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This book presents a detailed, self-contained theory of continuous mappings. It is mainly addressed to students who have already studied these mappings in the setting of metric spaces, as well as multidimensional differential calculus. The needed background facts about sets, metric spaces and linear algebra are developed in detail, so as to provide a seamless transition between students' previous studies and new material.  In view of its many novel features, this book will be of interest also to mature readers who have studied continuous mappings from the subject's classical texts and wish to become acquainted with a new approach. The theory of continuous mappings serves as infrastructure for more specialized mathematical theories like differential equations, integral equations, operator theory, dynamical systems, global analysis, topological groups, topological rings and many more. In light of the centrality of the topic, a book of this kind fits a variety of applications, especially those that contribute to a better understanding of functional analysis, towards establishing an efficient setting for its pursuit.

Mathematics. --- Functional analysis. --- Topology. --- Functional Analysis. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Functional calculus --- Math --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Calculus of variations --- Functional equations --- Integral equations --- Continuation methods.

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Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.

Continuation methods. --- Boundary value problems --- Differentiable dynamical systems. --- Numerical analysis. --- Bifurcation theory. --- Computer programs. --- Differential equations, Nonlinear --- Stability --- Mathematical analysis --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Continuation (Mathematics) --- Continuation techniques --- Differential equations, Partial --- Numerical solutions --- Complex analysis --- Numerical analysis --- Mathematics. --- Engineering mathematics. --- Engineering. --- Complex Systems. --- Dynamical Systems and Ergodic Theory. --- Applied and Technical Physics. --- Applications of Mathematics. --- Mathematical and Computational Engineering. --- Engineering, general. --- Construction --- Industrial arts --- Technology --- Engineering --- Engineering analysis --- Math --- Science --- Mathematics --- Statistical physics. --- Dynamical systems. --- Dynamics. --- Ergodic theory. --- Physics. --- Applied mathematics. --- 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 --- Physics --- Statics --- Mathematical statistics --- Statistical methods