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Series: Dissertation Defense

This thesis is mainly concerned with problems in the areas of the
Calculus of Variations and Partial Differential Equations (PDEs). The
properties of the functional to minimize play an important role in
the existence of minimizers of integral problems. We will introduce the
important concepts of quasiconvexity and polyconvexity. Inspired by
finite element methods from Numerical Analysis, we introduce a
perturbed problem which has some surprising uniqueness properties.

Series: Dissertation Defense

We consider several possibilities on how to select a Filippov sliding
vector field on a co-dimension 2 singularity manifold, intersection of two
co-dimension 1 manifolds, under the assumption of general attractivity. Of specific
interest is the selection of a smoothly varying Filippov sliding vector field. As a
result of our analysis and experiments, the best candidates of the many possibilities
explored are based on so-called barycentric coordinates: in particular, we choose
what we call the moments solution. We then examine the behavior of the moments vector
field at the first order exit points, and show that it aligns smoothly with the exit
vector field. Numerical experiments illustrate our results and contrast the present
method with other choices of Filippov sliding vector field. We further generalize
this construction to co-dimension 3 and higher.

Series: Dissertation Defense

We derive at-the-money call-price and implied volatility asymptotic
expansions in time to maturity for a selection of exponential Levy models, restricting our
attention to asset-price models whose log returns structure is a Levy process. We consider
two main problems. First, we consider very general Levy models that are in the domain of
attraction of a stable random variable. Under some relatively minor assumptions, we give
first-order at-the-money call-price and implied volatility asymptotics. In the case where
our Levy process has Brownian component, we discover new orders of convergence by showing
that the rate of convergence can be of the form t^{1/\alpha} \ell( t ) where \ell is a slowly
varying function and \alpha \in (1,2). We also give an example of a Levy model which
exhibits this new type of behavior where \ell is not asymptotically constant. In the case of
a Levy process with Brownian component, we find that the order of convergence of the call
price is \sqrt{t}. Second, we investigate the CGMY process whose call-price asymptotics are
known to third order. Previously, measure transformation and technical estimation methods were
the only tools available for proving the order of convergence. We give a new method that relies
on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using
only the characteristic function of the Levy process. While this method does not provide a
less technical approach, it is novel and is promising for obtaining second-order call-price
asymptotics for at-the-money options for a more general class of Levy processes.

Series: Dissertation Defense

We demonstrate new results in additive combinatorics, including a proof of
the following conjecture by J. Solymosi: for every epsilon > 0, there
exists delta > 0 such that, given n^2 points in a grid formation in R^2, if
L is a set of lines in general position such that each line intersects at
least n^{1-delta} points of the grid, then |L| < n^epsilon. This result
implies a conjecture of Gy. Elekes regarding a uniform statistical version
of Freiman's theorem for linear functions with small image sets.

Series: Dissertation Defense

This dissertation investigates the problem of estimating a kernel over a
large graph based on a sample of noisy observations of linear measurements
of the kernel. We are interested in solving this estimation problem in the
case when the sample size is much smaller than the ambient dimension of the
kernel. As is typical in high-dimensional statistics, we are able to design
a suitable estimator based on a small number of samples only when the
target kernel belongs to a subset of restricted complexity. In our study,
we restrict the complexity by considering scenarios where the target kernel
is both low-rank and smooth over a graph. The motivations for studying such
problems come from various real-world applications like recommender systems
and social network analysis.
We study the problem of estimating smooth kernels on graphs. Using standard
tools of non-parametric estimation, we derive a minimax lower bound on the
least squares error in terms of the rank and the degree of smoothness of
the target kernel. To prove the optimality of our lower-bound, we proceed
to develop upper bounds on the error for a least-square estimator based on
a non-convex penalty. The proof of these upper bounds depends on bounds for
estimators over uniformly bounded function classes in terms of Rademacher
complexities. We also propose a computationally tractable estimator based
on least-squares with convex penalty. We derive an upper bound for the
computationally tractable estimator in terms of a coherence function
introduced in this work. Finally, we present some scenarios wherein this
upper bound achieves a near-optimal rate.

Series: Dissertation Defense

The work in this dissertation is mainly focused on three subjects which are
essentially related to linear systems on metric graphs and its application: (1)
rank-determining sets of metric graphs, which can be employed to actually compute
the rank function of arbitrary divisors on an arbitrary metric graph, (2) a
tropical convexity theory for linear systems on metric graphs, and (3) smoothing of
limit linear series of rank one on refined metrized complex (an intermediate object
between metric graphs and algebraic curves),

Series: Dissertation Defense

In this work, we numerically studied the effect of the vorticity on the
enhancement of heat transfer in a channel flow. Based on the model we
proposed, we find that the flow exhibits different properties depending on
the value of four dimensionless parameters. In particularly, we can
classify the flows into two types, active and passive vibration, based on
the sign of the incoming vortices. The temperature profiles according to
the flow just described also show different characteristics corresponding
to the active and passive vibration cases. In active vibration cases, we
find that the heat transfer performance is directly related to the strength
of the incoming vortices and the speed of the background flow. In passive
vibration cases, the corresponding heat transfer process is complicated and
varies dramatically as the flow changes its properties. Compared to the
fluid parameters, we also find that the thermal parameters have much less
effect on the heat transfer enhancement. Finally, we propose a more
realistic optimization problem which is to minimize the maximum temperature
of the solids with a given input energy. We find that the best heat
transfer performance is obtained in the active vibration case with zero
background flow.

Series: Dissertation Defense

Robertson and Seymour proved that graphs are well-quasi-ordered by the
minor relation and the weak immersion relation. In other words, given
infinitely many graphs, one graph contains another as a minor (or a
weak immersion, respectively). An application of these theorems is
that every property that is closed under deleting vertices, edges, and
contracting (or "splitting off", respectively) edges can be
characterized by finitely many graphs, and hence can be decided in
polynomial time.
In this thesis we are concerned with the topological minor relation.
We say that a graph G contains another graph H as a topological
minor if H can be obtained from a subgraph of G by repeatedly
deleting a vertex of degree two and adding an edge incident with the
neighbors of the deleted vertex. Unlike the relation of minor and weak immersion,
the topological minor relation does not well-quasi-order graphs in general.
However, Robertson conjectured in the late 1980's that for every positive integer
k, the topological minor relation well-quasi-orders graphs that do not contain a
topological minor isomorphic to the path of length k with each edge duplicated.
This thesis consists of two main results. The first one is a structure theorem for
excluding a fixed graph as a topological minor, which is analogous to a cornerstone
result of Robertson and Seymour, who gave such structure for graphs that exclude a
fixed minor. Results for topological minors were previously obtained by Grohe and
Marx and by Dvorak, but we push one of the bounds in their theorems to the
optimal value. This improvement is needed for the next theorem.
The second main result is a proof of Robertson's conjecture. As a
corollary, properties on certain graphs closed under deleting
vertices, edges, and "suppressing" vertices of degree two can be
characterized by finitely many graphs, and hence can be decided in
polynomial time.

Series: Dissertation Defense

We consider a class of dynamical systems with random switching with the following specifics: Given a finite collection of smooth vector fields on a finite-dimensional smooth manifold, we fix an initial vector field and a starting point on the manifold. We follow the solution trajectory to the corresponding initial-value problem for a random, exponentially distributed time until we switch to a new vector field chosen at random from the given collection. Again, we follow the trajectory induced by the new vector field for an exponential time until we make
another switch. This procedure is iterated. The resulting two-component process whose first component records the position on the manifold, and whose second component records the driving vector field at any given time, is a Markov process.
We identify sufficient conditions for its invariant measure to be unique and absolutely continuous. In the one-dimensional case, we show that the invariant densities are smooth away from critical points of the vector fields and derive asymptotics for the invariant densities at critical points.

Series: Dissertation Defense

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose elements give invariants of alternating links which strictly refine the first four stable coefficients. We conjecture that all stable coefficients are elements of this algebra, and give experimental evidence for the fifth and sixth stable coefficient. We illustrate our results in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.