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Series: Dissertation Defense

Classical Hardy, Sobolev, and Hardy-Sobolev-Maz'ya inequalities are well known
results that have been studied for awhile. In recent years, these results have been
been generalized to fractional integrals. This Dissertation proves a new Hardy
inequality on general domains, an improved Hardy inequality on bounded convex
domains, and that the sharp constant for any convex domain is the same as that known
for the upper halfspace. We also prove, using a new type of rearrangement on the
upper halfspace, based in part on Carlen and Loss' concept of competing symmetries,
the existence of the fractional Hardy-Sobolev-Maz'ya inequality in the case p = 2,
as well as proving the existence of minimizers, at least in limited cases.

Series: Dissertation Defense

Real world networks typically consist of a large number of
dynamical units with a complicated structure of interactions. Until recently
such networks were most often studied independently as either graphs or as
coupled dynamical systems. To integrate these two approaches we introduce
the concept of an isospectral graph transformation which allows one to
modify the network at the level of a graph while maintaining the eigenvalues
of its adjacency matrix. This theory can then be used to rewire dynamical
networks, considered as dynamical systems, in order to gain improved
estimates for whether the network has a unique global attractor. Moreover,
this theory leads to improved eigenvalue estimates of Gershgorin-type.
Lastly, we will discuss the use of Schwarzian derivatives in the theory of
1-d dynamical systems.

Series: Dissertation Defense

Series: Dissertation Defense

A stochastic differential equation with vanishing martingale term is
studied.
Specifically, given a domain D, the asymptotic scaling properties of both
the exit time from the domain and the exit distribution are considered under
the additional (nonstandard) hypothesis that the initial condition also has
a scaling limit. Methods from dynamical systems are applied to get more
complete estimates than the ones obtained by the probabilistic large
deviation theory.
Two situations are completely analyzed. When there is a unique critical
saddle
point of the deterministic system (the system without random effects), and
when
the unperturbed system escapes the domain D in finite time. Applications to
these results are in order. In particular, the study of 2-dimensional
heteroclinic networks is closed with these results and shows the existence
of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare
events are further studied using these results as building blocks.

Series: Dissertation Defense

We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute the unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero or infinity and derive analogues of classical probability theory results such as central limit theorem and large deviation principle.

Series: Dissertation Defense

A graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In this thesis, we study the structure of critical graphs on higher surfaces. One major result in this area is Carsten Thomassen's proof that there are finitely many 6-critical graphs on a fixed surface. This proof involves a structural theorem about a precolored cycle C of length q. In general terms, he proves that a coloring \phi of C can be extended inside the cycle, or there exists a subgraph H with at most 5^{q^3} vertices such that \phi cannot be extended to a 5-coloring of H. In Chapter 2, we provide an alternative proof that reduces the number of vertices in H to be cubic in q. In Chapter 3, we find the nine 6-critical graphs among all graphs embeddable on the Klein bottle. Finally, in Chapter 4, we prove a result concerning critical graphs related to an analogue of Steinberg's conjecture for higher surfaces. We show that if G is a 4-critical graph embedded on surface \Sigma, with Euler genus g and has no cycles of length four through ten, then |V(G)| \leq 2442g + 37.

Series: Dissertation Defense

Series: Dissertation Defense

A construction is given for which the Hausdorﬀ measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorﬀ measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorﬀ and box dimensions diﬀer—in particular, it does not depend on any self-similarity or regularity conditions on the space or an embedding in an ambient space. The only restriction on the space is that it have positive s-dimensional Hausdorﬀ measure, where s is the Hausdorﬀ dimension of the space, assumed to be ﬁnite.

Series: Dissertation Defense

Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful
generalization of Weierstrass's Theorem. We prove a new surprisingly simple
representation for the Müntz orthogonal polynomials on the interval of orthogonality,
and in particular obtain new formulas for some of the classical orthogonal
polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong
asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior
follows, as well as estimates for the smallest and largest zeros. This is the first
time that such asymptotics have been obtained for general Müntz exponents. We also
look at the asymptotic behavior outside the interval and the asymptotic properties of
the associated Christoffel functions.

Series: Dissertation Defense

We improve the lower bound for the L_\infty norm of the discrepancy function. This result makes a partial step towards resolving the Discrepancy Conjecture. Being a theorem in the theory of irregularities of distributions, it also relates to corresponding results in approximation theory (namely, the Kolmogorov entropy of spaces of functions with bounded mixed derivatives) and in probability theory (namely, Small Ball Inequality - small deviation inequality for the Brownian sheet). We also provide sharp bounds for the exponential Orlicz norm and the BMO norm of the discrepancy function in two dimensions. In the second part of the thesis we prove that any sufficiently smooth one-dimensional Calderon-Zygmund convolution operator can be recovered through averaging of Haar shift operators. This allows to generalize the estimates, which had been previously known for Haar shift operators, to Calderon-Zygmund operators. As a result, the A_2 conjecture is settled for this particular type of Calederon-Zygmund operators.