QSpace Community: Queen's UniversityInformation
http://hdl.handle.net/1974/6
Queen's UniversityInformationFri, 03 Jul 2015 22:06:09 GMT2015-07-03T22:06:09ZThe Channel Imagehttps://qspace.library.queensu.ca:443/retrieve/122/Jeffery_hall.jpg
http://hdl.handle.net/1974/6
Linearization and Stability of Nonholonomic Mechanical Systems
http://hdl.handle.net/1974/13149
Title: Linearization and Stability of Nonholonomic Mechanical Systems
Authors: Yang, Steven
Abstract: The stability of an equilibrium point of a nonlinear system is typically analyzed in two ways: (1) stability of its linearization, and (2) Lyapunov stability. An unconstrained simple mechanical system is a type of nonlinear system with a special structure, and so the methods for stability analysis can be specialized for this particular class of nonlinear systems. For a simple mechanical system subject to velocity constraints, the situation becomes more complicated. If the constraints are holonomic, then the problem can simply be reduced to that of an unconstrained simple mechanical system by restricting analysis to a certain submanifold of the configuration space. If the constraints are nonholonomic, this approach cannot be taken. In this report we study the differences and additional complexities that arise in these nonholonomic mechanical systems, and derive results with regards to linearization and stability of its equilibria.Tue, 23 Jun 2015 04:00:00 GMThttp://hdl.handle.net/1974/131492015-06-23T04:00:00ZOptimal Quantization and Approximation in Source Coding and Stochastic Control
http://hdl.handle.net/1974/13147
Title: Optimal Quantization and Approximation in Source Coding and Stochastic Control
Authors: Saldi, NACI
Abstract: This thesis deals with non-standard optimal quantization and approximation problems in source coding and stochastic control.
The first part of the thesis considers randomized quantization. Adapted from stochastic control, a general representation of randomized quantizers that is probabilistically equivalent to common models in the literature is proposed via mixtures of joint probability measures induced by deterministic quantizers. Using this general model, we prove the existence of an optimal randomized quantizer for the generalized distribution preserving quantization problem. A Shannon theoretic version of this source coding problem is also considered, in which an optimal (minimum distortion) coding of stationary and memoryless source is studied under the requirement that the quantizer's output distribution also be stationary and memoryless possibly different than source distribution. We provide a characterization of the achievable rate region where the rate region includes both the coding rate and the rate of common randomness shared between the encoder and the decoder.
In the second part of the thesis, we consider the quantization problems in stochastic control from viewpoints of information transmission and computation. The first problem studies the finite-action approximation (via quantization of the action space) of deterministic stationary policies of a discrete time Markov decision process (MDP), while the second problem considers finite-state approximations (via quantization of the state space) of discrete time Markov decision process. Under certain continuity conditions on the components of the MDP, we establish that optimal policies for the finite models can approximate with arbitrary precision optimal deterministic stationary policies for the original MDP. Combining these results leads to a constructive scheme for obtaining near optimal solutions via well known algorithms developed for finite state/action MDPs. For both problems, we also obtain explicit bounds on the approximation error in terms of the number of representation points in the quantizer, under further conditions.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-06-19 12:20:57.086Tue, 23 Jun 2015 04:00:00 GMThttp://hdl.handle.net/1974/131472015-06-23T04:00:00ZExamining the Probability that the Number of Points on an Elliptic Curve over a Finite Field is Prime
http://hdl.handle.net/1974/13089
Title: Examining the Probability that the Number of Points on an Elliptic Curve over a Finite Field is Prime
Authors: Wheeler, Emilie
Abstract: This project examines the work in the article "The Probability
that the Number of Points on an Elliptic Curve over a Finite
Field is Prime", in which authors Galbraith and McKee ask the
question `What is the probability that a randomly chosen elliptic
curve over Fp has kq points, where k is small and q is prime?' I
performed my own computations and will compare them to their
results.Wed, 27 May 2015 04:00:00 GMThttp://hdl.handle.net/1974/130892015-05-27T04:00:00ZMathematics Problems and Thinking Mathematically in Undergraduate Mathematics
http://hdl.handle.net/1974/13045
Title: Mathematics Problems and Thinking Mathematically in Undergraduate Mathematics
Authors: Matthews, Asia R
Abstract: Mathematics is much more than a formal system of procedures and formulae; it is also a way of thinking built on creativity, precision, reasoning, and representation. I present a model for framing the process of doing mathematics within a constructivist ideology, and I discuss two fundamental parts to this process: mathematical thinking and the design of undergraduate mathematics problems. I highlight the mathematical content and the structuredness of the problem statement and I explain why the initial work of re-formulating an ill-structured problem is especially important in learning mathematics as a mental activity. Furthermore, I propose three fundamental processes of mathematical thinking: Discovery (acts of creation), Structuring (acts of arranging), and Justification (acts of reflection). In the empirical portion of the study, pairs of university students, initially characterized by certain affective variables, were observed working on carefully constructed problems. Their physical and verbal actions, considered as proxies of their mental processes, were recorded and analyzed using a combination of qualitative and quantitative measurement. The results of this research indicate that ill-structured problems provide opportunities for a concentration of Discovery and Structuring. Though all of the identified processes of mathematical thinking were observed, students who are highly metacognitive appear to engage in more frequent and advanced mathematical thinking than their less metacognitive peers. This study highlights pedagogical opportunities, for both highly metacognitive students as well as for those who demonstrate fewer metacognitive actions, arising from the activity of doing ill-structured problems. The implications of this work are both theoretical, providing insight into the relationship between metacognition and student “performance,” and practical, by providing a simple tool for identifying processes of mathematical thinking.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-04-30 11:28:32.416Fri, 01 May 2015 04:00:00 GMThttp://hdl.handle.net/1974/130452015-05-01T04:00:00Z