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The primary concerns of mathematics educators are learning and teaching mathematics. It is, therefore, natural to ask "what implications and benefits might there be if learning were perceived as a risk-taking event?" (Atkinson, 1957, p. 266). The underlying motivation of this study is to analyze the risks students take in the mathematics classroom and how risk influences student creation of meaning and development of understanding. I define risk in the mathematics classroom to be any observable act that entails uncertain outcome. The research presented here focuses on a table of four students: Andrew, Carina, Kam, and Mark as they grapple with the mathematical uncertainties inherent in the Ticket Line Task. In analyzing student work and development of mathematical understanding, I identify risks that students take and the benefits they claim result from doing so. Contextualized Risk Theory (CRT) is introduced to improve our understanding of the risks students take in learning mathematics in a student-centered classroom where students exercise personal agency in mathematical problem solving. Findings include characterization of risks these students took, significant student mathematical activity, student enjoyment of their work, student development of personal understanding of purposes and meanings of specific mathematics, and students achieving mathematical success as defined by the researcher and the participants.

Mathematics students and teachers are familiar with the difficulty of learning and teaching concepts of continuity and limits. Research has expanded our knowledge of how students think about these concepts, including different conceptions and metaphors students use to reason about continuity and limits at a point. From the literature I have identified four potentially problematic conceptions (PPCs) students may use when reasoning about limit and continuity at a point. Questionnaires were administered to 861 BYU students in various mathematics courses to determine how prevalent and persistent the PPCs are among the students in each course. Interviews were conducted with nine first semester calculus to get an idea of how students reason about continuity and limit at a point and how that influences whether they use the PPCs. Students showed evidence of holding the four PPCs with a decrease in these conceptions typically after they took a course in analysis. Participants also did not understand the Formal definition of a Limit until they took a course in Analysis. Students were able to reason appropriately using many different conceptions of continuity. Considering limit conceptions, students using a Dynamic conception of Limit tended to be better able to reason about continuity and limit at a point. Students who did not use a Dynamic conception of limit tended to use the PPCs in general and incorrectly more often.

Recently, there has been an increasing interest in high school mathematics education, especially in the teaching and learning of calculus. For example, studies conducted by Bressoud (2010); Judson and Nishimori (2005); Koh and Divaharan (2011); and St. Jarre (2008) all looked at how to improve the understanding of calculus students and what roles the educator must take to ensure that their students are successful. The purpose of this study was to determine if there was a significant difference between instruction using computer algebra system (CAS) compared to instruction using the graphing calculator in high school calculus on students’ conceptual and procedural understanding. This study explored and compared two different types of instruction based on the use of two different types of technology, CAS and graphing calculator. The total population for this study consisted of 333 students. There were 187 students classified as using the graphing calculator and 146 students classified as using CAS. The data for this study were collected from four Advanced Placement (AP) calculus AB courses from high schools in Florida. The study used observations and two sets of calculus tasks in order to gather data. The research questions for this study looked at comparing the grades of students categorized based on the type of instruction received during the learning of calculus. The statistical procedure that was used was a simple oneway analysis of variance (ANOVA). The results indicated that there was no significant difference between the two types of instruction on the students’ procedural knowledge, iii however, there was statistical significance on the students’ conceptual understanding in favor of the CAS students. The study introduces a framework on how to obtain information about the effects of different types of instruction on students’ understanding of calculus. The results of this study contribute in assisting teachers and future researchers on how to analyze student work in order to obtain information about the students’ conceptual and procedural understanding of first semester calculus.

In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.

In the first part of the dissertation, we construct two-dimensional TQFTs which categorify the evaluations of circles in Kuperberg’s 𝐵₂ spider. We give a purely combinatorial evaluation formula for these TQFTs and show that it is compatible with the trace map on the corresponding commutative Frobenius algebras. Furthermore, we develop a theory of Θ-foams and their combinatorial evaluations to lift the ungraded evaluation of the Θ-web, thus paving a way for categorifying 𝐵₂ webs to 𝐵₂ foams. In the second part of the dissertation, we study the calculus of unoriented 𝔰𝔩₃ webs and foams. We focus on webs with a small number of boundary points. We obtain reducible collections and consider bilinear forms on these collections given by pairings of webs. We give web categories stable under the action of certain endofunctors and derive relations between compositions of these endofunctors.

Individanpassad marknadsföring blir ett allt mer kraftfullt verktyg för företag. Den fortsatta framväxten av digitala medier leder till en ökad insamling av konsumenters persondata som används för att skräddarsy individuella erbjudanden. Företagens användande av individanpassad marknadsföring ger dock upphov till problematik gällande konsumentens integritet. Marknadsförare måste beakta flertalet faktorer för att undvika en integritetskränkande respons hos konsumenterna och istället frambringa marknadsföring som upplevs gynnande. Genom fokusgruppsintervjuer ämnar denna studie att undersöka diverse faktorer som påverkar hur konsumenter upplever individanpassad marknadsföring. Med teoretisk grund rotad i privacy paradox och privacy calculus, undersöker denna studie faktorerna som påverkar konsumenters inställning ur ett integritetsperspektiv. Resultaten belyser hur en majoritet av intervjudeltagarna anser att individanpassad marknadsföring medför vissa gynnande aspekter, dock i ganska låg utsträckning, samt hur allt för personlig och direkt marknadsföring upplevs integritetskränkande i större grad.

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