L'éclatement est une transformation jouant un rôle important en géométrie, car il permet de résoudre des singularités, de relier des variétés birationnellement équivalentes, et de construire des variétés possédant des propriétés inédites. Ce mémoire présente d'abord l'éclatement tel que développé en géométrie algébrique classique. Nous l'étudierons pour le cas des variétés affines et (quasi-)projectives, en un point, et le long d'un idéal et d'une sous-variété. Nous poursuivrons en étudiant l'extension de cette construction à la catégorie différentiable, sur les corps réels et complexes, en un point et le long d'une sous-variété. Nous conclurons cette section en explorant un exemple de résolution de singularité. Ensuite nous passerons à la catégorie symplectique, où nous ferons la même chose que pour le cas différentiable complexe, en portant une attention particulière à la forme symplectique définie sur la variété. Nous terminerons en étudiant un théorème dû à François Lalonde, où l'éclatement joue un rôle clé dans la démonstration. Ce théorème affirme que toute 4-variété fibrée par des 2-sphères sur une surface de Riemann, et différente du produit cartésien de deux 2-sphères, peut être équipée d'une 2-forme qui lui confère une structure symplectique réglée par des courbes holomorphes par rapport à sa structure presque complexe, et telle que l'aire symplectique de la base est inférieure à la capacité de la variété. La preuve repose sur l'utilisation de l'éclatement symplectique. En effet, en éclatant symplectiquement une boule contenue dans la 4-variété, il est possible d'obtenir une fibration contenant deux sphères d'auto-intersection -1 distinctes: la pré-image du point où est fait l'éclatement complexe usuel, et la transformation propre de la fibre. Ces dernières sont dites exceptionnelles, et donc il est possible de procéder à l'inverse de l'éclatement - la contraction - sur chacune d'elles. En l'accomplissant sur la deuxième, nous obtenons une variété minimale, et en combinant les informations sur les aires symplectiques de ses classes d'homologies et de celles de la variété originale nous obtenons le résultat. / The blow-up is a transformation which plays an important role in geometry, because it can be used to resolve singularities, relate birationally equivalent varieties, and construct varieties with new properties. This thesis first presents blowing-up as developped in classical algebraic geometry. We will study it in the case of affine and (quasi-)projective varieties, on a point and along an ideal and a subvariety. Then a discussion about its extension to the differential category will be carried out, over the real and complex fields, on a point and along a submanifold. An example of a resolution of singularity will then follow. Subsequently we will discuss blowing-up in the symplectic category, where we will do the same as for complex manifolds, paying careful attention to the symplectic form. To conclude, we will study a theorem by François Lalonde, where the symplectic blow-up plays a major part in proof. This theorem states that any 4-variety fibered by 2-spheres over a Riemann surface, and different than the Cartesian product of two 2-spheres, can be equiped with a 2-form giving it a symplectic structure ruled by curves that are holomorphic with respect to its almost-complex structure, and such that the symplectic area of the base is smaller that the capacity of the variety. In the proof, we blow up a ball in the 4-variety, and obtain a fibration containing two distinct spheres with a self-intersection equal to -1: the pre-image of the point where the usual complex blow-up is done, and the proper transform of the fiber. These two are exceptional, so it is possible to do the inverse operation - the blow down - on each of them. By blowing down the latter, we get a minimal variety, and by combining information about the symplectic area of its homology classes and of those of the original variety, we obtain the result.

No description available.

Ingeniero Civil Matemático / Este trabajo consiste principalmente en dos resultados matemáticos, basados en el estudio de ecuaciones dispersivas no lineales, la estabilidad de ciertas soluciones de las mismas, como así también la posible explosión en tiempo finito. En una primera parte, Capítulo 1, presentamos una breve introducción a los tópicos tratados en esta memoria. Se hace especial énfasis en la descripción de los conceptos de ecuación dispersiva, buen colocamiento, 2-solitones, estabilidad y explosión. En el Capítulo 2 probaremos que las soluciones de tipo 2-soliton de la ecuación de sine-Gordon (SG) son orbitalmente estables en el espacio de energía, el espacio natural para resolver este problema. Las soluciones que estudiamos son los 2-kink, kink-antikink y breather de SG. Con el objetivo de probar este resultado, utilizaremos las transformaciones de Bäcklund implementadas gracias al Teorema de la Función Implícita. Estas transformaciones nos permitirán reducir el problema de estabilidad para cada una de la soluciones, al caso de la solución cero. Probaremos estos resultados siguiendo el espíritu de un paper de M. A. Alejo y C. Muñoz, que trata el caso de la ecuación de Korteweg-de Vries modificada. Sin embargo, más adelante veremos que el caso de la ecuación de SG presenta varias nuevas dificultades dado el carácter vectorial de sus soluciones. Este resultado mejora los anteriores probados por M. A. Alejo et al., y entrega una primera demostración rigurosa de la estabilidad de los 2-solitones de la ecuación de SG en el espacio de energía. En el Capítulo 3 nuestro principal objetivo será estudiar nuevas propiedades de blow-up dispersivo para el sistema de Schrödinger-Korteweg-de Vries. Más precisamente, probaremos explosión para datos iniciales en H^2-(R)xH^{3/2-}(R), como consecuencia de mostrar previamente una nueva propiedad de persistencia del flujo asociado al sistema, establecida sobre ciertos espacios de Sobolev con pesos fraccionarios cuidadosamente escogidos. / Este trabajo ha sido parcialmente financiado por los proyectos Fondecyt Regular 1150202 y CMM Conicyt PIA AFB170001

In this work we deal with the nonlinear o.d.e u"+ku = εu³ which represents a spring-mass system with no damping but perturbed by external force εu³. We want to know how the spring constant k and the perturbed term act on the equation. So we study this equation by the way: (I) u" + ku = 0 (II)u" = u³ (III) u" + ku = εu³ During the period of calculating, we find that k, ε and energy constant E(0) play important roles in the properties of the solutions of the equation. Finally we give the relation about them.

ABSTRACT Injection stretch blow moulded PET bottles are the most widely used container type for carbonated soft drinks. PET offers excellent clarity, good mechanical and barrier properties, and ease of processing. Typically, these bottles have a petaloid-shaped base, which gives good stability to the bottle and it is the most appropriate one for beverage storage. However, the base is prone to environmentally induced stress cracking and this a major concern to bottle manufacturers. The object of this study is to explain the occurrence of stress cracking, and to prevent it by optimising both the geometry of the petaloid base and the processing parameters during bottle moulding. A finite element model of the petaloid shape is developed in CATIA V5 R14, and used to predict the von Mises stress in the bottle base for different combinations of three key dimensions of the base: foot length, valley width, and clearance. The combination of dimensions giving the minimum stress is found by a statistical analysis approach using an optimisation and design of experiments software package ECHIP-7. A bottle mould was manufactured according to the optimum base geometry and PET bottles are produced by injection stretch blow moulding (ISBM). In order to minimise the stresses at the bottom of the bottle, the ISBM process parameters were reviewed and the effects of both the stretch rod movement and the temperature profile of the preform were studied by means of the process simulation software package (Blow View version 8.2). Simulated values of the wall thickness, stress, crystallinity, molecular orientation and biaxial ratio in the bottle base were obtained. The process parameters, which result in low stress and uniform material in the bottle base, are regarded as optimum operating conditions. In the evaluation process of the optimum bottle base, bottles with standard (current) and optimized (new) base were produced under the same process conditions via a two-stage ISBM machine. In order to compare both the bottles, environmental stress crack resistance, top load strength, burst pressure strength, thermal stability test as well as crystallinity studies ¬¬¬via modulated differential scanning calorimetry (MDSC) and morphology studies via environmental scanning electron microscopy (ESEM) and optical microscopy were conducted. In this study carried out, the new PET bottle with the optimised base significantly decreased the environmental stress cracking occurrence in the bottom of the bottle. It is found that the bottle with optimised base is stronger than the bottle with standard base against environmental stress cracking. The resistance time against environmental stress cracking are increased by about % 90 under the same operating process conditions used for standard (current) bottles; and by % 170 under the optimised process conditions where the preform re-heating temperature is set to 105 oC.

Let Md be the moduli space of algebraic maps (morphisms) of degree d from P^1 to a fixed Grassmannian. The main purpose of this thesis is to provide an explicit construction of a compactification of Md satisfying the following property: the compactification is a smooth projective variety and the boundary is a simple normal crossing divisor. The main tool of the construction is blowing-up. We start with a smooth compactification given by Quot scheme, which we denote by Qd. The boundary Qd\Md is singular and of high codimension. Next, we give a filtration of the boundary Qd\Md by closed subschemes: Zd,0 subset Zd,1 subset ... Zd,d-1=Qd\Md. Then we blow up the Quot scheme Qd along these subschemes succesively, and prove that the final outcome is a compactification satisfying the desired properties. The proof is based on the key observation that each Zd,r has a smooth projective variety which maps birationally onto it. This smooth projective variety, denoted by Qd,r, is a relative Quot scheme over the Quot-scheme compactification Qr for Mr. The map from Qd,r to Zd,r is an isomorphism when restricted to the preimage of Zd,r\ Zd,r-1. With the help of the Qd,r's, one can show that the final outcome of the successive blowing-up is a smooth compactification whose boundary is a simple normal crossing divisor.

Blow fly (Diptera: Calliphoridae) interactions in decomposition ecology are well studied; however, the non-consumptive effects (NCE) of predators on the behavior and development of prey species have yet to be examined. The effects of these interactions and the resulting cascades in the ecosystem dynamics are important for species conservation and community structures. The resulting effects can impact the time of colonization (TOC) of remains for use in minimum post-mortem interval (mPMI) estimations. The development of the predacious blow fly, Chrysomya rufifacies (Macquart) was examined and determined to be sensitive to muscle type reared on, and not temperatures exposed to. Development time is important in forensic investigations utilizing entomological evidence to help establish a mPMI. Validation of the laboratory-based development data was done through blind TOC calculations and comparisons with known TOC times to assess errors. A range of errors was observed, depending on the stage of development of the collected flies, for all methods tested with no one method providing the most accurate estimation. The NCE of the predator blow fly on prey blow fly, Cochliomyia macellaria (Fabricius) behavior and development were observed in the laboratory. Gravid female adult attraction was significantly greater to resources with predatory larvae rather than prey larvae and oviposition occurred on in the presence of heterospecific (predatory) and conspecific larvae equally. However, the life stages necessary for predation to occur never overlapped and so these results may not be as surprising as they seem. Conversely, exposing prey larvae to predator cues through larval excretions/secretions led to larger prey larvae and faster times to pupariation when appropriate life stages overlapped. Differences in size and development times of prey larvae in the presence of predatory cues could lead to errors when estimating the mPMI. These data also partially explain the ability of C. macellaria to survive in the presence of Ch. rufifacies. Colonization of a resource with late instar Ch. rufifacies enhanced development and size of resulting larvae indicating that lag colonization, rather than being a primary colonizer, could become an alternate strategy for C. macellaria to survive the selective pressures of the predator, Ch. rufifacies. The differing effects of temperature on Ch. rufifacies and C. macellaria may also lend an advantage to C. macellaria over the predacious Ch. rufifacies in an environment with variable temperatures unlike what Ch. rufifacies is adapted for.

L'éclatement est une transformation jouant un rôle important en géométrie, car il permet de résoudre des singularités, de relier des variétés birationnellement équivalentes, et de construire des variétés possédant des propriétés inédites. Ce mémoire présente d'abord l'éclatement tel que développé en géométrie algébrique classique. Nous l'étudierons pour le cas des variétés affines et (quasi-)projectives, en un point, et le long d'un idéal et d'une sous-variété. Nous poursuivrons en étudiant l'extension de cette construction à la catégorie différentiable, sur les corps réels et complexes, en un point et le long d'une sous-variété. Nous conclurons cette section en explorant un exemple de résolution de singularité. Ensuite nous passerons à la catégorie symplectique, où nous ferons la même chose que pour le cas différentiable complexe, en portant une attention particulière à la forme symplectique définie sur la variété. Nous terminerons en étudiant un théorème dû à François Lalonde, où l'éclatement joue un rôle clé dans la démonstration. Ce théorème affirme que toute 4-variété fibrée par des 2-sphères sur une surface de Riemann, et différente du produit cartésien de deux 2-sphères, peut être équipée d'une 2-forme qui lui confère une structure symplectique réglée par des courbes holomorphes par rapport à sa structure presque complexe, et telle que l'aire symplectique de la base est inférieure à la capacité de la variété. La preuve repose sur l'utilisation de l'éclatement symplectique. En effet, en éclatant symplectiquement une boule contenue dans la 4-variété, il est possible d'obtenir une fibration contenant deux sphères d'auto-intersection -1 distinctes: la pré-image du point où est fait l'éclatement complexe usuel, et la transformation propre de la fibre. Ces dernières sont dites exceptionnelles, et donc il est possible de procéder à l'inverse de l'éclatement - la contraction - sur chacune d'elles. En l'accomplissant sur la deuxième, nous obtenons une variété minimale, et en combinant les informations sur les aires symplectiques de ses classes d'homologies et de celles de la variété originale nous obtenons le résultat. / The blow-up is a transformation which plays an important role in geometry, because it can be used to resolve singularities, relate birationally equivalent varieties, and construct varieties with new properties. This thesis first presents blowing-up as developped in classical algebraic geometry. We will study it in the case of affine and (quasi-)projective varieties, on a point and along an ideal and a subvariety. Then a discussion about its extension to the differential category will be carried out, over the real and complex fields, on a point and along a submanifold. An example of a resolution of singularity will then follow. Subsequently we will discuss blowing-up in the symplectic category, where we will do the same as for complex manifolds, paying careful attention to the symplectic form. To conclude, we will study a theorem by François Lalonde, where the symplectic blow-up plays a major part in proof. This theorem states that any 4-variety fibered by 2-spheres over a Riemann surface, and different than the Cartesian product of two 2-spheres, can be equiped with a 2-form giving it a symplectic structure ruled by curves that are holomorphic with respect to its almost-complex structure, and such that the symplectic area of the base is smaller that the capacity of the variety. In the proof, we blow up a ball in the 4-variety, and obtain a fibration containing two distinct spheres with a self-intersection equal to -1: the pre-image of the point where the usual complex blow-up is done, and the proper transform of the fiber. These two are exceptional, so it is possible to do the inverse operation - the blow down - on each of them. By blowing down the latter, we get a minimal variety, and by combining information about the symplectic area of its homology classes and of those of the original variety, we obtain the result.

Techn. Universiẗat, Diss., 2001--Berlin.

Techn. Universiẗat, Diss., 2003--Berlin.