MSc, MA, & PhD defences

Protein sequence analysis is arguably a challenging modern bioinformatics problem covering various areas and applications such as sequence annotation, metagenomics, and comparative genomics. Proteomic information is primarily obtained through high-throughput techniques such as mass spectrometry and microarrays. Manually extracting and analyzing this information can be exceedingly time-consuming, potentially taking weeks or months. Consequently, mathematicians and computer science researchers collaborate with scholars and scientists to computationally analyze protein-related information. By utilizing contemporary network approaches, they can expedite their projects and enhance accuracy.

Proteins are essential for biological processes, such as functionality, structure, and regulation of the body’s tissues. They play a central role in various functions derived from enzymatic catalysis, transporting molecules from one organ to another to manage cells. Understanding unknown properties of proteins, and their functions based on measurable features is crucial for disease research, precision medicine, and therapeutics. Proteins mostly are built from 20 different amino acids making a one-dimension sequence, called the primary protein sequence. When a new protein is discovered, the primary sequence family classification is seeking to examine the probability of its similarity based on its properties to a prior labeled family class with similar function and general behavior. Given the emergence of sequencing technologies and the resulting large-scale protein databases with unknown properties, protein family classification is an open problem in the bioinformatics research area. Recent advances in computer science and digital technologies have opened new gates to researchers in various scientific domains. Bioinformatics, as an intermediary research field, takes advantage of these advancements from conventional machine learning methods to large language models, and biostatistics. Conventional alignment-free techniques such as K-mer are commonly used for protein family classification, but they are limited to defining parameters that are computationally expensive, or difficult to estimate. Moreover, in the classification step, they depend on heuristic methods that can lead to a rough approximation of alignment distances. The latter limitation has been offset via the training step in machine learning models. Although machine learning techniques such as Support Vector Machines have been utilized for protein family classification, they are dependent on domain experts to generate features which could be a time-consuming and challenging task. Deep learning (DL) algorithms have shown promising results in proteomics; however, their application is limited to the availability of massive data sets for training. Since the required data comes from experiments, it can be highly complex or incomplete. Moreover, optimizing parameters in DL models can be time-consuming and costly. As an alternative, few-shot models can learn and generalize from a few observations. To rank the similarity between inputs, these networks employ a unique ranking structure, not requiring extensive training. To address the mentioned limitations, in this research, we designed and implemented a deep few-shot learning network for protein family classification and our result showed outperformance to state-of-the art baseline models. To the best of our knowledge, this is the first deep network tailored for primary sequence family classification that can highly perform with a very limited number of observations.

Please contact Dr. Yogendra Chaubey at yogen.chaubey@concordia.ca for more details

Graphs are an essential mathematical tool used to model real-life complex systems such as social and transportation networks. Understanding diffusion processes on graphs is crucial for modelling phenomena such as the propagation of information within a network of individuals or the flux of goods and/or people through a transportation network. Accurately simulating these diffusion processes can be, in general, computationally demanding since it requires the solution of large systems of ordinary differential equations.

Motivated by this challenge, we propose to construct surrogate models able to approximate the state of a graph at a given time from the knowledge of the diffusivity parameters. Specifically, we consider recently introduced high-dimensional approximation methods based on sparse polynomial expansions, which are known to produce accurate, sample-efficient approximations when the function to be approximated has holomorphic regularity. Hence, to justify our methodology, we will theoretically show that solution maps arising from a certain class of parametric graph diffusion processes are indeed holomorphic. Then, we will numerically illustrate that it is possible to efficiently compute accurate sparse polynomial surrogate models from a few random samples, hence empirically showing the validity of our approach.

Please contact Dr. Simone Brugiapaglia at simone.brugiapaglia@concordia.ca or Dr. Pawel Gora at pawel.gora@concordia.ca for more details.

The famous Rogers-Shephard inequality states that, for any convex body 𝐾 ⊂ ℝ𝑛, we have a volumetric inequality 𝑉𝑛 (𝐾 − 𝐾) ≤ (2𝑛/𝑛) 𝑉𝑛 (𝐾) that compares the volume of the difference body of 𝐾, 𝐾 − 𝐾, with the volume of K. Using Cauchy's surface area formula, a particular case of the more general Kubota's Formulae for Quermassintegrals, we extend this classical inequality to extrapolate an upper bound 𝐶𝐾 =𝑆(𝐾−𝐾) / 𝑆(𝐾) for the surface area of the difference body within the Euclidean space ℝ𝑛. We accompany this upper bound with a lower bound that we derive from the classical Brunn-Minkowski inequality, 𝑉𝑛 (𝐴 + 𝐵) 1/𝑛 ≥ 𝑉𝑛(𝐴)1/𝑛 + 𝑉𝑛(𝐵)1/𝑛. Embracing a geometric perspective, we delve into the nuanced relationships between convex bodies and their respective surface areas, scrutinizing the patterns and properties of the difference body. This includes the validation of the upper bound in ℝ3 for certain classes of convex bodies, including smooth bodies and polytopes. We then analyse and establish a pattern for the formation of the difference body of pyramidal structures in ℝ3. Finally, we draw a conclusion on the effect of symmetry on 𝐶𝐾′𝑠 proximity to either bound. More specifically, we observe how deviations from symmetry, mainly when 𝐾 is a simplex, often considered the most asymmetric convex body, draws the constant closer to the upper bound.

Please contact Dr.Alina Stancu at alina.stancu@concordia.ca for more details

The famous Rogers-Shephard inequality states that, for any convex body 𝐾 ⊂ ℝ𝑛, we have a volumetric inequality 𝑉𝑛 (𝐾 − 𝐾) ≤ (2𝑛/𝑛) 𝑉𝑛 (𝐾) that compares the volume of the difference body of 𝐾, 𝐾 − 𝐾, with the volume of K. Using Cauchy's surface area formula, a particular case of the more general Kubota's Formulae for Quermassintegrals, we extend this classical inequality to extrapolate an upper bound 𝐶𝐾 =𝑆(𝐾−𝐾) / 𝑆(𝐾) for the surface area of the difference body within the Euclidean space ℝ𝑛. We accompany this upper bound with a lower bound that we derive from the classical Brunn-Minkowski inequality, 𝑉𝑛 (𝐴 + 𝐵) 1/𝑛 ≥ 𝑉𝑛(𝐴)1/𝑛 + 𝑉𝑛(𝐵)1/𝑛. Embracing a geometric perspective, we delve into the nuanced relationships between convex bodies and their respective surface areas, scrutinizing the patterns and properties of the difference body. This includes the validation of the upper bound in ℝ3 for certain classes of convex bodies, including smooth bodies and polytopes. We then analyse and establish a pattern for the formation of the difference body of pyramidal structures in ℝ3. Finally, we draw a conclusion on the effect of symmetry on 𝐶𝐾′𝑠 proximity to either bound. More specifically, we observe how deviations from symmetry, mainly when 𝐾 is a simplex, often considered the most asymmetric convex body, draws the constant closer to the upper bound.

Please contact Dr.Alina Stancu at alina.stancu@concordia.ca for more details

Hybrid Models For Claim Frequency And Severity   

Thesis Proposal

Rate making in insurance refers to the pricing of insurance premiums through calculations, by actuaries, and adjustments in various factors. Fair pricing of insurance products is of utmost importance for insurance companies to be able to face market competition and stay in business. Therefore, poor rate making, which could be the result of poor prediction of risks, would be dangerous for insurers. Insurance data is characterized by an imbalance between the number of policyholders that claim and does that do not. The majority of premium payers do not incur accidents and thus do not claim their losses, resulting in a large number of “zero–claims". However, it is very important for the company to identify the customers who are more likely in future to file a claim, because every claim incurs a cost to the enterprise. Chapter 1 proposes a sampling technique devised to improve the identification of the possible future losses by better tracking of the non–zero claims. Generalized Linear Models have long been used by actuaries to accomplish the rate making task. The method is parametric and is based on certain assumptions about the distribution of the data. Insurance data with its probability mass at zero, do not fall exactly into the framework of GLMs. However, in recent years, various new machine learning algorithms have provided improvement by being more effective predictors than GLMs. What these algorithms lack is interpretability. Chapter 2 uses simple algorithms like regression trees, in combination with GLMs, to create a pre–processed GLM that is more effective than a standalone GLM. The endeavor of improving the classical GLM continues in Chapter 3. Here another combination of trees and GLM is used to produce results with more predictive capability that anyone of these algorithms as stand-alones. The new results are interpretable as well as improved.

Please contact Dr. Jose Garrido at jose.garrido@concordia.ca for more information.

Outlier detection plays a crucial role in data analysis and is employed in various domains such as finance, healthcare, and anomaly detection. This Thesis presents a novel approach for detecting outliers using quantiles and depth functions, and we apply it to an air quality dataset. Quantiles provide a statistical measure of the distribution of data, while depth functions assess the centrality of observations relative to the entire dataset. Combining these two techniques, we propose a robust and effective method to identify outliers in multidimensional datasets. Our approach is particularly useful in scenarios where traditional outlier detection methods may be inadequate or fail to capture the complex patterns present in the data. By considering multiple quantiles, we can identify outliers that deviate from different aspects of the data distribution. Additionally, we incorporate depth functions, which measure the centrality of observations within a dataset, to further refine our outlier detection process. To evaluate the effectiveness of our approach, we apply it to a real-world air quality dataset. The data is about the New York Air Quality Measurements of 1973 for five months from May to September recorded daily. It contains 153 observations of 6 variables. By applying our method, we can identify outliers representing unusual air quality patterns, potentially indicating anomalies or errors in the data collection process. Our experimental findings support the proposed approach and effectively detect outliers in the air quality dataset. Compared to traditional outlier detection techniques, our method achieves higher accuracy and provides more detailed insights into the nature of the outliers. Furthermore, we show that the identified outliers can be valuable in understanding the factors contributing to air pollution and in improving the quality of air quality monitoring systems. The findings of this research contribute to the advancement of outlier detection methodologies and provide valuable insights for practitioners in identifying and handling outliers in real-world applications.

Please contact Dr. Melina Mailhot at melina.mailhot@concordia.ca for more details

This thesis explores hedging strategies for European-type derivatives under the non-causal AR (1) Cauchy model. Recently, such non-causal models have raised much attention in the finance literature due to their ability to replicate bubbles often observed in the cryptocurrency market, as well as their tractability for pricing standard European options. However, these discrete-time models are incomplete, meaning that it is impossible to perfectly replicate a derivative's payoff in such a market. This thesis explores the use of quadratic hedging approaches to manage the risk of a derivative trader.

Please contact Dr. Patrice Gaillardetz at patrice.gaillardetz@concordia.ca and/or Dr. Yang Lu at yang.lu@concordia.ca for more details.

This work is devoted to the stationary solutions of the 2D Euler equations describing the time-independent flows of an ideal incompressible fluid. There exists an infinite-dimensional set of such solutions; however, they do not form a smooth manifold in the space of all divergence-free vector fields tangent to the boundary of the flow domain. This circumstance hinders the efforts to understand the structure of the set of stationary flows, and to further study other classes of solutions such as the time-periodic or quasiperiodic flows. The previous authors considered the solutions in the Fréchet space of smooth functions and used powerful methods such as the Nash-Moser-Hamilton implicit function theorem. However, in their approach they overlook a surprising feature of the stationary flows which make the picture much more transparent and opens the way to further progress. This is the observation that the particle trajectories in the flow described by arbitrary solutions of the Euler equations in domains with analytic boundary are analytic curves, even if the velocity field has a finite regularity (say, belongs to the Sobolev or Hölder space). In particular, for any stationary solution, the flow lines are analytic curves, despite limited regularity of the velocity field.

To study the stationary flows we change the viewpoint and consider the flow field as a family of analytic flow lines non-analytically depending on parameter. We quantify the analyticity by introducing spaces of functions which have an analytic continuation to some strip containing the real axis such that on the boundary of the strip the function belongs to the Sobolev space. Further, we introduce the class of Sobolev functions of two variables which are analytic (in the above sense) with respect to one variable. Such functions describe the families of flow lines of stationary flows. These partially analytic functions form a complex Banach space. The stationary solutions satisfy (in the new coordinates) a quasilinear elliptic equation whose local solvability is proved by using the Banach Analytic Implicit Function Theorem (BAIF Theorem), thus we prove that the set of stationary flows is an analytic manifold in the complex Banach space of the flows (i.e. families of flowlines).

In our previous work we realized this idea in the case of stationary flows in a periodic channel with analytic boundaries. In the present work we study a more complicated case of flows in a domain close to the disc, having one stagnation point. We use polar coordinates centered at the (unknown) stagnation point. This results in an elliptic quasilinear equation in the annulus which is degenerate at one component of the boundary. This makes the analysis more difficult. We introduce function spaces which are adaptations of the Kondratev spaces to the partially analytic setup and prove that the problem is Fredholm in those spaces. Further we use the BAIF Theorem, and prove that in our spaces, the set of stationary flows is locally a complex-analytic manifold.

Kindly contact Dr. Alexey Kokotov at alexey.kokotov@concordia.ca for more details.

In recent years, multiple global events have drawn society's attention to fairness-related issues and various societal movements resulted from them. For many fields, the impact was immediate and substantial, but for others it has been much more timid. Insurance is one of the latter. More specifically, the way fairness is implemented in algorithms used to calculate insurance premiums has not changed in decades due in part to the lack of modernization from regulators and in part to the complexity of the issue. Nonetheless, in preparation for society's growing expectations, researchers have developed many ways to implement algorithmic fairness. An exposition is made on this concept, including qualitative and quantitative definitions of fairness as well as approaches to its implementation found in the literature. In particular, the method developed by Lindholm et al. (2022) is discussed in detail and followed up by the introduction of our own novel approach. This approach is demonstrated on simulated data, and it is shown that it can significantly reduce unfairness according to pre-determined metrics.

Please contact Dr. Melina Mailhot at melina.mailhot@concordia.ca or Dr. Patrice Gaillardetz at patrice.gaillardetz@concordia.ca for more details.

This dissertation explores three stochastic models: additive functionals of reflected jump-diffusion processes, two-time scale dynamical systems forced by α-stable Lévy noise, and a variant of the Optimistic Policy Iteration algorithm in Reinforcement Learning. The connecting thread between all three projects is in showing convergence of these objects whose results have direct applied implications.

A large deviation principle is established for general additive processes of reflected jump-diffusions on a bounded domain, both in the normal and oblique setting. A characterization of the large deviation rate function, which quantifies the rate of exponential decay for the rare event probabilities of the additive processes, is provided. This characterization relies on a solution of a partial integro-differential equation with boundary constraints that is numerically solved with its implementation provided. It is then applied to a few practical examples, in particular, a reflected jump-diffusion arising from applications to biochemical reactions.

We derive the weak convergence of the functional central limit theorem for a fast-slow dynamical system driven by two independent, symmetric, and multiplicative α-stable noise processes. To do this, a strong averaging principle is established by solving an auxiliary Poisson equation where the regularity properties of the solution are essential to the proof. The latter allows for the order of convergence to the averaged process of 1-1/α to be established and subsequently used to show weak convergence of the scaled deviations of the slow process from its average. The theory is then applied to a Monte Carlo simulation of an illustrative example.

In the Optimistic Policy Iteration algorithm, Monte Carlo simulations of trajectories for some known environments are used to evaluate a value function and greedily update the policy which we show converges to its optimal value almost surely. This is done for undiscounted costs and without restricting which states are used for updating. We employ the greedy lookahead policies used in previous results thereby extending current research to discount factor α=1. The first-visit variation of this algorithm follows as a corollary, and we further extend previous known results when the first state is picked for updating.

Kindly contact Dr. Lea Popovic at lea.popovic@concordia.ca for more details.

This thesis focuses on - (1) the study of the existence and exactness of Absolutely Continu-ous Invariant Measures (acim) for piecewise convex maps with a countable number of branches,(2) developing the Ulam’s method for approximation of density function and (3) the study of the existence of Absolutely Continuous Invariant Measures (acim) for piecewise concave maps using conjugation.

For the first topic, we investigate the existence and uniqueness of acim for two classes Tpc∞(I), Tpc∞,0(I) of piecewise convex maps τ : I = [0, 1] → [0, 1] with countable number of branches. We provide sufficient conditions under which these maps have a unique acim, and we also give examples where multiple acims exist. Our results are based on the study of the Frobenius-Perron operator associated with the map, and we use analytical techniques to understand the properties of this operator.

The main result of this Ph.D. thesis is the generalization of the existence of absolutely continu-ous invariant measure for piecewise convex maps of an interval with countable number of branches. We consider two classes Tpc∞(I), Tpc∞,0(I) of piecewise convex maps τ : I = [0, 1] → [0, 1] with countable number of branches. For the first class Tpc∞(I), we consider piecewise convex maps τ : I = [0, 1] → [0, 1] with countable number of branches and arbitrary countable number of limit points of partition points separated from 0. For the second class Tpc∞,0(I), we consider piece-wise convex maps τ : I = [0, 1] → [0, 1] with countable number of branches with the partition points converging to 0. In the thesis, we study absolutely continuous invariant measures (acim) of τ ∈ Tpc∞(I) and τ ∈ Tpc∞,0(I) respectively. We also consider non-autonomous dynamical systems of maps in Tpc∞(I) or Tpc∞,0(I) and study the existence of acims for the limit map. Moreover, we study exactness of τ ∈ Tpc∞(I) and τ ∈ Tpc∞,0(I) respectively. We give a similar result for piecewise concave maps as well.

We also discuss the approximation of acims for piecewise convex maps with countable number of branches using Ulam’s method. We also investigate the existence and uniqueness of ACIMs for two classes, Tpcv∞ (I) and Tpcv∞,1(I), of piecewise concave mapping σ. We use conjugation of piecewise convex map τ with countable number of branches that is defined in chapter 3 which implies that σ preserves a normalized absolutely continuous invariant measure whose density is an increasing function.

Kindly contact Dr. Pawel Gora at pawel.gora@concordia.ca for more details.

We consider three projects motivated by model uncertainty in multiple fields of actuarial science. The first project investigates the impact of model combination methods on extreme precipitation quantiles. Non-parametric methods rely on different assumptions than parametric methods such as Bayesian Model Averaging (BMA), potentially leading to very different results. Precipitation projections from 24 experts in Montreal and Quebec City are studied, and the difference in results between non-parametric and Bayesian approaches is used to illustrate the uncertainty of model combination. This shows how an actuary using multiple expert models should consider more than one combination method to properly assess the impact of climate change on loss distributions, seeing as a single method can lead to overconfidence in projections.

Second, we propose an improvement to BMA to correct the known issue of convergence to a single model and adapt the algorithm to a heteroscedastic context. This improvement stems from including data uncertainty within the BMA algorithm, which considers data as the only observable data. The heteroscedastic adaptation is necessary to use BMA in an actuarial context, where loss is not homoscedastic between different insureds. A simulation study is used to demonstrate the efficiency of the proposed method, and it is applied to granular reserve models using a simulated dataset. This also builds on the work of Avanzi et al. (2022), who used BMA on aggregate reserve models.

Finally, we aim to generalize the uncertainty integration approach by allowing for non-constant weight attribution. This flexibility will allow for recognizing which model works best on different parts of the data in a transparent way. Such an approach can provide a useful tool for actuaries in pricing as well as reserving.

Please contact Dr. Melina Mailhot at melina.mailhot@concordia.ca for more information.

The aim of this thesis is to extend and elaborate on the initial sections of Neal Koblitz’s article titled "A New Proof of Certain Formulas for p-Adic L-Functions." Koblitz’s article focuses on the construction of p-adic L-functions associated with Dirichlet’s character and the computation of their values at s = 1. He employs measure theoretic methods to construct the p-adic L-functions and compute the Leopoldt formula Lp(1,χ).

To begin, we devote the first section (1.1) to providing a comprehensive proof of Dirichlet’s theorem for prime numbers. This is done because the theorem serves as a noteworthy example of how Dirichlet L-functions became relevant in the field of Number Theory.

In the second chapter, we introduce the complex version of Dirichlet L-functions and Riemann Zeta functions. We explore their analytical properties, such as functional equations and analytic continuation. Subsequently, we construct the field of p-adic numbers and equip it with the p-adic norm to facilitate analysis. We introduce measures and perform p-adic integrations.

Finally, we delve into the concept of p-adic interpolation for the Riemann Zeta function, aiming to establish the p-adic Zeta function. To accomplish this, we employ Mazur’s measure theoretic approach, utilizing the tools introduced in the third chapter. The thesis concludes by incorporating Koblitz’s work on this subject.

For more information, please contact Dr. Adrian Iovita at adrian.iovita@concordia.ca

This thesis consists of two parts. In the first part, we present a positive characteristic analogue of Shimura's theorem on the special values of modular forms at CM points. More precisely, we show using Hayes' theory of Drinfeld modules that the special value at a CM point of an arithmetic Drinfeld modular form of arbitrary rank lies in the Hilbert class field of the CM field up to a period, independent of the chosen modular form. This is achieved via Pink's realisation of Drinfeld modular forms as sections of a sheaf over the compactified Drinfeld modular curve.

In the second part of the thesis, we present various computational and algorithmic aspects both for the classical theory (over C) and function field theory. First, we implement the rings of quasimodular forms in SageMath and give some applications such as the symbolic calculation of the derivative of a classical modular form. Second, we explain how to compute objects associated with a Drinfeld modules such as the exponential, the logarithm, and Potemine's set of basic J-invariants. Lastly, we present a SageMath package for computing with Drinfeld modular forms and their expansion at infinity using the nonstandard A-expansion theory of López and Petrov.

Kindly contact Dr. Giovanni Rosso at giovanni.rosso@concordia.ca for more details.

This thesis is comprised of three parts which collectively serve as a study of stochastic epidemiological models, in particular, the susceptible, infected, recovered/removed (SIR) model.

I

We propose a unified stochastic SIR model driven by Lévy noise. The structural model allows for time-dependency, nonlinearity, discontinuity, demography and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.

II

This part is two-fold; we investigate the parameter estimation and forecasting of two forms of astochastic SIR model driven by small Lévy noises, and we provide theoretical results on parameter estimation of time-dependent drift for Lévy noise-driven stochastic differential equations. A novel algorithm is introduced for approximating the least-squares estimators, which lack attainable closed-forms; moreover, the presented results ensure the consistency of these approximated Estimators.

III

We apply the previous results to study the COVID-19 pandemic using data from New York City, New York. This application yields parameter estimation and predictive analysis, including the unknown period for a periodic transmission function and importation/exportation of infection.

Kindly contact Dr. Wei Sun at wei.sun@concordia.ca for more details

Under the current regulations, banks and insurance companies have the option to use their own internal models to monitor their risk. To this end, Value-at-Risk (VaR) and the Expected Shortfall (ES) are typically used as the risk measures to compute their capital requirements. Nevertheless, both present flaws, such as the lack of coherence for VaR and lack of elicitability for ES. Recently, expectile has attracted much attention as a potential alternative to VaR and ES. However, the literature on expectile is mainly focused on its statistical inference, and just few traditional backtesting procedures have been proposed. This thesis proposes a traditional backtesting procedure for the expectile and considers its application on financial data.

For more information, please contact Dr. Melina Mailhot at melina.mailhot@concordia.ca or Dr. Yang Lu at yang.lu@concordia.ca

The development of big data technologies and in-vehicle devices has contributed to the growth of Usage-Based Insurance (UBI) in recent years. These in-vehicle devices, such as GPS and sensors, collect certain variables that can represent the driving behaviour of policyholders. This collected data, called telematic data, consist of several variables that have strong relationship with the likelihood of having an accident. Consequently, one can use telematic data to improve risk assessments and personalize car insurance premiums.

In this thesis, a synthetic car insurance dataset emulated from a Canadian-based insurance company is used to investigate the use of telematic data in predicting the likelihood of having an accident. More precisely four machine learning techniques—logistic regression, random forests, gradient boosting trees, and feed-forward neural networks—are employed to predict the risk of having an accident. Actuaries often use white box machine learning methods like logistic regression for risk assessment due to their interpretability. However, these methods are unable to detect non-linear relationships between variables accurately. Therefore, more complex machine learning techniques such as random forests, gradient boosting trees, and feed-forward neural networks are used to achieve more accurate risk assessment for accidents.

In addition, two variable importance assessment methods—Shapley decomposition and marginal performance loss upon feature removal—are employed to provide insights into the feature contributions in the overall predictive performance of the models.

For more information, please contact Dr. Jose Garrido at jose.garrido@concordia.ca or Dr. Frédéric Godin at frederic.godin@concordia.ca .

The Bézier curve is a continuous parametric curve that is used in numerous applications, such as automobile design and road design. I begin by surveying the fundamental properties of the Bézier curve and provide a variety of examples and calculations. I also survey, but more succinctly, the notion of the Bézier surface, which also has numerous applications in computer-aided design software, such as aircraft design. I end by presenting the special types of points that a cubic Bézier curve (with four control points) can have. The configuration of planar control points that I present can be used by computer engineers to produce various shapes with respect to the necessary design requirements and to minimize fluctuations of the curve itself.

For more information, please contact Dr. Alina Stancu at alina.stancu@concordia.ca

The long standing Gaussian product inequality (GPI) conjecture states that E[|X1|y1|X2|y2···|Xn|yn] ≥ E[|X1|y1]E[|X2|y2] ··· E[|Xn|yn] for any centered Gaussian random vector (X1,...,Xn) and any non-negative real numbers yj,j=1,...,n. First, we complete the picture of bivariate Gaussian product relations by proving a novel “oppositeGPI” when -1<y1<0 and y2>0: E [|X1|y1|X2|y2] ≤ E [|X1|y1] E [|X2|y2]. Next, we investigate the three dimensional inequality E [X12X22m2Xn2m3] ≥ E [X12] E [X22m2] E [Xn2m3] for any natural numbers m2, m3.We show that this inequality is implied by a combinatorial inequality which we verify directly for small values of m2 and arbitrary m3.

Then, we complete the proof through the discovery of a novel moment ratio inequality which implies this three-dimensional GPI. We then extend these three-dimensional results to the case where the exponents in the GPI can be real numbers rather than simply even integers. Finally, we describe two computational algorithms involving sums-of-squares representations of polynomials that can be used to resolve the GPI conjecture. To exhibit the power of these novel methods, we apply them to prove new four-and five-dimensional GPIs.

For more details, please contact Dr. Wei Sun at wei.sun@concordia.ca.

The prime number theorem for arithmetic progressions tells us that there are asymptotically as many primes congruent to 1 mod 4 as there are congruent to 3 mod 4. That being said, Chebyshev noticed that (numerically) there almost always seems to be slightly more primes in the 3 class. This simple fact has a highly non-trivial explanation. Rubinstein and Sarnak proved that under the assumption of some natural (yet still unproven) conjectures, there is a way to prove that there are more primes congruent to 3 than congruent to 1 more than half of the time (in an appropriate sense).

Many other sets of integers demonstrate a bias towards a certain residue class modulo some number q Recently, Gorodetsky showed that the sums of two squares exhibit a Chebyshev-type bias, and that in this case the conjectures one must assume to prove the existence of the bias are weaker. In this thesis, we present two papers which demonstrate some bias in arithmetic progressions for sets of integers that are represented by a given binary quadratic form.

In the first paper, we examine a bias towards the zero residue class for the integers represented by binary quadratic forms. In many cases, we are able to prove that the bias comes from a secondary term in the associated asymptotic expansion (unlike Chebyshev's bias, which lives somewhere at the level of O(x^{1/2+\epsilon).) In some other cases, we are unable to prove that a bias exists, even though it is present numerically. We thus make a conjecture on the general situation which includes the cases we could not prove. Many interesting results on the distribution of the integers represented by a quadratic form are proven along the way, and the paper finishes with some numerical data that is illustrative of the generic data for any quadratic form.

In the second paper, we examine a different kind of bias. We ask for the distribution of pairs of sums of two squares in arithmetic progressions, i.e. how many numbers are the sum of two squares, congruent to a mod q, and are such that the next largest sum of two squares is congruent to b mod q? We prove that when q=1 mod 4, we have equidistribution among the q^2 possible pairs of residue classes. That being said, there exist bizarre numerical biases, most notably a negative bias towards repetition. The main purpose of the second paper is to provide a conjecture which explains the bias, via a secondary and tertiary term in the associated asymptotic expansion. We then support this conjecture with both numerical and theoretical evidence. The paper contains many partial results in the direction of the conjecture, as well as some theorems on the sums of two squares that are of independent interest. For example, we provide an integral representation for the number of integers not exceeding x which are the sum of two squares. This integral representation is akin to Li(x) for primes, in that it has a O(x^{1/2+\epsilon}) error term under the Generalized Riemann Hypothesis.

Please contact Dr. Chantal David at chantal.david@concordia.ca for more details.

Research from the past two decades (Lithner, 2004; Barbé et al., 2005; Bergé, 2008; Hardy, 2009; Brandes, 2017; Broley, 2020) has dug into the college calculus institution to determine the type of student activity engendered by this institution. This research has found students in college-level calculus courses routinize techniques and rely on non-mathematical knowledge to justify the validity of their techniques, such as the authority of their teachers and textbooks (Hardy, 2009). This makes sense in light of research that examined the curricular documents in college-level calculus courses (Lithner, 2004; Brandes, 2017): the tasks to be accomplished and the related techniques lend themselves well to routinization and can be replicated without knowing why the techniques are apt or why they work. Broley (2020) investigated the activity of students who had recently passed Analysis 1, the first course in the calculus stream in which the knowledge to be taught might be expected to shift students’ focus away from rote procedure and toward the mathematical theory that undergirds it; but the results question the extent to which such a transition occurs. To accomplish tasks that appear in both Analysis 1 and in differential calculus, participants activated routines acquired in calculus and could not mobilize the related mathematical knowledge from Analysis 1 (Broley and Hardy, 2022).

This research has come to understand college-level calculus students’ practices through a focus on the institutional nature of calculus courses, using Chevallard's (1985, 1992, 1999) Anthropological Theory of the Didactic. These investigations into the type of practices students develop in college calculus found them to be non-mathematical.

But no equivalent research has checked into students’ learning in other domains of mathematics. This leaves unanswered a rather big question: to what extent can results about students' practices be explained by difficulties inherent to the mathematics at stake in college calculus? Do students develop such practices in other introductory mathematics courses?

We propose to contribute to this research by taking a step toward answering this last question. Is students’ behavior in calculus replicated in linear algebra courses of the same level? Two courses in the college linear algebra stream seem appropriate venues for investigation: a college linear algebra course on matrix and vector algebra, which we'll call Linear Algebra 1 (LA1), and a follow-up linear algebra course, Linear Algebra 2 (LA2), where the knowledge to be taught might be expected to shift focus away from rote procedure and toward the mathematical theory that frames it. We wish to tackle the two so we may attend, as Broley (2020) has, to the potential transition from non-mathematical to mathematical bodies of knowledge.

For our research, we adopted the methodology and theoretical framework previously used by (Lithner, 2004; Barbé et al., 2005; Bergé, 2008; Hardy, 2009; Brandes, 2017; Broley, 2020) to investigate students’ practices in these two courses. We have three objectives. First, to identify the knowledge students are expected to learn in LA1 and LA2, as indicated by problems on the final exams in these courses. Second, to identify the nature of the knowledge students from these courses mobilize when they solve linear algebra tasks; is it mathematical or non-mathematical? These first two objectives will serve to achieve our third: to determine if students' practices, as observed by research on the learning done in college calculus, are replicated in linear algebra.

In the first part of the thesis, we discuss the boundedness of inhomogeneous singular integral operators suitable for local Hardy spaces as well as their commutators. First, we consider the equivalence of different localizations of a given convolution operator by giving minimal conditions on the localizing functions; in the case of the Riesz transforms this results in equivalent characterizations of $h^1$. Then, we provide weaker integral conditions on the kernel of the operator and sufficient and necessary cancellation conditions to ensure the boundedness on local Hardy spaces for all values of $p$. Finally, we introduce a new class of atoms and use them to establish the boundedness of the commutators of inhomogeneous singular integral operators with $\bmo$ function.

In the second part of the thesis, we investigate periodic solutions of a class of stochastic partial differential equations driven by degenerate noises with regime-switching. First, we consider the existence and uniqueness of solutions to the equations. Then, we discuss the existence and uniqueness of periodic measures for the equations. In particular, we establish the uniqueness of periodic measures by proving the strong Feller property and irreducibility of semigroups associated with the equations. Finally, we use the stochastic fractional porous medium equation as an example to illustrate the main results.

Introduction: Common measures of adiposity (such as body mass index [BMI]) are only proxies. In contrast, dual-energy X-ray absorptiometry (DXA) is more precise to measure body composition. Therefore, this thesis utilized an unsupervised machine learning technique to group individuals based on similarities in their fat and muscle mass body-composition from DXA. Associations between the newly developed body-composition phenotypes with cardiometabolic risks were compared to phenotypes using a median split.

Methods: Data were from National Health and Nutrition Examination Survey (NHANES: 1999-2006 cycles, n=5,566; split into 70/30% training and test datasets), a representative U.S. population. The K-means cluster phenotypes based on partitioning observations from deciles of fat-mass and muscle-mass adjusted for age and sex were identified. Model fit was assessed using the silhouette and elbow method. Performance of logistic regression models to identify unfavorable cardiometabolic risks using either the K-means or the 50th percentile cut-off phenotypes was assessed with the area under the receiver operating characteristic (ROC-AUC). Analyses were performed separately for males and females and incorporated weighting and the complex sampling design.

In the first part of this thesis, we study the contracting planar flow on the space of smooth, strictly convex, simple closed curves by an  -power of their curvature for 0 < a < 1. We show that under certain initial conditions, properly renormalized solutions of the flow converge to limiting shapes that solve different classes of Lp-Minkowski problem for p < 0. To prove our results, we introduce a new method of switching between rescalings.

In the second part, we study, with a focus on surfaces of revolution, a nonhomogeneous flow in ℝ3 that is a generalization of the flow we studied in the first part where ai > 0 for i = 1, 2, X is the embedding of a smooth, closed, strictly convex surface in ℝ3 , 𝜅1, 𝜅2 are the principal curvatures and ν is the normal vector to the surface in ℝ3.

We show that convex solutions to equation (2) remain convex, as long as they exist, and shrink to a point in finite time. Under suitable rescaling, solutions converge to a certain convex hypersurface.

For the third part of the thesis, we propose to study the properties and the asymptotic behaviour of the following flow which is a generalization of the flow considered in ℝ3 where ai > 0 for i = 1, 2, · · · , n.

Body mass index cannot distinguish between fat-mass and muscle-mass, which may result in obesity misclassification. A dual-energy x-ray absorptiometry (DXA)-derived phenotype classification based on fat-mass and muscle-mass has been proposed for adults (>18 yo). We extend this research by developing children’s fat-mass and muscle-mass reference curves and determining their utility in identifying cardiometabolic risk. Children’s (≤17 yo) DXA data in NHANES, a US national health survey (n=6,120) were used to generate sex- and age-specific deciles of appendicular skeletal muscle index and fat mass index (kg/m2) with the Lambda Mu Sigma (LMS) method. The final curves were selected through goodness of fit (AIC, Q-tests, detrended Q-Q plot). Four phenotypes (high [H] or low [L], adiposity [A] and muscle mass [M]: HA-HM, HA-LM, LA-HM, LA-LM) were identified using the literature’s guidelines above/below the median compared to same-sex and same-age peers.

The curves and their corresponding phenotypes were applied to QUALITY data, a longitudinal cohort (n=630, 8-10 yo in 2005) to assess whether the phenotypes correctly identified cardiometabolic risk using multiple linear regression at baseline, follow-up one (2008-2010), or follow-up two (2015-2017). Models were adjusted for age, sex, and Tanner’s stage. Chained equation was used to impute missing values in QUALITY. Compared to LA-HM, LA-LM was associated with lower glucose at baseline; HA-HM was associated with lower HDL-c and higher LDL-c, triglycerides, and HOMA-IR; HA-LM was associated with elevated triglycerides and HOMA-IR at all timepoints (all p<0.05). These phenotypes allowed for discrimination of cross-sectional cardiometabolic risks, but further longitudinal exploration is recommended.

In this dissertation, we consider moduli spaces of meromorphic quadratic differentials in homological coordinates and applications of underlying deformation theory of Ahlfors-Rauch type.

At first, we derive variational formulas for objects associated with generalized $SL(2)$ Hitchin's spectral covers: Prym matrix, Prym bidifferential, Bergman tau-function. The resulting formulas are antisymmetric versions of Donagi-Markman residue formula. Then we adapt the framework of topological recursion to the case of double covers to compute higher-order variations.

Another application of the deformation theory lies within the symplectic geometry of the monodromy map of the Schrödinger equation on a Riemann surface with a meromorphic potential having second order poles. We discuss the conditions for the base projective connection, which induces its own set of Darboux homological coordinates, to imply the Goldman Poisson structure on the character variety. Using this result, we perform generalized WKB expansion of the generating function of monodromy symplectomorphism (the Yang-Yang function) and compute its leading asymptotics.

Finally, we relate these two studies by showing how the variational analysis on Hitchin's spectral covers could be applied towards the computation of higher asymptotics of the WKB expansion.

Climate change is leading to an increase in the severity and prevalence of natural catastrophes. From a statistical and actuarial perspective, it is desirable to measure the potential impact of changes in different aspects of these extreme events. Sensitivity analysis is used to measure and characterize uncertainty of a model based on these changes, where a baseline model includes a number of covariates mapped to an output via an aggregation function. Given a defined stress on the baseline distribution, a type of sensitivity analysis used in actuarial mathematics, Reverse Sensitivity Testing, is suitable for several types of models (including black box models) and uses different risk measures along with the Kullback–Leibler divergence (KL divergence) as a measure of discrepancy between the baseline probability measure and the stressed probability measure.

An expansion of Reverse Sensitivity Testing is provided to include both a coherent and elicitable risk measure; expectiles. Since the KL divergence is considered to be a pessimistic divergence for extreme values, the Renyi divergence, which is a broader divergence, is included as an extension ideal for extreme events given a user specified order parameter. Both the KL divergence and the Renyi divergence are implemented on a standard normal random variable, a numerical example, and then applied to extreme loss data from a natural catastrophe that hit Western Canada in 2020.

Some of the most fundamental questions about $L$-functions are concerned with the location of their zeros, in particular at the central point, or on the critical line. Following the work of Montgomery and then of Katz and Sarnak, number theorists have learned that it is very fruitful to study families of $L$-functions rather than individual $L$-functions. Given a family of $L$-functions, it is common to classify it according to its symmetry type. The symmetry type can be either symplectic, orthogonal, or unitary, which refers to the corresponding ensemble from random matrix theory that models most accurately the distribution of the zeros of the family.

This thesis presents two papers studying the zeros of the family of Dirichlet twists of the $L$-function of an elliptic curve $E$ over $\mathbb{F}_q[t]$. In the first paper (Chapter 2), we show that the one-level density (the study of the low-lying zeros) for this family agrees with the conjecture of Katz and Sarnak based on random matrix theory, for test functions with limited support on the Fourier transform. For quadratic twists, the support of the Fourier transforms of the test functions is restricted to the interval (-1,1), and we observe an orthogonal symmetry. For higher order twists, the support is restricted to (-1/2,1/2) and we observe a unitary symmetry. For quadratic twists, the support is large enough to obtain a positive proportion of $L$-functions which do not vanish at the central point.

In the second paper (Chapter 3), we are taking the opposite point of view, and we construct certain elliptic curves $\mathbb{F}_q[t]$ with infinitely many twists of high order vanishing at the central point, generalizing a construction of Li and Donepudi-Li for Dirichlet $L$-functions. This construction only works when $E/\mathbb{F}_q[t]$ is a constant curve. In the general case where $E$ is not a constant curve, we performed extensive numerical computations that are compatible with the conjectures of David-Fearnley-Kisilevsky and Mazur-Rubin over number fields, which predict that such vanishing should be rare.

The last chapter presents an algorithm to construct the factorizations of the monic polynomials, a description of the zeros of $L(E\otimes\chi,u)$, and a family of quadratic twists such that the rank of $L(\chi,u)$ goes to infinity.

We consider several classical problems in analytic number theory from the point of view of multiple Dirichlet series.

In Chapter 1, we review the required background and give an overview of the thesis.

In Chapter 2, we introduce multiple Dirichlet series and the relevant ideas used in later chapters.

In Chapter 3, we study the double character sum   

𝑆(𝑋,𝑌) =         Σ                        Σ                 (  m  )
                m≤X,m odd         n≤Y,n odd           n

The asymptotic formula for this sum was obtained by Conrey, Farmer and Soundararajan. We recover this formula using our approach with an improved error term.

In Chapter 4, we use multiple Dirichlet series to give a conditional proof of the ratios conjecture for the family of real Dirichlet L-functions in some region of the shifts. As an application of our result, we compute the one-level density in our family for test functions whose Fourier transform is supported in $(-2,2)$, including all lower-order terms.

In Chapter 5, we elaborate on an ongoing work of the author with Siegfred Baluyot. We compare two methods to come up with conjectural asymptotic formulas for moments in the family of real Dirichlet L-functions -- the recipe developed by Conrey, Farmer, Keating, Rubinstein and Snaith, and the multiple Dirichlet series approach introduced by Diaconu, Goldfeld and Hoffstein. We consider shifted moments and show that the two methods are essentially equivalent, in that they give rise to the same terms which come from the functional equation of the individual L-functions.
 

Room S-LB 921-4 and Online via Zoom

For more information, please contact Dr. Marco Bertola

The Painlevé II hierarchy is a sequence of nonlinear ODEs, with the Painlevé II equation as first member. Each member of the hierarchy admits a Lax pair in terms of isomonodromic deformations of a rank 2 system of linear ODEs, with polynomial coefficient for the homogeneous case. It was recently proved that the Tracy-Widom formula for the Hastings-McLeod solution of the homogeneous PII equation can be extended to analogue solutions of the homogeneous PII hierarchy using Fredholm determinants of higher order Airy kernels. These integral operators are used in the theory of determinantal point processes with applications in statistical mechanics, random matrix theory, random growth models and other fields. From this starting point, this PhD thesis explored the following directions.

We found a formula of Tracy-Widom type connecting the Fredholm determinants of matrix-valued analogue of the higher order Airy kernels with particular solution of a matrix-valued PII hierarchy. The result is achieved by using a matrix-valued Riemann-Hilbert problem to study these Fredholm determinants and by deriving a block-matrix Lax pair for the relevant hierarchy.

We also found another generalization of the Tracy-Widom formula, this time relating the Fredholm determinants of finite-temperature versions of higher order Airy kernels to parti- cular solutions of an integro-differential Painlevé II hierarchy. In this setting, a suitable operator-valued Riemann-Hilbert problem is used to study the relevant Fredholm determinant. The study of its solution produces in the end an operator-valued Lax pair that naturally encodes an integro-differential Painlevé II hierarchy.

In this thesis, we study the space of functions of bounded mean oscillation (BMO) on shapes. We prove the boundedness of important nonlinear operators, such as maximal functions and rearrangements, on this space and analyze how the bounds are affected by the underlying geometry of the shapes.

We provide a general definition of BMO on a domain in R^n, where mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We prove many properties inherent to BMO that are valid for any choice of basis; in particular, BMO is shown to be complete. Many shapewise inequalities, which hold for every shape in a given basis, are proven with sharp constants. Moreover, a sharp norm inequality, which holds for the BMO norm that involves taking a supremum over all shapes in a given basis, is obtained for the truncation of a BMO function. When the shapes exhibit some product structure, a product decomposition is obtained for BMO.

We consider the boundedness of maximal functions on BMO on shapes in R^n. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, the collection of functions of bounded lower oscillation. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO.

We obtain boundedness and continuity results for rearrangements on BMO. This allows for an improvement of the known bound for the basis of cubes. We show, by example, that the decreasing rearrangement is not continuous on BMO, but that it is both bounded and continuous on VMO, the subspace of functions of vanishing mean oscillation. Boundedness for the symmetric decreasing rearrangement is then established for the basis of balls in R^n.

Part I: Some Fluctuation Results on Draw-down Times for Spectrally Negative Levy Processes

In this thesis, we first introduce and review some fluctuation theory of Levy processes, especially spectrally negative Levy processes with and without tax. Then we consider a more realistic model by associate draw-down times, which is a function of the running maximum, into these fluctuation results. Particularly, we present the expressions for the classical two-sided exit problems for these processes with draw-down times in terms of scale functions. Also, we find the expressions for the discounted present values of tax payments with draw-down time in place of ruin time. Finally, we obtain the solution to the occupation times of the general spectrally negative Levy processes spends in draw-down interval killed on either exiting a fix upper level or a draw-down lower level.

Part II: On Estimation of Entropy and Residual Entropy for Nonnegative Random Variables

Entropy has become more and more essential in statistics and machine learning. A large number of its applications can be found in data transmission, cryptography, signal processing, network theory, bioinformatics, and so on. Therefore, the question of entropy estimation comes naturally. Generally, if we consider the entropy of a random variable knowing that it has survived up to time t, then it is defined as the residual entropy. In this paper we focus on entropy and residual entropy estimation for nonnegative random variable. We first present a quick review on properties of popular existing estimators. Then we propose some candidates for entropy and residual entropy estimator along with simulation study and comparison among estimators.

We first study bound-state solutions of the Klein--Gordon equation for vector potentials which in one spatial dimension have the form V(x) = vf(x), where f(x)≤0 is the shape of a finite symmetric centra=l potential that is monotone non-decreasing on [0, ꝏ) and vanishes as x→ ꝏ , and v>0 is the coupling parameter.

We characterize the graph of spectral functions of the form v= G(E) which represent solutions of the eigen-problem in the coupling parameter v for a given E:  they are concave, and at most uni-modal with a maximum near the lower limit E = -m of the energy E ԑ (-m, m). This formulation of the spectral problem immediately extends to central potentials in d > 1 spatial dimensions. Secondly, for each of the dimension cases, d=1 and d ≥ 2, a comparison theorem is proven, to the effect that if two potential shapes are ordered f_1(r) ≤ f_2(r), then so are the corresponding pairs of spectral functions G_1(E) ≤ G_2(E) for each of the existing eigenvalues. These results remove the restriction to positive energies necessitated by earlier comparison theorems for the Klein--Gordon equation by Hall and Aliyu. Corresponding results are obtained when scalar potentials S(x) are also included.

We then weaken the condition for the ground states by proving that if ∫^x₀[ƒ₂(t) - ƒ₁(t]φ(t)dt ≥ 0, the corresponding coupling parameters remain ordered, where φ_i= 1, or φ_i is the bound state solution of the Klein--Gordon equation with potential V_i, i = 1, 2.