We study two related inequalities that arise in Harmonic Analysis: restriction type inequalities and isoperimetric inequalities. The (Lp, Lq) Restriction type inequalities have been the subject of much interest since they were first conceived in the 1960s. The classical restriction type inequality involving surfaces of non-vanishing curvature is only fully resolved in two dimensions and there have been a lot of recent developments to establish the conjectured (p,q) range in higher dimensions. However, it also interesting to consider what can be said for curves where the curvature does vanish. In particular we build upon a restriction result for homogeneous polynomial surfaces, using what is considered the natural weight - the one induced by the affine curvature of the surface. This is known to hold with a non-universal constant which depends in some way on the coefficients of the polynomial. In this dissertation we shall quantify that relationship. Restriction estimates (for curves or surfaces) using the affine curvature weight can be shown to lead to an affine isoperimetric inequality for such curves or surfaces. We first prove, directly, this inequality for polynomial curves, where the constant depends only on the degree of the underlying polynomials. We then adapt this method, to prove an isoperimetric inequality for a wide class of curves, which includes curves for which a restriction estimate is not yet known. Next we state and prove an analogous result of the relative affine isoperimetric inequality, which applies to unbounded convex sets. Lastly we demonstrate that this relative affine isoperimetric inequality for unbounded sets is in fact equivalent to the classical affine isoperimetric inequality.

The first part of this thesis establishes a series of geometric ineqalities related to fractional integration in some geometric settings, including bilinear and multilinear forms. In the second part of this thesis, we study some kinds of rearrangement inequalities. In particular, some applications of rearrangement inequalities will be given, for instance, the determination of the extremals of some geometric problems. By competing symmetries and rearrangement inequalities, we prove the sharp versions of geometric inequalities introduced in the first part in Euclidean spaces. Meanwhile, there are the corresponding conformally equivalent formulations in unit sphere and in hyperbolic space. The last part is about collaborative work on the regularity of the Hardy-Littlewood maximal functions. We give a simple proof to improve Tanaka's result of the paper entitled "A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function". Our proof is based on the behaviour of the local maximum of the non-centered Hardy-Littlewood maximal function.

In Hamiltonian systems with symmetry, many previous studies have centred their attention on compact symmetry groups, but relatively little is known about the effects of noncompact groups. This thesis investigates the properties of the system of N point vortices on the hyperbolic plane H2, which has noncompact symmetry SL (2, R).The Poisson Hamiltonian structure of this dynamical system is presented and the relative equilibria conditions are found. We also describe the trajectories of relative equilibria with momentum value not equal to zero. Finally, stability criteria are found for a number of cases, focusing on N = 2, 3. These results are placed in context with the study of point vortices on the sphere, which has compact symmetry.

Dans cette thèse, nous examinons comment un assistant de preuve peut être utilise pour étudier les fondements de la géométrie. Nous débutons en nous concentrant sur les façons d’axiomatiser la géométrie euclidienne et leurs relations. Ensuite, nous exposons une nouvelle preuve de l’indépendance de l’axiome des parallèles des autres axiomes de la géométrie euclidienne du premier ordre. Cela nous amène à affiner la classification des plans de Hilbert de Pejas en considérant les propriétés de décidabilité. Mais, notre intuition nous amène souvent à négliger leur utilisation. Un assistant de preuve nous permet d’utiliser un outil parfait qui ne possède aucune intuition : un ordinateur. De plus, les assistants de preuve nous laissent exploiter les capacités de calcul des ordinateurs. Nous démontrons comment utiliser de méthodes algébriques de déduction automatique en géométrie synthétique. Enfin, nous présentons une procédure spécifique destinée à automatiser des preuves d’incidence. / In this thesis, we investigate how a proof assistant can be used to study the foundations of geometry. We start by focusing on ways to axiomatize Euclidean geometry and their relationship to each other. Then, we expose a new proof that Euclid’s parallel postulate is not derivable from the other axioms of first-order Euclidean geometry. This leads us to refine Pejas’ classification of parallel postulates. We do so by considering decidability properties when classifying the postulates. However, our intuition often guides us to overlook uses of such properties. A proof assistant allows us to use a perfect tool which possesses no intuition: a computer. Moreover, proof assistants let us leverage the computational capabilities of computers. We demonstrate how we enable the use of algebraic automated deduction methods thanks to the arithmetization of geometry. Finally, we present a specific procedure designed to automate proofs of incidence properties.

The three geodesics theorem of Lusternik and Schnirelmann asserts that for every Riemannian metric on the 2-sphere, there exist at least three embedded closed geodesics. In the process of determining the geodesics as critical points of the energy or length functional, a suitable method of curve shortening is needed. It has been suggested to use the so-called curve shortening flow as it continuously deforms smooth embedded curves while naturally preserving their embeddedness. In the 1980s, the investigation of the curve shortening flow began and a proof of the Lusternik-Schnirelmann theorem using the flow was sketched. We build upon these results. After introducing the curve shortening flow, we prove the well-known result that the geodesic curvature of a smooth embedded closed curve on a smooth closed two-dimensional Riemannian manifold decreases smoothly to zero, provided the curve evolves forever under the flow. From this, we prove subconvergence to an embedded closed geodesic, using mainly local arguments. After introducing, in the form of Lusternik-Schnirelmann theory, the topological machinery employed in the process of determining critical points of certain functions, we turn to the three geodesics theorem which we prove under a few assumptions. For the round metric on the 2-sphere, we deformation retract a suitable space of unparametrized curves onto a simpler space of which we determine the homology groups relative to a subspace which deformation retracts onto the subspace of point curves. As this yields three subordinate homology classes, proving the validity of Lusternik-Schnirelmann theory for the curve shortening flow and the length functional on our space of curves completes the proof.

Made available in DSpace on 2016-12-08T17:19:26Z (GMT). No. of bitstreams: 1 Darlan Silva Siewerdt.pdf: 4084524 bytes, checksum: 6cfc1de54400889b3bfa578652d76489 (MD5) Previous issue date: 2015-02-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nowadays, pressure vessel has been used in large industrial scale. Its manufacture requires a lot of special care and demand knowledge of standards and suitable materials for every type of application. Given this, it becomes mandatory an adequate welding process and inspection to ensure reliability of the welded joint. In this context, non-destructive testing of Xrays has been made in order to evaluate these welds and verify the presence of discontinuities. Thus, it was found that the porosity is a major discontinuities which generate rework welding, increasing costs. This thesis aims to contribute to the research of porosity in steel weld joints of SA 516 Grade 70 used pressure vessels, through the study of the correlation between the procedure of submerged arc welding and the formation of such discontinuities. In a second stage all the experimental welding procedure is done, varying the welding parameters: voltage, current, welding speed and flow, as the design of experiment. On the sequence, tests of X-ray and visual inspection are performed to determining the amount of porosity and other discontinuities in each sample. Finally, from these evidences is carried out a study of the porosities formation, using factorial design and determined the influence of each parameter in its porosities formation. Besides, an optimized welding procedure was determined, where the chosen samples were submitted to bend test, macrograph, metallography and Vickers hardness to qualify the procedure developed. The thesis concluded that the welding speed and the type of flow are the most important factors that determine the amount and type of porosity formed in welding of steel for pressure vessels. / Nos dias atuais, vasos de pressão têm sido utilizados em grande escala industrial. Sua fabricação requer uma série de cuidados especiais e exigem o conhecimento de normas e materiais adequados para cada tipo de aplicação. Diante disto, torna-se mandatório um processo de soldagem e inspeção adequado para garantir a confiabilidade da junta soldada. Neste contexto, ensaios não destrutivos de raios X têm sido feitos com o intuito de avaliar estas soldas e verificar a presença de descontinuidades. Assim, verificou-se que a porosidade é uma das principais descontinuidades que geram retrabalhos de solda, aumentando os custos. Esta Dissertação de Mestrado pretende contribuir na pesquisa da porosidade em soldas de aço SA 516 Grau 70 utilizado vasos de pressão, através do estudo da correlação entre o procedimento de soldagem por arco submerso e a formação de tais descontinuidades. Num segundo momento é feito todo o procedimento experimental de soldagem, variando os seguintes parâmetros: tensão, corrente, velocidade de soldagem e fluxo, conforme o delineamento do experimento elaborado para então realizar ensaios de raios X e inspeção visual, determinando a quantidade de porosidade e outras descontinuidades em cada amostra. Finalmente, a partir destas evidências realiza-se um estudo da formação de porosidades, através do projeto fatorial e determina-se a influência de cada parâmetro na sua formação. Paralelamente foi elaborado um procedimento otimizado de soldagem, onde as amostras escolhidas passaram por ensaios de dobramento, macrografia, e dureza vickers para qualificar o procedimento. Pode-se concluir que a velocidade de soldagem e o tipo de fluxo são os fatores que mais influenciam na quantidade e tipo de porosidade formada na soldagem de aços para vasos de pressão.

This thesis study is concerned with genetical investigation of Balikesir Balya Pb-Zn mineralization through the mineralogic-petrographic and geochemical examination of the core samples obtained from a total of 9 holes drilled by EczacibaSi ESAN Madencilik. The Pb-Zn mineralization in Balya is mainly of vein-type. Wall rocks hosting mineralizations are dacite, dacite porphyry and microdiorite. Major types of alteration are silicification, carbonatization and calc-silicate alteration, each of which is further subdivided into early and late stages and overprinted by argillic alteration of probable supergene origin. The ore minerals are mainly Zn- and Pb-sulphides and are hosted by the rocks with late calc-silicate alteration that underwent pervasive late silicification and late carbonatization. Sulfide mineralization is spatially and temporally associated with the late silicification and carbonatization stages. Lateral-vertical correlation of drill logs suggest that thickness of the ore zone tends to decrease towards north. The volcanic rocks hosting the mineralization have calc-alkaline nature. Major, trace and rare earth element (REE) geochemistry suggests either crustal contamination or subduction signature in the mantle source of the volcanics. Multi element patterns and discrimination diagrams collectively point to a post-collisional setting for their generation. Alteration geochemistry reveals that Fe2O3 and CaO are enriched during calc-silicate alteration in contrast to depletion of SiO2. Al2O3 and TiO2 are almost constant during late calc-silicate alteration. Enrichment of Fe2O3 and Na2O, and depletion of K2O characterize the silicified zones. Carbonatization is accompanied by strong enrichment of CaO and depletion of SiO2, Al2O3 and K2O.

This study is concerned with volcanostratigraphic and petrologic evolution of the S&uuml / phan, which is a 4050 m high Quaternary stratovolcano in eastern Anatolia. The eruptive products of S&uuml / phan Stratovolcano, including transitional mildly alkaline to calc-alkaline rocks, are lavas, domes and pyroclastics ranging in composition from basalts to rhyolites. Ar-Ar age data from different levels of the volcanostratigrafic succession yield a range of 0.76-0.06 Ma. Textural features, wide temperature ranges obtained for intermediate members, and the linear trends of whole-rock geochemistry are strongly suggestive of magma mixing in the evolution of S&uuml / phan volcanics. Presence of crystal clots in many lavas suggests that cogenetic plutonic rocks were also involved in the mixing process. Comparison of whole-rock, melt inclusion and glass chemistry data of S&uuml / phan to data from experimental studies reported in literature indicate that the melt inclusions describe true liquid lines of descent from a common hydrous parent at pressures of ~500 MPa. EC-AFC modeling of trace element and isotopic compositions reveals 2-8% crustal contamination in the differentiated lavas. REE modeling indicates that primitive rocks of S&uuml / phan volcanics were products of mixing of melts from spinel and garnet lherzolite sources, with contributions of 60% and 40%, respectively, in the mixture. A two-stage petrogenetic model is proposed for Suphan stratovolcano. Mantle- derived melts stall and undergo chemical differentiation in a deep hot zone in lower to mid-crust / variably evolved melts ascending from this zone are arrested and mixed at a shallow level where they construct a sub-volcanic magma reservoir beneath Suphan.

The two-dimension packing problem is concerned with the arrangement of items without overlaps inside a container. In particular we have considered the case when the items are circular objects, some of the general examples that can be found in the industry are related with packing, storing and transportation of circular objects. Although there are several approaches we want to investigate the use of formulation space search. Formulation space search is a fairly recent method that provides an easy way to escape from local optima for non-linear problems allowing to achieve better results. Despite the fact that it has been implemented to solve the packing problem with identical circles, we present an improved implementation of the formulation space search that gives better results for the case of identical and non-identical circles, also considering that they are packed inside different shaped containers, for which we provide the needed modifications for an appropriate implementation. The containers considered are: the unit circle, the unit square, two rectangles with different dimension (length 5, width 1 and length 10 width 1), a right-isosceles triangle, a semicircle and a right-circular quadrant. Results from the tests conducted shown several improvements over the best previously known for the case of identical circles inside three different containers: a right-isosceles triangle, a semicircle and a circular quadrant. In order to extend the scope of the formulation space search approach we used it to solve mixed-integer non-linear problems, in particular those with zero-one variables. Our findings suggest that our implementation provides a competitive way to solve these kind of problems.

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