The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.

The model-free analysis of NMR relaxation data, which is widely used for the study of protein dynamics, consists of the separation of the Brownian rotational diffusion from internal motions relative to the diffusion frame and the description of these internal motions by amplitude and timescale. Through parametric restriction and the addition of the Rex parameter a number of model-free models can be constructed. The model-free problem is often solved by initially estimating the diffusion tensor. The model-free models are then optimised and the best model is selected. Finally, the global model of all diffusion and model-free parameters is optimised. These steps are repeated until convergence. This thesis will investigate all aspects of the model-free data analysis chain. (For complete abstract open document)

Doctor of philosophy (Ph D) / This thesis deals with numerical investigations on some rheological problems of fibre suspensions: the fibre level simulation of non-dilute fibre suspensions in shear flow; the numerical simulation of complex fibre suspension flows and simulating the particle motion in viscoelastic flows. These are challenging problems in rheology. Two numerical approaches were developed for simulating non-dilute fibre suspensions from the fibre level. The first is based on a model that accounts for full hydrodynamic interactions between fibres, which are approximately calculated as a superposition of the long-range and short-range hydrodynamic interactions. The long-range one is approximated by using slender body theory and includes infinite particle interactions. The short-range one is approximated in terms of the normal lubrication forces between close neighbouring fibres. The second is based on a model that accounts only for short-range interactions, which comprise the lubrication forces and normal contact and friction forces. These two methods were applied to simulate the microstructure evolution and rheological properties of non-dilute fibre suspensions. The Brownian configuration method was combined with the highly stable finite element method to simulate the complex flow of fibre suspensions. The method is stable and robust, and can provide both micro and macro information. It does not require any closure approximations in calculating the fibre stress tensor and is more efficient and variance reduction, compared to CONNFFESSITT, for example. The flow of fibre suspensions past a sphere in a tube and the shear induced fibre migration were successfully simulated using this method The completed double layer boundary element method was extended to viscoelastic flow cases. A point-wise solver was developed to solve the constitutive equation point by point and the fixed least square method was employed to interpolate and differentiate data locally. The method avoids volume meshing and only requires the boundary mesh on particle surfaces and data points in the flow domain. A sphere settling in the Oldroyd-B fluid and a prolate spheroid rotating in shear flow of the Oldroyd-B fluid were simulated. Based on the simulated orbit of a prolate spheroid in shear flow, a constitutive model for the weakly viscoelastic fibre suspensions was proposed and its predictions were compared with some available experimental results. All simulated results are in general agreement with experimental and other numerical results reported in literature. This indicates that these numerical methods are useful tools in rheological research.

O estudo apresenta um modelo de precificação de derivativos financeiros baseado em processos de mistura de movimentos brownianos. A partir de uma modelagem probabilística, são apresentados ajustes ao modelo tradicional de Black-Scholes-Merton para contemplar situações em que o retorno do ativo-objeto não segue uma distribuição normal. O trabalho discute ainda um mecanismo de estimação de parâmetros da mistura de normais. O resultado da pesquisa possibilita a análise de preço de opções em situações mais gerais. / The study presents a model for pricing financial derivatives based on a mixture of Brownian motion processes. From a probabilistic modeling, the research focuses on adjustments to the traditional Black- Scholes- Merton model to address situations where the return of the underlying asset does not follow a normal distribution. The paper also discusses a mechanism to estimate parameters of a mixture of normal distributions. The result of the study allows an analysis of option price in more general situations.

Orientador: Pedro José Catuogno / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T19:39:42Z (GMT). No. of bitstreams: 1 Neves_EduardodeAmorim_D.pdf: 2575933 bytes, checksum: c52665fabd4cb103ecdff3106990155e (MD5) Previous issue date: 2013 / Resumo: Esta tese está dividida em duas partes. Na primeira parte, faremos uma abordagem probabilística para a teoria de aplicações L-harmônicas em variedades diferenciáveis, passaremos para esse contexto os Teoremas de Liouville, Picard, Elworthy e Dirichlet. Na segunda parte do trabalho, o objetivo é generalizar e caracterizar o conceito de difusão, martingale e movimento Browniano em variedades que estejam munidas por uma família de métricas e conexões que variam diferenciavelmente com o tempo / Abstract: This thesis is divided into two parts. In the first part, we will make a probabilistic approach to the theory of L-harmonic applications on manifolds; we generalize to this context Theorems of Liouville, Picard, Elworthy and Dirichlet. In the second part of the work, the goal is to generalize and characterize the concept of diffusion, martingale and Brownian motion on manifolds that are provided by a family of metrics and connections which depends smoothly on time / Doutorado / Matematica / Doutor em Matemática

Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:49:45Z (GMT). No. of bitstreams: 1 SilvaJunior_RinaldoVieirada_M.pdf: 1586291 bytes, checksum: 8d01bdf434ecba2fb62a57725c46dd4a (MD5) Previous issue date: 2005 / Resumo: Nesta dissertação fizemos um estudo da teoria de difusão em variedades Finsler, onde abor-damos o transporte paralelo estocástico, desenvolvimento estocástico de Cartan e Movimento Browniano. O objetivo principal é obter uma descrição mais geométrica dos objetos citados acima ainda que por enquanto em coordenadas locais e assim termos um paralelo entre o cálculo estocástico em variedades Riemannianas e variedades Finsler / Abstract: In this work we study diffusion theory in Finsler manifolds. It includes the stochastic par-allel transport, stochastic Cartan development and Brownian motion. The main objective is to provide a geometric description of the objects mentioned and 50 to draw a compari-50n between stochastic calculus in Riemannian manifolds and stochastic calculus in Finsler manifolds / Mestrado / Matematica / Mestre em Matemática

Orientadores: Paulo Regis Caron Ruffino, Pedro Jose Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T04:01:14Z (GMT). No. of bitstreams: 1 Vieira_MarceloGoncalvesOliveira_M.pdf: 798971 bytes, checksum: f8f23c9cc6bf1a5acb8f987a59bb2b28 (MD5) Previous issue date: 2005 / Resumo: Este trabalho aborda a homotopia monotônica, uma variante apropriada de homotopia, de trajetórias de sistemas de controle. Primeiro é introduzido um conceito de regularidade para funções de controle e depois é considerada a definição de homotopia monotônica de trajetórias regulares de um sistema de controle 'sigma' evoluindo sobre uma variedade M. Em seguida são mostrados que o conjunto 'gama' ('sigma', x) de classes de homotopia monotônica das trajetórias regulares do sistema 'sigma' a partir de um estado fixo tem um estrutura de variedade diferenciável. Outro resultado importante é a caracterização para trajetórias monotonicamente homotópicas (contidas no conjunto dos pontos acessíveis a partir de x) via os levantamentos das mesmas à variedade 'gama' ('sigma', x). Finalmente, são feitas considerações sobre homotopia monotônica e trajetórias de um sistema estocástico / Abstract: This work accosts the monotonic homotopy, an appropriate variant of homotopy of trajectories of control systems. It is introduced a concept of regularity for control functions and it is considered the de¯nition of monotonic homotopy of regular trajectories of a given control system 'sigma' on a manifold M. Then it is shown that the set 'gama' ('sigma', x) of monotonic homotopy classes of regular trajectories of 'sigma' starting at a given fixed point x has a differentiable manifold structure with the same dimension of M. Another important result is the caracterization of monotonic homotopy of trajectories (on acessible points set starting at x) via the lifts of same trajectories on the manifold 'gama' ('sigma', x). Finally, we make same considerations about monotonic homotopy and trajectories of an stochastic system / Mestrado / Matematica / Mestre em Matemática

Optical tweezers are a sensitive position and force transducer widely employed in physics and biology. In a focussed laser, forces due to radiation pressure enable to trap and manipulate small dielectric particles used as probes for various experiments. For sensitive biophysical measurements, microspheres are often used as a handle for the molecule of interest. The force range of optical traps well covers the piconewton forces generated by individual biomolecules such as kinesin molecular motors. However, cellular processes are often driven by ensembles of molecular machines generating forces exceeding a nanonewton and thus the capabilities of optical tweezers. In this thesis I focused, fifirst, on extending the force range of optical tweezers by improving the trapping e fficiency of the probes and, second, on applying the optical tweezers technology to understand the mechanics of molecular motors. I designed and fabricated photonically-structured probes: Anti-reflection-coated, high-refractive-index, core-shell particles composed of titania. With these probes, I significantly increased the maximum optical force beyond a nanonewton. These particles open up new research possibilities in both biology and physics, for example, to measure hydrodynamic resonances associated with the colored nature of the noise of Brownian motion. With respect to biophysical applications, I used the optical tweezers to study the mechanics of single kinesin-8. Kinesin-8 has been shown to be a very processive, plus-end directed microtubule depolymerase. The underlying mechanism for the high processivity and how stepping is affected by force is unclear. Therefore, I tracked the motion of yeast (Kip3) and human (Kif18A) kinesin-8s with high precision under varying loads. We found that kinesin-8 is a low-force motor protein, which stalled at loads of only 1 pN. In addition, we discovered a force-induced stick-slip motion, which may be an adaptation for the high processivity. Further improvement in optical tweezers probes and the instrument will broaden the scope of feasible optical trapping experiments in the future.

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Les travaux présentés dans cette thèse s'intéressent à la géométrie fractale de processus stochastiques à travers le prisme d'un outil appelé l'analyse 2-microlocale. Ce dernier est issu d'une autre branche des mathématiques, l'analyse fonctionnelle et l'étude des équations aux dérivées partielles, et s'est avéré être pertinent pour décrire la géométrie fine de fonctions déterministes ou de processus aléatoires, généralisant notamment les exposants de Hölder classiques. Nous envisageons ainsi dans ce manuscrit différentes classes de processus, traitant en premier lieu le cas des martingales continues et de l'intégrale stochastique d'Ito. La régularité 2-microlocale de ces derniers fait notamment apparaître un autre concept, la pseudo frontière 2-microlocale, étroitement lié à son aîné. Nous appliquons également ce formalisme d'étude à une classe de processus gaussiens : le mouvement brownien multifractionnaire. Nous caractérisons ainsi sa régularité 2-microlocale et hölderienne, et déterminons dans un deuxième temps la forme générale de la dimension fractale de ses trajectoires. Dans notre étude portant sur les processus de Lévy, nous combinons le formalisme 2-microlocale à l'analyse multifractale, permettant alors de mettre en évidence des comportements géométriques n'étant pas captés par les outils usuels. Nous obtenons également en corollaire le spectre multifractal des processus fractionnaires de Lévy. Enfin, dans une dernière partie, nous nous intéressons à la définition et aux propriétés de certains processus de Markov multiparamètres, pouvant être plus généralement indicés par des ensembles. / The work presented in this thesis concerns the study of the fractal geometry of stochastic processes using the formalism of 2-microlocal analysis. The latter has been introduced in another branch of mathematics -functional analysis- but has also proved to be relevant to describe the geometry of deterministic functions or random processes, extending in particular the classic Hölder exponents. Several classes of processes are investigated in this manuscript, beginning with continuous martingales and Ito integrals. In particular, the characterisation of the 2-microlocal regularity of the latter leads to the introduction of a closely related concept: the pseudo 2-microlocal frontier. We also investigate using this formalism a class of Gaussian processes called multifractional Brownian motion and obtain a fine description of its Hölder and 2-microlocal behaviours. In addition, we characterize entirely the Hausdorff and Box dimensions of its graph. In our study of Lévy processes, we combine the 2-microlocal formalism and multifractal analysis to describe their regularity, exhibiting in particular some subtle geometrical behaviours which are not captured by classic tools. Furthermore, as a corollary of this result, we also determine the multifractal spectrum of another family of processes: the fractional Lévy processes. Lastly, we also define a class of multiparameter and set-indexed Markov processes and study its properties.