On the parabolic kernel of the Schrödinger operator
Peter Li,Shing-Tung Yau +1 more
TLDR
Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x, t)=0 sur une variete riemannienne generale as discussed by the authors.Abstract:
Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x,t)=0 sur une variete riemannienne generale. Introduction. Estimations de gradients. Inegalites de Harnack. Majorations et minorations des solutions fondamentales. Equation de la chaleur et noyau de Green. Operateur de Schrodingerread more
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Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Posted Content
The entropy formula for the Ricci flow and its geometric applications
TL;DR: In this article, a monotonic expression for Ricci flow, valid in all dimensions and without curvature assumptions, is presented, interpreted as an entropy for a certain canonical ensemble.
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On the mathematical foundations of learning
TL;DR: A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference) and relations of the theory of learning to the mainstream of mathematics are emphasized.
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On the geometry of metric measure spaces. II
TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.
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Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
TL;DR: In this article, the authors provide an overview of the properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion.
References
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Comparison theorems in Riemannian geometry
Jeff Cheeger,David G. Ebin +1 more
TL;DR: In this article, Toponogov's theorem and its generalizations are studied for complete manifolds of nonnegative curvature and compact manifold of nonpositive curvature, respectively.
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Differential equations on riemannian manifolds and their geometric applications
Shiu-Yuen Cheng,Shing-Tung Yau +1 more
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Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
TL;DR: In this paper, it was shown that E(xλ 9 JC2, 0 is a positive (symmetric) C function of JC1? x2, t which for fixed t and (say) %2> ι s * the domain of all positive powers of Δ as a function of xλ.
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The isoperimetric inequality
TL;DR: For a survey of generalizations of the isoperimetric inequality, see as mentioned in this paper, where the main focus is on geometric versions and generalisations of the inequality, with emphasis on recent contributions.