A Counterexample to a Theorem of Chase

On (po-)torsion free and principally weakly (po-)flat $S$-posets

In this paper, we first consider (po-)torsion free and principally weakly (po-)flat $S$-posets, specifically  we discuss when (po-)torsion freeness implies principal weak (po-)flatness. Furthermore, we give a counterexample to show that Theorem 3.22 of Shi is incorrect. Thereby we present a correct version of this theorem. Finally, we characterize pomonoids over which all cyclic $S$-posets are ...

Picard ' S Theorem by Douglas Bridges

This paper deals with the numerical content of Picard's Thsorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.

Hastings’s Additivity Counterexample via Dvoretzky’s Theorem

The goal of this note is to show that Hastings’s counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky’s theorem.

Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.

Why Learning How to Chase Butterflies Matters: A Response to Recent Commentaries