In this thesis, we give a topological interpretation of knot contact homology, by considering intersections of a particular class of chains of open strings with the knot itself. In doing so, we provide evidence toward a differential graded algebra structure on the algebra generated by chains of open strings.

Fix an integer d > 0. In 2008, Chantal David and Tom Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This dissertation is an extension of that work, investigating the frequency with which a principally polarized abelian surface A over Q with real multiplication by Q adjoin a squared-root of 5 has a nontrivial p-torsion point defined over K. Averaging by height, the main result shows that A picks up a nontrivial p-torsion point over K for only finitely many p. The proof of our main theorem primarily rests on three lemmas. The first lemma uses the reduction-exact sequence of an abelian survace defined over an unramified extension K of Qp to give a mod p2 condition for detecting when A has a nontrival p-torsion point defined over K. The second lemma employs crystalline Dieudonne theory to count the number of isomorphism classes of lifts of abelian surfaces over Fp to Z/pp that satisfy the conditions from our first lemma. Finally, the third lemma addresses the issue of the assumption in the first lemma that K is an unramified extension of Qp. Specifically, it shows that if A has a nontrival p-torsion point over a ramified extension K of Qp and p - 1 > d then this p-torsion point is actually defined over the maximal unramified subextension of K. We then combine these algebraic results to reduce the main analytic calculation toa series of straightforward estimates.

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A reliability-based optimization (RBO) methodology was developed and applied to fiber-reinforced polymer (FRP) bridge decks. Commercially available software was used to optimize a FRP bridge deck panel by weight with structural reliability, stress, and deflection constraints. A methodology using optimization software, finite element analysis, and probabilistic analysis software was developed to examine the effects of load and resistance uncertainties in FRP bridge deck optimization. Eight modular deck designs were considered for use in the RBO methodology. Investigations into random variable sensitivities, design variable sensitivities, wheel positions, and buckling were conducted to minimize computational effort. Five models were eventually optimized with deterministic methods and the RBO methodology. Ply thicknesses were treated as design variables. Material parameters, design variables, and load were taken as random variables in the reliability calculations. A comparison of RBO designs was made with the best candidate chosen based on deck panel weight.

<p>p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Times} span.s1 {font: 11.5px Helvetica}</p> <p>This paper consists of mainly two parts. First it is a survey of some results on automorphisms of Riemann surfaces and Fuchsian groups. A theorem of Hurwitz states that the maximal automorphism group of a compact Riemann surface of genus g has order at most 84(g-1). It is well-known that the Klein quartic is the unique genus 3 curve that attains the Hurwitz bound. We will show in the second part of the paper that, in fact, the Klein curve is the unique non-singular curve in ℂP² that attains the Hurwitz bound. The last section concerns automorphisms of surfaces with cusps or punctured surfaces.</p> / Master of Science (MS)

<p>Students entering university for the first time face many new challenges and obstacles, which often include first year calculus. Many students come to university liking mathematics and hoping to study in a math intensive field, but encounter a roadblock when they stalt taking university-level calculus. Why do these first year mathematics courses cause problems? What challenges do the students face, and what can be done to help them overcome these obstacles and achieve their goals? In this study we set out to try to answer these questions by interviewing 16 students about their experiences taking first year calculus at McMaster University. Here we discuss our findings regarding the students' study habits, problem solving techniques and views of mathematics, as well as a number of other factors which could have caused them problems. We will also discuss how the views of the students are not always in alignment with those of the faculty and staff, as well as possible ways in which the teaching of calculus could be improved at McMaster University.</p> / Master of Science (MS)

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