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Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds. Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists. Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, with many results old and new. .

Homotopy theory. --- Algebraic topology. --- Manifolds (Mathematics) --- Invariant manifolds. --- Deformations, Continuous --- Mathematics. --- Manifolds (Mathematics). --- Complex manifolds. --- Algebraic Topology. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Topology --- Geometry, Differential --- Invariants --- Cell aggregation --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Analytic spaces

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This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples,  many of them well known in the literature of numerical computation in dynamical systems.  A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and  applications of computational dynamical systems.

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Invariant manifolds. --- Mathematics. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Numerical analysis. --- Statistical physics. --- Dynamical systems. --- Dynamical Systems and Ergodic Theory. --- Statistical Physics, Dynamical Systems and Complexity. --- Numerical Analysis. --- Partial Differential Equations. --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Mathematical statistics --- Mathematical analysis --- Partial differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Math --- Science --- Statistical methods --- Invariants --- Manifolds (Mathematics) --- Differentiable dynamical systems. --- Differential equations, partial. --- Complex Systems. --- Statistical Physics and Dynamical Systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics

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From Kinetic Models to Hydrodynamics serves as an introduction to the asymptotic methods necessary to obtain hydrodynamic equations from a fundamental description using kinetic theory models and the Boltzmann equation.  The work is a survey of an active research area, which aims to bridge time and length scales from the particle-like description inherent in Boltzmann equation theory to a fully established “continuum” approach typical of macroscopic laws of physics. The author sheds light on a new method—using invariant manifolds—which addresses a functional equation for the nonequilibrium single-particle distribution function.  This method allows one to find exact and thermodynamically consistent expressions for: hydrodynamic modes; transport coefficient expressions for hydrodynamic modes; and transport coefficients of a fluid beyond the traditional hydrodynamic limit.  The invariant manifold method paves the way to establish a needed bridge between Boltzmann equation theory and a particle-based theory of hydrodynamics.  Finally, the author explores the ambitious and longstanding task of obtaining hydrodynamic constitutive equations from their kinetic counterparts. The work is intended for specialists in kinetic theory—or more generally statistical mechanics—and will provide a bridge between a physical and mathematical approach to solve real-world problems.

Friction. --- Hydrodynamics. --- Kinetics. --- Hydrodynamics --- Dynamics --- Invariant manifolds --- Molecular theory --- Statistical mechanics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Applied Physics --- Operations Research --- Applied Mathematics --- Dynamics. --- Mathematical models. --- Dynamical systems --- Kinetics --- Mathematics. --- Mathematical physics. --- Physics. --- Statistical physics. --- Dynamical systems. --- Mathematical Physics. --- Mathematical Methods in Physics. --- Mathematical Applications in the Physical Sciences. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Modeling and Industrial Mathematics. --- Statistical Physics, Dynamical Systems and Complexity. --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Complex Systems. --- Physical mathematics --- Mathematical statistics --- Models, Mathematical --- Simulation methods --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Statistical methods

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Model reduction and coarse-graining are important in many areas of science and engineering. How does a system with many degrees of freedom become one with fewer? How can a reversible micro-description be adapted to the dissipative macroscopic model? These crucial questions, as well as many other related problems, are discussed in this book. Specific areas of study include dynamical systems, non-equilibrium statistical mechanics, kinetic theory, hydrodynamics and mechanics of continuous media, (bio)chemical kinetics, nonlinear dynamics, nonlinear control, nonlinear estimation, and particulate systems from various branches of engineering. The generic nature and the power of the pertinent conceptual, analytical and computational frameworks helps eliminate some of the traditional language barriers, which often unnecessarily impede scientific progress and the interaction of researchers between disciplines such as physics, chemistry, biology, applied mathematics and engineering. All contributions are authored by experts, whose specialities span a wide range of fields within science and engineering.

Invariant manifolds --- Statistical physics --- Dynamics --- Mathematical models --- Models, Mathematical --- Simulation methods --- Invariants --- Manifolds (Mathematics) --- 51-74 --- 531.1 --- 536.75 --- 681.3*I61 --- 681.3*J6 --- 681.3*J6 Computer-aided engineering: computer-aided design; CAD; computer-aided manufacturing; CAM --- Computer-aided engineering: computer-aided design; CAD; computer-aided manufacturing; CAM --- 681.3*I61 Simulation theory: model classification; continuous simulation; discrete simulation (Simulation and modeling) --- Simulation theory: model classification; continuous simulation; discrete simulation (Simulation and modeling) --- 536.75 Entropy. Statistical thermodynamics. Irreversible processes --- Entropy. Statistical thermodynamics. Irreversible processes --- 531.1 Kinematics. Mathematical-mechanical geometry of motion --- Kinematics. Mathematical-mechanical geometry of motion --- Mathematics in engineering science and technology --- Engineering. --- Systems theory. --- Chemistry --- Statistical physics. --- Theoretical, Mathematical and Computational Physics. --- Complex Systems. --- Engineering, general. --- Systems Theory, Control. --- Math. Applications in Chemistry. --- Statistical Physics and Dynamical Systems. --- Mathematics. --- System theory. --- Physics --- Mathematical statistics --- Construction --- Industrial arts --- Technology --- Systems, Theory of --- Systems science --- Science --- Statistical methods --- Philosophy --- Mathematical physics. --- Dynamical systems. --- Chemometrics. --- Chemistry, Analytic --- Analytical chemistry --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Physical mathematics --- Measurement