- As We Live and Breathe the KdV
*Chandler Haight, Diana Son*Faculty Advisor:

*Svetlana Roudenko, Kai Yang***Abstract:**We characterize and investigate breather solutions to the following Korteweg-de Vries (KdV) type equations via numerical simulation, ranging from the integrable systems such as the modified KdV equations and Gardner's equations, to the Hamiltonian systems such as the generalized KdV equations with absolute value potential (\(|u|^{p-1}u_x\)). Our numerical simulation results suggest that the absolute potential structure \(|u|^{p-1}u_x\) is the key factor in forming the breather solutions. Moreover, these breather solutions are stable under all kinds of perturbations, such as the initial data or the extra nonlinearity in the potential term. These numerical results are consistent with the theoretical orbital and asymptotic stability results for the integrable systems, e.g., mKdV equation and Gardner equation in other work indicating the future studies of the breather solutions for the Hamiltonian systems. - Constructing an Environmentally Friendly Portfolio
*Dan Blisker*Faculty Advisor:

*Jorge Hernandez***Abstract:**This paper outlines how a modified version of the Black-Litterman asset allocation model is integrated into quadratic programming to predict the performance of stock prices.We then use our work to construct a portfolio that maximizes expected returns and minimizes variance given a constraint on the carbon emissions of our portfolio. The first section will explain what the Black-Litterman model is and how it is an improvement over the Markowitz model for portfolio optimization.The next section will show how we modify Black-Litterman's standard calculation of variance and covariance with corresponding GARCH methods. Next, we present how the model is implemented with quadratic programming in Python.Finally, we demonstrate how to construct a portfolio with below threshold carbon emissions and superior economic performance compared to a default strategy of buying and holding the S&P 500. - Construction of an exponential basis on split intervals
*Mark Leal, Ismael Morell, and Vladyslav Drezels*Graduate Mentors

*: Aleh Asipchuk*Faculty Advisor:

*Laura De Carli***Abstract:**We considered the space \(L^2(I)\), where \(I\) is a union of intervals of total length 1. We constructed an exponential Riesz basis on such space with some restrictions on the gaps between the intervals. Also, we prove a stability theorem when I is a union of two intervals.A subset of the findings have been published.

- Domination of Cartesian Product of Complete Graphs
*Liam Busch, Grant Silewski, Hanzhang Yin*Graduate Mentors

*: Justin Wisby*Faculty Advisor:

*Walter Carballosa Torres***Abstract:**Let \(G=(V(G),E(G))\) be a finite undirected graph. A set \(S\) of vertices in \(V\) is said to be total \(k\)-dominating if every vertex in \(V\) is adjacent to at least \(k\) vertices in \(S\). The \(k\)-domination number, \(\gamma_{k}(G)\), is the minimum cardinality of a \(k\)-dominating set in \(G\). In this work we study the \(k\)-domination number of Cartesian products of two complete graphs, which is a lower bound of the total \(k\)-domination number of Cartesian product of any two graphs with the same number of vertices. We obtain new lower and upper bounds for the \(k\)-domination number of Cartesian product of two complete graphs. Some asymptotic behaviors are obtained as a consequence of the bounds we found. - Periodic Nonlinear Schrödinger Equation
*Samuel Kilgore, Beckett Sanchez*Graduate Mentors

*: Iryna Petrenko*Faculty Advisor:

*Svetlana Roudenko, Oscar Riano***Abstract:**We consider the nonlinear Schrödinger equation \(iu_t+\Delta u+\lambda\mathcal{N}(u)u=0\), such that \(x\in \mathbb{T}^N,\,t\in \mathbb{R}\), on a periodic domain, where the nonlinear term \(\mathcal{N}(u)\) can be expressed in several ways i.e. power nonlinearity, logarithmic, exponential and sine functions. We investigate solutions to this model and show the local in time existence and uniqueness of solutions and continuous dependence on a class of initial data. - A Mathematical Model for Tumor Growth
*Khadeja Ghannam and Alberto Sales*Graduate Mentors

*: Ginelle Gonzalez and Eduardo Perez*Faculty Advisor:

*Stephen Tennenbaum***Abstract:**Systems of differential equations can be used to model populations of various cell types in the human body. A mathematical model observing the immune, normal, and tumor cell interactions is presented in this paper. Our main objective is to perform stability and bifurcation analysis on this model. After introducing and removing different terms and parameters from an existing model, our observations revealed vital cell behaviors in the body depending on the patient's biological attributes. Analysis of the system's stability and bifurcations around its equilibrium points is carried out to observe the behavior of three distinct cases: cured, deceased, and coexistence. We add a crowding sensitivity parameter, \(m\), to the tumor equation and remove a term from the immune cell equation in our system. We found that Hopf bifurcations arise when changing our competition parameters from destructive to advantageous and changing our crowding parameter from insensitive to sensitive. - Blow-Up Criteria in the cNLS
*Gia Azcoitia, Hannah Wubben*Graduate Mentors

*: Alex David Rodriguez*Faculty Advisor:

*Svetlana Roudenko***Abstract:**We consider the conditions for the blow up criteria to the solutions of the combined nonlinear Schr\"{o}dinger equation, \(i\partial_t u + \Delta u + \epsilon_1|u|^{p+1}u + \epsilon_2|u|^{q+1} = 0\) in one dimension. Specifically for \(\epsilon_2 = 1\) and \(\epsilon_1\) is defocusing (\(\epsilon_1 < 0\)). We study this criteria by examining the variance of solutions giving us information on its spread. We show that the variance must be tending towards zero in finite time, thus the solution becomes a singularity leading to blow up. In addition, we examine initial data with a quadratic phase \(e^{\frac{ibx^2}{4}}\) where \(b < 0\) leads to blow up solutions. While results are known for well posedness and scattering for this equation, the criteria for blow up solutions had yet to be studied. - A Robust PPO-based Single Asset Trading Strategy
*Neila Bennamane, Richard Solomon*Graduate Mentors

*: Giancarlo Sanchez, Luis Caicedo Torres***Abstract:**Algorithmic trading has taken over the majority of the trading worldwide. Reinforcement Learning, as a decision making tool, has been successful in a variety of fields such as video games and robotics. It has now beaten pro players at Atari, Go, Starcraft and is well on its way to solve the autonomous driving problem. The financial markets can be seen as an environment where a Reinforcement Learning model can be trained to make optimal decisions. This work reviews the performance of a policy-based RL algorithm in trading in a Bitcoin environment. We train on synthetic data, such as fractional geometric brownian motion, as a data augmentation technique in order to make trading strategies more robust. We present results along with relevant benchmark statistics to assess the performance of our model. - Cohomology of Singular K3 Surfaces
*Steppan Konoplev and Vaughan Russell*Graduate Mentors

*: Jose Medel*Faculty Advisor:

*Gueo Grantcharov and Anna Fino***Abstract:**We compute the cohomology of singular K3 surfaces which appear in Iano-Fletcher's paper by blowing them up at points several times until they're smooth. The cohomologies of smooth K3 surfaces is well-known, and we have a theorem allowing us to compute the cohomology of the original surface from the cohomology of a chain of blow-ups, so we can apply the theorem to undo each blowup until arriving at the cohomology of the original surface. - What is the Full Price of a Plastic Spoon? An Externalities Estimation Equation
*Kevin S. Sterling*Faculty Advisor:

*Stephen Tennenbaum***Abstract:**An equation to estimate negative externalities is derived. Since all physical objects in the economy require energy to produce and eventually decay, the price of the externalities emitted can now be calculated. The equation can be used to price externalities for items created primarily from one resource (such as plastic) to n-resource mix, such as coins, phones, or houses. The externalities produced by a range of common items are compared. Plastic is used as the main example. Using conservative estimates, a plastic spoon causes roughly $0.04 in externalities. Thus, the full price of a plastic spoon with the externality included is about $0.08. Thus, the externality is equal to about 100% of the market price of a spoon. - Doubly Non-Local Cahn Hilliard Equation with Fractional Time Derivative
*Saja Gherri, Samantha Roberts*Graduate Mentors

*: Melissa De Jesus*Faculty Advisor:

*Ciprian G. Gal***Abstract:**We consider the doubly non-local Cahn Hilliard equation (dnCHE) with a fractional time derivative. This modified model seeks to account for non-linear motion. We first present a preliminary section composed of useful functions, identities, and calculations which are often referenced, as well as a brief overview of fractional calculus. We establish both the existence and uniqueness of our solution to this new model. Then, we employ a forward Euler scheme to portray numerical approximations of our solution. We aim to show the convergence of our solution to that of the dnCHE, as our fractional order approaches 1. - Existence and scattering results for an inhomogeneous version of the nonlinear Schrödinger equation
*Troy Roberts*Graduate Mentors

*: Justin Wisby*Faculty Advisor:

*Svetlana Roudenko, Oscar Riano***Abstract:**In this work, we construct for every \(\alpha > 0\) and \(\lambda \in \mathbb{C}\) a class of initial data \(u_0\) such that there exists a unique, local solution of the inhomogeneous nonlinear Schr\"odinger equation \(iu_t + \Delta u + \lambda v(x) |u|^\alpha u = 0\) on \(\mathbb{R}^N\). In particular, we consider a class of weights \(v(x)\) with polynomial-like behavior. This equation has strong motivations in optics, such as modeling laser propagation in an inhomogeneous medium. In addition, we construct for every \(\alpha > \frac{2}{N}\) a class of initial data for which there exists a global solution that scatters as \(t \to \infty\). Finally, we consider the periodic setting in which we find local well-posedness for every \(\alpha \in \mathbb{R}\).

Here we catalogue the technical reports that the AMRPU students completed during their time at FIU.