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In Bose-Einstein condensates from physics and competing species system from population dynamics, it is observed that different condensates (or species) tend to be separated. This is known as the phase separation phenomena. These pose a new class of free boundary problems of nonlinear partial differential equations. Besides its great difficulty in mathematics, the study of this problem will help us get a better understanding of the phase separation phenomena. This thesis is devoted to the study of the asymptotic behavior of singularly perturbed partial differential equations and some related free boundary problems arising from Bose-Einstein condensation theory and competing species model. We study the free boundary problems in the singular limit and give some characterizations, and use this to study the dynamical behavior of competing species when the competition is strong. These results have many applications in physics and biology.   It was nominated by the Graduate University of Chinese Academy of Sciences as an outstanding PhD thesis.

Differential equations, Partial. --- Heat -- Convection. --- Partial differential operators. --- Boundary value problems --- Differential equations, Partial --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Asymptotic theory --- Asymptotic theory. --- Asymptotic theory in partial differential equations --- Asymptotic theory of boundary value problems --- Mathematics. --- Functional analysis. --- Partial differential equations. --- Partial Differential Equations. --- Functional Analysis. --- Asymptotic expansions --- Differential equations, partial. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations

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There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution. Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions. The main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables. This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.

Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematics. --- Approximation theory. --- Partial differential equations. --- Partial Differential Equations. --- Approximations and Expansions. --- Differential equations, Elliptic --- Boundary value problems --- Inhomogeneous materials --- Asymptotic theory. --- Mathematical models. --- Heterogeneous materials --- Inhomogeneous media --- Media, Inhomogeneous --- Materials --- Matter --- Asymptotic theory of boundary value problems --- Asymptotic theory of elliptic differential equations --- Asymptotic expansions --- Differential equations, partial. --- Math --- Science --- Partial differential equations --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems

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The book is dedicated to the construction of particular solutions of systems of ordinary differential equations in the form of series that are analogous to those used in Lyapunov’s first method. A prominent place is given to asymptotic solutions that tend to an equilibrium position, especially in the strongly nonlinear case, where the existence of such solutions can’t be inferred on the basis of the first approximation alone. The book is illustrated with a large number of concrete examples of systems in which the presence of a particular solution of a certain class is related to special properties of the system’s dynamic behavior. It is a book for students and specialists who work with dynamical systems in the fields of mechanics, mathematics, and theoretical physics.

Differential equations, Nonlinear. --- Geometry, Differential. --- Nonlinear wave equations. --- Differential equations --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Asymptotic theory --- Asymptotic theory. --- 517.91 Differential equations --- Mathematics. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Physics. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Mathematical Methods in Physics. --- Differential Equations. --- Differentiable dynamical systems. --- Mathematical physics. --- Physical mathematics --- Physics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics

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Optimization, simulation and control are very powerful tools in engineering and mathematics, and play an increasingly important role. Because of their various real-world applications in industries such as finance, economics, and telecommunications, research in these fields is accelerating at a rapid pace, and there have been major algorithmic and theoretical developments in these fields in the last decade. This volume brings together the latest developments in these areas of research and presents applications of these results to a wide range of real-world problems. The book is composed of invited contributions by experts from around the world who work to develop and apply new optimization, simulation, and control techniques either at a theoretical level or in practice. Some key topics presented include: equilibrium problems, multi-objective optimization, variational inequalities, stochastic processes, numerical analysis, optimization in signal processing, and various other interdisciplinary applications. This volume can serve as a useful resource for researchers, practitioners, and advanced graduate students of mathematics and engineering working in research areas where results in optimization, simulation and control can be applied.

Optimization. --- Simulation. --- Statistical hypothesis testing -- Asymptotic theory. --- Mathematical optimization --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Mathematical optimization. --- Control theory. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematics. --- System theory. --- Mathematical models. --- Systems Theory, Control. --- Mathematical Modeling and Industrial Mathematics. --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Dynamics --- Machine theory --- Systems theory. --- Models, Mathematical --- Systems, Theory of --- Systems science --- Science --- Philosophy