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As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.
Differential equations, Nonlinear --- Solitons --- Numerical solutions --- Solitons. --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- Numerical analysis --- Numerical solutions. --- Differential equations, Nonlinear - Numerical solutions
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'Optimal Solution of Nonlinear Equations' is a text/monograph designed to provide an overview of optimal computational methods for the solution of nonlinear equations, fixed points of contractive and noncontractive mapping and for the computation of the topological degree. It is of interest to any reader working in the area of information-based complexity.
Differential equations, Nonlinear --- Mathematical optimization. --- Fixed point theory. --- Topological degree. --- Degree, Topological --- Degree (Topology) --- Degree theory --- Algebraic topology --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Fixed point theorems (Topology) --- Nonlinear operators --- Coincidence theory (Mathematics) --- Numerical analysis --- Numerical solutions. --- Fixed point theory --- Mathematical optimization --- Topological degree --- Differential equations, Nonlinear - Numerical solutions --- Analyse numérique. --- Équations. --- Théories non linéaires. --- Equations --- Nonlinear theories --- Topologie algébrique. --- Degré topologique. --- Analyse numérique. --- Équations. --- Théories non linéaires. --- Topologie algébrique --- Degré topologique.
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This book provides a modern investigation into the bifurcation phenomena of physical and engineering problems. Systematic methods - based on asymptotic, probabilistic, and group-theoretic standpoints - are used to examine experimental and computational data from numerous examples (soil, sand, kaolin, concrete, domes). For mathematicians, static bifurcation theory for finite-dimensional systems, as well as its implications for practical problems, is illuminated by the numerous examples. Engineers may find this book, with its minimized mathematical formalism, to be a useful introduction to modern bifurcation theory. This second edition strengthens the theoretical backgrounds of group representation theory and its application, uses of block-diagonalization in bifurcation analysis, and includes up-to-date topics of the bifurcation analysis of diverse materials from rectangular parallelepiped sand specimens to honeycomb cellular solids. Reviews of first edition: "The present book gives a wide and deep description of imperfect bifurcation behaviour in engineering problems. … the book offers a number of systematic methods based on contemporary mathematics. … On balance, the reviewed book is very useful as it develops a modern static imperfect bifurcation theory and fills the gap between mathematical theory and engineering practice." (Zentralblatt MATH, 2003) "The current book is a graduate-level text that presents an overview of imperfections and the prediction of the initial post-buckling response of a system. ... Imperfect Bifurcation in Structures and Materials provides an extensive range of material on the role of imperfections in stability theory. It would be suitable for a graduate-level course on the subject or as a reference to research workers in the field." ( Applied Mechanics Reviews, 2003) "This book is a comprehensive treatment of the static bifurcation problems found in (mainly civil/structural) engineering applications.... The text is well written and regularly interspersed with illustrative examples. The mathematical formalism is kept to a minimum and the 194 figures break up the text and make this a highly readable and informative book. ... In summary a comprehensive treatment of the subject which is very well put together and of interest to all researchers working in this area: recommended." (UK Nonlinear News, 2002).
Bifurcation theory. --- Differential equations, Nonlinear -- Numerical solutions. --- Electronic books. -- local. --- Engineering mathematics --- Bifurcation theory --- Structural analysis (Engineering) --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Mathematical Theory --- Mathematical models --- Differential equations, Nonlinear --- Numerical solutions. --- Engineering. --- Dynamics. --- Ergodic theory. --- System theory. --- Applied mathematics. --- Engineering mathematics. --- Structural mechanics. --- Control engineering. --- Control. --- Appl.Mathematics/Computational Methods of Engineering. --- Systems Theory, Control. --- Dynamical Systems and Ergodic Theory. --- Structural Mechanics. --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Automation --- Programmable controllers --- Architectural engineering --- Engineering, Architectural --- Structural mechanics --- Structures, Theory of --- Structural engineering --- Engineering --- Engineering analysis --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Construction --- Industrial arts --- Technology --- Philosophy --- Numerical analysis --- Stability --- Numerical solutions --- Systems theory. --- Differentiable dynamical systems. --- Mechanics. --- Mechanics, Applied. --- Control and Systems Theory. --- Mathematical and Computational Engineering. --- Solid Mechanics. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Applied mechanics --- Engineering, Mechanical --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Mathematical models.
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