# M.Sc. & Ph.D. Defences

 Upcoming Defence Title: Slope Conditions for Stability of ACIMs of Dynamical Systems Speaker: Mr. Zhengyang Li (Ph.D.) Date: Monday, June 10, 2013 Time: 12:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: (Click here to view)
 Previous Defences Title: Experimenting With Discursive and Non-Discursive Styles of Teaching Absolute Value Inequalities to Mature Students Speaker: Ms. Maria Tutino (M.T.M.) Date: Friday, April 12, 2013 Time: 12:30 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: This research is a follow-up of Sierpinska, Bobos, and Pruncut’s 2011 study, which experimented with three teaching approaches to teaching absolute value inequalities (AVI), visual, procedural, and theoretical, presented over an audio lecture with slides. The study demonstrated that participants treated with the visual approach were more likely to engage in theoretical thinking than those treated with the other two approaches. In the present experiment, two groups of participants enrolled in prerequisite mathematics courses at a large, urban North American University were taught AVI with the visual approach using two different teaching styles: discursive (permitting and actively encouraging teacher-student interactions during the lecture) and non-discursive (not allowing teacher-student interactions during the lecture). In Sierpinska et al.’s study, the non-discursive style was used in all three approaches (the lectures were recorded and the teacher was not present in person). In the present study, a live teacher was lecturing in both treatments. Another difference was that in Sierpinska et al.’s study lectures were delivered individually to each participant, while in the present study, all participants in a group were treated simultaneously. Therefore, in the discursive approach, not only teacher-student but also student-student interactions during the lecture were possible. The aim of this research was to explore the conjecture that the discursive approach is more likely to promote theoretical thinking in students. The group exposed to the discursive approach was, therefore, my experimental group and the other played the role of the control group. The conjecture was not confirmed, but the two approaches seem to have provoked different aspects of theoretical thinking. The experimental group was found to be more reflective, while the control group tended to be more systemic in their thinking. Some striking results, not predicted by Sierpinska et al.’s study, were also found with respect to reflective thinking, definitional thinking, proving behavior, and analytic thinking.
 Title: Some Support Properties for a Class of Ʌ-Fleming-Viot Processes Speaker: Ms. Huili Liu (Ph.D.) Date: Thursday, March 28, 2013 Time: 9:30 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: (Click here to view)
 Title: An Analysis of Students’ Difficulties Learning Group Theory Speaker: Ms. Julie Lewis (M.Sc.) Date: Monday, March 25, 2013 Time: 10:30 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Research in mathematics education and anecdotal data suggest that undergraduate students often find their introductory courses to group theory course particularly difficult.  Research in this area, however, is scarce.  In this thesis, I consider students’ difficulties in their first group theory course and conjecture that they have two distinct sources. The first source of difficulties would pertain to a conceptual understanding of what group theory is and what it studies. The second would relate to the modern abstract formulation of the topics learned in a group theory course and the need to interpret and write meaningful statements in modern algebra. To support this hypothesis, the group concept is explored through a historical perspective which examines the motivations behind developing group theory and its practical uses. Modern algebra is also viewed in a historical context in terms of three defining characteristics of algebra; namely, symbolism, justifications and the study of objects versus relationships. Finally, a pilot study was conducted with 4 students who had recently completed a group theory course and their responses are analyzed in terms of their conceptual understanding of group theory and modern algebra. The analysis supports the hypothesis of the two sources. Based on the results, I propose a remediation strategy and point in the direction of future research.
 Title: Centro-Affine Normal Flows and Their Applications Speaker: Mr. Mohammad Najafi Ivaki (Ph.D.) Date: Monday, December 3, 2012 Time: 11:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: (Click here to view)
 Title: Rankin L-Functions and the Birch and Swinnerton-Dyer Conjecture Speaker: Mr. Reza Sadoughianzadeh (M.Sc.) Date: Tuesday, September 11, 2012 Time: 1:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: (Click here to view)
 Title: Teaching the Singular Value Decomposition of Matrices: A Computational Approach Speaker: Mr. Zoltan Lazar (M.T.M.) Date: Monday, September 10, 2012 Time: 1:15 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: In this thesis, I present a small experiment of teaching the singular value decomposition of matrices using a computational approach. The experiment took place in the summer of 2011 and consisted in two sessions of lectures of four hours each, in the computer lab, on the premises of Concordia University. The same four students (“Carrie”, ”Chal”, ”Desse” and “Nat”) attended both sessions. The underlying methodology was to introduce theoretical results, let participants explore them using mathematical software and then generalize and formalize them. Participants’ responses to test questions were collected and analyzed, and the results are presented in the fourth chapter. The goals of the experiment were to assess the participants’ preparedness for this topic and their level of acceptance of this teaching technique. One of the immediate conclusions is that without a good understanding of the “building blocks” concepts of linear algebra the topic of singular value decomposition of matrices could prove challenging for undergraduate students. The participants showed interest in the teaching method, but mentioned that more time would be required to really benefit from the numerical advantages and from the vast applications of the singular value decomposition.
 Title: Statistical Analysis of Volatility Surfaces Speaker: Mr. Dimitris Lianoudakis (M.Sc.) Date: Friday, September 7, 2012 Time: 12:30 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Option prices can be represented by their corresponding implied volatilities. Implied volatility is dependent on both the strike price and the time to maturity. This dependence creates a mapping known as the implied volatility surface (IVS). The volatility surface is known to practitioners as being synonymous with option prices. These surfaces change dynamically and have distinct features that can be modeled and broken down into a small number of factors. Using time series data of option prices on the S&P500 index, we study the dynamics of the implied volatility surface and deduce a factor model which best represents the surface. We explore the different methods of smoothing the IVS and derive the local volatility function. Using standard dimension reduction techniques and more recent non-linear manifold statistics, we aim to identify and explain these distinct features and show how the surface can be represented by a small number of these prominent factors. A thorough analysis is conducted using principal component analysis (PCA) and common principal component analysis (CPC). We introduce a new form of dimension reduction technique known as principal geodesic analysis (PGA) and give an example. We try to set up a geometric framework for the volatility surface with the aim of applying PGA.
 Title: Galois theory for schemes Speaker: Ms. Shan Gao (M.Sc.) Date: Wednesday, August 29, 2012 Time: 3:40 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: The main object of study of this thesis is the \'etale fundamnetal group of a connected scheme. This profinite group is defined as the automorphism group of a fiber functor defined on the category of finite \'etale covers of our base scheme.
 Title: Pricing Ratchet EIA under Heston’s Stochastic Volatility with Deterministic Interest Speaker: Mr. Dezhao Han (M.Sc.) Date: Wednesday, August 29, 2012 Time: 1:30 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Since its introduction in 1995, Equity-indexed annuities (EIAs) are paid increasing attention. In 2008 it represents 42% of the annuities sold by agents, but after the 2008 financial crisis the number shrank to 25% in 2010. Thus pricing and hedging EIAs is an interesting topic. Researches have been done on pricing different kinds of EIAs, their hedging strategies as well as their optimal stopping strategies. However, the underlying asset is always assumed to follow the geometric Brownian motion that is the Black-Sholes (BS) model. The BS model is plagued by its assumption of constant volatility, while stochastic volatility models become increasingly popular. In this paper we assume the asset price follows Heston’s stochastic volatility model with deterministic interest, and introduce two methods to price the Ratchet EIA. The first method is called JTPDF (joint transition probability density function) method. Given the JTPDF of the asset price and stochastic variance in Heston’s framework, pricing Ratchet EIA is a problem on solving multiple integrals. We solve the multiple integral using Quasi Monte Carlo method and the importance sampling technique. We call the other method CE (conditional expectation) approach. Conditioning on the path of volatility, we first price the Rachet EIA analytically in BS framework. Then the price in Heston’s framework can be evaluated by simulating the path of volatility. Greeks for the Ratchet EIA can also be calculated by the JTPDF or CE methods. At the end, we did some sensitivity tests for Ratchet EIAs’ prices and Greeks. Keywords: stochastic volatility, equity-indexed annuity, high-dimensional integrals, simulating Heston’s stochastic volatility, Greeks of Ratchet EIA
 Title: On the stability of the absolutely continuous invariant measure of certain class of maps with deterministic perturbation Speaker: Mr. Ivo Pendev (M.Sc.) Date: Tuesday, August 28, 2012 Time: 1:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Keller showed the instability of the absolutely continuous invariant measure (acim) for the family of W-shaped maps. This instability is the result of the invariant neighborhood of the fixed turning point at 1/2. The construction of these maps, for which the renowned Lasota-Yorke inequality fails to prove stability (due to the magnitude of the slopes in the limiting map),  has recently been generalized. In the Eslami-Misiurewicz paper, a map was defined, whose third iterate has a fixed turning point at 1/2, raising the question of the stability of the map. The goal of this thesis is to show the stability of this map. We define a family of deterministic perturbation of the map and we express their invariant densities as an infinite sum with the purpose of showing that the normalized invariant densities are uniformly bounded.  This result is used to show the stability of absolutely continuous invariant measure of this transformation.
 Title: Relating modulus and Poincaré inequalities on modified Sierpiński carpets Speaker: Mr. Andrew Fenwick (M.A.) Date: Tuesday, August 28, 2012 Time: 11:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: This thesis investigates the question of whether a doubling metric measure space supports a Poincaré inequality and explains the relationship between the existence of such an inequality and the non-triviality of the respective modulus.  It discusses in detail a general class of modified Sierpiński carpets presented by Mackay, Tyson, and Wildrick [14], which are the first examples of spaces that support Poincaré inequalities for a renormalized Lebesgue measure that are also compact subsets of Euclidean space with empty interior. It describes the intricate relationship between the sequence used in the construction of a modified Sierpiński carpet and the validity of Poincaré inequalities on that space.
 Title: Computations on the Birch and Swinnerton-Dyer conjecture for elliptic curves over pure cubic extensions Speaker: Ms. Céline Maistret (M.Sc..) Date: Monday, August 6, 2012 Time: 10:30 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: The Birch and Swinnerton-Dyer conjecture remains an open problem. In this thesis, we propose to give numerical evidence toward this conjecture when restricted to elliptic curves over pure cubic extensions. We present the general conjecture for elliptic curves over number fields and detail each arithmetic invariants involved. Assuming the conjecture holds, for given elliptic curves E over specific number fields K, we compute the order of the Shafarevich-Tate group of E(K).
 Title: Concentration of Measure and Ricci Curvature Speaker: Mr. Ryan Benty (M.A.) Date: Wednesday, July 25, 2012 Time: 11:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: In 1917, Paul Levy proved his classical isoperimetric inequality on the N-dimensional sphere.  In the 1970's, Mikhail Gromov extended this inequality to all Riemannian manifolds with Ricci curvature bounded below by that of The N-sphere.  Around the same time, the Concentration of Measure phenomenon was being put forth and studied by Vitali Milman.  The relation between Concentration of Measure and Ricci curvature was realized shortly thereafter. Elaborating on several articles, we begin by explicitly presenting a proof of the Concentration of Measure Inequality for the N-sphere as the archetypical space of positive curvature, followed by a complete proof extending this result to all Riemannian manifolds with Ricci curvature bounded below by that of the N-sphere in the process, we present a detailed technical proof of the Gromov-Levy isoperimetric inequality. Following Yann Ollivier, we note and prove a Concentration of Measure inequality on the discrete Hamming cube, and discuss his extension of Ricci curvature to general metric spaces, particularly discrete metric measure spaces.  We show that this “coarse” Ricci curvature on the Hamming cube is positive and present Ollivier's Concentration of Measure inequality for all spaces admitting positive coarse Ricci curvature. In addition, we calculate the coarse Ricci curvature for several discrete metric spaces.
 Title: An upper bound for the average number of amicable pairs Speaker: Mr. James Park (Ph.D.) Date: Wednesday, June 7, 2012 Time: 1:30 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Amicable numbers have been known since Pythagoras and are defined to be two different numbers so related that the sum of the proper divisors of each is equal to the other number. In 2009 Silverman and Stange provided an elliptic curve analogue to amicable numbers. Let E be an elliptic curve over Q. They defined a pair (p, q) of rational primes to be an amicable pair for E if E has good reduction at these primes and the number of points on the reductions Ep and Eq satisfy #Ep(Fp) = q and #Eq(Fq) = p. Let QE(X) denote the number of amicable pairs (p, q) for E/Q with p<= X. Then they conjectured that QE(X) ~ X/(\log X) -2 if E does not have complex multiplication. In this thesis I will provide an upper bound for the average of QE(X) over the family of all elliptic curves which is very close to the conjectural asymptotic of Silverman and Stange.
 Title: Determinant of Pseudo-Laplacians Speaker: Mr. Tayeb Aissiou (Ph.D.) Date: Wednesday, May 30, 2012 Time: 10:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Let X be a compact Riemannian manifold of dimension two or three and let P be a point of X. We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of (symmetric) Laplace operator with domain, consisting of smooth functions with compact supports which do not contain P, to the zeta-regularized determinant of the self-adjoint Laplacian on X.
 Title: Normal Form Analysis of a Mean-Field Inhibitory Neuron Model Speaker: Ms. Loukia Tsakanikas (M.Sc.) Date: Monday, April 23, 2012 Time: 2:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: In neuroscience one of the open problems is the creation of the alpha rhythm detected by the electroencephalogram (EEG). One hypothesis is that the alpha rhythm is created by the inhibitory neurons only. The mesoscopic approach to understand the brain is the most appropriate to mathematically modelize the EEG records of the human scalp. In this thesis we use a local, mean-field potential model restricted to the inhibitory neuron population only to reproduce the alpha rhythm. We perform extensive bifurcation analysis of the system using AUTO. We use Kuznetsov's method that combines the center manifold reduction and normal form theory to analytically compute the normal form coefficients of the model. The bifurcation diagram is largely organized around a codimension 3 degenerate Bogdanov-Takens point. Alpha rhythm oscillations are detected as periodic solutions.
 Title: Secondary School Mathematics Teachers use of Technology through the Lens of Instrumentation Theory Speaker: Mr. Nicolas Boileau (M.T.M.) Date: Wednesday, April 11, 2012 Time: 3:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: The primary goal of our research was to gain a sense of the technology that secondary school mathematics teachers in Montreal, Quebec are currently using, how they are using it, and some of the reasons why they use the technology that they do, in the ways that they do. The secondary goal was to test the effectiveness of our approach in obtaining this information. The approach consisted of interviewing two local secondary school mathematics teachers. The interview questions prodded at what Instrumentation Theory  suggests to be some of the fundamental aspects of one’s interactions with technology; the ‘artifact’ (the particular technology), the subject (the user of that technology), the ‘task’ that the subject tries to complete with the artifact, the ‘instrumented techniques’ that she/he employs to complete the task (which reveal some of their ‘schemes of use’), and the process through which the subject and the artifact interact and ‘shape’ each other, called ‘an instrumental genesis’. The teachers’ responses to the interview questions revealed that, although they both used most of the same technology (with a few exceptions), significant differences existed between the ways that they used some of them, why certain technologies were used, and why others were not. The two teachers also differed in their views on the value of their instrumented techniques. These findings are discussed in light of the literature review, demonstrating some of the effectiveness of our approach. We believe that our approach was useful as it allowed us to elicit detailed descriptions of these two teachers’ uses of technology and because it facilitated the analysis of the data (as the questions were based on the same theoretical framework that was then used to analyze the teachers’ responses). We conclude with some suggestions for future research. One of the suggestions addresses ways in which our approach could be improved to give researchers who might use it in the future more informative responses.
 Title: A Skew-Normal Copula-Driven Generalized Linear Mixed Model for Longitudinal Data Speaker: Mr. Mohamad Elmasri (M.Sc.) Date: Tuesday, April 10, 2012 Time: 1:30 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Arellano-Valle et al. [2005] studied the effect of generalizing the presumed normality structures of a linear mixed model (LMM) random effect and error to a skew-normal distribution, to adequately fit a broader range of longitudinal data. Closed forms of marginal distributions were explicitly indicated along with a maximum likelihood formation. Working with the results found by Arellano-Valle et al.[2005], this paper extends to an even wider set of models for longitudinal data based on the exponential family of distributions. This is achieved by combining a skew-normal copula and a general exponential family distribution, where a generalized linear model (GLM) framework is applied. Some special cases are discussed, in particular, the exponential and gamma distribution. Simulations with multiple link functions are shown. A real data example is also analyzed.
 Title: The Use of Blogs in Ontario Secondary Mathematics Education Speaker: Ms. Christy Lyons (M.T.M.) Date: Monday, April 2 , 2012 Time: 11:45 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Blogging was such a popular activity in the mid-2000s that “blog” was chosen as the word of the year in 2004 by Merriam-Webster.  This thesis examines whether this popularity for blogging transferred to the mathematics classroom; if so, in what ways are mathematics teachers using blogs in relation to their teaching activities and if not, in which ways they are willing to use them.  Previous research shows that teachers use classroom blogs to promote collaboration, to allow conversation with outside experts, as a positive tool for ESL students because of access to online translators, to provide a voice for quiet students who might not speak up in class, to encourage critical thinking skills and as a venue for reflective thinking or metacognition.  Twenty-one Ontario Secondary Mathematics teachers were surveyed to determine their personal, professional and classroom blogging views and habits.  Although only one teacher reported on having operated a classroom blog, teachers showed an interest for blogs and blogging activities (e.g., seventy-six percent were willing to read blogs for professional development and forty-five percent would be willing to operate blogs in their classrooms in the future).  On the whole, interest for blogs and blogging activities seems to grow slowly but firmly. Based on teachers’ responses and previous research, I discuss the possible benefits of blogging-related activities in secondary classrooms.
 Title: Students’ Understanding of Real, Rational, and Irrational Numbers Speaker: Ms. Deidre Maher Arbour (M.T.M.) Date: Friday, March 30, 2012 Time: 1:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: This thesis presents a study of the understanding of real, rational, and irrational numbers by 30 fourth semester college science students in the Montreal region.  The written answers to a set of seven questions were analyzed to determine the students’ interpretations of mathematical signs according to C. S. Pierce’s classifications and to describe their modes of thought according to Vygotsky’s theory of concept development.  From these interpretations, we are able to reconstruct a facsimile of what the students’ concept images are as they pertain to the sets in question.  Finding the concept images to be idiosyncratic and rarely in agreement with what conventional mathematics holds to be true, we examine the way the number systems are approached in school and in the field of mathematics and use this, along with the analyses, to make pedagogical recommendations.
 Title: Motivating Adult Students Taking a Basic Algebra Course in a University Setting Speaker: Ms. Carol Beddard (M.T.M.) Date: Wednesday, March 28, 2012 Time: 2:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Understanding the motivation of students learning mathematics and using this understanding to strengthen motivation can serve to improve mathematics instruction for students, especially students who may dislike math.  Motivation is modeled as arising from an interaction of needs, goals and dimensions of the self, resulting in behavior that is regulated by feedback and external factors. In this study, the motivation of university students in a preparatory algebra class is investigated.  The study uses a Conjecture-Driven design in the teaching situation of MATH 200 – Fundamental Concepts of Algebra which is a required course for many students.  That they have to take a basic algebra course at the university level is indicative of some previous difficulties with mathematics which in turn can be linked to negative affect towards math.  The conjecture was that motivation would be lacking among this group of students but that a class that is taught from a motivational standpoint would result in better attitudes towards math.  Based on an a priori profile of motivational characteristics of the students in the course, the hypothetical student, the course was taught with the aim of improving motivation.  Observations, course evaluations, a questionnaire and a survey were used to:  (1) create a profile of observed motivational characteristics, the realistic student, and (2) to describe the effect of the course on student motivation. It was found that a classroom that addressed students’ needs for autonomy, competence and relatedness, promoted an understanding of why a procedure was used (rather than just how to apply the procedure), and that at all times respected the dignity of students, was motivational.  In this classroom, the students reported improved affect towards mathematics across the dimensions of emotions, attitudes, beliefs and values.
 Title: The Minimizer of Dirichlet Integral Speaker: Mr. Ruomeng Lan (M.Sc.) Date: Wednesday, March 21, 2012 Time: 11:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: In this thesis, we consider the minimizer of the Dirichlet integral, which is used to compute the magnetic energy. We know that the Euler equations describe a motion of an inviscid incompressible fluid. We show that the infimum of the Dirichlet integral, by the action of area-preserving diffeomorphisms, is a stream function corresponding to some velocity field, which is a solution to the stationary Euler equation. According to this result, we study the properties and behaviors of the steady incompressible flow numerically. We utilize three distinct numerical methods to simulate the minimizer of the Dirichlet integral. In all cases the singularity formation was observed. Every hyperbolic critical point of the original function gives rise to a singularity of the minimizer. Title: Heuristic Results for Ratio Conjectures of LE(1, χ) Speaker: Mr. Jungbae Nam (M.Sc.) Date: Thursday, January 12, 2012 Time: 10:30 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: Let LE(s, χ) be the Hasse-Weil L-function of an elliptic curve E defined over Q and twisted by a Dirichlet character χ of order k and of conductor fχ. Keating and Snaith introduced the way to study L-functions through random matrix theory of certain topological groups. Conrey, Keating, Rubinstein, and Snaith and David, Fearnley, and Kisilevsky developed their ideas in statistics of families of critical values of LE(1, χ) twisted by Dirichlet characters of conductors ≤ X and proposed conjectures regarding the number of vanishings in their families and the ratio conjectures of moments and vanishings which are strongly supported by numerical experiments. In this thesis, we review and develop their works and propose the ratio conjectures of moments and vanishings in the family of LE(1, χ) twisted by Dirichlet characters of conductors fχ ≤ X and order of some odd primes, especially 3, 5, and 7 inspired by the connections of L-function theory and random matrix theory. Moreover, we support our result on the ratio conjectures of moments and vanishings of the families for some certain elliptic curves by numerical experiments.
 Title: Analysis of the Dynamic Traveling Salesman Problem with Different Policies Speaker: Mr. Santiago Ravassi (M.Sc.) Date: Thursday, December 8 , 2011 Time: 1:30 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: We propose and analyze new policies for the traveling salesman problem in a dynamic and stochastic environment (DTSP). The DTSP is defined as follows: demands for service arrive in time according to a Poisson process, are independent and uniformly distributed in a Euclidean region of bounded area, and the time service is zero; the objective is to reduce the time the server visits all the present demands for the first time. We start by analyzing the nearest neighbour (NN) policy since it has the best performance for the dynamic vehicle routing problem (DTRP), a closely related problem to the DTSP. We further introduce the random start policy whose efficiency is similar to that of NN, and we observe that when the random start policy is delayed, it behaves as the DTRP with NN policy. Finally, we introduce the partitioning policy and show that it reduces the expected time demands are swept from the region for the first time relative to other policies.
 Title: On Existence and Stability of Absolutely Continuous Invariant Measures in Some Chaotic Dynamical Systems Speaker: Mr. Peyman Eslami (Ph.D.) Date: Friday, September 9, 2011 Time: 10:30 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: (Click here to view)
 Title: APOS Theory as a Framework to Study the Conceptual Stages of Related Rates Problems Speaker: Mr. Mathew Tziritas (M.T.M) Date: Wednesday, September 7, 2011 Time: 10:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: A study was done in an attempt to use the APOS theory of learning and teaching mathematics to develop and test a teaching cycle for the improvement of students’ conceptual understanding of related rates problems. “APOS” is an acronym that stands for Action, Process, Object, and Schema, and refers to both a theory of teaching and learning and a research methodology in mathematics education. APOS theory originated in the research of an American mathematician and mathematics educator Ed Dubinsky on undergraduate students’ learning of mathematics (Calculus, Linear Algebra, Abstract Algebra). “Related rates problems” refers to problems in Calculus that require finding the rate of change of one value, given the rate of change of a related value. Part of APOS research methodology is a “genetic decomposition” of the concepts to be learned by the students in terms of the mental constructions that such learning requires. In the present study, the genetic decomposition focused on the mental constructions required for student success during the initial conceptual stages of related rates problems learning. The decomposition was constructed using the author’s knowledge of the subject. The genetic decomposition was used to construct an Action – Discussion – Exercise (ACE) teaching cycle which was then tested on two groups of students. Finally, students were asked to solve related rates problems during an individual interview with the author. Data from students’ involvement in the ACE cycle as well as their work during the interview process were then used to suggest changes to the genetic decomposition and the ACE cycle. These suggestions constitute the results of the study. Their purpose is to improve the starting point for further iterations of experimentation of teaching related rates problems.
 Title: Parameter Estimation in a Two-Dimensional Commodity Speaker: Ms. Wenxi Liu (M.Sc.) Date: Wednesday, August 31, 2011 Time: 2:00 p.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: We consider the problem of estimating the parameters of an unobservable model for the spot price of a commodity. Using the observable time-series of the term-structure of futures prices and a filter-based implementation of the expectation maximization (EM) algorithm, we calculate the maximum likelihood parameter estimates (MLEs). New finite-dimensional filters are derived that allow the EM algorithm to be implemented without calculating Kalman smoother estimates. The method is applied to a two-factor commodity price model.
 Title: Theory and Applications of Generalized Linear Models in Insurance Speaker: Mr. Jun Zhou (Ph.D.) Date: Monday, August 29, 2011 Time: 10:00 a.m. Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.) Abstract: (Click here to view)

Speaker: Mr. Oscar Quijano Xacur (M.Sc.)

Date: Tuesday, August 23, 2011

Time: 1:30 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Property and Casualty Premiums Based on Tweedie Families of Generalized Linear Models

Abstract: We consider the problem of estimating accurately the pure premium of a property and casualty insurance portfolio when the individual aggregate losses can be assumed to follow a compound Poisson distribution with gamma jump size. The Generalized Linear Models (GLMs) with a Tweedie response distribution are analyzed as a method for this estimation. This approach is compared against the standard practice in the industry of combining estimations obtained separately for the frequency and severity by using GLMs with Poisson and gamma responses respectively. We show that one important difference between these two methods is the variation of the scale parameter of the compound Poisson-gamma distribution when it is parametrized as an exponential dispersion model. We conclude that both approaches need to be considered during the process of model selection for the pure premium.

Speaker: Mr. Petr Zorin (M.Sc.)

Date: Tuesday, August 23, 2011

Time: 11:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: The Discrete Spectra of Dirac Operators

Abstract: A single particle is bound by an attractive central potential and obeys the Dirac equation in d dimensions. The Coulomb potential is one of the few examples for which exact analytical solutions are available. A geometrical approach called 'the potential envelope method' is used to study the discrete spectra generated by potentials V(r) that are smooth transformations V(r) = g(-1/r) of the soluble Coulomb potential. When g has definite convexity, the method leads to energy bounds. This is possible because of the recent comparison theorems for the Dirac equation. The results are applied to study soft-core Coulomb potentials used as models for confined atoms. The estimates are compared with accurate eigen values found by numerical methods.

Speaker: Ms. Anne Mackay (M.Sc.)

Date: Monday, August 22, 2011

Time: 1:00 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Pricing and Hedging Equity-Linked Products Under Stochastic Volatility Models

Abstract: Equity-indexed annuities (EIAs) are becoming increasingly interesting for investors as market volatility increases. Simultaneously, they represent a higher risk for insurers, which amplifies the need for hedging strategies that perform well when index returns present unexpected changes in their volatility. In this thesis, we introduce hedging strategies that aim at reducing the risk of the financial guarantees embedded in EIAs.

We first derive closed-form expressions for the price and the Greeks of a point-to-point EIA under the Heston model, which assumes stochastic volatility. To do so, we rely on the similarity between the payoff of a European call option and that of the EIA. We use the Greeks to develop dynamic hedging strategies that aim at reducing equity and volatility risk. Using Monte Carlo simulations to derive the distribution of the resulting hedging errors, we compare the performance of hedging strategies that use the Greeks derived under the Heston model to other strategies based on Greeks developed under Black-Scholes.

We show that, when the market is Hestonian, the performance of hedging strategies developed in a Black-Scholes framework are significantly affected by the calibration of the model and the volatility risk premium. We further show that the performance of a simple delta hedging strategy using Heston Greeks is also reduced by the presence of a volatility risk premium, and that this performance can be improved by incorporating gamma or vega hedging to the strategy. We conclude by recommending the use of a delta-vega hedging strategy to reduce model calibration and volatility risk.

Speaker: Ms. Mengjue Tang (M.Sc.)

Date: Tuesday, July 26, 2011

Time: 10:30 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: A Comparison of Two Nonparametric Density Estimators in the Context of Actuarial Loss Model

Abstract: In this thesis, I will introduce two estimation methods for estimating loss function in actuarial science. Both of them are related to nonparametric density estimation (kernel smoothing). One is deriving from kernel smoothing which is called semi-parametric transformation kernel smoothing while another one derives from Hille's lemma and perturbation idea which is quite similar to kernel smoothing. As the increasing frequently used of nonparametric density estimation in many areas, actuaries are more likely to use this kind of simple method when doing decision-making. There are now existing many nonparametric density estimation methods, but which one is better? In order to compare the two methods which are introduced in this thesis, I conduct simulation study on both of them and try to find out which one is preferable and easier to apply. Also the second method which derives from Hille's lemma gives us a new idea about how to estimate loss function when we are doing decision-making in actuarial science.

Speaker: Mr. Ramin Okhrati (Ph.D.)

Date: Monday, July 25, 2011

Time: 10:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Credit Risk Modeling under Jump Processes and under a Risk Measure-Based Approach

Speaker: Ms. Li Ma (Ph.D.)

Date: Monday, June 27, 2011

Time: 2:00 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Generalized Feynman-Kac Transformation and Fukushima's Decomposition for Nearly Symmetric Markov Processes

Date: Wednesday, June 22, 2011

Time: 10:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Comparison theorems for the principal eigenvalue of the Laplacian

Abstract: We study the Faber - Krahn inequality for the Dirichlet eigenvalue problem of the Laplacian, first in $\mathbb{R}^N$, then on a compact smooth Riemannian manifold $M$. For the latter, we consider two cases. In the first case, the compact manifold has a lower bound on the Ricci curvature, in the second, the integral of the reciprocal of an isoperimetric estimator function of the Riemannian manifold is convergent. In all cases, we show that the first eigenvalue of a domain in $\mathbb{R}^N$, respectively $M$, is minimal for the ball of the same volume, respectively, for a geodesic ball of the same relative volume in an appropriate manifold $M^\ast$. While working with the isoperimetric estimator, the manifold $M^\ast$ need not have constant sectional curvature. In $\mathbb{R}^N$, we also consider the Neumann eigenvalue problem and present the Szeg\"o - Weinberger inequality. In this case, the principal eigenvalue of the ball is maximal among all principal eigenvalues of domains with same volume.

Speaker: Mr. Alexandre Laurin (M.Sc.)

Date: Friday, April 1, 2011

Time: 1:30 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: On Duncan's characterization of McKay's monstrous E_8

Abstract: McKay's Monstrous $E_8$ observation has provided further evidence, along with the evidence provided by the study of Monstrous Moonshine, that the Monster is intimately linked with a wide spectrum of other mathematical objects and, one might even say, with the natural organization of the universe. Although these links have been observed and facts about them proved, we have yet to understand exactly where and how they originate. We here review a set of conditions, due to Duncan, imposed on arithmetic subgroups of $PSL2(R)$ that return McKay's Monstrous $E_8$ diagram. The purpose is to compare these with Conway, McKay and Sebbar's (CMS) conditions that return the complete set of Monstrous Moonshine groups in order to gain some insight on their meaning. By way of doing this review of Duncan's conditions, we will also review and elaborate on Conway's method for understanding groups like $\Gamma_0(N)$.

Speaker: Mr. Jun Li (Ph.D.)

Date: Monday, November 29, 2010

Time: 1:30 p.m.

Room: H 762 (Concordia University, Hall Building, 1455 de Maisonneuve Blvd. W.)

Title: Some Contributions to Nonparametric Estimation of Density and Related Functionals for Biased Data

Abstract: Length biased sampling as a special case of biased sampling occurs naturally in many statistical applications. One aspect regarding length biased data in which people are interested is estimating the underlying true density with the observed samples. Since most length biased data are nonnegative, the true density has a support with a non-negative finite end point. The current proposed kernel density estimators with symmetric kernels may have large bias at the lower boundary. In this thesis, we propose some new smooth density estimators with weights generating from Poisson distribution or nonnegative asymmetric kernels for length biased data to take care of the edge effect. Besides density estimators, we also consider smooth estimators of distribution function and functions related to distribution and density function, such as hazard function and mean residual life function. Our methods are easily to extend to the general biased data as well.

Speaker: Ms. Di Xu (M.Sc.)

Date: Friday, November 26, 2010

Time: 10:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: The Range Time for Jump Diffusion with Two-Sided Exponential Jumps

Abstract: The range time for a stochastic process is the stopping time when the difference between its running maximum and running minimum first exceeds a certain level. It has been studied by several authors for random walks and diffusion processes. In this presentation we consider a jump diffusion process with two-sided exponential jumps. By a martingale approach, we first solve the two-sided exit problem for this jump diffusion process. Using solutions to the exit problem, we then obtain several results concerning the range time related to joint distributions for the jump diffusion.

Speaker: Ms. Janine Bachrachas (M.Sc.)

Date: Monday, November 22, 2010

\Time: 1:30 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: On the Mean Curvature Flow

Abstract: We present a self-contained expository review on the mean curvature flow for smooth embedded hypersurfaces in the (n+1)-dimensional Euclidean space. We start by addressing the short time existence of solutions to the flow, followed by the long time existence in the case of compact convex hypersurfaces and entire graphs. Although the results presented here are part of the classical literature originated in the 80’s, we derive all necessary calculations and gather the simplest possible approach in view of later developments of the area.

Speaker: Ms. Yafang Wang (Ph.D.)

Date: Friday, October 29, 2010

Time: 3:30 p .m.

Room: LB 921-4 (Concordia University, Hall Building, 1400 de Maisonneuve Blvd. W.)

Title: The Distribution of the Discounted Compound PH-Renewal Process

Abstract: The family of phase--type (PH) distributions has many useful properties such as closure under convolution and mixtures, as well as rational Laplace transforms. PH distributions are widely used in applications of stochastic models such as in queuing systems, biostatics and engineering. They are also applied to insurance risk, such as in ruin theory.  In this thesis, we extend the work of Wang (2007), that discussed the moment generating function (mgf) of discounted compound sums with PH inter--arrival times under a nonzero net interest rate. Here we focus on the distribution of the discounted compound sums. This represents a generalization of the classical risk model for which the net interest rate is zero.  A differential equation system is derived for the mgf of a discounted compound sum with PH inter--arrival times and any claim severity if its mgf exists. For some PH inter-arrival times, we can further simplify this differential equation system. If the matrix is order of 2, an ordinary differential equation is developed for PH inter-arrival times. By inverting the corresponding Laplace transforms, the density functions and cumulative distribution functions are also obtained. In addition, the series and transformation methods for solving differential equations are discussed, when the mean of inter-arrival times is small.  Applications such as stop-loss premiums, and risk measures such as VaR and CTE are investigated. These are compared for different inter-arrival times. Some numerical examples are given to illustrate the results.  Finally asymptotic results have been discussed, when the mean inter-arrival time goes to zero. We obtain normality to approximate compound renewal processes. The asymptotic normal distribution is also derived for the discounted compound renewal sum at a fixed time.

Speaker: Mr. Xinghua Zhou (M.Sc.)

Date: Thursday, September 2, 2010

Time: 10:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Stochastic Flow and FBSDE Approaches to Quadratic Term Structure Models

Abstract: We study the stochastic flow method and Forward-Backward Stochastic Differential Equation (FBSDE) approach to Quadratic Term Structure Models (QTSMs). Applying the stochastic flow approach, we get a closed form solution for the zero-coupon bond price under a one-dimensional QTSM. However, in the higher dimensional cases, the stochastic flow approach is difficult to implement. Therefore, we solve the n-dimensional QTSMs by implementing the FBSDE approach, which shows that the zero-coupon bond price under QTSM provided some Riccati type equations have global solutions.

Speaker: Ms. Wenxia Li (M.Sc.)

Date: Friday, August 27, 2010

Time: 10:30 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Optimal Surrender and Asset Allocation Strategies for Equity-Indexed Insurance Investors

Abstract: Equity-indexed annuity (EIA) products is getting more and more popular since first introduced in 1995. An EIA investor may consider surrendering the contract before maturity and invest in the stock index in order to earn the full stock growth. We consider an EIA policyholder who seeks the optimal surrender strategy and asset allocation strategy after surrender in order to maximize his expected discounted utility at the maturity of the contract or his time of death, whichever comes first. The optimal value functions satisfy Hamilton-Jacobi-Bellman equations from which the optimal strategies are derived.

Speaker: Mr. Ferenc Balogh (Ph.D.)

Date: Tuesday, July 20, 2010

Time: 11:00 a.m.

Room: H 443 (Concordia University, Hall Building, 1455 de Maisonneuve Blvd. W.)

Title: Orthogonal Polynomials, Equilibrium Measures and Quadrature Domains Associated with Random Matrix Models

Abstract: Motivated by asymptotic questions related to the spectral theory of complex random matrices, this work focuses on the asymptotic analysis of orthogonal polynomials with respect to quasi-harmonic potentials in the complex plane. The ultimate goal is to develop new techniques to obtain strong asymptotics (asymptotic expansions valid uniformly on compact
subsets) for planar orthogonal polynomials and use these results to understand the limiting behavior of spectral statistics of matrix models as their size goes to infinity. For orthogonal polynomials on the real line the powerful Riemann--Hilbert approach is the main analytic tool to derive asymptotics for the eigenvalue correlations in Hermitian matrix models. As yet, no such method is available to obtain asymptotic information about planar orthogonal polynomials, but some steps in this direction have been taken.  The results of this thesis concern the connection between the asymptotic behavior of orthogonal polynomials and the corresponding equilibrium measure. It is conjectured that this connection is established via a quadrature identity: under certain conditions the weak-star limit of the normalized zero counting measure of the orthogonal polynomials is a quadrature measure for the support of the equilibrium measure of the corresponding two-dimensional electrostatic variational problem of the underlying potential.  Several results are presented on equilibrium measures, quadrature domains, orthogonal polynomials and their relation to matrix models. In particular, complete strong asymptotics are obtained for the simplest nontrivial quasi-harmonic potential by a contour integral reduction method and the Riemann-Hilbert approach, which confirms the above conjecture for this special case.

Speaker: Mr. Farhat Abohalfya (Ph.D.)

Date: Tuesday, May 11, 2010

Time: 1:00 p.m.

Room: H 443 (Concordia University, Hall Building, 1455 de Maisonneuve Blvd. W.)

Title: On RG-spaces and the Space of Prime d-ideals in C(X)

Abstract: Let A be a commutative semiprime ring with identity. Then A has at least two epimorphic regular extensions namely, the universal epimorphic regular extension T(A), and the epimorphic hull H(A). We are mainly interested in the case of C(X), the ring of real-valued continuous functions defined on a Tychonoff space X. It is a commutative semiprime ring with identity and it has another important epimorphic regular extension namely, the minimal regular extension G(X). In our study we show in chapter 5 that the spectrum of the ring H(A) with the spectral topology is homeomorphic to the space of the prime ξ-ideals in A with the patch topology. In the case of C(X), the spectrum of the epimorphic hull H(X) with the spectral topology is homeomorphic to the space of prime d-ideals in C(X) with the patch topology.

A Tychonoff space X which satisfies the property that G(X) = C(Xδ) is called an RG-space. We shall introduce a new class of topological spaces namely the class of almost k-Baire spaces, and as a special case of this class we shall have the class of almost Baire spaces. We show that every RG-space is an almost Baire space but it need not be a Baire space. However, in the case of RG-spaces of countable pseudocharacter, RG-spaces have to be Baire spaces. Furthermore, in this case every dense set in RG-spaces has a dense interior.

The Krull z-dimension and the Krull d-dimension will play an important role to determine which of the extensions H(X) and G(X) has the form of a ring of real-valued continuous functions on some topological space. In [31] the authors gave some techniques to prove that there is no RG-space with infinite Krull z-dimension, but there was an error that we found in the proof of theorem 3.4. In this study, we will give an accurate proof which applies to many spaces but the general theorem will remain open. And we will use the same techniques to prove that if C(X) has an infinite chain of prime d-ideals then H(X) cannot be isomorphic to a ring of real-valued continuous functions.

Speaker: Ms. Noushin Sabetghadam Haghighi (Ph.D.)

Date: Thursday, April 8, 2010

Time: 3:00 p.m.

Room: LB 649 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: On Larcher Subgroups and Fourier Coefficients of Modular Forms

Abstract: This work consists of two parts, both revolving around Monstrous moonshine. First we compute the signature of Generalized Larcher subgroups. These subgroups were first introduced by Larcher to prove his result about the cusp widths of any congruence subgroup. They also played a significant role in the classification of torsion-free low genus congruence subgroups. In the second part, we establish universal recurrence formulae satisfied by the Fourier coefficients of meromorphic modular forms on moonshine-type subgroups.

Speaker: Ms. Huan Yi Li (M.Sc.)

Date: Wednesday, April 7 , 2010

Time: 10:30 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Analyzing Equity-Indexed Annuities Using Lee-Carter Stochastic Mortality Model

Abstract: Equity-indexed annuity (EIA) insurance products have become more and more popular since being introduced in 1995. Some of the most important characteristics of these products are that they allow the policyholders to benefit from the equity market’s potential growth and ensure that the principals can grow with a minimum guaranteed interest rate. In this thesis, we show how to derive the closed-form pricing formula of a point-to-point (PTP) financial guarantee, using the Black-Scholes framework. Furthermore, the PTP equity-indexed annuity is discussed in details as well. We will show how to construct the replicating portfolio for both the PTP financial guarantee and the PTP equity-indexed annuity. Because in the real financial market, companies cannot trade continuously, which violates the assumptions of the complete-market, the replicating portfolio will generate hedging errors. The distributions of the present values hedging errors for both the financial guarantee and EIA will be shown. In addition, the distribution of the present values of hedging errors will be showed. We will talk about the impacts on the hedging errors caused by the stochastic mortality rates in the end of the thesis.

Speaker: Mr. Colin Grabowski (M.Sc.)

Date: Wednesday, March 31 , 2010

Time: 11:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Local Torsion on Elliptic Curves

Abstract: Let E be an elliptic curve over Q. Let p be a prime of good reduction for E. We say that p is a local torsion prime if E has p-torsion over Qp, and more generally, we say that p is a local torsion prime of degree d if E has p-torsion over an extension of degree d of Qp.

We study in this thesis local torsion primes by presenting numerical evidence, and by computing estimates for the number of local torsion primes on aver- age over all elliptic curves over Q.

Speaker: Mr. Amir Reza Raji-Kermany (Ph.D.)

Date: Thursday, February 4, 2010

Time: 10:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Mathematical Models for Interactions Among Evolutionary Forces in Finite and Infinite Populations

Abstract: Mathematical modeling in population genetics plays an important role in understanding the effects of different evolutionary forces on the evolution of populations. The complexity of these models increases as we include more factors affecting the genetic composition of the population under consideration. In this thesis, we focus on interactions among evolutionary forces in finite and infinite populations. In the first part of the thesis we study the effect of migration between two populations of equal sizes with mutations occurring between two alleles at the locus under study.
Stochastic changes in the frequencies of one of the alleles in the population is described by a two-dimensional diffusion process. The stationary distribution of this process is characterized by identifying the joint moments under the stationary measure. The second part of this thesis is devoted to studying the effect of recombination on the distribution of types in an infinite haploid population with selection and mutation. In particular, we study the frequency of an allele promoting recombination in such a population. The dynamics of this system are studied in a deterministic framework where the distribution of types is described by a system of ordinary differential equations. We provide numerical solutions to this system. Our results suggest that even if there is no epistatic interaction among loci under selection, an increased rate of deleterious mutations provides a sufficient condition for recombination to be favored in the population.

Speaker: Mr. Zhaoyang Wu (M.Sc.)

Date: Monday, January 25, 2010

Time: 11:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Predicting Stock Index Based on Grey Theory, Arima Model and Wavelet Methods

Abstract: In this thesis, we develop a new forecasting method by merging traditional statistical methods with innovational non-statistical theories for the purpose of improving prediction accuracy of stock time series. The method is based on a novel hybrid model which combines the grey model, the ARIMA model and wavelet methods. First of all, we improve the traditional GM (1, 1) model to the GM (1, 1, u, v) model by introducing two parameters: the grey coefficient u and the grey dimension degree v. Then we revise the normal G-ARMA model by merging the ARMA model with the GM (1, 1, u, v) model. In order to overcome the drawback of directly modeling original stock time series, we introduce wavelet methods into the revised-ARMA model and name this new hybrid model WG-ARMA model. Finally, we obtain the WPG-ARMA model by replacing the wavelet transform with the wavelet packets decomposition. To keep consistency, all the proposed models are merged into a single model by estimating parameters simultaneously based on the total absolute error (TAE) criterion. To verify prediction performance of the models, we present case studies for the models based on the leading Canadian stock index: S&P/TSX Composite Index on the daily bases. The experimental results give the rank of predictive ability in terms of the TAE, MPAE and DIR metrics as following :WPG-ARMA,WG-ARMA,G-ARMA,GM(1,1,u,v),ARIMA.

Date: Tuesday, September 15, 2009

Time: 11:00 a.m.

Room: H 769 (Concordia University, Hall Building, 1455 de Maisonneuve Blvd. W.)

Title: On Fontaine Sheaves

Abstract: In this thesis we focus our research on constructing two new types of Fontaine sheaves, Armax and Amax in the third chapter and the fourth one respectively and in proving some of their main properties, most important the localization over small affines. This pair of new sheaves plays a crucial role in generalizing a comparison isomorphism theorem of Faltings for the ramified case. In the first chapter we introduce the concept of p-adic Galois representation and provide and analyze some examples. The second chapter is an overview of the Fontaine Theory. We define the concept of semi-linear representation and study the period rings introduced by Fontaine while understanding their importance in classifying the p-adic Galois representations.

Speaker: Ms. Klara Kelecsenyi (Ph.D.)

Date: Thursday, September 3, 2009

Time: 2:00 p.m.

Room: H 760 (Concordia University, Hall Building, 1455 de Maisonneuve Blvd. W.)

Title: Popularization of Mathematics as Intercultural Communication – An Exploratory Study

Abstract: Popularization of mathematics seems to have gained importance in the past decades. Besides the increasing number of popular books and lectures, there are national and international initiatives, usually supported by mathematical societies, to popularize mathematics. Despite this apparent attention towards it, studying popularization has not become an object of research; little is known about how popularizers choose the mathematical content of popularization, what means they use to communicate it, and how their audiences interpret popularized mathematics. This thesis presents a framework for studying popularization of mathematics and intends to investigate various questions related to the phenomenon, such as:

- What are the institutional characteristics of popularization?
- What are the characteristics of the mathematical content chosen to be popularized?
- What are the means used by popularizers to communicate mathematical ideas?
- Who are popularizers and what do they think about popularization?
- Who are audience members of a popularization event?
- How audience members interpret popularization?

The thesis presents methodological challenges of studying popularization and suggests some ideas on the methods that might be appropriate for further studies. Thus it intends to offer a first step for developing suitable means for studying popularization of mathematics.

Speaker: Mr. Jeremy Porter (M.Sc.)

Date: Thursday, September 3, 2009

Time: 11:00 a.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: On a Conjecture for the Distributions of Primes Associated with Elliptic Curves

Abstract: For an elliptic curve E and fixed integer r, Lang and Trotter have conjectured an asymptotic estimate for the number of primes p bounded by x such that the trace of Frobenius equals r. Using similar heuristic reasoning, Koblitz has conjectuerd an asymptotic estimate for the number of primes p bounded by x such that the order of the group of points of E over the finite field of prime characteristic p is also prime. These estimates have been proven correct for elliptic curves “on average”; however, beyond this the conjectures both remain open.

In this thesis, we combine the condition of Lang and Trotter with that of Koblitz to conjecture an asymptotic for the number of primes p bounded by x such that both the order of the group of points of E over the finite field of characteristic p is prime, and the trace of Frobenius equals r. In the case where E is a Serre curve, we will give an explicit construction for the estimate. As support for the conjecture, we will also provide several examples of Serre curves for which we computed the number of primes p bounded by large x such that the order of the group of points of E over the finite field of characteristic p is prime and the trace of Frobenius equals r, and compared this count with the conjectured estimates.

Speaker: Ms. Valerie Hudon (Ph.D.)

Date: Friday, August 28, 2009

Time: 10:30 a.m.

Title: Study of the Coadjoint Orbits of the Pointcare Group in 2 + 1 Dimensions and Theiry Coherent States

Abstract: The first main objective of this thesis is to study the orbit structure of the (2+1)-Poincaré group (the symmetry group of relativity in two space and one time dimensions) by obtaining an explicit expression for the coadjoint action. From there, we compute and classify the coadjoint orbits. We obtain a degenerate orbit, the upper and lower sheet of the two-sheet hyperboloid, the upper and lower cone and the one-sheet hyperboloid. They appear as two-dimensional coadjoint orbits and, with their cotangent planes, as four-dimensional coadjoint orbits. We also confirm a link between the four-dimensional coadjoint orbits and the orbits of the action of SO(2,1) on the dual of R^(2,1).

The second main objective of this thesis is to use the information obtained about the structure to induce a representation and build the coherent states on two of the coadjoint orbits, namely the upper sheet of the two-sheet hyperboloid and the upper cone. We obtain coherent states on the hyperboloid for the principal section. The Galilean and the affine sections only allow us to get frames. On the cone, we obtain a family of coherent states for a generalized principal section and a frame for the basic section.

Speaker: Mr. Baohua He (M.Sc.)

Date: Friday, August 21, 2009

Time: 2:00 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Smoothing Parameter Selection for a New Regression Estimator for Non-Negative Data

Abstract: In this thesis, cross-validation based smoothing parameter election technique is applied to Chaubey, Laib and Sen’s (2008) estimator, which is a new regression estimation for nonnegative random variables. The estimator is based on a generalization of Hille's lemma and a perturbation idea. A second order expansion for mean squared error (MSE) of the estimator is derived and the theoretical optimal values of the smoothing parameters are discussed and calculated. Simulation results and graphical illustrations on the new estimator comparing with Fan's (1992, 2003) local linear regression estimators are provided.

Date: Friday, June 19, 2009

Time: 1:30 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Wavelet-Based Noise Reduction of CDNA Microarray Images

Abstract: Microarray experiments have greatly advanced our understanding of how genes function by enabling us to examine the activity of thousands of genes simultaneously. In cDNA microarray experiments, information regarding gene activity is extracted from a pair of red and green channel images. These images are often of poor quality since they are corrupted with noise arising from different sources, including the imaging system itself. Inferences based on noisy microarray images can be highly misleading. Many noise reduction algorithms have been proposed for natural images. Among these various methods, those that have been developed in the wavelet transform domain are found to be most successful. Unfortunately, the existing wavelet-based methods are not very efficient for reducing noise in cDNA microarray images because they are only capable of processing the red and green channel images separately. In doing so, they ignore the correlation that exists between the wavelet coefficients of the images in the two channels. This thesis deals with the problem of developing novel wavelet-based methods for reducing noise in cDNA microarray images for the purpose of obtaining accurate information regarding gene activity. Two types of wavelet transforms have been used. The proposed methods use joint statistical models that take into account the inter-channel dependencies for estimation of the noise-free images of the two channels. The performance of the proposed methods is compared with that of other methods through extensive experimentations which are carried out on a large set of microarray images. Results show that the new methods lead to improved noise reduction performance and more accurate estimation of the level of gene activity. Thus, it is expected that these methods will play a significant role in improving the reliability of results obtained from real microarray images.

Speaker: Ms. Yuliya Klochko (Ph.D.)

Date: Monday, May 4, 2009

Time: 9:30 a.m.

Room: LB 646 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Genus One Polyhedral Surfaces, Spaces of Quadratic Differentials On Tori and Determinants of Laplacians

Abstract: This thesis presents a formula for the determinant of the Laplacian on an arbitrary compact polyhedral surface of genus one. The formula generalizes the well-known Ray-Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic curve is also considered. We study the determinant of the Laplacian as a functional on the moduli space of meromorphic quadratic differentials with L simple poles and L simple zeroes and derive formulas form variations of this functional with respect to natural coordinates on this space. We also give a new proof of Troyanov's theorem stating the existence of a conformal flat conical metric on a compact Riemann surface of arbitrary genus with a prescribed divisor of conical points.

Speaker: Ms. Olga Veres (Ph.D.)

Date: Wednesday, April 8, 2009

Time: 12:15 p.m.

Room: H 771 (Concordia University, Hall Building, 1455 de Maisonneuve Blvd. W.)

Title: On the Complexity of Polynomial Factorization Over P-adic Fields

Abstract: Let p be a rational prime and Φ (x) be a monic irreducible polynomial in Zp[x]. Based on the work of Ore on Newton polygons (Ore, 1928) and MacLane's characterization of polynomial valuations (MacLane, 1936), Montes described an algorithm for the decomposition of the ideal pOK over an algebraic number field (Montes, 1999). We give a simplified version of the Montes algorithm with a full Maple implementation which tests the irreducibility of Φ (x) over Qp. We derive an estimate of the complexity of this simplified algorithm in the worst case, when Φ (x) is irreducible over Qp. We show that in this case the algorithm terminates in at most O((deg Φ)^3+-epsilon v_p(disc Φ)^2+\epsilon) bit operations. Lastly, we compare the "one-element" and "two-element" variations of the Zassenhaus "Round Four" algorithm with the Montes algorithm.

Date: Friday, April 3, 2009

Time: 2:30 p.m.

Room: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve Blvd. W.)

Title: Students’ Models of the Knowledge to be Learned About Limits in College Level Calculus Courses. The Influence of Routine Tasks and the Role Played By Institutional Norms

Abstract: This thesis presents a study of instructors' and students' perceptions of the knowledge to be learned about limits of functions in a college level Calculus course, taught in a North American college institution. I have analyzed these perceptions from an anthropological perspective combining elements of the Anthropological Theory of Didactics, developed in mathematics education, with a framework for the study of institutions - the Institutional Analysis and Development framework - developed in political science. The analysis of these perceptions is based on empirical data: final examinations from the past six years (2001-2007), used in the studied College institution, and specially designed interviews with 28 students. While a model of the instructors' perceptions could be formulated mostly in mathematical terms,

a model of the students' perceptions had to include an eclectic mixture of mathematical, social, cognitive and didactic norms. The analysis that I carry out shows that these students' perceptions have their source in the institutional emphasis on routine tasks and on the norms that regulate the institutional practices. Finally, I describe students' thinking about various tasks on limits from the perspective of Vygotsky's theory of concept development. Based on the 28 interviews that I have carried out, I will discuss the role of institutional practices on students' conceptual development.