Newton S Method Applied To Two Quadratic Equations In Mathbb C 2 Viewed As A Global Dynamical System
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Newton S Method Applied To Two Quadratic Equations In Mathbb C 2 Viewed As A Global Dynamical System
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Author : John H. Hubbard
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Newton S Method Applied To Two Quadratic Equations In Mathbb C 2 Viewed As A Global Dynamical System written by John H. Hubbard 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 2008 with Mathematics categories.
The authors study the Newton map $N:\mathbb{C}^2\rightarrow\mathbb{C}^2$ associated to two equations in two unknowns, as a dynamical system. They focus on the first non-trivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things: The Russakovksi-Shiffman measure does not change the points of indeterminancy. The lines joining pairs of roots are invariant, and the Julia set of the restriction of $N$ to such a line has under appropriate circumstances an invariant manifold, which shares features of a stable manifold and a center manifold. The main part of the article concerns the behavior of $N$ at infinity. To compactify $\mathbb{C}^2$ in such a way that $N$ extends to the compactification, the authors must take the projective limit of an infinite sequence of blow-ups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blow-up: the Farey blow-up. This construction is studied in its own right in chapter 4, where they show among others that the real oriented blow-up of the Farey blow-up has a topological structure reminiscent of the invariant tori of the KAM theorem. They also show that the cohomology, completed under the intersection inner product, is naturally isomorphic to the classical Sobolev space of functions with square-integrable derivatives. In chapter 5 the authors apply these results to the mapping $N$ in a particular case, which they generalize in chapter 6 to the intersection of any two conics.
Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System
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Author : John Hamal Hubbard
language : en
Publisher:
Release Date : 2008
Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System written by John Hamal Hubbard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Differentiable dynamical systems categories.
Studies the Newton map $N: \mathbb{C} DEGREES2\rightarrow\mathbb{C} DEGREES2$ associated to two equations in two unknowns, as a dynamical system. This title focuses on the first non-trivial case: two simultaneous quadratics, to intersect two conics. It proves among other things: the Russakovksi-Shiffman measure does not change the points of
Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System
DOWNLOAD
Author : John H. Hubbard
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System written by John H. Hubbard 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 2008 with Mathematics categories.
Introduction Fundamental properties of Newton maps Invariant 3-manifolds associated to invariant circles The behavior at infinity when $a=b=0$ The Farey blow-up The compactification when $a=b=0$ The case where $a$ and $b$ are arbitrary Bibliography
Newton Methods For Nonlinear Problems
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Author : Peter Deuflhard
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-01-13
Newton Methods For Nonlinear Problems written by Peter Deuflhard 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 2005-01-13 with Mathematics categories.
This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite and in infinite dimension. Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.
Averaging Methods In Nonlinear Dynamical Systems
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Author : Jan A. Sanders
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Averaging Methods In Nonlinear Dynamical Systems written by Jan A. Sanders 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 2013-04-17 with Mathematics categories.
In this book we have developed the asymptotic analysis of nonlinear dynamical systems. We have collected a large number of results, scattered throughout the literature and presented them in a way to illustrate both the underlying common theme, as well as the diversity of problems and solutions. While most of the results are known in the literature, we added new material which we hope will also be of interest to the specialists in this field. The basic theory is discussed in chapters two and three. Improved results are obtained in chapter four in the case of stable limit sets. In chapter five we treat averaging over several angles; here the theory is less standardized, and even in our simplified approach we encounter many open problems. Chapter six deals with the definition of normal form. After making the somewhat philosophical point as to what the right definition should look like, we derive the second order normal form in the Hamiltonian case, using the classical method of generating functions. In chapter seven we treat Hamiltonian systems. The resonances in two degrees of freedom are almost completely analyzed, while we give a survey of results obtained for three degrees of freedom systems. The appendices contain a mix of elementary results, expansions on the theory and research problems.
Primal Dual Interior Point Methods
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Author : Stephen J. Wright
language : en
Publisher: SIAM
Release Date : 1997-01-01
Primal Dual Interior Point Methods written by Stephen J. Wright and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-01 with Interior-point methods categories.
In the past decade, primal-dual algorithms have emerged as the most important and useful algorithms from the interior-point class. This book presents the major primal-dual algorithms for linear programming in straightforward terms. A thorough description of the theoretical properties of these methods is given, as are a discussion of practical and computational aspects and a summary of current software. This is an excellent, timely, and well-written work. The major primal-dual algorithms covered in this book are path-following algorithms (short- and long-step, predictor-corrector), potential-reduction algorithms, and infeasible-interior-point algorithms. A unified treatment of superlinear convergence, finite termination, and detection of infeasible problems is presented. Issues relevant to practical implementation are also discussed, including sparse linear algebra and a complete specification of Mehrotra's predictor-corrector algorithm. Also treated are extensions of primal-dual algorithms to more general problems such as monotone complementarity, semidefinite programming, and general convex programming problems.
Introduction To Matrix Analytic Methods In Stochastic Modeling
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Author : G. Latouche
language : en
Publisher: SIAM
Release Date : 1999-01-01
Introduction To Matrix Analytic Methods In Stochastic Modeling written by G. Latouche and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-01-01 with Mathematics categories.
Presents the basic mathematical ideas and algorithms of the matrix analytic theory in a readable, up-to-date, and comprehensive manner.
Handbook Of Variational Methods For Nonlinear Geometric Data
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Author : Philipp Grohs
language : en
Publisher: Springer Nature
Release Date : 2020-04-03
Handbook Of Variational Methods For Nonlinear Geometric Data written by Philipp Grohs and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-03 with Mathematics categories.
This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art. Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance. Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities. The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations. Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.
Mathematics Of Complexity And Dynamical Systems
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Author : Robert A. Meyers
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-10-05
Mathematics Of Complexity And Dynamical Systems written by Robert A. Meyers 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 2011-10-05 with Mathematics categories.
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.
Advances In Energy System Optimization
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Author : Valentin Bertsch
language : en
Publisher: Springer
Release Date : 2017-03-16
Advances In Energy System Optimization written by Valentin Bertsch and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-16 with Mathematics categories.
The papers presented in this volume address diverse challenges in energy systems, ranging from operational to investment planning problems, from market economics to technical and environmental considerations, from distribution grids to transmission grids and from theoretical considerations to data provision concerns and applied case studies. The International Symposium on Energy System Optimization (ISESO) was held on November 9th and 10th 2015 at the Heidelberg Institute for Theoretical Studies (HITS) and was organized by HITS, Heidelberg University and Karlsruhe Institute of Technology.
