Seminars and Colloquia by Series

Asymptotic behavior of Müntz orthogonal polynomials

Series
SIAM Student Seminar
Time
Friday, October 30, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ulfar StefanssonSchool of Mathematics, Georgia Tech
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.

Shuffling biological sequences

Series
SIAM Student Seminar
Time
Friday, October 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tianjun YeGeorgia Tech
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.

Approximations of Short Term Options Pricing Under Stochastic Volatility Models with Jumps

Series
SIAM Student Seminar
Time
Friday, October 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Allen HoffmeyerSchool of Mathematics, Georgia Tech
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.

Frames and integral operators

Series
SIAM Student Seminar
Time
Friday, October 2, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shannon BishopGeorgia Tech
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.

Dynamics of Functions with an Eventual Negative Schwarzian Derivative

Series
SIAM Student Seminar
Time
Friday, September 25, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Benjamin WebbSchool of Mathematics, Georgia Tech
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

Viscosity and Principal-Agnet Problem

Series
SIAM Student Seminar
Time
Friday, September 11, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ruoting GongGeorgia Tech
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.

Linear algebra method in combinatorics

Series
SIAM Student Seminar
Time
Friday, April 10, 2009 - 12:30 for 2 hours
Location
Skiles 269
Speaker
Tianjun YeSchool of Mathematics, Georgia Tech
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.

Small random perturbation of ODE around hyperbolic points

Series
SIAM Student Seminar
Time
Friday, April 3, 2009 - 12:30 for 2 hours
Location
Skiles 269
Speaker
Sergio AlmadaSchool of Mathematics, Georgia Tech
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.

Longest Increasing Subsequence for Finite Alphabets

Series
SIAM Student Seminar
Time
Friday, March 27, 2009 - 12:30 for 2 hours
Location
Skiles 255
Speaker
Huy HuynhSchool of Mathematics, Georgia Tech
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.

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