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| − | ===Radu Precup=== | + | ===Tóth János== |
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| − | Harnack type inequalities and localization of solutions
| + | The concept of reaction extent or the progress of a reaction, advancement of the reaction, conversion, etc. was introduced around 100 years ago. |
| | + | Most of the literature provides a definition for the exceptional case of a single reaction step or gives an implicit definition that cannot be made explicit. |
| | + | Starting from the standard definition we extend the classic definition of the reaction extent in explicit form for an arbitrary number of species |
| | + | and of reaction steps and arbitrary kinetics. Then, we study the mathematical properties (evolution equation, continuity, monotony, differentiability, etc.) of the defined quantity, and connect them to the formalism of modern reaction kinetics. |
| | + | Our approach tries to adhere to the customs of chemists and be mathematically correct simultaneously. |
| | | | |
| − | 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.
| + | We also show how to apply this concept to exotic reactions: reactions with more than one stationary state, oscillatory reactions, and reactions showing chaotic behavior. With the new definition, one can calculate not only the time evolution of the concentration of each reacting species but also the number of occurrences of the individual reaction events. |
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| − | ===Jue Wang===
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| − | Locating Objects of Interest from Screening Ultrasound
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| − | 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.
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| − | ===Matthew Johnston===
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| − | A Mathematical Model of COVID-19 Spread by Vaccination Status
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| − | 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.
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| − | ===Xiang Xu===
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| − | Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals
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| − | 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.
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| − | ===Lorand Parajdi===
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| − | On the controllability of some systems modeling cell dynamics related to leukemia
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| − | 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.
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| − | ===Adrian Tudorascu===
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| − | On the convexity condition for the semi-geostrophic system
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| − | 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.
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Wednesdays at 4pm EDT
zoom link (pass mathAdi2)
The 2021-2022 Applied Mathematics Seminar is organized by Adrian Tudorascu and Casian Pantea.
The talks are on zoom until further notice. The regular time for the Seminar is on Wednesday at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
If you'd like to suggest speakers for the fall semester please contact Adrian or Casian.
Schedule
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.
=Tóth János
The concept of reaction extent or the progress of a reaction, advancement of the reaction, conversion, etc. was introduced around 100 years ago.
Most of the literature provides a definition for the exceptional case of a single reaction step or gives an implicit definition that cannot be made explicit.
Starting from the standard definition we extend the classic definition of the reaction extent in explicit form for an arbitrary number of species
and of reaction steps and arbitrary kinetics. Then, we study the mathematical properties (evolution equation, continuity, monotony, differentiability, etc.) of the defined quantity, and connect them to the formalism of modern reaction kinetics.
Our approach tries to adhere to the customs of chemists and be mathematically correct simultaneously.
We also show how to apply this concept to exotic reactions: reactions with more than one stationary state, oscillatory reactions, and reactions showing chaotic behavior. With the new definition, one can calculate not only the time evolution of the concentration of each reacting species but also the number of occurrences of the individual reaction events.