Electroencephalography (EEG) has evolved over the years to be one of the primary diagnostic technologies providing information concerning the dynamics of spontaneous and stimulated electrical brain activity. The core question of EEG is to acquire the precise location and strength of the sources inside the human brain by knowledge of an electrical potential measured on the scalp. But in what way is the source recovered? Leaving aside the biological mechanisms on the cellular level responsible for the recorded EEG signals, we pay attention to the mathematical aspects of the narrative. Our goal is to provide a brief and concise introduction of the mathematical terminology associated with the modality of EEG. We start from the very beginning, presenting step by step the mathematical formulation behind EEG in a simple and clear manner, keeping the mathematical notation to a minimum. Whilst we serve only the key relations for the described problems, we focus specifically on the limitations of each modelling approach. In this fashion, the reader can appreciate the beauty of the formulas presented and discover every single piece of information encoded within these formulas.

A livestock population can be characterized by different population genetic parameters, such as linkage disequilibrium and recombination rate between pairs of genetic markers. The population structure, which may be caused by family stratification, has an influence on the estimates of these parameters. An expectation maximization algorithm has been proposed for estimating these parameters in half-sibs without phasing the progeny. It, however, overlooks the fact that the underlying likelihood function may have two maxima. The magnitudes of the maxima depend on the maternal allele frequencies at the investigated marker pair. Which maximum the algorithm converges to depends on the chosen start values. We present a stepwise procedure in which the relationship between the two modes is exploited. The expectation maximization algorithm for the parameter estimation is applied twice using different start values, followed by a decision process to assess the most likely estimate. This approach was validated using simulated genotypes of half-sibs. It was also applied to a dairy cattle dataset consisting of multiple half-sib families and 39,780 marker genotypes, leading to estimates for 12,759,713 intrachromosomal marker pairs. Furthermore, the proper order of markers was verified by studying the mean of estimated recombination rates in a window adjacent to the investigated locus as well as in a window at its most distant chromosome end. Putatively misplaced markers or marker clusters were detected by comparing the results with the revised bovine genome assembly UMD 3.1.1. In total, 40 markers were identified as candidates of misplacement. This outcome may help improving the physical order of markers which is also required for refining the bovine genetic map.

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22Single nucleotide polymorphisms (SNPs) which capture a significant impact on 23 a trait can be identified with genome-wide association studies. High linkage disequi-24 librium (LD) among SNPs makes it difficult to identify causative variants correctly. 25Thus, often target regions instead of single SNPs are reported. Sample size has not 26 only a crucial impact on the precision of parameter estimates, it also ensures that a 27 desired level of statistical power can be reached. We study the design of experiments 28 for fine-mapping of signals of a quantitative trait locus in such a target region. 29A multi-locus model allows to identify causative variants simultaneously, to state 30 their positions more precisely and to account for existing dependencies. Based on 31 the commonly applied SNP-BLUP approach, we determine the z-score statistic for 32 locally testing non-zero SNP effects and investigate its distribution under the al-33 ternative hypothesis. This quantity employs the theoretical instead of observed 34 dependence between SNPs; it can be set up as a function of paternal and maternal 35 LD for any given population structure. 36 We simulated multiple paternal half-sib families and considered a target region of 37 1 Mbp. A bimodal distribution of estimated sample size was observed, particu-38 larly if more than two causative variants were assumed. The median of estimates 39 constituted the final proposal of optimal sample size; it was consistently less than 40 sample size estimated from single-SNP investigations which was used as a baseline 41 approach. The second mode pointed to inflated sample sizes and could be explained 42 by blocks of varying linkage phases leading to negative correlations between SNPs. 43Optimal sample size increased almost linearly with number of signals to be identified 44 but depended much stronger on the assumption on heritability. For instance, three 45 times as many samples were required if heritability was 0.1 compared to 0.3. These 46 results enable the resource-saving design of future experiments for fine-mapping of 47