arXiv subjects

Alison Becker

Publications and source records attributed to Alison Becker.

3 recordsLinked to original sources

arXiv2026

Graded n-Absorbing Ideals and their Combinatorial Structure

Graded $n$-absorbing ideals generalize graded prime ideals by extending absorption properties to products of $(n+1)$ homogeneous elements. We study several generalizations of graded prime ideals, including graded $n$-absorbing, graded weakly $n$-absorbing, graded strongly $n$-absorbing, and graded $n$-absorbing primary ideals, as well as related graded $n$-absorbing subgroups. Our primary result establishes a combinatorial model for graded $n$-absorbing principal monomial ideals in polynomial rings with the standard grading. By identifying principal monomial ideals with exponential vectors in $\mathbb{N}^{m}$, we show that the graded $n$-absorbing principal monomial ideals correspond precisely to lattice points in the simplex $\{\alpha \in \mathbb{N}^{m}: \vert \alpha \vert \leq n\}.$ Consequently, the Hasse diagram of principal monomial ideals is realized as the 1-skeleton of the Cayley graph of $\mathbb{N}^{m}$, yielding a geometric and combinatorial interpretation of graded $n$-absorption.

math.AC↗

arXiv2020

Algebraic Relations Via a Monte Carlo Simulation

The conjugation action of the complex orthogonal group on the polynomial functions on $n \times n$ matrices gives rise to a graded algebra of invariant polynomials. A spanning set of this algebra is in bijective correspondence to a set of unlabeled, cyclic graphs with directed edges equivalent under dihedral symmetries. When the degree of the invariants is $n+1$, we show that the dimension of the space of relations between the invariants grows linearly in $n$. Furthermore, we present two methods to obtain a basis of the space of relations. First, we construct a basis using an idempotent of the group algebra referred to as Young symmetrizers, but this quickly becomes computationally expensive as $n$ increases. Thus, we propose a more computationally efficient method for this problem by repeatedly generating random matrices using a Monte Carlo algorithm.

math.RT↗

arXiv2013

The hunt for new pulsars with the Green Bank Telescope

The Green Bank Telescope (GBT) is the largest fully steerable radio telescope in the world and is one of our greatest tools for discovering and studying radio pulsars. Over the last decade, the GBT has successfully found over 100 new pulsars through large-area surveys. Here I discuss the two most recent---the GBT 350 MHz Drift-scan survey and the Green Bank North Celestial Cap survey. The primary science goal of both surveys is to find interesting individual pulsars, including young pulsars, rotating radio transients, exotic binary systems, and especially bright millisecond pulsars (MSPs) suitable for inclusion in Pulsar Timing Arrays, which are trying to directly detect gravitational waves. These two surveys have combined to discover 85 pulsars to date, among which are 14 MSPs and many unique and fascinating systems. I present highlights from these surveys and discuss future plans. I also discuss recent results from targeted GBT pulsar searches of globular clusters and Fermi sources.

astro-ph.HE↗