arXiv:2604. 02995v3 Announce Type: replace-cross Abstract: We introduce the penalised Saito functional $\mathfrak S_{\lambda,\beta}(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ of $n$ lines and a prescribed pair $d_1+d_2=n-1$.
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 $\varepsilon$.
arXiv:2504. 19451v4 Announce Type: replace-cross Abstract: This paper presents two concrete applications of Artificial Intelligence to algorithmic and analytic number theory.
arXiv:2609. 17578v1 Announce Type: cross Abstract: Let $a_1,\ldots,a_m$ be distinct elements of a field $K$, and let $Z(X)=\prod_{i=1}^m (X-a_i)$.
arXiv:2605. 29151v2 Announce Type: replace-cross Abstract: We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli.
arXiv:2607. 23500v1 Announce Type: cross Abstract: Razborov's flag algebra method is a powerful tool for proving asymptotic inequalities in extremal graph theory, often reducing the task to finding a finite certificate by semidefinite programming.
arXiv:2610. 00545v1 Announce Type: new Abstract: We study adversarial online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed sets.
arXiv:2609. 29881v1 Announce Type: cross Abstract: Choosing Shellsort gaps is a well-known open problem.
arXiv:2608. 11211v1 Announce Type: new Abstract: Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists.
arXiv:2504. 19952v2 Announce Type: replace-cross Abstract: We present two general lower bounds for stopping times of sequential tests between arbitrary composite nulls $\mathcal P$ and alternatives $\mathcal Q$.
arXiv:2608. 13549v1 Announce Type: new Abstract: The per-instance Jaccard score, or intersection over union (IoU), is standard in multi-label classification and binary segmentation.
We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_{q}$-norm over an $\ell_{p}$-ball of radius $R$, where $1\leq p,q\leq \infty$. For $p<q$, we obtain error $\widetilde{O}_{p,q}(GR/T^{1/p-(1/q-1/2)_{+}})$ after $T$ oracle queries, efficiently realizing the nearly optimal rates of (MBG+26), thereby resolving the nonsmooth end of the COLT 2015 open problem (Guz15b).
arXiv:2609. 20701v1 Announce Type: cross Abstract: We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_{q}$-norm over an $\ell_{p}$-ball of radius $R$, where $1\leq p,q\leq \infty$.