Physical Sciences and Mathematics Commons^™

Knots And Links In Overtwisted Contact Manifolds, Rima Chatterjee

LSU Doctoral Dissertations

Suppose $(\M,\xi)$ be an overtwisted contact 3-manifold. We prove that any Legendrian and transverse link in $(\M,\xi)$ having overtwisted complement can be coarsely classified by their classical invariants. Next, we defined an invariant called the support genus for transverse links and extended the definition of support genus of Legendrian knots to Legendrian links and prove that any coarse equivalence class of Legendrian and transverse loose links has support genus zero. Further, we show that the converse is not true by explicitly constructing an example. We also find a relationship between the support genus of the transverse link and its Legendrian …

Stochastic Navier-Stokes Equations With Markov Switching, Po-Han Hsu

LSU Doctoral Dissertations

This dissertation is devoted to the study of three-dimensional (regularized) stochastic Navier-Stokes equations with Markov switching. A Markov chain is introduced into the noise term to capture the transitions from laminar to turbulent flow, and vice versa. The existence of the weak solution (in the sense of stochastic analysis) is shown by studying the martingale problem posed by it. This together with the pathwise uniqueness yields existence of the unique strong solution (in the sense of stochastic analysis). The existence and uniqueness of a stationary measure is established when the noise terms are additive and autonomous. Certain exit time estimates …

Nonlinear Potential Analysis On Sobolev Multiplier Spaces, Keng Hao Ooi

LSU Doctoral Dissertations

We characterize preduals and Kothe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. As a maneuver, we make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results. After characterizing the preduals, we establish a capacitary strong type inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated …

Towards A General Framework For Practical Quantum Network Protocols, Sumeet Khatri

LSU Doctoral Dissertations

The quantum internet is one of the frontiers of quantum information science. It will revolutionize the way we communicate and do other tasks, and it will allow for tasks that are not possible using the current, classical internet. The backbone of a quantum internet is entanglement distributed globally in order to allow for such novel applications to be performed over long distances. Experimental progress is currently being made to realize quantum networks on a small scale, but much theoretical work is still needed in order to understand how best to distribute entanglement and to guide the realization of large-scale quantum …

A Conjecture On The Irregularity Function For Local Geometric Langlands Parameters And The Formal Frenkel-Gross Connection, Andrew Alaniz

LSU Doctoral Dissertations

For a simple complex algebraic group $G$, M. Kamgarpour and D. Sage have shown that the adjoint irregularity of an irregular singular flat $G$-bundle on the formal punctured disc is bounded from below by the rank of $G$, moreover the rank is realized by the formal Frenkel-Gross connection. This is a geometric analog of a conjecture of Gross and Reeder on the swan conductor of arithmetic local Langlands parameters. In this work, we explore an interesting combinatorial problem which arises when trying to consider the minimal value of the irregularity function with respect to an arbitrary representation of $G$.

Some Results On Seymour’S Second-Neighborhood Conjecture And On Decompositions Of Graphs, Farid Bouya

LSU Doctoral Dissertations

This dissertation consists of two parts. In the first part, I examine Seymour’s Second-Neighborhood Conjecture, which states that every orientation of every simple graph has at least one vertex v such that the number of vertices of out-distance 2 from v is at least as large as the number of vertices of out-distance 1 from it. I present alternative statements of this conjecture using the language of linear algebra, the last one being completely in terms of the inverse of some matrix. In the second part of this dissertation, comprising of Chapters 2 and 3, I examine two conjectures on …

Evolution Of Computational Thinking Contextualized In A Teacher-Student Collaborative Learning Environment., John Arthur Underwood

LSU Doctoral Dissertations

The discussion of Computational Thinking as a pedagogical concept is now essential as it has found itself integrated into the core science disciplines with its inclusion in all of the Next Generation Science Standards (NGSS, 2018). The need for a practical and functional definition for teacher practitioners is a driving point for many recent research endeavors. Across the United States school systems are currently seeking new methods for expanding their students’ ability to analytically think and to employee real-world problem-solving strategies (Hopson, Simms, and Knezek, 2001). The need for STEM trained individuals crosses both the vocational certified and college degreed …

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

LSU Doctoral Dissertations

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit …

On Selected Subclasses Of Matroids, Tara Elizabeth Fife

LSU Doctoral Dissertations

Matroids were introduced by Whitney to provide an abstract notion of independence.

In this work, after giving a brief survey of matroid theory, we describe structural results for various classes of matroids. A connected matroid $M$ is unbreakable if, for each of its flats $F$, the matroid $M/F$ is connected%or, equivalently, if $M^*$ has no two skew circuits. . Pfeil showed that a simple graphic matroid $M(G)$ is unbreakable exactly when $G$ is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids. A laminar family is …

Supporting The Algebra I Curriculum With An Introduction To Computational Thinking Course, Michelle M. Laskowski

LSU Master's Theses

The Louisiana Workforce Commission predicts a 33.6% increase in computer science and mathematical occupations by 2022 and the Bureau of Labor Statistics foresees a 16% increase in computer scientists from 2018-2028. Despite these opportunities for job and financial security, the number of Louisiana students enrolled in a nationally accredited computing course is less than 1%, compared to national leaders California and Texas which have 3% and 3.8% of students respectively. Furthermore, the international assessments of mathematical literacy, PISA and TIMMS, both report American students continue to fall further behind their international peers in mathematics achievement.

This thesis rejects these statistics …

Alpha Capture Reaction Rates For Nucleosynthesis Within An Ab Initio Framework, Alison Constance Dreyfuss

LSU Doctoral Dissertations

Clustering in nuclear systems has broad impacts on all phases of stellar burning, and plays a significant role in our understanding of nucleosynthesis, or how and where nuclei are produced in the universe. The role of alpha particles in particular is extremely important for nuclear astrophysics: ⁴He was one of the earliest elements produced in the Big Bang, it is one of the most abundant elements in the universe, and helium burning -- in particular, the triple-alpha process -- is one of the most important ``engines'' in stars. To better understand nucleosynthesis and stellar burning, then, it is important …

Quantum Cluster Algebras At Roots Of Unity, Poisson-Lie Groups, And Discriminants, Kurt Malcolm Trampel Iii

LSU Doctoral Dissertations

This dissertation studies quantum algebras at roots of unity in regards to cluster structure and Poisson structure. Moreover, quantum cluster algebras at roots of unity are rigorously defined. The discriminants of these algebras are described, in terms of frozen cluster variables for quantum cluster algebras and Poisson primes for specializations of quantum algebras. The discriminant is a useful invariant for representation theoretic and algebraic study, whose laborious computation deters direct evaluation. The discriminants of quantum Schubert cells at roots of unity will be computed from the two distinct approaches. These methods can be applied to many other quantum algebras.

Design Of Metamaterials For Optics, Abiti Adili

LSU Doctoral Dissertations

First part of this dissertation studies the problem of designing metamaterial crystals with double negative effective properties for applications in optics by investigating the conditions necessary for generating novel dispersion properties in a metamaterial crystal with subwavelength microstructure. This provides novel optical properties created through local resonances tied to the geometry of the media in subwavelength regime.

In the second part, this dissertation studies the representation formula used to describe band structures in photonic crystals with plasmonic inclusions. By using layer potential techniques, a magnetic dipole operator describing the tangential component of the electrical field generated by magnetic distribution is …

Dehn Functions Of Bestvina-Brady Groups, Yu-Chan Chang

LSU Doctoral Dissertations

In this dissertation, we prove that if the flag complex on a finite simplicial graph is a 2-dimensional triangulated disk, then the Dehn function of the associated Bestvina--Brady group depends on the maximal dimension of the simplices in the interior of the flag complex. We also give some examples where the flag complex on a finite simplicial graph is not 2-dimensional, and we establish a lower bound for the Dehn function of the associated Bestvina--Brady group.

Reflection Positivity: A Quantum Field Theory Connection, Joseph W. Grenier

LSU Doctoral Dissertations

At the heart of constructive quantum field theory lies reflection positivity. Through its use one may extend results for a Euclidean field theory to a relativistic theory. In this dissertation we connect functorial and constructive quantum field theories through reflection positivity. In 2014 Santosh Kandel constructed examples of $d$-dimensional functorial QFTs when $d$ is even. We define functorial reflection positivity and show that this functorial theory is a reflection positive theory. We go on to show that every reflection positive theory produces a reflection positive Hilbert space. Iterated doubles are then introduced and used as a starting point to produce …

Unavoidable Immersions And Intertwines Of Graphs, Matthew Christopher Barnes

LSU Doctoral Dissertations

The topological minor and the minor relations are well-studied binary relations on the class of graphs. A natural weakening of the topological minor relation is an immersion. An immersion of a graph H into a graph G is a map that injects the vertex set of H into the vertex set of G such that edges between vertices of H are represented by pairwise-edge-disjoint paths of G. In this dissertation, we present two results: the first giving a set of unavoidable immersions of large 3-edge-connected graphs and the second on immersion intertwines of infinite graphs. These results, along with …

Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, Maitreyee Chandramohan Kulkarni

LSU Doctoral Dissertations

This dissertation describes a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.

Invariant Of Noncommutative Algebras And Poisson Geometry, Bach Van Nguyen

LSU Doctoral Dissertations

In this dissertation, we describe the structure of discriminant of noncommutative algebras using the theory of Poisson quantization and ring theoretic properties of Poisson algebra. In particular, under appropriate conditions, we express the discriminant of specialization of K[q^{+-1}]-algebras as product of Poisson prime elements in some Poisson central subalgebra. In addition, we provide methods for computing noncommutative discriminant in various settings using results obtained for specialization of K[q^{+-1}]-algebras. Further, to demonstrate, we explicitly compute the discriminant of algebra of quantum matrices and quantum Schubert cell algebras specializing at roots of unity. This dissertation is part of the collaboration with Trampel …

Templates For Representable Matroids, Kevin Manuel Grace

LSU Doctoral Dissertations

The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors.

Chapter 1 gives a brief introduction …

Two Results In Drawing Graphs On Surfaces, Joshua E. Fallon

LSU Doctoral Dissertations

In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian …

Mixed Categories Of Sheaves On Toric Varieties, Sean Michael Taylor

LSU Doctoral Dissertations

In [BGS96], Beilinson, Ginzburg, and Soergel introduced the notion of mixed categories. This idea often underlies many interesting "Koszul dualities." In this paper, we produce a mixed derived category of constructible complexes (in the sense of [BGS96]) for any toric variety associated to a fan. Furthermore, we show that it comes equipped with a t-structure whose heart is a mixed version of the category of perverse sheaves. In chapters 2 and 3, we provide the necessary background. Chapter 2 concerns the categorical preliminaries, while chapter 3 gives the background geometry. This concerns both some basics of toric varieties as well …

On Structures Of Large Rooted Graphs, Shilin Wang

LSU Doctoral Dissertations

A rooted graph is a pair (G,R), where G is a graph and R⊆V(G). There are two research topics in this thesis. One is about unavoidable substructures in sufficiently large rooted graphs. The other is about characterizations of rooted graphs excluding specific large graphs.

The first topic of this thesis is motivated by Ramsey Theorem, which states that K_n and ¯(K_n ) are unavoidable induced subgraphs in every sufficiently large graph. It is also motivated by a classical result of Oporowski, Oxley, and Thomas, which determines unavoidable large 3-connected minors. We first determine unavoidable induced subgraphs, and unavoidable subgraphs in …

Operator Semigroups Induced By First-Order Differential Equations, Amy Adair

Honors Theses

No abstract provided.

The Graphs And Matroids Whose Only Odd Circuits Are Small, Kristen Nicole Wetzler

LSU Doctoral Dissertations

This thesis is motivated by a graph-theoretical result of Maffray, which states that a 2-connected graph with no odd cycles exceeding length 3 is bipartite, is isomorphic to K_4, or is a collection of triangles glued together along a common edge. We first prove that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is M(K_4) or F_7, or M is the cycle matroid of a graph consisting of a collection of triangles glued together along a common edge. This result implies that a 2-connected loopless graph G …

Characterizations Of Some Classes Of Graphs That Are Nearly Series-Parallel, Victoria Fontaine

LSU Doctoral Dissertations

A series-parallel graph can be built from a single-edge graph by a sequence of series and parallel extensions. The class of such graphs coincides with the class of graphs that do not have the complete graph K₄ as a minor. This dissertation considers a class M₁ of graphs that are close to being series-parallel. In particular, every member of the class has the property that one can obtain a series-parallel graph by adding a new edge and contracting it out, or by splitting a vertex into two vertices whose neighbor sets partition the neighbor set of the original …

Electromagnetic Resonant Scattering In Layered Media With Fabrication Errors, Emily Anne Mchenry

LSU Doctoral Dissertations

In certain layered electromagnetic media, one can construct a waveguide that supports a harmonic electromagnetic field at a frequency that is embedded in the continuous spectrum. When the structure is perturbed, this embedded eigenvalue moves into the complex plane and becomes a “complex resonance” frequency. The real and imaginary parts of this complex frequency have physical meaning. They lie behind anomalous scattering behaviors known collectively as “Fano resonance”, and people are interested in tuning them to specific values in optical devices. The mathematics involves spectral theory and analytic perturbation theory and is well understood [16], at least on a theoretical …

The Elvis Problem A Minimal Time Problem With Constant Dynamics, Emily Ribando-Gros

Honors Theses

No abstract provided.

Resonance Varieties Of Pure Braid-Like Groups, Jared Able

Honors Theses

No abstract provided.

Present Value Calculations In Personal Injury, Nicholas J. Klinka

Honors Theses

No abstract provided.

Superstable Semigroups, Stephanie Olsen

Honors Theses

No abstract provided.