/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Convex bodies: the brunn–minkowski theory

A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric spaces Eigenspace representations Solutions to exercises Bibliography Symbols frequently used Index.

Geometric Analysis on Symmetric Spaces

Entanglement entropy obeys a ‘first law’, an exact quantum generalization of the ordinary first law of thermodynamics. In any CFT with a semiclassical holographic dual, this first law has an interpretation in the dual gravitational theory as a constraint on the spacetimes dual to CFT states. For small perturbations around the CFT vacuum state, we show that the set of such constraints for all ball-shaped spatial regions in the CFT is exactly equivalent to the requirement that the dual geometry satisfy the gravitational equations of motion, linearized about pure AdS. For theories with entanglement entropy computed by the Ryu-Takayanagi formula S = $ \mathcal{A} $
 /(4G
 N), we obtain the linearized Einstein equations. For theories in which the vacuum entanglement entropy for a ball is computed by more general Wald functionals, we obtain the linearized equations for the associated higher-curvature theories. Using the first law, we also derive the holographic dictionary for the stress tensor, given the holographic formula for entanglement entropy. This method provides a simple alternative to holographic renormalization for computing the stress tensor expectation value in arbitrary higher derivative gravitational theories.

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Gravitation from entanglement in holographic CFTs

GROUPS AND GEOMETRIC ANALYSIS Integral Geometry, Invariant Differential Operators, and Spherical Functions (Pure and Applied Mathematics: A Series of Monographs and Textbooks)

We prove a relation between the fc-dimensional totally geodesic Radon transforms on the various constant curvature spaces using the geodesic correspondence between the spaces. Then we use this relation to obtain improved support theorems for these transforms.

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Support theorems for totally geodesic Radon transforms on constant curvature spaces

Abstract An r-hyperconvex body is a set in the d-dimensional Euclidean space 𝔼d that is the intersection of a family of closed balls of radius r. We prove the analogue of the classical Blaschke–Santaló inequality for r-hyperconvex bodies, and we also establish a stability version of it. The other main result of the paper is an r-hyperconvex version of the reverse isoperimetric inequality in the plane.

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Inequalities for hyperconvex sets

A correspondence among the totally geodesic Radon transforms— as well as among their duals—on the constant curvature spaces is established, and is used here to obtain various range characterizations.

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Radon transform on spaces of constant curvature

/pdf/the-invertibility-of-the-radon-transform-on-abstract-5eigk95vny.pdf

Árpád Kurusa

Papers

Support theorems for totally geodesic Radon transforms on constant curvature spaces

Inequalities for hyperconvex sets

Radon transform on spaces of constant curvature

The Invertibility of the Radon Transform on Abstract Rotational Manifolds of Real Type.

A characterization of the Radon transform's range by a system of PDEs