This thesis develops an algorithm for the Schubert calculus of the Grassmanian. Specifically, we state a puzzle-based, synthesis algorithm for a triple intersection of Schubert varieties. Our algorithm is a reformulation of the synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace their combinatorial approach, based on specialized Lebesgue measures, with an approach based on the puzzles of Knutson, Tao and Woodward. The use of puzzles in our algorithm is beneficial for several reasons, foremost among them being the larger body of work exploiting puzzles. To understand the algorithm, we study the necessary Schubert calculus of the Grassmanian to define synthesis. We also discuss the puzzle-based Littlewood-Richardson rule, which connects puzzles to triple intersections of Schubert varieties. We also survey three combinatorial objects related to puzzles in which we include a puzzle-based construction, by King, Tollu, and Toumazet, of the well known Horn inequalities.

This study explored the development of student’s conceptual understanding of limit and derivative when specific computational tools were utilized. Fourteen students from a secondary Advanced Placement Calculus AB course explored the limit and derivative concepts from calculus using computational tools in the Maple computer algebra system. Students worked in pairs utilizing the pair-programming collaborative model. Four groups of student pairs constructed computational tools and used them to explore the limit and derivative concepts. The remaining four student pairs were provided similar tools and asked to perform identical explorations. A multiple embedded case design was utilized to explore ways students in two classes, a programming class P and a non-programming class N, constructed understandings focusing upon their interactions with each other and with the computational tools. The Action-Process-Object-Schema (APOS) conceptual model and Constructionist framework guided design and construction of the tools, outlined developmental goals and milestones, and provided interpretive context for analysis. Results provided insights into the effective design and use of computational tools in fostering conceptual understanding. The study found the additional burden of programming redirected students’ attention away from the intended conceptual understandings. The study additionally found, however, that pre-constructed tools effectively promote conceptual understanding of the limit concept when coupled with a mature conceptual model of development. Four themes influencing development of these understandings emerged: An instructional focus on skills over concepts, the instructional sequence, the willingness and ability of students to adopt and utilize computational tools, and the ways cognitive conflict was mediated.

Since the World Wide Web Consortium (W3C) has recommended XQuery as the standard XML query language, the interest in using existing relational technology to query the XML data has dramatically increased. The most significant challenge of the relational approach is how to fully support XQuery semantics in XQuery-to-SQL translation. To eliminate the implicit semantics of XQuery, an XQuery fragment must be defined with simple syntax and explicit semantics. XQ is proposed as an XQuery fragment to express XML queries. <br /><br /> In this thesis, XQ is intensively investigated. It is encoded by <em>System F</em>, a second-order lambda calculus with a considerable expressive power and a strong normalization property. Since XML data is defined as inductive data types, XML tree and XML forest, in <em>System F</em>, all basic XML operators in XQ have been successfully encoded. Also, the semantics of XQ are represented in <em>System F</em> where XQ's semantics environment is encoded by an <em>Environment</em> data type with the corresponding operators. The successful encoding of XQ by <em>System F</em> ensures the termination of XQ query evaluation. <br /><br /> Moreover, an extension of XQ by a new tree operator Xtree and a vertical Vfor clause is proposed in this thesis to express some <em>undefinable</em> XQ queries. It is demonstrated that this extension still allows XQ to retain its XQ-to-SQL translation property that ensures the polynomial evaluation time complexity, and its <em>System F</em> encodable property that ensures the termination of query evaluation.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

This thesis develops an algorithm for the Schubert calculus of the Grassmanian. Specifically, we state a puzzle-based, synthesis algorithm for a triple intersection of Schubert varieties. Our algorithm is a reformulation of the synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace their combinatorial approach, based on specialized Lebesgue measures, with an approach based on the puzzles of Knutson, Tao and Woodward. The use of puzzles in our algorithm is beneficial for several reasons, foremost among them being the larger body of work exploiting puzzles. To understand the algorithm, we study the necessary Schubert calculus of the Grassmanian to define synthesis. We also discuss the puzzle-based Littlewood-Richardson rule, which connects puzzles to triple intersections of Schubert varieties. We also survey three combinatorial objects related to puzzles in which we include a puzzle-based construction, by King, Tollu, and Toumazet, of the well known Horn inequalities.

Autonomous mobile sensor networks consist of a number of autonomous mobile robots equipped with various sensors and tasked with a common mission. This thesis considers the topology control of such an ad hoc mobile sensor network. In particular, I studied the problem of controlling the size, with respect to a distance metric, of the network for general interactive forcing among agents. Developed is a stability result, allowing one to design force laws to control the spread of the network. Many of the current results assume a known and/or fixed topology of the graph representing the communication between the nodes, i.e. the graph laplacian is assumed constant. They also assume fixed and known force-laws. Hence, the results are limited to time-invariant dynamics. The research considers stability analysis of sensor networks, unconstrained by specific forcing functions or algorithms, and communication topologies. Since the graph topologies are allowed to change as the agents move about, the system dynamics become discontinuous in nature. Filippov’s calculus of differential equations with discontinuous right hand sides is used to formally characterize the multi-agent system with the above attributes. Lyapunov’s Stability Theory, applied to discontinuous systems, is then used to derive bounds on the norm of the system states given bounds on its initial states and input. The above derived stability results lend themselves to the derivation of methods for the design of algorithms or force-laws for mobile sensor networks. The efficacy of the derived results is illustrated through several examples where it is shown how they may be used for synthesizing a topology managing strategy. Examples are given of designing force-laws that limit the network in a desired area.

Thesis (Ph. D.)--University of Queensland, 2002. / Includes bibliographical references.

published_or_final_version / Mathematics / Master / Master of Philosophy

Not available

This dissertation examines the relationship between an instructional style called Interactive-Engagement (IE) and gains on a measure of conceptual knowledge called the Calculus Concept Inventory (CCI). The data comes from two semesters of introductory calculus courses (Fall 2010 and Spring 2011), consisting of a total of 482 students from the first semester and 5 instructors from the second semester. The study involved the construction and development of a videocoding protocol to analyze the type of IE episodes which occurred during classes. The counts of these episodes were then studied along with student gains, measured in a number of different ways. These methods included a traditionally used measure of gain, called normalized gain, which is computed at the instructor level. Additionally, gains were further investigated by constructing hierarchical linear models (HLMs) which allowed us to consider individual student characteristics along with the measures of classroom interactivity. Another framework for computing ability estimates, called Item Response Theory (IRT), was used to compute gains, allowing us to determine whether the method of computing gains affected our conclusions. The initial investigation using instructor-level gain scores indicated that the total number of interactions in a classroom and a particular type of interaction called "encouraging revisions" were significantly associated with normalized gain scores. When individual-level gain scores were considered, however, these instructor-level variables were no longer significantly associated with gains unless a variable indicating whether a student had taken calculus or precalculus in high school or in college was included in the model. When IRT was used to create an alternative measure of gain, the IE variables were not significant predictors of gains, regardless of whether prior mathematics courses were included, suggesting that the method of calculating gain scores is relevant to our findings.