### Initial-Boundary Value Problems in Fluid Dynamics Modeling

- Series
- Dissertation Defense
- Time
- Monday, August 10, 2009 - 15:00 for 2 hours
- Location
- Skiles 255
- Speaker
- Kun Zhao – School of Mathematics, Georgia Tech

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- Series
- Dissertation Defense
- Time
- Monday, August 10, 2009 - 15:00 for 2 hours
- Location
- Skiles 255
- Speaker
- Kun Zhao – School of Mathematics, Georgia Tech

- Series
- Dissertation Defense
- Time
- Thursday, July 2, 2009 - 13:30 for 2.5 hours
- Location
- Skiles 255
- Speaker
- Turkay Yolcu – School of Mathematics, Georgia Tech

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the
Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite entropy, we use a variational scheme to
discretize the equation in time and construct approximate solutions. Moreover, De
Giorgi's interpolation method reveals to be a powerful tool for proving convergence
of our algorithm. Finally, we analyze uniqueness and stability of our solution in L^1.

- Series
- Dissertation Defense
- Time
- Wednesday, July 1, 2009 - 15:30 for 3 hours
- Location
- Skiles 255
- Speaker
- Alan J. Michaels – School of Electrical and Computer Engineering, Georgia Tech

This disseratation provides the conceptual development, modeling and simulation, physical implementation and measured hardware results for a procticable digital coherent chaotic communication system.

- Series
- Other Talks
- Time
- Wednesday, July 1, 2009 - 12:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Pablo Laguna – School of Physics, Georgia Tech

This will be an informal seminar with a discussion on some mathematical problems in relativistic astrophysics, and discuss plans for future joint seminars between the Schools of Mathematics and Physics.

- Series
- Graph Theory Seminar
- Time
- Thursday, June 11, 2009 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Daniel Kral – ITI, Charles University, Prague

We study several parameters of cubic graphs with large girth. In particular, we prove that every n-vertex cubic graph with sufficiently large girth satisfies the following:

- has a dominating set of size at most 0.29987n (which improves the previous bound of 0.32122n of Rautenbach and Reed)
- has fractional chromatic number at most 2.37547 (which improves the previous bound of 2.66881 of Hatami and Zhu)
- has independent set of size at least 0.42097n (which improves the previous bound of 0.41391n of Shearer), and
- has fractional total chromatic number arbitrarily close to 4 (which answers in the affirmative a conjecture of Reed). More strongly, there exists g such that the fractional total chromatic number of every bridgeless graph with girth at least g is equal to 4.

The presentation is based on results obtained jointly with Tomas Kaiser, Andrew King, Petr Skoda and Jan Volec.

- Series
- Graph Theory Seminar
- Time
- Thursday, June 4, 2009 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Zdenek Dvorak – Simon Fraser University

Richter and Salazar conjectured that graphs that are critical for a fixed crossing number k have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of k. We disprove these conjectures for every k >170, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree, and explore the structure of such graphs.

- Series
- Combinatorics Seminar
- Time
- Thursday, May 21, 2009 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Joshua Cooper – Department of Mathematics, University of South Carolina

We consider the Ulam "liar" and "pathological liar" games, natural and well-studied variants of "20 questions" in which the adversarial respondent is permitted to lie some fraction of the time. We give an improved upper bound for the optimal strategy (aka minimum-size covering code), coming within a triply iterated log factor of the so-called "sphere covering" lower bound. The approach is twofold: (1) use a greedy-type strategy until the game is nearly over, then (2) switch to applying the "liar machine" to the remaining Berlekamp position vector. The liar machine is a deterministic (countable) automaton which we show to be very close in behavior to a simple random walk, and this resemblance translates into a nearly optimal strategy for the pathological liar game.

- Series
- Dissertation Defense
- Time
- Monday, May 11, 2009 - 13:00 for 2 hours
- Location
- Skiles 255
- Speaker
- Evan Borenstein – School of Mathematics, Georgia Tech

- Series
- PDE Seminar
- Time
- Tuesday, May 5, 2009 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Giuseppe Savare – Università degli Studi di Pavia, Italy

Some interesting nonlinear fourth-order parabolic equations, including the "thin-film" equation with linear mobility and the quantum drift-diffusion equation, can be seen as gradient flows of first-order integral functionals in the Wasserstein space of probability measures. We will present some general tools of the metric-variational approach to gradient flows which are useful to study this kind of equations and their asymptotic behavior. (Joint works in collaboration with U.Gianazza, R.J. McCann, D. Matthes, G. Toscani)

- Series
- Analysis Seminar
- Time
- Wednesday, April 29, 2009 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Francisco Marcellan – Universidad Carlos III de Madrid

In this contribution we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-Type inner product

\langle p, q\rangle_S = \int^\infty_0 p(x)q(x)x^\alpha e^{-x} dx + IP(0)^t AQ(0), \alpha > -1,

where p and q are polynomials with real coefficients,

A = \pmatrix{M_0 & \lambda\\ \lambda & M_1},
IP(0) = \pmatrix{p(0)\\ p'(0)}, Q(0) = \pmatrix{q(0)\\ q'(0)},

and A is a positive semidefinite matrix.

First, we analyze some algebraic properties of these polynomials. More precisely, the connection relations between the polynomials orthogonal with respect to the above inner product and the standard Laguerre polynomials are deduced. On the other hand, the symmetry of the multiplication operator by x^2 yields a five term recurrence relation that such polynomials satisfy.

Second, we focus the attention on their outer relative asymptotics with respect to the standard Laguerre polynomials as well as on an analog of the Mehler-Heine formula for the rescaled polynomials.

Third, we find the raising and lowering operators associated with these orthogonal polynomials. As a consequence, we deduce the holonomic equation that they satisfy. Finally, some open problems will be considered.