perturbation problems a speci?c approach,that takes advantage of such a perturbative setting, seems the most appropriate. These abstract tools are provided by perturbation methods in critical point theory. Actually, it turns out that such a framework can be used to handle a large variety of equations, usually considered di?erent in nature. Theaimofthismonographistodiscusstheseabstractmethodstogetherwith their applications to several perturbation problems, whose common feature is to n involve semilinear Elliptic Partial Di?erential Equations on R with a variational structure.
A Primer To The Theory Of Critical Phenomena
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Author : Jurgen M. Honig
language : en
Publisher: Elsevier
Release Date : 2016-09-01
A Primer To The Theory Of Critical Phenomena written by Jurgen M. Honig and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-09-01 with Science categories.
A Primer to the Theory of Critical Phenomena