In the work of Cromwell and Dynnikov, grid equivalence is given by the grid moves commutation, (de-)stabilization and cyclic permutation. This paper gives a proof that cyclic permutation is a sequence of (de-)stabilization and commutation grid moves.

Statistical inference procedures that require no distributional assumptions make up the area of nonparametric statistics. The permutation test is a common nonparametric test that can be used to compare measures of center for two data sets, but it is yet to be explored for three-dimensional rotation data. A permutation test for such data is developed and the statistical power of this test is considered under various scenarios. The test is then used in an application comparing movement around joints in the foot and ankle for humans, chimpanzees, and baboons.

We find examples of positive integers $n$ such that $\varphi \left(n^3\right)\sigma \left(n^3\right)$ is a perfect square.

A new proof is given that the symmetric group of any set $X$ with three or more elements, finite or infinite, has cardinality strictly greater than that of $X$. Use of the axiom of choice is avoided throughout.

We study adjacency matrices of zero-divisor graphs of ${\mathbb{Z}}_n$ for various $n$. We find their determinant and rank for all $n$, develop a method for finding nonzero eigenvalues, and use it to find all eigenvalues for the case $n=p^3$, where $p$ is a prime number. We also find upper and lower bounds for the largest eigenvalue for all $n$.

Edge-flip distance between triangulations of polygons is equivalent to rotation distance between rooted binary trees. Both distances measure the extent of similarity of configurations. There are no known polynomial-time algorithms for computing edge-flip distance. The best known exact universal upper bounds on rotation distance arise from measuring the maximum total valence of a vertex in the corresponding triangulation pair obtained by a duality construction. Here we describe some properties of the distribution of maximum vertex valences of pairs of triangulations related to such upper bounds.

This paper provides a novel proof of a generalization of Pappus’ centroid theorem on $n$-dimensional tubes using Stokes’ theorem on manifolds.

We investigate the level sets of extremal Sobolev functions. For $\Omega \subset {ℝ}^n$ and $1\le p<2n∕\left(n-2\right)$, these functions extremize the ratio $\parallel \nabla u{\parallel }_{L^2\left(\Omega \right)}∕\parallel u{\parallel }_{L^p\left(\Omega \right)}$. We conjecture that as $p$ increases, the extremal functions become more “peaked” (see the introduction below for a more precise statement), and present some numerical evidence to support this conjecture.

The school choice problem (SCP) looks at assignment mechanisms matching students in a public school district to seats in district schools. The Gale–Shapley deferred acceptance mechanism applied to the SCP, known as the student optimal stable matching (SOSM), is the most efficient among stable mechanisms yielding a solution to the SCP. A more recent mechanism, the efficiency adjusted deferred acceptance mechanism (EADAM), aims to address the well-documented tension between efficiency and stability illustrated by SOSM. We introduce two alternative efficiency adjustments to SOSM, both of which necessarily sacrifice stability. Our discussion focuses on the mathematical novelty of new efficiency modifications rather than any practical superiority of implementation or outcome. That is, our contribution lies in process rather than outcome. Yet we argue that the demonstration of multiple processes yielding common outcomes is, in itself, a measure of the quality of that outcome. More specifically the consistency of outcome from different processes strengthens the argument that Pareto dominations of SOSM can be supported as “fair” despite the resulting priority violations.

Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs and that they can be told apart by their chromatic polynomials, by showing that the latter give distinct results when evaluated at 3. He conjectured that the six different signed Petersen graphs also have distinct zero-free chromatic polynomials, and that both types of chromatic polynomials have distinct evaluations at any positive integer. We developed and executed a computer program (running in SAGE) that efficiently determines the number of proper $k$-colorings for a given signed graph; our computations for the signed Petersen graphs confirm Zaslavsky’s conjecture. We also computed the chromatic polynomials of all signed complete graphs with up to five vertices.

In 1992, Elkies, Kuperberg, Larsen, and Propp introduced a bijection between domino tilings of Aztec diamonds and certain pairs of alternating-sign matrices whose sizes differ by one. In this paper we first study those smaller permutations which, when viewed as matrices, are paired with the matrices for doubly alternating Baxter permutations. We call these permutations snow leopard permutations, and we use a recursive decomposition to show they are counted by the Catalan numbers. This decomposition induces a natural map from Catalan paths to snow leopard permutations; we give a simple combinatorial description of the inverse of this map. Finally, we also give a set of transpositions which generates these permutations.

Benford’s law states that many data sets have a bias towards lower leading digits (about 30% are 1s). It has numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It’s important to know which common probability distributions are almost Benford. We show that the Weibull distribution, for many values of its parameters, is close to Benford’s law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.

In this paper we examine finite unions of unit squares in same plane and consider the ratio of perimeter to area of these unions. In 1998, T. Keleti published the conjecture that this ratio never exceeds 4. Here we study the continuity and differentiability of functions derived from the geometry of the union of those squares. Specifically we show that if there is a counterexample to Keleti’s conjecture, there is also one where the associated ratio function is differentiable.

We prove that the well-covered dimension of the Levi graph of a point-line configuration with $v$ points, $b$ lines, $r$ lines incident with each point, and every line containing $k$ points is equal to $0$, whenever $r>2$.