QSpace Community: Queen's UniversityInformationQueen's UniversityInformationhttp://hdl.handle.net/1974/62015-05-22T17:34:25Z2015-05-22T17:34:25ZMathematics Problems and Thinking Mathematically in Undergraduate MathematicsMatthews, Asia Rhttp://hdl.handle.net/1974/130452015-05-01T15:20:53Z2015-05-01T04:00:00ZTitle: 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 11:28:32.4162015-05-01T04:00:00ZAsymptotic Liberation in Free ProbabilityVazquez Becerra, Josue Danielhttp://hdl.handle.net/1974/127112015-01-28T06:16:55Z2015-01-26T05:00:00ZTitle: Asymptotic Liberation in Free Probability
Authors: Vazquez Becerra, Josue Daniel
Abstract: Recently G. Anderson and B. Farrel presented the notion of asymptotic liberation on sequences of families of random unitary matrices. They showed that asymptotically liberating sequences of families of random unitary matrices, when used for conjugation, delivers asymptotic freeness, a fundamental concept in free probability theory. Furthermore, applying the Fibonacci-Whitle inequality together with some combinatorial manipulations, they established sufficient conditions on a sequence of families of random unitary matrices in order to be asymptotically liberating.
On the other hand, a theorem by J. Mingo and R. Speicher states that given a graph
$G=(E,V,s,r)$ there exists an optimal rational number $ \mathfrak{r}_{G}$,
depending only on the structure of $G$, such that for any collection of
$n\times n$ complex matrices
$\{ A_{e}=( A_{e} (i_{s(e)},i_{r(e)}) ) \mid e\in E \}$ we have%
\begin{equation*}
\left\vert
\sum_{ i_{v_{1}},\ldots i_{v_{m}} =1 }^{n}
\left( \prod_{e\in E} A_{e} (i_{s(e)},i_{r(e)}) \right)
\right\vert
\leq
n^{\mathfrak{r}_{G}}\prod\limits_{e\in E}\left\Vert A_{e}\right\Vert
\end{equation*}
where $V=\left\{ v_{1},\ldots,v_{m}\right\} $ and $\left\Vert \cdot
\right\Vert $ denotes the operator norm.
In this report we show how to use the latter inequality to prove the same result as G. Anderson and B. Farrell
regarding sufficient conditions for asymptotic liberation.2015-01-26T05:00:00ZThe evolution of altruistic behaviour in homogeneous deme-structured populationsCabral, Michaelhttp://hdl.handle.net/1974/127102015-01-28T06:21:53Z2015-01-26T05:00:00ZTitle: The evolution of altruistic behaviour in homogeneous deme-structured populations
Authors: Cabral, Michael2015-01-26T05:00:00ZOptimal Binary Signaling for Correlated Sources over the Orthogonal Gaussian Multiple-Access ChannelMitchell, Tysonhttp://hdl.handle.net/1974/126372014-12-04T06:10:59Z2014-12-03T05:00:00ZTitle: Optimal Binary Signaling for Correlated Sources over the Orthogonal Gaussian Multiple-Access Channel
Authors: Mitchell, Tyson
Abstract: Optimal binary communication, in the sense of minimizing symbol error rate, with nonequal probabilities has been derived in [1] under various signalling configurations for the single-user case with a given average energy E. This work extends a subset of the results in [1] to a two-user orthogonal multiple access Gaussian channel (OMAGC) transmitting a pair of correlated sources, where the modulators use a single phase or basis function and have given average energies E1 and E2, respectively. These binary modulation schemes fall in one of two categories: (1) transmission sig- nals are both nonnegative, or (2) one transmission signal is positive and the other negative. To optimize the energy allocations for the transmitters in the two-user OMAGC, the maximum a posteriori detection rule, probability of error, and union error bound are derived. The optimal energy allocations are determined numerically and analytically. Both results show that the optimal energy allocations coincide with corresponding results from [1]. It is demonstrated in Chapter 3 that three parameters are needed to describe the source. The optimized OMAGC is compared to three other schemes with varying knowledge about the source statistics, which influence the optimal energy allocation. A gain of at least 0.73 dB is achieved when E1 = E2 or 2E1 =E2. When E1 ≫ E2 again of at least 7 dB is observed.2014-12-03T05:00:00Z