arXiv subjects

Steven Damelin

Publications and source records attributed to Steven Damelin.

4 recordsLinked to original sources

arXiv2024

On the Packing Functions of some Linear Sets of Lebesgue Measure Zero

We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension $d\in(0,1)$. Our main result is a proof that Minkowski measurability is a sufficient condition for the existence of best packing asymptotics on monotone rearrangements of these sets. For each such set, the main result provides an explicit constant of proportionality $p_d,$ depending only on the Minkowski dimension $d,$ that relates its packing limit and Minkowski content. We later use the Digamma function to study the limiting value of $p_d$ as $d\to 1^-.$ For sharpness, we use renewal theory to prove that the packing constant of the $(1/2,1/3)$ Cantor set is less than the product of its Minkowski content and $p_d$. We also show that the measurability hypothesis of the main theorem is necessary by demonstrating that a monotone rearrangement of the complementary intervals of the 1/3 Cantor set has Minkowski dimension $d=\log2/\log3\in(0,1),$ is not Minkowski measurable, and does not have convergent first-order packing asymptotics. The aforementioned characterization of Minkowski measurability further motivates the asymptotic study of an infinite multiple subset sum problem.

math.CA↗

arXiv2019

Power Weighted Shortest Paths for Clustering Euclidean Data

We study the use of power weighted shortest path distance functions for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoretically and experimentally, that this leads to higher clustering accuracy. We also present a fast algorithm for computing these distances.

cs.LG↗

arXiv2018

On a condition equivalent to the Maximum Distance Separable conjecture

We denote by $\mathcal{P}_q$ the vector space of functions from a finite field $\mathbb{F}_q$ to itself, which can be represented as the space $\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $\mathcal{O}_n \subset \mathcal{P}_q$ the set of polynomials that are either the zero polynomial, or have at most $n$ distinct roots in $\mathbb{F}_q$. Given two subspaces $Y,Z$ of $\mathcal{P}_q$, we denote by $\langle Y,Z \rangle$ their span. We prove that the following are equivalent. A) Let $k, q$ integers, with $q$ a prime power and $2 \leq k \leq q$. Suppose that either: 1) $q$ is odd 2) $q$ is even and $k \not\in \{3, q-1\}$. Then there do not exist distinct subspaces $Y$ and $Z$ of $\mathcal{P}_q$ such that: 1') $dim(\langle Y, Z \rangle) = k$ 2') $dim(Y) = dim(Z) = k-1$. 3') $\langle Y, Z \rangle \subset \mathcal{O}_{k-1}$ 4') $Y, Z \subset \mathcal{O}_{k-2}$ 5') $Y\cap Z \subset \mathcal{O}_{k-3}$. B) The MDS conjecture is true for the given $(q,k)$.

cs.IT↗

arXiv2016

On the Structure of the Littlewood Polynomials and their Zero Sets

In fractal geometry, the main objects of study have been geometric objects with a global dimension that need not be integer valued. More recently, locally fractal objects, ones in which the dimension is a local property rather than a global one, have become of interest. We explore one such object, the zero set of Littlewood polynomials, its connection to more traditional fractal objects, and develop a method for computing local approximations.

math.CV↗