Tootfinder

Warped products over one-dimensional base spaces and the RCD condition
Christian Ketterer
https://arxiv.org/abs/2506.10809 https://ar…

Warped products over one-dimensional base spaces and the RCD condition
We prove the Riemannian curvature-dimension condition $\mathsf{RCD}(KN,N+1)$ for an $N$-warped product $B\times_f^N F$ over a one-dimensional base space $B$ with a Lipschitz function $f: B\rightarrow \mathbb R_{\geq 0}$, provided (1) $f$ is a $Kf$-concave function, (2) $f$ satisfies a sub-Neumann boundary condition $\frac{\partial f}{\partial n}\geq 0$ on $\partial B\backslash f^{-1}(0)$ and $F$ is a compact metric measure space satisfying (3) the condition $\mathsf{RCD}(K_F (N-1), N)$ with $K_…

Improved girth approximation in weighted undirected graphs
Avi Kadria, Liam Roditty, Aaron Sidford, Virginia Vassilevska Williams, Uri Zwick
https://arxiv.org/abs/2507.13869

Improved girth approximation in weighted undirected graphs
Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ…