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Series: Dissertation Defense

Series: Dissertation Defense

The first part, consists on a result in the area of commutators. The classic result by Coifman, Rochber and Weiss, stablishes a relation between a BMO function, and the commutator of such a function with the Hilbert transform. The result obtained for this thesis, is in the two parameters setting (with obvious generalizations to more than two parameters) in the case where the BMO function is matrix valued. The second part of the thesis corresponds to domination of operators by using a special class called sparse operators. These operators are positive and highly localized, and therefore, allows for a very efficient way of proving weighted and unweighted estimates. Three main results in this area will be presented: The first one, is a sparse version of the celebrated $T1$ theorem of David and Journé: under some conditions on the action of a Calderón-Zygmund operator $T$ over the indicator function of a cube, we have sparse control.. The second result, is an application of the sparse techniques to dominate a discrete oscillatory version of the Hilbert transform with a quadratic phase, for which the notion of sparse operator has to be extended to functions on the integers. The last resuilt, proves that the Bochner-Riesz multipliers satisfy a range of sparse bounds, we work with the ’single scale’ version of the Bochner-Riesz Conjecture directly, and use the ‘optimal’ unweighted estimates to derive the sparse bounds.

Series: Dissertation Defense

Series: Dissertation Defense

I will discuss two topics in Dynamical Systems. A uniformly hyperbolic dynamical system preserving Borel probability measure μ is called fair dice like or FDL if there exists a finite Markov partition ξ of its phase space M such that for any integers m and j(i), 1 ≤ j(i) ≤ q one has μ ( C(ξ, j(0)) ∩ T^(-1) C(ξ, j(1)) ∩ ... ∩ T^(-m+1)C(ξ, j(m-1)) ) = q^(-m) where q is the number of elements in the partition ξ and C(ξ, j) is element number j of ξ. I discuss several results about such systems concerning finite time prediction regarding the first hitting probabilities of the members of ξ. Then I will discuss a natural modification to all billiard models which is called the Physical Billiard. For some classes of billiard, this modification completely changes their dynamics. I will discuss a particular example derived from the Ehrenfests' Wind-Tree model. The Physical Wind-Tree model displays interesting new dynamical behavior that is at least as rich as some of the most well studied examples that have come before.

Series: Dissertation Defense

Series: Dissertation Defense

Constrained low rank approximation is a general framework for data analysis, which usually has the advantage of being simple, fast, scalable and domain general. One of the most known constrained low rank approximation method is nonnegative matrix factorization (NMF). This research studies the design and implementation of several variants of NMF for text, graph and hybrid data analytics. It will address challenges including solving new data analytics problems and improving the scalability of existing NMF algorithms. There are two major types of matrix representation of data: feature-data matrix and similarity matrix. Previous work showed successful application of standard NMF for feature-data matrix to areas such as text mining and image analysis, and Symmetric NMF (SymNMF) for similarity matrix to areas such as graph clustering and community detection. In this work, a divide-and-conquer strategy is applied to both methods to improve their time complexity from cubic growth with respect to the reduced low rank to linear growth, resulting in DC-NMF and HierSymNMF2 method. Extensive experiments on large scale real world data shows improved performance of these two methods.Furthermore, in this work NMF and SymNMF are combined into one formulation called JointNMF, to analyze hybrid data that contains both text content and connection structure information. Typical hybrid data where JointNMF can be applied includes paper/patent data where there are citation connections among content and email data where the sender/receipts relation is represented by a hypergraph and the email content is associated with hypergraph edges. An additional capability of the JointNMF is prediction of unknown network information which is illustrated using several real world problems such as citation recommendations of papers and activity/leader detection in organizations.The dissertation also includes brief discussions of relationship among different variants of NMF.

Series: Dissertation Defense

Series: Dissertation Defense

Let P be a graph with a vertex v such that P-v is a forest and let Q be an outerplanar graph. In 1993 Paul Seymour asked if every two-connected graph of sufficiently large path-width contains P or Q as a minor.mDefine g(H) as the minimum number for which there exists a positive integer p(H) such that every g(H)-connected H-minor-free graph has path-width at most p(H). Then g(H) = 0 if and only if H is a forest and there is no graph H with g(H) = 1, because path-width of a graph G is the maximum of the path-widths of its connected components.Let A be the graph that consists of a cycle (a_1,a_2,a_3,a_4,a_5,a_6,a_1) and extra edges a_1a_3, a_3a_5, a_5a_1. Let C_{3,2} be a graph of 2 disjoint triangles. In 2014 Marshall and Wood conjectured that a graph H does not have K_{4}, K_{2,3}, C_{3,2} or A as a minor if and only if g(H) >= 2. In this thesis we answer Paul Seymour's question in the affirmative and prove Marshall and Wood's conjecture, as well as extend the result to three-connected and four-connected graphs of large path-width. We introduce ``cascades", our main tool, and prove that in any tree-decomposition with no duplicate bags of bounded width of a graph of big path-width there is an ``injective" cascade of large height. Then we prove that every 2-connected graph of big path-width and bounded tree-width admits a tree-decomposition of bounded width and a cascade with linkages that are minimal. We analyze those minimal linkages and prove that there are essentially only two types of linkage. Then we convert the two types of linkage into the two families of graphs P and Q. In this process we have to choose the ``right'' tree decomposition to deal
with special cases like a long cycle. Similar techniques are used for three-connected and four-connected graphs with high path-width.

Series: Dissertation Defense

Series: Dissertation Defense

This dissertation concerns isoperimetric and functional inequalities in discrete spaces. The majority of the work concerns discrete notions of curvature. There isalso discussion of volume growth in graphs and of expansion in hypergraphs. [The dissertation committee consists of Profs. J. Romberg (ECE), P. Tetali (chair of the committee), W.T. Trotter, X. Yu and H. Zhou.]