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Series: Analysis Seminar

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.

Series: ACO Seminar

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)

Series: Dissertation Defense

Series: Analysis Seminar

Let A be a Hilbert space operator. If A = UP is the polar decomposition of A,
and 0 < \lambda < 1, the \lambda-Aluthge transform of A is defined to be
the operator \Delta_\lambda = P^\lambda UP^{1-\lambda}. We will discuss the recent progress on
the convergence of the iteration. Infinite and finite dimensional cases will be discussed.

Series: Combinatorics Seminar

We develop an information-theoretic foundation for compound Poisson
approximation and limit theorems (analogous to the corresponding
developments for the central limit theorem and for simple Poisson
approximation). First, sufficient conditions are given under which the
compound Poisson distribution has maximal entropy within a natural
class of probability measures on the nonnegative integers. In
particular, it is shown that a maximum entropy property is valid
if the measures under consideration are log-concave, but that it
fails in general. Second, approximation bounds in the (strong)
relative entropy sense are given for distributional approximation
of sums of independent nonnegative integer valued random variables
by compound Poisson distributions. The proof techniques involve the
use of a notion of local information quantities that generalize the
classical Fisher information used for normal approximation, as well
as the use of ingredients from Stein's method for compound Poisson
approximation. This work is joint with Andrew Barbour (Zurich),
Oliver Johnson (Bristol) and Ioannis Kontoyiannis (AUEB).

Friday, April 24, 2009 - 15:00 ,
Location: Skiles 269 ,
Thang Le ,
School of Mathematics, Georgia Tech ,
Organizer: John Etnyre

These are two hour lectures.

We will develop general theory of quantum invariants based on sl_2 (the simplest Lie algebra): The Jones polynomials, the colored Jones polynomials, quantum sl_2 groups, operator invariants of tangles, and relations with the Alexander polynomial and the A-polynomials. Optional: Finite type invariants and the Kontsevich integral.

Series: Stochastics Seminar

It is of interest that researchers study competing risks in which subjects may fail from any one of k causes. Comparing any two competing risks with covariate effects is very important in medical studies. In this talk, we develop omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. The omnibus tests are derived under the additive risk model by a weighted difference of estimates of cumulative cause-specific hazard rates. Simultaneous confidence bands for the difference of two conditional cumulative incidence functions are also constructed. A simulation procedure is used to sample from the null distribution of the test process in which the graphical and numerical techniques are used to detect the significant difference in the risks. In addition, we conduct a simulation study, and the simulation result shows that the proposed procedure has a good finite sample performance. A melanoma data set in clinical trial is used for the purpose of illustration.

Series: Algebra Seminar

Let S be a group or semigroup acting on a variety V, let x be a point on V, and let W be a subvariety of V. What can be said about the structure of the intersection of the S-orbit of x with W? Does it have the structure of a union of cosets of subgroups of S? The Mordell-Lang theorem of Laurent, Faltings, and Vojta shows that this is the case for certain groups of translations (the Mordell conjecture is a consequence of this). On the other hand, Pell's equation shows that it is not true for additive translations of the Cartesian plane. We will see that this question relates to issues in complex dynamics, simple questions from linear algebra, and techniques from the study of linear recurrence sequences.

Thursday, April 23, 2009 - 13:00 ,
Location: Skiles 255 ,
Per-Gunnar Martinsson ,
Dept of Applied Mathematics, University of Colorado ,
Organizer: Haomin Zhou

Note special day

Linear boundary value problems occur ubiquitously in many areas of
science and engineering, and the cost of computing approximate
solutions to such equations is often what determines which problems
can, and which cannot, be modelled computationally. Due to advances in
the last few decades (multigrid, FFT, fast multipole methods, etc), we
today have at our disposal numerical methods for most linear boundary
value problems that are "fast" in the sense that their computational
cost grows almost linearly with problem size. Most existing "fast"
schemes are based on iterative techniques in which a sequence of
incrementally more accurate solutions is constructed. In contrast, we
propose the use of recently developed methods that are capable of
directly inverting large systems of linear equations in almost linear
time. Such "fast direct methods" have several advantages over
existing iterative methods:
(1) Dramatic speed-ups in applications involving the repeated solution
of similar problems (e.g. optimal design, molecular dynamics).
(2) The ability to solve inherently ill-conditioned problems (such as
scattering problems) without the use of custom designed preconditioners.
(3) The ability to construct spectral decompositions of differential
and integral operators.
(4) Improved robustness and stability.
In the talk, we will also describe how randomized sampling can be used
to rapidly and accurately construct low rank approximations to matrices.
The cost of constructing a rank k approximation to an m x n matrix A
for which an O(m+n) matrix-vector multiplication scheme is available
is O((m+n)*k). This cost is the same as that of the well-established
Lanczos scheme, but the randomized scheme is significantly more robust.
For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)),
which should be compared to the O(m*n*k) cost of existing deterministic
methods.

Series: Graph Theory Seminar

A well know theorem of Kuratowski states that a graph is planar graph iff it contains no TK_5 or TK_{3,3}. In 1970s Seymour conjectured that every 5-connected nonplanar graph contains a TK_5. In the talk we will discuss several special cases of the conjecture, for example the graphs containing K_4^- (K_4 withour an edge). A related independent paths theorem also will be covered.