# Summer Undergraduate Applied Mathematics Institute

### May 30 - July 22, 2023

# Projects

► *Assessing the Robustness of VBMC for Extracting Parameters in Differential Equations*, **Anna Rittenhouse, Grace Rojo, Alia Valentine**

**Advisors:** Rachel Kurchin, David Offner, Jerry Wang

**Abstract:** We study a challenging problem related to parameter inference for a non-linear dynamical system of significant physical interest. In particular, we investigate a recently proposed flavor of Bayesian Monte Carlo -- Variational Bayesian Monte Carlo (VBMC) -- in the context of inferring parameters for a system of coupled anharmonic oscillators. Whereas the magnitude of the force on a harmonically oscillating mass is given by $k|x|$ (where $x$ is the distance of the mass from equilibrium and $k$ is the spring constant), we study a variation on this system where the magnitude of the force is $k|x|^a$. Though the change is a small one, all bets are off on anything resembling an analytical solution (or even an approximation) when $a\neq 1$.

► *Colorings avoiding disjoint rainbow triangles*, **tahda queer, Cyrus Young, Wohua Zhou**

**Advisor:** Juergen Kritschgau

**Abstract:** Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow triangles. Rainbow problems have been studied extensively in the context of anti-Ramsey theory, and more recently, in the context of Tur\'{a}n problems. B. Li. et al. *European J. Combin. 36 (2014)* found that a graph must contain a rainbow triangle if $e(G)+c(G) \geq \binom{n}{2}+ n$. L. Li. and X. Li. *Discrete Applied Mathematics 318 (2022)* conjectured a lower bound on $e(G)+c(G)$ such that $G$ must contain a proper $mK_3$.

In this paper, we provide a construction that disproves the conjecture, and we introduce results that guarantee the existence of a proper $mK_3$ in general graphs and complete graphs.

► *Doubling measures along sets of natural numbers: analysis meets number theory*, **Zoe Markman, Teresa Pollard, Joahua Zeitlin**

**Advisors:** Theresa Anderson, Elisa Bellah

**Abstract:** Using a wide array of machinery from diverse fields across mathematics, we provide a construction of a measure on the real line which is doubling on all n-adic intervals for any finite list of natural number n, yet not doubling overall. In particular, we extend previous results in the area, where only two coprime numbers $n$ were allowed, by using substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, $A_p$ weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.

► *Leaky positive semi-definite zero forcing*, **Ian Farish, Olivia Elias, Emrys King, Josh Kyei**

**Advisor:** Ryan Moruzzi

**Abstract:** Zero forcing on a graph is an iterative graph coloring process where, starting with an initial set of blue vertices, we try to force other non-blue vertices blue according to a color change rule. The zero forcing number of a graph was first defined in 2008 by a AIM (American Institute of Mathematics) working group as a method of bounding the maximum nullity of a graph. Motivated by monitoring an electrical network, leaks are introduced into the graph hindering the ability of vertices to force. Though we lose the connection with the maximum nullity of a graph, leaks in a graph present interesting new avenues of research pertaining to the area of zero forcing. Our work involved determining the leaky forcing number of graphs for a variation of zero forcing known as positive semidefinite zero forcing.

# Students

**Olivia Elias**, University of Colorado**Ian Farish**, California State Polytechnic University**Emrys King**, Pomona College**Josh Kyei**, Morehouse College**Zoe Markman**, Swarthmore College**Teresa Pollard**, New York University**tahda queer**, CUNY**Anna Rittenhouse**, Clark Atlanta University**Grace Rojo**, Massachusetts Institute of Technology**Alia Valentine**, Michigan State University**Cyrus Young**, University of California, Irvine**Joahua Zeitlin**, Yale University**Wohua Zhou**, California State University, East Bay

_{(from left to right) tahda queer, Wohua Zhou, Josh Zeitlin, Juergen Kritschgau, Cyrus Young, Ian Farish, Elisa Bellah, Josh Kyei, Emrys King, Teresa Pollard, Zoe Markman, Olivia Elias, Ryan Moruzzi, Alia Valentine, Rachel Kurchin, David Offner, Anna Rittenhouse, Grace Rojo, Jerry Wang, Adam Hill, Michael Young}*(from left to right) Elisa Bellah, Teresa Pollard, Josh Zeitlin, Zoe Markman, Theresa Anderson*

*(from left to right) Jerry Wang, David Offner, Alia Valentine, Grace Rojo, Anna Rittenhouse, Rachel Kurchin*

*(from left to right) Wohua Zhou, tahda queer, Juergen Kritschgau, Cyrus Young*

*(from left to right) Ian Farish, Josh Kyei, Emrys King, Olivia Elias, Ryan Moruzzi*

# Faculty

**Theresa Anderson**

Assistant Professor

**E-mail: **tanders2@andrew.cmu.edu

**Elisa Bellah**

Postdoctoral Associate

**E-mail: **ebellah@andrew.cmu.edu

**Juergen Kritschgau**

Postdoctoral Associate

**Rachel Kurchin**

Assistant Research Professor

**E-mail: **rkurchin@andrew.cmu.edu

**Ryan Moruzzi**

Assistant Professor

California State University, East Bay

**E-mail: **ryan.moruzzi@csueastbay.edu

**David E. Offner**

Associate Teaching Professor

**E-mail: **doffner@andrew.cmu.edu

**Jerry Wang**

Assistant Professor

**E-mail: **gjwang@cmu.edu