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Articles 151 - 180 of 255
Full-Text Articles in Mathematics
3d Image Reconstruction And Level Set Methods, Spencer R. Patty
3d Image Reconstruction And Level Set Methods, Spencer R. Patty
Theses and Dissertations
We give a concise explication of the theory of level set methods for modeling motion of an interface as well as the numerical implementation of these methods. We then introduce the geometry of a camera and the mathematical models for 3D reconstruction with a few examples both simulated and from a real camera. We finally describe the model for 3D surface reconstruction from n-camera views using level set methods.
Stability For Traveling Waves, Joshua W. Lytle
Stability For Traveling Waves, Joshua W. Lytle
Theses and Dissertations
In this work we present some of the general theory of shock waves and their stability properties. We examine the concepts of nonlinear stability and spectral stability, noting that for certain classes of equations the study of nonlinear stability is reduced to the analysis of the spectra of the linearized eigenvalue problem. A useful tool in the study of spectral stability is the Evans function, an analytic function whose zeros correspond to the eigenvalues of the linearized eigenvalue problem. We discuss techniques for numerical Evans function computation that ensure analyticity, allowing standard winding number arguments and rootfinding methods to be …
The Constrained Isoperimetric Problem, Minh Nhat Vo Do
The Constrained Isoperimetric Problem, Minh Nhat Vo Do
Theses and Dissertations
Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio FR(S) = |S|/|∂S|. We consider different measures for subsets of R^2,R^3,Z^2,Z^3 and describe the properties that must be satisfied for sets S that maximize the Folner ratio. We give explicit examples.
An Algebra Isomorphism For The Landau-Ginzburg Mirror Symmetry Conjecture, Jared Drew Johnson
An Algebra Isomorphism For The Landau-Ginzburg Mirror Symmetry Conjecture, Jared Drew Johnson
Theses and Dissertations
Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. …
Maximal Unramified Extensions Of Cyclic Cubic Fields, Ka Lun Wong
Maximal Unramified Extensions Of Cyclic Cubic Fields, Ka Lun Wong
Theses and Dissertations
Maximal unramified extensions of quadratic number fields have been well studied. This thesis focuses on maximal unramified extensions of cyclic cubic fields. We use the unconditional discriminant bounds of Moreno to determine cyclic cubic fields having no non-solvable unramified extensions. We also use a theorem of Roquette, developed from the method of Golod-Shafarevich, and some results by Cohen to construct cyclic cubic fields in which the unramified extension is of infinite degree.
A Multi-Frequency Inverse Source Problem For The Helmholtz Equation, Sebastian Ignacio Acosta
A Multi-Frequency Inverse Source Problem For The Helmholtz Equation, Sebastian Ignacio Acosta
Theses and Dissertations
The inverse source problem for the Helmholtz equation is studied. An unknown source is to be identified from the knowledge of its radiated wave. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. In particular, we prove that data obtained from finitely many frequencies is not sufficient. On the other hand, if the frequency varies within an open interval of the positive real line, then the source is determined uniquely. An algorithm is based on an incomplete Fourier transform of the measured data and we establish an error estimate under certain regularity assumptions on the …
Three-Dimensional Galois Representations And A Conjecture Of Ash, Doud, And Pollack, Vinh Xuan Dang
Three-Dimensional Galois Representations And A Conjecture Of Ash, Doud, And Pollack, Vinh Xuan Dang
Theses and Dissertations
In the 1970s and 1980s, Jean-Pierre Serre formulated a conjecture connecting two-dimensional Galois representations and modular forms. The conjecture came to be known as Serre's modularity conjecture. It was recently proved by Khare and Wintenberger in 2008. Serre's conjecture has various important consequences in number theory. Most notably, it played a key role in the proof of Fermat's last theorem. A natural question is, what is the analogue of Serre's conjecture for higher dimensional Galois representations? In 2002, Ash, Doud and Pollack formulated a precise statement for a higher dimensional analogue of Serre's conjecture. They also provided numerous computational examples …
The Weak Cayley Table And The Relative Weak Cayley Table, Melissa Anne Mitchell
The Weak Cayley Table And The Relative Weak Cayley Table, Melissa Anne Mitchell
Theses and Dissertations
In 1896, Frobenius began the study of character theory while factoring the group determinant. Later in 1963, Brauer pointed out that the relationship between characters and their groups was still not fully understood. He published a series of questions that he felt would be important to resolve. In response to these questions, Johnson, Mattarei, and Sehgal developed the idea of a weak Cayley table map between groups. The set of all weak Cayley table maps from one group to itself also has a group structure, which we will call the weak Cayley table group. We will examine the weak Cayley …
Mean Square Estimate For Primitive Lattice Points In Convex Planar Domains, Ryan D. Coatney
Mean Square Estimate For Primitive Lattice Points In Convex Planar Domains, Ryan D. Coatney
Theses and Dissertations
The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1;m2) in ZxZ : m1^2 + m2^2 <= x }, the number of integer lattice points inside a circle of radius sqrt(x). Gauss showed that P(x) = N(x)- pi * x satisfi es P(x) = O(sqrt(x)). Later Hardy and Landau independently proved that P(x) = Omega_(x1=4(log x)1=4). It is conjectured that inf{e in R : P(x) = O(x^e )}= 1/4. I. K atai showed that the integral from 0 to X of |P(x)|^2 dx = X^(3/2) + O(X(logX)^2). Similar results to those of the circle have been obtained for regions D in R^2 which contain the origin and whose boundary dD satis fies suff cient smoothness conditions. Denote by P_D(x) the similar error term to P(x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on dD, P_D(x) = Omega_(x1=4(log x)1=4) and that the integral from 0 to X of |P_D(x)|^2 dx = O(X^(3/2)). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, {(m1;m2) in Z^2 : gcd(m1;m2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.
Understanding Streaming In Dictyostelium Discoideum: Theory Versus Experiments, J. C. Dallon, Brittany Dalton, Chelsea Malani
Understanding Streaming In Dictyostelium Discoideum: Theory Versus Experiments, J. C. Dallon, Brittany Dalton, Chelsea Malani
Faculty Publications
Recent experimental work involving Dictyostelium discoideum seems to contradict several theoretical models. Experiments suggest that localization of the release of the chemoattractant cyclic adenosine monophosphate to the uropod of the cell is important for stream formation during aggregation. Yet several mathematical models are able to reproduce streaming as the cells aggregate without taking into account localization of the chemoattractant. A careful analysis of the experiments and the theory suggests the two major features of the system which are important to stream formation are random cell motion and chemotaxis to regions of higher cell density. Random cell motion acts to reduce …
A Toolkit For The Construction And Understanding Of 3-Manifolds, Lee R. Lambert
A Toolkit For The Construction And Understanding Of 3-Manifolds, Lee R. Lambert
Theses and Dissertations
Since our world is experienced locally in three-dimensional space, students of mathematics struggle to visualize and understand objects which do not fit into three-dimensional space. 3-manifolds are locally three-dimensional, but do not fit into 3-dimensional space and can be very complicated. Twist and bitwist are simple constructions that provide an easy path to both creating and understanding closed, orientable 3-manifolds. By starting with simple face pairings on a 3-ball, a myriad of 3-manifolds can be easily constructed. In fact, all closed, connected, orientable 3-manifolds can be developed in this manner. We call this work a tool kit to emphasize the …
The Minimum Rank, Inverse Inertia, And Inverse Eigenvalue Problems For Graphs, Mark Condie Kempton
The Minimum Rank, Inverse Inertia, And Inverse Eigenvalue Problems For Graphs, Mark Condie Kempton
Theses and Dissertations
For a graph G we define S(G) to be the set of all real symmetric n by n matrices whose off-diagonal zero/nonzero pattern is described by G. We show how to compute the minimum rank of all matrices in S(G) for a class of graphs called outerplanar graphs. In addition, we obtain results on the possible eigenvalues and possible inertias of matrices in S(G) for certain classes of graph G. We also obtain results concerning the relationship between two graph parameters, the zero forcing number and the path cover number, related to the minimum rank problem.
Wild Low-Dimensional Topology And Dynamics, Mark H. Meilstrup
Wild Low-Dimensional Topology And Dynamics, Mark H. Meilstrup
Theses and Dissertations
In this dissertation we discuss various results for spaces that are wild, i.e. not locally simply connected. We first discuss periodic properties of maps from a given space to itself, similar to Sharkovskii's Theorem for interval maps. We study many non-locally connected spaces and show that some have periodic structure either identical or related to Sharkovskii's result, while others have essentially no restrictions on the periodic structure. We next consider embeddings of solenoids together with their complements in three space. We differentiate solenoid complements via both algebraic and geometric means, and show that every solenoid has an unknotted embedding …
A Lift Of Cohomology Eigenclasses Of Hecke Operators, Brian Francis Hansen
A Lift Of Cohomology Eigenclasses Of Hecke Operators, Brian Francis Hansen
Theses and Dissertations
A considerable amount of evidence has shown that for every prime p &neq; N observed, a simultaneous eigenvector v_0 of Hecke operators T(l,i), i=1,2, in H^3(Γ_0(N),F(0,0,0)) has a “lift” v in H^3(Γ_0(N),F(p−1,0,0)) — i.e., a simultaneous eigenvector v of Hecke operators having the same system of eigenvalues that v_0 has. For each prime p>3 and N=11 and 17, we construct a vector v that is in the cohomology group H^3(Γ_0(N),F(p−1,0,0)). This is the first construction of an element of infinitely many different cohomology groups, other than modulo p reductions of characteristic zero objects. We proceed to show that v …
Applications Of Descriptive Set Theory In Homotopy Theory, Samuel M. Corson
Applications Of Descriptive Set Theory In Homotopy Theory, Samuel M. Corson
Theses and Dissertations
This thesis presents new theorems in homotopy theory, in particular it generalizes a theorem of Saharon Shelah. We employ a technique used by Janusz Pawlikowski to show that certain Peano continua have a least nontrivial homotopy group that is finitely presented or of cardinality continuum. We also use this technique to give some relative consistency results.
Planar Cat(K) Subspaces, Russell M. Ricks
Planar Cat(K) Subspaces, Russell M. Ricks
Theses and Dissertations
Let M_k^2 be the complete, simply connected, Riemannian 2-manifold of constant curvature k ± 0. Let E be a closed, simply connected subspace of M_k^2 with the property that every two points in E are connected by a rectifi able path in E. We show that E is CAT(k) under the induced path metric.
Asian Spread Option Pricing Models And Computation, Sijin Chen
Asian Spread Option Pricing Models And Computation, Sijin Chen
Theses and Dissertations
In the commodity and energy markets, there are two kinds of risk that traders and analysts are concerned a lot about: multiple underlying risk and average price risk. Spread options, swaps and swaptions are widely used to hedge multiple underlying risks and Asian (average price) options can deal with average price risk. But when those two risks are combined together, then we need to consider Asian spread options and Asian-European spread options for hedging purposes. For an Asian or Asian-European spread call option, its payoff depends on the difference of two underlyings' average price or of one average price and …
Multiscale Modeling Of Cellular Systems In Biology, J. C. Dallon
Multiscale Modeling Of Cellular Systems In Biology, J. C. Dallon
Faculty Publications
Here we review eight different multiscale modeling efforts dealing with cellular systems in biology. The first two models focus on collagen based tissue, one dealing with the biomechanical properties of the tissue and the other focusing on how the dermis is remodeled in scar tissue formation. The next two models deal with first avascular tumor growth and then the role of the vasculature in tumor growth. We then consider two models which use the Immersed Boundary method to model tissue properties and cell-cell adhesion. Finally we conclude with two models with treatments of the Cellular Potts Model. The first models …
Hokua – A Wavelet Method For Audio Fingerprinting, Steven S. Lutz
Hokua – A Wavelet Method For Audio Fingerprinting, Steven S. Lutz
Theses and Dissertations
In recent years, multimedia identification has become important as the volume of digital media has dramatically increased. With music files, one method of identification is audio fingerprinting. The underlying method for most algorithms is the Fourier transform. However, due to a lack of temporal resolution, these algorithms rely on the short-time Fourier transform. We propose an audio fingerprinting algorithm that uses a wavelet transform, which has good temporal resolution. In this thesis, we examine the basics of certain topics that are needed in understanding audio fingerprinting techniques. We also look at a brief history of work done in this field. …
Growth And Geodesics Of Thompson's Group F, Jennifer L. Schofield
Growth And Geodesics Of Thompson's Group F, Jennifer L. Schofield
Theses and Dissertations
In this paper our goal is to describe how to find the growth of Thompson's group F with generators a and b. Also, by studying elements through pipe systems, we describe how adding a third generator c affects geodesic length. We model the growth of Thompson's group F by producing a grammar for reduced pairs of trees based on Blake Fordham's tree structure. Then we change this grammar into a system of equations that describes the growth of Thompson's group F and simplify. To complete our second goal, we present and discuss a computer program that has led to some …
Four-Body Problem With Collision Singularity, Duokui Yan
Four-Body Problem With Collision Singularity, Duokui Yan
Theses and Dissertations
In this dissertation, regularization of simultaneous binary collision, existence of a Schubart-like periodic orbit, existence of a planar symmetric periodic orbit with multiple simultaneous binary collisions, and their linear stabilities are studied. The detailed background of those problems is introduced in chapter 1. The singularities of simultaneous binary collision in the collinear four-body problem is regularized in chapter 2. We use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions in both the decoupled case and the coupled case. All the solutions are found and more importantly, we find a crucial first integral which describes …
The Expectation Of Transition Events On Finite-State Markov Chains, Jeremy Michael West
The Expectation Of Transition Events On Finite-State Markov Chains, Jeremy Michael West
Theses and Dissertations
Markov chains are a fundamental subject of study in mathematical probability and have found wide application in nearly every branch of science. Of particular interest are finite-state Markov chains; the representation of finite-state Markov chains by a transition matrix facilitates detailed analysis by linear algebraic methods. Previous methods of analyzing finite-state Markov chains have emphasized state events. In this thesis we develop the concept of a transition event and define two types of transition events: cumulative events and time-average events. Transition events generalize state events and provide a more flexible framework for analysis. We derive computable, closed-form expressions for the …
The Orbifold Landau-Ginzburg Conjecture For Unimodal And Bimodal Singularities, Natalie Wilde Bergin
The Orbifold Landau-Ginzburg Conjecture For Unimodal And Bimodal Singularities, Natalie Wilde Bergin
Theses and Dissertations
The Orbifold Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasihomogeneous singularity W and a group G of symmetries of W, there is a dual singularity WT and dual group GT such that the orbifold A-model of W/G is isomorphic to the orbifold B-model of WT/GT. The Landau-Ginzburg A-model is the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan, and the B-model is the Orbifold Milnor ring of WT . The unorbifolded conjecture has been verified for Arnol'd's list of simple, unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group by Priddis, Krawitz, Bergin, Acosta, et …
Some Congruence Properties Of Pell's Equation, Nathan C. Priddis
Some Congruence Properties Of Pell's Equation, Nathan C. Priddis
Theses and Dissertations
In this thesis I will outline the impact of Pell's equation on various branches of number theory, as well as some of the history. I will also discuss some recently discovered properties of the solutions of Pell's equation.
Evans Function Computation, Blake H. Barker
Evans Function Computation, Blake H. Barker
Theses and Dissertations
In this thesis, we review the stability problem for traveling waves and discuss the Evans function, an emerging tool in the stability analysis of traveling waves. We describe some recent developments in the numerical computation of the Evans function and discuss STABLAB, an interactive MATLAB based tool box that we developed. In addition, we verify the Evans function for shock layers in Burgers equation and the p-system with and without capillarity, as well as pulses in the generalized Kortweg-de Vries (gKdV) equation. We conduct a new study of parallel shock layers in isentropic magnetohydrodynamics (MHD) obtaining results consistent with stability.
Numerical Solutions For Stochastic Differential Equations And Some Examples, Yi Luo
Numerical Solutions For Stochastic Differential Equations And Some Examples, Yi Luo
Theses and Dissertations
In this thesis, I will study the qualitative properties of solutions of stochastic differential equations arising in applications by using the numerical methods. It contains two parts. In the first part, I will first review some of the basic theory of the stochastic calculus and the Ito-Taylor expansion for stochastic differential equations (SDEs). Then I will discuss some numerical schemes that come from the Ito-Taylor expansion including their order of convergence. In the second part, I will use some schemes to solve the stochastic Duffing equation, the stochastic Lorenz equation, the stochastic pendulum equation, and the stochastic equations which model …
Properties Of The Zero Forcing Number, Kayla Denise Owens
Properties Of The Zero Forcing Number, Kayla Denise Owens
Theses and Dissertations
The zero forcing number is a graph parameter first introduced as a tool for solving the minimum rank problem, which is: Given a simple, undirected graph G, and a field F, let S(F,G) denote the set of all symmetric matrices A=[a_{ij}] with entries in F such that a_{ij} doess not equal 0 if and only if ij is an edge in G. Find the minimum possible rank of a matrix in S(F,G). It is known that the zero forcing number Z(G) provides an upper bound for the maximum nullity of a graph. I investigate properties of the zero forcing number, …
Fusion Of The Parastrophic Matrix And Weak Cayley Table, Nathan C. Perry
Fusion Of The Parastrophic Matrix And Weak Cayley Table, Nathan C. Perry
Theses and Dissertations
The parastrophic matrix and Weak Cayley Tables are matrices that have close ties to the character table. Work by Ken Johnson has shown that fusion of groups induces a relationship between the character tables of the groups. In this paper we will demonstrate a similar induced relationship between the parastrophic matrices and Weak Cayley Tables of the fused groups.
Topics In Analytic Number Theory, Kevin James Powell
Topics In Analytic Number Theory, Kevin James Powell
Theses and Dissertations
The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 …
Alternating Links And Subdivision Rules, Brian Craig Rushton
Alternating Links And Subdivision Rules, Brian Craig Rushton
Theses and Dissertations
The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every nonsingular, prime alternating link and all torus links, and explore some of their properties and applications. Several examples are exhibited with color coding of tiles.