Em 1968, J. Milnor introduziu a monodromia local de Picard-Lefschetz de uma hipersuperfície complexa com singularidade isolada. Em seguida, E. Brieskorn perguntou se esta monodromia é sempre finita. Em 1972, Lê Dúng Trâng provou que a resposta é positiva no caso de germes de curvas planas analíticas irredutíveis. Na época, já eram conhecidos exemplos de curvas planas com dois ramos e monodromia finita. Em 1973, N. A?Campo produziu o primeiro exemplo de germe de curva plana com dois ramos e monodromia infinita. Portanto, a questão mais simples, e ainda em aberto, que se coloca neste contexto, é a determinação da finitude da monodromia para germes de curvas planas com dois ramos. O presente trabalho, consiste em determinar, em várias situações, o polinômio mínimo da monodromia de germes de curvas analíticas planas com dois ramos, cujos gêneros são menores ou iguais a dois, o que permite decidir a sua finitude / In 1968, J. Milnor introduced the Picard-Lefschetz monodromy of a complex hypersurface with an isolated singularity. Subsequently, E. Brieskorn asked if this monodromy is always finite. In 1972, Lê Dúng Trâng proved that the answer is positive in the case of irreducible analytic germs of plane curves. At this time, examples of plane curves with two branches and finite monodromy were known. In 1973, N. A?Campo produced the first example of a germ of plane curve with two branches and infinite monodromy. Therefore, the simplest and still open problem in this context is to determine whether the monodromy of a plane curve with two branches is finite or infinite. The present work consists in determining, in several situations, the minimal polynomial of the monodromy for germs of plane analytic curves with two branches, whose genera are less or equal than two, wich allows us to decide its finiteness

ABSTRACT One of the most important tasks in the chemical industry is the separation of multicomponent liquid mixtures into one or more high-purity products. Several technologies are feasible for this task, either alone or in combination, such as distillation, extraction, crystallization, ect. Among these, distillation is by far the most widely spread and has a long history in chemical technology. However, until recently, there has been no systematic approach for understanding the separation of complex mixtures where azeotropes and multiple liquid phases may occur. There has been a growing interest in the use of residue curve and column profiles for the preliminary design of distillation columns. Residue curves and column profile are not only used to predict the composition changes in the distillation column but also to determine the feasibility of the proposed separation. Recently, theory underlying column profile maps has been developed by Tapp, Holland and co-workers. However there has been no direct experimental validation of the predictions of the column profile map theory. The main aim of this thesis is to experimentally verify some of the predictions of column profile map theory. A simple experimental batch apparatus has been developed to measure residue curve maps (RCMs) by Tapp and co-workers, the apparatus was modified so that it could be used to measure column profile maps (CPMs) in this thesis. CPM theory has shown that CPMs are linear transforms of the residues curve maps (RCMs). A stable node which was the apex of a mass balance triangle (MBT) was introduced inside the MBT, this was done by transforming the RCMs to CPMs using the appropriate distillate composition xd and reflux ratio R. It was also shown that the saddle point which was on the boundary of the triangle of the RCM can be shifted inside the MBT by transforming the RCM to CPM. This is again in accordance with theoretical predictions of CPM theory. iv Residue curves (RCs) and pinch point curves (PPCs) are used to determine the operation leaves and hence the feasible region for distillation columns operating at a specific distillate and bottoms composition for all fixed reflux ratio. The operating leaves were expanded beyond the pinch point curve by varying the reflux ratio from a higher reflux to a lower reflux ratio. This showed that one can effectively cross the pinch point curve hence expanding the operating leave. Finally the importance of experimentally measuring CPMs is demonstrated. Two thermodynamic models were used to predict the profiles of a complex system. The binary vapor-liquid equilibrium (VLE) diagrams and the residue curves produced from using these two thermodynamic models did not predict the same topology. The composition of the profiles were not the same because there were multiple liquid phases involved in this system, which made it difficult for the researchers to measure the correct profiles. Column profile maps were simulated using the different thermodynamic models, they also showed that there is some discrepancy between the predictions of the two models.

Mineral nutrients have marked effects on plant health by providing the building blocks for plant growth, as well as for mitigating abiotic and biotic stress factors, particularly disease development. Even if mineral nutrition field studies are conducted to study pest management, they are at the mercy of complex soil, water, and climatic conditions not amenable to strict experimental control. Therefore, a hydroponic method of growing lettuce was developed and growth curves were established for the macronutrients nitrogen (N), phosphorus (P), and potassium (K). Lettuce plants were grown at varying levels of each nutrient: 2.5, 5, 10, 20, 40, 80, 160, and 320 mg N/L; 0.5, 1, 2, 4, 8, 16, 32 and 64 mg P/L; and 0, 2.5, 5, 10, 20, 40, 80 and 160 mg K/L. Due to inadequate results lettuce was grown again at 0, 10, 20, 40, 80, 160, 320 and 640 mg L K. Optimal levels of N, P, and K were 160 mg/L, 4.0 mg/L, and 80 mg/L respectively. C:N ratios were also looked at for the N experiment. The overall result was consistent with results from similar studies. Unlike similar hydroponic studies done with other plants, micronutrient levels did not become deficient at high phosphorus levels suggesting phosphorus toxicity. These growth curves can be used to test lettuce resilience to various biotic and abiotic stresses.

Soit f un germe de fonction holomorphe dans deux variables qui s'annule à l'origine. L'ensemble zéro de cette fonction définit un germe de courbe analytique. Bien que la classification topologique d'un tel germe est bien connue depuis les travaux de Zariski, la classification analytique est encore largement ouverte. En 2012, Hefez et Hernandes ont résolu le cas irréductible et ont annoncé le cas de deux components. En 2015, Genzmer et Paul ont résolu le cas des fonctions topologiquement quasi-homogènes. L'objectif principal de cette thèse est d'étudier la première classe topologique de fonctions non quasi-homogènes. Dans le deuxième chapitre, nous décrivons l'espace local des modules des feuillages de cette classe et nous donnons une famille universelle de formes normales analytiques. Dans le même chapitre, nous prouvons l'unicité globale de ces formes normales. Dans le troisième chapitre, nous étudions l'espace des modules de courbes, qui est l'espace des modules des feuillages à une équivalence analytique des séparatrices associées près. En particulier, nous présentons un algorithme pour calculer sa dimension générique. Le quatrième chapitre présente une autre famille universelle de formes normales analytiques, qui est globalement unique aussi. En effet, il n'ya pas de modèle canonique pour la distribution de l'ensemble des paramètres sur les branches. Ainsi, avec cette famille, nous pouvons voir que la famille précédente n'est pas la seule et qu'il est possible de construire des formes normales en considérant une autre distribution des paramètres. Enfin, pour la globalisation, nous discutons dans le cinquième chapitre une stratégie basée sur la théorie géométrique des invariants et nous expliquons pourquoi elle ne fonctionne pas jusqu'à présent. / Let f be a germ of holomorphic function in two variables which vanishes at the origin. The zero set of this function defines a germ of analytic curve. Although the topological classification of such a germ is well known since the work of Zariski, the analytical classification is still widely open. In 2012, Hefez and Hernandes solved the irreducible case and announced the two components case. In 2015, Genzmer and Paul solved the case of topologically quasi-homogeneous functions. The main purpose of this thesis is to study the first topological class of non quasi-homogeneous functions. In chapter 2, we describe the local moduli space of the foliations in this class and give a universal family of analytic normal forms. In the same chapter, we prove the global uniqueness of these normal forms. In chapter 3, we study the moduli space of curves which is the moduli space of foliations up to the analytic equivalence of the associated separatrices. In particular, we present an algorithm to compute its generic dimension. Chapter 4 presents another universal family of analytic normal forms which is globally unique as well. Indeed, there is no canonical model for the distribution of the set of parameters on the branches. So, with this family, we can see that the previous family is not the only one and that it is possible to construct normal forms by considering another distribution of the parameters. Finally, concerning the globalization, we discuss in chapter 5 a strategy based on geometric invariant theory and explain why it does not work so far.

Space curves have a variety of uses within mathematics, and much attention has been paid to calculating quantities related to such objects. The quantities of curvature and energy are of particular interest to us. While the notion of curvature is well-known, the Mobius energy is a much newer concept, having been first defined by Jun O'Hara in the early 1990s. Foundational work on this energy was completed by Freedman, He, and Wang in 1994, with their most important result being the proof of the energy's conformal invariance. While a variety of results have built those of Freedman, He, and Wang, two topics remain largely unexplored: the interaction of curvature and Mobius energy and the generalization of the Mobius energy to curves with a varying thickness. In this thesis, we investigate both of these subjects. We show two fundamental results related to curvature and energy. First, we show that any simple, closed, twice-differentiable curve can be transformed in an energy-preserving and length-preserving way that allows us to make the pointwise curvature arbitrarily large at a point. Next, we prove that the total absolute curvature of a twice-differentiable curve is uniformly bounded with respect to conformal transformations. This is accomplished mainly via an analytic investigation of the effect of inversions on total absolute curvature. In the second half of the thesis, we define a generalization of the Mobius energy for simple curves of varying thickness that we call the "nonuniform energy." We call such curves "weighted knots," and they are defined as the pairing of a curve parametrization and positive, continuous weight function on the same domain. We then calculate the first variation formulas for several different variations of the nonuniform energy. Variations preserving the curve shape and total weight are shown to have no minimizers. Variations that "slide" the weight along the curve are shown to preserve energy is special cases.

The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q). From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form E: y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E: y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E: y2 = x3 + ax2 + bx as in Tate and Silverman's Elliptic Curves, and study analogous homomorphisms on E: y2 = x3 + c and relevant facts.

Lactation curves of dairy cows have been studied by dairy scientists for many years. The effects of various hereditary end non-hereditary influences on the lactation curve have been observed. Factors have been developed to standardize production to a common basis, correcting for differences in age, length of lactation, milking per day, gestation and environment.

Vibrio parahaemolyticus is a leading cause of seafood-borne illness with a newly emerged pandemic strain. Previous studies compared the pandemic and non-pandemic strains to understand the evolution of the pandemic strain but no definitive explanation for its emergence has been discovered. This study investigated the molecular characteristics of the pandemic strain and growth characteristics at different temperatures. The hypothesis tested was that pandemic strains of V. parahaemolyticus have modifications to their proteome that give a selective advantage over the other V. parahaemolyticus strains at temperatures normally encountered in the environment. Molecular typing techniques; automated ribotyping, pandemic specific PCR and multilocus sequence typing (MLST), were compared to determine the best method for pandemic strain determination. MLST was the best method because it was the most informative and accurate. Furthermore, nine Florida outbreak strains were identified as pandemic. Using representatives of both strains, growth curves were produced at four temperatures. The five pandemic strains had a significantly faster growth rate at 12°C than five non-pandemic strains. Temperature specific proteomic comparisons were completed using liquid chromatography followed by tandem mass spectroscopy. The proteome differences between these two groups at 12°C included three proteins (DnaA, DnaJ-related protein and DnaK-related protein) with functions related to cold stress. DnaA was expressed in the non-pandemic strain and not the pandemic strain, while the reverse was true for DnaJ-related and DnaK-related proteins. Western blot analysis and LC-MS/MS analysis on additional strains did not support the initial LC-MS/MS results. Growth studies using expression recombinants were employed to investigate these proteins on growth at 12°C. The overexpression of DnaA and DnaJ-related proteins did not significantly alter the growth rates compared to the control strain, but the overexpression recombinant strains DnaK-5 has a significantly slower growth rate than the control strain, the opposite direction as expected. The pandemic strain grows faster at lower temperatures, but the reason has not been determined. A theory is offered in which the pandemic growth advantage related to regulation of cold stress, leading to a shorter lag phase and faster growth rate after acclimation to the lower temperatures. Further experiments to investigate this theory are discussed.

Doctor of Philosophy (PhD) / In this work, we investigate methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves. Our principal tool is the theory of correspondences, in which homomorphisms of Jacobians are represented by divisors on products of curves. We give families of hyperelliptic curves of genus three, five, six, seven, ten and fifteen whose Jacobians have explicit isogenies (given in terms of correspondences) to other hyperelliptic Jacobians. We describe several families of hyperelliptic curves whose Jacobians have complex or real multiplication; we use correspondences to make the complex and real multiplication explicit, in the form of efficiently computable maps on ideal class representatives. These explicit endomorphisms may be used for efficient integer multiplication on hyperelliptic Jacobians, extending Gallant--Lambert--Vanstone fast multiplication techniques from elliptic curves to higher dimensional Jacobians. We then describe Richelot isogenies for curves of genus two; in contrast to classical treatments of these isogenies, we consider all the Richelot isogenies from a given Jacobian simultaneously. The inter-relationship of Richelot isogenies may be used to deduce information about the endomorphism ring structure of Jacobian surfaces; we conclude with a brief exploration of these techniques.

Let E be a one-parameter family of elliptic curves over a number field. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any non-degenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E.<br />The first such conjecture -- commonly associated with Chowla -- asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. We prove the conjecture for homogeneous polynomials of degree 3.<br />The second conjecture used states that any non-constant homogeneous polynomial yields to a square-free sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods.