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A Variation Of The Carleman Embedding Method For Second Order Systems., Charles Nunya Dzacka

Electronic Theses and Dissertations

The Carleman Embedding is a method that allows us to embed a finite dimensional system of nonlinear differential equations into a system of infinite dimensional linear differential equations. This technique works well when dealing with first-order nonlinear differential equations. However, for higher order nonlinear ordinary differential equations, it is difficult to use the Carleman Embedding method. This project will examine the Carleman Embedding and a variation of the method which is very convenient in applying to second order systems of nonlinear equations.

Biological Simulations And Biologically Inspired Adaptive Systems, Edgar Alfredo Duenez-Guzman

Doctoral Dissertations

Many of the most challenging problems in modern science lie at the interface of several fields. To study these problems, there is a pressing need for trans-disciplinary research incorporating computational and mathematical models. This dissertation presents a selection of new computational and mathematical techniques applied to biological simulations and problem solving: (i) The dynamics of alliance formation in primates are studied using a continuous time individual-based model. It is observed that increasing the cognitive abilities of individuals stabilizes alliances in a phase transition-like manner. Moreover, with strong cultural transmission an egalitarian regime is established in a few generations. (ii) A …

Some New Problems In Changepoint Analysis, Jonathan Woody

All Dissertations

Climatological studies have often neglected changepoint effects when modeling
various physical phenomena. Here, changepoints are plausible whenever a station location moves or its instruments are changed. There is frequently meta-data to
perform sound statistical inferences that account for changepoint
information. This dissertation focuses on two such problems in changepoint analysis.
The first problem we investigate involves assessing trends
in daily snow depth series. Here, we introduce a stochastic storage model. The model allows for seasonal features, which permits the
analysis of daily data. Changepoint times are shown to greatly influence estimated trends in one snow depth series and are accounted …

Decoding Of Multipoint Algebraic Geometry Codes Via Lists, Nathan Drake

All Dissertations

Algebraic geometry codes have been studied greatly since their introduction by Goppa . Early study had focused on algebraic geometry codes CL(D;G) where G was taken to be a multiple of a single point. However, it has been shown that if we allow G to be supported by more points, then the associated code may have better parameters. We call such a code a multipoint code and if G is supported by m points, then we call it an m-point code. In this dissertation, we wish to develop a decoding algorithm for multipoint codes. We show how we can embed …

Qualitative Models Of Neural Activity And The Carleman Embedding Technique., Azamed Yehuala Gezahagne

Electronic Theses and Dissertations

The two variable Fitzhugh Nagumo model behaves qualitatively like the four variable Hodgkin-Huxley space clamped system and is more mathematically tractable than the Hodgkin Huxley model, thus allowing the action potential and other properties of the Hodgkin Huxley system to be more readily be visualized. In this thesis, it is shown that the Carleman Embedding Technique can be applied to both the Fitzhugh Nagumo model and to Van der Pol's model of nonlinear oscillation, which are both finite nonlinear systems of differential equations. The Carleman technique can thus be used to obtain approximate solutions of the Fitzhugh Nagumo model and …

Sparse Representations In Power Systems Signals, Jack Cooper

All Theses

This thesis seeks to detect transient disturbances in power system signals in a sparse framework. To this end, an overcomplete wavelet packet dictionary and damped sinusoid dictionary are considered, and for each dictionary Matching Pursuit is compared with Basis Pursuit. Previous work in developing waveform dictionary theory and sparse representation is reviewed, and simulations are run on a test signal in both noisy and noiseless environments. The solutions are viewed as time-frequency plane tilings to compare the accuracy and sparsity of these algorithms in properly resolving optimal representations of the disturbances. The advantages and disadvantages of each combination of dictionary …

Quality Representation In Multiobjective Programming, Stacey Faulkenberg

All Dissertations

In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective programming problems. This manuscript presents two methods for generating discrete representations with equidistant points for multiobjective programs with solution sets determined by convex cones. The Bilevel Controlled Spacing (BCS) method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing. The Constraint Controlled Spacing (CCS) method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. Both methods (under certain assumptions) are proven to produce (weakly) nondominated points. Along …

Change-Point Analysis: Asymptotic Theory And Applications, Michael Robbins

All Dissertations

The problem of undocumented change-points in data sets appears in many areas of science. Mathematical fundamentals of asymptotic methods used in change-point analysis are discussed, and several important maximally selected change-point statistics are introduced. First, the likelihood ratio method is applied to abstract data models within the setting of precipitation series. Basic inference as to the legitimacy and effectiveness of asymptotic methods at detecting undocumented change-points is provided. Next, maximally selected chi-square statistics are discussed in detail and applied to data on tropical cyclone behavior, where a widely available and widely analyzed data set on Atlantic basin cyclones is studied. …

Variations On Graph Products And Vertex Partitions, Jobby Jacob

All Dissertations

In this thesis we investigate two graph products called double vertex graphs and complete double vertex graphs, and two vertex partitions called dominator partitions and rankings.
We introduce a new graph product called the complete double vertex graph and study its properties. The complete double vertex graph is a natural extension of the Cartesian product and a generalization of the double vertex graph.
We establish many properties of complete double vertex graphs, including results involving the chromatic number of a complete double vertex graph and the characterization of planar complete double vertex graphs. We also investigate the important problem of …

Asymptotics Of Families Of Polynomials And Sums Of Hurwitz Class Numbers, Timothy Flowers

All Dissertations

In a note in the American Mathematical Monthly in 1960, Strodt mentions a way to prove both the Euler-Maclaurin summation formula and the Boole summation formula using operators. In a 2009 article in the Monthly, Borwein, Calkin, and Manna expand on this idea. Therein, they define Strodt operators and Strodt polynomials and show that the classical Bernoulli polynomials and Euler polynomials are examples of Strodt polynomials.
It is well known that both Bernoulli polynomials and Euler polynomials on a fixed interval are asymptotically sinusoidal. Borwein, Calkin, and Manna show that a similar result holds for the uniform Strodt polynomials. We …

Discrete Dynamics Over Finite Fields, Jang-Woo Park

All Dissertations

A dynamical system consists of a set V and a map f : V → V . The primary goal is to characterize points in V according to their limiting behaviors under iteration of the map f . Especially understanding dynamics of nonlinear maps is an important but difficult problem, and there are not many methods available. This work concentrates on dynamics of certain nonlinear maps over finite fields. First we study monomial dynamics over finite fields. We show that determining the number of fixed points of a boolean monomial dynamics is #P–complete problem and consider various cases in which …

Multiobjective Optimization For Complex Systems, Melissa Gardenghi

All Dissertations

Complex systems are becoming more and more apparent in a variety of disciplines, making solution methods for these systems valuable tools. The solution of complex systems requires two significant skills. The first challenge of developing mathematical models for these systems is followed by the difficulty of solving these models to produce preferred solutions for the overall systems. Both issues are addressed by this research.
This study of complex systems focuses on two distinct aspects. First, models of complex systems with multiobjective formulations and a variety of structures are proposed. Using multiobjective optimization theory, relationships between the efficient solutions of the …

Factoring Polynomials And Groebner Bases, Genhua (Yinhua) Guan

All Dissertations

Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most
computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied
in the last few decades by many mathematicians and computer scientists. The main approaches include Berlekamp's method
(1967) based on the kernel of Frobenius map, Niederreiter's method (1993) via an ordinary differential equation,
Zassenhaus's modular approach (1969), Lenstra, Lenstra and Lovasz's lattice reduction (1982), and Gao's method via a partial differential equation (2003). These methods and their recent improvements due to van Hoeij (2002) and
Lecerf et al …

Local Adaptive Smoothing In Kernel Regression Estimation, Qi Zheng

All Theses

We consider nonparametric estimation of a smooth regression function of one variable. In practice it is quite popular to use the data to select one global smoothing parameter. Such global selection procedures cannot sufficiently account for local sparseness of the covariate nor can they adapt to local curvature of the regression function. We propose a new method to select local smoothing parameters which takes into account sparseness and adapts to local curvature of the regression function. A Bayesian method allows the smoothing parameter to adapt to the local sparseness of the covariate and provides the basis for a local cross …

Portfolio Selection Problem Under Uncertainty And Risk, Dimitri Nowak

All Theses

The purpose of this research is the investigation of a portfolio problem in an uncertain environment. Given possible investments with random performance depending on uncertain environmental settings the objective is to establish a methodology for construction of a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution is preferable for a least risk decision maker.

Optimization Models For Designing Spatially Compact Ecological Reserve Systems, Lakmali Weerasena

All Theses

Over the past decades, a number of mathematical models and solution techniques have been developed to preserve reserve sites for species and their natural habitats. Two optimization models for designing spatially compact ecological reserve systems are addressed here as zero-one integer programming problems. These formulations have a bicriteria objective function that is a combination of both boundary length and distance. The two formulations cluster the sites into a relatively small number of compact groups while preserving a required number of sites that contain a certain species using a given amount of resources. Two general types of approaches have been developed …

Development Of Scoring Rubrics And Pre-Service Teachers Ability To Validate Mathematical Proofs, Timothy J. Middleton

Mathematics & Statistics ETDs

The basic aim of this exploratory research study was to determine if a specific instructional strategy, that of developing scoring rubrics within a collaborative classroom setting, could be used to improve pre-service teachers facility with proofs. During the study, which occurred in a course for secondary mathematics teachers, the primary focus was on creating and implementing a scoring rubric, rather than on direct instruction about proofs. In general, the study had very mixed results. Statistically, the quantitative data indicated no significant improvement occurred in participants' ability to validate proofs. However, the qualitative results and the considerable improvement by some participants …

An Adaptive Method For Calculating Blow-Up Solutions, Charles F. Touron

Mathematics & Statistics Theses & Dissertations

Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as "blow-up." The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting …

Finite Sample Properties Of Minimum Kolmogorov-Smirnov Estimator And Maximum Likelihood Estimator For Right-Censored Data, Jerzy Wieczorek

Dissertations and Theses

MKSFitter computes minimum Kolmogorov-Smirnov estimators (MKSEs) for several different continuous univariate distributions, using an evolutionary optimization algorithm, and recommends the distribution and parameter estimates that best minimize the Kolmogorov-Smirnov (K-S) test statistic. We modify this tool by extending it to use the Kaplan-Meier estimate of the cumulative distribution function (CDF) for right-censored data. Using simulated data from the most commonly-used survival distributions, we demonstrate the tool's inability to consistently select the correct distribution type with right-censored data, even for large sample sizes and low censoring rates. We also compare this tool's estimates with the right-censored maximum likelihood estimator (MLE). While …

Patch Models And Applications On The Spread Of Avian Influenza, Kimberly Rude

Theses, Dissertations and Culminating Projects

The avian influenza virus (AIV) is an infectious disease that predominantly affects birds. Economic losses due to large-scale deaths of domestic poultry as a result of past outbreaks have been devastating. Additionally, there is major concern about the spread of the virus to humans. The virus has spread to humans in the past, but has not yet been known to spread beyond one human. Since influenza viruses are known to mutate easily, there is serious concern that the virus could mutate into a strain that can be transmitted easily to and among humans.

There has been much speculation that migratory …

Analytical Upstream Collocation Solution Of A Quadratic Forced Steady-State Convection-Diffusion Equation, Eric Paul Smith

Boise State University Theses and Dissertations

In this thesis we present the exact solution to the Hermite collocation discretization of a quadratically forced steady-state convection-diffusion equation in one spatial dimension with constant coeffcients, defined on a uniform mesh, with Dirichlet boundary conditions. To improve the accuracy of the method we use \upstream weighting" of the convective term in an optimal way. We also provide a method to determine where the forcing function should be optimally sampled. Computational examples are given, which support and illustrate the theory of the optimal sampling of the convective and forcing term.

On Elliptic Curves, Modular Forms, And The Distribution Of Primes, Ethan Smith

All Dissertations

In this thesis, we present four problems related to elliptic curves, modular forms, the distribution of primes, or some combination of the three. The first chapter surveys the relevant background material necessary for understanding the remainder of the thesis. The four following chapters present our problems of interest and their solutions. In the final chapter, we present our conclusions as well as a few possible directions for future research.
Hurwitz class numbers are known to have connections to many areas of number theory. In particular, they are intimately connected to the theory of binary quadratic forms, the structure of imaginary …

New Directions In Multivariate Public Key Cryptography, Raymond Heindl

All Dissertations

Most public key cryptosystems used in practice are based on integer factorization or discrete logarithms (in finite fields or elliptic curves). However, these systems suffer from two potential drawbacks. First, they must use large keys to maintain security, resulting in decreased efficiency. Second, if large enough quantum computers can be built, Shor's algorithm will render them completely insecure.
Multivariate public key cryptosystems (MPKC) are one possible alternative. MPKC makes use of the fact that solving multivariate polynomial systems over a finite field is an NP-complete problem, for which it is not known whether there is a polynomial algorithm on quantum …

Migration And Mixing Between Populations In Disease Models, David Burger

Theses, Dissertations and Culminating Projects

The goal of this thesis is to model the spread of disease between populations and find ways to prevent its continued epidemic. This thesis studies disease spread as a function of migration in epidemiological models. The models are constructed using the compartmental approach, and we compare discrete and continuous time approximations. In the discrete model, we will look at ways that induced migration can cause an epidemic case to turn into a dieout case. It will be shown that migration can only effect the size of an outbreak, but cannot create or destroy one. For the continuous cases, we will …

Intersections And Representations Of Graphs, John Light

All Dissertations

Given two graphs G and H sharing the same vertex set, the edge-intersection spectrum of G and H is the set of possible
sizes of the intersection of the edge sets of both graphs. For example,
the spectrum of two copies of the cycle C₅ is {0, 2, 3, 5}, and the spectrum of two copies of the star K_1,r is {1, r}. The intersection spectrum was initially studied for designs by Lindner and Fu and others and was originally extended to graphs by Eric Mendelsohn. Several examples are studied, both when G and H are isomorphic and …

Construction Of A Dimension Two Rank One Drinfeld Module, Catherine Trentacoste

All Theses

Consider F_r[t] where r = pm for some prime p and m in the natural numbers. Let f(t) be an irreducible square-free polynomial with even degree in F_r[t] so that the leading coeffcient is not a square
mod F_r. Let A = L = F_r[t][\sqrt{f(t)}].
We will examine the basic set-up required for a dimension two rank one Drinfeld module over L along with an explanation of our choice of f(t). In addition we will show the construction for the exponential function.

A Dual Algorithm For The Weighted Euclidean Distance Min-Max Location Problem In R^2 And R^3, Andrea Smith

All Theses

A dual approach algorithm is given for the solution of the weighted min-max location problem with Euclidean distance in R^2 and R^3. Each subproblem is solved using a directional search procedure and by taking advantage of its geometric structure. An algebraic replacement rule is employed to update the subproblem.

Binary Quadratic Forms Over F[T] And Principal Ideal Domains, Jeff Beyerl

All Theses

This paper concerns binary quadratic forms over F[T]. It develops theory analogous to the theory of binary quadratic forms over the integers. Most although not all of the results are almost identical, while some of the proofs require different techniques.
In particular, the form class group is determined when the form takes values in a principal ideal domain, and the ideal class group (and class group isomorphism) is determined when the form takes values in F[T].

Ethnomathematics In The Dominican Republic: A Mathematics Education Approach To Knowledge And Emancipation, Sofia Pablo-Hoshino

Honors Capstone Projects - All

This focus of this project was to look at the extent to which ethnomathematics is being used in the mathematics curriculum in theDominican Republic. Broadly described, ethnomathematics emphasizes that the culture, history, and experiences of the students are significant and therefore should be infused into the mathematics curriculum.

I read articles and books as well as traveled to theDominican Republicto conduct qualitative research. I interviewed 32 professionals consisting of mathematics teachers, mathematics professors, and mathematics education professors from theSantiagoandSanto Domingoareas. I then volunteered at the Dominican Republic Education and Mentoring (DREAM) Project where I was able to observe and participate …

Modeling Hiv Drug Resistance, Mingfu Zhu

All Dissertations

Despite the development of antiviral drugs and the optimization of therapies, the emergence of drug resistance remains one of the most challenging issues for successful treatments of HIV-infected patients. The availability of massive HIV drug resistance data provides us not only exciting opportunities for HIV research, but also the curse of high dimensionality.
We provide several statistical learning methods in this thesis to analyze sequence data from different perspectives. We propose a hierarchical random graph approach to identify possible covariation among residue-specific mutations. Viral progression pathways were inferred using an EM-like algorithm in literature, and we present a normalization method …