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The Grothendieck Constant is Less Than $\frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$
Computer Science > Data Structures and Algorithms [Submitted on 2 Jun 2026] Title:The Grothendieck Constant is Less Than $\fracπ{2 \log (1+ \sqrt{2})} - 10^{-5}$ View PDF HTML (experimental)Abstract:We prove that the Grothendieck constant $K_G < $\frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$. This improves on the work of braverman et.
The Grothendieck Constant is Less Than $\frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$
Computer Science > Data Structures and Algorithms [Submitted on 2 Jun 2026 (v1), last revised 6 Jun 2026 (this version, v2)] Title:The Grothendieck Constant is Less Than $\fracπ{2 \log (1+ \sqrt{2})} - 10^{-5}$ View PDF HTML (experimental)Abstract:We prove that the Grothendieck constant $K_G 0$. Submission history From: Pravesh K Kothari [view email][v1] Tue, 2 Jun 2026 17:59:53 UTC (829 KB)
An Upper Bound on Grothendieck's Constant
Announce Type: cross Abstract: We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-500}$. As a corollary of our result, we prove the bound...
Incremental BPE Tokenization
arXiv:2605.30813v1 Announce Type: new Abstract: We propose a novel algorithm for incremental Byte Pair Encoding (BPE) tokenization. The algorithm processes each input byte in worst-case $\mathcal{O}(\log^2 t)$ time, leading to an overall complexity of $\mathcal{O}(n \log^2 t)$, where $n$ is the input length and $t$ is the maximum token length. The algorithm incrementally maintains BPE tokenization results for every prefix of the input text, implementing the standard BPE merge procedure...
Complementary Time-Space Tradeoff for Self-Stabilizing Leader Election: Polynomial States Meet Sublinear Time
arXiv:2505.23649v3 Announce Type: replace Abstract: We study the self-stabilizing leader election (SS-LE) problem in the population protocol model, assuming exact knowledge of the population size $n$. Burman, Chen, Chen, Doty, Nowak, Severson, and Xu [BCC+21a] (PODC) showed that this problem can be solved in $O(n)$ expected time with $O(n)$ states. Recently, G\k{a}sieniec, Grodzicki, and Stachowiak [GGS25] (PODC) proved that $n+O(\log n)$ states suffice to achieve $O(n \log n)$ time both in...
Fast Bounded-Independence Functions and Their Duals
arXiv:2606.07009v1 Announce Type: new Abstract: We continue the study of {\em fast} functions, computable by linear-size circuits, that share useful properties of random functions. Motivated by cryptographic applications, we generalize and improve on previous results in this area, obtaining the following results: - For any constant $t$, we construct a fast $t$-wise independent hash function with algebraic degree $\log_2 t$ (over $\mathbb F_2$), simultaneously optimizing both asymptotic...
Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
arXiv:2606.02596v1 Announce Type: new Abstract: The curvature exponent $\alpha$ in $h_k \propto \sigma_k^\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\alpha \approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). We prove the Spectral Alignment Decomposition: $\alpha = 2 + d\log\Phi_k / d\log\sigma_k$, where $\Phi_k$ measures alignment between Kronecker factor...
Randomized Least Squares Value Iteration itself is Joint Differentially Private
arXiv:2606.01952v1 Announce Type: new Abstract: As reinforcement learning (RL) increasingly applies to sensitive domains, such as health care and recommendation systems, privacy-preserving techniques have become essential to protect users' sensitive information. We investigate privacy-preserving RL under an episodic setting, focusing on algorithms based on randomized exploration, such as Randomized Least Squares Value Iteration (RLSVI).
Measles is still spreading in the U.S., with more than 2,000 cases this year
Measles cases in the U.S. reached 2,030 on Friday, the Centers for Disease Control and Prevention reported. That’s just a few hundred shy of the 2,288 logged in all of 2025, a record-breaking year that saw more measles diagnoses than any year since 1991. There have been 30 new outbreaks this year, compared to 48 last year, the CDC said.
AZD5582 robustly reactivates latently infected cells and clears the majority of those reactivated from the SIV reservoir
AZD5582 (AZD) is a latency reversing agent used to support the "shock-and-kill" strategy in HIV-1 cure research. Previous studies in ART-suppressed rhesus macaques have shown that AZD can promote reactivation of latently infected cells, resulting in 2-3 log increases in on-ART viral load and significant reductions in SIV reservoir size over 5-10 doses. To quantify the impact of AZD on the reservoir, we developed an ensemble of mechanistic viral dynamic models and fit them to longitudinal...