# College of Science

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# Browsing College of Science by Department "Mathematics and Statistics"

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Item Analysis and applications of the weighted central direction method(2016-12) Choi, HaekyoungIterative methods yield an approximated solution to a given problem by producing a sequence of points that converges to the exact solution. Due to the effectiveness of these methods, they become one of the core mathematical procedures widely used in many major mathematical areas such as Differential Equations, Linear Algebra, and Matrix Analysis. The Methods of Alternating Projection, which we will use in this thesis, form a class of iterative methods based on the relevant projection algorithm introduced by John von Neumann. This outstanding algorithm received considerable attention by mathematicians which contributed to a number of different algorithms to solve several problems. Recently, two former graduate students at Texas A&M University-Corpus Christi, Melina Wijaya and Zulema Cervantes, introduced a couple of new algorithms in this area. Wijaya presented the Weighted Direction algorithm, and Cervantes introduced the Weighted Central Direction algorithm by combining the Weighted Direction and the Central Direction. The algorithm with the combination of two directions achieved a faster convergence than the algorithm with the Weighted Direction only. However, the Weighted Central Direction algorithm needs a parameter which depends on the size of the problems. This thesis carries two objectives. Firstly, study the role of the wide angle condition which guarantees the convergence in the Weighted Direction algorithm in order to analyze if this condition is necessary for the convergence of the algorithm. Secondly, improve the algorithms developed by Wijaya and Cervantes in order to obtain a faster convergence by finding an adequate ratio between the Weighted Direction and the Central Direction that is independent of the size of the problems.Item Analytic solutions for the harmonic potentials involving concentric layered dielectric spheres(2018-05) Cisneros, Frank; Palaniappan, Devanayagam; Sadovski, Alexey L.; Zimmer, BeateThe mathematical problem of a conducting spherical core of radius 𝑎 concentrically covered by a dielectric phase of radius 𝑏 placed in an arbitrary external electric field is investigated. The vector field equations for the electric field (Maxwell equations) and the boundary conditions are transformed to a scalar boundary value problem (BVP) in terms of the harmonic potential functions. The harmonic potentials denoted by Φ𝐼(𝑟,𝜃,𝜙) and Φ𝐼𝐼(𝑟,𝜃,𝜙) where (𝑟,𝜃,𝜙) are spherical coordinates, satisfy the Laplace equations in the regions 𝑏<𝑟 and 𝑎 < 𝑟 < 𝑏, respectively. General analytical solutions for the potentials in the two phases are determined in infinite series form using spherical harmonics methods. Exact closed form solutions are also derived via an alternative approach. The latter solutions contain integrals involving harmonic functions. Our general solutions are applicable for arbitrary external potentials disturbed by a conducting spherical core with a dielectric coating. Several illustrative examples are investigated and exact solutions for them are constructed using our general solutions. The non-dimensional parameter 𝑘= 𝜀𝐼/(𝜀𝐼+𝜀𝐼𝐼) , where 𝜀𝐼 is the dielectric constant for the region 𝑟 > 𝑏 and 𝜀𝐼𝐼 is the dielectric constant for the region 𝑎 < 𝑟 < 𝑏, influences the potential patterns in the case of externally imposed constant and linear fields. Our results for the source induced field indicate that the force is positive or negative depending on 𝑘 < 0.5 or 𝑘 > 0.5. Furthermore, the force is greater than zero when the core radius 𝑎 approaches the value of the outer radius 𝑏. We believe that our mathematical results are of interest where coated dielectric objects are exposed to external electric fields.Item Calculating Kendall's Tau with multiple measurements(2018-05) Geller, Adam; Geller, Adam; Sterba-Boatwright, Blair; Sterba-Boatwright, Blair; Jin, Lei; Guardiola, Jose; Jin, Lei; Guardiola, JoséRelationships between time series of environmental variables are commonly calculated using non-parametric methods, such as Kendall’s τ, because of ”non-detects”, i.e., left-censored data that falls below a measurement limit. However, these methods are not well-adapted to situations where variables have multiple contemporaneous measurements. In this thesis, we define a new method, τ ̃, in an attempt to calculate correlations using each of the multiple measurements instead of daily means. We investigate τ ̃ using two methods: simulations that approximate a null distribution for τ ̃ and closed form calculations for a specific special case. We also apply τ ̃ to an actual data set. The results of our investigation shows that τ ̃ may handle certain things, such as outliers, better than current methods. However, its requirements for distributional assumptions about the data make it a less practical option for real data. Further work could explore ways to avoid the prerequisite need for distribution knowledge and could also further investigate τ ̃ under noise sampled from asymmetric distributions.Item Calculation of laplace and helmholtz potentials in two-phase problems(2017-05) Alarki, Rabab Abdulwahed; Palaniappan, Devanayagam; Sadovski, Alexey L.; Zimmer, BeateIn this thesis, two phase models in a magnetostatics context using the Maxwell-Maxwell (MM) model and the Maxwell-London (ML) model are investigated. The vector equations are transformed in terms of scalar potentials leading to mixed boundary value problems for Laplace-Laplace and Laplace Helmholtz equations in the respective cases. Exact analytic solutions for the exterior and interior potentials Fe(r;q;f) and Fi(r;q;f), where r;q;f are the spherical coordinates, are obtained as infinite series and in closed forms for the MM model. The general solutions are found as a theorem. Several illustrative examples for specific externally imposed magnetic fields including a magnetic monopole and dipole are discussed based on our analytic solutions. It is shown that the magnetic permeability parameter k = me me+mi , where me and mi are magnetic permeabilities in the exterior and interior phases, has a significant impact on the magnetic induction fields and the forces acting on the sphere. A new relation for the multipole coefficients of the external phase is derived as well. Exact solutions for the ML model involving a superconducting sphere are derived in terms of the magnetic flux density functions Ye(r;q) and Yi(r;q) in the respective phases. The general solutions are established as a theorem for this model as well. The non-dimensional penetration depth parameter l is found to dictate the induction fields in ML model. Our results are of interest in various topics in mathematical physics where two phase models are used.Item Cognitive demand and the level of alignment in the mathematics items of the staar test and the textbook tasks in the intended curriculum for 7th, 8th, and algebra 1 students in south Texas(2020-05) Viera, Justene; Dogbey, James; Postelnicu, Valentina; Ekici, Celil; Sadovski, Alexey L.This study examined the cognitive demand and the level of alignment between the mathematics items on the STAAR Test and the corresponding textbook tasks in the intended curricular for students in grades 7, 8, and Algebra 1 in South Texas. One of the primary drives for the present study was based on the notion that when there is a strong alignment between students assessment and the intended mathematics curriculum, educators and policymakers are better placed to evaluate students’ performance in relation to the expected students learning at the respective grade levels, and thus, able to make appropriate policy decisions regarding students learning. The study, therefore, employed the Mathematical Task Framework developed by Stein and Smith (1998) with its four levels - memorization, procedures without connections, procedures with connections, and doing mathematics (in order from the lowest level to the highest) to examine the the cognitive demand between the curricular tasks and the assessment items on linear equations, inequalities, and functions on the STAAR Test and selected mathematics textbooks specifically developed for students in grades 7, 8, and Algebra 1 in South Texas. The results of the cognitive demand analysis conducted showed that each grade level textbook contained tasks at the four levels of cognitive demand. Each textbook contained the majority of its tasks at the memorization, procedures without connections, and procedures with connections levels of cognitive demand. There were few tasks at the doing mathematics level of cognitive demand. The STAAR Assessments contained items only at the procedures without connections and procedures with connections levels of cognitive demand. Because the STAAR Assessment is a timed standardized test, none of the assessments contained items at the doing mathematics level of cognitive demand; items at this level require larger amounts of time than the assessment could permit. The results indicated a weak alignment with the linear equations STAAR Assessment items and the McGraw-Hill textbook tasks (curriculum) for 7th grade. Linear equations assessment items were properly aligned with the McGraw-Hill textbook tasks for 8th grade and Algebra 1. Linear functions were properly aligned for all three grade levels. Linear inequalities were poorly aligned for 7th and 8th grade because the 7th grade STAAR Assessments did not contain any linear inequatity items nor the textbook. And the 8th grade STAAR Assessment contained 2 linear inequality items but the textbook did not contain tasks. For Algebra 1, the linear inequality STAAR Assessment items were properly aligned with McGraw-Hill textbook tasks. Overall, the McGraw-Hill TEKS textbook series did a good job preparing students for the corresponding STAAR Assessments. It is the hope of the researcher that the results from this study will inform assessment practices, textbook development, classroom practices, as well as other educational policies at the local and state levels.Item A comparative analysis of uniformity tests in circular statistics(2018-12) Ruiz Mora, Armando Marcial; Ruiz Mora, Armando Marcial; Guardiola, Jose; Jin, Lei; Guardiola, Jose; Jin, LeiGuardiola, Jose; Jin, Lei; Sterba-Boatwright, Blair; Sterba-Boatwright, BlairIn the context of circular statistics, data may not behave uniformly around the circle, exhibiting a preferred direction, thus the need to find appropriate methods to detect departures from a uniform distribution. First, we discuss some uniformity tests for circular data. Then, a likelihood ratio test (LRT) is proposed using standard statistical theory. The cardioid distribution seems particularly adequate as an alternative hypothesis for the LRT when data show a smooth transition in a unimodal preferred direction. Second, when the data are not uniformly distributed, we apply different circular regression methods to devise the patterns of dependence on some independent variables. When the distributional assumptions for parametric regression analysis are violated, a bootstrap method is proposed to test the regression coefficients. Finally, we have applied the uniformity tests and the circular regression methods to analyze wind directional data. Numerical results are summarized in tables for comparison purposes. We observe consistent results that the wind direction is not uniformly distributed via the uniformity tests. For the regression methods, we notice that wind speed is not significant to predict direction while the pattern of wind direction depends on the circular variable time of day.Item Connections in High School Mathematics Textbooks that Support Understanding of Key Features of Graphs of Functions(2016-07) Malachite, Anna M.; Dogbey, JamesThis study aims to evaluate a particular high school mathematics series of textbooks for its potential to adequately support teachers in emphasizing function-related connections. The investigation was limited to topics surrounding analysis of graphs of functions, specifically, domain and range; minima, maxima, and vertex; slope and increasing or decreasing intervals; and, intercepts and zeros. The chosen textbooks are McGraw-Hill’s Algebra 1, Algebra 2, and Precalculus textbooks with a copyright date of 2016. Each textbook was reviewed in sequential order. Items relating to the topics of interest were placed in a connection category and evaluated for nine different qualities. Data collected about the types of connections and the quality of those items reveals that the chosen textbooks have numerous connections among representations, but fewer connections to the real-world and across disciplines. Furthermore, a greater number of contextual items are included in Algebra 1 and Algebra 2 than in Precalculus. Future investigations may aim to compare the series evaluated in this study to another series of mathematics textbooks. Additionally, teacher interactions with the textbook might provide more useful insights.Item The Development of the Concept of Rates of Change and its Impact on Students' Understanding of Functions(2016-07) Bannerjee, PragatiThe focus of the study was to identify similarities and differences of grade seven through grade twelve students' understanding of rates of change and functions while keeping in mind that upperclassmen study participants will have more mature mathematical thinking in comparison to the participants from middle school. In addition, the study explored how students' understandings and difficulties relate to their teachers' expectations for prerequisite knowledge, and expected outcomes for students at each grade level. In contrast to the prior studies, the current study provides a comprehensive picture of progression in students' understanding of rates of change (or slope) by the use of the study instrument that included focused problems on mostly all unique conceptualizations of slope. A cross-sectional study was conducted in a public high school in South-Texas with 187 Grades 7- 12 students enrolled in Math-7 to Calculus (AB), and 8 teachers. All student participants completed the Diagnostic Test on various representations of Rates of Change (DTRC) in physical and functional situations. Fourteen students and six teachers were interviewed. To analyze student performance by grade, and by current math course, Welch's Test for Analysis of Variance including Games-Howell Simultaneous Tests for Difference of Means (One-way ANOVA) was employed besides computing descriptive statistics. The student performance on DTRC increased very little in both cases, across grades and by math course. Results from DTRC data and students' interviews showed that students had difficulty computing unit rate in contextual problems, and when required to compare and extract information from multiple representations within the same problem. A large proportion of study participants including Grades 11, and Grades 12 had difficulty in estimating average rate of change from a data table or when presented with a graph of a non-linear function. There was a lack of evidence in the students' work to explore the rate of change of a dependent variable with respect to an independent variable. The data from the interview points towards a need for enhancing teacher horizon knowledge on how slope is connected to the ideas of unit rate, average rate of change, and to the instantaneous rate of change in functional situations. The same instrument and procedures may be used with a greater cross-section of the general population to have more variability in participant's socio-economic status to allow enhancement in interpretation and generalization of the findings of the current study.Item Estimation and comparison of the image noise levels via subsampling(2019-08) Le, Dung Anh; Jin, Lei; Sterba-Boatwright, Blair; Huang, YuxiaAs the amount of digital data has increased critically in the last decade, image data has become more and more important. Noise is a random signal which always present in an image during image acquisition, coding and, transmission. Image noise leads to pixels representing incorrectly the color or the exposure of the scene. The noise level is an important component for measuring the quality of an image. In the literature, very little work has been done in statistical inference for image noise. Most existing methods deal with the point estimation of the noise level, but the sampling distribution of these point estimates are unknown in general. In this project, we propose sub-sampling methods for image data to approximate the sampling distribution for the point estimates. Also, we develop some methods to compare the noise level of two images. First, we review different models of image noise including independent noise, dependent noise and bivariate noise models. Usually the probability models for image noise are not simple and the variance estimates are complicated. The estimates themselves require sampling distributions for statistical inference. Second, we approximate the sampling distributions via subsampling methods. In statistics, bootstrap and resampling methods are widely used to construct the sampling distributions and estimate the variances of different statistics. Here, we generalize resampling methods for one dimensional data to deal with two dimensional images. From these, confidence intervals for the noise level are constructed. Also, some methods for comparing the image variance in paired images are evaluated for different types of noise such as independent noise and dependent bivariate Gaussian noise. The results of the estimation noise level and hypothesis tests on variance comparison are provided. It seems that the proposed subsampling methods provide reasonable results while both the F-test and the Pitman t-test may not work well.Item Existence of a unique solution to a system of equations modeling compressible fluid flow with capillary stress effects(2018-05) Cosper, Lane; Denny, Diane; Palaniappan, Devanayagam; Sadovski, Alexey L.The purpose of this thesis is to prove the existence of a unique solution to a system of partial differential equations which models the flow of a compressible barotropic fluid under periodic boundary conditions. The equations come from modifying the compressible Navier-Stokes equations. The proof utilizes the method of successive approximations. We will define an iteration scheme based on solving a linearized version of the equations. Then convergence of the sequence of approximate solutions to a unique solution of the nonlinear system will be proven. The main new result of this thesis is that the density data is at a given point in the spatial domain over a time interval instead of an initial density over the entire spatial domain. Further applications of the mathematical model are fluid flow problems where the data such as concentration of a solute or temperature of the fluid is known at a given point. Future research could use boundary conditions which are not periodic.Item A feed-forward neural network framework for the solutions of linear and nonlinear ordinary differential equations(2022-05) Venkatachalapathy, Pavithra; Mallikarjunaiah, S. M.; Palaniappan, Devanayagam; Vasilyeva, M.Ordinary Differential Equations (ODEs) play a key role in describing the physical, chemical, and biological processes. The methods for obtaining the solutions to such differential equations are widely studied topic among scientific community. Certain simplified ODEs are tractable by well known analytical techniques while many other demand sophisticated numerical methods. In this thesis we propose a method for solving ordinary differential equations using a framework of Ar tificial Neural Networks (ANN). The unsupervised type of feed-forward ANN is used to find the approximate numerical solutions to the given ODEs up to the desired accuracy. The mean squared loss function is the sum of two terms: the first term satisfies the differential equation; the second term satisfies the initial or boundary conditions. The total loss function is minimized by using general type of quasi-Newton optimization methods to get the desired network output. The approximation capability of the proposed method is verified for varieties of initial or boundary value problems, including linear, nonlinear, singular second-order ODEs, and a system of cou pled nonlinear ODEs with Dirichlet, Neumann and mixed type boundary conditions. Point-wise comparison of our approximations shows strong agreement with available exact solutions and/or Runge-Kutta based numerical solutions. We remark that our proposed algorithm minimizes the learnable network parameters in a given initial or boundary value problems. We believe that the method developed in this thesis can be applied to approximate the solutions to partial differential equations on complex domains.Item Fuzzy ranking from fuzzy pairwise comparisons with applications(2015-08) Wang, YunfengOne method of ranking items is to score all of them against a standard scale. Sometimes it is difficult to create or use such a scale. As an alternative, it is possible to make side-by-side comparisons of some or all of the pairs of items. Then the problem is to convert the collective pairwise comparisons into a ranking. This problem has been studied previously in many contexts. This thesis addresses several methods, including where fuzzy comparisons are made for some, but not all, of the pairs. The Colley method and PageRank algorithm both use pairwise comparisons for some pairs to rank all items in a set. This thesis shows how those pairwise comparisons can be fuzzy. It also shows how Saaty's method for ranking alternatives can be completed when not all comparisons are used.Item A hemodynamic modelling of the blood circulation(2016-12) Shrestha, Maha LaxmiA mathematical model of the blood flow through an axisymmetric stenosis vessel was developed using the Navier-Stokes equations. A numerical approach has been used to analyze behavior of the blood flow because an analytical solution of such a problem is impossible. The method of finite elements was applied to find the velocity and the pressure of the blood flow in stenotic vessels of the cardio-vascular system. The solution of the Navier-Stokes equations was done using Matlab and COSMOL Multiphysics. The findings of the modeling demonstrated that velocity and wall shear stress significantly increase due to the stenosis in the part of the vessel that is blocking blood flow and that this increase depends on the size of the blockage. The modeling shows that increased velocity, vessel wall shear stress, and significant variations in blood pressure may lead to abnormality of the blood flow, which in turn may be a cause of the heart attacks or/and strokes. The analysis of this research has been compared with the existing results published in research papers. Future investigations of the effect of a more general boundary for a stenosis are planned.Item How scaffolding rate of change problems promotes positive mathematical transferability(2021-05) Bogan, Jennifer C.; Tintera, George; Postelnicu, Valentina; Dogbey, JamesThis thesis reports on action research conducted by the author while teaching rate of change problems to first-semester Calculus students at a public university in the United States, with the purpose to ensure transfer of learning, and increase students’ conceptual understanding of rate of change and their performance on solving rate of change problems. The action research cycles (planning, action/implementation, evaluation) involved scaffolding to promote positive mathematical transferability through conceptual understanding. The result is a Lesson Guide consisting of a learning task designed using scaffolding and abiding by principles developed and supported by the research literature on the theory of transfer, an assessment task, and a self-evaluation tool for students. Future action research cycles may be informed by the Lesson Guide developed in this study. Calculus I instructors can use the Lesson Guide in the teaching of rate of change problems. This study contributes to the progressive perspective of the transfer of learning in mathematics education and adds to the growing number of theses conducted with action research methodologyItem Inferning networks for graph partitioning(2019-05) Nguyen, Quan; Zimmer, Beate; Porter, Reid; Jin, LeiImage analysis, pattern recognition, and computer vision pose very interesting and challenging problems, especially in this time when billions of images are generated and uploaded every single day. In many computer vision systems, segmentation often is the first step in which it localizes the objects presented in the image by partitioning the image into meaningful segments. More precisely, image segmentation is the task of assigning discrete labels to pixels, where the label represents the segment (or cluster) to which a pixel belongs. We express an image as a graphical model, also known as an affinity graph, whose vertices denote pixels or regions and whose edges denote adjacency. Each edge in the affinity graph has a weight that quantifies the similarity between the adjoining vertices. To partition the graph, we select a threshold and discard all edges with weights below the threshold and then form segments as path-connected regions. We can define an energy function of a segmentation by adding up the weights of all edges between different segments, which is referred to as the correlation clustering energy. Partitioning the graph to obtain a segmentation is seen as inference in a graphical model and can be formulated as the minimization of the energy function over all possible segmentations, an NP-hard problem. We train a deep neural network to produce the affinity graph, whose segmentation minimizes a natural learning objective called the Rand error. For a graph with a ground truth segmentation, the Rand error measures the pairwise misclassification error between a predicted segmentation and the ground truth segmentation over all possible vertex pairs. We describe an efficient mini-batch learning algorithm based on Kruskal’s algorithm and discuss two formulations of the loss function and two graph encoding schemes used in training the neural net. We present a novel concept, namely that during the training process we select the optimal threshold that minimizes the correlation clustering energy function over a restricted set of segmentations given by different thresholds. We present experiments on a synthetic dataset that illustrate that adding this extra inference step to the training of the neural net causes it to learn different affinities that lead to a 5% reduction in the Rand error on a validation set of similar synthetic images.Item Kalman filtering and application to storm surges(2019-08) Slaten, Jesse; Sadovski, Alexey; Chen, Baohua; Jin, LeiOver the last decades with the advancement of computational power and access to data, the ability to create advanced forecasts and hind-casts too has grown. Recently, with an increase of the global population moving closer towards coastal areas. There is a much stronger presence to understand severe weather systems and there impact on the local population as well as the economy. With that there is still much work to be done within the field of weather forecasting specifically in tandem with real time decision making.This project will consider forecasting in the event of extreme weather systems. Precisely we will focus on the role of storm surge and investigating novel techniques in trying to increase accuracy in sea-level prediction models and decreases error associated with tidal gauge systems along the coast of Texas. Under the thought experiment that this will be used for some decision making process. Along with including an optimal warning time derived from the prediction methods.Tidal prediction methods have either been of two differing methods, statistical or deterministic. For typical usage most tidal predictions are given by deterministic methods, i.e. used by solving hydrodynamic equations in tandem with their astronomical constituents. Statistical methods have since been developed with the added benefit that we can include live measurements to improve tidal predictions based off a time series of observations.For high-impact weather systems we do not have the ability to solve the same hydrodynamic equations so quickly and readily as to aid emergency services and workers. Accurate models are desirable not only from a human life standpoint but from an economic standpoint as well. It may be the case that community is not endanger but it may affect local businesses that rely on precise forecasts. In these cases we must insist on the exact time closures must be necessary in order to minimize any economic impact or loss.In these cases we choose to employ a combination of both statistical and deterministic methods. The Kalman filter is one such method of combining these two methods in order to increase the accuracy in model.Item Mathematical results for slipping flows past a cylinder or a sphere embedded in a porous medium(2015-07) Alshehri, NadaFlow past a circular cylinder or a sphere embedded in a porous medium is investigated mathematically by treating porous matrix as an incompressible fluid. Closed form analytic solutions for the fourth order scalar boundary-value problems for Ψ (r, θ) - known as the Stokes stream function - are obtained by using Navier-slip conditions. Our exact results for Ψ (r, θ) capture flow fields prevailing in the vicinity of a cylinder/sphere suspended in a uniform or a shear flow field. All the physical quantities computed from our solutions depend on two key parameters, namely, δ (the effective permeability) and ζ (slip coefficient). Flow separation and velocity overshoot behavior are found for certain values of δ and ζ. The force acting on the cylinder/sphere is calculated in each case. It is observed that the presence of slip decreases the force on the boundary. Our results show that in the limit δ -> 0, there is no solution to the two-dimensional boundary-value problem, confirming the familiar Stokes paradoxItem Mathematics of microorganism swimming in micropolar fluids(2021-12) Harinarasimha Prasad, Shankar Narasimha; Palaniappan, Devanayagam; Chen, Lea-Der; Mallikarjunaiah, S. M.Typical representation of the swimming motion of microorganisms in fluid environments model the microorganisms as spherical squirmers in a viscous fluid (Newtonian Fluid) with prescribed surface velocities on the squirmer surface. There are numerous fluids in nature that deviate from the classical Newtonian fluid, such as human and animal blood. Modeling swimming mechanisms in these non-classical fluid settings can be very useful but are mathematically challenging. In this thesis, we develop and analyze a mathematical model for the swimming of microorganisms in micropolar fluids - the fluids that depart from the classical Newtonian fluid due to the microrotational effect. Specifically, micropolar fluid continuum equations involve both the velocity and internal spin vector fields resulting in antisymmetric and couple stresses. The mathematical problem of swimming in micropolar fluids is analyzed via a spherical squirmer model in the absence of inertial effects and assuming steady motion. The idealized configuration allows exact analytical solutions for the velocity and spin fields surrounding the squirmer via Stokes’s stream function formulation. Effects of normal and tangential modes induced on the surface of the squirmer are explained for the two-Mode squirmer. Closed-form expressions for the physical quantities involving the n-Mode general case are also reported. Our exact solutions to the boundary value problem (BVP) for the sixth-order partial differential equation (PDE) contain previously derived results for Stokes and Brinkman fluid squirmer models. It is observed that the propulsion speed, calculated using the force-free condition, depends on the first surface velocity mode only. Surprisingly, the swimming speed in a micropolar fluid is the same as that of the spherical microorganism swimming velocity in Newtonian (Stokes) fluids. The power dissipation and swimming efficiency results derived using non-zero spin boundary conditions on the squirmer surface, however, reveal the microrotational effects due to the inclusion of higher surface velocity modes. The two-mode analytical results are further utilized to inspect the structure of flow fields surrounding the spherical. Our exact mathematical results presented herein may be of interest in understanding microorganisms swimming mechanisms in fluids that exhibit angular momentum due to internal micro-rotation.Item Mispecification bootstrap tests of the capital asset pricing model(2017-12) Bo, Nhieu; Jin, Lei; Friday, H. Swint; Sterba-Boatwright, BlairThe development of the Capital Asset Pricing Model (CAPM) marks the birth of asset pricing framework in finance. The CAPM is a simple and powerful tool to describe the linear relationship between risk and expected return. According to the CAPM, all pricing errors should be jointly equal to zero. Many empirical studies were conducted to test the validity of the model in various stock markets. Traditional methods such as Black, Jensen, and Scholes (1972), Fama and MacBeth (1973) and cross-sectional regression have some limitations and encounter difficulties because they often involve estimation of the covariance matrix between all estimated price errors. It becomes even more difficult when the number of assets becomes larger. Our research is motivated by the objective to overcome the limitations of the traditional methods. In this study, we propose to use bootstrap methods which can capture the characteristics of the original data without any covariance estimation. The principle philosophy of bootstrap procedures is to treat the data sample as the population to draw bootstrap re-samples. The bootstrap methods comprise two general steps. First, we use historical monthly returns to estimate the parameters using both ordinary least square and the Cochrane-Orcutt method. Next, we implement model-based procedures to generate bootstrap samples. Following the idea of the block bootstrap, we consider all assets at a point in time as one block under different bootstrap schemes to capture the dependence structure between different assets. With the assumption of no serial correlation in the CAPM, we conduct the independent bootstrap over time scale. Furthermore, we introduce the block bootstrap with blocks over time to capture the temporal dependence. The bootstrap tests were applied to the CAPM in the US and Vietnam (VN) stock markets, providing some interesting results.Item Network television dynamics: a conceptual mathematical model(2015-07) Maceyko, Aimee EIn this thesis we will be expanding on modifications we previously made relating to Edwards and Buckmire’s model of box office dynamics to network television. We will introduce the number of viewers with a negative reaction to the product as a function with respect to time. The interaction of the main parameters of the box office dynamics as they translate to the network television problem: viewership, revenue, and audience perception for television programming will be presented as a conceptual model of a system of three 1st order differential equations. The eigenvalue method, Routh-Hurwitz stability criterion, and control theory will be used to solve the problem and the stability of the solution will be checked. Finally, based on the numerical solution and its stability, recommendations will be presented.