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Articles 61 - 81 of 81
Full-Text Articles in Applied Mathematics
Discrete Dynamics Over Finite Fields, Jang-Woo Park
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 …
Variations On Graph Products And Vertex Partitions, Jobby Jacob
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
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 …
Quality Representation In Multiobjective Programming, Stacey Faulkenberg
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 …
Factoring Polynomials And Groebner Bases, Genhua (Yinhua) Guan
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 …
On Elliptic Curves, Modular Forms, And The Distribution Of Primes, Ethan Smith
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 …
Intersections And Representations Of Graphs, John Light
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 K1,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 …
New Directions In Multivariate Public Key Cryptography, Raymond Heindl
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 …
Modeling Hiv Drug Resistance, Mingfu Zhu
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 …
Fast Fourier Transform Algorithms With Applications, Todd Mateer
Fast Fourier Transform Algorithms With Applications, Todd Mateer
All Dissertations
This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the 'Fast Fourier Transform' as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the 'Fast Fourier Transform' as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the 'Fast Fourier Transform' that can also be understood by someone who has an …
Numerical Analysis Of A Fractional Step Theta-Method For Fluid Flow Problems, John Chrispell
Numerical Analysis Of A Fractional Step Theta-Method For Fluid Flow Problems, John Chrispell
All Dissertations
The accurate numerical approximation of viscoelastic fluid flow poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. Specifically, the viscoelastic modeling equations have a hyperbolic constitutive equation coupled to a parabolic conservation of momentum equation. An appealing approximation approach is to use a fractional step $\theta$-method. The $\theta$-method is an operator splitting technique that may be used to decouple mathematical equations of different types as well as separate the updates of distinct modeling equation variables when modeling mixed systems of partial …
Homomorphisms Of Graphs, Samuel Lyle
Homomorphisms Of Graphs, Samuel Lyle
All Dissertations
Understanding the structure of graphs is fundamental to advances in many areas of graph theory, as well as in many applications. In many cases, an analysis of the structure of graphs follows one of two approaches; either many structural properties are considered over a restricted class of graphs, or a particular structural property is considered over many classes of graphs. Both approaches will be considered in this dissertation.
Graphs which do not contain a clique of size r, i.e., Kr-free graphs, are of fundamental importance in the area of extremal graph theory. Many results have been obtained …
Portfolio Selection Under Various Risk Measures, Hariharan Kandasamy
Portfolio Selection Under Various Risk Measures, Hariharan Kandasamy
All Dissertations
Portfolio selection has been a major area of study after Markowitz's ground-breaking paper. Risk quantification for portfolio selection is studied in the literature extensively and many risk measures have been proposed.
In this dissertation we study portfolio selection under various risk measures. After exploring important risk measures currently available we propose a new risk measure, Unequal Prioritized Downside Risk (UPDR). We illustrate the formulation of UPDR for portfolio selection as a mixed-integer program. We establish conditions under which UPDR can be formulated as a linear program.
We study single-period portfolio selection using two risk measures simultaneously. We propose four alternate …
Issues In Model Selection, Minimax Estimation, And Censored Data Analysis, Meng Zhao
Issues In Model Selection, Minimax Estimation, And Censored Data Analysis, Meng Zhao
All Dissertations
In this dissertation, we address several research problems in statistical inference. We obtain results in the following four directions: linear model selection, minimax estimation of linear functionals, Bayes type estimators for the survival functions based on right censored data, and estimation of survival functions based on doubly censored data.
Automorphic Decompositions Of Graphs, Robert Beeler
Automorphic Decompositions Of Graphs, Robert Beeler
All Dissertations
Let G and H be graphs. A G-decomposition D of a graph H is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. It is well known that a graceful labelling (or more generally a rho-valuation) of a graph G induces a cyclic G-decomposition of a complete graph. We will extend these notions to that of a general valuation in a cyclic group. Such valuations yield decompositions of circulant graphs. We will show that every graph has a valuation and hence is a …
Estimates Related To The Arithmetic Of Elliptic Curves, Bryan Faulkner
Estimates Related To The Arithmetic Of Elliptic Curves, Bryan Faulkner
All Dissertations
This dissertation presents results related to two problems in the arithmetic of elliptic curves.
Feng and Xiong equate the nontriviality of the Selmer groups associated with congruent number curves to the presence of certain types of partitions of graphs associated with the prime factorization of n. The triviality of the Selmer groups associated to the congruent number curve implies that the curve has rank zero which in turn implies n is noncongruent. We extend the ideas of Feng and Xiong in order to compute the Selmer groups of congruent number curves.
We prove an average version of a generalization of …
Numerical Approximation Of Shear-Thinning And Johnson-Segalman Viscoelastic Fluid Flows, Jason Howell
Numerical Approximation Of Shear-Thinning And Johnson-Segalman Viscoelastic Fluid Flows, Jason Howell
All Dissertations
In this work computational approaches to the numerical simulation of steady-state viscoelastic fluid flow are investigated. In particular, two aspects of computing viscoelastic flows are of interest: 1) the stable computation of high Weissenberg number Johnson-Segalman fluids and 2) low-order approaches to simulating the flow of fluids obeying a power law constitutive model.
The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge. For the nonlinear Johnson-Segalman constitutive model, defect-correction and continuation methods are examined …
Permutation Decoding Of Codes From Graphs And Designs, Padmapani Seneviratne
Permutation Decoding Of Codes From Graphs And Designs, Padmapani Seneviratne
All Dissertations
Permutation decoding is a technique, developed by Jessie McWilliams in 1960's. It involves finding a set of automorphisms of the code, called a PD-set. If such a set exists and if the generator matrix of the code is in standard form then a simple algorithm using this set can be followed to correct the maximum number of errors of which the code is capable. Primarily this method was used originally on cyclic codes and Golay codes.
In this dissertation we study binary codes formed from an adjacency matrix of some classes of graphs and apply the permutation decoding method to …
Planning, Scheduling, And Timetabling In A University Setting, Christine Kraft
Planning, Scheduling, And Timetabling In A University Setting, Christine Kraft
All Dissertations
Methods and procedures for modeling university student populations, predicting course enrollment, allocating course seats, and timetabling final examinations are studied and proposed. The university enrollment model presented uses a multi-dimensional state space based on student demographics and the Markov property, rather than longitudinal data to model student movement. The procedure for creating adaptive course prediction models uses student characteristics to identify groups of undergraduates whose specific course enrollment rates are significantly different than the rest of the university population. Historical enrollment rates and current semester information complete the model for predicting enrollment for the coming semester. The course prediction model …
Equitable Efficiency In Multiple Criteria Optimization, Vijay Singh
Equitable Efficiency In Multiple Criteria Optimization, Vijay Singh
All Dissertations
Equitable efficiency in multiple criteria optimization was introduced mathematically in the middle of nineteen-nineties. The concept tends to strengthen the notion of Pareto efficiency by imposing additional conditions on the preference
structure defining the Pareto preference. It is especially designed to solve multiple criteria problems having commensurate criteria where different criteria values can be compared directly.
In this dissertation we study some theoretical and practical aspects of equitably efficient solutions. The literature on equitable efficiency is not very extensive and provides very limited number of ways of generating such solutions. After introducing
some relevant notations, we develop some scalarization based …
Random Vectors Over Finite Fields, Shannon Lockard
Random Vectors Over Finite Fields, Shannon Lockard
All Dissertations
The study of random objects is a useful one in many applications and areas of mathematics. The Probabilistic Method, introduced by Paul Erdos and his many collaborators, was first used to study the behavior of random graphs and later to study properties of random objects. It has developed as a powerful tool in combinatorics as well as finding applications in linear algebra, number theory, and many other areas. In this dissertation, we will consider random vectors, in particular, dependency among random vectors. We will randomly choose vectors according to a specified probability distribution. We wish to determine how many vectors …