Title: The Birth of Oscillations: Bifurcations in Dynamical Systems

Abstract : In this presentation, I will talk about the dynamic behavioral changes of ordinary differential equations, focusing on the Bifurcation phenomenon. We begin with some examples of one-dimensional systems and will examine how equilibria change under parameter variation. Next, we will discuss the emergence of oscillatory behavior via limit cycles in two-dimensional systems and will discuss Hopf Bifurcation – which is a fundamental mechanism for the birth of periodic solutions.

Title: An introduction to the theory of Sobolev spaces.

Abstract:

The goal of this talk will be to give an expository presentation (at a level appropriate for any graduate student in mathematics) on the key aspects pertaining to the theory of Sobolev spaces. In particular, this talk will aim to demonstrate the motivating utility that Sobolev spaces provide for modern analysis, as well as their utility in the theory underpinning the modern study of partial differential equations. For example, this talk will cover the notion of weak differentiability, regularity and the approximation of Sobolev functions by smooth functions, Sobolev-type inequalities, and a few other notable properties. Also, some prerequisite material will be briefly reviewed.

Title: LOW–ORDER RAVIART–THOMAS APPROXIMATIONS OF AXISYMMETRIC DARCY FLOW

Abstract: We study the lowest–order mixed finite element method for the axisymmetric Darcy problem using Raviart–Thomas elements. In contrast to the Cartesian setting, the method is non–conforming in the sense that the discretely divergence–free functions are not solenoidal. We derive several estimates that measure the inconsistency of the method and derive error estimates of the discrete pressure and velocity solutions. We show that if the domain is convex, then the errors converge with optimal order modulo logarithmic terms. Numerical experiments are presented, and they indicate that the estimates are sharp.

Title: Using Data Fusion to Analyze Dynamic Stability of Atmospheric Entry Vehicles

Abstract: 

Atmospheric entry and descent is one of the most important stages of a space mission. After a successful entry, the vehicle decelerates with the aerodynamic forces exerted on the body. As the vehicle decelerates, it starts to oscillate due to the aerodynamic nature of the blunt bodies. However, a safe landing often requires parachute deployment, which can be done within a certain oscillation frequency and amplitude. Unless these oscillations are addressed during the design stage, they may lead to catastrophic events, such as unsuccessful parachute deployment or tumbling during descent. As a result, identifying the dynamic characteristics of the vehicle is crucial for the safety and success of the mission. In this study, we are proposing a data fusion algorithm to estimate the dynamic stability coefficients of an atmospheric entry vehicle by using two different datasets representing the numerical and experimental results. The algorithm uses the trajectory of the vehicle as the input data and estimates the dynamic stability coefficients based on the angle of attack oscillations. The proposed algorithm consists of three steps: (i) the reconstruction of the trajectory based on the sparse observation points representing the experimental data, (ii) the numerical and experimental data fusion, and (iii) the dynamic stability coefficient estimation by using the Markov Chain Monte Carlo method.

Title: Analyzing Political Bias in Language Models: Impact of Entities on Bias Identification

Abstract:  Pretrained language models (LMs) are widely used for tasks like detecting political leaning in text documents. However, the complexity of political ideologies and inherent biases in the training data pose significant challenges. LMs are trained on diverse data sources—including news, forums, books, and encyclopedias—that often contain socially biased opinions and perspectives. These biases can lead LMs to unknowingly favor certain ideologies or overemphasize specific entities, propagating social biases when used for downstream tasks such as hate speech detection, polarization, misinformation detection etc. Fine-tuning these models with politically biased datasets can further shift their biases. However, the mechanisms driving this bias shift remain underexplored. We aim to systematically investigate the causes of political bias shifts in LMs post fine-tuning and develop methods to measure political biases in LMs. 

We particularly focus on language models’ emphasis on person names and location names in classifying political leaning of text documents. Based on our experiments with 37,554 news articles, we find that (i) all language models are somehow biased towards new data during inference, and (ii) some models weigh heavily on names (celebrities or civilians) to understand political bias. We also evaluate prompt-based querying or extracting political leaning on a small data sample.

Our work highlights the importance of addressing and mitigating biases in LMs to improve the fairness and accuracy of political leaning identification and other socially oriented NLP tasks without relying on human-annotated data.

Title: Forecasting dengue fever in Brazil: An assessment of climate conditions

Abstract:  Local climate conditions play a major role in the biology of the Aedes aegypti mosquito, the main vector responsible for transmitting dengue, zika, chikungunya and yellow fever in urban centers. For this reason, a detailed assessment of periods in which changes in climate conditions affect the number of human cases may improve the timing of vector-control efforts. In this work, a new machine-learning algorithms was developed to analyze climate time series and their connection to the occurrence of dengue epidemic years for seven Brazilian state capitals. The method described in this talk explores the impact of two key variables—frequency of precipitation and average temperature—during a wide range of time windows in the annual cycle. The results indicate that each Brazilian state capital considered has its own climate signatures that correlate with the overall number of human dengue-cases. However, for most of the studied cities, the winter preceding an epidemic year shows a strong predictive power. Understanding such climate contributions to the vector’s biology could lead to more accurate prediction models and early warning systems. 

Abstract:  Human mobility plays a crucial role in the spatial and temporal spread of infectious diseases, and many studies have investigated its impact on disease propagation extensively. However, integrating mobility data into metapopulation models remains elusive. In this study, we aimed to explore how human mobility influences the spread of diseases within a metapopulation. We focused specifically on short-range trips between counties across the fifty states in the United States. To achieve this, we equipped the SIR-network model with human mobility data by incorporating human movement trajectories between counties, which allowed us to trace the evolution of the disease across the counties. This approach resulted in more realistic epidemic patterns, which could not be captured by a theoretically fully connected mobility network as previously studied. Our findings suggest that mobility data-driven dynamics are much richer than traditional homogeneous deterministic models. We then identified the most high-risk counties for infectious diseases, called hotspot counties, in each state of the U.S. We also observed that population flow metrics, serving as a proxy for mobility, were positively correlated with reported COVID-19 cases during the pre-lockdown period. Furthermore, we demonstrated that reducing human mobility can significantly influence the overall epidemic size, as shown through scenario modeling, which led to identifying a more cost-effective control strategy. Finally, we showed how human movement contributes to disease transmission in the metapopulation, offered valuable insights for targeted interventions, and helped public health officials manage future outbreaks like COVID-19 more effectively. 

Abstract:  We consider two modifications of the Arrow-Hurwicz (AH) iteration for solving the incompressible steady Navier-Stokes equations for the purpose of accelerating the algorithm: grad-div stabilization, and Anderson acceleration. AH is a classical iteration for general saddle point linear systems and it was later extended to Navier-Stokes iterations in the 1970s which has recently come under study again. We apply recently developed ideas for grad-div stabilization and divergence-free nite element methods along with Anderson acceleration of xed point iterations to AH in order to improve its convergence. Analytical and numerical results show that each of these methods improves AH convergence, but the combination of them yields an efficient and effective method that is competitive with more commonly used solvers.

Spring 2024

SIAM Student Chapter Bi-weekly Seminer 

Speaker: Haridas Das, Ph.D. Student

Department of Mathematics, Oklahoma State University

Room: MSCS 114, Date: May 2nd,  2024, Time: 1:30 pm -2: 30 pm

Title: Understanding the Dynamics of Mpox Transmission with Data-Driven Methods and a Deterministic Model

Abstract:   Mpox (formerly monkeypox) is an infectious disease that spreads mostly through direct contact with infected animals or people's blood, bodily fluids, or cutaneous or mucosal lesions. In light of the global outbreak that occurred in 2022–2023, in this paper, we analyzed global Mpox univariate time series data and provided a comprehensive analysis of disease outbreaks across the world, including the USA with Brazil and three continents: North America, South America, and Europe. The novelty of this study is that it delved into the Mpox time series data by implementing the data-driven methods and a mathematical model concurrently — an aspect not typically addressed in the existing literature. The study is also important because implementing these models concurrently improved our predictions' reliability for infectious diseases.

Abstract:  Given a discrete data set, there are many ways to extract a continuous function. We discuss one type of method, that being polynomial interpolation. We will begin by discussing Lagrange interpolation and the resulting Runge's phenomenon, which makes high degree interpolants numerically unstable. To rectify this issue, we will use spline interpolation. Throughout, we will apply the techniques we learn to a numerical solution of a two-point boundary problem for a second-order, linear ODE.

Abstract:  Elliptic operators are ubiquitous; they appear frequently in classical mechanics, quantum mechanics, and even in general relativity. This talk aims to explain ellipticity (somewhat generally) from scratch. I will start with the classic definition of ellipticity for a single linear partial differential equation (PDE) in flat Euclidean space. Then we extend this definition to a system of linear PDEs via the notion of "principal symbols." Then the next step is to define ellipticity for linear operators on vector bundles over manifolds. In this stage, the geometric interpretation of symbols & ellipticity will be addressed. I will extend the notion of ellipticity one more time and explain the Nirenberg-Douglis ellipticity. If time permits, I will give an example from general relativity.