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This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The presentation is self-contained and begins with the Radon transform and radial sound speeds as motivating examples. The required geometric background is developed in detail in the context of simple manifolds with boundary. An in-depth analysis of various geodesic X-ray transforms is carried out together with related uniqueness, stability, reconstruction and range characterization results. Highlights include a proof of boundary rigidity for simple surfaces as well as scattering rigidity for connections. The concluding chapter discusses current open problems and related topics. The numerous exercises and examples make this book an excellent self-study resource or text for a one-semester course or seminar.

Inverse problems (Differential equations) --- Inversions (Geometry) --- Geometry, Differential.

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Fourier, Transformations de --- Fourier transformations. --- Inversions (géométrie) --- Inversions (Geometry) --- Plancherel, Formule de. --- Théorèmes taubériens

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This monograph covers the existing results regarding Green’s functions for differential equations with involutions (DEI).The first part of the book is devoted to the study of the most useful aspects of involutions from an analytical point of view and the associated algebras of differential operators. The work combines the state of the art regarding the existence and uniqueness results for DEI and new theorems describing how to obtain Green’s functions, proving that the theory can be extended to operators (not necessarily involutions) of a similar nature, such as the Hilbert transform or projections, due to their analogous algebraic properties. Obtaining a Green’s function for these operators leads to new results on the qualitative properties of the solutions, in particular maximum and antimaximum principles.

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations. --- Involutes (Mathematics) --- 517.91 Differential equations --- Differential equations --- Mathematics. --- Physics. --- Ordinary Differential Equations. --- Mathematical Methods in Physics. --- Curves --- Inversions (Geometry) --- Differential Equations. --- Mathematical physics. --- Physical mathematics --- Physics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics