arXiv subjects

Jacob Martin

Publications and source records attributed to Jacob Martin.

3 recordsLinked to original sources

arXiv2026

Leading term strandings for webs

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, which constrains the boundary data of such strandings enough to yield a sufficient criterion for a set of webs to form a web basis. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis, giving a non-recursive construction of Fontaine's $\mathfrak{sl}_n$ web bases. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we show a leading term stranding for any $\mathfrak{sl}_3$ web can be reached from an arbitrary stranding via a sequence of operations called strand reversals.

math.CO↗

arXiv2026

Negative Latin-Square-Type Partial Difference Sets in Non-Elementary Abelian 2-Groups

Using cubic cyclotomic classes, character theory, and product constructions motivated by generalized Denniston partial difference sets, we construct negative Latin-square-type partial difference sets in $\mathbb{F}_{2^6}^{+}\times\mathbb{Z}_4^4$, $\mathbb{Z}_4^4\times\mathbb{F}_{2^{10}}^{+}$, $\mathbb{F}_{2^{10}}^{+}\times\mathbb{Z}_8^4$, $\mathbb{Z}_8^4\times\mathbb{F}_{2^{14}}^{+}$, and $\mathbb{Z}_4^4\times\mathbb{Z}_{16}^2$. The first four constructions replace cubic cyclotomic partitions by partitions of non-elementary abelian 2-groups having the same character-value patterns. To the best of our knowledge, these are the first partial difference sets with the stated parameters in the indicated groups.

math.CO↗

arXiv2025

Modeling smooth and localized mortality patterns across age, time, and space to uncover small-area inequalities

Small-area mortality estimation is inherently difficult, as random fluctuations from low death counts can obscure real geographic differences. We introduce a flexible model that borrows strength across age, space, and time to estimate mortality schedules and trends in very small populations. The approach ensures smooth patterns across these dimensions while allowing localized breaks from the spatial structure, capturing broad trajectories as well as sharp local contrasts. We implement our model within a Penalized Spline framework and estimate it using Generalized Linear Array Model techniques, resulting in a computationally fast, interpretable, and parsimonious method. Crucially, it can readily incorporate sudden mortality shocks, such as the Covid-19 pandemic, making it highly versatile for real-world demographic and epidemiological challenges. We demonstrate its application by estimating life expectancy and age-specific mortality inequalities in over 4,800 small areas across the Greater London Authority from 2002 to 2024.

stat.AP↗