Digital Commons Network^™

Random Walks In A Sparse Random Environment, Anastasios Matzavinos, Alexander Roitershtein, Youngsoo Seol

Mathematics and Statistics Faculty Publications

We introduce random walks in a sparse random environment on ℤ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a “locally strong” perturbation of a simple random walk by a random potential induced by “rare impurities,” which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, …

Efficient Algorithms And Applications In Topological Data Analysis, Junyi Tu

USF Tampa Graduate Theses and Dissertations

Topological Data Analysis (TDA) is a new and fast growing research field developed over last two decades. TDA finds many applications in computer vision, computer graphics, scientific visualization, molecular biology, and material science, to name a few. In this dissertation, we make algorithmic and application contributions to three data structures in TDA: contour trees, Reeb graphs, and Mapper. From the algorithmic perspective, we design a parallel algorithm for contour tree construction and implement it in OpenCL. We also design and implement critical point pairing algorithms to compute persistence diagrams directly from contour trees, Reeb graphs, and Mapper. In terms of …

On A Desert Island With Unit Sticks, Continued Fractions And Lagrange, Victor J. Ricchezza, H. L. Vacher

Numeracy

GLY 4866, Computational Geology, provides an opportunity, welcomed by our faculty, to teach quantitative literacy to geology majors at USF. The course continues to evolve although the second author has been teaching it for some 20 years. This paper describes our experiences with a new lab activity that we are developing on the core issue of measurement and units. The activity is inspired by a passage in the 2008 publication of lectures that Joseph Louis Lagrange delivered at the Ecole Normale in 1795. The activity envisions that young scientists are faced with the need to determine the dimensions of a …

On Spectral Properties Of Single Layer Potentials, Seyed Zoalroshd

USF Tampa Graduate Theses and Dissertations

We show that the singular numbers of single layer potentials on smooth curves asymptotically behave like O(1/n). For the curves with singularities, as long as they contain a smooth sub-arc, the resulting single layer potentials are never trace-class. We provide upper bounds for the operator and the Hilbert-Schmidt norms of single layer potentials on smooth and chord-arc curves. Regarding the injectivity of single layer potentials on planar curves, we prove that among single layer potentials on dilations of a given curve, only one yields a non-injective single layer potential. A criterion for injectivity of single layer potentials on …

Some Results Concerning Permutation Polynomials Over Finite Fields, Stephen Lappano

USF Tampa Graduate Theses and Dissertations

Let p be a prime, p a power of p and 𝔽_q the finite field with q elements. Any function φ: 𝔽_q → 𝔽_q can be unqiuely represented by a polynomial, 𝔽_φ of degree < q. If the map x ↦ F_φ(x) induces a permutation on the underlying field we say F_φ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F( …

Hamiltonian Formulations And Symmetry Constraints Of Soliton Hierarchies Of (1+1)-Dimensional Nonlinear Evolution Equations, Solomon Manukure

USF Tampa Graduate Theses and Dissertations

We derive two hierarchies of 1+1 dimensional soliton-type integrable systems from two spectral problems associated with the Lie algebra of the special orthogonal Lie group SO(3,R). By using the trace identity, we formulate Hamiltonian structures for the resulting equations. Further, we show that each of these equations can be written in Hamiltonian form in two distinct ways, leading to the integrability of the equations in the sense of Liouville. We also present finite-dimensional Hamiltonian systems by means of symmetry constraints and discuss their integrability based on the existence of sufficiently many integrals of motion.

Putnam's Inequality And Analytic Content In The Bergman Space, Matthew Fleeman

USF Tampa Graduate Theses and Dissertations

In this dissertation we are interested in studying two extremal problems in the Bergman space. The topics are divided into three chapters.

In Chapter 2, we study Putnam’s inequality in the Bergman space setting. In [32], the authors showed that Putnam’s inequality for the norm of self-commutators can be improved by a factor of 1 for Toeplitz operators with analytic symbol φ acting on the Bergman space A2(Ω). This improved upper bound is sharp when φ(Ω) is a disk. We show that disks are the only domains for which the upper bound is attained.

In Chapter 3, we consider the …

A Statistical Analysis Of Hurricanes In The Atlantic Basin And Sinkholes In Florida, Joy Marie D'Andrea

USF Tampa Graduate Theses and Dissertations

Beaches can provide a natural barrier between the ocean and inland communities, ecosystems, and resources. These environments can move and change in response to winds, waves, and currents. When a hurricane occurs, these changes can be rather large and possibly catastrophic. The high waves and storm surge act together to erode beaches and inundate low-lying lands, putting inland communities at risk. There are thousands of buoys in the Atlantic Basin that record and update data to help predict climate conditions in the state of Florida. The data that was compiled and used into a larger data set came from two …

Quandle Coloring And Cocycle Invariants Of Composite Knots And Abelian Extensions, W Edwin Clark, Masahico Saito, Leandro Vendramin

Mathematics and Statistics Faculty Publications

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of …

Generalized Phase Retrieval: Isometries In Vector Spaces, Josiah Park

USF Tampa Graduate Theses and Dissertations

In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp.

Resonant Solutions To (3+1)-Dimensional Bilinear Differential Equations, Yue Sun

USF Tampa Graduate Theses and Dissertations

In this thesis, we attempt to obtain a class of generalized bilinear differential equations in (3+1)-dimensions by D_p-operators with p = 5, which have resonant solutions. We construct resonant solutions by using the linear superposition principle and parameterizations of wave numbers and frequencies. We test different values of p in Maple computations, and generate three classes of generalized bilinear differential equations and their resonant solutions when p = 5.

On The Number Of Colors In Quandle Knot Colorings, Jeremy William Kerr

USF Tampa Graduate Theses and Dissertations

A major question in Knot Theory concerns the process of trying to determine when two knots are different. A knot invariant is a quantity (number, polynomial, group, etc.) that does not change by continuous deformation of the knot. One of the simplest invariant of knots is colorability. In this thesis, we study Fox colorings of knots and knots that are colored by linear Alexander quandles. In recent years, there has been an interest in reducing Fox colorings to a minimum number of colors. We prove that any Fox coloring of a 13-colorable knot has a diagram that uses exactly five …

Leonard Systems And Their Friends, Jonathan Spiewak

USF Tampa Graduate Theses and Dissertations

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let

\text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$.

A {\em Leonard system} on $V$ is a sequence

\[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\]

where

$\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$;

$\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$

are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and

for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and

nonzero when $\vert i-j \vert = 1$.

%

Leonard systems arise in connection …

An Efficient Scheme For Numerical Solution Of Burgers’ Equation Using Quintic Hermite Interpolating Polynomials, Shelly Arora, Inderpreet Kaur

Mathematics and Statistics Faculty Publications

A numerical scheme combining the features of quintic Hermite interpolating polynomials and orthogonal collocation method has been presented to solve the well-known non-linear Burgers’ equation. The quintic Hermite collocation method (QHCM) solves the non-linear Burgers’ equation directly without converting it into linear form using Hopf–Cole transformation. Stability of the QHCM has been checked using Eucledian and Supremum norms. Numerical values obtained from QHCM are compared with the values obtained from other techniques such as orthogonal collocation method, orthogonal collocation on finite elements and pdepe solver. Numerical values have been plotted using plane and surface plots to demonstrate the results graphically. …

Counter Machines And Crystallographic Structures, Natasha Jonoska, Mile Krajcevski, Gregory Mccolm

Mathematics and Statistics Faculty Publications

One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages DCL_d,d=0,1,2,…, within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class DCL_d. An intersection of d languages in DCL₁ defines DCL_d. We prove that there is …

Remembering Lynn Steen: A Steen-Numeracy Citation Index (2008-2015), H. L. Vacher

Numeracy

This editorial memorializes Lynn Arthur Steen (1941-2015) with a bibliographic resource that indexes all of his writings (papers, books, edited volumes, and papers contained therein) that are cited in papers in Numeracy. The citation index contains 67 cited works, each accompanied with a list of linked Numeracy papers that cite them. All told, there are 68 such citing papers (called sources); they cite the 67 cited works a total of 290 times and are listed alphabetically in a source index with links. The paired citation and source indexes provide a vehicle for easy browsing by which those familiar with …

Size And Concentration Analysis Of Gold Nanoparticles With Ultraviolet-Visible Spectroscopy, Sanim Rahman

Undergraduate Journal of Mathematical Modeling: One + Two

Spherical gold nanoparticles (GNPs) are synthesized by Turkevich Method and PEL 35 UV-VIS spectrophotometer recorded the wavelength and absorption of the Surface Plasmon Resonance (SPR) peak. The diameter and concentration of solute GNPs are calculated. The concentration of GNPs is done with Beer’s law. The average diameter of GNPs is done via the ratio of SPR peak to the absorbance at 450 nm. The diameters are compared to SEM scan of synthesized GNPs and Sigma-Aldrich values.

Electrical Efficiency Of A Solar Cell, Johnnie Cairns

Undergraduate Journal of Mathematical Modeling: One + Two

Our goal was to determine the electrical efficiency of a solar cell, specifically a CuInGaSe2 solar cell. Solar cells work by converting energy from the sun in the form of photons to electrical energy in the form of electrons. However, the solar energy converted into electrical energy is limited by a property of solar cells called the quantum efficiency. The quantum efficiency of a solar cell is the fraction of photons hitting the cell that are converted into electrons; quantum efficiency varies as a function of wavelength. Another key component which also varies as a function of wavelength is spectral …

Maximum Efficiency Of A Wind Turbine, Marisa Blackwood

Undergraduate Journal of Mathematical Modeling: One + Two

The concept of a wind turbine is not new technology, however, in a day and age where renewable energy is popular, the interest in wind turbines has increased dramatically. This project investigates multiple aspects of a wind turbine and will derive the maximum power efficiency of an ideal wind turbine, first introduced by Albert Betz in 1919. This project will also calculate the size of a wind turbine at maximum efficiency given certain parameters and determine the optimum outlet velocity as a function of wind speed to maximize the mechanical energy produced. These equations will be derived by using concepts …

Hydrolysis Of Acetic Anhydride In A Cstr, Veronica N. Coraci

Undergraduate Journal of Mathematical Modeling: One + Two

To find the optimal reactor volume and temperature for the hydrolysis of acetic anhydride at the lowest possible cost with a 90% conversion of acetic anhydride, a formula for the total cost of the reaction was created. Then, the first derivative was taken to find a value for the temperature. This value was then inputted into the second derivative of the equation to find the sign of the value which would indicate whether that point was a minima or maxima value. The minima value would then be the lowest total cost for the optimum reaction to take place.

Analysis Of Adiabatic Batch Reactor, Erald Gjonaj

Undergraduate Journal of Mathematical Modeling: One + Two

A mixture of acetic anhydride is reacted with excess water in an adiabatic batch reactor to form an exothermic reaction. The concentration of acetic anhydride and the temperature inside the adiabatic batch reactor are calculated with an initial temperature of 20°C, an initial temperature of 30°C, and with a cooling jacket maintaining the temperature at a constant of 20°C. The graphs of the three different scenarios show that the highest temperatures will cause the reaction to occur faster.

Optimization Of An Agitated Extractor, Evan Zapf

Undergraduate Journal of Mathematical Modeling: One + Two

In this scenario, we use calculus to determine the optimal operating specifications of a chemical extraction process. The results are achieved by first developing an expression that yields the total annual cost of the process. Factoring in electricity, vessel, agitator, and solvent costs, an annual cost expression was fabricated. AGIEX company manufactures a certain amount of a product, A, which has an impurity that it wishes be reduced. By engineering an optimally sized vessel to perform the extraction of impurity, the company will save costs by ensuring no inefficient spending on power or excess solvent occurs. A specific amount of …

Spintronic Circuits: The Building Blocks Of Spin-Based Computation, Roshan Warman

Undergraduate Journal of Mathematical Modeling: One + Two

In the most general situation, binary computation is implemented by means of microscopic logical gates known as transistors. According to Moore’s Law, the size of transistors will half every two years, and as these transistors reach their fundamental size limit, the quantum effects of the electrons passing through the transistors will be observed. Due to the inherent randomness of these quantum fluctuations, the basic binary logic will become uncontrollable. This project describes the basic principle governing quantum spin-based computing devices, which may provide an alternative to the conventional solid-state computing devices and circumvent the technological limitations of the current implementation …

Temperature Of The Central Processing Unit, Ivan Lavrov

Undergraduate Journal of Mathematical Modeling: One + Two

Heat is inevitably generated in the semiconductors during operation. Cooling in a computer, and in its main part – the Central Processing Unit (CPU), is crucial, allowing the proper functioning without overheating, malfunctioning, and damage. In order to estimate the temperature as a function of time, it is important to solve the differential equations describing the heat flow and to understand how it depends on the physical properties of the system. This project aims to answer these questions by considering a simplified model of the CPU + heat sink. A similarity with the electrical circuit and certain methods from electrical …

Optimum Gear Ratios For An Electric Vehicle, Scott Parkinson

Undergraduate Journal of Mathematical Modeling: One + Two

The goal of this project is to determine the optimal gear ratios for a vehicle containing a four-speed transmission. This vehicle is required to reach a speed of 30 m/s in the minimum time possible. Equations for the velocity at each shift point were found. An equation for the total time that the vehicle took to reach 30 m/s was then derived and equations for the times spent in each gear were found through integration of the provided formula for acceleration. The optimal gear ratios were then found by taking the partial derivatives of the total time equation with respect …