Seminars and Colloquia by Series

Parabolic systems and an underlying Lagrangian

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.

Digital Chaotic Communications

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.

Brown bag seminar on mathematical challenges in astrophysics

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.

Cubic graph with large girth

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 presented bounds are based on a simple probabilistic argument.

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

Crossing-critical graphs with large maximum degree

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.

Liar Games, Optimal Codes, and Deterministic Simulation of Random Walks

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.

Nonlinear 4th order diffusion equations by optimal transport

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)

Laguerre-Sobolev type orthogonal polynomials. Algebraic and analytic properties

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.

Optimal Query Complexity Bounds for Finding Graphs

Series
ACO Seminar
Time
Wednesday, April 29, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Jeong Han Kim – Yonsei University and NIMS, South Korea
We consider the problem of finding an unknown graph by using two types of queries with an additive property. Given a graph, an additive query asks the number of edges in a set of vertices while a cross-additive query asks the number of edges crossing between two disjoint sets of vertices. The queries ask sum of weights for the weighted graphs. These types of queries were partially motivated in DNA shotgun sequencing and linkage discovery problem of artificial intelligence. For a given unknown weighted graph G with n vertices, m edges, and a certain mild condition on weights, we prove that there exists a non-adaptive algorithm to find the edges of G using O\left(\frac{m\log n }{\log m}\right) queries of both types provided that m \geq n^{\epsilon} for any constant \epsilon> 0. For a graph, it is shown that the same bound holds for all range of m. This settles a conjecture of Grebinski for finding an unweighted graph using additive queries. We also consider the problem of finding the Fourier coefficients of a certain class of pseudo-Boolean functions. A similar coin weighing problem is also considered. (This is joint work with S. Choi)

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