Related papers: Finiteness of Disjoint Covering Systems with Preci…

We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for…

It is widely recognized that no tractable necessary and sufficient conditions exist for determining whether a system is, in general, differentially flat. However, specific cases do provide such conditions. For instance, driftless systems…

We complete the determination of saturated fusion systems on maximal class 3-groups of rank two.

The concept of a covering system was first introduced by Erd\H{o}s in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of covering systems with certain restrictions on the moduli.…

Kneser-Haken Finiteness asserts that for each compact 3-manifold M there is an integer c(M) such that any collection of k>c(M) closed, essential, 2-sided surfaces in M must contain parallel elements. We show here that if M is closed then…

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for…

Given an implicational base, a well-known representation for a closure system, an inconsistency binary relation over a finite set, we are interested in the problem of enumerating all maximal consistent closed sets (denoted by MCCEnum for…

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two…

We study open zooming systems and potentials with uniqueness of equilibrium states. The uniqueness is established for a certain class of zooming potentials when the map is topologically exact, including the null one. Also, with equilibrium…

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions. In earlier work, Stanley proved a…

A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if…

We give a new proof of the fact that any finite quadratic module can be decomposed into indecomposable ones. For any indecomposable finite quadratic module, we construct a lattice, and a positive definite lattice, both of which are of the…

We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique…

We consider graphs on monomials in $n$ variables of a fixed degree $d$ where two monomials are adjacent if and only if their least common multiple has degree $d+1$. We prove that when $n = 3$ and $d$ is divisible by $3$ as well as when…

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a…

We show that an adjoint of a loopless matroid is connected if and only if it itself is connected. Our first goal is to study the adjoint of modular matroids. We prove that a modular matroid has only one adjoint (up to isomorphism) which can…

We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in R^4 that looks like a union of two almost orthogonal planes at infinity is a cone. The main point…

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.

Let $\mathcal{F}$ be a set of finite groups. A finite group $G$ is called an \emph{$\mathcal{F}$-cover} if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. An $\mathcal{F}$-cover is called \emph{minimal} if no proper…

Given a permutation w, we look at the range of how often a simple reflection s_k appears in reduced decompositions of w. We compute the minimum and give a sharp upper bound on the maximum. That bound is in terms of 321- and 3412-patterns in…