In this thesis, we solve for (equivariant) Gromov-Witten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2-bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g-2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of non-toric threefolds. Secondly, we compute the C-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Grornov-Witten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the Gromov-Witten theory of Nakamura's G- Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the Cromov-Witten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)].

In this thesis, we solve for (equivariant) Gromov-Witten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2-bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g-2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of non-toric threefolds. Secondly, we compute the C-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Grornov-Witten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the Gromov-Witten theory of Nakamura's G- Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the Cromov-Witten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)].

In this thesis, we solve for (equivariant) Gromov-Witten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2-bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g-2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of non-toric threefolds. Secondly, we compute the C-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Grornov-Witten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the Gromov-Witten theory of Nakamura's G- Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the Cromov-Witten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)]. / Science, Faculty of / Mathematics, Department of / Graduate

O enriquecimento isotópico de nitrogênio-15 por troca química no sistema NO/HNO3 foi estudado através de duas teorias distintas. Os fatores de fracionamento isotópicos, obtidos pela teoria de contracorrente e os estimados pela teoria da eqüipartição isotópica, foram confrontados através de um modelo. Construiu-se uma coluna de contracorrente, em escala de laboratório, e parâmetros tais como: número de placas teóricas, altura equivalente de uma placa teórica, tipo de enchimento, altura total da coluna, produção de H15NO3/semana, obtidos em condições de equilíbrio dinâmico isotópico, foram estudados comparativamente aos da literatura / Nitrogen-15 isotope enrichment by chemical exchange in NO/HNO3 system was studied using two different theories. The isotope fractionation factors obtained by the countercurrent theory was compared to those estimated by the isotope equipartition theory were confronted through a model. It was built a column in countercurrent at laboratory scale and parameters such as: number of theoretical plates, height equivalent to a H15NO3week, obtained under isotope dynamic equilibrium conditions, were studied in comparison to those in the literature

O enriquecimento isotópico de nitrogênio-15 por troca química no sistema NO/HNO3 foi estudado através de duas teorias distintas. Os fatores de fracionamento isotópicos, obtidos pela teoria de contracorrente e os estimados pela teoria da eqüipartição isotópica, foram confrontados através de um modelo. Construiu-se uma coluna de contracorrente, em escala de laboratório, e parâmetros tais como: número de placas teóricas, altura equivalente de uma placa teórica, tipo de enchimento, altura total da coluna, produção de H15NO3/semana, obtidos em condições de equilíbrio dinâmico isotópico, foram estudados comparativamente aos da literatura / Nitrogen-15 isotope enrichment by chemical exchange in NO/HNO3 system was studied using two different theories. The isotope fractionation factors obtained by the countercurrent theory was compared to those estimated by the isotope equipartition theory were confronted through a model. It was built a column in countercurrent at laboratory scale and parameters such as: number of theoretical plates, height equivalent to a H15NO3week, obtained under isotope dynamic equilibrium conditions, were studied in comparison to those in the literature

No description available.

L’étude des propriétés d’équilibre d’un système Coulombien quantique à plusieurs composantes présente un intérêt théorique fondamental, au-delà de ses nombreuses applications. Le mélange hydrogène-hélium est omniprésent dans la nébuleuse interstellaire ou les planètes géantes, et c’est aussi le constituant majoritaire du Soleil, où les interactions entre électrons et noyaux sont purement électrostatiques en première approximation.Ce travail est dévolu à l’équation d’état de ce mélange vu comme un plasma quantique constitué de protons, de noyaux d’Hélium et d’électrons. Dans ce cadre, nous développons des méthodes numériques pour estimer des intégrales de chemin représentant des ingrédients essentiels. En outre, nous construisons une nouvelle version de la diagrammatique à la Mayer resommée bien adaptée à nos objectifs.Tout d’abord, nous améliorons le double développement basse température et basse densité, dit SLT, pour l’hydrogène pur, grâce à de meilleures estimations des termes à trois corps, les résultats étant par ailleurs comparés à la fameuse équation d’état OPAL. Les densités plus élevées sont atteintes de manière non-perturbative, en utilisant des fonctions de partition d’entités recombinées suffisamment précises. Ainsi l’ionisation par pression est décrite sur une base théorique robuste. Nous étudions également d’autres quantités d’équilibre, comme l’énergie interne et la vitesse du son. Dans la dernière partie, nous calculons l’équation d’état du mélange hydrogène-hélium en incluant les effets d’écran associés aux ions He+, ainsi que des corrections à la Debye déterminées de manière auto-cohérente. Nos résultats nous permettent de comprendre le contenu physique d’approches ad-hoc et de déterminer leurs régimes de validité. Nous obtenons aussi une description plus fiable du mélange, qui devrait être précise le long de l'adiabate du Soleil. / The study of the thermodynamic properties of a multi-component quantum Coulomb system is of fundamental theoretical interest and has, beyond that, a wide range of applications. The Hydrogen-Helium mixture can be found in the interstellar nebulae and giant planets, however the most prominent example is the Sun. Here the interaction between the electrons and the nuclei is almost purely electrostatic.In this work we study the equation of state of the Hydrogen-Helium mixture starting from first principles, meaning the fundamental Coulomb interaction of its constituting particles. In this context we develop numerical methods to study the few-particle clusters appearing in the theory by using the path integral language. To capture the effects of the long-range Coulomb interaction between the fundamental particles, we construct a new version of Mayer-diagrammatic, which is appropriate for our purposes. In a first step, we ameliorate the scaled-low-temperature (SLT) equation of state, valid in the limit of low density and low temperature, by taking three-body terms into account and we compare the predictions to the well-established OPAL equation of state. Higher densities are accessed by direct inversion of the density equations and by the use of cluster functions that include screening effects. These cluster functions put the influence of screening on the ionization, unto now treated ad-hoc, on a theoretically well-grounded basis. We also inspect other equilibrium quantities such as the speed of sound and the inner energy. In the last part we calculate the equation of state of the Hydrogen-Helium mixture including the charged He+ ions in the screening process. Our work gives insights in the physical content of previous phenomenological descriptions and helps to better determine their range of validity. The equation of state derived in this thesis is expected to be very precise as well as reliable for conditions found in the Sun.