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A mass transportation approach to quantitative isoperimetric inequalities
A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved
The power of quantum neural networks
This work is the first to demonstrate that well-designed quantum neural networks offer an advantage over classical neural networks through a higher effective dimension and faster training ability, which is verified on real quantum hardware.
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies
We obtain a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of
The Optimal Partial Transport Problem
Given two densities f and g, we consider the problem of transporting a fraction $${m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}$$ of the mass of f onto g minimizing a transportation cost. If the cost
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations
In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting
Hölder Continuity and Injectivity of Optimal Maps
Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y).