Question: Foundress associations of the desert leaf-cutter ant Acromyrmex versicolor exhibit a single forager-foundress prior to worker eclosion. A prior specialized task, removing soil from the nest during excavation, allocates later forager risk. Experiment reveals that a queen refusing such prior assignment is not replaced by her cofoundresses; rather, these latter suicidally punish the shir...

It is shown that the graph Γn that has the set of all n× n symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if n≥ 3. Eigenvalues of the graph Γn are calculated as well.

A celebrated 1942 result of Schoenberg characterizes all entry-wise functions which preserve positivity of matrices of any size. I will present a characterization of polynomials which preserve positivity when applied entry-wise on matrices of a fixed dimension. All put in historical context and motivated by recent demands of statistics of large data and optimization theory. A sketch of the proo...

Wemake improvements to the current upper bounds on pairwise distance preservers and additive spanners. A distance preserver is a sparse subgraph that exactly preserves all distances in a given pair set P . We show that every undirected unweighted graph has a distance preserver on O(max{n|P |, n|P |}) edges, and we conjecture that O(n|P |+ n) is possible. An additive subset spanner is a sparse s...

Norm preserver maps of Jordan product on the algebra Mn of n×n complex matrices are studied, with respect to various norms. A description of such surjective maps with respect to the Frobenius norm is obtained: Up to a suitable scaling and unitary similarity, they are given by one of the four standard maps (identity, transposition, complex conjugation, and conjugate transposition) on Mn, except ...