Events
Department of Mathematics and Statistics
Texas Tech University
Serre defined and studied an intersection multiplicity for finitely
generated modules over a regular local ring by using the Euler
characteristic, and showed it satisfies many properties that one would
expect from an intersection theory. In this talk we discuss a new
notion of lifting modules over a noetherian local ring to a regular
local ring. We show how it can be used to prove a new case of Serre's
long standing conjecture on the positivity of the Euler
characteristic, and then provide characterizations of these liftable
modules. This is joint work with Nawaj KC, and with Benjamin Katz,
Nawaj KC, Kesavan Mohana Sundaram, and Ryan Watson.
Abstract: In this work, we propose a new class of time integration methods, referred to as two-derivative exponential Runge--Kutta (TDexpRK) methods for stiff semilinear parabolic PDEs. Specifically, we construct TDexpRK integrators that inherit the favorable properties of both two-derivative Runge--Kutta (TDRK) methods and explicit exponential Runge--Kutta (ExpRK) methods. In particular, TDexpRK methods treat the stiff linear part exactly via exponential operator, while handling the nonlinear term with a two-derivative correction weighted by exponential $\varphi$-functions of the linear operator. The structure of our TDexpRK schemes enables a local error expansion involving only four stiff order conditions for methods up to fifth order, which is significantly fewer than the sixteen conditions required for ExpRK integrators. Based on this analysis, we rigorously prove convergence up to fifth-order accuracy, with an error bound that remains uniform with respect to the stiffness of the linear operator. As a result, we obtain high-order, explicit, stiffly accurate TDexpRK schemes that exhibit unconditional linear stability and require only a few stages per step (e.g., 2-stage 4th-order scheme). Numerical experiments on PDEs in one and two spatial dimensions confirm the superior accuracy and efficiency of the proposed methods compared with existing exponential Runge--Kutta/Rosenbrock schemes from the literature.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this
Direct Link that embeds meeting and ID and passcode.
- Choice #2: Using this link, Join the Meeting, where you will have to input the ID and Passcode by hand:
* Meeting ID: 949 9288 2213
* Passcode: Applied
TTU Math Circle Spring Flyer 6:30-7:30 PM Thursdays in the basement of Math, room 010
As a well-developed branch of mathematics, graph theory provides unique tools to quantifiably assess various properties of complex networks. Applied to brain circuits, network-level analyses can illustrate disruptions to brain organization that yield both mechanistic and diagnostic insight. Previously, graph theory has been used with functional magnetic resonance imaging (fMRI) datasets to quantify connections among different brain regions, readily capturing the macroscopic-scaled differences in brain networks between healthy and Alzheimer’s subjects. Here, we applied graph theory on the microscopic scale, using miniscope-based calcium imaging recordings from the freely behaving wild type and Shank3fx mice (a mouse model of autism), and compared their functional connections among individual neurons in the prefrontal cortical microcircuits during social behavior tasks. We demonstrated that Shank3fx mice displayed reduced neural activity, a less integrated network, and fewer network changes in the prefrontal microcircuits between the presence and absence of social targets. Furthermore, we employed machine learning to test whether graph-theoretic metrics extracted from the prefrontal microcircuits could be predictive of genotype and genotype-associated social behavior difference between Shank3fx and WT mice. Our results indicate a strong link between altered prefrontal microcircuits and social behavior differences in an ASD mouse model, highlighting prefrontal microcircuitry as a potential diagnostic and therapeutic target for ASD.
The Biomath seminar may be attended virtually Friday at 11:00 AM CST (UTC-6) via this Zoom link.
Meeting ID: 938 8653 3169
Passcode: 883472
abstract 2 PM CST (UTC-6)
Zoom link available from Dr. Brent Lindquist upon request.
