L’objectif principal de cette thèse est divisée en deux parties. La première partie est consacrée à l’étude du problème, { −Δu + u = 0 dans Ω, ∂u/∂n + g(u) = μ sur ∂Ω, (0.1) où Ω est un ouvert régulier borné de ℝᴺ, g(·) est une fonction continue qui vérifie la condition du signe s · g(s) ≥ 0, dans certains modèles on ajoute l’hypothèse g(·) croissante, et finalement μ est une mesure bornée sur ∂Ω. Certains de nos résultats sont valables lorsque Ω := ℝᴺ+ . On commencera par montrer l’existence de solution de (0.1) lorsque μ est une fonction de L1(∂Ω), et cela sans ajouter une hypothèse supplémentaire sur g(·). Puis, on étudiera (0.1) lorsque μ est une mesure de Radon sur ∂Ω, dans ce contexte, le problème (0.1) pourra ne pas admettre une solution, et des conditions apparaissent sur g(·) et sur μ pour assurer l’existence d’une solution. On montrera l’existence de solutions lorsque, g(·) est une non-linéarité sous-critique en dimension N supérieure ou égale à trois, et lorsque g(·) satisfait l’hypothèse de singularité faible sur le bord en dimension N égale à deux (voir Chapitre 2 pour définitions). / The main aim of this thesis is divided into two parts. The first part is devoted to the study of the problem, { −Δu + u = 0 in Ω, ∂u/∂n + g(u) = μ on ∂Ω, (0.3) where Ω is a bounded regular domain of ℝᴺ, g(·) is a continuous function that satisfies the sign condition s · g(s) ≥ 0, in some model case we will assume that g(·) is increassing, and finally μ is a bounded measure on ∂Ω. Some of our results remain true when Ω := ℝᴺ+ . We will start by proving the existence of a solution of (0.3) when μ is an L1(∂Ω) function, and this independently of the nonlinearity g(·) that satisfies the previous hypothesis. Then, we will study (0.3) when μ is a Radon measure on ∂Ω. In such a context, some new conditions appear on g(·) and μ that assure the existence of a solution. We will prove the existence of a solution when g(·) is a sub-critical nonlinearity in dimension N larger or equal to three, and when g satisfies the weak singularity assumption on the boundary in case N equals two (see Chapter 2 for the definitions).

O objetivo principal deste trabalho é a obtenção dos modos e as freqüências naturais de vigas de variação linear e em forma de cunha, com condições de contorno clássicas e não-clássicas, descritas pelo modelo estrutural de Euler-Bernoulli. A forma dos modos foi determinado com o uso das funções cilíndricas. No caso forçado se considera uma força harmônica e se resolve o problema pelo método espectral, utuilizando o software simbólico Maple V5. Realiza-se uma análise comparativa dos resultados obtidos com os resultados existentes na literatura para vigas uniformes.

Maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions involves inverting the ratio R_nu = I_{nu+1} / I_nu of modified Bessel functions. Computational issues when using approximative or iterative methods were discussed in Tanabe et al. (Comput Stat 22(1):145-157, 2007) and Sra (Comput Stat 27(1):177-190, 2012). In this paper we use Amos-type bounds for R_nu to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of R is evaluated at values tending to 1 (from the left). We show that previously introduced rational bounds for R_nu which are invertible using quadratic equations cannot be used to improve these bounds. / Series: Research Report Series / Department of Statistics and Mathematics

Finite mixtures of von Mises-Fisher distributions allow to apply model-based clustering methods to data which is of standardized length, i.e., all data points lie on the unit sphere. The R package movMF contains functionality to draw samples from finite mixtures of von Mises-Fisher distributions and to fit these models using the expectation-maximization algorithm for maximum likelihood estimation. Special features are the possibility to use sparse matrix representations for the input data, different variants of the expectationmaximization algorithm, different methods for determining the concentration parameters in the M-step and to impose constraints on the concentration parameters over the components. In this paper we describe the main fitting function of the package and illustrate its application. In addition we compare the clustering performance of finite mixtures of von Mises-Fisher distributions to spherical k-means. We also discuss the resolution of several numerical issues which occur for estimating the concentration parameters and for determining the normalizing constant of the von Mises-Fisher distribution. (authors' abstract)

Orientador: Michel Zamboni Rached / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-22T19:43:51Z (GMT). No. of bitstreams: 1 GarayAvendano_RogerLeonardo_M.pdf: 3532246 bytes, checksum: c57f4a9f337b3e0e75f701a896e690a0 (MD5) Previous issue date: 2013 / Resumo: Embora a propagação de feixes ópticos seja um tema muito investigado, existe ainda uma grande variedade de estudos a se efetuar, principalmente no desenvolvimento de métodos que permitam, de forma analítica, a descrição exata da enorme diversidade de feixes com propriedades distintas. A principal contribuição desta dissertação é a proposta de uma metodologia matemática para a obtenção de feixes escalares e eletromagnéticos não paraxiais puramente propagantes como soluções analíticas exatas da equação de onda e das equações de Maxwell. Tal método baseia-se em uma solução analítica para as integrais que descrevem superposições de feixes de Bessel de ordem zero (não evanescentes) com qualquer tipo de função espectral. Exemplos de feixes não paraxiais são apresentados para a validação do método proposto neste trabalho, os quais provam a grande eficiência em termos do pouco esforço computacional quando são comparados com os métodos de outros autores / Abstract: Although the propagation of optical beams has been vastly studied, there is still a huge amount of research topics to be exploited, mainly regarding the developing of exact analytic methods. The main contribution of this work is the development of a mathematical methodology to obtain nonparaxial propagating scalar and electromagnetic beams as exact analytic solutions of the wave equation and Maxwell's equations, respectively. This method is based on a very general solution to the continuous superposition of zero order Bessel beams (non-evanescent) with any kind of spectral function. Examples of non paraxial beams are shown to validate the method proposed in this work, which proves to be very efficient, based on low computational effort when compared to other author's methods / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

>Magister Scientiae - MSc / This dissertation consists of two main sections. The first section focuses on generating zero order Bessel beams using axicons. An axicon with an opening angle y = 5⁰ was illuminated with a Gaussian beam of width ω₀ = 1.67 mm from a cw fiber laser with central wavelength λ = 1064 nm to generate zero order Bessel beams with a central spot radius r₀ = 8.3 ± 0.3 μm and propagation distance ½zmax = 20.1 ± 0.5 mm. The central spot size of a Bessel beam changes slightly along the propagation distance. The central spot radius r₀ can be varied by changing the opening angle of the axicon, y, and the wavelength of the beam. The second section focuses on applications of the generated Bessel beams in laser microdrilling. A Ti:Sapphire pulsed femtosecond laser (λ = 775 nm, ω₀ = 2.5 mm, repetition rate kHz, pulse energy mJ, and pulse duration fs) was used to generate the Bessel beams for drilling stainless steel thin sheets of thickness 50 μm and 100 μm and microscopic glass slides 1 mm thick. The central spot radius was r₀ = 15.9 ± 0.3 μm and ½zmax = 65.0 ± 0.5 mm. The effect of the Bessel beam shape on the quality of the holes was analysed and the results were discussed. It was observed that Bessel beams drill holes of better quality on transparent microscopic glass slides than on stainless steel sheet. The holes drilled on stainless steel sheets deviated from being circular on both the top and bottom surface for both thicknesses. However the holes maintained the same shape on both sides of each sample, indicating that the walls are close to being parallel. The holes drilled on the glass slides were circular and their diameters could be measured. The measured diameter (15.4±0.3 μm) of the hole is smaller than the diameter of the central spot (28.2 ± 0.1 μm) of the Bessel beam. Increasing the pulse energy increased the diameter of the drilled hole to a value close to the measured diameter of the central spot.

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22823号 / 理博第4633号 / 新制||理||1666(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.

This work aims to implement 3D microstructures that generate light with orbital angular momentum towards applications in Biophotonics. Over the past few decades, 3D printing has established itself as the most versatile technology with effortless adaptability. Parallel to this, the concept of miniaturiza tion has seen tremendous growth irrespective of the field and has become an estab lished trend motivated by the need for compact, portable and multi-function devices. Therefore, when these two concepts get together, i.e., 3D printing of miniaturized objects, it could lead to an exciting path with endless opportunities. When it comes to optics, miniaturized 3D printing offers the potential to create compact optical micro-systems and exhibits a way to manufacture freeform µ-optics. In particular, two-photon lithography (TPL) is a cutting edge 3D printing technology that has re cently demonstrated groundbreaking solutions for optics as it offers high resolution with a great degree of flexibility. With a TPL 3D printer, it is possible to fabricate complex µ-optical elements and employ them for compelling applications. In recent years, light with orbital angular momentum (OAM), or ”twisted” light, has captured the interests of several researchers due to its inspiring applications. Tra ditionally, to generate OAM beams, one would require bulk, table-top optics, restrict ing their applications to over-the-table setup. An alternative approach of OAM beam generation is through µ-structures over the fiber, as they can open up new opportu nities, especially in Bioscience, and facilitate in-vivo operations. In particular, this probe-like setup can be used for processes such as optical trapping, high-resolution microscopy, etc. Hence, I propose the development of a novel approach with un precedented capabilities for generating OAM beams right from single-mode optical fibers, by transforming its Gaussian-like output beam by using complex 3D printed microstructures. In this document, I will showcase designs and results on generating Bessel beams (both zeroth- and high-order) and high-NA converging beams (with and without OAM) for optical trapping from the fiber. Remarkably, I achieved the first-ever fiber-based high-order Bessel beam generation and the first-ever fiber optical tweezers with OAM.