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This thesis revolves around a new concept of independence of algebras. The independence nicely fits into the framework of universal products, which have been introduced to classify independence relations in quantum probability theory; the associated product is called (r,s)-product and depends on two complex parameters r and s. Based on this product, we develop a theory which works without using involutive algebras or states. The following aspects are considered: 1. Classification: Universal products are defined on the free product of algebras (the coproduct in the category of algebras) and model notions of independence in quantum probability theory. We distinguish universal products according to their behaviour on elements of length two, calling them (r,s)-universal products with complex parameters r and s respectively. In case r and s equal 1, Muraki was able to show that there exist exactly five universal products (Murakiâ€™s five). For r equals s nonzero we get five one parameter families (q-Murakiâ€™s five). We prove that in the case r not equal to s the (r,s)-product, a two parameter deformation of the Boolean product, is the only universal product satisfying our set of axioms. The corresponding independence is called (r,s)-independence. 2. Dual pairs and GNS construction: By use of the GNS construction, one can associate a product of representations with every positive universal product. Since the (r,s)-product does not preserve positivity, we need a substitute for the usual GNS construction for states on involutive algebras. In joint work with M. Gerhold, the product of representations associated with the (r,s)-product was determined, whereby we considered representations on dual pairs instead of Hilbert spaces. This product of representations is - as we could show - essentially different from the Boolean product. 3. Reduction and quantum LĂ©vy processes: U. Franz introduced a category theoretical concept which allows a reduction of the Boolean, monotone and antimonotone independence to the tensor independence. This existing reduction could be modified in order to apply to the (r,s)-independence. Quantum LĂ©vy processes with (r,s)-independent increments can, in analogy with the tensor case, be realized as solutions of quantum stochastic differential equations. To prove this theorem, the previously mentioned reduction principle in the sense of U. Franz and a generalization of M. SchĂĽrmannâ€™s theory for symmetric Fock spaces over dual pairs are used. As the main result, we obtain the realization of every (r,s)-LĂ©vy process as solution of a quantum stochastic differential equation. When one, more generally, defines LĂ©vy processes in a categorial way using U. Franzâ€™s definition of independence for tensor categories with inclusions, compatibility of the inclusions with the tensor category structure plays an important role. For this thesis such a compatibility condition was formulated and proved to be equivalent to the characterization proposed by M. Gerhold. 4. Limit distributions: We work with so-called dual semigroups in the sense of D. V. Voiculescu (comonoids in the tensor category of algebras with free product). The polynomial algebra with primitive comultiplication is an example for such a dual semigroup. We use a "weakened" reduction which we call reduction of convolution and which essentially consists of a cotensor functor constructed from the symmetric tensor algebra. It turns dual semigroups into commutative bialgebras and also translates the convolution exponentials. This method, which can be nicely described in the categorial language, allows us to formulate central limit theorems for the (r,s)-independence and to calculate the correponding limit distributions (convergence in moments). We calculate the moments appearing in the central limit theorem for the (r,s)-product: The even moments are homogeneous polynomials in r and s with the Eulerian numbers as coefficients; the odd moments vanish. The moment sequence that we get from the central limit theorem for an arbitrary universal product is the moment sequence of a probability measure on the real line if and only if r equals s greater or equal to 1. In this case we present an explicit formula for the probability measure.

A slice is an intersection of a hyperplane and a self-similar set. The main purpose of this work is the mathematical description of slices. A suitable tool to describe slices are branching dynamical systems. Such systems are a generalisation of ordinary discrete dynamical systems for multivalued maps. Simple examples are systems arising from Bernoulli convolutions and beta-representations. The connection between orbits of branching dynamical systems and slices is demsonstrated and conditions are derived under which the geometry of a slice can be computed. A number of interesting 2-d and 3-d slices through 3-d and 4-d fractals is discussed.

Today the process of improving technology and software allows to create, save and explore massive data sets in little time. "Big Data" are everywhere such as in social networks, meteorology, customersâ€™ behaviour â€“ and in biology. The Omics research field, standing for the organism-wide data exploration and analysis, is an example of biological research that has to deal with "Big Data" challenges. Possible challenges are for instance effcient storage and cataloguing of the data sets and finally the qualitative analysis and exploration of the information. In the last decade largescale genome-wide association studies and high-throughput techniques became more effcient, more profitable and less expensive. As a consequence of this rapid development, it is easier to gather massive amounts of genomic and proteomic data. However, these data need to get evaluated, analysed and explored. Typical questions that arise in this context include: which genes are active under sever al physical states, which proteins and metabolites are available, which organisms or cell types are similar or different in their enzymesâ€™or genesâ€™ behaviour. For this reason and because a scientist of any "Big Data" research field wants to see the data, there is an increasing need of clear, intuitively understandable and recognizable visualization to explore the data and confirm thesis. One way to get an overview of the data sets is to cluster it. Taxonomic trees and functional classification schemes are hierarchical structures used by biologists to organize the available biological knowledge in a systematic and computer readable way (such as KEGG, GO and FUNCAT). For example, proteins and genes could be clustered according to their function in an organism. These hierarchies tend to be rather complex, and many comprise thousands of biological entities. One approach for a space-filling visualization of these hierarchical structured data sets is a treemap. Existing algorithms for producing treemaps struggle with large data sets and have several other problems. This thesis addresses some of these problems and is structured as follows. After a short review of the basic concepts from graph theory some commonly used types of treemaps and a classification of treemaps according to information visualization aspects is presented in the first chapter of this thesis. The second chapter of this thesis provides several methods to improve treemap constructions. In certain applications the researcher wants to know, how the entities in a hierarchical structure are related to each other (such as enzymes in a metabolic pathway). Therefore in the 3 third chapter of this thesis, the focus is on the construction of a suitable layout overlaying an existing treemap. This gives rise to optimization problems on geometric graphs. In addition, from a practical point of view, options for enhancing the display of the computed layout are explored to help the user perform typical tasks in this context more effciently. One important aspect of the problems on geometric graphs considered in the third chapter of the thesis is that crossings of edges in a network structure are to be minimized while certain other properties such as connectedness are maintained. Motivated by this, in the fourth chapter of this thesis, related combinatorial and computational problems are explored from a more theoretical point of view. In particular some light is shed on properties of crossing-free spanning trees in geometric graphs.