arXiv subjects

Elias Williams

Publications and source records attributed to Elias Williams.

3 recordsLinked to original sources

arXiv2025

Small 3-fold blocking sets in $\mathrm{PG}(2,p^n)$

A $t$-fold blocking set of the finite Desarguesian plane $\mathrm{PG}(2,p^n)$, $p$ prime, is a set of points meeting each line of the plane in at least $t$ points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. $1$-fold blocking sets not containing a line) in $\mathrm{PG}(2,p^n)$ is an open question when $n\geq 5$ is odd. For $n>1$ the conjectured lower bound for this size is $(p^n+p^{n(s-1)/s}+1)$, where $p^{n/s}$ is the size of the largest proper subfield of $\mathbb{F}_{p^n}$. Since the union of $t$ pairwise disjoint nontrivial blocking sets is a $t$-fold blocking set, it is conjectured that when $p^{n/s}$ is large enough w.r.t. $t$, then the minimum size of a $t$-fold blocking set in $\mathrm{PG}(2,p^n)$ is $t(p^n+p^{n(s-1)/s}+1)$. If $n$ is even, then the decomposition of the plane into disjoint Baer subplanes gives a $t$-fold blocking set of this size. However, for odd $n$, the existence of such sets is an unsolved problem in most cases. In this paper, we construct $3$-fold blocking sets of conjectured size. These blocking sets are obtained as the disjoint union of three linear blocking sets of R\'edei type, and they lie on the same orbit of the projectivity $(x:y:z)\mapsto (z:x:y)$.

math.CO↗

arXiv2025

Covering spiky annuli by planks

Answering Tarski's plank problem, Bang showed in 1951 that it is impossible to cover a convex body $K \subset \mathbb{R}^d$ with $d \geq 1$ by planks whose total width is less than the minimal width $w(K)$ of $K$. In 2003, A. Bezdek asked whether the same statement holds if one is required to cover only the annulus obtained from $K$ by removing a homothetic copy contained within. He showed that if $K$ is the unit square, then saving width in a plank covering is not possible, provided that the homothety factor is sufficiently small. White and Wisewell in 2006 characterized polygons that possess this property. We generalize the constructive part of their classification to spiky convex bodies: a body $K$ is spiky at a boundary point $x$ with supporting hyperplane $H$ and corresponding outer normal $u$, if both $K$ and its tangent cone at $x$ intersect $H$ only at $x$. We show that if $K$ is a convex disc or a convex body in 3-space that is spiky in a minimal width direction, then for every $\varepsilon \in (0,1)$ it is possible to cut a homothetic copy $\varepsilon K$ from the interior of $K$ so that the remaining annulus can be covered by planks whose total width is strictly less than $w(K)$.

math.MG↗

arXiv2025

Benchmarking and optimizing organism wide single-cell RNA alignment methods

Many methods have been proposed for removing batch effects and aligning single-cell RNA (scRNA) datasets. However, performance is typically evaluated based on multiple parameters and few datasets, creating challenges in assessing which method is best for aligning data at scale. Here, we introduce the K-Neighbors Intersection (KNI) score, a single score that both penalizes batch effects and measures accuracy at cross-dataset cell-type label prediction alongside carefully curated small (scMARK) and large (scREF) benchmarks comprising 11 and 46 human scRNA studies respectively, where we have standardized author labels. Using the KNI score, we evaluate and optimize approaches for cross-dataset single-cell RNA integration. We introduce Batch Adversarial single-cell Variational Inference (BA-scVI), as a new variant of scVI that uses adversarial training to penalize batch-effects in the encoder and decoder, and show this approach outperforms other methods. In the resulting aligned space, we find that the granularity of cell-type groupings is conserved, supporting the notion that whole-organism cell-type maps can be created by a single model without loss of information.

cs.LG↗