Applied Mathematics Commons^™

Stability Of A Circular System With Multiple Asymmetric Laplacians, Ivo Herman, Dan Martinec, J. J. P. Veerman, Michael Sebek

Mathematics and Statistics Faculty Publications and Presentations

We consider an asymptotic stability of a circular system where the coupling Laplacians are different for each state used for synchronization. It is shown that there must be a symmetric coupling in the output state to guarantee the stability for agents with two integrators in the open loop. Systems with agents having three or more integrators cannot be stabilized by any coupling. In addition, recent works in analysis of a scaling in vehicular platoons relate the asymptotic stability of a circular system to a string stability. Therefore, as confirmed by simulations in the paper, our results have an application also …

A Survey Of Mathematical Models Of Dengue Fever, Iurii Bakach

College of Graduate Studies: Theses & Dissertations

In this paper, we compare and contrast five models of Dengue fever. We evaluate each model using different scenarios and identify the strenghts and wecknesses of each of the model

Solutions Of Inequality Constrained Spline Optimization Problems With The Active Set Method, Joshua A. Holloway

College of Graduate Studies: Theses & Dissertations

We solve the problem of finding a near-interpolant curve, subject to constraints, which minimizes the bending energy of the curve. Using B-splines as our tools, we give a brief overview of spline properties and develop several different cases of inequality constrained optimization problems of this type. In particular, we develop the active set method and use it to solve these problems, emphasizing the fact that this algorithm will converge to a solution in finite iterations. Our solution will solve an open problem regarding near-interpolant spline curves. Furthermore, we supplement this with an iterative technique for better choosing data sites so …

Discrete Nonlinear Planar Systems And Applications To Biological Population Models, Shushan Lazaryan, Nika Lazaryan, Nika Lazaryan

Theses and Dissertations

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential.

We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation …

A Comparison Of Obesity Interventions Using Energy Balance Models, Marcella Torres

Theses and Dissertations

An energy balance model of human metabolism developed by Hall et al. is extended to compare body composition outcomes among standard and proposed obesity interventions. Standard interventions include a drastic diet or a drastic diet with endurance training. Outcomes for these interventions are typically poor in clinical studies. Proposed interventions include a gradual diet and the addition of resistance training to preserve lean mass and metabolic rate. We see that resistance training, regardless of dietary strategy, achieves these goals. Finally, we observe that the optimal obesity intervention for continued maintenance of a healthy body composition following a diet includes a …

Law And The Art Of Modeling: Are Models Facts?, Rebecca Haw Allensworth

Vanderbilt Law School Faculty Publications

In 2013, the Supreme Court made the offhand comment that empirical models and their estimations or predictions are not 'findings of fact" deserving of deference on appeal. The four Justices writing in dissent disagreed, insisting that an assessment of how a model works and its ability to measure what it claims to measure are precisely the kinds of factual findings that the Court, absent clear error cannot disturb. Neither side elaborated on the controversy or defended its position doctrinally or normatively. That the highest Court could split 5-4 on such a crucial issue without even mentioning the stakes or the …

Dynamic Meshing Around Fluid-Fluid Interfaces With Applications To Droplet Tracking In Contraction Geometries, Ahmad Baniabedalruhman

Dissertations, Master's Theses and Master's Reports - Open

The dynamic meshing procedure in an open source three-dimensional solver for calculating immiscible two-phase flow is modified to allow for simulations in two-dimensional planar and axisymmetric geometries. Specifically, the dynamic mesh refinement procedure, which functions only for the partitioning of three-dimensional hexahedral cells, is modified for the partitioning of cells in two-dimensional planar and axisymmetric flow simulations. Moreover, the procedure is modified to allow for computing the deformation and breakup of drops or bubbles that are very small relative to the mesh of the flow domain. This is necessary to avoid mass loss when tracking small drops or bubbles through …

Some Combinatorial Applications Of Sage, An Open Source Program, Jessica Ruth Chowning

Masters Theses

"In this thesis, we consider the usefulness of Sage, an online and open-source program, in analyzing permutation puzzles such as the Rubik's cube and a specific combinatorial structure called the projective plane. Many programs exist to expedite calculations in research and provide previously-unavailable solutions; some require purchase, while others, such as Sage, are available for free online. Sage is asked to handle a small permutation puzzle called Swap, and then we explore how it calculates solutions for a Rubik's cube. We then discuss projective planes, Sage's library of functions for dealing with projective planes, and how they relate to the …

Application Of Loglinear Models To Claims Triangle Runoff Data, Netanya Lee Martin

Masters Theses

"In this thesis, we presented in detail different aspects of Verrall's chain ladder method and their advantages and disadvantages. Insurance companies must ensure there are enough reserves to cover future claims. To that end, it is useful to estimate mean expected losses. The chain ladder technique under a general linear model is the most widely used method for such estimation in property and casualty insurance. Verrall's chain ladder technique develops estimators for loss development ratios, mean expected ultimate claims, Bayesian premiums, and Bühlmann credibility premiums. The chain ladder technique can be used to estimate loss development in cases where data …

A Heterogeneous Multiscale Method For Poroelasticity, Paul M. Delgado

Open Access Theses & Dissertations

In this Thesis, we develop and analyze a heterogeneous multiscale model for coupled fluid flow and solid deformation in porous media based on operator splitting and finite volume method. The splitting method results in two elliptic multiscale PDE's in the form of a reaction diffusion equation and a linear elasticity equation. We extend our previous multiscale method from 1D to higher dimensions and develop new approaches for the inclusion of mixed boundary conditions and source terms. We derive an error estimate for our multiscale method and analyze the stability of our splitting method. We also test the effectiveness of our …

Optimization Schemes For The Inversion Of Bouguer Gravity Anomalies, Azucena Zamora

Open Access Theses & Dissertations

Data sets obtained from measurable physical properties of the Earth structure have helped advance the understanding of its tectonic and structural processes and constitute key elements for resource prospecting. 2-Dimensional (2-D) and 3-D models obtained from the inversion of geophysical data sets are widely used to represent the structural composition of the Earth based on physical properties such as density, seismic wave velocities, magnetic susceptibility, conductivity, and resistivity. The inversion of each one of these data sets provides structural models whose consistency depends on the data collection process, methodology, and overall assumptions made in their individual mathematical processes. Although sampling …

Left-Orderability, Cyclic Branched Covers And Representations Of The Knot Group, Ying Hu

LSU Doctoral Dissertations

A group G is called left-orderable if one can find a total order on G, which is preserved under left multiplication. In this paper we first give a sufficient condition for the fundamental group of the nth cyclic branched cover of the three sphere over a prime knot K to be left-orderable, in terms of representations of the knot group. Then we make use of this criterion to study the left-orderability of fundamental groups of cyclic branched covers over two-bridge knots and satellite knots.

Well-Quasi-Ordering By The Induced-Minor Relation, Chanun Lewchalermvongs

LSU Doctoral Dissertations

Robertson and Seymour proved Wagner's Conjecture, which says that finite graphs are well-quasi-ordered by the minor relation. Their work motivates the question as to whether any class of graphs is well-quasi-ordered by other containment relations. This dissertation is concerned with a special graph containment relation, the induced-minor relation. This dissertation begins with a brief introduction to various graph containment relations and their connections with well-quasi-ordering. In the first chapter, we discuss the results about well-quasi-ordering by graph containment relations and the main problems of this dissertation. The graph theory terminology and preliminary results that will be used are presented in …

A New Method In Distribution Theory With A Non-Smooth Framework, Yunyun Yang

LSU Doctoral Dissertations

In this work, we present a complete treatment of the theory of thick distributions and its asymptotic expansion. We also present several applications of thick distributions in mathematical physics, function spaces, and measure theory. We also discuss regularization using different surfaces. In the last chapter we present some recent applications of distributions in clarifying the moment terms in the heat kernel expansion, and in explaining the relation between the heat kernel expansion and the cylinder kernel expansion.

Analysis Of Nonlinear Dispersive Model Equations, Jacob Grey

LSU Doctoral Dissertations

In this work we begin with a brief survey of the classical fluid dynamics problem of water waves, and then proceed to derive well known evolution equations via a Hamiltonian Variational approach. This method was first introduced in the seminal work of Walter Craig, et al. \cite{CG}. The distinguishing feature of this scheme is that the Dirichlet-Neumann operator of the fluid domain appears explicitly in the Hamiltonian. In the second and third chapters, we utilize the Hamiltonian perturbation theory introduced in \cite{CG} to derive the Benjamin-Bona-Mahony (BBM) and Benjamin-Bona-Mahony-Kadomtsev-Petviashvili (BBM-KP)equations. Finally, we briefly review the existence theory for their corresponding …

An Examination Of Mathematical Models For Infectious Disease, David M. Jenkins

Williams Honors College, Honors Research Projects

Starting with the original 1926 formulation of the SIR (Susceptible-Infected-Removed) model for infectious diseases, mathematical epidemiology continued to grow. Many extensions such as the SEIR, MSIR, and MSEIR models were developed using SIR as a basis to model diseases in a variety of circumstances. By taking the original SIR model, and reducing the system of three first-order equations to a single first-order equation, analysis shows that the model predicts two possible situations. This analysis is followed by discussion of an alternative use of the SIR model which allows for one to track the amount of sustainable genetic variation in a …

Wavelets, Coorbit Theory, And Projective Representations, Amer Hasan Darweesh

LSU Doctoral Dissertations

Banach spaces of functions, or more generally, of distributions are one of the main topics in analysis. In this thesis, we present an abstract framework for construction of invariant Banach function spaces from projective group representations. Coorbit theory gives a unified method to construct invariant Banach function spaces via representations of Lie groups. This theory was introduced by \Fch\, and \Gro\, in \cite{FG,FG1, FG2,FG3} and then extended in \cite{CO2}. We generalize this concept by constructing coorbit spaces using projective representation which is first studied by O. Christensen in \cite{O1}. This allows us to describe wider classes of function spaces as …

Topological Dynamics On Compact Phase Spaces, Lieth Abdalateef Majed

LSU Doctoral Dissertations

Our main focus will be to investigate the various facets of what are commonly called dynamical systems or flows, which are triples $(S,X,\pi)$, where $X$ is a compact Hausdorff space and $\pi:S \times X \longrightarrow X$ is a separately continuous action of a semigroup $S$ on $X$. Historically, as was introduced by R.Ellis 1960, the enveloping semigroup, which is a closure of the set of continuous functions on a compact space $X$, was discovered to be an important tool to study dynamical systems. Soon, a realization of the existence of a universal compactification of a phase semigroup with an extended …

Excluding Two Minors Of The Petersen Graph, Adam Beau Ferguson

LSU Doctoral Dissertations

In this dissertation, we begin with a brief survey of the Petersen graph and its role in graph theory. We will then develop an alternative decomposition to clique sums for 3-connected graphs, called T-sums. This decomposition will be used in Chapter 2 to completely characterize those graphs which have no P_3 minor, where P_3 is a graph with 7 vertices, 12 edges, and is isomorphic to the graph created by contracting three edges of a perfect matching of the Petersen Graph. In Chapter 3, we determine the structure of any large internally 4-connected graph which has no P_2 minor, where …

Partial Cosine-Funk Transforms At Poles Of The Cosine-Λ Transform On Grassmann Manifolds, Christopher Adam Cross

LSU Doctoral Dissertations

The cosine-λ transform, denoted C^λ, is a family of integral transforms we can define on the sphere and on the Grassmannian manifolds of p-dimensional subspaces in Kⁿ where K is R, C or the skew field H of quaternions. We treat the Grassmannians as the symmetric spaces SO(n)/S(O(p) × O(q)), SU(n)/S(U(p) × U(q)) and Sp(n)/(Sp(p) × Sp(q)) and we work by analogy with the case of the cosine-λ transform on the sphere, which is also a symmetric space.

The family C^λ extends meromorphically in λ to the complex plane with poles at (among other values) λ …

Domination Numbers Of Semi-Strong Products Of Graphs, Stephen R. Cheney

Theses and Dissertations

This thesis examines the domination number of the semi-strong product of two graphs G and H where both G and H are simple and connected graphs. The product has an edge set that is the union of the edge set of the direct product of G and H together with the cardinality of V(H), copies of G. Unlike the other more common products (Cartesian, direct and strong), the semi-strong product is neither commutative nor associative.

The semi-strong product is not supermultiplicative, so it does not satisfy a Vizing like conjecture. It is also not submultiplicative so it shares these two …

Applications Of Stability Analysis To Nonlinear Discrete Dynamical Systems Modeling Interactions, Jonathan L. Hughes

Theses and Dissertations

Many of the phenomena studied in the natural and social sciences are governed by processes which are discrete and nonlinear in nature, while the most highly developed and commonly used mathematical models are linear and continuous. There are significant differences between the discrete and the continuous, the nonlinear and the linear cases, and the development of mathematical models which exhibit the discrete, nonlinear properties occurring in nature and society is critical to future scientific progress. This thesis presents the basic theory of discrete dynamical systems and stability analysis and explores several applications of this theory to nonlinear systems which model …

Graphs Of Classroom Networks, Rebecca Holliday

College of Graduate Studies: Theses & Dissertations

In this work, we use the Havel-Hakimi algorithm to visualize data collected from students to investigate classroom networks. The Havel-Hakimi algorithm uses a recursive method to create a simple graph from a graphical degree sequence. In this case, the degree sequence is a representation of the students in a classroom, and we use the number of peers with whom a student studied or collaborated to determine the degree of each. We expand upon the Havel-Hakimi algorithm by coding a program in MATLAB that generates random graphs with the same degree sequence. Then, we run another algorithm to find the isomorphism …

The Pascal Matrix Function And Its Applications To Bernoulli Numbers And Bernoulli Polynomials And Euler Numbers And Euler Polynomials, Tian-Xiao He, Jeff Liao, Peter Shiue

Scholarship

A Pascal matrix function is introduced by Call and Velleman in [3]. In this paper, we will use the function to give a united approach in the study of Bernoulli numbers and Bernoulli polynomials. Many well-known and new properties of the Bernoulli numbers and polynomials can be established by using the Pascal matrix function. The approach is also applied to the study of Euler numbers and Euler polynomials.

On The Dynamics Of Internal Waves Interacting With The Equatorial Undercurrent, Alan Compelli, Rossen Ivanov

Articles

The interaction of the nonlinear internal waves with a nonuniform current with a specific form, characteristic for the equatorial undercurrent, is studied. The current has no vorticity in the layer, where the internal wave motion takes place. We show that the nonzero vorticity that might be occuring in other layers of the current does not affect the wave motion. The equations of motion are formulated as a Hamiltonian system.

Knots, Skein Theory And Q-Series, Mustafa Hajij

LSU Doctoral Dissertations

The tail of a sequence {P_n(q)} of formal power series in Z[q^{-1}][[q]], if it exists, is the formal power series whose first $n$ coefficients agree up to a common sign with the first n coefficients of P_n. The colored Jones polynomial is link invariant that associates to every link in S^3 a sequence of Laurent polynomials. In the first part of this work we study the tail of the unreduced colored Jones polynomial of alternating links using the colored Kauffman skein relation. This gives a natural extension of a result by Kauffman, Murasugi, and Thistlethwaite regarding the highest and lowest …

Improved Full-Newton-Step Infeasible Interior-Point Method For Linear Complementarity Problems, Mustafa Ozen

College of Graduate Studies: Theses & Dissertations

In this thesis, we present an improved version of Infeasible Interior-Point Method (IIPM) for monotone Linear Complementarity Problem (LCP). One of the most important advantages of this version in compare to old version is that it only requires feasibility steps. In the earlier version, each iteration consisted of one feasibility step and some centering steps (at most three in practice). The improved version guarantees that after one feasibility step, the new iterated point is feasible and close enough to central path. Thus, the centering steps are eliminated. This improvement is based on the Lemma(Roos, 2015). Thanks to this lemma, proximity …

Enumerating Graphs Using Integrals From Quantum Field Theory, William A. Coggins

College of Graduate Studies: Theses & Dissertations

Enumerating graphs is a relatively new subfield of mathematics. In this thesis, we will discuss a enumerative method that derives from Quantum Field Theory. We begin with the basic ideas of Calculus and extend them into a enumerative method that will allow us to classify graphs embedded on surfaces.

Evaluating The Long-Term Effects Of Logging Residue Removals In Great Lakes Aspen Forests, Michael I. Premer

Dissertations, Master's Theses and Master's Reports

Commercial aspen (Populus spp.) forests of the Great Lakes region are primarily managed for timber products such as pulp fiber and panel board, but logging residues (topwood and non-merchantable bolewood) are potentially important for utilization in the bioenergy market. In some regions, pulp and paper mills already utilize residues as fuel in combustion for heat and electricity, and progressive energy policies will likely cause an increase in biomass feedstock demand. The effects of removing residues, which have a comparatively high concentration of macronutrients, is poorly understood when evaluating long-term site productivity, future timber yields, plant diversity, stand dynamics, and …

Toric Varieties, Monoid Schemes And Cdh Descent, Guillermo Cortiñas, C. Haesemeyer, Mark E. Walker, Charles Weibel

Department of Mathematics: Faculty Publications

We give conditions for the Mayer–Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.