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

Let X be an Atkin-Lehner quotient of the modular curve X_0(N) whose Jacobian J_f is a simple quotient of J_0(N)^{new} over Q. We give analytic methods for determining the rational points of X using quadratic Chabauty by explicitly computing two p-adic Gross--Zagier formulas for the newform f of level N and weight 2 associated with J_f when f has analytic rank 1. Combining results of Gross-Zagier and Waldspurger, one knows that for certain imaginary quadratic fields K, there exists a Heegner divisor in J_0(N)(K) whose image is finite index in J_f(Q) under the action of Hecke. We give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna, assuming some hypotheses on the prime p and on K. This value is proportional to the logarithm of the Heegner divisor on J_f with respect to the differential form f dq/q. We also compute the p-adic height of the Heegner divisor on J_f using a p-adic Gross-Zagier formula of Perrin-Riou. Additionally, we give algorithms for the geometric quadratic Chabauty method of Edixhoven and Lido. Our algorithms describe how to translate their algebro-geometric method into calculations involving Coleman-Gross heights, logarithms, and divisor arithmetic. We achieve this by leveraging a map from the Poincaré biextension to the trivial biextension.

La membrane cellulaire est un environnement très dynamique où diffusent plusieurs familles de protéines. Pour assurer la communication entre neurones, certains récepteurs membranaires diffusant librement doivent être stabilisés aux jonctions de communication neuronales: les synapses. Des changements dans le comportement diffusif de ces récepteurs induisent des variations de leurs populations à la synapse. La capacité d'observer ces protéines diffuser individuellement est donc importante pour mieux comprendre certains mécanismes cellulaires associés au fonctionnement du système nerveux. Alors que les fluorophores organiques (Alexa, Atto, ...) et génétiquement encodées (GFP, RFP, ...) photoblanchissent rapidement, les nanocristaux semi-conducteurs fluorescents (points quantiques: QDs) sont photo-stables, ce qui permet de suivre les protéines marquées sur de plus longues périodes. Même si la structure cristalline du QD dépasse rarement les 10nm de diamètre, la taille de la sonde fonctionnalisée, une fois équipée d'un enrobage hydrophile et de protéines d'hameçonnage, est d'environ 30 nm, limitant leur accessibilité dans la fente synaptique (environ 20 nm). De plus, la multivalence des versions commerciales des QDs fonctionnalisés pose problème, en leur permettant de lier plusieurs récepteurs à la fois. Dans ce mémoire, nous présentons donc différentes approches pour la biofonctionnalisation de petit QDs monovalents, ainsi que les principales stratégies pour marquer des récepteurs synaptiques uniques à partir de QDs. De plus, nous présentons une étude de la diffusion de récepteurs synaptiques diffusant à l'intérieur et à l'extérieur de synapses et comparons les résultats en fonction des 2 types de sondes utilisés : QDs enrobés d'anticorps et QDs enrobés de streptavidine. Ces travaux devraient contribuer à l'optimisation d'une approche pour suivre les dynamiques des récepteurs synaptiques dans des modèles de plasticité synaptique.

Le sujet de cette thèse est l'étude des progressions arithmétiques dans les nombres entiers. Plus précisément, nous nous intéressons à borner inférieurement v(N), la taille du plus grand sous-ensemble des nombres entiers de 1 à N qui ne contient pas de progressions arithmétiques de 3 termes. Nous allons donc construire de grands sous-ensembles de nombres entiers qui ne contiennent pas de telles progressions, ce qui nous donne une borne inférieure sur v(N). Nous allons d'abord étudier les preuves de toutes les bornes inférieures obtenues jusqu'à présent, pour ensuite donner une autre preuve de la meilleure borne. Nous allons considérer les points à coordonnés entières dans un anneau à d dimensions, et compter le nombre de progressions arithmétiques qu'il contient. Pour obtenir des bornes sur ces quantités, nous allons étudier les méthodes pour compter le nombre de points de réseau dans des sphères à plusieurs dimensions, ce qui est le sujet de la dernière section. / The subject of this thesis is the study of arithmetic progressions in the integers. Precisely, we are interested in the size v(N) of the largest subset of the integers from 1 to N that contains no 3 term arithmetic progressions. Therefore, we will construct a large subset of integers with no such progressions, thus giving us a lower bound on v(N). We will begin by looking at the proofs of all the significant lower bounds obtained on v(N), then we will show another proof of the best lower bound known today. For the proof, we will consider points on a large d-dimensional annulus, and count the number of integer points inside that annulus and the number of arithmetic progressions it contains. To obtain bounds on those quantities, it will be interesting to look at the theory behind counting lattice points in high dimensional spheres, which is the subject of the last section.

Le sujet de cette thèse est l'étude des progressions arithmétiques dans les nombres entiers. Plus précisément, nous nous intéressons à borner inférieurement v(N), la taille du plus grand sous-ensemble des nombres entiers de 1 à N qui ne contient pas de progressions arithmétiques de 3 termes. Nous allons donc construire de grands sous-ensembles de nombres entiers qui ne contiennent pas de telles progressions, ce qui nous donne une borne inférieure sur v(N). Nous allons d'abord étudier les preuves de toutes les bornes inférieures obtenues jusqu'à présent, pour ensuite donner une autre preuve de la meilleure borne. Nous allons considérer les points à coordonnés entières dans un anneau à d dimensions, et compter le nombre de progressions arithmétiques qu'il contient. Pour obtenir des bornes sur ces quantités, nous allons étudier les méthodes pour compter le nombre de points de réseau dans des sphères à plusieurs dimensions, ce qui est le sujet de la dernière section. / The subject of this thesis is the study of arithmetic progressions in the integers. Precisely, we are interested in the size v(N) of the largest subset of the integers from 1 to N that contains no 3 term arithmetic progressions. Therefore, we will construct a large subset of integers with no such progressions, thus giving us a lower bound on v(N). We will begin by looking at the proofs of all the significant lower bounds obtained on v(N), then we will show another proof of the best lower bound known today. For the proof, we will consider points on a large d-dimensional annulus, and count the number of integer points inside that annulus and the number of arithmetic progressions it contains. To obtain bounds on those quantities, it will be interesting to look at the theory behind counting lattice points in high dimensional spheres, which is the subject of the last section.

published_or_final_version / Chinese Medicine / Master / Master of Philosophy

Le but de cette thèse est d'utiliser plusieurs résultats profonds de géométrie diophantienne et de géométrie algébrique pour obtenir des applications à la factorisation des polynômes lacunaires. Dans la première partie, on décrit un algorithme qui détermine une représentation des points de torsion d'une sous-variété de Gn m définie par des polynômes lacunaires. La complexité de cet algorithme est quasilinéaire en le logarithme du degré des polynômes définissant cette sous-variété. Dans la seconde partie, on s'intéresse à des systèmes surdéterminés d'équations polynomiales. On décrit un algorithme qui permet d'écrire les zéros communs de trois polynômes à deux variables comme une réunion finie d'intersections complètes en dehors d'un ouvert de A2. La complexité de cet algorithme est encore quasi-linéaire en le logarithme du degré des polynômes en entrée mais cet algorithme dépend de la validité de la conjecture de Zilber qui est encore à ce jour un problème ouvert.

We examine a class of planar area preserving mappings and give a geometric condition that guarantees the existence of homoclinic points. To be more precise, let $f,g:R \to R$ be $C^1$ functions with domain all of $R$. Let $F:R^2 \to R^2$ denote a horizontal reflection in the curve $x=-f(y)$, and let $G:R^2 \to R^2$ denote a vertical reflection in the curve $y=g(x)$. We consider maps of the form $T=G \circ F$ and show that a simple geometric condition on the fixed point sets of $F$ and $G$ leads to the existence of a homoclinic point for $T$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-17 14:22:35.72

Physical properties are fundamental to all chemical, biochemical and environmental industries. One of these properties is the normal boiling point of a compound. However, experimental values in literature are quite limited and measurements are expensive and time consuming. For this reason, group contribution estimation methods are generally used. Group contribution is the simplest form of estimation requiring only the molecular structure as input. Consequently, the aim of this project was the development of a reliable group contribution method for the estimation of normal boiling points of non-electrolytes applicable for a broad range of components. A literature review of the available methods for the prediction of the normal boiling points from molecular structure only, was initially undertaken. From the review, the Cordes and Rarey (2002) method suggested the best scientific approach to group contribution. This involved defining the structural first-order groups according to its neighbouring atoms. This definition also provided knowledge of the neighbourhood and the electronic structure of the group. The method also yielded the lowest average absolute deviation and probability of prediction failure. Consequently, the proposed group contribution method was then developed using the Cordes and Rarey method as a starting point. The data set included experimental data for approximately 3000 components, 2700 of which were stored in the Dortmund Data Bank (DDB) and about 300 stored in Beilstein. The mathematical formalism was modified to allow for separate examination and regression of individual contributions using a meta-language filter program developed specifically for this purpose. The results of this separate examination lead to the detection of unreliable data, the re-classification of structural groups, and introduction of new structural groups to extend the range of the method. The method was extended using steric parameters, additional corrections and group interaction parameters. Steric parameters contain information about the greater neighbourhood of a carbon. The additional corrections were introduced to account for certain electronic and structural effects that the first-order groups could not capture. Group interactions were introduced to allow for the estimation of complex multifunctional compounds, for which previous methods gave extraordinary large deviations from experimental findings. Several approaches to find an improved linearization function did not lead to an improvement of the Cordes and Rarey method. The results of the new method are extensively compared to the work of Cordes and Rarey and currently-used methods and are shown to be far more accurate and reliable. Overall, the proposed method yielded an average absolute deviation of 6.50K (1.52%) for a set of 2820 components. For the available methods, Joback and Reid produced an average absolute deviation of 21.37K (4.67%) for a set of 2514 components, 14.46K (3.53%) for 2578 components for Stein and Brown, 13.22K (3.15%) for 2267 components for Constantinou and Gani, 10.23 (2.33%) for 1675 components for Marrero and Pardillo and 8.18K (1.90%) for 2766 components for Cordes and Rarey. This implies that the proposed method yielded the lowest average deviation with the broadest range of applicability. Also, on an analysis of the probability of prediction failure, only 3% of the data was greater than 20K for the proposed method. This detailed comparison serves as a very valuable tool for the estimation of prediction reliability and probable error. Structural groups were defined in a standardized form and the fragmentation of the molecular structures was performed by an automatic procedure to eliminate any arbitrary assumptions. / Thesis (M.Sc.Eng.)-University of Natal, Durban, 2004.

A morphism $\sigma$ is \emph{(strongly) unambiguous} with respect to a word $\alpha$ if there is no other morphism $\tau$ that maps $\alpha$ to the same image as $\sigma$. Moreover, $\sigma$ is said to be \emph{weakly unambiguous} with respect to a word $\alpha$ if $\sigma$ is the only \emph{nonerasing} morphism that can map $\alpha$ to $\sigma(\alpha)$, i.\,e., there does not exist any other nonerasing morphism $\tau$ satisfying $\tau(\alpha) = \sigma(\alpha)$. In the first main part of the present thesis, we wish to characterise those words with respect to which there exists a weakly unambiguous \emph{length-increasing} morphism that maps a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions. \par The second main part of the present thesis studies the question of whether, for any given word, there exists a strongly unambiguous \emph{1-uniform} morphism, i.\,e., a morphism that maps every letter in the word to an image of length $1$. This problem shows some connections to previous research on \emph{fixed points} of nontrivial morphisms, i.\,e., those words $\alpha$ for which there is a morphism $\phi$ satisfying $\phi(\alpha) = \alpha$ and, for a symbol $x$ in $\alpha$, $\phi(x) \neq x$. Therefore, we can expand our examination of the existence of unambiguous morphisms to a discussion of the question of whether we can reduce the number of different symbols in a word that is not a fixed point such that the resulting word is again not a fixed point. This problem is quite similar to the setting of Billaud's Conjecture, the correctness of which we prove for a special case.