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William Chen

Publications and source records attributed to William Chen.

80 records · Page 5Linked to original sources
arXiv2022

Genetic Algorithms For Extractive Summarization

Most current work in NLP utilizes deep learning, which requires a lot of training data and computational power. This paper investigates the strengths of Genetic Algorithms (GAs) for extractive summarization, as we hypothesized that GAs could construct more efficient solutions for the summarization task due to their relative customizability relative to deep learning models. This is done by building a vocabulary set, the words of which are represented as an array of weights, and optimizing those set of weights with the GA. These weights can be used to build an overall weighting of a sentence, which can then be passed to some threshold for extraction. Our results showed that the GA was able to learn a weight representation that could filter out excessive vocabulary and thus dictate sentence importance based on common English words.

cs.CL↗
arXiv2021

Nonabelian level structures, Nielsen equivalence, and Markoff triples

In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show that for all but finitely many primes $p$, the group of Markoff automorphisms acts transitively on the nonzero $\mathbb{F}_p$-points of the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$. This yields a strong approximation property for the Markoff equation, the finiteness of congruence conditions satisfied by Markoff numbers, and the connectivity of a certain infinite family of Hurwitz spaces of $\text{SL}_2(\mathbb{F}_p)$-covers of elliptic curves. With possibly finitely many exceptions, this resolves a conjecture of Bourgain, Gamburd, and Sarnak, first posed by Baragar in 1991. Since their methods are effective, this reduces the conjecture to a finite computation.

math.NT↗
arXiv2020

Audrey: A Personalized Open-Domain Conversational Bot

Conversational Intelligence requires that a person engage on informational, personal and relational levels. Advances in Natural Language Understanding have helped recent chatbots succeed at dialog on the informational level. However, current techniques still lag for conversing with humans on a personal level and fully relating to them. The University of Michigan's submission to the Alexa Prize Grand Challenge 3, Audrey, is an open-domain conversational chat-bot that aims to engage customers on these levels through interest driven conversations guided by customers' personalities and emotions. Audrey is built from socially-aware models such as Emotion Detection and a Personal Understanding Module to grasp a deeper understanding of users' interests and desires. Our architecture interacts with customers using a hybrid approach balanced between knowledge-driven response generators and context-driven neural response generators to cater to all three levels of conversations. During the semi-finals period, we achieved an average cumulative rating of 3.25 on a 1-5 Likert scale.

cs.CL↗
arXiv2016

Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism $u^2-64$. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of $u^2-64$. We discuss these results in the context of the moduli space of stable rank two holomorphic bundles with fixed odd determinant over a Riemann surface.

math.GT↗
arXiv2016

Damped Infinite Energy Solutions of the 3D Euler and Boussinesq Equations

We revisit a family of infinite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in finite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup. Further, we show that similar infinite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system.

math.AP↗
arXiv2008

An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing "deficiency" with multiplicity of $θ$ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai-Edmonds Structure Theorem. Consequently, the matching polynomial of a vertex transitive graph has simple roots.

math.CO↗
arXiv2007

Concrete and abstract structure of the sandpile group for thick trees with loops

We answer a question of Laszlo Babai concerning the abelian sandpile model. Given a graph, the model yields a finite abelian group of recurrent configurations which is closely related to the combinatorial Laplacian of the graph. We explicitly describe the group elements and operations in the case of thick trees with loops--that is, graphs which are obtained from trees by setting arbitrary edge multiplicities and adding loops at vertices. We do this both concretely (by describing the so-called recurrent and identity configurations) and abstractly (by computing the group's abstract structure), and define maps identifying the two.

math.CO↗