Vanderbilt Mathematics
Analysis & Biomathematics Seminar
2005--2006

Seminars are listed in reverse chronological order. The top of the list is subject to change, since more seminars are still being planned. All seminars are held at 2:10p.m. in 1431 Stevenson Center unless otherwise noted. For further information on events in the department, you may also consult the colloquia schedule, the weekly calendar and past calendars.

Previous semesters:
Wednesday, November 30, 2:10 SC 1431
Philip Crooke, Vanderbilt University
Title: In Silico Pulmonology
Abstract: In this talk we look at mathematical models for mechanical ventilation. Mechanical ventilation is mechanically assisted breathing using an electrically powered device (ventilator) that forces oxygenated air into the lungs and then allows time for passive exhalation of air. This type of medical procedure can often lead to lung injury (due to high alveolar pressures and/or the repetitive opening and closing of alveoli). The main role of these models is to provide the clinician with strategies that minimize the deleterious side effects of mechanical ventilation. In these models the compliance of the patient's lung play a key role; in particular, we assume that compliance is a function of lung volume.
The following problems: will be discussed: The models used in the above work are differential equations. If time permits, we will look at sleep apnea and how attributed finite automata can be used to analyze nasal pressures in sleeping individuals to detect airway obstructions.
Friday, November 18, 2:10 SC 1431
Changfeng Gui, University of Connecticut
Title: Entire Solutions in Phase Transitions
Abstract: Entire solutions often play an important role in the study of partial differential equations since they arise naturally in the blow-up analysis of singularities. In this talk, I will survey some existence and symmetry results on various entire solutions related to phase transition, including the Allen-Cahn model and multi-phase model.
Wednesday, November 9, 2:10 SC 1431
Chunming Li, Institute of Imaging Science, Vanderbilt University
Title: Level Set Evolution without Reinitialization: A New Variational Formulation
Abstract: Level set methods have been extensively applied in computational geometry, fluid mechanics, computer vision, and material science. Traditional level set methods typically require periodic reinitialization of the level set function to a signed distance function to maintain stable evolution and produce usable results. The reinitialization procedure is computationally expensive, and has an undesirable side effect of moving the zero level set away from its original location. Moreover, it is still a serious problem as when and how to apply the reinitialization. So far, the reinitialization procedure has been applied in an ad-hoc manner.
In this talk, I will present a new variational level set method without the need of reinitialization. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired position and shape. The level set evolution is the gradient flow that minimizes the overall energy functional, and thereby is able to simultaneously drive the motion of the zero level set while maintaining the level set function as an approximate signed distance function throughout the entire evolution.
Our level set method not only eliminates the need of reinitialization, but also presents several advantages over the traditional methods. First, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and the algorithm is computationally more efficient. Second, a significantly larger time step can be used to speed up the curve evolution. Third, the computation is more accurate in terms of the location of the zero level set. In addition, the level set function can be initialized as general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function.
Wednesday, November 2, 2:10 SC 1431
Erik Boczko, Vanderbilt University
Title: An inverse problem in nano-biology
Abstract: In many situations and applications it is of interest to understand the dynamics of interaction between a biopolymer fiber and ligand. For instance it is of great interest to understand how proteins interact with DNA. For instance, do you imagine that a transcription factor docks to its cis regulatory element like a space shuttle, guided by some potential? We have concieved an assay to record and reconstruct these dynamics. The assay technique is based upon the solid foundation of wavepropagation in elastic media and well studied inverse problems in partial differential equations. We have conceived a design for a nonoscale device that has the potential to realize this theory.
Wednesday, October 26, 2:10 SC 1431
Akram Aldroubi, Vanderbilt University
Title: Best Shift -Invariant space models
Given a set of functions $F=\{f_1, \dots,f_m\} \subset L^2(R^d)$, we study the problem of finding the shift-invariant space $V$ with $n$ generators $\{\phi_1,\dots,\phi_n\}$ such that

\begin{displaymath}V = \hbox{argmin}_{V' \in \mathcal V_n}\sum \limits_{i=1}^m w_i\Vert f_i-P_{V'}f_i\Vert^2, \end{displaymath}

where $w_i$s are positive weights, and $\mathcal V_n$ is the set of all shift-invariant spaces that can be generated by $n$ or less generators. The Eckard-Young Theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an infinite set of finite dimensional problems that can be solved using an extension of the Eckard-Young Theorem. We prove that the finite dimensional solution can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-Rays), from the observation of a set of $m$ signals or images $f_1, \dots,f_m$, which may be theoretical samples, or experimental data.
Wednesday, October 19, 2:10, SC 1431
Peter Hinow, Vanderbilt University
Title: Quantification of p53-DNA binding kinetics in living cells using fluorescence recovery after photobleaching
The tumor suppressor protein p53 plays a key role in guarding genomic stability of mammalian cells and preventing malignant transformation. We investigated the intracellular diffusion of a p53-GFP fusion and GFP using fluorescence recovery after photobleaching (FRAP). To interpret the data we use two mathematical models, one for free diffusion and one for diffusion of particles that are binding to a spatially homogeneous immobile structure. We find that p53-GFP undergoes binding and unbinding while GFP diffuses freely, in agreement with the role of p53 as a transcription factor. We estimate the diffusion constants to be Dp53-GFP = 15.4 um2s-1 and DGFP = 41.6 um2s-1 . The reaction rates of the binding and unbinding of p53-GFP are estimated to be k1=0.3 s-1 and k2=0.4 s-1, respectively. The emphasis in this talk will be on the close interplay between mathematical modeling, statistical analysis and designing photobleaching experiments.
Wednesday, October 12, 2:10, SC 1431
Xu Zhen, Office of Disease Control and Emergency Response,
Chinese Center for Disease Control and Prevention, Beijing, China.
Title: Overview of SARS epidemcs in Mainland China, 2003 and 2004.
The talk will focus on:
  1. the epidemiological analysis on time and population distribution in different regions in China;
  2. the analysis on the possibility of SARS reemergence in China; and the introduction of basic strategy and policy in SARS control in China,
  3. As well as a brief introduction of SARS epidemics in 2004 and the comparison between the epidemics in 2003 and in 2004.
Wednesday, September 21, 3:10, SC 1432.
Alexander Powell, Vanderbilt University
Title: Uncertainty principles for time-frequency expansions.
A Gabor system consists of translates and modulates of a fixed window function, and can be used to give signal decompositions in various settings. The Balian-Low theorem (BLT) is an uncertainty principle for Gabor orthonormal bases which places strong restrictions on the time-frequency localization of window functions in this case. We shall discuss some of our recent work on the BLT. For example, we shall show that the BLT is sharp, address variants for non-symmetric weight pairs, and consider complementary results when the Gabor structure is removed.
Wednesday, September 14, 3:10, SC 1432.
Christoph Walker, Vanderbilt University
Title: Solvability of a Mathematical Model of Prion Proliferation
We will consider the model of prion proliferation introduced by Glenn Webb in the previous talk. The model consists of an ordinary differential equation coupled with a non-local partial differential equation. For bounded degradation and splitting rates we prove global existence and uniqueness of classical solutions. We also provide sufficient conditions for global existence of weak solutions for unbounded rates. In both cases asymptotical stability of the disease-free steady state is shown.
Wednesday, September 7, 3:10, SC 1432.
Glenn Webb, Vanderbilt University.
Title: Mathematical Models of Prion Proliferation.
Abstract: Prions are infectious proteins that are hypothesized to be the causative agent of diseases such as Creutzfeld-Jakob disease in humans, scrapie in sheep, and bovine spongiform encephalopathies in cows (mad cow disease). This hypothesis is controversial, because prion populations are capable of proliferation even though prions do not contain DNA or RNA. A mathematical population model is analyzed to explain prion proliferation. The prion population is structured by the length of prion polymers. The model consists of a nonlinear system of coupled ordinary and partial differential equations.
Wednesday, August 31, 3:10pm, SC 1432
Rico Zacher, Martin-Luther-Universität, Halle, Germany.
Title: A priori estimates for weak solutions of fractional evolution equations in divergence form
Abstract: Let $ J=[0,T]$ and $ \Omega\subset\mathbb{R}^n$ be a bounded domain. We consider fractional evolution equations of the form
$\displaystyle \partial_t^\alpha u-D_i(a^{ij}D_j u)=f,\quad t\in J,\,x\in\Omega.$ (1)

Here $ \partial_t^\alpha$ stands for the Riemann-Liouville fractional derivation operator of order $ \alpha\in (0,1)$ defined by
$\displaystyle \partial_t^\alpha u(t)=\partial_t \int_0^t g_{1-\alpha}(t-\tau)u(\tau)\,d\tau, $

where $ \partial_t$ is the usual derivation operator and $\displaystyle g_\beta(t)=\,\frac{t^{\beta-1}}{\Gamma(\beta)}\,,\quad t>0,\quad\beta>0. $
The coefficients $ a^{ij}$ are bounded measurable functions that satisfy an ellipticity condition $ a^{ij}(t,x)\xi_i \xi_j\ge \lambda_0 \vert\xi\vert^2,\,t\in J,\, x\in\Omega,\,\xi\in \mathbb{R}^n$, for some $ \lambda_0>0$. The function $ f$ is assumed to be sufficiently smooth.
We present several a priori estimates for weak solutions of (1) which generalize the well-known De Giorgi-Nash-Moser theory for parabolic equations in divergence form. These results include, among others, a boundedness estimate for subsolutions and a weak Harnack inequality for nonnegative supersolutions. The approach relies on Moser's iteration technique and an abstract lemma of Bombieri and Giusti.