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||This book reports recent mathematical developments in the Priority Program 'Analysis, Modeling and Simulation of Multiscale Problems', which has been supported by the
Deutsche Forschungsgemeinschaft from 2000 to 2007. Multiscale problems occur in many fields of science, such as microstructures in materials, sharp-interface models,
many-particle systems, and motions on different spatial and temporal scales in quantum mechanics or in molecular dynamics. Based on exemplary models from these fields,
recently developed mathematical tools are described in a comprehensive manner. Applied analysis covers modern applications of weak convergence theories such as Gamma-convergence,
Young measures and Wigner measures.Using this it is shown how new effective models on the large temporal or spatial scales can be derived rigorously, like modulation equations for
waves in dispersive systems. Special numerical tools are developed for simulating multiscale problems efficiently, for instance exponential integrators for Hamiltonian systems,
averaging techniques, wavelet multiresolution, and adaptive computation of microstructures via relaxation.