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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Fu, Lei; Fritz, Claudia; Scavone, Gary;

    This experiment explores how changes in soundpost height affect the perceptual qualities of the violin and whether there is a threshold of change below which players and luthiers do not perceive differences. A violin installed with a height-adjustable carbon fibre soundpost was employed. The experiment was designed as a sequence of playing tests. An experimenter was present to change the soundpost height. Thirteen professional violinists and six luthiers participated. The experiment involved two phases. During the first phase, subjects played and described their feelings about the violin with different soundpost settings in order to find their optimal soundpost height. During the second phase, the experimenter randomly increased, decreased or did not change the soundpost height in ten trials within a range of approximately ±0.1 mm around their optimal height. For each trial, subjects were asked to play the violin, comparing it with the previous setting, and to decide whether they were the same or different. Initial results indicate that each subject's optimal soundpost height varies within an interquartile range of 0.3 mm and the smallest height variation that could be recognized above chance level is about 0.04 mm.

    image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ Hyper Article en Lig...arrow_drop_down
    image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Greyson, Devon; Morrison, Heather; Waller, Andrew;

    This article is a summary of recent Open Access activity in Canada, focusing on policies and mandates, repositories, and initiatives in libraries.

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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ E-LISarrow_drop_down
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Ilcinkas, David; Pelc, Andrzej;

    The behavior of rational selfish agents has been classically studied in the framework of strategic games in which each player has a set of possible actions, players choose actions simultaneously and the payoff for each player is determined by the matrix of the game. However, in many applications, players choose actions asynchronously, and simultaneity of this process is not guaranteed: it is possible that a player learns the action of another player before making its choice. Delays of choices are controled by the adversary and each player can only secure the worst-case payoff over the adversary's decisions. In this paper we consider such asynchronous versions of arbitrary two-person strategic games and we study how the presence of the asynchronous adversary influences the behavior of the players, assumed to be selfish but rational. We concentrate on deterministic (pure) strategies, and in particular, on the existence and characteristics of pure Nash equilibria in such games. It turns out that the rational behavior of players changes significantly if the decision process is asynchronous. We show that pure Nash equilibria often exist in the asynchronous version of the game even if there were no such equilibria in the synchronous game. We also show that a mere threat of asynchrony in the game may make social optimum a rational choice while it was not rational in the synchronous game.

    image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ Hyper Article en Lig...arrow_drop_down
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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Holbrook, J.A.;

    This paper addresses indicators of the performance of the Vancouver human health biotechnology cluster.

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    image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ Simon Fraser Univers...arrow_drop_down
      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Branfireun, B. A.; Roulet, N. T.;

    The fate and transport of methylmercury (MeHg) were studied in a small boreal catchment. Hydrological processes largely govern the magnitude of the flux of MeHg. Seasonal and inter-annual variability in hydrology produce variable source strengths of MeHg throughout the catchment. The mass flux of MeHg within, and from the catchment is dependent on the mass flux of water and the relative placement of landscape units in the catchment hydrological cascade. Hydrology also governs the maintenance of the methylating environments in the catchment. Specifically, hydrological processes maintain zones of anoxia in both the catchment uplands and peatlands that support obligate anaerobic sulphate-reducing bacteria. In addition, groundwater flow paths are an essential control on the delivery of sulphate to these bacteria that facilitate in situ mercury methylation. Keywords: methylmercury, methylation, hydrology, boreal catchment, peatland, Ontario, Canada

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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Barré, Sylvain; Pichot, Mikael;

    We prove the Haagerup property for an infinite discrete group constructed using surgery on a Euclidean Tits building of type A ˜ 2 .

    image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ Hyper Article en Lig...arrow_drop_down
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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Karouby, Johanna;

    L'étude de l'Univers primordial adresse quelques-unes des questions les plus fondamentales de la physique théorique. Cette thèse a pour objet l'exploration de trois aspects principaux de la cosmologie primordiale. Dans un premier temps, nous discutons d'une alternative au paradigme scientique qu'est le modèle du Big Bang. A savoir, nous explorons un modèle d'univers à rebond qui évite la singularité initiale du Big Bang. Nous commencerons dans l'introduction par revoir les éléments de base nécessaires à la compréhension de la cosmologie. A la suite de quoi, nous montrerons un modèle spécifique d'Univers à rebond contenant des champs additionnels particuliers en complément des champs présents habituellement. Ces nouveaux champs proviennent de ce qui s'appelle le modèle "Lee-Wick" de la physique des particules. En particulier, nous prouvons qu'un univers à rebond dans ce contexte est instable lorsque l'on ajoute une composante de radiation en plus de la matière. Dans la seconde partie, nous considérons la production de particules via un phénomène de résonance paramétrique durant la phase de "préchauffement", a la fin de l'inflation cosmologique. Plus précisément, nous prouvons que dans le cas où l'inflation a une limite de vitesse, les termes cinétiques non-canoniques décrivant n'importe quel Lagrangien effectif n'améliorent pas la production de particules. Finalement, le dernier sujet aborde concerne les défauts topologiques pendant la transition de phase de la chromodynamique quantique. A savoir, nous étudions les cordes cosmiques provenant des champs de pions présents dans le modèle standard de la physique des particules et trouvons un méchanisme pour les stabiliser. Nous prouvons alors qu'un bain thermique de photons en contact avec ces cordes réduit la variété du vide à un cercle. Cela a pour effet d'autoriser la présence de "cordes pioniques" topologiquement stables. The study of the Early Universe raises some of the most fundamental questions in theoretical physics. This thesis explores three main aspects of early universe cosmology. The first part discusses alternatives to the Big Bang scenario which is the current paradigm of cosmology. Namely, it discusses bouncing universe models where the initial Big Bang singularity is replaced by a finite size universe. After reviewing the necessary cosmology background in the introduction, we show a specific model of a bouncing universe that contains additional "Lee-Wick fields", partners to the standard fields. In particular we prove that a Lee-Wick matter bounce is unstable when one adds radiation to matter. In the second part of this thesis, we consider particle production via parametric resonance during preheating, at the end of cosmological inflation. Specifically, we prove that in the case of a speed-limited inflaton, non-canonical kinetic terms used to described any effective Lagrangian do not enhance particle production. Finally, the last topic involves topological defects during the Quantum Chromodynamics phase transition. Namely, we study cosmic strings coming from pion fields present in the Standard Model of particle physics and find a mechanism to stabilize them. We show how a thermal bath of photons reduces the effective vacuum manifold to a circle and thus allows the presence of topologically stable pion strings.

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    Authors: Peters, James;

    Cet article présente des présentations de groupe libres de formes triangulaires à trajectoire plane et de leurs nombres de Betti correspondants. Les chemins sont des cartes dans la théorie de l'homotopie. Les arêtes des triangles de chemins sont des chemins homotopes. Les principaux résultats de cet article sont que (1) chaque triangle de chemin a une présentation de groupe libre et (2) chaque triangle de chemin de chemin a une réalisation géométrique. This paper introduces free group presentations of planar path triangular shapes and their corresponding Betti numbers. Paths are maps in homotopy theory. The edges of path triangles are homotopic paths. Main results in this paper are that (1)every path triangle has a free group presentation and (2) every path path triangle has a geometric realization.

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    Authors: Sturdee, Miriam Amber; Alexander, Jason Mark; Coulton, Paul; Carpendale, Sheelagh;

    Almost all research output includes tables, diagrams, photographs and even sketches, and papers within HCI typically take advantage of including these figures in their files. However the space given to non-diagrammatical or tabular figures is often small, even in papers that primarily concern themselves with visual output. The reason for this might be the publishing models employed in most proceedings and journals: Despite moving to a digital format which is unhindered by page count or physical cost, there remains a somewhat arbitrary limitation on page count. Recent moves by ACM SIGCHI and others to remove references from the maximum page count suggest that there is movement on this, however images remain firmly within the limits of the text. We propose that images should be celebrated – not penalised – and call for not only the adoption of the Pictorials format in CHI, but for images to be removed from page counts in order to encourage greater transparency of process in HCI research.

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    Authors: St-Hilaire, F.; Wu, J.; Roulet, N. T.; Frolking, S.; +3 Authors

    We developed the McGill Wetland Model (MWM) based on the general structure of the Peatland Carbon Simulator (PCARS) and the Canadian Terrestrial Ecosystem Model. Three major changes were made to PCARS: 1. the light use efficiency model of photosynthesis was replaced with a biogeochemical description of photosynthesis; 2. the description of autotrophic respiration was changed to be consistent with the formulation of photosynthesis; and 3. the cohort, multilayer soil respiration model was changed to a simple one box peat decomposition model divided into an oxic and anoxic zones by an effective water table, and a one-year residence time litter pool. MWM was then evaluated by comparing its output to the estimates of net ecosystem production (NEP), gross primary production (GPP) and ecosystem respiration (ER) from 8 years of continuous measurements at the Mer Bleue peatland, a raised ombrotrophic bog located in southern Ontario, Canada (index of agreement [dimensionless]: NEP=0.80, GPP=0.97, ER=0.97; systematic RMSE [g C m?2 d?1]: NEP=0.12, GPP=0.07, ER=0.14; unsystematic RMSE [g C m?2 d?1]: NEP=0.15, GPP=0.27, ER=0.23). Simulated moss NPP approximates what would be expected for a bog peatland, but shrub NPP appears to be underestimated. Sensitivity analysis revealed that the model output did not change greatly due to variations in water table because of offsetting responses in production and respiration, but that even modest temperature increases could lead to converting the bog from a sink to a source of CO2. General weaknesses and further developments of MWM are discussed.

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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Fu, Lei; Fritz, Claudia; Scavone, Gary;

    This experiment explores how changes in soundpost height affect the perceptual qualities of the violin and whether there is a threshold of change below which players and luthiers do not perceive differences. A violin installed with a height-adjustable carbon fibre soundpost was employed. The experiment was designed as a sequence of playing tests. An experimenter was present to change the soundpost height. Thirteen professional violinists and six luthiers participated. The experiment involved two phases. During the first phase, subjects played and described their feelings about the violin with different soundpost settings in order to find their optimal soundpost height. During the second phase, the experimenter randomly increased, decreased or did not change the soundpost height in ten trials within a range of approximately ±0.1 mm around their optimal height. For each trial, subjects were asked to play the violin, comparing it with the previous setting, and to decide whether they were the same or different. Initial results indicate that each subject's optimal soundpost height varies within an interquartile range of 0.3 mm and the smallest height variation that could be recognized above chance level is about 0.04 mm.

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    Authors: Greyson, Devon; Morrison, Heather; Waller, Andrew;

    This article is a summary of recent Open Access activity in Canada, focusing on policies and mandates, repositories, and initiatives in libraries.

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    Authors: Ilcinkas, David; Pelc, Andrzej;

    The behavior of rational selfish agents has been classically studied in the framework of strategic games in which each player has a set of possible actions, players choose actions simultaneously and the payoff for each player is determined by the matrix of the game. However, in many applications, players choose actions asynchronously, and simultaneity of this process is not guaranteed: it is possible that a player learns the action of another player before making its choice. Delays of choices are controled by the adversary and each player can only secure the worst-case payoff over the adversary's decisions. In this paper we consider such asynchronous versions of arbitrary two-person strategic games and we study how the presence of the asynchronous adversary influences the behavior of the players, assumed to be selfish but rational. We concentrate on deterministic (pure) strategies, and in particular, on the existence and characteristics of pure Nash equilibria in such games. It turns out that the rational behavior of players changes significantly if the decision process is asynchronous. We show that pure Nash equilibria often exist in the asynchronous version of the game even if there were no such equilibria in the synchronous game. We also show that a mere threat of asynchrony in the game may make social optimum a rational choice while it was not rational in the synchronous game.

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    Authors: Holbrook, J.A.;

    This paper addresses indicators of the performance of the Vancouver human health biotechnology cluster.

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    Authors: Branfireun, B. A.; Roulet, N. T.;

    The fate and transport of methylmercury (MeHg) were studied in a small boreal catchment. Hydrological processes largely govern the magnitude of the flux of MeHg. Seasonal and inter-annual variability in hydrology produce variable source strengths of MeHg throughout the catchment. The mass flux of MeHg within, and from the catchment is dependent on the mass flux of water and the relative placement of landscape units in the catchment hydrological cascade. Hydrology also governs the maintenance of the methylating environments in the catchment. Specifically, hydrological processes maintain zones of anoxia in both the catchment uplands and peatlands that support obligate anaerobic sulphate-reducing bacteria. In addition, groundwater flow paths are an essential control on the delivery of sulphate to these bacteria that facilitate in situ mercury methylation. Keywords: methylmercury, methylation, hydrology, boreal catchment, peatland, Ontario, Canada

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    Authors: Barré, Sylvain; Pichot, Mikael;

    We prove the Haagerup property for an infinite discrete group constructed using surgery on a Euclidean Tits building of type A ˜ 2 .

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    Authors: Karouby, Johanna;

    L'étude de l'Univers primordial adresse quelques-unes des questions les plus fondamentales de la physique théorique. Cette thèse a pour objet l'exploration de trois aspects principaux de la cosmologie primordiale. Dans un premier temps, nous discutons d'une alternative au paradigme scientique qu'est le modèle du Big Bang. A savoir, nous explorons un modèle d'univers à rebond qui évite la singularité initiale du Big Bang. Nous commencerons dans l'introduction par revoir les éléments de base nécessaires à la compréhension de la cosmologie. A la suite de quoi, nous montrerons un modèle spécifique d'Univers à rebond contenant des champs additionnels particuliers en complément des champs présents habituellement. Ces nouveaux champs proviennent de ce qui s'appelle le modèle "Lee-Wick" de la physique des particules. En particulier, nous prouvons qu'un univers à rebond dans ce contexte est instable lorsque l'on ajoute une composante de radiation en plus de la matière. Dans la seconde partie, nous considérons la production de particules via un phénomène de résonance paramétrique durant la phase de "préchauffement", a la fin de l'inflation cosmologique. Plus précisément, nous prouvons que dans le cas où l'inflation a une limite de vitesse, les termes cinétiques non-canoniques décrivant n'importe quel Lagrangien effectif n'améliorent pas la production de particules. Finalement, le dernier sujet aborde concerne les défauts topologiques pendant la transition de phase de la chromodynamique quantique. A savoir, nous étudions les cordes cosmiques provenant des champs de pions présents dans le modèle standard de la physique des particules et trouvons un méchanisme pour les stabiliser. Nous prouvons alors qu'un bain thermique de photons en contact avec ces cordes réduit la variété du vide à un cercle. Cela a pour effet d'autoriser la présence de "cordes pioniques" topologiquement stables. The study of the Early Universe raises some of the most fundamental questions in theoretical physics. This thesis explores three main aspects of early universe cosmology. The first part discusses alternatives to the Big Bang scenario which is the current paradigm of cosmology. Namely, it discusses bouncing universe models where the initial Big Bang singularity is replaced by a finite size universe. After reviewing the necessary cosmology background in the introduction, we show a specific model of a bouncing universe that contains additional "Lee-Wick fields", partners to the standard fields. In particular we prove that a Lee-Wick matter bounce is unstable when one adds radiation to matter. In the second part of this thesis, we consider particle production via parametric resonance during preheating, at the end of cosmological inflation. Specifically, we prove that in the case of a speed-limited inflaton, non-canonical kinetic terms used to described any effective Lagrangian do not enhance particle production. Finally, the last topic involves topological defects during the Quantum Chromodynamics phase transition. Namely, we study cosmic strings coming from pion fields present in the Standard Model of particle physics and find a mechanism to stabilize them. We show how a thermal bath of photons reduces the effective vacuum manifold to a circle and thus allows the presence of topologically stable pion strings.

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    Authors: Peters, James;

    Cet article présente des présentations de groupe libres de formes triangulaires à trajectoire plane et de leurs nombres de Betti correspondants. Les chemins sont des cartes dans la théorie de l'homotopie. Les arêtes des triangles de chemins sont des chemins homotopes. Les principaux résultats de cet article sont que (1) chaque triangle de chemin a une présentation de groupe libre et (2) chaque triangle de chemin de chemin a une réalisation géométrique. This paper introduces free group presentations of planar path triangular shapes and their corresponding Betti numbers. Paths are maps in homotopy theory. The edges of path triangles are homotopic paths. Main results in this paper are that (1)every path triangle has a free group presentation and (2) every path path triangle has a geometric realization.

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