QSpace Community: Queen's UniversityInformation
http://hdl.handle.net/1974/6
Queen's UniversityInformationFri, 29 May 2015 14:22:04 GMT2015-05-29T14:22:04ZThe Channel Imagehttps://qspace.library.queensu.ca:443/jspui/retrieve/122/Jeffery_hall.jpg
http://hdl.handle.net/1974/6
Examining 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/1974/130452015-05-01T04:00:00ZAsymptotic Liberation in Free Probability
http://hdl.handle.net/1974/12711
Title: 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.Mon, 26 Jan 2015 05:00:00 GMThttp://hdl.handle.net/1974/127112015-01-26T05:00:00ZThe evolution of altruistic behaviour in homogeneous deme-structured populations
http://hdl.handle.net/1974/12710
Title: The evolution of altruistic behaviour in homogeneous deme-structured populations
Authors: Cabral, MichaelMon, 26 Jan 2015 05:00:00 GMThttp://hdl.handle.net/1974/127102015-01-26T05:00:00Z