A counterexample on relative regular neighborhoods

A Counterexample to Thiagarajan's Conjecture on Regular Event Structures

We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular prime event structures correspond exactly to those obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded \-cliques are recognizable by finite t...

Regular Neighborhoods Are Not Topologically Invariant

A counterexample to monotonicity of relative mass in random walks

For a finite undirected graph G = (V, E), let pu,v(t) denote the probability that a continuoustime random walk starting at vertex u is in v at time t. In this note we give an example of a Cayley graph G and two vertices u, v ∈ G for which the function ru,v(t) = pu,v(t) pu,u(t) t ≥ 0 is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

Codimension Two Pl Embeddings of Spheres with Nonstandard Regular Neighborhoods

Abstract. For a given polyhedron K ⊂ M the notation RM (K) denotes a regular neighborhood of K in M . We study the following problem: find all pairs (m,k) such that if K is a compact k-polyhedron and M a PL m-manifold, then RM (fK) ∼= RM (gK), for each two homotopic PL embeddings f, g : K → M . We prove that R Sk+2 (S) 6∼= S × D for each k ≥ 2 and some PL sphere S ⊂ S (even for any PL sphere S ...

A note on semi-regular locales

Semi-regular locales are extensions of the classical semiregular spaces. We investigate the conditions such that semi-regularization is a functor. We also investigate the conditions such that semi-regularization is a reflection or coreflection.