In this thesis, we work on the upper bound for nonlinearity of t-resilient Boolean functions given by Sarkar and Maitra, which is based on divisibility properties of spectral weights of resilient functions and study construction methods that achieve the upper bound. One of the construction methods, introduced by Maity and Johansson, starts with a bent function and complements some values of its truth table corresponding to a previously chosen set of inputs, S, which satisfies three criteria. In this thesis, we show that a fourth criterion is needed for t-resiliency of the resulting function, and prove that three criteria of Maity and Johansson do not guarantee resiliency. We also work on other constructions, one by Sarkar and Maitra, which uses a Maiorana-McFarland like technique to satisfy the upper bound and the other by Tarannikov, which satisfies the nonlinearity bound using a technique with low computational complexity. However, these methods have tendency to maximize the order of resiliency for a given number of variables, therefore one cannot construct functions for all possible resiliency values given the number of variables, using this method. We further go into details and compute the auto-correlation functions of the constructed Boolean functions to find the absolute indicator and sum-of-squared-errors for each of them. We also provide a comparison of Boolean functions constructed by other techniques given in the literature, together with the ones studied in this thesis.

The aim of the thesis is to define the dependency of the classification of patients affected by neurodegenerative diseases on the choice of the parcellation atlas. Part of this thesis is the application of the functional connectivity analysis and the calculation of graph metrics according to the method published by Olaf Sporns and Mikail Rubinov [1] on fMRI data measured at CEITEC MU. The application is preceded by the theoretical research of parcellation atlases for brain segmentation from fMRI frames and the research of mathematical methods for classification as well as classifiers of neurodegenerative diseases. The first chapters of the thesis brings a theoretical basis of knowledge from the field of magnetic and functional magnetic resonance imaging. The physical principles of the method, the conditions and the course of acquisition of image data are defined. The third chapter summarizes the graph metrics used in the diploma thesis for analyzing and classifying graphs. The paper presents a brief overview of the brain segmentation methods, with the focuse on the atlas-based segmentation. After a theoretical research of functional connectivity methods and mathematical classification methods, the findings were used for segmentation, calculation of graph metrics and for classification of fMRI images obtained from 96 subjects into the one of two classes using Binary classifications by support vector machines and linear discriminatory analysis. The data classified in this study was measured on patiens with Parkinson’s disease (PD), Alzheimer’s disease (AD), Mild cognitive impairment (MCI), a combination of PD and MCI and subjects belonging to the control group of healthy individuals. For pre-processing and analysis, the MATLAB environment, the SPM12 toolbox and The Brain Connectivity Toolbox were used.