In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation. A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that the constant times the Hausdorff measure is greater than or equal to the packing measure when a subset of the realization is evaluated. Self—affine Sierpinski carpets, which have been analyzed by McMullen with respect to their Hausdorff dimension and capacity dimension, are analyzed with respect to their packing dimension. Conditions under which the Hausdorff measure of the construction object is positive and finite are given.

Pose estimation uses computer vision to predict how a body moves. The likeliness off different movements is predicted with a neural network and the most likely pose is predicted. With DeepLabCut, an open source software package for 3D animal pose estimation, information about animals behaviour and movement can be extracted. In this report the pose estimation of horses four hooves is used. By looking at the position of the hooves different gaits can be identified. Horses used for riding in the major disciplines in Sweden have three different gaits, walk, trot and canter. Walk is a four-stoke gait, trot is two-stoke and canter is three-stoke. This can be used to classify the different gaits. By looking at the hooves movement in vertical position over time and fitting a sinewave to the data it is possible to see the phase difference in the hooves movement. For walk and trot there was a significant pattern which was easy to identify and corresponded well to the theory of horses movement. For canter our pre-trained model lacked in accuracy, so the output data were insufficient. Therefore it was not possible to find a significant pattern for canter which corresponds to the theory of horses movements. The Fourier Transform were also tested to classify the gaits and when plotted it was possible to detect the different gaits, but not significant enough to be reliable for different horses in different sizes running in different paces. It was also possible to add the data for all four hooves together and fit a sinewave to the added data, and then compare it with the sinewaves for each hoof separately. Depending on the gait the frequency of the sinewaves differed between the hooves separately and added together and the gaits could be identified.

Title:Preparation of rider on horse from performance level Z to S in dressage Objectives: Compilation of helpful methodologic material, awareness of fundamental value, correct position of the riders sed, choice of good horse, gaits of horses, characteristic performance levels, characteristics and preparation for the degree Z, characterization and preparation for the degree L, characterization and preparation for the degree S. Methods: Analysis of the literature, comparing information targeted in the content of the work of literature, analysis of my own experiences from my riding praxis, analysis of independent observations of horse riders, analysis of experiences from lessons with my riding students, compilation of information into a complete whole and methodological manual. Results: Preparation of methodological material, which may a rider or trainer of horse riding (dressage) use in their training hours. Keywords: dressage, rider, performance stage, horse, gaits of horses, walk, trot, canter, scale of training, exercises in dressage