RESEARCH PROJECTS

Often, when solving large linear systems, iterative methods based on Krylov subspaces are used, such methods include GMRES, CG, MINRES, SYMMLQ. When such methods demonstrate slow convergence, preconditioning techniques are used to accelerate convergence. These techniques are often informed by the physics of the underlying model or the structure of the coefficient matrix of the linear system. In this project will explore such methods generally and for problems derived from acoustic and electromagnetic wave scattering problems.
The Korteweg-De Vries (KdV) equation is a mathematical model of shallow-water waves. In this project, we will study one and two soliton solutions to both the KdV and modified KdV equations. We will derive a one-soliton solution by using the traveling wave setting, then we will apply the Miura transformation and a bilinear method to obtain a two-soliton solution.
Soliton solution of the sine-Gordon equation: Sine-Gordon equation is a very important nonlinear equation in physics. It admits kink, anti-kink, and breather solutions. We will use several methods to construct these solutions.
The Toda lattice is a simple model for one-dimensional crystals in solid-state physics Why is this model exactly solvable? In this project, we will derive the equations for the Toda lattice and construct its one and two soliton solutions. Then by introducing the Lax pair operators, we will derive the conservation laws of the Toda lattice system and show why it is exactly solvable.
A graph coloring is a function from the vertices of a graph to a set of colors such A graph coloring is a function from the vertices of a graph to a set of colors such that adjacent vertices get distinct colors. The chromatic polynomial is a function of a graph G which counts the number of colorings of G with a given number of colors. A major open problem is to determine which polynomials arise as chromatic polynomials of graphs. The goal of the project is to study the coefficients of chromatic polynomials of graphs, and other variations of graphs, to determine new properties of these polynomials.

In this summer project, we will explore patterns in an infinite sequence of number triangles related to Pascal's Triangle. The results we discover may be part of a larger, publishable project. Some of the techniques we will use are generating functions and recurrence relations. Creativity and curiosity are critical! For more information, students are welcome to contact :

Dr. Kronholm at brandt.kronholm@utrgv.edu

or stop by:
room EMAGC 3.502
Thursdays at 3:00.

Where we are currently investigating these number triangles.

Optimal quantization has broad application in signal processing and data compression. It also has many applications in our daily life. While driving it is important that the cell phones are working, i.e., the cell phones always get signal, so that for any emergency we can get help. The signal of a cell phone depends on how far we are from the tower. While driving in a remote area sometimes we can see that there is no signal in our cell phones. This happens probably we are far away from the tower, or there is no tower nearby to catch the signal. In optimal quantization, our goal is to end the exact locations of the towers to be installed so that while driving we can get the best signal as much as possible on our cell phones. By the best signal, it is meant that the distortion error, also known as noise, is minimum. Next, suppose that we need to install n mobile towers in a state of our country for some positive integer n, so that wherever we move around the state we can get the best signal in our cell phones. In quantization theory, our goal is to investigate such exact locations.
Topics in Elementary Algebraic Geometry - The program aims at attracting students to the world of pure mathematics, especially to the field of algebraic geometry. The roots of algebraic geometry date back to the mathematics from 5th century BC. On the other hand, modern algebraic geometry is still developing rapidly through interaction with the analytical method. In this program, we will learn and understand the basics of modern languages (sheaf, variety, divisors. . . ) in algebraic geometry. This will provide enough background for undergraduate students to continue studying the advanced topics. We will eventually be able to prove some well-known results.

Need help?

Josef Sifuentes

Josef Sifuentes, Ph.D.

Office: EMAGC 3.614
Phone: (956) 665-2197
Email: josef.sifuentes@utrgv.edu