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Related Articles from SNS
The Preisach Extremum Stack is a Shannon-Minimal Sufficient Statistic for Rate-Independent Functionals
Announce Type: new Abstract: Let R denote the class of all computable, causal functionals that are rate-independent in the classical sense (invariant under monotone time reparametrizations), and let Pi_n be the Preisach extremum stack of an input sequence u_{0:n}. We prove a characterization theorem establishing that every F in R satisfies Fu = f(Pi_n) for a computable f, and derive two information-theoretic results. First, under any probability measure on u_{0:n}, the equality I(u_{0:n};...
Optimizing Explicit Unit-Distance Lower-Bound Certificates
arXiv:2606.03419v4 Announce Type: replace-cross Abstract: The 2026 disproof of Erd\H{o}s's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report treats Sawin's parameter...
Optimizing Explicit Unit-Distance Lower-Bound Certificates
arXiv:2606.03419v3 Announce Type: replace-cross Abstract: The 2026 disproof of Erd\H{o}s's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report starts from Sawin's...
Optimizing Explicit Unit-Distance Lower-Bound Certificates
arXiv:2606.03419v1 Announce Type: cross Abstract: The 2026 disproof of Erd\H{o}s's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes finite parameters whose choice is not fully optimized. This report formulates the...
Optimizing Explicit Unit-Distance Lower-Bound Certificates
arXiv:2606.03419v2 Announce Type: replace-cross Abstract: The 2026 disproof of Erd\H{o}s's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report starts...
Strong Polarization and Entropy
arXiv:2606.02567v1 Announce Type: cross Abstract: We show that for any set of $n$ unit vectors $v_1,\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\ldots,p_n$ satisfying $\sum_j p_j = 1$, there exists a unit vector $u$ such that \[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequality for products of powers of linear functionals and a...
On the sharp linear convergence rate of the circumcentered--reflection method on subspaces
arXiv:2606.07888v1 Announce Type: cross Abstract: For two subspaces $U,V\subseteq\RR^n$, the circumcentered--reflection method (CRM) of Behling, Bello-Cruz, and Santos~\cite{BBS2018} computes the projection onto $U\cap V$ using only the reflections across $U$ and $V$, with known linear-convergence rate $c_F$, the cosine of the Friedrichs angle. We prove that, when CRM is initialized in $V$, it contracts at the strictly smaller rate...
WWDC 2026 bonus live blog: Tech Talk with Craig Federighi
One of the funnier parts of the keynote. Fresh off the WWDC keynote presentation, The Verge has been invited to an "on-the-record technical deep dive into the bold new architecture enabling Apple Intelligence capabilities." Apple SVP of Software Engineering Craig Federighi and his team will be there, and so will we.
Shantell Sans
The Story of Shantell Sans Shantell Sans mixes variable axes for Weight, Italic, Informality, and Bounce to deliver a wide array of font styles, from friendly, readable, everyday typographic workhorses to striking, high-energy, experimental styles meant especially for animation. This is the story behind its inspiration and creation. Shantell Martin, Artist One of my first relationships with words was back in elementary school.
An extremal problem for completely unclustered Burrows-Wheeler images
arXiv:2606.01267v1 Announce Type: cross Abstract: The Burrows--Wheeler transform is usually viewed as a clustering transform: it tends to group equal letters into long runs. We study the opposite extremal regime, where the BWT output is completely unclustered, that is, has as many equal-letter runs as positions. Known results imply, on the one hand, that the number of runs in the BWT of a Lyndon word can increase by at most a factor of two, and, on the other hand, that over every alphabet of...