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 Creator:
 Wu, YueJia
 Description:
 Objectives. 1. To determine the presence of mesenchymal stem cells (MSC) in rat embryo limb buds and adult rat knee joints. 2. To determine the pharmacological effects of recombinant human GDF5 (rhGDF5) on rat MSCs. 3. To determine the effect of rhGDF5 on load bearing in a rodent model of osteoarthritis. Methods. FluorescentActivated Cell Sorting (FACS) was used to identify and purify MSCs from E18 rat embryo and adult rat knee cartilage. Quantitative RealTime Polymerase Chain Reaction (qRTPCR) was performed to assess the RNA profile of MSCs. qRTPCR was specifically used to quantify the expression of chondrogenesis markers including Aggrecan (Agc), Type II collagen (Col2), and Sox9 and osteogenesis markers including Runx2 and Type I collagen. An Alamar Blue (AB) assay was utilized to assess effects of rhGDF5 on MSC proliferation. An Alkaline Phosphatase (ALP) assay was used to study the effects of rhGDF5 on MSC differentiation. Neuroprobe migration chambers were used to assess rhGDF5 effects on chondrocyte chemotaxis. In situ hybridization (ISH) was conducted to identify MSC (expression of chondrogenic RNA biomarkers) in rat embryos and in adult knee joints from normal and osteoarthritic rats. Differential weight bearing analysis was assessed using Bioseb's DWB system. Results. MSCs were obtained from E18 rat embryo limb buds and adult rat knee joint cartilage. qRTPCR showed that MSC comprised preosteochondrocytes expressing Agc, Col2a1, Sox9, and GDF5 RNAs. rhGDF5 treated MSC showed a dosedependent increase in AB intensity from Day 1 to Day 14, followed by an increase in rhGDF5 mediated ALP activity on Day 14 and Day 21. The increased ALP activity coincided with increased Agc, Col2a1, Sox9, and GDF5 RNA expression indicating that rhGDF5 selectively promoted chondrogenesis. Greater chemotaxis effect of rhGDF5 was observed with MSC that presumably were more differentiated after 23 days in culture with rhGDF5. Following 23 days of treatment with rhGDF5, MSC appeared as condensations and precartilaginous structures. ISH confirmed the presence of GDF5, Sox9, and Runx2 in rat embryo hind limbs and in the experimentallyinduced osteoarthritic joint of adult animals. DWB analysis revealed significant differential joint loading in the Medial Meniscus Tear (MMT) injured knee, but no reverse differential weight bearing was detected in rhGDF5 treated rats after a 9week treatment period. Conclusion. Mesenchymal stem cells are present in rat embryonic limb buds and adult rat knee joint. Our studies show that rhGDF5 may promote an increase in metabolic activity of MSCs prior to commitment to the chondrocyte lineage. The positive chemotaxic effect of GDF5 suggests a role in MSC cell homing, cell condensation, and formation of precartilaginous structures. Taken together, GDF5 is a selective chondrocyte differentiation factor. The presence of MSC in the damaged joint suggests that rhGDF5 may act to differentiate endogenous MSC to chondrocytes.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Danielson, Rachel
 Description:
 Higher education personnel face a myriad of tasks in the course of their professional duties. The timing and priority of these tasks are sometimes controlled by the staff themselves, but are often determined by campus or external constituents. Five enrollment management staff members were interviewed to identify specific occupational pressures, situations, and challenges in their places of employment. The goal of this study was to determine the common workplace experiences of enrollment management staff and how those circumstances and conditions may affect productivity, professionalism, and purpose.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 West, Philip
 Description:
 If one imagines the Gaussian primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture. If the frog's journey terminates for a given hop size, it implies that a prime free "moat" greater than the hop size completely surrounds the origin. In the Chauvenet Prize winning paper "A Stroll Through the Gaussian Primes", Ellen Gethner, Stan Wagon, and Brian Wick [4] explored this problem and by computational methods proved the existence of a square root of 26 moat. Additionally they proved that primefree neighborhoods of arbitrary radius k surrounding a Gaussian prime exist. In their concluding remarks, Gethner et al. note that "Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1."
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Gehani, Rohit
 Description:
 The existence of cancer stem cells (CSC) postulates that tumors are organized as a cellular hierarchy and that tumor initiation, growth and cellular heterogeneity are driven by a subset of cells with stem cell like properties. The CSCs are endowed with the ability to selfrenew and thereby to proliferate indefinitely. At a functional level, CSCs are characterized by their ability to regenerate in vivo from a single cell into the full spectrum of histology of the tumor of origin and to form spheroid colonies in vitro in an anchorage independent environment. The specific aims for this project include setting up assays that will enable the quantification and the characterization of CSCs and evaluating cell surface markers to enrich for CSC. Additionally, in our in vivo assays, we will compare different mice strains as hosts to our in vivo assays. In order to assess the frequency of CSCs within a solid tumor, we established an in vitro and in vivo limited dilution assay (LDA). A known number of cells were seeded in a low attachment well with stem cell growth media, and the number of spheroids that grew was counted by Optronix GelCount. In an in vivo LDA, immune deficient mice were inoculated with a specific number of cells, and the number of tumor bearing mice was counted. In order to optimize our methods, immortalized colorectal cancer (CRC) cell lines COLO205 and T84 were used as tools to develop LDAs. We were able to determine that the sphere and tumorforming efficiency of COLO205 was superior to T84, resulting in the primary use of COLO205 cells for assay development. We also aimed to compare the tumor initiating capacity of the COLO205 cell line in order to investigate the effect of the mouse immune system on the readout of the in vivo tumor formation assay. The result of this experiment was inconclusive; no tumors grew, which is a defect that we could pinpoint to a loss of viability of sorted tumor cells by FACS. We worked with four colorectal cancer models derived from patient tumors and continuously passaged in immune deficient mice. Two models were established from primary tumors obtained from the Amgen Tissue Bank. These tumors were dissociated and cultured in serumfree stem cell growth media on low attachment plates. The other two models were obtained from a company that had continuously passaged the tumors in immune deficient mice. We used these two models to test mouse immune background and effects of irradiation on tumor initiation. Antibodies to extracellular proteins epithelial cell adhesion molecule (EpCAM) and CD133 were used to enrich for a homogenous population of CSCs by sorting protocols. We could not confirm that either of these is a good marker but did find that FACS sorting may affect tumorigenicity and viability of CSCs in our CRC models. The outcome of the CSC enrichment procedure will be important to enable further characterization at the molecular and cellular level of CSC and the identification of potential therapeutic targets that could selectively eliminate CSCs. iv
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Fard, Roxana
 Description:
 Bone graft substitutes are commonly used as an alloplastic source for complex bone repair. Human mesenchymal stem cells (hMSCs) have become an idealistic source for bone repair and regeneration due to their potential to differentiate into osteogenic precursors. The purpose of engineering synthetic bone grafts it to successfully find a substitute that is biocompatible, bioresorbable, and has osteoconductive characteristics. The purpose of this study is to construct a bone biocomposite with an optimal amount of biphasic hydroxyapatite/βtricalcium phosphate (HATCP) powder to promote hMSC proliferation with sufficient mechanical stiffness. Results have indicated an increase in metabolic proliferation over a 2week time period. The constructs seeded with hMSCs exhibited a 3 to 9 fold or greater increase in proliferation depending on the formulation of the construct. This work demonstrates that higher volumes of HATCP promote hMSC proliferation in the constructs while maintaining sufficient mechanical stiffness. Optimizing the components of the scaffolds will allow for the most innovative biomimetic bone composite for mesenchymal stem cell differentiation into osteoblasts in an in vivo model.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Cochran, Dana
 Description:
 We examine properties of the Blum Medial Axis (BMA) to improve its effectiveness for automated shape understanding. The BMA has been used to represent objects since the 1960s. Here we examine measurements on the BMA including shape tubularity (ST), erosion thickness (ET), and the extended distance function (EDF). We make connections between the three and use the EDF to find significant paths in 2D shapes. Three shapes are used: a whale tail with only one significant path, a symmetrical flower with a few identical significant paths, and a chicken shape with a few distinct significant paths. We analyze values of the EDF in order to automatically determine the significant directions within a shape. We also establish a relationship between the ST and ET measures that aid in shape identification in future work.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Ruiz, Michael
 Description:
 The study of prime numbers has been an area of interest in mathematics
since antiquity. One natural question one may ask is "How many primes
are there less than or equal to some positive integer?" The first attempts to
answering this were in the late 1700s, culminating in the celebrated Prime
Number Theorem. We investigate how this may be generalized to primes in
an imaginary quadratic number rings in a given sector of the complex plane.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Suarez, Ricardo
 Description:
 Clifford algebras were originally introduced to generalize the set of quaternions to higher dimensions. Since these algebras are vector spaces, we can apply methods from Linear Algebra to better understand them. In particular, we can view its elements as matrices by using techniques of Representation Theory. Although such results are already known in the context of spinors, we propose to derive these representations through projection maps. By using properties of these projections, we can reduce the dimensions of the matrix representations and often obtain minimal representations. More specifically, we find such representations for Clifford algebras having up to four generators. Besides giving matrix representations over R, we explore the possibilities of matrix representations over C and H as well.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Ananda, Dev
 Description:
 Let d(u,v) denote the distance between two vertices u and v on a graph G and let diam(G) denote the diameter of such a graph G. A connected graph G has a radio labeling f if, for all vertices u, v of G,
d(u,v) + ƒ(u)  ƒ(v) ≥ diam(G) + 1 (1)
The span of the labeling function f is the maximum integer assigned by f. The radio number of a graph G, rn(G), is the minimum possible span obtained over all possible radio labelings of the graph G. A path graph Pn has n consecutive vertices along n — 1 consecutive edges. A grid graph is defined as the Cartesian product of two path graphs, and a square grid graph is obtained by taking the product of identical path graphs Pn Pn. In this paper, the radio number of all even grid graphs is determined using bounding techniques alone, while establishing fundamental guidelines for odd grids and distancemaximizing labelings, in general.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Romero, Alba
 Description:
 The moon landing in 1969, achieved despite of the complexity and the lack of computer power at the time, raises questions about various computational aspects of the mission. We present a model of the lunar docking problem, in which the optimal rendezvous orbits to dock the lunar module back onto the orbiting command module are presented through analytical equations and analysis for the optimal trajectories.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 MorrisRivera, Marc
 Description:
 The attributable risk is the fraction of outcomes in the population that would be reduced if the probability for a (higher probability) group were reduced to the probability for another (lower probability) group (Fleiss, et. al). We will use methods of weighted least squares (Koch, et. al) and maximum likelihood (Dobson and Barnett) to compute odds ratios and the pooled odds ratio, and weighted least squares to compute attributable risks and the pooled attributable risk, which we introduce here using a Mantel Haenszel like definition. Although odds ratios and attributable risks can be calculated using specialized statistical software, we will show how these can be obtained using Microsoft. Excel array formulas and V B A programming.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Ahmed, Nausheen
 Description:
 The two different language based projects were designed to introduce the concepts of parabolas and integrals. Integration was introduced to college students, and eighth grade students worked with parabolas. We incorporated a story and acting exercises to help students be active participants in the learning process. To make it interesting to children, the story was illustrated with colorful pictures in the eighth grade activities. The goals for this research were to evaluate if there was a significant difference in the overall performance of the test groups versus the control groups, and if there was a significant improvement in learning on the conceptual problems. Our data shows that the project was successful and students involved in the story based activities performed better.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Koressa, Marcel
 Description:
 As computer and Internet technologies evolve, they play an increasingly significant role in our lives, especially in education. In particular, more students are using computers to attend online classes or to do computer based homework. Various online homework systems have become more advanced and available to the mathematics instructors and students. These online systems provide various advantages for the students, such as more practice problems, time flexibility, improvement of self discipline, and improved computer skills. However, the effectiveness of online math classes and the computerbased homework differ from one person to another, depending on many factors. We designed this study to explore these factors. We prepared survey questions for a group of students in order to evaluate the effectiveness of the computerbased math homework and their satisfaction with online math activities. Additionally, the survey provides the participants' feedback on features and benefits of online instructions, including e mails, tutoring and videos. Moreover, we evaluate the factors that determine the students' success in using computers for their homework. These factors could be the result of the students' experience with technology or learning experiences using various computer software.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Kerissa, Hoda
 Description:
 Tutoring Centers are designed to assist students in becoming independent learners with study skills and strategies that will help them succeed in classes across the curriculum. Those are not intended to replace direct instructions classroom, but to enhance studying and learning habits. In addition, tutoring helps students who work off campus and those who may have difficulty asking instructors questions during the class sessions by providing unstructured learning environment. In this study, we evaluated the effectiveness of the tutoring services at two college centers and one university center. We collected data on their ability to provide students with improvement of their learning skills and achievement of their academic goals. We created a survey to get an accurate evaluation for specific tutoring services given in algebra. The survey included anonymous questions and we expected the students to rate the algebra tutoring services honestly, as well as share their thoughts about the centers. The survey included questions about learning and the influences that affect the students' experience at the tutoring center. Also, we evaluated the students' background and attitudes towards alggebra. For the statistical analysis of data we used Excel software. This study has provided interesting results and we made conclusions from our data analysis. We included some recommendations as well as the students' comments about how to improve the tutoring services at the end of this paper.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Vader, Bryan E.
 Description:
 Alcohol biosensor devices have the prospect to positively impact medicine and law enforcement by giving a noninvasive method to acquire continuous alcohol readings. We propose to develop a nonparametric estimation algorithm that estimates the joint mixing distribution of the parameters of a heat equation model via a maximum likelihood method. This model is assumed to estimate the diffusion of alcohol through transdermal layers while taking into account measurement error. These parameters are assumed to be random due to natural irregularity in an individual's body conditions and the variability of population data. This is superior to parametric estimation methods since it can capture unusual fluctuations of a subject’s condition as well as environmental factors. This will help to ascertain a precise relation between blood alcohol concentration and transdermal alcohol concentration.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 LaGrange, Thomas L.
 Description:
 Markov Chains are a category of stochastic processes with an associated dependence structure. Their inception resulted from A. A. Markov’s desire to disprove a rival’s assertion concerning the application of the Law of Large Numbers (LLN) to dependent variables. Markov’s use of a classic Russian novel to illustrate a dependent relationship between vowels and consonants serves as motivation for investigating the extent to which the properties of these processes can be applied to other means of communication. This work summarizes a history of the Law of Large Numbers and the Markovian properties associated with three different aspects of communication.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Costa, Matthew
 Description:
 Derivations of wave equations, various presentations of their solutions and MATLAB models are presented. Thereafter, basic ocean wave forecasting will be discussed along with it’s applications.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Zechlin, Mollie J.
 Description:
 Public key cryptography, is the basis of m odem cryptography, allows us to send and receive messages over public channels secretly, without requiring a meeting beforehand. Most public key cryptosystems, such as the Diffie Hellman Key Exchange, rely on the difficulty of solving the Discrete Logarithm Problem (DLP). We can translate public key cryptosystems that rely on the DLP to Elliptic Curve cryptosystems as the Elliptic Curve Discrete Logarithm Problem (ECDLP) is believed to be more difficult and therefore harder
to break. There are certain precautions we need to take when using Elliptic Curve Cryptography to safeguard against particular attacks on the cryptosystem.
Therefore, picking a curve that is secure enough is crucial to a good cryptosystem.
Unfortunately, there are only a handful of secure elliptic curves that are publicly known and used. The goal of this thesis is to generate more elliptic curves that are useful for our security systems.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Lee, Seungju
 Description:
 Packaging/repackaging is essential to ensure pharmaceutical product’s stability (i.e. efficacy and safety). Before processing to package/repackage products into container/closures, there must be evidence to verify its stability. The most common verification methods are realtime study and statistical analysis. Realtime study can be very accurate, but it is also timeconsuming, which is inadequate for industry use. On the other hand, statistical analysis is appropriate for industry use since it can take shortterm data and obtain the results quickly. In particular, we apply regression analysis on pharmaceutical product container/closure data along its decisionmaking framework to determine a container height 95% confidence interval and estimate the shelf life. Using a freeware program R, we are able to conduct the analysis more quickly and efficiently.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Chen, Vickie V.
 Description:
 Recent progress has been made toward understanding the density that k integers are Gwise relatively prime as a limiting form of a uniform distribution motivate this work. Fix a positive integer k and let G be a simple graph with k vertices that are arbitrary integers. We say that these integers are Gwise relatively prime if for any pair of vertices joined by an edge, the corresponding integers are relatively prime. Observe that if G is a complete graph, then this reduces to the notion of integers being pairwise relatively prime. From this
foundation, we can compute the density that k integers are Gwise relatively prime. The main objective of this thesis is to extend the notion of Gwise relative primality to rings of algebraic integers and to rings of polynomials over a finite field.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Ferguson, Vincent
 Description:
 Exploring the Effects of Wolbachiainfected Mosquitoes on the Spread of the Zika Virus
by Vincent Ferguson
Recent efforts to stop the spread of Zika, a vectorborne disease, through pesticides and insecticides have not been successful. Research has shown that Zika can be spread via the Aedes aegypti mosquito and human sexual contact. We examine how the use of Wolbachia as a form of control can reduce the transmission rate between infectious mosquitoes and susceptible humans. Through analysis of our mathematical model, we find the reproductive number of Zika before and after including Wolbachia into our calculations. With this model, we are able to run simulations to show how fast Zika can spread in a population. Using local sensitivity analysis, we study how changes in different parameter values affect the reproductive number.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Lieberman, David A.
 Description:
 This paper aims to generalize results on single variable polynomial rings over commutative rings with zerodivisors to the case of polynomial rings in arbitrarily many variables. Given a commutative ring R, we give necessary and sufficient conditions for the ring of polynomials with coefficients in R in arbitrarily many variables to be a PVMR and Krull ring. In answering these questions, we make use of the t and v operations on ideals as a means of characterizing these rings. We also give conjectures on necessary and
sufficient conditions for an arbitrary polynomial ring to be a Dedekind ring, a UFR, and integrally closed.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Vo, Don
 Description:
 This thesis is an attempt to develop mathematically consistent procedures to perform ThreeDimensional (3D) matrix math. 3D matrix operations for Rightmultiplication, Leftmultiplication, LeftIdentity, RightIdentity, LeftInverse, RightInverse, and Eigenvalue/Eigenvector matrix are explained.The innovative technique used here is to separate ThreeDimensional (3D) matrix arrays into multiple Twodimensional (2D) matrix arrays to simplify performing 3D array mathematics. A n x m x A: 3D matrix is separated into k, n x m 2D matrices. Then 2D array math techniques are used to produce a 3D Matrix result.
The purpose of the present study is to adjust the accepted procedures used to perform 2D matrix operations for application to 3D matrices. These new procedures will be explained with a select group of examples. In this project techniques to successfully perform the basic 3D Matrix manipulations that are derived from the rules for 2D matrix operation are explored. The process of performing these 3D matrix operations follow from the definition for Multiplication of a 3D matrix box. Multiplication is performed by taking Top face to Bottom face matrices of the first box and multiply with the Front face to Back face matrices of the second box which equals the Top face to Bottom face of the result matrix. And similarly define operation to calculate identity, inverse, and eigenvector matrix. Followed by numerical example for each of these operations.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Walden, George
 Description:
 Tortuosity is an intuitive term used to describe paths on the plane that exhibit multiple twists and turns. It is used in a variety of applications as a measure of how much a path deviates from a straight line, especially to analyze images. However, unlike curvature, tortuosity does not have a good mathematical definition yet. Nor is there a consensus on how to measure tortuosity.
This paper examines the information obtained from curvature and how it applies to the measure of tortuosity. Methods from the literature are examined and a new method is proposed using multiple measures instead of a single value.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Stafford, Emery
 Description:
 In this paper, we introduce the reader to the graph labeling question in the setting of
quantum mechanics. We formally define an edge weight of a graph G and use it to develop the general labeling number. We derive some elementary results about this number and give an example of a quantum chromatic labeling function of a simple path graph of two vertices that is not replicable in the classical setting. We define a quantum labeling number for any graph and show that it is at most as large as the classical labeling number. Finally we posit a relationship between the quantum labeling numbers as seen in different tensor products of operator spaces common to Quantum Information Theory.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Silva, Jennifer Ann
 Description:
 Teaching Algebraic Manipulations to Students with Disabilities
A nonsymbolic methodology was designed, and field tested for solving algebra and geometry problems within a special education high school mathematics classroom and then compared to the achievements of general education students given a different methodology, but the same curriculum. The results show that the achievements of the special education population were comparable to those with the general education population, suggesting minimizing difficult abstract mathematical concepts and algebraic manipulations. We conclude that designing differentiated instruction using alternate methodologies and offering individualized instruction benefits all students.
Keywords: Education, Mathematics, Special Education, Teaching, High School, Specialized Academic Instruction, Individualized Education Plan, Algebra, Geometry, Algebraic Manipulations, Symbolic Representations.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Bennett, David
 Description:
 Paraffin wax tea light candles and ethanolfuel candles are ignited within an enclosed, plexiglass and glass environment in an attempt to create a model of burn time before selfextinction of various candles regardless of type, size, and the volume and shape of the enclosed container. A lower limit of final percent O sub 2 is found that is consistent with theory, but these experiments do not always reach this limit, leading to nonrepeatability. Many factors, such as the relevant fluid dynamical and flame dynamical systems, known combustion models such as the those given by the National Institute of Standards and Technology and the Boussinesq model of buoyant combustion, and other current combustion studies have been considered and controlled for without success.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Tran, Chinh, Grzegorczyk, Ivona, and Berg, Gary
 Description:
 A tiling is a covering of the plane with nonoverlapping figures that have no holes between them. For centuries, civilizations have incorporated both the complexity as well as aesthetic properties of tilings into their art and everyday lives. No matter where we look today, from religious to modern day buildings, to the typical household decorative arts, we can see some form of tiling being displayed. In this research, we will study tilings of the plane by various polygons. We want to determine whether, it is possible to tile the plane with a given set of specified figures.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Topolinski, Katrina, Grzegorczyk, Ivona, and Berg, Gary
 Description:
 There are many components in the movement of horses that can be mathematically modeled. In this research we have classified footprint patterns of various strides and found an analytical family of curves describing simple jumps. This approach is different from the standard center of mass method typically used in physics, as it is more visual and practically useful. We will show curve fitting for four different horses and the equations for the curves describing the jumps. We standardize the curves as quartics with prescribed inflection points that contain basic information about each horse. We will discuss the properties of this family of functions and various mathematical methods of comparing the horses.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Yi, Cory, Wyels, Cynthia, and Berg, Gary
 Description:
 An unproven claim is that all trees may be gracefully labeled. However there are some special classes of trees that are proven to have graceful labelings. A path is the simplest form of a tree, and it has been proven that all paths can be gracefully labeled. The focus of this study is on the characteristics of gracefully labeled paths and a method for producing graceful labelings of P sub n with given properties. We report on progress towards a proof that labelings of paths of any size may assign the label 1 to any node and be completed as graceful labelings. Representations such as the Edge Tree Diagram and the MatrixEntry Choosing methods are developed. We also prove that certain assignments of labels to the first two vertices of a path guarantee that a labeling may not be completed to form a graceful labeling. Finally we develop a computer program to generate gracefully labeled paths to assist in examining the results and identifying conjectures worthy of further study.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Grzegorczyk, Ivona, Nikjeh, Esmaail, and Berg, Gary
 Description:
 There are many areas of mathematics that are difficult for students to learn. My extensive teaching experience tells me that students have difficulty finding the Least Common Denominator (LCD). For this reason many students don't like fractions and consequently don't like mathematics. In this study, we designed and tested a new method called the Ladder Method for finding LCD that can be used for subtracting and adding fractions. To evaluate effectiveness of this method and students' improvement of understanding of fractions, we designed a pretest and a posttest. We collected and analyzed data for two groups of basic algebra students. The study group used the Ladder Method activities and the control group used the prime factorization method for finding LCD. Finally via hypothesis testing our results show that the new method significantly improves students' performance and understanding of fractions.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Department:
 Mathematics
 Creator:
 Milne, Susan
 Description:
 Our paper summarizes a study that used gamebased learning activities (GBLA) That we
designed to teach underprepared university algebra students how to simplify polynomials. The activities were designed to visually, verbally and kinetically reinforce the concepts of combining like terms to add, subtract and simplify polynomials. We compared the achievement and attitudes of a study group who played the game, worked in groups to create their own polynomials and then completed a worksheet, to that of a control group who was taught by traditional lecture. Our analysis shows that GBLA improved the study group’s performance and the final results were better than the group using traditional method. We observed a positive disposition of the participants in the learning activity.
Key words: game play, learning activities, developmental mathematics, remedial mathematics, polynomials, algebra, like terms, simplifying polynomials, adding polynomials, subtracting polynomials.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Perkins, Elisabeth
 Description:
 For this Master's Thesis Project, I created a website that educates people about nine different population models. I wrote several Java applets that allow users to explore both single population and multiple population models. Additionally, I explained each model from a combined ecological and mathematical perspective, and gave examples in the text that pair with examples in the programs. Most of the models use differential equations. This website explains how differential equations work, along with why and how the differential equations are created for the various models. In order to create graphs of the populations, the fourth order RungeKutta method was used to find approximations of the solutions of the systems of differential equations. One model incorporates randomness through stochastic differential equations, and uses a discrete approximation technique to generate the data points.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Jones, Mattie
 Description:
 In this thesis, we researched various methods leading to recognition of a person's face by ways of statistical and mathematical analysis and comparisons of facial features to that of a known person. Many approaches to overcome inherent face recognition challenges have been developed over the years. One of the most accurate and rapid ways to identify faces is to use what is called the eigenface [1] technique, that was created as a linear combination model using the mathematical software called MatLab.
The linear combination model has been recognized for several years and uses the process of Principal Components Analysis (PCA). This system was able to successfully recognize all randomly generated photos (mugshots) with 97.7 percent accuracy using 21 eigenfaces. The eigenface technique uses a highly effective combination of linear algebra and statistical analysis (PCA) to generate an identifying set of base faces, the eigenfaces, against which the inputs are tested and matched. Although using a sophisticated statistical model is a means to recognize a person by facial patterns, it is also critically important to acknowledge that the collected data is imperfect and requires some manipulation to be both clean and normalized. The objective is to represent a face as a linear combination of images from our data base. Recently, Random Projection (RP) has emerged as a powerful method for dimensionality reduction. In this paper, I will compare and contrast Random Projection (RP) with PCA using a well known face database. The experimental results illustrate that although PCA represents faces in a lowdimensional subspace, the overall performance is comparable to that of Random Projection, having higher computational requirements and being data dependent.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Terzian, Tammy
 Description:
 In 2002 Manindra Agrawal, Neeraj Kayal, and Nitin Saxena discovered an algorithm to test a number for primality that is both deterministic and runs in polynomial time. The AKS algorithm hinges on a calculated value they call r, which is defined for a given integer n >1 as the least value for which the order of n modulo r is greater than log 2 over 2 n. This r has a proven upper bound of log 5 over 2 n. In this paper, we prove that 2 + log 2 over 2 n is a lower bound of the value r, and if n is a square, there is a lower bound of 1 + 2 log 2 over 2 n . We also present data suggesting that 3 log 2 over 2 n is a smaller upper bound of r. If this is indeed an upper bound, the AKS Primality Test is shown to have a time complexity of O(log 6 + ε over 2 n) in bit operations for any small ε greater than 0. Data also suggests a number n is a square if and only if its corresponding r is greater than 3 log 2 over 2 n.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Gautreau, Rebecca
 Description:
 Graph pebbling is a mathematical game in which pebbles are placed on the vertices of a graph. The game is made up of a series of pebbling steps that consist of removing two pebbles from one vertex, discarding a pebble, and placing the other on an adjacent vertex. The goal of the game is to reach a particular vertex by performing a series of pebbling steps. This paper will focus on implementing particular pebbling strategies on specific types of graphs to determine the minimum amount of pebbles needed to reach a vertex via pebbling moves. We will use these strategies as a basis for determining the lower bound of the pebbling number of our graph.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Dahme, Marissa
 Description:
 The overall goal of this thesis is to provide a glimpse into the inner life of music that can be seen through the use of mathematics. Chapter 1 provides the reader with some background knowledge needed including necessary definitions for intervals and transformations that will be used in subsequent chapters. The main focus of the thesis will be on musical canons, specifically rhythmic canons and crab canons. Chapter 2 is a summary of work by Hall and Klingsberg on counting the number of asymmetrical rhythms which is then applied to rhythmic canons. Chapter 3 includes a brief summary of musical canons, an analysis of Bach's "2 Canon a 2" from Musical Offering, as well as a definition for isomorphic crab canons essential for the work in Chapter 4. Chapter 4 gives two separate algorithms for creating crab canons suggesting that a certain mathematical structure can be used to compose musical crab canons. Chapter 5 provides the reader with possibilities for further research. A brief musical background and prior knowledge of combinatorics techniques are not necessary, but of course helpful, in the reading of this thesis.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Black, Lorrene
 Description:
 Many students arrive in college thinking of mathematics as a set of procedures to memorize and apply. This is in direct opposition to mathematicians‟ conviction that mathematical learning consists of developing an understanding of mathematical concepts and the ability of using them in various contexts. This thesis explores the efficacy of using art projects to supplement the standard lecture model in Intermediate Algebra courses.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands
 Creator:
 Ta, Jacquelyne Lan
 Description:
 The Global Positioning System (GPS) is a satellitebased navigation
system made up of a network of 24 satellites placed into orbit by the United States Department of Defense (DOD) and managed by the United States Air Force 50th Space Wing. GPS was originally intended for military applications. It was designed
to assist soldiers and military vehicles, planes, and ships in accurately determining their locations worldwide. However, in the 1980s, the government made the system available for civilian
use.
Civilian applications are evolving and expanding constantly. Today,
the uses of GPS have extended to include both the commercial
and scientific worlds. Commercially, GPS is used as a navigation and positioning tool in airplanes, boats, cars, and most outdoor recreational activities such as hiking, fishing, and kayaking. In the scientific community, GPS plays an important role in the earth sciences. Meteorologists use GPS for weather forecasting and global climate studies. Geologists use GPS for surveying and earthquake studies to measure tectonic motions during and between earthquakes.
The GPS is vast, expensive and involves a lot of technical ingenuity,
but the fundamental concepts at work are quite simple and intuitive. The objective of this thesis paper is to give the readers a working familiarity with both the basic theoretical and practical aspects of how the GPS works. Chapter 1 introduces a condensed GPS program history that involves three competing concepts from the Transit, Timation, and Project 621B programs.
Chapter 2 examines the GPS system consisting of three segments: space segment, control segment, and user segment. The three segments contribute to overall accuracy, reliability, and functionality. Chapter 3 outlines the basic mathematical methods used to calculate user’s position based on a system with no errors. Finally, Chapter 4 and 5 covers two important math theories needed to handle satellite and receiver clock errors:
NewtonRaphson Method and the Least Squares Method.
 Resource Type:
 Thesis
 Campus Tesim:
 Channel Islands