Events
Department of Mathematics and Statistics
Texas Tech University
N/A
N/A
We are interested in classifying various types of canonical domains which are conformally equivalent to any domain arbitrarily given in the complex plane. Among the techniques of conformal uniformization, extremum problems play a significant role. The central object of the theory is that univalent analytic functions in a given domain form a normal family which guarantees the existence of an extremal element for a reasonable extremum problem. We give an extremal problem which uniformized a given finitely connected subdomain $\Omega$ in $z$-sphere with $\infty \in \Omega $ onto a domain bounded by rectangles with sides parallel to real and imaginary axis having the conformal module $m$.We investigate the clustering dynamics of a network of inhibitory interneurons, where each neuron is connected to some set of its neighbors. We use phase model analysis to study the existence and stability of cluster solutions. In particular,we describe cluster solutions which exist for any type of oscillator, coupling and connectivity. We derive conditions for the stability of these solutions in the case where each neuron is coupled to its two nearest neighbors on each side. We apply our analysis to show that changing the connection weights in the network can change the stability of solutions in the inhibitory network. Numerical simulations of the full network model confirm and supplement our theoretical analysis. Our results support the hypothesis that cluster solutions may be related to the formation of neural assemblies.
 | Wednesday
Nov. 13
3:00 PM
Math 111
|
| Algebra and Number Theory
No Seminar
|
We propose a first order energy stable linear semi-implicit method for solving the Allen-Cahn-Ohta-Kawasaki equation. By introducing a new nonlinear term in the Ohta- Kawasaki free energy functional, all the system forces in the dynamics are localized near the interfaces which results in the desired hyperbolic tangent profile. In our numerical method, the time discretization is done by some stabilization technique in which some extra nonlocal but linear term is introduced and treated explicitly together with other linear terms, while other nonlinear and nonlocal terms are treated implicitly. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are proved for this method in both semi-discretization and full discretization levels. Numerical experiments indicate the force localization and desire hyperbolic tangent profile due to the new nonlinear term. We test the first order temporal convergence rate of the proposed scheme. We also present hexagonal bubble assembly as one type of equilibrium for the Ohta-Kawasaki model. Additionally, the two-third law between the number of bubbles and the strength of long-range interaction is verified which agrees with the theoretical studies.Math Circle Fall Poster