Series: PDE Seminar
Couette flows are shear flows with a linear velocity profile. Known by Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time, for L^2 vorticity. It is interesting to know if the perturbed Euler flow near Couette tends to a nearby shear flow. Such problems of nonlinear inviscid damping also appear for other stable flows and are important to understand the appearance of coherent structures in 2D turbulence. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H^s (s<3/2) norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H^s (s<3/2) norm. We also showed that in (vorticity) H^s (s>3/2) neighborhood of Couette flows, the only steady structures (including travelling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H^s (s>3/2) space might be simpler. Similar results will also be discussed for the problem of nonlinear Landau damping in 1D electrostatic plasmas.
Series: Research Horizons Seminar
Hosted by: Huy Huynh and Yao Li
I will discuss some theorems and conjectures in the relatively new field of arithmetic dynamics, focusing in particular on some methods from number theory which can be used to study the orbits of points in algebraic dynamical systems.
Series: Analysis Working Seminar
Monday, February 1, 2010 - 13:00 , Location: Skiles 255 , Manu O. Platt , Biomedical Engineering (BME), Georgia Tech , Organizer:
Tissue remodeling involves the activation of proteases, enzymes capable of degrading the structural proteins of tissue and organs. The implications of the activation of these enzymes span all organ systems and therefore, many different disease pathologies, including cancer metastasis. This occurs when local proteolysis of the structural extracellular matrix allows for malignant cells to break free from the primary tumor and spread to other tissues. Mathematical models add value to this experimental system by explaining phenomena difficult to test at the wet lab bench and to make sense of complex interactions among the proteases or the intracellular signaling changes leading to their expression. The papain family of cysteine proteases, the cathepsins, is an understudied class of powerful collagenases and elastases implicated in extracellular matrix degradation that are secreted by macrophages and cancer cells and shown to be active in the slightly acidic tumor microenvironment. Due to the tight regulatory mechanisms of cathepsin activity and their instability outside of those defined spaces, detection of the active enzyme is difficult to precisely quantify, and therefore challenging to target therapeutically. Using valid assumptions that consider these complex interactions we are developing and validating a system of ordinary differential equations to calculate the concentrations of mature, active cathepsins in biological spaces. The system of reactions considers four enzymes (cathepsins B, K, L, and S, the most studied cathepsins with reaction rates available), three substrates (collagen IV, collagen I, and elastin) and one inhibitor (cystatin C) and comprise more than 30 differential equations with over 50 specified rate constants. Along with the mathematical model development, we have been developing new ways to quantify proteolytic activity to provide further inputs. This predictive model will be a useful tool in identifying the time scale and culprits of proteolytic breakdown leading to cancer metastasis and angiogenesis in malignant tumors.
Series: CDSNS Colloquium
We improved a computational model of leukemia development from stem cells to terminally differentiated cells by replacing the probabilistic, agent-based model of Roeder et al. (2006) with a system of deterministic, difference equations. The model is based on the relatively recent theory that cancer originates from cancer stem cells that reside in a microenvironment, called the stem cell niche. Depending on a stem cell’s location within the stem cell niche, the stem cell may remain quiescent or begin proliferating. This emerging theory states that leukemia (and potentially other cancers) is caused by the misregulation of the cycle ofproliferation and quiescence within the stem cell niche.Unlike the original agent-based model, which required seven hours per simulation, our model could be numerically evaluated in less than five minutes. The results of our numerical simulations showed that our model closely replicated the average behavior of the original agent-based model. We then extended our difference equation model to a system of age-structured partial differential equations (PDEs), which also reproduced the behavior of the Roeder model. Furthermore, the PDE model was amenable to mathematical stability analysis, which revealed three modes of behavior: stability at 0 (cancer dies out), stability at a nonzero equilibrium (a scenario akin to chronic myelogenous leukemia), and periodic oscillations (a scenario akin to accelerated myelogenous leukemia).The PDE formulation not only makes the model suitable for analysis, but also provides an effective mathematical framework for extending the model to include other aspects, such as the spatial distribution of stem cells within the niche.
Series: Combinatorics Seminar
One of the biggest hurdles in high performance computing today is the analysis of massive quantities of data. As the size of the datasets grows to petascale (and beyond), new techniques are needed to efficiently compute meaningful information from the raw data. Graph-based data (which is ubiquitous in social networks, biological interaction networks, etc) poses additional challenges due to the difficulty of parallelizing many common graph algorithms. A key component in success is the generation of "realistic" random data sets for testing and benchmarking new algorithms. The R-MAT graph generator introduced by Chakrabarti, Faloutsos, and Zhan (2004) offers a simple, fast method for generating very large directed graphs. One commonly held belief regarding graphs produced by R-MAT is that they are "scale free"; in other words, their degree distribution follows a power law as is observed in many real world networks. These properties have made R-MAT a popular choice for generating graphs for use in a variety of research disciplines including graph theoretic benchmarks, social network analysis, computational biology, and network monitoring. However, despite its wide usage and elegant, parsimonius design, our recent work provides the first rigorous mathematical analysis of the degree distributions of the generated graphs. Applying results from occupancy problems in probability theory, we derive exact expressions for the degree distributions and other parameters. We also prove that in the limit (as the number of vertices tends to infinity), graphs generated with R-MAT have degree distributions that can be expressed as a mixture of normal distributions. This talk will focus on the techniques used in solving this applied problem in terms of classical "ball and urn" results, including a minor extension of Chistyakov's theorem.
Friday, January 29, 2010 - 14:00 , Location: Skiles 269 , Mohammad Ghomi , School of Mathematics, Georgia Tech , Organizer:
We prove that convex hypersurfaces M in R^n which are level sets of functions f: R^n --> R are C^1-regular if f has a nonzero partial derivative of some order at each point of M. Furthermore, applying this result, we show that if f is algebraic and M is homeomorphic to R^(n-1), then M is an entire graph, i.e., there exists a line L in R^n such that M intersects every line parallel L at precisely one point. Finally we will give a number of examples to show that these results are sharp.
Series: SIAM Student Seminar
In 2006, my coadvisor Xiaoming Huo and his colleague published an annal of statistics paper which designs an asymptotically powerful testing algorithm to detect the potential curvilinear structure in a messy point cloud image. However, such an algorithm involves a membership threshold and a decision threshold which are not well defined in that paper because the distribution of LSP was unknown. Later on, Xiaoming's student Chen, Jihong found some connections between the distribution of LSP and the so-called Erdos-Renyi law. In some sense, the distribution of LSP is just a generalization of the Erdos-Renyi law. However this JASA paper of Chen, Jihong had some restrictions and only partially found out the distribution of LSP. In this talk, I will show the result of the JASA paper is actually very close to the distribution of LSP. However, these is still much potential work to do in order to strengthen this algorithm.
Series: Job Candidate Talk
Tropical geometry can be thought of as geometry over the tropical semiring, which is the set of real numbers together with the operations max and +. Just as ordinary linear and polynomial algebra give rise to convex geometry and algebraic geometry, tropical linear and polynomial algebra give rise to tropical convex geometry and tropical algebraic geometry. I will introduce the basic objects and problems in tropical geometry and discuss some relations with, and applications to, polyhedral geometry, computational algebra, and algebraic geometry.