Abstract:
Two families of stochastic interacting particle systems, the interacting Brownian motions and Bessel processes, are defined as extensions of Dyson's Brownian motion models and the eigenvalue processes of the Wishart and Laguerre processes where the parameter $\beta$ from random matrix theory is taken as a real positive number. These are systems where several particles evolve as individual Brownian motions and Bessel processes while repelling mutually through a logarithmic potential. These systems are also special cases of Dunkl processes, a broad family of multivariate stochastic processes defined by using Dunkl operators. In this thesis, the steady state under an appropriate scaling and and the freezing ($\beta\to\infty$) regime of the interacting Brownian motions and Bessel processes are studied, and it is proved that the scaled steady-state distributions of these processes converge in finite time to the eigenvalue distributions of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrices. It is also shown that the scaled final positions of the particles in these processes are fixed at the zeroes of the Hermite and Laguerre polynomials in the freezing limit. These results are obtained as the consequence of two more general results proved in this thesis. The first is that Dunkl processes in general converge in finite time to a scaled steady-state distribution that only depends on the type of Dunkl process considered. The second is that in the freezing limit, their scaled final position is fixed to a set of points called the peak set, which is the set of points which maximizes their steady-state distribution.

Abstract:
Dunkl processes are generalizations of Brownian motion obtained by using the differential-difference operators known as Dunkl operators as a replacement of spatial partial derivatives in the heat equation. Special cases of these processes include Dyson's Brownian motion model and the Wishart-Laguerre eigenvalue processes, which are well-known in random matrix theory. It is known that the dynamics of Dunkl processes is obtained by transforming the heat kernel using Dunkl's intertwining operator. It is also known that, under an appropriate scaling, their distribution function converges to a steady-state distribution which depends only on the coupling parameter $\beta$ as the process time $t$ tends to infinity. We study scaled Dunkl processes starting from an arbitrary initial distribution, and we derive expressions for the intertwining operator in order to calculate the asymptotics of the distribution function in two limiting situations. In the first one, $\beta$ is fixed and $t$ tends to infinity (approach to the steady state), and in the second one, $t$ is fixed and $\beta$ tends to infinity (strong-coupling limit). We obtain the deviations from the limiting distributions in both of the above situations, and we find that they are caused by the two different mechanisms which drive the process, namely, the drift and exchange mechanisms. We find that the deviation due to the drift mechanism decays as $t^{-1}$, while the deviation due to the exchange mechanism decays as $t^{-1/2}$.

Abstract:
Esta pesquisa objetivou comparar a densidade relativa de fragmentos ósseos mandibulares de suínos com a de penetr metros de alumínio e comparar a densidade entre dois penetr metros. Utilizou-se dois penetr metros de alumínio de diferentes fabrica es, denominados de P1 e P2, constituídos por 16 degraus, com 0,3 mm de espessura e 5 fragmentos da cortical vestibular de mandíbulas secas de suínos. Os fragmentos e os penetr metros foram radiografados com filmes Ultra-speed, tempo de exposi o de 0,32 segundos e distancia focal de 25 cm. As radiografias foram processadas em uma camara escura pelo método de tempo e temperatura, e posteriormente digitalizadas. As imagens foram analisadas por meio da ferramenta histograma do programa Image Tool, de acordo com áreas selecionadas nos fragmentos ósseos e nos penetr metros. A análise dos resultados pelos testes estatísticos ANOVA e Tukey, mostrou que n o houve diferen as estatisticamente significantes dos valores da densidade entre os fragmentos ósseos e o degrau 3 (3,6 mmEq/Al) do penetr metro P1 (p > 0,05). Observou-se que houve diferen as estatisticamente significantes entre os valores da densidade dos fragmentos ósseos e dos degraus 1 (3 mmEq/Al), 2 (3,3 mmEq/Al) e 3 (3,6 mmEq/Al) do penetr metro P2 (p < 0,05). Finalmente, constatou-se que houve diferen as estatisticamente significantes dos valores da densidade relativa dos degraus 1 (3 mmEq/Al), 2 (3,3 mmEq/Al) e 3 (3,6 mmEq/Al), entre os penetr metros P1 e P2 (p < 0,05). Concluiu-se que foi possível atribuir valores em milímetros equivalentes de alumínio à densidade relativa dos fragmentos ósseos, e que existe diferen a da densidade em pixels entre penetr metros constituídos pelo mesmo metal, porém com fabrica es diferentes, tornando-se necessário a sua padroniza o quando utilizado como material de referência para estudos de densidade óssea.

Abstract:
Trata-se do relato de uma experiência de aprendizagem enquanto acadêmico do quarto ano do Curso de Gradua o em Enfermagem da Universidade Federal de Goiás, nas aulas práticas da Disciplina Enfermagem Materno Infanto-Juvenil, no qual pretende-se compartilhar o significado de cuidar de uma crian a de nove anos de idade, de sexo feminino, portadora de insuficiência renal cr nica, que em oito dias evoluiu para óbito. Vivenciar esse processo n o foi uma tarefa fácil, pois me proporcionou sentimentos intensos de alegria e terror. Essa experiência contribuiu para nosso crescimento pessoal e profissional, por esse motivo compartilhamos.

Abstract:
We consider the interacting Bessel processes, a family of multiple-particle systems in one dimension where particles evolve as individual Bessel processes and repel each other via a log-potential. We consider two limiting regimes for this family on its two main parameters: the inverse temperature beta and the Bessel index nu. We obtain the time-scaled steady-state distributions of the processes for the cases where beta or nu are large but finite. In particular, for large beta we show that the steady-state distribution of the system corresponds to the eigenvalue distribution of the beta-Laguerre ensembles of random matrices. We also estimate the relaxation time to the steady state in both cases. We find that in the freezing regime beta->infinity, the scaled final positions of the particles are locked at the square root of the zeroes of the Laguerre polynomial of parameter nu-1/2 for any initial configuration, while in the regime nu->infinity, we prove that the scaled final positions of the particles converge to a single point. In order to obtain our results, we use the theory of Dunkl operators, in particular the intertwining operator of type B. We derive a previously unknown expression for this operator and study its behaviour in both limiting regimes. By using these limiting forms of the intertwining operator, we derive the steady-state distributions, the estimations of the relaxation times and the limiting behaviour of the processes.

Abstract:
We consider a one-dimensional system of Brownian particles that repel each other through a logarithmic potential. We study two formulations for the system and the relation between them. The first, Dyson's Brownian motion model, has an interaction coupling constant determined by the parameter beta > 0. When beta = 1,2 and 4, this model can be regarded as a stochastic realization of the eigenvalue statistics of Gaussian random matrices. The second system comes from Dunkl processes, which are defined using differential-difference operators (Dunkl operators) associated with finite abstract vector sets called root systems. When the type-A root system is specified, Dunkl processes constitute a one-parameter system similar to Dyson's model, with the difference that its particles interchange positions spontaneously. We prove that the type-A Dunkl processes with parameter k > 0 starting from any symmetric initial configuration are equivalent to Dyson's model with the parameter beta = 2k. We focus on the intertwining operators, since they play a central role in the mathematical theory of Dunkl operators, but their general closed form is not yet known. Using the equivalence between symmetric Dunkl processes and Dyson's model, we extract the effect of the intertwining operator of type A on symmetric polynomials from these processes' transition probability densities. In the strong coupling limit, the intertwining operator maps all symmetric polynomials onto a function of the sum of their variables. In this limit, Dyson's model freezes, and it becomes a deterministic process with a final configuration proportional to the roots of the Hermite polynomials multiplied by the square root of the process time, while being independent of the initial configuration.