Some of our math honors students are giving talks on their research on  Friday, April 15, 2022 at 3 pm.   There will be nine 12 – 15 minute talks on

 zoom and it would be great if you can attend this zoom session to listen to their research work.

https://ncsu.zoom.us/j/96201139617?pwd=eFVadDNFOE1nQ3dyRVpvMk8yM0pOUT09

Meeting ID: 962 0113 9617
Passcode: 915393

1. Erick Boniface

 

Title: Choosing a good variable ordering for Cylindrical Algebraic Decomposition

 

Abstract: Cylindrical Algebraic Decomposition (CAD) is a key algorithm for a fundamental problem in real algebraic geometry, with numerous real-life applications in diverse areas such as real quantifier elimination problems, stability analysis, robust control system design, reachability analysis, parametric optimization, parameter estimation, robot motion planning, computer vision, epidemic modeling, etc. The variable ordering of the input for a CAD calculation significantly impacts the computation time, and thus, numerous authors have worked to solve the problem of finding a good variable ordering. We propose a novel approach for finding a good variable ordering. In addition, this approach may be implemented to improve other key algorithms in algebraic geometry. (Faculty mentor: Dr. H. Hong)

 

2. Soren Davis

 

Title: Blind Digital Signatures with RSA

 

Abstract: We investigate the idea of a digital signature, which cryptographically authenticates a message as coming from a particular sender, then we extend that idea to discuss blind signature schemes, which allows an author to get a message signed by a third party without that party ever being able to correlate the message with its source. Both of these can be concretely accomplished using the RSA cryptosystem, which relies on certain computational properties of modular arithmetic operations. (Faculty mentor: Dr. E. Stitzinger)

 

3. Nicholas Gawron

 

Title: Decorrelation Detection in a Financial Time Series Data Set

 

Abstract: For this NSF funded REU project, we partnered with Wellington Management Company and researched how dimension estimation can determine whether seemingly diversified index fund portfolios in fact have latent commonalities that increase risk for investors. Principal Component Analysis (PCA) is a widely used tool for dimension estimation, but during periods of financial crisis, PCA is unable to adequately perform dimension estimation, because it relies on linear relationships and is sensitive to outliers. We investigate robust and nonlinear techniques including Robust PCA and autoencoders, assessing their usefulness for three tasks: dimension estimation, data reconstruction, and correlation visualization. We find several methods that are resistant to market shocks and develop a novel method to estimate dimension using autoencoders. We additionally provide insight into future time series modeling. (Faculty mentor: Dr. F. Wang)

 

4. Matthew Hefferie

 

Title: Global Sensitivity Analysis Across Scales

 

Abstract: This presentation will discuss properties that carry across homogenization. There is a focus on the temperature of a one-dimensional medium with periodic structure with uncertainty parameters in the source. The sensitivity is then studied for both the homogonous and non-homogonous equations, with the quantity of interest being the integral of the temperatures of the structure. The uncertainty parameters are contained in the source term. Monte Carlo approximations were used to calculate the Sobol indices for both cases. In this presentation, we show that the Sobol indices of the non-homogenized equations will converge to those of the homogenized solution. Using the homogenized equation allows for a cheaper computation of the Sobol indices and could be expanded upon to more complicated problems. (Faculty mentor: Dr. A. Alexanderian)

 

5. Jordan Jackson

 

Title: Backward Error of the Polynomial Eigenvalue Problem Expressed in the Chebyshev Basis

 

Abstract: Given a matrix polynomial P(λ) of degree k expressed in a polynomial basis, the associated polynomial eigenvalue problem (PEP) consists of finding λ0 ∈ C and nonzero x, y ∈ C_n such that P(λ_0)x = 0 and yT P(λ_0) = 0. The PEP is then solved through the use of a linearization L(λ) of P(λ), where L(λ) is a matrix polynomial of degree 1 with the same eigenvalues and multiplicities as P(λ). Finding exact eigenpairs of L(λ) allows us to recover exact eigenpairs of P(λ). Our goal is to find a linearization of P(λ), within a given family, that minimizes the backward error η for the computed eigenpairs of P(λ) from L(λ). The backward error tells us the distance of the “closest” matrix polynomial P′(λ) to P(λ) for which the computed eigenpair is exact. We analyze these linearizations for matrix polynomials expressed in the Chebyshev basis. In order to accomplish this goal, we construct bounds on the term ηP/ηL for the given linearizations L(λ). These bounds ideally give us criteria under which we may choose L(λ) to minimize ηP. (Faculty mentor: Dr. M. Cachadina)

6. Nicholas Labaza

Title: Exploring Digital Signatures and Zero Knowledge Proofs

 

Abstract: Digital Signature schemes and Zero knowledge proofs are two of the important applications of modern cryptosystems. These tools allow an individual to prove or verify something about themselves without giving away extra information. Many of the popular cryptosystems have well studied digital signature schemes, but not all have standard zero knowledge protocols. We looked at the RSA and lattice based GGH cryptosystems and their digital signatures and created and explored a possible zero knowledge protocol for each. (Faculty mentor: Dr. E. Stitzinger)

 

7. Madhusudan Madhavan

Title: Computational Studies on Zermelo’s Navigation Problem: Preliminary Results

 

Abstract: Zermelo’s Navigation Problem has been a relatively recent but important problem in the field of optimal control. An intriguing aspect of this problem is its broad applicability to various physical fields, such as navigation in quantum fields and three-dimensional obstacle avoidance, along with its seemingly straightforward underlying concept [1, 3]. In this project, we examine modifications of Zermelo’s Navigation Problem for an aircraft in a nonlinear wind-field, computationally solve the relevant optimal control problem, and analyze the preliminary impact of parameters on the optimal trajectory. Particularly, we focus on both sinusoidal and real-world wind fields in a maximum-range formulation that incorporates a form of drag and limited acceleration. To enable computational studies, we discretize the optimal control problem and apply unconstrained optimization routines to the multivariable system in MATLAB. We then examine the solutions for the analytic wind field with perturbed model parameters, such as final time and initial velocity. (Faculty mentor: Dr. A. Alexanderian)

 

 

8. Hang Nguyen

 

Title: Analysis of Bactrocera dorsalis Count and Environmental Factors Affecting Bactrocera dorsalis and Fopius arisanus Growth

 

Abstract: The purpose of this research is to analyze the changes in daily temperature, humidity, and Bactrocera dorsalis (Oriental Fruit Fly) count in three regions of Senegal between October 2017 and March 2019 to inform a delayed differential equations model of the population dynamics between B. dorsalis and a parasitoid called Fopius arisanus. After accounting for missing measurements using cold deck imputation from publicly available weather data, we tested for differences in daily temperature, humidity, and fly count each month. To factor in autocorrelation between measurements, we used 99% z confidence intervals of pairwise differences for environmental variables in our hypothesis tests. We compared the fly counts using the Kolmogorov-Smirnov Test for the distance between the count distributions. Our analyses show that for all months, there are statistically significant differences in temperature and humidity between the three regions. For most months there is sufficient evidence at alpha = 0.05 that the count distributions for each region are different. In addition, we found that the mean daily temperature and counts both peak during the short rainy season from July to September in the North and June to October in the South. The results lead us to conclude that environmental variables and fruit fly counts are different between the three regions most of the time, and that there is a relationship between temperature and fruit fly growth. This motivates us to explore different models of B. dorsalis and F. arianus dynamics using temperature for each region in Senegal. (Faculty mentor: Dr. J. Banks)

9. Thomas Steckmann

Title: A Lax Dynamics Solver for Quantum Computing Operators

 

Abstract: Developing applications for Quantum Computers requires first compiling an algorithm into the native set of physical operations allowed on the quantum computer — called gates. For applications in physics, one such algorithm is the time evolution operator which describes the dynamics of a physical system over time. Compiling this operator efficiently is a task of increasing importance, as modern quantum hardware is error prone and reducing the total number of gates increases the accuracy of the resulting simulation.  The difficulty is in implementing the matrix exponential of the non-commuting Hamiltonian operator, which is a descriptor of the energy in the system and governs the time dynamics. In this work, we apply a method called Cartan Decomposition on the minimal Lie Algebra which spans the basis vectors of the Hamiltonian. However, although a general method to construct the form of the decomposition is well known, solving the exact parameters of the decomposition is not well studied, and currently relies on a generic minimization algorithm. Here, we develop a framework for solving the Cartan decomposition using a parameter flow in Lax Dynamics, which produces a first order differential equation that can be numerically integrated to diagonalize the Hamiltonian within a required subspace. (Faculty mentor: Dr. M. Chu)