For modelling microstructures of materials the Voronoi diagram is one of the most commonly used models. In this thesis we study a generalization of Voronoi diagrams known as the Laguerre-Voronoi diagram. In particular, we consider the stereological problem of estimating the 3D cell volume distribution of such a diagram from one of its 2D planar sections. This problem is not in general solvable for all Laguerre-Voronoi diagrams. We consider a specific class of Laguerre-Voronoi diagrams generated using an algorithm which allows to control the volume distribution of its cells. Inspired by related stereological problems, an estimator for the distribution of areas observed in a planar section is proposed. This estimator can be used for estimating the cell volume distribution from a sample of observed areas in a planar section of a 3D Laguerre-Voronoi diagram. Given that the problem is motivated by a materials science application we consider Laguerre-Voronoi diagrams with a lognormal cell volume distribution, which is commonly used in this field. The simulations show that the proposed method works well in the sense that on average the estimated parameters of the lognormal cell volume distribution are close to the actual parameter values. While the focus is on the lognormal distribution, generalization of the estimator considering other parametric distributions is briefly discussed. The thesis is concluded with a case study: the 3D grain volume distribution of a real steel microstructure is estimated from observed areas obtained from 2D image data.

A variation on the random Kuramoto model was analyzed, a mean-field model which describes the behavior of coupled oscillators in a random environment. The analytic results are based on the behavior of the system in the infinite volume limit. In the analysis critical values were found for the parameters in the model which can be used to determine when the oscillators synchronize. Numerical simulations were used to verify the analytic results.