Le modèle de Ruijsenaars-Schneider trigonométrique (tRS) quantique est un problème à N corps relativiste intégrable qui généralise le modèle de Calogero-Moser- Sutherland trigonométrique (tCMS). Les fonctions propres du modèle tRS sont les polynômes de Macdonald. La limite non relativiste qui relie les modèles tRS et tCMS est la même qui lie les polynômes de Macdonald et de Jack, les fonctions propres du modèle tCMS. Le but de ce mémoire est d'explorer la possibilité d'étendre le succès obtenu avec l'extension supersymétrique du modèle tCMS au modèle tRS. Le cas échéant, les superpolynômes de Macdonald pourraient être définis. Dans l'approche considérée, obtenir un coproduit diagonal de l'algèbre de Hecke est essentiel, mais n'a pas été possible pour TV > 2. On présente donc les résultats partiels connus pour le cas supersymétrique à deux et trois variables ainsi que la nature des obstacles qui, jusqu'à maintenant, ont empêché d'obtenir la généralisation voulue.

Tableau d'honneur de la Faculté des études supérieures et postdorales, 2014-2015 / La théorie des superpolynômes symétriques ([DLM03, DLM06]) est généralisée avec l’introduction d’une nouvelle base de superfonctions qui dépend de deux paramètres q et t. Cette nouvelle base, que l’on appelle les polynômes de Macdonald dans le superespace (ou simplement, les superpolynômes de Macdonald), généralise toutes les autres bases de superfonctions connues. Celles-ci sont retrouvées via différentes spécialisations (ou limites) de q et t. On démontre que les superpolynômes de Macdonald sont uniquement déterminés par les deux propriétés suivantes. Premièrement, ils se décomposent de façon triangulaire dans la base des superfonctions monomiales (par rapport à l’ordre de dominance entre les superpartitions). Deuxièmement, ils sont orthogonaux par rapport à un produit scalaire donné dans la base des superfonctions sommes de puissances et qui dépend de q, t. L’étape clef pour démontrer ce résultat est la connexion avec la théorie des polynômes non symétriques de Macdonald. En fait, il est montré que les superpolynômes de Macdonald sont également donnés par un processus de symétrisation particulier des polynômes non symétriques de Macdonald. Cette connexion peut être alors exploitée pour obtenir une famille d’opérateurs qui est diagonale dans la base des superpolynômes de Macdonald ainsi qu’une seconde relation d’orthogonalité donnée par l’évaluation d’un terme constant. Ces deux éléments, i.e. famille d’opérateurs et orthogonalité (analytique), permettent de relier les superpolynômes de Macdonald à un problème de mécanique quantique supersymétrique généralisant le modèle Ruijsenaars-Schneider (RS). L’hamiltonien de ce modèle est défini par l’anticommutateur d’une supercharge qui est le générateur de la transformation supersymétrique. La structure algébrique sous-jacente à ce modèle est l’algèbre de Poincaré supersymétrique (i.e. une algèbre de Lie graduée). Tous les états propres de l’hamiltonien sont donnés par le produit de la fonction d’onde de l’état du vide par les superpolynômes de Macdonald. L’intégrabilité du modèle est également démontrée. / The theory of symmetric superpolynomials ([DLM03, DLM06]) is further extended with the introduction of a family of superpolynomials that depends upon two parameters, denoted by q and t. This new basis, that can be called Macdonald polynomials in superspace (or simply stated, Macdonald superpolynomials), generalizes all the previously discovered bases of superpolynomials. These are obtained by the evaluation (or by a limiting process) of the parameters q and t. It is proved that the Macdonald superpolynomials are uniquely defined by the two following properties. First, they decompose triangularly in the monomial basis (with respect to a certain ordering between superpartitions). Second, they are orthogonal with respect to a given scalar product evaluated in the power sum basis and which depends on q and t. The crucial step to prove this result is the connection between Macdonald superpolynomials and the theory of non-symmetric Macdonald polynomials. More precisely, it is showed that the Macdonald superpolynomials can be expressed by a certain symmetrizer acting on the non-symmetric analogue. Using this connection, a family of eigen-operators is obtained, which is diagonalized by the Macdonald superpolynomals basis. In addition, another orthogonality relation that involves a constant term evaluation (referred to as the analytic orthogonality) is obtained. These two elements, i.e. the eigen-operators and the orthogonality (analytic), link the Macdonald superpolynomials to a supersymmetric quantum mechanic model that generalizes the Ruijsenaars-Schneider (RS) model. The Hamiltonian of this model is naturally written as an anticommutator of a supercharge which is the generator of supersymmetric transformation. The underlying algebra of this model is the super Poincaré algebra (i.e. a graded Lie algebra). All the quantum states of the Hamiltonian are given as a product of the ground state function times Macdonald superpolynomials. Finally, the integrability of the supersymmetric RS model is demonstrated.

This thesis offers a re-evaluation of the fiction of George MacDonald (1824-1905), both fantasy and non-fantasy. The general trend in MacDonald studies is to focus primarily on his works of fantasy, either ignoring the rest (which includes non-fantasy fiction, sermons, poetry, and criticism) or using them to illuminate the fantasies. The overall critical consensus is that these works, particularly MacDonald’s non-fantasy fiction, possess little inherent value. Though many critics acknowledge similarities between MacDonald’s fantasy fiction and his non-fantasy fiction, MacDonald has been the victim of a critical double standard that treats fantasy and realism as completely irreconcilable, and allows certain features to be acceptable, even desirable, in one form that are completely unacceptable in the other. The thesis begins by looking at MacDonald’s writings about the imagination and about literature, from which a clear theory of literature emerges, one with strong opinions about the function and purpose of literature, as well as about what makes good literature. By re-examining MacDonald’s fiction, its plots, characterization and narration, in the light of his own theories, the reasons underlying the artistic choices made throughout his fiction take on a more deliberate and calculated appearance. Furthermore, by placing MacDonald in his proper context, and looking at the diversity of generic options available to the Victorian writer, the critical double standard underlying much MacDonald scholarship, based on a strict fantasy/realism separation, crumbles. What emerges from this analysis is a different MacDonald—a careful craftsman who consciously and skillfully uses the tools of his trade to produce a unique and specific reading experience.

Os contos de fada de George MacDonald são obras de grande influência no âmbito da literatura infantil, particularmente no caso de autores como Lewis Carroll e C. S. Lewis. De fato, muitos apreciaram suas histórias enigmáticas, que misturam humor e reverência, nonsense e sentido, luz e escuridão. Apesar disso, no contexto do português brasileiro, poucos dos seus contos de fada estão traduzidos e há ainda menos crítica literária disponível. Aqui procuro ajudar a preencher ambas essas lacunas ao produzir uma tradução comentada de três de seus contos de fada. O objetivo é realizar um estudo que encontre algumas das características literárias mais importantes nesses contos e, então, produzir uma tradução que leve em conta as percepções proporcionadas por esse estudo. Este trabalho contém, então, um exame crítico dos contos de fada de MacDonald, uma consideração sobre como traduzi-los, os três contos de fada em formato bilíngue (incluindo a fonte em inglês e a tradução para o português) e, também, uma seção comentando minhas escolhas de tradução e como elas se relacionam com as análises feitas. / George MacDonald\'s fairy tales are works of great influence in the realm of children\'s literature, particularly so for authors such as Lewis Carroll and C. S. Lewis. Indeed, many have enjoyed his enigmatic stories, which blend humor and reverence, nonsense and meaning, light and darkness. In spite of that, in the context of Brazilian Portuguese, few of his fairy tales are translated, and there is even less literary criticism available. Here, I aim to help fill both of those gaps by producing a commented translation of three of his fairy tales. The goal is to carry out a study which finds out some of the most important literary characteristics in these tales, and then produce a translation that is mindful of the insights provided by that study. This work contains, then, a critical appraisal of MacDonald\'s fairy stories, some consideration on how to translate them, the three fairy tales presented as bilingual texts (including the English source and the translation to Portuguese), and also a section commenting on my translation choices and how they relate to the analyses made.

Os contos de fada de George MacDonald são obras de grande influência no âmbito da literatura infantil, particularmente no caso de autores como Lewis Carroll e C. S. Lewis. De fato, muitos apreciaram suas histórias enigmáticas, que misturam humor e reverência, nonsense e sentido, luz e escuridão. Apesar disso, no contexto do português brasileiro, poucos dos seus contos de fada estão traduzidos e há ainda menos crítica literária disponível. Aqui procuro ajudar a preencher ambas essas lacunas ao produzir uma tradução comentada de três de seus contos de fada. O objetivo é realizar um estudo que encontre algumas das características literárias mais importantes nesses contos e, então, produzir uma tradução que leve em conta as percepções proporcionadas por esse estudo. Este trabalho contém, então, um exame crítico dos contos de fada de MacDonald, uma consideração sobre como traduzi-los, os três contos de fada em formato bilíngue (incluindo a fonte em inglês e a tradução para o português) e, também, uma seção comentando minhas escolhas de tradução e como elas se relacionam com as análises feitas. / George MacDonald\'s fairy tales are works of great influence in the realm of children\'s literature, particularly so for authors such as Lewis Carroll and C. S. Lewis. Indeed, many have enjoyed his enigmatic stories, which blend humor and reverence, nonsense and meaning, light and darkness. In spite of that, in the context of Brazilian Portuguese, few of his fairy tales are translated, and there is even less literary criticism available. Here, I aim to help fill both of those gaps by producing a commented translation of three of his fairy tales. The goal is to carry out a study which finds out some of the most important literary characteristics in these tales, and then produce a translation that is mindful of the insights provided by that study. This work contains, then, a critical appraisal of MacDonald\'s fairy stories, some consideration on how to translate them, the three fairy tales presented as bilingual texts (including the English source and the translation to Portuguese), and also a section commenting on my translation choices and how they relate to the analyses made.

Dans cette thèse, j’étudie trois sujets reliés : les modèles de tenseurs, les nombres de Hurwitz et les polynômes de Macdonald-Koornwinder. Les modèles de tenseurs généralisent les modèles de matrices en tant qu’une approche à la gravité quantique en dimension arbitraire (les modèles de matrices donnent une version bidimensionnelle). J’étudie un modèle particulier qui s’appelle le modèle quartique mélonique. Sa spécialité est qu’il s’écrit en termes d’un modèle de matrices qui est lui-même aussi intéressant. En utilisant les outils bien établis, je calcule les deux premiers ordres de leur 1=N expansion. Parmi plusieurs interprétations, les nombres de Hurwitz comptent le nombre de revêtements ramifiés de surfaces de Riemann. Ils sont connectés avec de nombreux sujets en mathématiques contemporaines telles que les modèles de matrices, les équations intégrables et les espaces de modules. Ma contribution principale est une formule explicite pour les nombres doubles avec 3-cycles complétées d’une part. Cette formule me permet de prouver plusieurs propriétés intéressantes de ces nombres. Le dernier sujet de mon étude est les polynôme de Macdonald et Koornwinder, plus précisément les identités de Littlewood. Ces polynômes forment les bases importantes de l’algèbre des polynômes symétriques. Un des problèmes intrinsèques dans la théorie des fonctions symétriques est la décomposition d’un polynôme symétrique dans la base de Macdonald. La décomposition obtenue (notamment si les coefficients sont raisonnablement explicites et compacts) est nommée une identité de Littlewood. Dans cette thèse, j’étudie les identités démontrées récemment par Rains et Warnaar. Mes contributions incluent une preuve d’une extension d’une telle identité et quelques progrès partiels vers la généralisation d’une autre. / In this thesis, I study three related subjects: tensor models, Hurwitz numbers and Macdonald-Koornwinder polynomials. Tensor models are generalizations of matrix models as an approach to quantum gravity in arbitrary dimensions (matrix models give a 2D version). I study a specific model called the quartic melonic tensor model. Its specialty is that it can be transformed into a multi-matrix model which is very interesting by itself. With the help of well-established tools, I am able to compute the first two leading orders of their 1=N expansion. Among many interpretations, Hurwitz numbers count the number of weighted ramified coverings of Riemann surfaces. They are connected to many subjects of contemporary mathematics such as matrix models, integrable equations and moduli spaces of complex curves. My main contribution is an explicit formula for one-part double Hurwitz numbers with completed 3-cycles. This explicit formula also allows me to prove many interesting properties of these numbers. The final subject of my study is Macdonald-Koornwinder polynomials, in particular their Littlewood identities. These polynomials form important bases of the algebra of symmetric polynomials. One of the most important problems in symmetric function theory is to decompose a symmetric polynomial into the Macdonald basis. The obtained decomposition (in particular, if the coefficients are explicit and reasonably compact) is called a Littlewood identity. In this thesis, I study many recent Littlewood identities of Rains and Warnaar. My own contributions include a proof of an extension of one of their identities and partial progress towards generalization of one another.

This thesis examines geomorphic processes that impact thermal stratification within floodplain lakes. Previously, thermal stratification was not thought prevalent in these numerous small water bodies, despite being found in small reservoirs. Thermal stratification leads to long term anoxia in the bottom waters and the release of metals and nutrients from the sediments. This can result in water quality problems such as algal blooms (including potentially toxic cyanobacteria), loss of fish habitat and deterioration of wetland habitat. Four floodplain lakes of the Macdonald River Valley NSW were profiled and tested over 3?? years to assess to what extent, under what conditions, and with what chemical and physical effects, they thermally stratified and destratified. The morphometry, bed facies, vegetation, hydrology and local weather patterns for each lake were also investigated. Physical and chemical analyses found elevated nutrients in all four lakes and higher concentrations of total phosphorous and total nitrogen in the bottom waters. These results were not dependant on stable thermal stratification and long-term anoxia in the bottom waters. Inflow samples had much higher concentrations of nutrients than the lake samples, so the catchment is the likely source. Samples were analysed for phytoplankton and significant populations of potentially toxic cyanobacteria were present. Stratification depended upon whether the lake was "sheltered" from wind stirring or from incident radiation, either topographically by steep slopes surrounding the lakes, or by submerged and emergent vegetation. Aquatic vegetation limits the amount of transfer of turbulent kinetic energy from wind shear and the amount of incident radiation received. At one lake, sheltered areas reached temperatures much higher than less sheltered areas. Topographic sheltering can make a difference of up to three degrees Celsius across the lake surface. Destratification depended on wind speed, wind direction and orientation of the lake to the dominant wind direction. An extension to the classification of fluvial lakes (from Timms 1992) is presented, with further subdivisions to describe the lakes as "sheltered" or "open" and the type of sheltering (vegetation or topography) present.

Though known for his fantastical writings George MacDonald is often considered to be a typical Victorian teacher of religious and moral seriousness. Approaches to MacDonald’s works normally seek to find his ‘message’ by expositing the moral, social, pedagogical, psychological or theological ‘content’ of his work. This study recasts MacDonald in the light of his shorter fairytales for the ‘childlike’ and argues that these seemingly small and insignificant works are a golden key to his artistic enterprise. This is not because of any particular ‘message’ that they carry but because of their peculiarly light mode of generating meaning and the relation of this lightness to theology. Whilst it is frequently disparaged, levity actually has strong parallels with the theological atmosphere of Christianity. Light modalities such as folly, ecstasy, play, vanity, carnival and Sabbath demonstrate that the Christian faith has greater affinities with lightness and whimsicality than its solemn defenders sometimes admit. MacDonald’s fairytales draw upon this surprising harmony between levity and faith to create environments in which readers can playfully reflect upon the nature of ultimate reality and begin to find their own place within that reality. By helping to remove the mask of ‘seriousness’ presented by things in the everyday world, fairytales engender a kind of ‘divine carelessness’ and help people to let go of the weighty cares and fears that keep them tightly bound to worldly things.

This thesis examines geomorphic processes that impact thermal stratification within floodplain lakes. Previously, thermal stratification was not thought prevalent in these numerous small water bodies, despite being found in small reservoirs. Thermal stratification leads to long term anoxia in the bottom waters and the release of metals and nutrients from the sediments. This can result in water quality problems such as algal blooms (including potentially toxic cyanobacteria), loss of fish habitat and deterioration of wetland habitat. Four floodplain lakes of the Macdonald River Valley NSW were profiled and tested over 3?? years to assess to what extent, under what conditions, and with what chemical and physical effects, they thermally stratified and destratified. The morphometry, bed facies, vegetation, hydrology and local weather patterns for each lake were also investigated. Physical and chemical analyses found elevated nutrients in all four lakes and higher concentrations of total phosphorous and total nitrogen in the bottom waters. These results were not dependant on stable thermal stratification and long-term anoxia in the bottom waters. Inflow samples had much higher concentrations of nutrients than the lake samples, so the catchment is the likely source. Samples were analysed for phytoplankton and significant populations of potentially toxic cyanobacteria were present. Stratification depended upon whether the lake was "sheltered" from wind stirring or from incident radiation, either topographically by steep slopes surrounding the lakes, or by submerged and emergent vegetation. Aquatic vegetation limits the amount of transfer of turbulent kinetic energy from wind shear and the amount of incident radiation received. At one lake, sheltered areas reached temperatures much higher than less sheltered areas. Topographic sheltering can make a difference of up to three degrees Celsius across the lake surface. Destratification depended on wind speed, wind direction and orientation of the lake to the dominant wind direction. An extension to the classification of fluvial lakes (from Timms 1992) is presented, with further subdivisions to describe the lakes as "sheltered" or "open" and the type of sheltering (vegetation or topography) present.

George MacDonald (1824-1905) is one of the most complex and significant Scottish writers of the nineteenth century, especially as a writer of children's fiction and literary fairy tales. His works, however, have seldom been studied as Scottish literature. This dissertation is the first full-length analysis of his writings for children in their Scottish context, focusing particularly on his use of Scottish folklore in his literary fairy tales. MacDonald wrote in the Scottish Romantic tradition of Robert Burns, Walter Scott, and James Hogg; by close reading his works alongside similar texts by his compatriots, such as Andrew Lang, MacDonald's own idiosyncratic contribution to that tradition becomes more apparent. His profound knowledge of and appreciation for Christian mysticism is in evidence throughout his work; his use of folklore was directly informed by his exploration of mystical ideas. Hogg is recast as a second Dante, and ‘bogey tales' become catalysts for spiritual awakening. MacDonald's fairy tales deal sensitively and profoundly with the theme of child death, a tragedy that held personal significance for him, and can thus be read as his attempt to come to terms with the reality of bereavement by using Scottish folklore to explain it in mystical terms. Traditional figures such as Thomas Rhymer, visionary poets, and doubles appear in his fairy tales as guides and pilgrims out of the material world toward mystical union with the Divine.