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Abstract We prove that among all 1-periodic configurations $$\Gamma $$ of points on the real line $$\mathbb{R}$$ the quantities $$\min _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma )^{2}}$$ and $$\max _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma )^{2}}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $$n$$ per period is sufficiently large (depending on $$\alpha $$). This solves the polarization problem for periodic configurations with a Gaussian weight on $$\mathbb{R}$$ for large $$n$$. The first result is shown using Fourier series. The second result follows from the work of Cohn and Kumar on universal optimality and holds for all $$n$$ (independent of $$\alpha $$).
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Abstract Let be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices , take to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices . We prove that this suggests that the graphGis, in a suitable sense, “m‐dimensional” by exhibiting an explicit 1‐Lipschitz embedding with good properties.more » « less
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Abstract We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by have diameter bounded by (a Bonnet–Myers theorem), that implies that has constant curvature (a Cheng theorem) and that there is a spectral gap (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin–Lu–Yau curvature. The von Neumann Minimax theorem features prominently in the proofs.more » « less
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Abstract We consider linear systems $Ax = b$ where $$A \in \mathbb{R}^{m \times n}$$ consists of normalized rows, $$\|a_i\|_{\ell ^2} = 1$$, and where up to $$\beta m$$ entries of $$b$$ have been corrupted (possibly by arbitrarily large numbers). Haddock, Needell, Rebrova & Swartworth propose a quantile-based Random Kaczmarz method and show that for certain random matrices $$A$$ it converges with high likelihood to the true solution. We prove a deterministic version by constructing, for any matrix $$A$$, a number $$\beta _A$$ such that there is convergence for all perturbations with $$\beta < \beta _A$$. Assuming a random matrix heuristic, this proves convergence for tall Gaussian matrices with up to $$\sim 0.5\%$$ corruption (a number that can likely be improved).more » « less
