Math 251-102 Fall 2022

Thursdays at 5pm EDT
zoom link (password euclid2022)

The 2022 WVU Junior Math Club is a mathematical enrichment program open to middle school and high school students in West Virginia and Southeast Pennsylvania. We meet on Zoom every Tuesday in Spring 2022 and we discuss various mathematical topics with special emphasis on competition-type problems. The club is supported by an MAA Dolciani grant and the West Virginia University School of Mathematical and Data Sciences. If you would like to participate, or for additional info please contact Casian Pantea at cpantea@math.wvu.edu.

Schedule (updated as we go)

date	topic	instructor	materials
February 3	Geometry 1	Pantea	Congruences, similarity and other interesting triangle-related stuff
February 7	Algebra 1	Voitiuk	Algebraic inequalities
February 3	Combinatorics 1	Goldwasser	Congruences, similarity and other interesting triangle-related stuff
February 24	Combinatorics 1	TBD	Counting sets, Pascal triangle, and other combinatorial identities

Abstracts

Veronica Ciocanel

Actin filaments are polymers that interact with myosin motor proteins and play important roles in cell motility, shape, and development. Depending on its function, this dynamic network of interacting proteins reshapes and organizes in a variety of structures, including bundles, clusters, and contractile rings. Motivated by observations from the reproductive system of the roundworm C. elegans, we use an agent-based modeling framework to simulate interactions between actin filaments and myosin motor proteins inside cells. We also develop techniques based on topological data analysis to understand time-series data extracted from these filament network interactions. These measures allow us to compare the filament organization resulting from myosin motors with different properties. Recently, we have also studied how different models of myosin regulation predict actin network organization during the cell cycle. This work also raises questions about how to assess the significance of features in common topological summaries.

Radu Precup

Harnack type inequalities and localization of solutions

First it is explained the problem of localization of solutions for boundary value problems in general. Next, the focus will be on radial solutions for phi-Laplace equations. Numerical solutions are obtained for some concrete equations.

Jue Wang

Locating Objects of Interest from Screening Ultrasound

A fast Enclosure Transform is developed to localize complex objects of interest from medical ultrasound images. This approach explores spatial constraints on regional features from a sparse image feature representation. Unrelated, broken ridge features surrounding an object are organized collaboratively, giving rise to the enclosureness of the object. Three enclosure likelihood measures are constructed, consisting of the enclosure force, potential energy, and encloser count. In the transform domain, the local maxima manifest the locations of interest objects, for which only the intrinsic dimension is known a priori. I will demonstrate two medical applications in detecting (1) suspicious breast masses in screening breast ultrasound, and (2) the location of the prostate on trans-abdominal ultrasound for verification of patient positioning in radiotherapy treatment of prostate cancer.

Matthew Johnston

A Mathematical Model of COVID-19 Spread by Vaccination Status

The novel coronavirus SARS-CoV-2, and the corresponding illness COVID-19, has afflicted hundreds of millions of people worldwide, killed at least 5 million, and devastated the world economy. The rapid development of effective and safe vaccines, which were authorized in the United States under Emergency Use Authorization in December 2020, have offered some hope that the end of the pandemic may be sight; however, at the same time we have seen the rise of variants, such as Delta, which are much more transmissible than the original strains. In this talk, I will introduce an n-stage vaccination model and corresponding system of differential equations which can simulate a disease outbreak by breaking the population down according to their vaccination status. This allows the mitigation effects of vaccination and accelerating effects of variants such as Delta to be uncoupled from one another, and offers valuable insight for the future course of the COVID-19 pandemic. I fit the model to 2021 data from the Virginia Department of Health.

Xiang Xu

Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals

The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential. More specifically, we provide precise estimates of both this singular potential and its gradient as the Q-tensor approaches its physical boundary.

Lorand Parajdi

On the controllability of some systems modeling cell dynamics related to leukemia

In this talk, I will present two control problems for a model of cell dynamics related to leukemia. The first control problem is in connection with classical chemotherapy, which indicates that the evolution of the disease under treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of leukemic cells only, and the control objective is that the disease state reaches a desired endpoint. The solvability of the two problems as well as their stability is proved by using a general method of analysis. Some numerical simulations are included to illustrate the theoretical results and prove their applicability. The results can possibly be used to design therapeutic scenarios such that an expected clinical evolution can be achieved.

Adrian Tudorascu

On the convexity condition for the semi-geostrophic system

We show that conservative distributional solutions to the Semi-Geostrophic system in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge- Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admissible vector field.