We show that, for any given dimension ≥ 2, the range of distinct possible designs for periodic frameworks with auxetic capabilities is infinite. We rely on a purely geometric approach to auxetic trajectories developed within our general theory of deformations of periodic frameworks.

The log-det distance between two aligned DNA sequences was introduced as a tool for statistically consistent inference of a gene tree under simple non-mixture models of sequence evolution. Here we prove that the log-det distance, coupled with a distance-based tree construction method, also permits consistent inference of species trees under mixture models appropriate to aligned genomic-scale sequences data. Data may include sites from many genetic loci, which evolved on different gene trees due to incomplete lineage sorting on an ultrametric species tree, with different time-reversible substitution processes. The simplicity and speed of distance-based inference suggests log-det based methods should serve as benchmarks for judging more elaborate and computationally-intensive species trees inference methods.