Lebesgue
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Related Articles from SNS
A Reproducible Certificate for the Brass--Sharifi Lower Bound in Lebesgue's Universal Cover Problem
arXiv:2606.04458v1 Announce Type: new Abstract: Brass and Sharifi proved the lower bound 0.832 for the convex form of Lebesgue's universal cover problem by combining geometric estimates with a computer search over placements of a disk, an equilateral triangle, and a regular pentagon. This paper gives a certificate-based reproduction of that computation. The certificate consists of a finite adaptive ledger, a terminal-route replay, three local lower-bound certificate families, compact...
A Reproducible Certificate for the Brass$-$Sharifi Lower Bound in Lebesgue's Universal Cover Problem
arXiv:2606.04458v2 Announce Type: replace Abstract: Brass and Sharifi proved the lower bound 0.832 for the convex form of Lebesgue's universal cover problem by combining geometric estimates with a computer search over placements of a disk, an equilateral triangle, and a regular pentagon. This paper gives a certificate-based reproduction of that computation. The finite record consists of an adaptive ledger, a terminal-route replay, three local lower-bound certificate families, compact...
Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. III
Announce Type: replace Abstract: We establish the last missing link allowing to describe those complete separable metric spaces $X$ in which the $k$ nearest neighbour classifier is universally consistent, both in combinatorial terms of dimension theory and via a fundamental property of real analysis. The following are equivalent: (1) The $k$-nearest neighbour classifier is universally consistent in $X$, (2) The strong Lebesgue--Besicovitch differentiation property holds in $X$ for every...
Generating Rectifiable Measures through Neural Networks
Announce Type: replace Abstract: We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural...
Bayesian learning for the stochastic shortest path problem
Announce Type: cross Abstract: Sequential decision-making problems are often modelled as a Markov decision process (MDP). We focus on the stochastic shortest path (SSP) problem, which is an infinite-horizon undiscounted MDP with absorbing terminal states. We develop a Bayesian framework to learn the optimal decision strategy through interactions with the decision-making task.