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Series: Research Horizons Seminar

Orthogonal polynomials are an important tool in many areas of pure and
applied mathematics. We outline one application in random matrix
theory. We discuss generalizations of orthogonal polynomials such as
the Muntz orthogonal polynomials investigated by Ulfar Stefansson.
Finally, we present some conjectures about biorthogonal polynomials,
which would be a great Ph.D. project for any interested student.

Wednesday, October 21, 2009 - 11:00 ,
Location: Skiles 269 ,
Klas Udekwu ,
Biology, Emory University ,
ikeche@gmail.com ,
Organizer:

Treatment of bacterial
infections with antibiotics is universally accepted as one of (if not THE) most
significant contributions of medical intervention to reducing mortality and
morbidity during last century. Surprisingly, basic knowledge about how
antibiotics kill or prevent the growth of bacteria is only just beginning to
emerge and the dose and term of antibiotic treatment has long been determined
by clinicians empirically and intuitively.
There is a recent drive to
theoretically and experimentally rationalize antibiotic treatment protocols with
the aim to them and to design protocols which maximize antibiotics’ efficacy
while preventing resistance emergence. Central to these endeavors are the
pharmacodynamics of the antibiotic(s) and bacteria, PD (the relationship
between the concentration of the antibiotic and the rate of growth/death of
bacteria), and the pharmacokinetics of the antibiotic, PK (the distribution and
change in concentration of the antibiotics in a treated host) of each
bacteria. The procedures for
estimating of PD and PK parameters are well established and standardized
worldwide. Although different PK
parameters are commonly employed for the design of antibiotic treatment
protocols most of these considerations, a single PD parameter is usually used,
the minimum inhibitory concentration (MIC). The Clinical and Laboratory
Standards Institute (CLSI)
approved method for estimating MICs defines testing conditions that are optimal
for the antibiotic, like low densities and exponential growth, rarely obtain
outside of the laboratory and virtually never in the bacteria infecting
mammalian hosts. Real
infections with clinical symptoms commonly involve very high densities of
bacteria, most of which are not replicating, and these bacteria are rarely
planktonic, rather residing as colonies or within matrices called biofilms
which sometimes include other species of bacteria. Refractoriness (non-inherited resistance) is the term used to
describe an observed inefficacy of antibiotics on otherwise
antibiotic-susceptible bacterial populations. This talk will focus on our
efforts to describe the pharmacodynamic relationship between Staphylococcus
aureus and antibiotics of six
classes in the light of antibiotic refractoriness.
I will begin by addressing
the effects of cell density on the MIC index, after which I intend to present
unpublished data descriptive of physiology-related effects on antibiotic
efficacy. Additionally, we will explore
the potential contribution of such in vitro results, to observed/predicted clinical
outcomes using standard mathematical models of antibiotic treatment which also
serve to generate testable hypotheses.

Series: PDE Seminar

Under the classical small-amplitude, long wave-length assumptions in which the
Stokes number is of order one, so featuring a balance between nonlinear and dispersive effects,
the KdV-equation
u_t+ u_x + uu_x + u_xxx = 0 (1)
and the regularized long wave equation, or BBM-equation
u_t + u_x + uu_x-u_xxt = 0 (2)
are formal reductions of the full, two-dimensional Euler equations for free surface flow. This
talk is concerned with the two-point boundary value problem for (1) and (2) wherein the wave
motion is specified at both ends of a finite stretch of length L of the media of propagation.
After ascertaining natural boundary specifications that constitute well posed problems, it is
shown that the solution of the two-point boundary value problem, posed on the interval [0;L],
say, converges as L converges to infinity, to the solution of the quarter-plane boundary value problem in
which a semi-infinite stretch [0;1) of the medium is disturbed at its finite end (the so-called
wavemaker problem). In addition to its intrinsic interest, our results provide justification for the use of the
two-point boundary-value problem in numerical studies of the quarter plane problem for
both the KdV-equation and the BBM-equation.

Tuesday, October 20, 2009 - 15:00 ,
Location: Skiles 269 ,
Daniel Bauer ,
Georgia State University ,
Organizer: Liang Peng

In recent literature, different mothods have been proposed on how to define
and model stochastic mortality. In most of these approaches, the so-called spot force
of mortality is modeled as a stochastic process. In contrast to such spot force
models, forward force mortality models infer dynamics on the entire
age/term-structure of mortality.
This paper considers forward models defined based on best-estimate forecasts of
survival probabilities as can be found in so-called best-estimate generation life
tables. We provide a detailed analysis of forward mortality models deriven by
finite-dimensional Brownian motion. In particular, we address the relationship to
other modeling approaches, the consistency problem of parametric forward models, and
the existence of finite dimensional realizations for Gaussian forward models. All
results are illustrated based on a simple example with an affine specification.

Series: Graph Theory Seminar

In 1865, Sylvester posed the following problem: For a region R in the plane,let q(R) denote the probability that four points chosen at random from Rform a convex quadrilateral. What is the infimum q* of q(R) taken over allregions R? The number q* is known as Sylvester's Four Point Problem Constant(Sylvester's Constant for short). At first sight, it is hard to imagine howto find reasonable estimates for q*. Fortunately, Scheinerman and Wilf foundthat Sylvester's Constant is intimately related to another fundamentalconstant in discrete geometry. The rectilinear crossing number of rcr(K_n)the complete graph K_n is the minimum number of crossings of edges in adrawing of K_n in the plane in which every edge is a straight segment. Itis not difficult to show that the limit as n goes to infinity ofrcr(K_n)/{n\choose 4} exists; this is the rectilinear crossing numberconstant RCR. Scheinerman and Wilf proved a surprising connection betweenthese constants: q* = RCR. Finding estimates of rcr(K_n) seems like a moreapproachable task. A major breakthrough was achieved in 2003 by Lovasz,Vesztergombi, Wagner, and Welzl, and simultaneously by Abrego andFernandez-Merchant, who unveiled an intimate connection of rcr(K_n) withanother classical problem of discrete geometry, namely the number of

Series: Analysis Working Seminar

In this working seminar we will give a proof of Seip's characterization of interpolating sequences in the Bergman space of analytic functions. This topic has connection with complex analysis, harmonic analysis, and
time frequency analysis and so hopefully many of the participants would
be able to get something out of the seminar. The initial plan will be
to work through his 1993 Inventiones Paper and supplement this with
material from his book "Interpolation and Sampling in Spaces of
Analytic Functions". Notes will be generated as the seminar progresses.

Series: Geometry Topology Seminar

We will introduce new constructions of infinite families of smooth structures on small 4-manifolds and infinite families of smooth knottings of surfaces.

Monday, October 19, 2009 - 13:00 ,
Location: Skiles 255 ,
Helga S. Huntley ,
University of Delaware ,
Organizer:

Biologists tracking crab larvae, engineers designing pollution mitigation strategies, and
climate scientists studying tracer transport in the oceans are among many who rely on
trajectory predictions from ocean models. State-of-the-art models have been shown to
produce reliable velocity forecasts for 48-72 hours, yet the predictability horizon for
trajectories and related Lagrangian quantities remains significantly shorter. We
investigate the potential for decreasing Lagrangian prediction errors by applying a
constrained normal mode analysis (NMA) to blend drifter observations with model velocity
fields. The properties of an unconstrained NMA and the effects of parameter choices are
discussed. The constrained NMA technique is initially presented in a perfect model
simulation, where the “true” velocity field is known and the resulting error can be
directly assessed. Finally, we will show results from a recent experiment in the East
Asia Sea, where real observations were assimilated into operational ocean model hindcasts.

Series: CDSNS Colloquium

Despite advances in treatment of chronic hepatitis B virus (HBV) infection,
liver transplantation remains the only hope for many patients with end-stage
liver disease due to HBV. A complication with liver transplantation,
however, is that the new liver is eventually reinfected in chronic HBV
patients by infection in other compartments of the body. We have formulated
a model to describe the dynamics of HBV after liver transplant, considering
the liver and the blood of areas of infection. Analyzing the model, we
observe that the system shows either a transcritical or a backward
bifurcation. Explicit conditions on the model parameters are given for the
backward bifurcation to be present, to be reduced, or disappear.
Consequently, we investigate possible factors that are responsible for
HBV/HCV infection and assess control strategies to reduce HBV/HCV
reinfection and improve graft survival after liver transplantation.

Series: Combinatorics Seminar

There has been substantial work on approximation algorithms for clustering
data under distance-based objective functions such as k-median, k-means, and
min-sum objectives. This work is fueled in part by the hope that
approximating these objectives well will indeed yield more accurate
solutions. That is, for problems such as clustering proteins by function, or
clustering images by subject, there is some unknown correct "target"
clustering and the implicit assumption is that clusterings that are
approximately optimal in terms of these distance-based measures are also
approximately correct in terms of error with respect to the target. In this
work we show that if we make this implicit assumption explicit -- that is, if
we assume that any c-approximation to the given clustering objective Phi is
epsilon-close to the target -- then we can produce clusterings that are
O(epsilon)-close to the target, even for values c for which obtaining a
c-approximation is NP-hard. In particular, for the k-median, k-means, and
min-sum objectives, we show that we can achieve this guarantee for any
constant c > 1.
Our results show how by explicitly considering the alignment between the
objective function used and the true underlying clustering goals, one can
bypass computational barriers and perform as if these objectives were
computationally substantially easier.
This talk is based on joint work with Avrim Blum and Anupam Gupta (SODA
2009), Mark Braverman (COLT 2009), and Heiko Roeglin and Shang-Hua Teng (ALT 2009).