Friday, November 20, 2009
Simulation Study of the Length of Longest Increasing Subsequence of Finite Alphabets
Fri, 11/20/2009 - 1:00pm, Skiles 255
Huy Huynh, Georgia Tech Organizer: Linwei Xin
Let X_1, X_2,...,X_n be a sequence of i.i.d random variables with values in a finite alphabet {1,...,m}. Let LI_n be the length of the longest increasing subsequence of X_1,...,X_n. We shall express the limiting distribution of LI_n as functionals of m and (m-1)- dimensional Brownian motions as well as the largest eigenvalue of Gaussian Unitary Ensemble (GUE) matrix. Then I shall illustrate simulation study of these results
Friday, November 13, 2009
From Gibbs free energy to the dynamical system with random perturbation
Fri, 11/13/2009 - 1:00pm, Skiles 255
Yao Li, School of Mathematics, Georgia Tech Organizer: Linwei Xin
Gibbs free energy plays an important role in thermodynamics and has strong connection with PDE, Dynamical system. The results about Gibbsfree energy in 2-Wasserstein metric space are known recently.First I will introduce some basic things, so the background knowledge isnot required. I will begin from the classic definition of Gibbs freeenergy functional and then move to the connection between Gibbs freeenergy and the Fokker-Planck equation, random perturbation of gradientsystems. Second, I will go reversely: from a dynamical system to thegeneralized Gibbs free energy functional. I will also talk about animportant property of the Gibbs free energy functional: TheFokker-Planck equation is the gradient flux of Gibbs free energyfunctional in 2-Wasserstein metric.So it is natural to consider a question: In topological dynamical systemand lattice dynamical system, could we give the similar definition ofGibbs free energy, Fokker-Planck equation and so on? If time allowed, Iwill basicly introduce some of my results in these topics.
Friday, November 6, 2009
Online Algorithms for Graphs and Partially Ordered Sets
Fri, 11/06/2009 - 1:00pm, Skiles 255
Mitch Keller, School of Mathematics, Georgia Tech Organizer: Linwei Xin
Suppose that Amtrak runs a train from Miami, Florida, to Bangor, Maine. The train makes stops at many locations along the way to drop off passengers and pick up new ones. The computer system that sells seats on the train wants to use the smallest number of seats possible to transport the passengers along the route. If the computer knew before it made any seat assignments when all the passengers would get on and off, this would be an easy task. However, passengers must be given seat assignments when they buy their tickets, and tickets are sold over a period of many weeks. The computer system must use an online algorithm to make seat assignments in this case, meaning it can use only the information it knows up to that point and cannot change seat assignments for passengers who purchased tickets earlier. In this situation, the computer cannot guarantee it will use the smallest number of seats possible. However, we are able to bound the number of seats the algorithm will use as a linear function of the minimum number of seats that could be used if assignments were made after all passengers had bought their tickets. In this talk, we'll formulate this problem as a question involving coloring interval graphs and discuss online algorithms for other questions on graphs and posets. We'll introduce or review the needed concepts from graph theory and posets as they arise, minimizing the background knowledge required.
Friday, October 30, 2009
Asymptotic behavior of Müntz orthogonal polynomials
Fri, 10/30/2009 - 1:00pm, Skiles 255
Ulfar Stefansson, School of Mathematics, Georgia Tech Organizer: Linwei Xin
After a brief introduction of the theory of orthogonal polynomials, where we touch on some history and applications, we present results on Müntz orthogonal polynomials. Müntz polynomials arise from consideration of the Müntz Theorem, which is a beautiful generalization of the Weierstrass Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials which holds on the interval of orthogonality, and in particular we get new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. We also look at the asymptotic behavior outside the interval, where we apply the method of stationary phase.
Friday, October 16, 2009
Shuffling biological sequences
Fri, 10/16/2009 - 1:00pm, Skiles 255
Tianjun Ye, Georgia Tech Organizer: Linwei Xin
This talk considers the following sequence shufling problem: Given a biological sequence (either DNA or protein) s, generate a random instance among all the permutations of s that exhibit the same frequencies of k-lets (e.g. dinucleotides, doublets of amino acids, triplets, etc.). Since certain biases in the usage of k-lets are fundamental to biological sequences, effective generation of such sequences is essential for the evaluation of the results of many sequence analysis tools. This talk introduces two sequence shuffling algorithms: A simple swapping-based algorithm is shown to generate a near-random instance and appears to work well, although its efficiency is unproven; a generation algorithm based on Euler tours is proven to produce a precisely uniforminstance, and hence solve the sequence shuffling problem, in time not much more than linear in the sequence length.
Friday, October 9, 2009
Approximations of Short Term Options Pricing Under Stochastic Volatility Models with Jumps
Fri, 10/09/2009 - 1:00pm, Skiles 255
Allen Hoffmeyer, School of Mathematics, Georgia Tech Organizer: Linwei Xin
This talk is based on a paper by Medvedev and Scaillet which derives closed form asymptotic expansions for option implied volatilities (and option prices). The model is a two-factor jump-diffusion stochastic volatility one with short time to maturity. The authors derive a power series expansion (in log-moneyness and time to maturity) for the implied volatility of near-the-money options with small time to maturity. In this talk, I will discuss their techniques and results.
Friday, October 2, 2009
Frames and integral operators
Fri, 10/02/2009 - 1:00pm, Skiles 255
Shannon Bishop, Georgia Tech, email Organizer: Linwei Xin
I will describe some interesting properties of frames and Gabor frames in particular. Then we will examine how frames may lead to interesting decompositions of integral operators. In particular, I will share some theorems for pseudodifferential operators and Fourier integral operators arising from Gabor frames.
Friday, September 25, 2009
Dynamics of Functions with an Eventual Negative Schwarzian Derivative
Fri, 09/25/2009 - 1:00pm, Skiles 255
Benjamin Webb, School of Mathematics, Georgia Tech Organizer: Linwei Xin
In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this talk we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience
Friday, September 11, 2009
Viscosity and Principal-Agnet Problem
Fri, 09/11/2009 - 1:00pm, Skiles 255
Ruoting Gong, Georgia Tech, email Organizer: Linwei Xin
We develop a stochastic control system from a continuous-time Principal-Agent model in which both the principal and the agent have imperfect information and different beliefs about the project. We attempt to optimize the agent’s utility function under the agent’s belief. Via the corresponding Hamilton-Jacobi-Bellman equation we prove that the value function is jointly continuous and satisfies the Dynamic Programming Principle. These properties directly lead to the conclusion that the value function is a viscosity solution of the HJB equation. Uniqueness is then also established.
Friday, September 4, 2009
Sharp A_p bounds for a certain class of singular integral operators
Fri, 09/04/2009 - 1:00pm, Skiles 255
Maria Reguera Rodriguez, Georgia Tech, email Organizer: Linwei Xin
In this talk we will review some of the classical weighted theory for singular integral operators, and discuss some recent progress on finding sharp bounds in terms of the A_p constant associated with the weight
Friday, April 10, 2009
Linear algebra method in combinatorics
Fri, 04/10/2009 - 12:30pm, Skiles 269
Tianjun Ye, School of Mathematics, Georgia Tech Organizer: Linwei Xin
Linear algebra method is a very useful method in combinatorics. Lovas Theorem (a very deep theorem about perfect graph) is proved by using this way. The idea is, if we want to come up with an upper bound on the size of a set of objects, associate them with elements in a vector space V of relatively low dimension, and show that these elements are linearly independent. Then we cannot have more objects in our set than the dimension of V. We will show we can use this way to solve some combinatorics problem, such as odd town problem and two-distance sets problem.
Friday, April 3, 2009
Small random perturbation of ODE around hyperbolic points
Fri, 04/03/2009 - 12:30pm, Skiles 269
Sergio Almada, School of Mathematics, Georgia Tech Organizer: Linwei Xin
Suppose b is a vector field in R^n such that b(0) = 0. Let A = Jb(0) the Jacobian matrix of b at 0. Suppose that A has no zero eigenvalues, at least one positive and at least one negative eigenvalue. I will study the behavior of the stochastic differential equation dX_\epsilon = b(X_\epsilon) + \epsilon dW as \epsilon goes to 0. I will illustrate the techniques done to deal with this kind of equation and make remarks on how the solution behaves as compared to the deterministic case.
Friday, March 27, 2009
Longest Increasing Subsequence for Finite Alphabets
Fri, 03/27/2009 - 12:30pm, Skiles 255
Huy Huynh, School of Mathematics, Georgia Tech Organizer: Linwei Xin
This is due to the paper of Dr. Christian Houdre and Trevis Litherland. Let X_1, X_2,..., X_n be a sequence of iid random variables drawn uniformly from a finite ordered alphabets (a_1,...,a_m) where a_1 < a_2 < ...< a_m. Let LI_n be the length of the longest increasing subsequence of X_1,X_2,...,X_n. We'll express the limit distribution of LI_n as functionals of (m-1)-dimensional Brownian motion. This is an elementary case taken from this paper.
Friday, March 6, 2009
An introduction to mathematical learning theory
Fri, 03/06/2009 - 12:30pm, Skiles 269
Kai Ni, School of Mathematics, Georgia Tech Organizer: Linwei Xin
In this talk, I will briefly introduce some basics of mathematical learning theory. Two basic methods named perceptron algorithm and support vector machine will be explained for the separable classification case. Also, the subgaussian random variable and Hoeffding inequality will be mentioned in order to provide the upper bound for the deviation of the empirical risk. If time permits, the Vapnik combinatorics will be involved for shaper bounds of this deviation.
Friday, February 27, 2009
Fredholm operators
Fri, 02/27/2009 - 12:30pm, Skiles 269
Weizhe Zhang, School of Mathematics, Georgia Tech Organizer: Linwei Xin
This talk will follow Peter Lax on the linear algebraic fact of the index of Fredholm operators such as the product formula and stability, all of which are totally elementary.