No description available.

In this paper, a relationship between the zeros and critical points of a polynomial p(z) is established. The relationship is used to prove Sendov's conjecture in some special cases.

We examine an early model for the interaction of HIV with CD4+ T cells in vivo and define possible parameters and effects of said parameters on the model. We then examine a newer, more simplified model for the interaction of HIV with CD4+ T cells that also considers four populations: uninfected T cells, latently infected T cells, actively infected T cells, and free virus. The stability of both the disease free steady state and the endemically infected steady state are examined utilizing standard methods and the Routh-Hurwitz criteria. We show that if N, the number of infectious virions produced per actively infected T cell, is less than a critical value, , then the uninfected state is the only steady state in the non negative orthant, and this state is stable. We establish an expression for . If , then the uninfected steady state is unstable, and the endemically infected state can be stable or unstable, depending on the value of the parameters utilized.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

No description available.

Over the past few years, White Americans have been exposed to moments of "racial reckonings" –where America was forced to realize racial injustice—often caused by police killings of Black Americans. The summer of 2020 sparked various discussions about White allies and their role in racial justice. Previous scholarship has explored White racial awakenings, also referred to as turning points, when White allies transition into allyship; however, little is known about how these narratives affect White allies' definitions and practices of White allyship. By interviewing self-identifying White allies in college, I examine types of turning point narratives and how they correspond to understandings of ally definitions and practices. I argue that experiencing turning points led White allies to similar definitions and practices of White allyship depending on whether these points occurred in routine or non-routine White experiences. Turning points that arise from typical White experiences (exposure to diversity and education) led White allies to common allyship failures and complicity with racialized structures. Alternatively, turning points resulting from uncommon White experiences (exposure to protests and diverse families) led allies to definitions and practices representing non-complicity with racialized structures. / Master of Science / Over the past few years, White Americans have been exposed to moments of race related dialogues, often caused by police killings of Black Americans. The summer of 2020 sparked various discussions about White allies and their role in racial justice. Previous scholarship has explored White racial awakenings, also referred to as turning points, when White allies transition into allyship; however, little is known about how these narratives affect White allies' definitions and practice of White allyship. By interviewing self-identifying White allies in college, I examine types of turning points and how they correspond ally definitions and practices. I argue that experiencing turning points led White allies to similar definitions and practices of White allyship depending on whether these points occurred in routine or non-routine White experiences. Turning points that arise from typical White experiences (exposure to diversity and education) led White allies to common allyship failures. Alternatively, turning points resulting from uncommon White experiences (exposure to protests and diverse families) led allies to beneficial definitions and practices of allyship.

Many real-world systems can be represented as network-graphs. Some of the networks have an inherent community structure based on interactions. The problem of identifying this grouping structure given a graph is termed as community detection problem which has certain existing algorithms. This thesis contributes by providing specific improvements to various community detection algorithms such as spectral clustering and extreme point algorithm. One of the main contributions is proposing a new sub-sampling method to make existing spectral clustering method scalable by reducing the computational complexity. Also, we have implemented extreme points algorithm for a general multiple communities detection case along with a sub-sampling based version to reduce the computational complexity. We have also developed spectral clustering algorithm for popularity-adjusted block model (PABM) model based graphs to make the algorithm exact thus improving its accuracy. / Master of Science / We live in an increasingly interconnected world, where agents constantly interact with each other. This general agent-interaction framework describes many important systems, such as social interpersonal systems, protein interaction systems, trade and financial systems, power grids, and the World Wide Web, to name a few. By denoting agents as nodes and their interconnections as links, any such system can be represented as a network. Such networks or graphs provide a powerful and universal representation for analyzing a wide variety of systems spanning a remarkable range of scientific disciplines. Networks act as conduits for many kinds of transmissions. For instance, they are influential in the dissemination of ideas, adoption of technologies, helping find jobs and spread of diseases. Thus networks play a critical role both in providing information and helping make decisions making them a crucial part of the Data and Decisions Destination Area. A well-known feature of many networks is community structure. Nodes in a network are often found to belong to groups or communities that exhibit similar behavior. The identification of this community structure, called community detection, is an important problem with many critical applications. For example, communities in a protein interaction network often correspond to functional groups. This thesis focuses on cutting-edge methods for community detection in networks. The main approach is efficient community detection via sub-sampling. This is applied to two different approaches. The first approach is optimization of a modularity function using a low-rank approximation for multiple communities. The second approach is a spectral clustering where we aim to formulate an algorithm for community detection by exploiting the eigenvectors of the network adjacency matrix.

This paper will be concerned with the convergence of complex power series.

Cette thèse poursuit deux directions dans le domaine des isolants et supraconducteurs topologiques.Dans la première partie de la thèse nous étudions des isolants en deux dimensions sur réseau, présentant un effet Hall quantique anormal (c'est-à-dire en l'absence d'un champ magnétique externe), induit par la présence d'un flux magnétique inhomogène dans la maille. Le système possède des phase isolantes caractérisés par un invariant topologique, le nombre de Chern, qui est lié à la conductance portée par le bord états. Nous montrons que les modèles à deux bandes admettent des phase à nombre de Chern arbitraire, ou, de façon équivalente, un nombre arbitraire d'états de bord, quand on augmente la portée des couplages sur réseau. Cette compréhension est rendue possible grâce à la démonstration d'une formule montrant que le nombre de Chern d'une bande dépend de certains propriétés d'un ensemble discret de points dans la zone de Brillouin, les points de Dirac en l'absence du gap. Ces idées sont rendues plus concrètes dans l'étude du modèle de Haldane et dans la création d'un modèle artificiel avec cinq phases de Chern dont les états de bord sont déterminés en détail. La deuxième partie de la thèse porte sur les supraconducteurs topologiques unidimensionnels qui exhibent des états exotiques d'énergie zéro: les états liés de Majorana. Nous étudions ici la présence de fermions de Majorana dans des fils de semiconducteurs à fort couplage spin-orbite sous l’effet de proximité d'un supraconducteur d'onde s. Nous montrons que la polarisation de spin des degrés de liberté électroniques dans la fonction d'onde Majorana dépend du poids relatif du couplage spin-orbite Dresselhaus et Rashba. Nous étudions également les fermions de Majorana dans des jonctions linéaires longues supraconducteur-normal et supraconducteur-normal-supraconducteur (SNS) où ils apparaissent comme des états étendus dans la jonction normale. En outre, la géométrie d'anneaux peut être mise en correspondance avec une jonction SNS, et, sous l'action de gradients dans la phase supraconductrice, des fermions Majorana étendus se forment encore à l'intérieur du fil normal. Enfin, un modèle à deux bandes avec des fermions de Majorana multiples est traité. Nous démontrons que les jonctions Josephson construites à partir de ce modèle maintiennent l'une des signatures remarquables des fermions de Majorana, à savoir la périodicité 4π de l'effet Josephson fractionnaire. / This thesis follows two threads in the field of topological insulators and superconductors. The first part of the thesis is devoted to the study of two-dimensional quantum anomalous Hall insulators on a lattice, in the absence of an external magnetic flux, but induced by an inhomogeneous flux in the unit cell. The system possesses several gapped phases characterized by a topological invariant, the Chern number, that is related to the conductance carried by the edge states. Here we show that two-band models admit an arbitrary large number of Chern phases or, equivalently, an arbitrary number of edge states, by adding hopping between distant neighbor sites. This result is based on a formula proving that the Chern number of a band depends on certain properties of a finite set of points in the Brillouin zone, i.e. the Dirac points for the gapless system. These ideas are made more concrete in the study of a modified Haldane model, and also by creating an artificial model with five Chern phases, whose edge states are determined in detail. The second part of the thesis focuses on one-dimensional topological superconductors with exotic zero-energy edge states: the Majorana bound states. Here we investigate the presence of Majorana fermions in spin-orbit coupled semiconducting wire in proximity to an s-wave superconductor. We show that the spin-polarization of the electronic degrees of freedom in the Majorana wave function depends on the relative weight of Dresselhaus and Rashba spin-orbit couplings. We also investigate Majorana fermions in linear superconductor-normal and long superconductor-normal-superconductor (SNS) junctions where they appear as extended states in the normal junction. Furthermore, ring geometries can be mapped to an SNS junction, and, we have shown that under the action of superconducting phases gradients, extended Majorana fermions can form again inside the normal wire. Finally a two-band model with multiple Majorana fermions is treated and we show that Josephson junctions built from this model maintain the 4π periodicity for the fractional Josephson effect, one of Majorana fermions signatures.