acase@tulane.edu / Let $G$ be a finite simple graph and let $I = I(G)$ be its edge ideal. Main goal in this thesis is to relate algebraic invariants of powers of edge ideals and combinatorial data of graphs. In particular, we focus on the Castelnuovo-Mumford regularity of an edge ideal and its powers. The first part of this thesis focuses on regularity of edge ideals. In that regard, we give new bounds on the regularity of $I$ when $G$ contains a Hamiltonian path and when $G$ is a Hamiltonian graph. Moreover, we explicitly compute the regularity of unicyclic graphs and characterize the unicyclic graphs with regularity $\nu(G)+1$ and $\nu(G)+2$ where $\nu(G)$ denotes the induced matching number of $G.$ The second problem is on the regularity of powers of edge ideals. Let $R=k[x_1, \ldots, x_n]$ be a polynomial ring and let $I \subset R$ be a homogeneous ideal. It is a celebrated result of Cutkosky, Herzog,Trung \cite{CHT}, Kodiyalam \cite{Kodi}, Trung and Wang \cite{TW} that regularity of $I^s$ is asymptotically a linear function for $s \gg 0,$ i.e., $as+b$ for integers $a,b$ and $s_0$ when $s \geq s_0.$ It is known that $a$ is equal to 2 when $I=I(G)$ is the edge ideal of a graph. We then turn on our focus on identifying $b$ and $s_0$ via combinatorial data of the graph $G.$ We explicitly compute the regularity of $I^s$ for all $s\geq 1$ when $G$ is a forest, a cycle and a unicyclic graph. We also present a lower bound on the regularity of powers of edge ideals in terms of the induced matching number of a graph. / 1 / Selvi Beyarslan

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In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions. First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations. The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.

archives@tulane.edu / 1 / Abu Thomas

Cylindrical algebraic decomposition is a powerful algorithmic technique in semi-algebraic geometry. Nevertheless, there is a disparity between what algorithms output and what the abstract definition of a cylindrical algebraic decomposition allows. Some work has been done in trying to understand what the ideal class of cylindrical algebraic decom- positions should be — especially from a topological point of view. We prove a special case of a conjecture proposed by Lazard in [22]; the conjecture relates a special class of cylindrical algebraic decompositions to regular cell complexes. Moreover, we study the properties that define this special class of cell decompositions, as well as their implications for the actual topology of the cells that make up the cell decompositions.

The Regularity Lemma of Szemeredi is a fundamental tool in extremal graph theory with a wide range of applications in theoretical computer science. Partly as a recognition of his work on the Regularity Lemma, Endre Szemeredi has won the Abel Prize in 2012 for his outstanding achievement. In this thesis we present both practical and theoretical applications of the Regularity Lemma. The practical applications are concerning the important problem of data clustering, the theoretical applications are concerning the monochromatic vertex partition problem. In spite of its numerous applications to establish theoretical results, the Regularity Lemma has a drawback that it requires the graphs under consideration to be astronomically large, thus limiting its practical utility. As stated by Gowers, it has been ``well beyond the realms of any practical applications', the existing applications have been theoretical, mathematical. In the first part of the thesis, we propose to change this and we propose some modifications to the constructive versions of the Regularity Lemma. While this affects the generality of the result, it also makes it more useful for much smaller graphs. We call this result the practical regularity partitioning algorithm and the resulting clustering technique Regularity Clustering. This is the first integrated attempt in order to make the Regularity Lemma applicable in practice. We present results on applying regularity clustering on a number of benchmark data-sets and compare the results with k-means clustering and spectral clustering. Finally we demonstrate its application in Educational Data Mining to improve the student performance prediction. In the second part of the thesis, we study the monochromatic vertex partition problem. To begin we briefly review some related topics and several proof techniques that are central to our results, including the greedy and absorbing procedures. We also review some of the current best results before presenting ours, where the Regularity Lemma has played a critical role. Before concluding we discuss some future research directions that appear particularly promising based on our work.

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With the given information that dendrimers have precisely controlled their sizes and spherical structures in the molecular level, the aim of this study is to show that dendrimer particles can become ordered into a self-assembled regular structure due to the nature of their regular sizes and shapes. For this project, melamine based generation 3 dendrimer was used for solution cast of thin films from the dendrimer-chloroform solutions with different casting conditions, i.e. various solution concentrations, casting temperatures, and substrates. As a result of these experiments, unique phenomena of highly ordered uniform 2-D contraction separations were observed during the solvent evaporation from the dendrimer films. The cast films from the concentration of 0.8 wt% and higher exhibit regular 2-D separation contraction patterns and make well-developed regularly arrayed structures due to the interaction between the contraction stresses and adhesion strength between films and substrates. From the DSC tests, both powder and cast film samples of a dendrimer show similar melting behaviors with different areas under the melting peaks. The results of these tests show that dendrimers, when they are in a descent environment that provides dendrimers with molecular mobility due to surface ionic bonding strength, can make a structural order and regularity in their macroscopic structures.

With the given information that dendrimers have precisely controlled their sizes and spherical structures in the molecular level, the aim of this study is to show that dendrimer particles can become ordered into a self-assembled regular structure due to the nature of their regular sizes and shapes. For this project, melamine based generation 3 dendrimer was used for solution cast of thin films from the dendrimer-chloroform solutions with different casting conditions, i.e. various solution concentrations, casting temperatures, and substrates. As a result of these experiments, unique phenomena of highly ordered uniform 2-D contraction separations were observed during the solvent evaporation from the dendrimer films. The cast films from the concentration of 0.8 wt% and higher exhibit regular 2-D separation contraction patterns and make well-developed regularly arrayed structures due to the interaction between the contraction stresses and adhesion strength between films and substrates. From the DSC tests, both powder and cast film samples of a dendrimer show similar melting behaviors with different areas under the melting peaks. The results of these tests show that dendrimers, when they are in a descent environment that provides dendrimers with molecular mobility due to surface ionic bonding strength, can make a structural order and regularity in their macroscopic structures.

We look at the algebraic properties of Gevrey, analytic and ultraanalytic function spaces, namely their closure under composition, division and inversion. We show that both Gevrey and ultra-analytic spaces, G s with 1 ≤ s < ∞ and 0 < s < 1 respectively, form algebras. Closure under composition, division and inversion is shown to hold for the Gevrey case. For the ultra-analytic case we show it is not closed under composition. We also show that if a function is in G s , with 0 < s < 1 on a compact set, then it is in G s everywhere.