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"This book offers an analytical rather than measure-theoretical approach to the derivation of the partial differential equations of nonlinear filtering theory. The basis for this approach is the discrete numerical scheme used in Monte-Carlo simulations of stochastic differential equations and Wiener's associated path integral representation of the transition probability density. Furthermore, it presents analytical methods for constructing asymptotic approximations to their solution and for synthesizing asymptotically optimal filters. It also offers a new approach to the phase tracking problem, based on optimizing the mean time to loss of lock. The book contains exercises and worked-out examples aimed at illustrating the methods of mathematical modeling and performance analysis of phase trackers. "

"This book shows how to apply theoretical mathematical models to unravel the mechanisms involved in processes found in mathematical physics and the biosciences. It is a unique collection of abstract methods that deploy nonlinear partial differential equations. "

"This book presents recent results on nonlinear parabolic-hyperbolic coupled systems such as the compressible navier-stokes equations, and liquid crystal system. It summarizes recently published research by the authors and their collaborators, but also includes new and unpublished material. All models under consideration are built on compressible equations and liquid crystal systems. This type of partial differential equations arises not only in many fields of mathematics, but also in other branches of science such as physics, fluid dynamics and material science."

"The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain.From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asymptotic behaviour, discretization schemes, coefficient identification, and to introduce relevant solving methods for each of them."

"The topic of the 2010 Abel Symposium, hosted at the Norwegian Academy of Science and Letters, Oslo, was Nonlinear Partial Differential Equations, the study of which is of fundamental importance in mathematics and in almost all of natural sciences, economics, and engineering.This area of mathematics is currently in the midst of an unprecedented development worldwide. Differential equations are used to model phenomena of increasing complexity, and in areas that have traditionally been outside the realm of mathematics. New analytical tools and numerical methods are dramatically improving our understanding of nonlinear models. Nonlinearity gives rise to novel effects reflected in the appearance of shock waves, turbulence, material defects, etc., and offers challenging mathematical problems. On the other hand, new mathematical developments provide new insight in many applications."

"The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields."

"M. Cinquini Cibrario: Equazioni non lineari e teoria delle caratteristiche.- J. Leray: La théorie de L. Garding des équations hyperboliques lineaires.- S.L. Sobolev: Lezioni sulle equazioni iperboliche non lineari.- A Weinstein: Equazioni alle derivate parziali singolari."