Events
Department of Mathematics and Statistics
Texas Tech University
The squeezing problem on $\mathbf{C}$ can be stated as follows.
Suppose that $\Omega$ is a multiply connected domain in the unit
disk $\mathbf{D}$ containing the origin $z=0$. How far can the
boundary of $\Omega$ be pushed from the origin by an injective
holomorphic function $f:\Omega\to \textbf{D}$ keeping the origin
fixed?
In this talk, we discuss recent results on this problem obtained
by Ng, Tang and Tsai (Math. Anal. 2020) and by Gumenyuk and Roth
(arXiv:2011.13734, 2020) and also prove few new results using a
method suggested in one of our previous papers. Several remaining
open problems also will be discussed.
In a paper from 1996, Knuth took a combinatorial approach to
Pfaffians, i.e. determinants of skew-symmetric matrices. It was
immediately noticed that this approach facilitates generalizations and
simplified proofs of several known identities involving Pfaffians. I
will discuss one particular case: a formula that expresses arbitrary
minors of a skew symmetric matrix in terms of Pfaffians; it first
appeared in a 1904 paper by Brill.
Join Zoom Meeting https://zoom.us/j/96217128540?pwd=NjU5dzE2RjZvV0prejhOOWVjVENadz09
Meeting ID: 962 1712 8540
Passcode: 474170
The subject of the talk are biological and biomedical problems that we model using systems of chemotactic equations with cross-diffusion terms. I will present recent results related to quasi-periodic behavior of biological system arising from chemotactic framework of Keller-Segel. Model will be derived from Eistein fundamental principal of Brownian motion with new law for expected value of the length of the free jump, depending on the spatial changes in the attractor (food).
In the first part of the talk, model representing traveling wave phenomena of bacteria dynamics in the presence of the diffusion in the media source of attraction (e.g. food, drug, etc.) will be presented. We use celebrated Keller-Segel framework using system of partial differential equation with cross-diffusion and reactive terms in case of traveling wave pattern stability. We use closed form type solution of Keller-Segel system when diffusion coefficient for source of the attractor is equal zero. Stability of the traveling band type solution for complete dynamical system with respect to parameters of equation initial and boundary data is investigated in detail. For this purpose Lyapunov type functional is constructed and non-linear Gronwall differential inequality for Energy functional is derived. Obtained result provide explicit estimate for differences between base-line solution with no diffusion in source term and solution of complete realistic system of equation.
Second one is light driven spatial Algae-Daphinia dynamics as primitive evolutionary model with chemotaxis. First constructed analytically time and spatially dependent solution of system of two equation, which model Algae-Daphinia dynamics and proved using maximum principle machinery that this solution is unique. Then I proved that this solution is stable depending on the relations between chemotactic and diffusion coefficients, and reactive terms, using Sobolev embedding theorems, and Energy functional.
Please virtually attend this seminar via this zoom link Wednesday the 27th at 4 PM.