Gradient flows on nonpositively curved metric spaces and harmonic maps

Comm. Anal. Geom., 6, no. 2, pp. 199-253. 1998
Gradient flows on nonpositively curved metric spaces and harmonic maps
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Abstract

The notion of gradient flows is generalized to a metric space setting without any linear structure. The metric spaces considered are a generalization of Hilbert spaces. The properties of such metric spaces are used to set up a finite-difference scheme of variational form.

The proof of the Crandall-Liggett generation theorem is adapted to show convergence. The resulting flow generates a strongly continuous semigroup of Lipschitz-continuous mappings, is Lipschitz continuous in time for positive time, and decreases the energy functional along a path of steepest descent.

In case the underlying metric space is a Hilbert space, the solutions resulting from this new theory coincide with those obtained by classical methods. As an application, the harmonic map flow problem for maps from a manifold into a nonpositively curved metric space is considered, and the existence of a solution to the initial boundary value problem is established.

Another publication from the same author: Uwe Mayer

Proceedings of the Sixteenth ACM Conference on Economics and Computation (EC '15). ACM, New York, NY, USA (2015)

Canary in the e-Commerce Coal Mine: Detecting and Predicting Poor Experiences Using Buyer-to-Seller Messages

Dimitriy Masterov, Uwe Mayer, Steve Tadelis

Reputation and feedback systems in online marketplaces are often biased, making it difficult to ascertain the quality of sellers. We use post-transaction, buyer-to-seller message traffic to detect signals of unsatisfactory transactions on eBay. We posit that a message sent after the item was paid for serves as a reliable indicator that the buyer may be unhappy with that purchase, particularly when the message included words associated with a negative experience. The fraction of a seller's message traffic that was negative predicts whether a buyer who transacts with this seller will stop purchasing on eBay, implying that platforms can use these messages as an additional signal of seller quality.

Another publication from the same category: Mathematics

Interfaces and Free Boundaries. 4, no. 1, pp. 89-109. 2002

A numerical scheme for axisymmetric solutions of curvature driven free boundary problems, with applications to the Willmore Flow

Uwe Mayer, Gieri Simonett

We present a numerical scheme for radially symmetric solutions to curvature driven moving boundary problems governed by a local law of motion, e.g. the mean curvature flow, the surface diffusion flow, and the Willmore flow. We then present several numerical experiments for the Willmore flow. In particular, we provide numerical evidence that the Willmore flow can develop singularities in finite time.

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