Published electronically January 12, 2021

DOI: 10.1137/20S1353344

**Authors:** John Nguyen and Aileen Zebrowski (University of Minnesota, Twin Cities) **Sponsor:** Dr. Kaitlin Hill (Wake Forest University)

**Abstract: **Permafrost is a thick layer of soil that is frozen throughout the year and covers significant portions of the northern hemisphere. Currently, there is a large amount of carbon trapped in the permafrost, and as permafrost melts, a significant portion of this carbon will be released into the atmosphere as either carbon dioxide or methane. We use empirical data to estimate that, on average, permafrost currently extends from the arctic to latitude 61_N. We propose an adaption to the Budyko energy balance model to study the impacts of receding permafrost. We track the steady-state latitude of both the permafrost line and the snow line as greenhouse gas emissions, and consequently, global mean temperature increases. Using the change in permafrost surface area, we are able to quantify the total carbon feedback of melting permafrost. Focusing our analysis on scenarios described in recent IPCC reports and the Paris Climate Agreement, we use change in the permafrost line latitude to estimate the amount of carbon dioxide released by the melted permafrost. Similarly, we use the snow line to calculate the minimum average global temperature that would cause the ice caps to completely melt. We find that our adaption of the Budyko model produces estimates of carbon dioxide emissions within the range of projections of models with higher complexity.

Published electronically January 19, 2021

DOI: 10.1137/19S1266885

**Authors:** Justin Park (MIT) **Sponsor:** Dr. Kasso Okoudjou (University of Maryland, College Park)

**Abstract: **We expand on a prior result about the cardinalities of harmonic tight frames generated from the discrete Fourier transform. Harmonic ﬁnite unit-norm tight frames (FUNTFs) constructed from the ﬁrst two rows of the M ×M discrete Fourier transform have previously been described and characterized as prime or divisible, where M ≥ 2 is an integer. We generalize the result to any choice of two rows b and c for which c−b has up to two distinct prime factors. These new results allow for much more ﬂexibility in constructing harmonic FUNTFs from M-th roots of unity.

Published electronically January 27, 2021

DOI: 10.1137/20S1363728

**Authors:** Elif Sensoy (Wilbur Wright College) **Sponsor:** Dr. Hellen Colman (Wilbur Wright College)

**Abstract: **We exhibit an algorithm with continuous instructions for two robots moving without collisions on a track shaped as a wedge of three circles. We show that the topological complexity of the configuration space associated with this problem is 3. The topological complexity is a homotopy invariant that can be thought of as the minimum number of continuous instructions required to describe the movement of the robots between any initial configuration to any final one without collisions. The algorithm presented is optimal in the sense that it requires exactly 3 continuous instructions.

Published electronically January 12, 2021

DOI: 10.1137/20S1360578

**Authors:** Foyez Alauddin (Trinity School, NYC)**Sponsor:** Gleb Pogudin (Polytechnique)

**Abstract: **Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. It has been recently used as a preprocessing step for new model order reduction methods, so it is important to keep the number of new variables small. Several algorithms have been designed to search for a quadratization with the new variables being monomials in the original variables. To understand the limitations and potential ways of improving such algorithms, we study the following question: can quadratizations with not necessarily new monomial variables produce a model of substantially smaller dimension than quadratization with only new monomial variables?

To do this, we restrict our attention to scalar polynomial ODEs. Our first result is that a scalar polynomial ODE x_ = p(x) = anxn + an1xn1 + : : : + a0 with n > 5 and an 6= 0 can be quadratized using exactly one new variable if and only if p(x an1 n_an ) = anxn + ax2 + bx for some a; b 2 C. In fact, the new variable can be taken as z := (x +an1 n_an)n1. Our second result is that two new non-monomial variables are enough to quadratize all degree 6 scalar polynomial ODEs. Based on these results, we observe that a quadratization with not necessarily new monomial variables can be much smaller than a monomial quadratization even for scalar ODEs.

The main results of the paper have been discovered using computational methods of applied nonlinear algebra (Gröbner bases), and we describe these computations.

Published electronically February 22, 2021

DOI: 10.1137/19S1300571

**Authors:** Justin Bennett, Karissa Gund, Jingteng iLu (Arizona State University), Xixu Hu (University of Science and Technology of China), and Anya Porter (Harvey Mud College) **Sponsor:** Yang Kuang (Arizona State University)

**Abstract: **Over time, tumor treatment resistance inadvertently develops when androgen de-privation therapy (ADT) is applied to metastasized prostate cancer (PCa). To combat tumor resistance, while reducing the harsh side effects of hormone therapy, the clinician may opt to cyclically alternates the patient’s treatment on and off. This method, known as intermittent ADT, is an alternative to continuous ADT that improves the patient’s quality of life while testosterone levels recover between cycles. In this paper, we explore the response of intermittent ADT to metastasized prostate cancer by employing a previously clinical data validated mathematical model to new clinical data from patients undergoing Abiraterone therapy. This cell quota model, a system of ordinary differential equations constructed using Droop’s nutrient limiting theory, assumes the tumor comprises of castration-sensitive (CS) and castration-resistant (CR) cancer sub-populations. The two sub-populations rely on varying levels of intracellular androgen for growth, death and transformation. Due to the complexity of the model, we carry out sensitivity analyses to study the effect of certain parameters on their outputs, and to increase the identifiability of each patient’s unique parameter set. The model’s forecasting results show consistent accuracy for patients with sufficient data, which means the model could give useful information in practice, especially to decide whether an additional round of treatment would be effective.

Published electronically February 25, 2021

Published electronically March 1, 2021

DOI: 10.1137/20S1367040

**Authors:** Vishnu Nittoor (The International School Bangalore)**Sponsor:** Dr. Tim Chartier (Davidson)

**Abstract: **This paper investigates the eﬀect of increasing competitive balance on the reliability of tournament rankings. Reliability of rankings, a previously qualitative property, is quantiﬁed in this paper by the closeness between ground truth rankings and the rankings of teams at the end of a tournament. Three metrics are used to measure this closeness: Spearman’s rank correlation coeﬃcient, Kendall’s tau, and a relatively unused algorithm in the ﬁeld of ranking: Levenshtein distance. Three tournament structures are simulated: round-robin, random pairings, and the Swiss system. The tournaments are simulated across multiple trials and over a varying number of games. It is found that the rate of growth of reliability of a tournament structure falls as the number of games increases. It is also found that there is a positive relationship between competitive imbalance and reliability. The marginal beneﬁt of increasing competitive imbalance falls as it is increased. Unexpectedly, in comparison to random pairings and Swiss pairings, the round-robin tournament structure is seen to achieve the highest reliability score across all metrics and number of games played. The diﬀerence in reliability between the tournament structures increases as competitive imbalance is increased. The further work suggested includes investigation of tournament outcome uncertainty in conjunction with reliability and competitive balance, a closer study into Levenshtein distance as a useful algorithm to quantify closeness between two rankings, and an inquiry into the speciﬁc factors that bottleneck reliability while the number of games played in a tournament increases.

Published electronically March 18, 2021

DOI: 10.1137/20S1361870

**Authors:** Srinath Mahankali (Stuyvesant High School)**Sponsor:** Yunan Yang (Courant Institute of Mathematical Sciences)

**Abstract: **Full-waveform inversion (FWI) is a method used to determine properties of the Earth from information on the surface. We use the squared Wasserstein distance (squared W2 distance) as an objective function to invert for the velocity of seismic waves as a function of position in the Earth, and we discuss its convexity with respect to the velocity parameter. In one dimension, we consider constant, piecewise increasing, and linearly increasing velocity models as a function of position, and we show the convexity of the squared W2 distance with respect to the velocity parameter on the interval from zero to the true value of the velocity parameter when the source function is a probability measure. Furthermore, we consider a two-dimensional model where velocity is linearly increasing as a function of depth and prove the convexity of the squared W2 distance in the velocity parameter on large regions containing the true value. We discuss the convexity of the squared W2 distance compared with the convexity of the squared L2 norm, and we discuss the relationship between frequency and convexity of these respective distances. We also discuss multiple approaches to optimal transport for non-probability measures by first converting the wave data into probability measures.

Published electronically March 18, 2021

DOI: 10.1137/20S137208110.1137/20S1372081

**Author:** Timothy Getscher (Woods Hole Oceanographic Institution)**Project Advisor****:** Kevin McIlhany (United States Naval Academy)

**Abstract: **This paper compares the advantages, limitations, and computational considerations of using Finite-Time

Lyapunov Exponents (FTLEs) and Lagrangian Descriptors (LDs) as tools for identifying barriers and mechanisms of fluid transport in two-dimensional time-periodic ows. These barriers and mechanisms of transport are often referred to as “Lagrangian Coherent Structures," though this term often changes meaning depending on the author or context. This paper will specifically focus on using FTLEs and LDs to identify stable and unstable manifolds of hyperbolic stagnation points, and the Kolmogorov-Arnold-Moser (KAM) tori associated with elliptic stagnation points. The background and theory behind both methods and their associated phase space structures will be presented, and then examples of FTLEs and LDs will be shown based on a simple, periodic, time-dependent double-gyre toy model with varying parameters.

Published electronically March 23, 2021

DOI: 10.1137/20S1367155

**Authors:** Manuel Santana (Utah State University), Abby Brauer (Lewis and Clark University), and Megan Krawick (Youngstown State University)**Project Advisor:** Jun Kitagawa (Michigan State University)

**Abstract: **Numerical methods for the optimal transport problem is an active area of research. Recent work of Kitagawa and Abedin shows that the solution of a time-dependent equation converges exponentially fast as time goes to inﬁnity to the solution of the optimal transport problem. This suggests a fast numerical algorithm for computing optimal maps; we investigate such an algorithm here in the 1-dimensional case. Speciﬁcally, we use a ﬁnite-diﬀerence scheme to solve the time-dependent optimal transport problem and carry out an error analysis of the scheme. A collection of numerical examples is also presented and discussed.