Here we catalogue the technical reports that the AMRPU students completed during their time at FIU.
- Topological Data Analysis for Time Series Forecasting
Catherine Fraga, and Jaclyn Pascual.
Graduate Mentors: Giancarlo Sanchez, Ginelle Gonzalez, and Luis Caicedo Torres,
Faculty Advisor Mirroslav Yotov
Abstract: In this paper we study the behavior of financial time series using Topological Data Analysis and Machine Learning methods with the goal of distinguishing geometric patterns within the underlying structure of the dataset to reveal classifications in the data that can help us make better predictions in forecasting models. These findings provide us with a clearer understanding of the chaotic behaviors that makes analyzing time series so difficult. We apply our methods to the stock data of Bitcoin in order to see the accuracy and stability of these methods. By observing the differences in the mean squared error given by a few methods, we can see how much TDA and ML contribute to the reduction of this error. - Algebraic De Rham Cohomology Calculations
Daniel Cabanez, and Vladyslav Drezels.
Graduate Mentor: Jose Medel,
Faculty Advisor Mirroslav Yotov
Abstract: In this paper we study the scheme theoretic description of the classically defined affine space, projective space, and the variety that describes the elliptic curve. Then we focus on one of the first cohomology theories introduced in algebraic geometry: the algebraic De Rham cohomology; and we calculate the algebraic De Rham cohomology of some of the fundamental objects in algebraic geometry. - Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity
Diana Son and Isaac Friedman.
PostdocMentors: Kai Yang and Oscar Riano.
Faculty Advisor: Svetlana Roudenko
Abstract: We study the short and long term behavior of solutions to the generalized Korteweg–de Vries equation with low power of nonlinearities. We first prove the local well-posedness of the equation with the small fractional powers of nonlinearity. To study the long term behavior in a general setting is out of reach currently via analytical methods, nevertheless, we investigate solutions’ long term behavior via numerical simulations. In our numerical approaches we include the comparison between the solutions of the equation with and without the absolute values in the nonlinearity; the asymptotic stability of the solitons as well as the confirmation of the soliton resolution conjecture. This result improved previously known results in gKdV (by Linares-Miayazi-Ponce).
The full paper can be found on arXiv.org
- Nonlinear Schrödinger Equation with Combined Nonlinearities
Gia Azcoitia and Hannah Wubben.
Graduate Mentor Alex Rodriguez
Postdoc Mentor: Oscar Riano
Faculty Advisor: Svetlana Roudenko
Abstract: We consider the nonlinear Schrödinger equation with finite number of terms of combined nonlinearities. We obtain the local well-posedness of this equation for any positive powers of α_1 ... α_n for a certain class of initial data, subset of H1 space. Furthermore, by the means of pseudo-conformal transformation, we obtain global solutions for the initial data with a quadratic phase e^{ib|x|^2} for sufficiently large b. Moreover, we show that such data scatters in H1. We also include some numerical simulations for several examples of time evolution of solutions, including computations of ground states in the focusing-defocusing problems. - A Dual Optimal Control Problem of an Epidemic and the Economy
Minato Hiraoka.
Faculty Advisor: Stephen Tennenbaum
Abstract: In this study, we design a dual optimal control problem, imposing a control variable for non-pharmaceutical interventions to minimize both the spread of infectious diseases and the economic impacts that arise from enforcing these restrictions. We propose a compartmental model that allows us to observe the economic impact that arises as a disease spreads over time. We develop a model that incorporates a modified SIR model and a separate model that describes the economy. In these we have applied the Solow-Swan growth model and the Cobb-Douglas production function. We determine this optimal control following Pontryagin’s Maximum Principle, and then provide numerical simulations to visualize the effects interventions such as social distancing have on mitigating the spread of an infectious disease and the economy. By interpreting our results, we are able to provide references and baselines for policymakers that can be taken to better equip society for an emerging epidemic. - Existence and Scattering Results for the Inhomogeneous Nonlinear Schrödinger Equation
Troy Roberts.
Postdoc Mentor: Oscar Riano,
Faculty Advisor Svetlana Roudenko
Abstract: In this work, we construct a class of initial conditions in an appropriated weighted space, for which there exist solutions to an inhomogeneous version of the nonlinear Schrödinger equation determined by a polynomial potential. As a further consequence of our results, we establish existence of global solutions and scattering. - Controllability of Finite-Dimensional Linear Systems
Minato Hiraoka.
Faculty Advisor: Louis Tebou
This project deals with the controllability of linear systems described by ordinary differential equations. The controllability problem discussed consists in finding a function (source) that steers the system from a given initial state to a desired final state. First, we study systems where the coefficient matrices are time dependent, and prove a necessary and sufficient condition for their controllability. Then, we analyze systems where the coefficient matrices are independent of time, and prove the necessary and sufficient Kalman rank condition for controllability. Afterwards, we use a duality method (Hilbert Uniqueness Method) introduced by J.L. Lions to prove controllability, as well. This Hilbert Uniqueness Method was instrumental in the development of the controllability of linear partial differential equations. Finally, we discuss some concrete examples of application.
- Machine Learning for Time Series Clustering
Monica Alvarez,
Graduate Mentors Giancarlo Sanchez, and Luis Caicedo Torres,
Faculty Advisor Mirroslav Yotov
Abstract. Price patterns arise naturally in financial markets. By grouping these patterns algorithmically and at different time frames, we can have better insight to the current market regime and possibly detect developing financial patterns. Clustering is a method that is used to classify data into groups based on distances and similarity measures in order to provide a better understanding of raw data. In this paper, we cluster time series sampled from Bitcoin’s price history using the k-means algorithm and Dynamic Time Warping as a similarity measure. We make inferences from multiple time series by exploring the approaches of different clustering methods in order to distinguish shape-based structures within the financial data of Bitcoin.
- Historical Modeling of Polity Dynamics Via Dynamical Systems
Peter J. Kauphusman.
Faculty Advisor Stephen Tennenbaum
Abstract. This paper aims to utilize dynamical systems in order to gain insight into the dynamics of historical polities through time. Mathematical modeling of polity dynamics provides structure into which phenomena foster mathematically interesting and stable information with respect to key assumptions of polity influences throughout time. Dynamic variables within the paper include polity territory and collective unity (termed asabiya from 14th century scholar Ibn-Khaldun), which are parameterized by civic mobility and morale, frontier length, percent of polity resource output, and ability to exert geopolitical pressure. Historically important influences of various polity dynamics can be drawn from those mathematical models that produce logically consistent mathematical results. Study of these systems can provide mathematical weight to historical assumptions regarding the dynamics of human polities through time until the technological advances of 1600CE+, elucidating which assumptions have more sway into the dynamics of proposed historical influences. Analysis of the models proposed within convey insight into the complexity and key parameters of polity stability and competition between neighbors, as well as how model structure influences the outcome of parameter influence. The application of dynamical systems to historical phenomena not only provides mathematically informative insights into humanity’s past, but also works toward greater mathematical literacy within the domain of historical studies. - Policies of Polities: Modeling Secrets to Obtain the Fittest Civilization
Alejandro Aponte,
Graduate Mentors: Ginelle Gonzalez and Justin Wisby,
Faculty Advisor Stephen Tennenbaum
Abstract. Popularized by Peter Turchin, Cliodynamics takes historical data and quantizes parts of history in an effort to fit mathematical models. We use cliodynamics to take a robust dataset and extract data relating to hundreds of governments and civilizations also known as polities. Using binary encoding, we interpret the data from the Seshat database to derive data that we can model. In this paper we aim to attempt to fit linear regression models. Analyzing common linear regressions, we find that there are no significant simple linear correlations found within our dataset controlling across population, duration, and land size. - Universal Behavior of the Bi-Laplacian Non-Linear Schrödinger Equation
Marcos Masip.
Graduate Mentor: Iryna Petrenko,
Postdoctoral Mentor: Oscar Riano
Faculty Advisor Svetlana Roudenko
In this project, we first proof the local well-posedness (existence, uniqueness and continuous dependence on the initial data) of the bi-Laplacian nonlinear Schrödinger equation. For that we define a new weighted space X, which is a subspace of H1, where the linear Schrödinger equation as well as the nonlinear equation have suitable solutions, and then use the contraction principle to obtain the uniqueness and continuous dependence. After that, we investigate the conservation of mass and energy, and extend the local well-posedness to global for the solutions with the positive quadratic phase initial data. This is the first such result for the bi-NLS with low power nonlinearities.