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This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high co-dimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Elliptic operators. --- Markov processes. --- Population biology --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Differential operators, Elliptic --- Operators, Elliptic --- Partial differential operators --- Mathematical models. --- 1-dimensional integral. --- Euclidean model problem. --- Euclidean space. --- Hlder space. --- Hopf boundary point. --- Kimura diffusion equation. --- Kimura diffusion operator. --- Laplace transform. --- Schauder estimate. --- WrightІisher geometry. --- adjoint operator. --- backward Kolmogorov equation. --- boundary behavior. --- degenerate elliptic operator. --- doubling. --- elliptic Kimura operator. --- elliptic equation. --- forward Kolmogorov equation. --- function space. --- general model problem. --- generalized Kimura diffusion. --- heat equation. --- heat kernel. --- higher dimensional corner. --- higher regularity. --- holomorphic semi-group. --- homogeneous Cauchy problem. --- hybrid space. --- hypersurface boundary. --- induction hypothesis. --- induction. --- inhomogeneous problem. --- irregular solution. --- long time asymptotics. --- long-time behavior. --- manifold with corners. --- martingale problem. --- mathematical finance. --- model problem. --- normal form. --- normal vector. --- null-space. --- off-diagonal behavior. --- open orthant. --- parabolic equation. --- perturbation theory. --- polyhedron. --- population genetics. --- probability theory. --- regularity. --- resolvent operator. --- semi-group. --- solution operator. --- uniqueness.

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Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications.

Geometry. --- Algorithms.