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The recent popularity of Bayesian inference is due to the practical advantages of the Bayesian approach. The Bayesian analysis makes it possible to reflect ones prior beliefs into the analysis. In this thesis, we explore some asymptotic results in Bayesian nonparametric inference for restricted and unrestricted space of distributions. This thesis is divided into two parts. In the first part, we employ the Dirichlet process in a hypothesis testing framework to propose a Bayesian nonparametric chi-squared goodness-of-fit test. Our suggested method corresponds to Lo's Bayesian bootstrap procedure for chi-squared goodness-of-fit test. Indeed, our bootstrap rectifies some shortcomings of regular bootstrap which only counts number of observations falling in each bin in contingency tables. We consider the Dirichlet process as the prior for the distribution of data and carry out the test based on the Kullback-Leibler distance between the Dirichlet process posterior and the hypothesized distribution. We prove that this distance asymptotically converges to the same chi-squared distribution as the classical frequentist's chi-squared test. Moreover, the results are generalized to the chi-squared test of independence for contingency tables. In the second part, our main focus is on Bayesian nonparametric inference for a restricted group of distributions called spherically symmetric distributions. We describe a Bayesian nonparametric approach to perform an inference for a bivariate spherically symmetric distribution. We place a Dirichlet invariant process prior on the set of all bivariate spherically symmetric distributions and derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of the Dirichlet invariant process for the symmetric distributions on the real line to bivariate spherically symmetric distribution where the underlying distribution is invariant under a finite group of rotations. Further, we obtain the Dirichlet invariant process posterior for the infinite transformation group and we prove that it approaches a certain Dirichlet process. Finally, we develop our approach to obtain the Bayesian nonparametric posterior distribution for functionals of the distribution's support when the support satisfies certain symmetry conditions. When symmetry holds with respect to the parallel lines of axes (for example, in two dimensional space x = a and y = b) we employ our approach to approximate the distribution of certain functionals such as area and perimeter for the support of the distribution. This suggests a Bayesian nonparametric bootstrapping scheme. The estimates can be derived based on posterior averaging. Then, our simulation results demonstrate that our suggested bootstrapping technique improves the accuracy of the estimates.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-26T20:13:44Z No. of bitstreams: 1 alcidesfariasndrade.pdf: 803619 bytes, checksum: 302375f647c6bebde2042805b53cddc2 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T20:49:09Z (GMT) No. of bitstreams: 1 alcidesfariasndrade.pdf: 803619 bytes, checksum: 302375f647c6bebde2042805b53cddc2 (MD5) / Made available in DSpace on 2017-08-07T20:49:09Z (GMT). No. of bitstreams: 1 alcidesfariasndrade.pdf: 803619 bytes, checksum: 302375f647c6bebde2042805b53cddc2 (MD5) Previous issue date: 2011-08-25 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Estudamos um defeito tipo deslocação com simetria esférica. Encontramos a métrica para um meio contendo uma única deformação deste tipo. Para isso, calculamos o vetor deslocamento por meio da teoria linear da elasticidade e usando o esquema da teoria geométrica de defeitos, na qual o meio é caracterizado por objetos geométricos tais como curvatura e torção, encontramos as componentes do tensor métrico. Calculamos também outras quantidades geométricas como os tensores de Riemann e Ricci, e o escalar de curvatura bem como as componentes do tensor momento-energia. Em todas estas quantidades aparecem funções δ, indicando divergência na superfície onde está localizado o defeito. Fora desta superfície, o meio possui uma geometria euclideana. Resolvemos as equações geodésicas radial e no plano para a região externa ao defeito e observamos que, mesmo localizado, ele exerce influência sobre o movimento nesta região. / We study a dislocation defect with spherical symmetry. We find the metric for a medium containing a single deformation of this kind. For this, we calculate the displacement vector through the linear theory of elasticity and using the scheme of geometric theory of defects, in which the medium is characterized by geometric objects such as curvature and torsion, we find the components of the metric tensor. We calculate also other geometrical quantities as the Riemann and Ricci tensor, and the scalar curvature as well as the energy-momentum tensor. In these quantities, δ-functions show up, indicating divergence in the surface where the defect is located. Outside this surface, the medium has an Euclidean geometry. We solve the geodesic equations for the outer region to the defect and we observed, despite being located, its influence on the movement in this region.

In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.

Šimon Knoška The spherically symmetric spacetimes represent one of the most important classes of solutions in general relativity. Therefore, it is very natural to study them also in the context of modified theories of gravity. We directly continue in the previous works in quadratic gravity, where the generalised solutions with the constant Ricci scalar were found in the form of power series expansion in the conformal coordinates. In this work, we have found an alternative expression of this solution in the Robinson-Trautman-like coordinates analogously in the form of power series expansion around the horizon. Al- though the prescribed recurrent power series solution is more complicated than that in the conformal-to-Kundt coordinates, it posses numerous advantages. Namely, the trans- formation to the Schwarzschild-like coordinates is considerable simple and the physical interpretation of the coordinates is more evident. These properties are demonstrated in the preliminary investigation of the geodesic motion of the test particles near the black hole with analysis of the effect of the so-called Bach parameter. In particular, we have observed that the Bach parameter together with the positive cosmological constant Λ > 0 has a significant impact on the global structure of the spacetime.