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Monday, November 9, 2009 - 13:00 ,
Location: Skiles 255 ,
Nicola Guglielmi ,
Università di L'Aquila ,
guglielm@univaq.it ,
Organizer: Sung Ha Kang

This is a joint work with Michael Overton (Courant Institute, NYU). The epsilon-pseudospectral abscissa and radius of an n x n matrix are respectively the maximum real part and the maximal modulus of points in its epsilon-pseudospectrum. Existing techniques compute these quantities accurately but the cost is multiple SVDs of order n, which makesthe method suitable to middle size problems. We present a novel approach based on computing only the spectral abscissa or radius or a sequence of matrices, generating a monotonic sequence of lower bounds which, in many but not all cases, converges to the pseudospectral abscissa or radius.

Series: CDSNS Colloquium

Many dynamical systems may be subject to stochastic excitations, so to find an efficient method to analyze the stochastic system is very important. As for the complexity of the stochastic systems, there are not any omnipotent methods. What I would like to present here is a brief introduction to quasi-non-integrable Hamiltonian systems and stochastic averaging method for analyzing certain stochastic dynamical systems. At the end, I will give some examples of the method.

Series: Combinatorics Seminar

This is joint work with Dr. Yi Zhao.

Graph tiling problems can be summarized as follows: given a graph H, what conditions do we need to find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. The most familiar example of a graph tiling problem is finding a matching in a graph. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove a degree condition that guarantees an arbitrary bipartite graph G will be tiled by an arbitrary bipartite graph H. We also prove this degree condition is best possible up to a constant. This answers a question of Zhao and proves an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

Friday, November 6, 2009 - 15:00 ,
Location: Skiles 269 ,
Meredith Casey ,
Georgia Tech ,
Organizer:

The goal of this talk is to describe simple constructions by which we can construct any compact, orientable 3-manifold. It is well-known that every orientable 3-manifold has a Heegaard splitting. We will first define Heegaard splittings, see some examples, and go through a very geometric proof of this therem. We will then focus on the Dehn-Lickorish Theorem, which states that any orientation-preserving homeomorphism of an oriented 2-manifold without boundary can by presented as the composition of Dehn twists and homeomorphisms isotopic to the identity. We will prove this theorm, and then see some applications and examples. With both of these resutls together, we will have shown that using only handlebodies and Dehn twists one can construct any compact, oriented 3-manifold.

Series: Other Talks

We consider the Stochastic Differential Equation
$dX_\epsilon=b(X_\epsilon)dt + \epsilon dW$ . Given a domain D, we
study how the exit time and the distribution of the process at the time
it exits D behave as \epsilon goes to 0. In particular, we cover the
case in which the unperturbed system $\frac{d}{dt}x=b(x)$ has a unique
fixed point of the hyperbolic type. We will illustrate how the behavior
of the system is in the linear case. We will remark how our results
give improvements to the study of systems admitting heteroclinic or
homoclinic connections. We will outline the general proof in two
dimensions that requires normal form theory from differential
equations. For higher dimensions, we introduce a new kind of non-smooth
stochastic calculus.

Series: SIAM Student Seminar

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.

Series: Stochastics Seminar

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence
$\sum_{i=1}^n S_i$, which we call the {\it integrated random walk}. We are interested in
the asymptotics of $$p_N:=\P \Bigl \{ \min \limits_{1 \le k \le N} \sum_{i=1}^k S_i \ge
0 \Bigr \}$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$
is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of
random walks that include double-sided exponential and double-sided geometric walks (not
necessarily symmetric). We also prove that $p_N \le c N^{-1/4}$ for lattice walks and
upper exponential walks, i.e., walks such that $\mbox{Law} (S_1 | S_1>0)$ is an
exponential distribution.

Series: School of Mathematics Colloquium

Real life networks are usually large and have a very complicated
structure. It is tempting therefore to simplify or reduce the associated
graph of interactions in a network while maintaining its basic structure
as well
as some characteristic(s) of the original graph. A key question is which
characteristic(s) to conserve while reducing a graph. Studies of
dynamical networks reveal that an important characteristic of a
network's structure is a spectrum of its adjacency matrix.
In this talk we present an approach which allows for the reduction of
a general
weighted graph in such a way that the spectrum of the graph's (weighted)
adjacency matrix is maintained up to some finite set that is known in
advance. (Here, the possible weights belong to the set of complex
rational functions, i.e. to a very general class of weights).
A graph can be isospectrally reduced to a graph on any subset of its
nodes, which could be an important property for various applications. It
is also possible to introduce a new equivalence relation in the set of
all networks. Namely, two networks are spectrally equivalent if each of
them can be isospectrally reduced onto one and the same (smaller) graph.
This result should also be useful for analysis of real networks.
As the first application of the isospectral graph reduction we
considered a problem of estimation of spectra of matrices. It happens
that our procedure allows for improvements of the estimates obtained by
all three classical methods given by Gershgorin, Brauer and Brualdi.
(Joint work with B.Webb)
A talk will be readily accessible to undergraduates familiar with
matrices and complex functions.

Series: Analysis Seminar

We will introduce a Bargmann transform from the space of square integrable functions on the n-sphere onto a suitable Hilbert space of holomorphic functions on a null quadric. On base of our Bargmann transform, we will introduce a set of coherent states and study their semiclassical properties. For the particular cases n=2,3,5, we will show the relation with two known regularizations of the Kepler problem: the Kustaanheimo-Stiefel and Moser regularizations.

Series: Other Talks

We will continue the study of derived functors between abelian categories. I will show why injective objects are needed for the construction. I will then show that, for any ringed space, the abelian category of all sheaves of Modules has enough injectives. The relation with Cech cohomology will also be studied.