Tootfinder

The Smith normal form of distance matrices of high dimensional trees
Carlos A. Alfaro, Jes\'us Uriel Medrano, Iv\'an T\'ellez T\'ellez
https://arxiv.org/abs/2510.04471

The Smith normal form of distance matrices of high dimensional trees
Graham-Lovász-Pollak \cite{GL,GP} obtained the celebrated formula $$\det({\sf D}(T_{n+1}))=(-1)^nn2^{n-1},$$ for the determinant of the distance matrix ${\sf D}(T_{n+1})$ for any tree $T_{n+1}$ with $n+1$ vertices. Later, Hou and Woo \cite{HW} extended this formula to the Smith normal form (SNF) obtaining that $\SNF({\sf D}(T_{n+1}))={\sf I}_2\oplus 2{\sf I}_{n-2}\oplus [2n]$, for any tree $T_{n+1}$ with $n+1$ vertices. A $k$-{\it tree} is either a complete graph on $k$ vertices or a graph o…

Identification in source apportionment using geometry
Bora Jin, Abhirup Datta
https://arxiv.org/abs/2510.03616 https://arxiv.org/pdf/2510.03616

Identification in source apportionment using geometry
Source apportionment analysis, which aims to quantify the attribution of observed concentrations of multiple air pollutants to specific sources, can be formulated as a non-negative matrix factorization (NMF) problem. However, NMF is non-unique and typically relies on unverifiable assumptions such as sparsity and uninterpretable scalings. In this manuscript, we establish identifiability of the source attribution percentage matrix under much weaker and more realistic conditions. We introduce the …