Session C: 1:45PM – 3:15PM
Sciences. Session C – Oral Presentations, Parlor A, Union
SESSION C (1:45PM – 3:15PM)
Location: Parlor A, A. Ray Olpin University Union
A Stable Numerical Scheme for a Model of Mutualism with Crowding Effects
Devan Hill, Southern Utah University
Chance Witt, Southern Utah University
Faculty Mentor Jianlong Han, Southern Utah University
SESSION C 1:45-2:00PM
Parlor A, Union
Science and Technology
We Study a Lotka-Volterra model of mutualism with crowding effects. After using nondimensionalization, we analyze the stability of the steady state solutions for the system. A nonstandard numerical scheme is proposed, and by using mathematical induction we prove that the numerical scheme is unconditionally stable. We also analyze the long term behavior of the numerical solution.
Mathematical Models of Tone in Thai Reduplication Patterns
Casey Miller, University of Utah
Faculty Mentor Aniello De Santo, University of Utah
SESSION C 2:05-2:20PM
Parlor A, Union
Science and Technology
Studying language mathematically allows us to define language processes in explicit terms, to determine their complexity. Formal characterizations give us an understanding of how language works in computational terms, insights into why some structures seem to be more favored than others, and insights into cognitive restrictions (Chandlee, 2017; Heinz, 2018; De Santo & Rawski, 2022). Mathematical formalization also gives us a way to model different processes with practical systems such as the Finite-State Transducer, for example, which has been used for applications such as machine translation. Recent work in the field of computational linguistics has argued that *sub*-regular characterizations are sufficient to model most phonological patterns-i.e., it takes significantly less computing power than previously thought to model such patterns (Chandlee, 2017; Heinz, 2018; Graf, 2019). In this work, I present a mathematical formalization of reduplication processes in Thai. Reduplication poses complications as many languages that feature reduplication patterns are also tonal, in which case tones and segments often act independently from each other. Importantly, Markowska, Heinz, & Rambow (2021) were able to model the tone reduplication patterns in Shupamen, a Bantu language, by using a synthesis of two-way finite-state transducers (Dolatian & Heinz, 2020). Thai is an interesting case because some linguists have argued that tone in Thai is a byproduct of throat position and thus is not completely independent from the segment. Additionally, tone shifts in Thai reduplication patterns ask interesting questions in regards to their complexity. The processes found in Thai provide a valuable contrast to the work done by Markowska, Heinz, & Rambow (2021). The formalization provided here adds further cross-linguistic insights to our broader understanding of how tone processes interact with reduplication patterns; such a formalization is also beneficial towards understanding more complex phonological processes, and offers insights for the language technologies being used and developed today.
Using Deep Reinforcement Learning To Generate Slice Surfaces from Knots in Braid Notation
Dylan Skinner, Brigham Young University
Faculty Mentor Mark Hughes, Brigham Young University
SESSION C 2:25-2:40PM
Parlor A, Union
Science and Technology
Deep reinforcement learning (DRL) has proven effective in recognizing patterns and finding solutions to problems that are difficult for humans. One problem in knot theory involves finding slice surfaces for knots with minimal genus. It is easy to find large genus slice surfaces bounded by a given knot, but in order to show that the slice genus of a knot is equal to a specified value, you must also prove that the knot does not bound a slice surface of a lesser genus. In this talk, I will outline an approach using DRL and braid notation of a knot to find small genus slice surfaces for a given knot, through a series of unknotted component addition/deletion, crossing addition/deletion, and relations in the braid group.
Ribbon Knots, Ribbon Doubles and Undoubles, and Symmetric Union Presentations
Moses Samuelson-Lynn, University of Utah
Faculty Mentor Edmund Karasiewicz, University of Utah
SESSION C 2:45-3:00PM
Parlor A, Union
Science and Technology
We define a ribbon knot as a knot that can be embedded into three-dimensional space such that it bounds a ribbon disk, that is, a surface that can be deformed in any topologically valid way, as well as passing any one section of the disc through another completely, with the line of intersection that forming a slit that does not touch the edges of the disc. Let K be a knot, not necessarily a ribbon knot, with crossing number k. We define an algorithm to create a ribbon knot from K which has crossing number at most 4k such that, which we call “ribbon doubling.” We also investigate the number of potential ribbon doubles for a knot, and potential restrictions on its crossing number. We propose a partial inverse operation for ribbon doubles, and show that it is not unique. Lastly, we propose a potential lower bound for the crossing number of the ribbon double of a knot. We then relate the concept of a ribbon double to that of a partial knot for symmetric union presentations, and propose a potential technique for selecting a unique ribbon double.