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Series: Dissertation Defense

The primary objective of this thesis is to make a quantitative study of
complex biological networks. Our fundamental motivation is to obtain the
statistical dependency between modules by injecting external noise. To
accomplish this, a
deep study of stochastic dynamical systems would be essential. The first
part is about the stochastic dynamical system theory. The classical
estimation of invariant measures of Fokker-Planck equations is improved by
the level set method. Further, we develop a discrete Fokker-Planck-type
equation to study the discrete stochastic dynamical systems. In the second
part, we quantify systematic measures including degeneracy, complexity and
robustness. We also provide a series of results on their properties and the
connection between them. Then we apply our theory to the JAK-STAT signaling
pathway network.

Series: Dissertation Defense

Series: Dissertation Defense

The fields of statistical physics, discrete probability, combinatorics, and theoretical computer science have converged around efforts to understand random structures and algorithms. Recent activity in the interface of these fields has enabled tremendous breakthroughs in each domain and has supplied a new set of techniques for researchers approaching related problems. This thesis makes progress on several problems in this interface whose solutions all build on insights from multiple disciplinary perspectives.
First, we consider a dynamic growth process arising in the context of DNA-based self-assembly. The assembly process can be modeled as a simple Markov chain. We prove that the chain is rapidly mixing for large enough bias in regions of Z^d. The proof uses a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile, which arises in the nanotechnology application. Moreover, we use intuition from statistical physics to construct a choice of the biases for which the Markov chain M_{mon} requires exponential time to converge.
Second, we consider a related problem regarding the convergence rate of biased permutations that arises in the context of self-organizing lists. The Markov chain M_{nn} in this case is a nearest-neighbor chain that allows adjacent transpositions, and the rate of these exchanges is governed by various input parameters. It was conjectured that the chain is always rapidly mixing when the inversion probabilities are positively biased, i.e., we put nearest neighbor pair x < y in order with bias 1/2 <= p_{xy} <=
1 and out of order with bias 1-p_{xy}. The Markov chain M_{mon} was known to have connections to a simplified version of this biased card-shuffling. We provide new connections between M_{nn} and M_{mon} by using simple combinatorial bijections, and we prove that M_{nn} is always rapidly mixing for two general classes of positively biased {p_{xy}}. More significantly,
we also prove that the general conjecture is false by exhibiting values for the p_{xy}, with 1/2 <= p_{xy} <= 1 for all x < y, but for which the transposition chain will require exponential time to converge.
Finally, we consider a model of colloids, which are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. This clustering has proved elusive to verify, since all local sampling algorithms are known to be inefficient at high density, and in fact a new nonlocal algorithm was recently shown to require exponential time in some cases. We characterize the high and low density phases for a general family of discrete interfering binary mixtures by showing that they exhibit a "clustering property" at high density and not at low density. The clustering property states that there will be a region that has very high area, very small perimeter, and high density of one type of molecule. Special cases of interfering binary mixtures include the Ising model at fixed magnetization and independent sets.

Series: Dissertation Defense

This dissertation has two principal components: the dimension of
posets with planar cover graphs, and the cartesian product of posets
whose cover graphs have hamiltonian cycles that parse into symmetric
chains. Posets of height two can have arbitrarily large dimension.
In 1981, Kelly provided an infinite sequence of planar posets that
shows that the dimension of planar posets can also be arbitrarily
large. However, the height of the posets in this sequence increases
with the dimension. In 2009, Felsner, Li, and Trotter conjectured
that for each integer h \geq 2, there exists a least positive
integer c_h so that if P is a poset having a planar cover graph
(hence P is a planar poset as well) and the height of P is h,
then the dimension of P is at most c_h. In the first principal
component of this dissertation we prove this conjecture. We also give
the best known lower bound for c_h, noting that this lower bound is
far from the upper bound. In the second principal component, we
consider posets with the Hamiltonian Cycle--Symmetric Chain Partition
(HC-SCP) property. A poset of width w has this property if its cover
graph has a Hamiltonian cycle which parses into w symmetric chains.
This definition is motivated by a proof of Sperner's Theorem that uses
symmetric chains, and was intended as a possible method of attack on
the Middle Two Levels Conjecture. We show that the subset lattices
have the HC-SCP property by showing that the class of posets with the
strong HC-SCP property, a slight strengthening of the HC-SCP property,
is closed under cartesian product with a two-element chain.
Furthermore, we show that the cartesian product of any two posets from
this class has the HC-SCP property.

Series: Dissertation Defense

Advisor: Liang Peng

In 1988, Owen introduced empirical likelihood as a nonparametric
method for constructing confidence intervals and regions.
It is well known that empirical likelihood has several attractive advantages
comparing to its competitors such as bootstrap: determining the
shape of confidence regions automatically; straightforwardly incorporating
side information expressed through constraints; being Bartlett correctable.
In this talk, I will discuss some extensions of the empirical likelihood
method to several interesting and important statistical inference situations
including: the smoothed jackknife empirical likelihood method for the
receiver operating characteristic (ROC) curve, the smoothed empirical
likelihood method for the conditional Value-at-Risk with the volatility
model being an ARCH/GARCH model and a nonparametric regression respectively. Then, I will
propose a method for testing nested stochastic models with discrete and
dependent observations.

Series: Dissertation Defense

Graduate Advisor: Eberhard Voit

Series: Dissertation Defense

The presented work deals with two distinct problems in the field
of Mathematical Physics, and as such will have two parts addressing each
problem.
The first part is dedicated to an 'almost' solution of Villani's conjecture,
a known conjecture related to a Statistical Mechanics model invented by Kac
in 1956, giving a rigorous explanation of some simple cases of the Boltzman
equation. In 2003 Villani conjectured that the time it will take the system
of particles in Kac's model to equalibriate is proportional to the number of
particles in the system. Our main result in this part is an 'almost proof'
of that conjecture, showing that for all practical purposes we can consider
it to be true.
The second part of the presentation is dedicated to a newly developed trace
inequality for the fractional Laplacian, connecting between the fractional
Laplacian of a function and its restriction to the intersection of the
hyperplanes x_n =...= x_n-j+1 = 0 , where 1 <= j < n.
The newly found inequality is
sharp and the functions that attain inequality in it are completely
classified.

Series: Dissertation Defense

In this work we provide several improvements in the study of phase
transitions of
interacting particle systems:
1. We determine a quantitative relation between non-extremality of the
limiting Gibbs
measure of a tree-based spin system, and the temporal mixing of the Glauber
Dynamics
over its finite projections. We define the concept of `sensitivity' of a
reconstruction
scheme to establish such a relation. In particular, we focus in the
independent sets
model, determining a phase transition for the mixing time of the Glauber
dynamics at
the same location of the extremality threshold of the simple invariant Gibbs
version
of the model.
2. We develop the technical analysis of the so-called spatial mixing
conditions for interacting
particle systems to account for the connectivity structure of the underlying
graph. This analysis leads to improvements regarding the location of the
uniqueness/non-uniqueness phase transition for the independent sets model
over amenable
graphs; among them, the elusive hard-square model in lattice statistics,
which has received
attention since Baxter's solution of the analogue hard-hexagon in 1980.
3. We build on the work of Montanari and Gerschenfeld to determine the
existence
of correlations for the coloring model in sparse random graphs. In
particular, we prove
that correlations exist above the `clustering' threshold of such model; thus
providing
further evidence for the conjectural algorithmic `hardness' occurring at
such point.

Series: Dissertation Defense

Advisor Chongchun Zeng

We study the normally elliptic singular perturbation problems
including both finite and infinite dimensional cases, which could
also be nonautonomous. In particular, we establish the existence
and smoothness of O(1) local invariant manifolds and provide various
estimates which are independent of small singular parameters.
We also use our results on local invariant manifolds to study
the persistence of homoclinic solutions under weakly dissipative and
conservative perturbations.

Series: Dissertation Defense

Classical partitioning problems, like the Max-Cut problem, ask for
partitions that optimize one quantity, which are important to such fields as
VLSI design, combinatorial optimization, and computer science. Judicious
partitioning problems on graphs or hypergraphs ask for partitions that
optimize several quantities simultaneously. In this dissertation, we work on
judicious partitions of graphs and hypergraphs, and solve or asymptotically
solve several open problems of Bollobas and Scott on judicious partitions,
using the probabilistic method and extremal techniques.