Mathématiques et statistiques // Mathematics and Statistics

Permanent URI for this collectionhttps://hdl.handle.net/10393/19587

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Analysis on Manifolds
(2025) Dionne, Benoit
The document is aimed at students planning to pursue their studies at the graduate level. The first part of the document is a rigorous introduction to integration, in particular to integration on manifolds. It includes a chapter on manifolds and differential forms. A chapter is devoted to applications and the link between modern differential geometry and classical vector calculus. The second part of the document is an introduction to modern differential geometry. There is a chapter on De Rham cohomology and a chapter on homology and cohomology, both simplicial and singular, with a proof of the relation between the simplicial and singular cohomology and de Rham cohomology. The last chapter is on Riemannian geometry and covers Cartan structural equations and geodesics, including a proof of Gauss-Bonnet theorem. The document ends with an introduction to non-euclidean geometries.
Partial Differential Equations
(2024) Dionne, Benoit
With an emphasis on teaching and learning, this book is for an advanced undergraduate course on partial differential equations or a beginning graduate course on partial differential equations. Divided into 13 chapters, this textbook covers most of the important topics found in books on partial differential equations, using most of the time the same approach. This book contains material on partial differential equations usually taught at the undergraduate level for science and engineering students. The reason for including this material is that many mathematics students do not have the chance to study partial differential equations at the undergraduate level. The sections covering the undergraduate material are accessible to readers with only a calculus background. The presentation in these sections is usually more formal and therefore appropriate for introducing partial differential equations for science and engineering students. This book is accessible to students with a minimal introduction to measure and integration, and functional analysis.
Benchmarking Differential Privacy and Existing Anonymization or De-identification Guidance
(2024-10-22) Kulik, Rafal
The project consisted of three phases: 1) Phase 1: Practical Assessment Framework of Differential Privacy 2) Phase 2: Experimentation Phase 3) Phase 3: Scope of Policies We have also conducted a survey of students, on their understanding and their attitude to different privacy concepts. The results of the survey can be found in the Appendices section. The first phase focuses on technical aspects related to differential privacy. We were able to discover the inconsistencies in the definitions of differential privacy and its parameters. We were also able to identify the basic techniques and the limitations of differential privacy. These discoveries then allowed us to determine suitable language and identify opportunities to overcome challenges of implementing differential privacy and better optimize its use within existing frameworks. In the second, exploratory phase we conducted 5 experiments to delve deeper into the intricacies of differential privacy. Phase 2 allowed us to answer fundamental questions about the relationship between the main techniques (pre-processing and post-processing) of differential privacy, differential privacy in a data privacy and data utility context, the relationship and combination of k-anonymity and differential privacy, and the comparison of privatized data to original data. We also concluded that some standard statistical techniques cannot be used in the context of assessing data privacy and data utility. Although there has been considerable development of privacy enhancing technologies that go beyond anonymization, their relationship to the concept of anonymization in data protection law is not always clear. Currently, there are no clear guidelines that explain how differential privacy may be aligned with the concept of anonymization in privacy law or how it might relate to the relative approach to anonymization developed in Canadian case law. This third part of this project will examine how differential privacy can be integrated with legal requirements in PIPEDA and in the proposed Bill C-27.
Calcul différentiel de plusieurs variables
(2024) Savage, Alistair; El basraoui, Abdelkrim
Notes pour le cours MAT 2522 Calcul différentiel de plusieurs variables à l’Université d’Ottawa. Nous nous pencherons principalement sur les fonctions à valeurs réelles à entrées multiples à valeurs réelles. De nombreux concepts seront discutés en utilisant le langage des vecteurs et de l’algèbre linéaire puisque c’est le cadre le plus naturel pour le calcul à plusieurs variables. Nous verrons comment une grande partie du calcul que vous avez appris dans les cours précédents se généralise en dimensions multiples. Cela nous permettra d’explorer des mathématiques nouvelles et intéressantes, comme l’intégration sur des surfaces et des régions tridimensionnelles.
Numerical Analysis
(2023) Dionne, Benoit
This book covers the material normally presented in a two-term course on numerical analysis, starting with the basic concept and ending with topics that are more appropriate for a course in numerical analysis for differential equations: numerical solution of systems of linear and nonlinear equations, polynomial interpolation, numerical approximation of functions, numerical computation of eigenvalues, numerical derivation and integration, numerical solution of ordinary and partial differential equations. With a focus on algorithms, it can be used as an introduction to numerical analysis for engineering and applied science students. With a focus on theory, it can be used as an introduction to numerical analysis for students in mathematics or physics. Most of the numerical methods presented in this book are accompanied by a MATLAB code.
Calcul différentiel et intégral
(2023) Dionne, Benoit
Le contenu de ce manuel couvre la grande majorité des sujets présentés dans les cours de calcul différentiel et intégral pour les étudiants en sciences et génie. Les seuls préalables sont les mathématiques normalement enseignées au secondaire en Ontario. Ce manuel peut être utilisé pour trois des variantes des cours de calcul différentiel et intégral que nous retrouvons dans la majorité des universités en Ontario : Calcul différentiel et intégral pour les étudiants en génie, Calcul différentiel et intégral pour les étudiants en sciences de la vie et Calcul différentiel et intégral pour les étudiants en administration. Des notes de cours pour le calcul différentiel et intégral pour les sciences de la vie sont disponibles sur ce site : https://benoitdionne.github.io/teaching/lecture_notes_fr.html
On computing the Newton polygons of plus and minus $p$-adic $L$-functions
(2023) Balakrishnan, Sai Sanjeev; Lei, Antonio; Palvannan, Bharathwaj
Let $p\geq3$ be a prime number and $E/\mathbb{Q}$ an elliptic curved with good supersingular reduction at $p$ and $a_p(E)=0$. In this article, we study the computation of the Newton polygons of Pollack's plus and minus $p$-adic $L$-functions attached to $E$. In particular, we furnish new examples where the assumption GCD in [Forum Math. Sigma, 7:Paper No. e25] holds. We also take this opportunity to rectify two imprecisions in the aforementioned article.
En avant les espaces vectoriels : une introduction à l’algèbre linéaire (première édition)
(2021) Giordano, Thierry; Jessup, Barry; Nevins, Monica; El basraoui, Abdelkrim
Conçu pour un cours de première année universitaire, ce manuel en algèbre linéaire adopte une approche peu commune : il présente les espaces vectoriels dès le début et traite des systèmes linéaires qu’après une introduction approfondie aux espaces vectoriels. Cette approche est fondée sur l’expérience des auteurs ayant observé au cours des 25 dernières années que les étudiantes et étudiants ont souvent besoin davantage de temps pour maîtriser les espaces vectoriels alors que les manuels traditionnels relèguent plutôt le sujet à la fin du cours. De cette façon, ces nouvelles notions au coeur de l’algèbre linéaire qui sont souvent considérées comme abstraites et difficiles dans un cours d’introduction peuvent ensuite être utilisées dans le reste du cours ainsi que différents contextes.
Vector Spaces First: An Introduction to Linear Algebra (4th Edition)
(2021) Giordano, Thierry; Jessup, Barry; Nevins, Monica
Created for a first-year university course, this linear algebra textbook takes an unusual approach: it introduces vector spaces at the outset and deals with linear systems only after a thorough introduction to vector spaces. This approach is based on the authors' experience over the past 25 years that students often need more time to master vector spaces while traditional textbooks relegate the topic to the end of the course. In this way, these new notions at the heart of linear algebra that are often considered abstract and difficult in an introductory course can then be used in the rest of the course as well as in different contexts.
The effect of landscape fragmentation on Turing-pattern formation
(2022) Zaker, Nazanin; Cobbold, Christina A.; Lutscher, Frithjof
Diffusion-driven instability and Turing pattern formation are a well-known mechanism by which the local interaction of species, combined with random spatial movement, can generate stable patterns of population densities in the absence of spatial heterogeneity of the underlying medium. Some examples of such patterns exist in ecological interactions between predator and prey, but the conditions required for these patterns are not easily satisfied in ecological systems. At the same time, most ecological systems exist in heterogeneous landscapes, and landscape heterogeneity can affect species interactions and individual movement behavior. In this work, we explore whether and how landscape heterogeneity might facilitate Turing pattern formation in predator–prey interactions. We formulate reaction-diffusion equations for two interacting species on an infinite patchy landscape, consisting of two types of periodically alternating patches. Population dynamics and movement behavior differ between patch types, and individuals may have a preference for one of the two habitat types. We apply homogenization theory to derive an appropriately averaged model, to which we apply stability analysis for Turing patterns. We then study three scenarios in detail and find mechanisms by which diffusion-driven instabilities may arise even if the local interaction and movement rates do not indicate it.
Reactivity of communities at equilibrium and periodic orbits
(2020) Lutscher, Frithjof; Wang, Xiaoying
Reactivity measures the transient response of a system following a perturbation from a stable state. For steady states, the theory of reactivity is well developed and frequently applied. However, we find that reactivity depends critically on the scaling used in the equations. We therefore caution that calculations of reactivity from nondimensionalized models may be misleading. The attempt to extend reactivity theory to stable periodic orbits is very recent. We study reactivity of periodically forced and intrinsically generated periodic orbits. For periodically forced systems, we contribute a number of observations and examples that had previously received less attention. In particular, we systematically explore how reactivity depends on the timing of the perturbation. We then suggest ways to extend the theory to intrinsically generated periodic orbits. We investigate several possible global measures of reactivity of a periodic orbit and show that there likely is no single quantity to consistently measure the transient response of a system near a periodic orbit.
Dynamical stabilization and traveling waves in integrodifference equations
(2020) Bourgeois, Adèle; LeBlanc, Victor; Lutscher, Frithjof
Integrodifference equations are discrete-time analogues of reactiondiffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.
Shigesada et al. (1986) and population spread in heterogeneous environments
(2020) Lutscher, Frithjof
Evolutionarily stable movement strategies in reaction-diffusion models with edge behavior
(2020) Maciel, Gabriel; Cosner, Chris; Cantrell, Robert Stephen; Lutscher, Frithjof
Many types of organisms disperse through heterogeneous environments as part of their life histories. For various models of dispersal, including reaction-advection-diffusion models in continuously varying environments, it has been shown by pairwise invasibility analysis that dispersal strategies which generate an ideal free distribution are evolutionarily steady strategies (ESS, also known as evolutionarily stable strategies) and are neighborhood invader strategies (NIS). That is, populations using such strategies can both invade and resist invasion by populations using strategies that do not produce an ideal free distribution. (The ideal free distribution arises from the assumption that organisms inhabiting heterogeneous environments should move to maximize their fitness, which allows a mathematical characterization in terms of fitness equalization.) Classical reaction diffusion models assume that landscapes vary continuously. Landscape ecologists consider landscapes as mosaics of patches where individuals can make movement decisions at sharp interfaces between patches of different quality. We use a recent formulation of reaction-diffusion systems in patchy landscapes to study dispersal strategies by using methods inspired by evolutionary game theory and adaptive dynamics. Specifically, we use a version of pairwise invasibility analysis to show that in patchy environments, the behavioral strategy for movement at boundaries between different patch types that generates an ideal free distribution is both globally evolutionarily steady (ESS) and is a global neighborhood invader strategy (NIS).
The Effect of Movement Behavior on Population Density in Patchy Landscapes
(2020) Zaker, Nazanin; Ketchemen, Laurence; Lutscher, Frithjof
Many biological populations reside in increasingly fragmented landscapes, where habitat quality may change abruptly in space. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. Several recent publications explore how such individual movement behavior affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We prove existence, uniqueness and global stability, and we classify their qualitative shape depending on movement behavior. We apply our results to study the question why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape.
The emergence of phase asynchrony and frequency modulation in metacommunities
(2019) Guichard, Frederic; Zhang, Yuxiang; Lutscher, Frithjof
Spatial synchrony can summarize complex patterns of population abundance. Studies of phase synchrony predict that limited dispersal can drive either in-phase or out-of-phase synchrony, characterized by a constant phase difference among populations. We still lack an understanding of ecological processes leading to the loss of phase synchrony. Here we study the role of limited dispersal as a cause of phase asynchrony defined as fluctuating phase differences among populations. We adopt a minimal predator-prey model allowing for dispersal-induced phase asynchrony, and show its dependence on species traits. We show that phase asynchrony in a homogeneous metacommunity requires a minimum of three communities and is characterized by the emergence of regional frequency modulation of population fluctuations. This frequency modulation results in spectral signatures in local time series that can be used to infer the causes and properties of metacommunity dynamics. Dispersal-induced phase asynchrony extends the application of ecological theories of synchrony to non stationary time series, and predicts observed spatiotemporal patterns in marine metacommunities.
How Spatial Heterogeneity Affects Transient Behavior in Reaction–Diffusion Systems for Ecological Interactions?
(2019) Wang, Xiaoying; Efendiev, Messoud; Lutscher, Frithjof
Most studies of ecological interactions study asymptotic behavior, such as steady states and limit cycles. The transient behavior, i.e., qualitative aspects of solutions as and before they approach their asymptotic state, may differ significantly from asymptotic behavior. Understanding transient dynamics is crucial to predicting ecosystem responses to perturbations on short timescales. Several quantities have been proposed to measure transient dynamics in systems of ordinary differential equations. Here, we generalize these measures to reaction-diffusion systems in a rigorous way and prove various relations between the non-spatial and spatial effects, as well as an upper bound for transients. This extension of existing theory is crucial for studying how spatially heterogeneous perturbations and the movement of biological species involved affect transient behaviors. We illustrate several such effects with numerical simulations.
Movement behaviour of fish, harvesting-induced habitat degradation and the optimal size of marine reserves
(2019) Alqawasmeh, Yousef; Lutscher, Frithjof
Many models have assessed how marine reserves protect fish populations and – under certain conditions – simultaneously increase yield. Only recently have models considered the effects of fishing-induced habitat damage by assuming reduced population growth in fishing areas. Even though it is understood that fish movement patterns affect the functioning and design of marine reserves, fishing-induced changes in movement patterns, as a response to decreased habitat quality, have not been studied in this context. Our work explores how harvesting-induced movement behaviour of fish can affect optimal yield and size of a marine reserve. Our model is based on reaction-diffusion equations and recent advances in their application to strongly heterogeneous environments with sharp transitions in environmental conditions. We model movement behaviour in response to harvesting and habitat destruction via increased diffusion rates and increased preference for protected areas, and implement reduced reproduction as an effect of habitat degradation. We find an alternative mechanistic explanation for the empirical observation that high fish mobility may not decrease fish density insidea reserve. We also find that movement-behavioural responses of fish to harvesting can decrease the economic value of protected areas and increase their conservation value. For maximum sustainable yield, we find that a low harvesting rate and small protected area are optimal when fish show a strong preference for protected areas as a response to fishing efforts. On the other hand, a high harvesting rate and a large protected area are optimal if fish respond to harvesting by a strong increase in movement rates in fishing areas.
Turing patterns in a predator–prey model with seasonality
(2019) Wang, Xiaoying; Lutscher, Frithjof
Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction-diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the 'summer' season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington-DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).
Analysis of Integrodifference Equations with a Separable Dispersal Kernel
(2019) Bramburger, Jason; Lutscher, Frithjof
Integrodifference equations are a class of infinite-dimensional dy- namical systems in discrete-time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dis- persal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite- dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.

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