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Generellt sätt misslyckas människor ofta att göra korrekta bedömningar baserade på principerna för sannolikhetsteori, vilket kan leda till det så kallade konjunktionsfelet. Konjunktionsfelet innebär att två kombinerade subkategorier anses mer troligt än en huvudkategori. “Nested-set”-hypotesen föreslår att konjunktionsfelet kan motverkas genom att ta hänsyn till relationen mellan kategorier och subkategorier. Denna uppsats ämnar undersöka om konjunktionsfelet minskar med ökad insikt om “nested-set”-strukturen med hjälp av få ledtrådar. Sextio deltagare fördelades randomiserat i två grupper med olika mycket ledtrådar om “nested-set”-strukturen. Resultaten indikerar att två ledtrådar om “nested-set”-strukturen är tillräcklig med information för att göra signifikant fler korrekta sannolikhetsbedömningar. En ledtråd var inte tillräcklig.  Konklusionen är att vi med hjälp av tillräckligt många ledtrådar kan förstå relationen mellan kategori- och subkategori och således fatta korrekta beslut utifrån sannolikhetslära. / Generally, people often fail to make accurate judgments according to the principles of probability theory which can lead to the so called conjunction fallacy. The conjunction fallacy consists of assigning a higher probability to two combined subcategories than one main category. The “nested-set” hypothesis proposes that the conjunction fallacy can be countered by accounting for the relationship between categories and subcategories. This thesis aims to investigate if the conjunction fallacy is reduced by giving participants few clues that reveal the set structure. Sixty participants were randomly divided into two groups with different clues about the “nested-set” structure. The results indicated that two clues about the nested set structure were enough information to make significantly more accurate probability estimates. One clue was not enough. Our conclusion is that with enough clues we are able to understand the relation between category and sub category and thus make correct decisions based on probability theory.

The purpose of this paper is to explore some of the properties of the Cantor set and to extend the idea of this set to metric spaces, in general, and to other sets of real numbers and sets in N-dimensional Euclidean space, in particular.

We present the results from three papers concerning partitions of vector spaces V over the set R of reals and of the set of lines in V.

The first theme which I focus on, is the life and work of the violin maker Václav Lanc. I explain his theories and methods of the violin building and setting up. I also approximate the luthiers craft and details of the string instruments construction. Then I write about the development of violin making in the Czech republic and also in the World and I emphasize the historical consequences. Finally I deal with the string musical instruments which are used at the basic music schools. The purpose is to minimize the problems, which teachers and pupils have with the daily maintenance and with setting up of the violin. At the end, for the youngest pupils, I dedicate the "Ten commandments" - How they should to look after their musical instruments. Keywords The building, the set up, the defects.

This thesis concerns two-point sets, which are subsets of the real plane which intersect every line in exactly two points. The existence of two-point sets was first shown in 1914 by Mazukiewicz, and since this time, the properties of these objects have been of great intrigue to mathematicians working in both topology and set theory. Arguably, the most famous problem about two-point sets is concerned with their so-called "descriptive complexity"; it remains open, and it appears to be deep. An informal interpretation of the problem, which traces back at least to Erdos, is: The term "two-point" set can be defined in a way that it is easily understood by someone with only a limited amount of mathemat- ical training. Even so, how hard is it to construct a two-point set? Can one give an effective algorithm which describes precisely how to do so? More formally, Erdos wanted to know if there exists a two-point set which is a Borel subset of the plane. An essential tool in showing the existence of a two-point set is the Axiom of Choice, an axiom which is taken to be one of the basic truths of mathematics.

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This dissertation serves as a unifying document for three related articles developed during my dissertation research. The projects involve the development of single-subject transcriptome (i.e. gene expression data) methodology for precision medicine and related applications. Traditional statistical approaches are largely unavailable in this setting due to prohibitive sample size and lack of independent replication. This leads one to rely on informatic devices including knowledgebase integration (e.g., gene set annotations) and external data sources (e.g., estimation of inter-gene correlation). Common statistical themes include multivariate statistics (such as Mahalanobis distance and copulas) and large-scale significance testing. Briefly, the first work describes the development of clinically relevant single-subject metrics of gene set (pathway) differential expression, N-of-1-pathways Mahalanobis distance (MD) scores. Next, the second article describes a method which overcomes a major shortcoming of the MD framework by accounting for inter-gene correlation. Lastly, the statistics developed in the previous works are re-purposed to analyze single-cell RNA-sequencing data derived from rare cells. Importantly, these works represent an interdisciplinary effort and show that creative solutions for pressing issues become possible at the intersection of statistics, biology, medicine, and computer science.

Volatility plays an important role in option pricing and risk management. It is crucial that volatility is modelled as accurately as possible in order to forecast with confidence. The challenge is in the selection of the ‘best’ model with so many available models and selection criteria. The Model Confidence Set (MCS) solves this problem by choosing a group of models that are equally good. A set of GARCH models were estimated for several JSE indices and the MCS was used to trim the group of models to a subset of equally superior models. Using the Mean Squared Error to evaluate the relative performance of the MCS, GARCH (1,1) and Random Walk, it was found that the MCS, with an equally weighted combination of models, performed better than the GARCH (1,1) and Random Walk for instances where volatility in the returns data was high. For instances of low volatility in the returns, the GARCH (1,1) had superior 5-day forecasts but the MCS had better performance for 10-days and greater. The EGARCH (2,1) volatility model was selected by the MCS for 5 out of the 6 indices as the most superior model. The Random Walk was shown to have better long term forecasting performance.

We identify three structures that lie in the parameter plane of the rational map F(z) = zⁿ + λ / zᵈ, for which z is a complex number, λ a complex parameter, n ≥ 4 is even, and d ≥ 3 is odd. There exists a Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the end of the arc. There exists as well a qualitatively different Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the center of the arc. Furthermore, there exist infinitely many arcs of each type. A parameter can travel along a continuous path from the Cantor set locus, along infinitely many arcs of the first type in a successively smaller region of the parameter plane, while passing through an arc of the second type, to the parameter at the center of the latter arc. This infinite sequence of Sierpindelbrot arcs is a Sierpinski Mandelbrot spiral.