arXiv subjects
Aaron Abrams
Publications and source records attributed to Aaron Abrams.
Configuration Spaces of Colored Graphs
This paper is intended to provide concrete examples of concepts discussed elsewhere in this volume, especially splittings of groups and non-positively curved cube complexes but also other things. The idea of the construction (configuration spaces) is not new, but this family of examples doesn't seem to be well-known. Nevertheless they arise in a variety of contexts; applications are discussed in the last section. Most proofs are omitted.
Spaces of polygonal triangulations and Monsky polynomials
Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a generalized triangulation $\T$ of a square must satisfy a single irreducible homogeneous polynomial relation $p(\T)$ depending only on the combinatorics of $\T$. The invariant $p(\T)$ is called the \emph{Monsky polynomial}; it captures algebraic, geometric, and combinatorial information about $\T$. We give an algorithm that computes a lower bound on the degree of $p(\T)$, and we present several examples in which the algorithm is used to compute the degree.
Finding good bets in the lottery, and why you shouldn't take them
We give a criterion under which the expected return on a ticket for certain large lotteries is positive. In this circumstance, we use elementary portfolio analysis to show that an optimal investment strategy includes a very small allocation for such tickets.
Evasive Random Walks and the Clairvoyant Demon
A pair of random walks $(R,S)$ on the vertices of a graph $G$ is {\it successful} if two tokens can be scheduled (moving only one token at a time) to travel along $R$ and $S$ without colliding. We consider questions related to P. Winkler's {\it clairvoyant demon problem}, which asks whether for random walks $R$ and $S$ on $G$, $Pr[\ (R,S) \mbox{ is successful }] >0$. We introduce the notion of an {\it evasive} walk on $G$: a walk $S$ so that for a random walk $R$ on $G$, $Pr[\ (R,S) \mbox{ is successful }]>0$. We characterize graphs $G$ having evasive walks, giving explicit constructions on such $G$. On a cycle, we show that with high probability the tokens must collide quickly. Finally we consider two variants of the problem for which, under certain assumptions on the graph $G$, we provide algorithms that schedule $(R,S)$ successfully with positive probability.
An iterated random function with Lipschitz number one
Consider the set of functions $f_{\theta}(x)=|\theta -x|$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0 \in \mathbb{R}$ and continues with $x_{k+1}=f_{\theta_{k+1}}(x_{k})$ with each $\theta _{k+1}$ picked from a fixed bounded distribution $\mu$ on $\mathbb{R}^+$. We prove the conjecture of G. Letac that if $\mu$ is not supported on a lattice, then this process has a unique stationary distribution $\pi_{\mu}$ and any distribution converges under iteration to $\pi_{\mu}$ (in the weak-$^*$ topology). We also give a bound on the rate of convergence in the special case that $\mu$ is supported on a two-point set. We hope that the techniques will be useful for the study of other Markov processes where the transition functions have Lipschitz number one.
Optimal estimators for threshold-based quality measures
We consider a problem in parametric estimation: given $n$ samples from an unknown distribution, we want to estimate which distribution, from a given one-parameter family, produced the data. Following Schulman and Vazirani, we evaluate an estimator in terms of the chance of being within a specified tolerance of the correct answer, in the worst case. We provide optimal estimators for several families of distributions on $\mathbb{R}$. We prove that for distributions on a compact space, there is always an optimal estimator that is translation-invariant, and we conjecture that this conclusion also holds for any distribution on $\mathbb{R}$. By contrast, we give an example showing it does not hold for a certain distribution on an infinite tree.
The number of possibilities for random dating
Let $G$ be a regular graph and $H$ a subgraph on the same vertex set. We give surprisingly compact formulas for the number of copies of $H$ one expects to find in a random subgraph of $G$.
The $k^{\text th}$ Upper Chromatic Number of the Line
Let $S \subseteq \mathbb{R}^n$, and let $k\in\mathbb{N}$. Greenwell and Johnson define ${\hat\chi\ }^{(k)}(S)$ to be the smallest integer $m$ (if such an integer exists) such that for every $k\times m$ array $D=(d_{ij})$ of positive real numbers, $S$ can be colored with the colors $C_1,\ldots,C_m$ such that no two points of $S$ which are a (Euclidean) distance $d_{ij}$ apart are both colored $C_j$, for all $1\leq i \leq k$ and $1\leq j \leq m$. If no such integer exists then we say that ${\hat\chi\ }^{(k)}(S)=\infty$. In this paper we show that ${\hat\chi\ }^{(k)}(\mathbb{R})$ is finite for all $k$.
Upper Chromatic Numbers: An Update
This is a survey written in 2000 about upper chromatic numbers
Yet Another Species of Forbidden-distances Chromatic Number
This 2001 paper introduces a new type of chromatic number for point sets.
Integer Area Dissections of Lattice Polygons via a Non-Abelian Sperner's Lemma
We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's Lemma plays a central role.
Integrality relations for polygonal dissections
Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the areas of the other triangles. In both cases, the integrality relations are invariant under deformation of the dissection. The trapezoid theorem implies and provides a new context for Monsky's Equidissection Theorem that a square cannot be dissected into an odd number of triangles of equal area. A corollary of these results is that the area polynomials for parallelograms introduced in previous work have all leading coefficients equal to $\pm 1$.
On Eigenvalue Gaps of Integer Matrices
Given an $n\times n$ matrix with integer entries in the range $[-h,h]$, how close can two of its distinct eigenvalues be? The best previously known examples have a minimum gap of $h^{-O(n)}$. Here we give an explicit construction of matrices with entries in $[0,h]$ with two eigenvalues separated by at most $h^{-n^2/16+o(n^2)}$. Up to a constant in the exponent, this agrees with the known lower bound of $\Omega((2\sqrt{n})^{-n^2}h^{-n^2})$ \cite{mahler1964inequality}. Bounds on the minimum gap are relevant to the worst case analysis of algorithms for diagonalization and computing canonical forms of integer matrices. In addition to our explicit construction, we show there are many matrices with a slightly larger gap of roughly $h^{-n^2/32}$. We also construct 0-1 matrices which have two eigenvalues separated by at most $2^{-n^2/64+o(n^2)}$.
An illustrated encyclopedia of area relations
To any combinatorial triangulation $T$ of a square, there is an associated polynomial relation $p_T$ among the areas of the triangles of $T$. With the goal of understanding this polynomial, we consider polynomials obtained from $p_T$ by choosing $l$ of its variables and specializing $p_T$ to these variables by zeroing out the remaining variables. We show that for fixed $l$, the set ${\mathcal E}_l$ of integer polynomials that appear as irreducible factors of such specializations is finite. We compute this area encyclopedia ${\mathcal E}_l$ for $l\leq 4$. We also show that in any dissection of a square into $l$ triangles, the areas of the triangles must satisfy a polynomial in ${\mathcal E}_l$. Our results are obtained by studying the rational map that associates to each drawing of $T$ the tuple of areas of the triangles in that drawing. By analyzing the ways of approaching the base locus, we derive restrictions on points of the closure of the image of this map.
Generalized Dissections and Monsky's Theorem
Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky's polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.
Germ order for one-dimensional packings
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set $D$ of positive integers, call a set $S$ "$D$-avoiding" if no two elements of $S$ differ by an element of $D$. We study the problem of determining, for fixed $D$, all $D$-avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.
Sums of twisted circulants
The rate of convergence of simple random walk on the Heisenberg group over $Z/nZ$ with a standard generating set was determined by Bump et al [1,2]. We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set. We also determine the rate of convergence of simple random walk on higher-dimensional versions of the Heisenberg group with a standard generating set. We obtain our results via Fourier analysis, using an eigenvalue bound for sums of twisted circulant matrices. The key tool is a generalization of a version of the Heisenberg Uncertainty Principle due to Donoho-Stark [4].