Die Arbeit besteht hauptsächlich aus zwei Teilen: einer Zusammenfassung kombinatorischer und algebraisch-geometrischer Themen (Gitterpolytope, torische (Gorenstein-)Varietäten, und Matroide), und einem Ergebnisteil. Letzerer besteht aus zwei Teilen. Im ersten Teil wird eine konstruktive Klassifikation von Multigraphen, deren graphisches Matroid ein Basispolytop erzeugt, das die Gorenstein-Eigenschaft erfüllt, erarbeitet. Im zweiten Teil wird ein Satz rekursiver Formeln, die die Ehrhartpolynome von symmetrischen Kantenpolytopen, die aus vollständig-biparten Graphen hervorgehen, zueinander in Beziehung stellen, vorgestellt. Außerdem wird Algorithmus, mit dem man solche Formel erzeugen kann, aufgezeigt.:1. Introduction 2. Notation 3. Preliminaries 3.1 Lattice Polytopes 3.2 Toric Varieties 3.3 Matroids 3.4 Gorenstein Toric Varieties 4. Results 4.1 Gorenstein Matroids 4.2 Recursive Formulas of Symmetric Edge Polytopes

The present thesis is located within the context of the diffusion approximation of Wright–Fisher models and the Kolmogorov equations describing their evolution. On the one hand, a full account of recombinational Wright–Fisher model is developed as well as their enhancement by other evolutionary mechanisms, including some information geometrical analysis. On the other hand, the thesis addresses several issues arising in the context of analytical solution schemes for such Kolmogorov equations, namely the inclusion of the entire boundary of the state space. For this, a hierarchical extension scheme is developed, both for the forward and the backward evolution, and the uniqueness of such extensions is proven. First, a systematic approach to the diffusion approximation of recombinational two- or more loci Wright–Fisher models is presented. As a point of departure a specific Kolmogorov backward equation for the diffusion approximation of a recombinational two-loci Wright–Fisher model is chosen, to which – with the help of some information geometrical methods, i. e. by calculating the sectional curvatures of the corresponding statistical manifold (which is the domain equipped with the corresponding Fisher metric) – one succeeds to identify the underlying Wright–Fisher model. Accompanying this, for all methods and tools involved a suitable introduction is presented. Furthermore, the considerations span a separate analysis for the two most common underlying models (RUZ and RUG) as well as a comparison of the two models. Finally, transferring corresponding results for a simpler model described by Antonelli and Strobeck, solutions of the Kolmogorov equations are contrasted with Brownian motion in the same domain. Furthermore, the perspective of the diffusion approximation of recombinational Wright–Fisher models is widened as the model underlying the Ohta–Kimura formula is subsequently extended by an integration of the concepts of natural fitness and mutation. Simultaneously, the corresponding extensions of the Ohta–Kimura formula are stated. Crucial for this is the development of a suitable fitness scheme, which is accomplished by a multiplicative aggregation of fitness values for pairs of gametes/zygotes. Furthermore, the model is generalised to have an arbitrary number of alleles and – in the following step – an arbitrary number of loci respectively. The latter involves an increased number of recombination modes, for which the concept of recombination masks is also implemented into the model. Another generalisation in terms of coarse-graining is performed via an application of schemata; this also affects the previously introduced concepts, specifically mask recombination, which are adapted accordingly. Eventually, a geometric analysis of linkage equilibrium states of the multi-loci Wright–Fisher models is carried out, relating to the concept of hierarchical probability distributions in information geometry, which concludes the considerations of recombinational Wright–Fisher models and their extensions. Subsequently, the discussion of analytical solution schemes for the Kolmogorov equations corresponding to the diffusion approximation of Wright–Fisher models is ushered in, which represents the second part of the thesis. This is started with the simplest setting of a 1-dimensional Wright–Fisher model, for which the solution strategy for the corresponding Kolmogorov forward equation given by M. Kimura is recalled. From this, one may construct a unique extended solution which also accounts for the dynamics of the model on lower-dimensional entities of the state space, i. e. configurations of the model where one of the alleles no longer exists in the population, utilising the concept of (boundary) flux of a solution; a discussion of the moments of the distribution confirms the findings. A similar treatment is then carried out for the corresponding Kolmogorov backward equation, yielding analogous results of existence and uniqueness for an extended solution. For the latter in particular, a corresponding account of the configuration on the boundary turns out to be crucial, which is also reflected in the probabilistic interpretation of the backward solution. Additionally, the long-term behaviour of solutions is analysed, and a comparison between such solutions of the forward and the backward equation is made. Next, it is basically aimed to transfer the results obtained in the previous chapter to the subsequent increasingly complicated setting of a Wright–Fisher model with 1 locus and an arbitrary number of alleles: With solution schemes for the interior of the state space (i. e. not encompassing the boundary) already existing in the literature, an extension scheme for a successive determination of the solution on lower-dimensional entities of the domain is developed. This scheme, again, makes use of the concept of the (boundary) flux of solutions, and one may therefore show that this extended solution fulfils additional properties regarding the completeness of the diffusion approximation with respect to the boundary. These properties may be formulated in terms of the moments of the distribution, and their connection to the underlying Wright–Fisher model is illustrated. Altogether, stipulating such a moments condition, existence and uniqueness of an extended solution on the entire domain are shown. Furthermore, the corresponding Kolmogorov backward equation is examined, for which similarly a (backward) extension scheme is presented, which allows extending a solution in a domain (perceived as a boundary instance of a larger domain) to all adjacent higher-dimensional entities of the larger domain along a certain path. This generalises the integration of boundary data observed in the previous chapter; in total, the existence of a solution of the Kolmogorov backward equation in the entire domain is shown for arbitrary boundary data. Of particular interest to the discussion are stationary solutions of the Kolmogorov backward equation as they describe eventual hit probabilities for a certain target set of the model (in accordance with the probabilistic interpretation of solutions of the backward equation). The presented backward extension scheme allows the construction of solutions for all relevant cases, reconfirming some results by R. A. Littler for the stationary case, but now providing a previously missing systematic derivation. Eventually, the hitherto missing uniqueness assertion for this type of solutions is established by means of a specific iterated transformation which resolves the critical incompatibilities of solutions by a successive blow-up while the domain is converted from a simplex into a cube. Then – under certain additional assumptions on the regularity of the transformed solution – the uniqueness directly follows from general principles. Lastly, several other aspects of the blow-up scheme are discussed; in particular, it is illustrated in what way the required extra regularity relates to reasonable additional properties of the underlying Wright–Fisher model.

The history of populations or species is of fundamental importance in a variety of areas. Gaining details about demographic, cultural, climatic or political aspects of the past may provide insights that improve the understanding of how populations have evolved over time and how they may evolve in future. Different types of resources can be informative about different periods of time. One especially important resource is genetic data, either from a single individual or a group of organisms. Environmental conditions and circumstances can directly affect the existence and success of a group of individuals. Since genetic material gets passed on from generation to generation, traces of past events can still be detected in today\''s genetic data. For many decades scientists have tried to understand the principles of how external influences can directly affect the appearance and features of populations, leading to theoretical models that can interpret modern day genetic variation in the light of past events. Among other influencing factors like migration, natural selection etc., population size changes can have a great impact on the genetic diversity of a group of organisms. For example, in the field of conservation biology, gaining insights into how the size of a population evolves may assist in detecting past or ongoing temporal reductions of population size. This seems crucial since the reduction in size also correlates with a reduction in genetic diversity which in turn might negatively affect the evolutionary potential of a population. Using computational and population genetics methods, sequences from whole genomes can be scanned for traces of such events and therefore assist in new interpretations of historical details of populations or groups of interest. This thesis focuses on the detection and interpretation of past population size changes. Two approaches to infer particular parameters from underlying demographic models are described. The first part of this thesis introduces two summary statistics which were designed to detect fluctuations in size from genome-wide Single Nucleotide Polymorphism (SNP) data. Demographic inferences from such data are inherently complicated due to recombination and ascertainment bias. Hence, two new statistics are introduced: allele frequency-identity by descent (AF-IBD) and allele frequency-identity by state (AF-IBS). Both make use of linkage disequilibrium information and exhibit defined relationships to the time of the underlying mathematical process. A fast and efficient Approximate Bayesian Computation framework based on AF-IBD and AF-IBS is constructed that can accurately estimate demographic parameters. These two statistics were tested for the biasing effects of hidden recombination events, ascertainment bias and phasing errors. The statistics were found to be robust to a variety of these tested biases. The inference approach was then applied to genome-wide SNP data to infer the demographic histories of two human populations: (i) Yoruba from Africa and (ii) French from Europe. Results suggest, that AF-IBD and AF-IBS are able to capture sufficient amounts of information from underlying data sets in order to accurately infer parameters of interest, such as the beginning, end and strength of periods of varying size. Additionally the results from empirical data suggest a rather stable ancestral population size with a mild recent expansion for Yoruba, whereas the French apparently experienced a rather long-lasting strong bottleneck followed by a drastic population growth. The second part of this thesis introduces a new way of summarizing information from the site frequency spectrum. Commonly applied site frequency spectrum based inference methods make use of allele frequency information from individual segregating sites. Our newly developed method, the 2 point spectrum, summarizes allele frequency information from all possible pairs of segregating sites, thereby increasing the number of potentially informative values from the same underlying data set. These additional information are then incorporated into a Markov Chain Monte Carlo framework. This allows for a high degree of flexibility and implements an efficient method to infer population size trajectories over time. We tested the method on a variety of different simulated data sets from underlying demographic models. Furthermore, we compared the performance and accuracy of our method to already established methods like PSMC and diCal. Results indicate that this non-parametric 2 point spectrum method can accurately infer the extent and times of past population size changes and therefore correctly estimates the history of temporal size fluctuations. Furthermore, the initial results suggest that the amount of required data and the accuracy of the final results are comparable with other publicly available non-parametric methods. An easy to use command line program was implemented and will be made publicly available. In summary, we introduced three highly sensitive summary statistics and proposed different approaches to infer parameters from demographic models of interest. Both methods provide powerful frameworks for accurate parameter inference from genome-wide genetic data. They were tested for a variety of demographic models and provide highly accurate results. They may be used in the settings as described above or incorporated into already existing inference frameworks. Nevertheless, the statistics should prove useful for new insights into populations, especially those with complex demographic histories.

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both \\mbox{R-cores} and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal.

The complexity and size of measured and simulated data in many fields of science is increasing constantly. The technical evolution allows for capturing smaller features and more complex structures in the data. To make this data accessible by the scientists, efficient and specialized visualization techniques are required. Maximum efficiency and value for the user can only be achieved by adapting visualization to the specific application area and the specific requirements of the scientific field. Part I: In the first part of my work, I address the visualization in the neurosciences. The neuroscience tries to understand the human brain; beginning at its smallest parts, up to its global infrastructure. To achieve this ambitious goal, the neuroscience uses a combination of three-dimensional data from a myriad of sources, like MRI, CT, or functional MRI. To handle this diversity of different data types and sources, the neuroscience need specialized and well evaluated visualization techniques. As a start, I will introduce an extensive software called \"OpenWalnut\". It forms the common base for developing and using visualization techniques with our neuroscientific collaborators. Using OpenWalnut, standard and novel visualization approaches are available to the neuroscientific researchers too. Afterwards, I am introducing a very specialized method to illustrate the causal relation of brain areas, which was, prior to that, only representable via abstract graph models. I will finalize the first part of my work with an evaluation of several standard visualization techniques in the context of simulated electrical fields in the brain. The goal of this evaluation was clarify the advantages and disadvantages of the used visualization techniques to the neuroscientific community. We exemplified these, using clinically relevant scenarios. Part II: Besides the data preprocessing, which plays a tremendous role in visualization, the final graphical representation of the data is essential to understand structure and features in the data. The graphical representation of data can be seen as the interface between the data and the human mind. The second part of my work is focused on the improvement of structural and spatial perception of visualization -- the improvement of the interface. Unfortunately, visual improvements using computer graphics methods of the computer game industry is often seen sceptically. In the second part, I will show that such methods can be applied to existing visualization techniques to improve spatiality and to emphasize structural details in the data. I will use a computer graphics paradigm called \"screen space rendering\". Its advantage, amongst others, is its seamless applicability to nearly every visualization technique. I will start with two methods that improve the perception of mesh-like structures on arbitrary surfaces. Those mesh structures represent second-order tensors and are generated by a method named \"TensorMesh\". Afterwards I show a novel approach to optimally shade line and point data renderings. With this technique it is possible for the first time to emphasize local details and global, spatial relations in dense line and point data. / In vielen Bereichen der Wissenschaft nimmt die Größe und Komplexität von gemessenen und simulierten Daten zu. Die technische Entwicklung erlaubt das Erfassen immer kleinerer Strukturen und komplexerer Sachverhalte. Um solche Daten dem Menschen zugänglich zu machen, benötigt man effiziente und spezialisierte Visualisierungswerkzeuge. Nur die Anpassung der Visualisierung auf ein Anwendungsgebiet und dessen Anforderungen erlaubt maximale Effizienz und Nutzen für den Anwender. Teil I: Im ersten Teil meiner Arbeit befasse ich mich mit der Visualisierung im Bereich der Neurowissenschaften. Ihr Ziel ist es, das menschliche Gehirn zu begreifen; von seinen kleinsten Teilen bis hin zu seiner Gesamtstruktur. Um dieses ehrgeizige Ziel zu erreichen nutzt die Neurowissenschaft vor allem kombinierte, dreidimensionale Daten aus vielzähligen Quellen, wie MRT, CT oder funktionalem MRT. Um mit dieser Vielfalt umgehen zu können, benötigt man in der Neurowissenschaft vor allem spezialisierte und evaluierte Visualisierungsmethoden. Zunächst stelle ich ein umfangreiches Softwareprojekt namens \"OpenWalnut\" vor. Es bildet die gemeinsame Basis für die Entwicklung und Nutzung von Visualisierungstechniken mit unseren neurowissenschaftlichen Kollaborationspartnern. Auf dieser Basis sind klassische und neu entwickelte Visualisierungen auch für Neurowissenschaftler zugänglich. Anschließend stelle ich ein spezialisiertes Visualisierungsverfahren vor, welches es ermöglicht, den kausalen Zusammenhang zwischen Gehirnarealen zu illustrieren. Das war vorher nur durch abstrakte Graphenmodelle möglich. Den ersten Teil der Arbeit schließe ich mit einer Evaluation verschiedener Standardmethoden unter dem Blickwinkel simulierter elektrischer Felder im Gehirn ab. Das Ziel dieser Evaluation war es, der neurowissenschaftlichen Gemeinde die Vor- und Nachteile bestimmter Techniken zu verdeutlichen und anhand klinisch relevanter Fälle zu erläutern. Teil II: Neben der eigentlichen Datenvorverarbeitung, welche in der Visualisierung eine enorme Rolle spielt, ist die grafische Darstellung essenziell für das Verständnis der Strukturen und Bestandteile in den Daten. Die grafische Repräsentation von Daten bildet die Schnittstelle zum Gehirn des Menschen. Der zweite Teile meiner Arbeit befasst sich mit der Verbesserung der strukturellen und räumlichen Wahrnehmung in Visualisierungsverfahren -- mit der Verbesserung der Schnittstelle. Leider werden viele visuelle Verbesserungen durch Computergrafikmethoden der Spieleindustrie mit Argwohn beäugt. Im zweiten Teil meiner Arbeit werde ich zeigen, dass solche Methoden in der Visualisierung angewendet werden können um den räumlichen Eindruck zu verbessern und Strukturen in den Daten hervorzuheben. Dazu nutze ich ein in der Computergrafik bekanntes Paradigma: das \"Screen Space Rendering\". Dieses Paradigma hat den Vorteil, dass es auf nahezu jede existierende Visualiserungsmethode als Nachbearbeitunsgschritt angewendet werden kann. Zunächst führe ich zwei Methoden ein, die die Wahrnehmung von gitterartigen Strukturen auf beliebigen Oberflächen verbessern. Diese Gitter repräsentieren die Struktur von Tensoren zweiter Ordnung und wurden durch eine Methode namens \"TensorMesh\" erzeugt. Anschließend zeige ich eine neuartige Technik für die optimale Schattierung von Linien und Punktdaten. Mit dieser Technik ist es erstmals möglich sowohl lokale Details als auch globale räumliche Zusammenhänge in dichten Linien- und Punktdaten zu erfassen.

The metabolism constitutes the universe of biochemical reactions taking place in a cell of an organism. These processes include the synthesis, transformation, and degradation of molecules for an organism to grow, to reproduce and to interact with its environment. A good way to capture the complexity of these processes is the representation as metabolic network, in which sets of molecules are transformed into products by a chemical reaction, and the products are being processed further. The underlying graph model allows a structural analysis of this network using established graphtheoretical algorithms on the one hand, and a visual representation by applying layout algorithms combined with information visualization techniques on the other. In this thesis we will take a look at three different aspects of graph visualization within the context of biochemical systems: the representation and interactive exploration of static networks, the visual analysis of dynamic networks, and the comparison of two network graphs. We will demonstrate, how established infovis techniques can be combined with new algorithms and applied to specific problems in the area of metabolic network visualization. We reconstruct the metabolic network covering the complete set of chemical reactions present in a generalized eucaryotic cell from real world data available from a popular metabolic pathway data base and present a suitable data structure. As the constructed network is very large, it is not feasible for the display as a whole. Instead, we introduce a technique to analyse this static network in a top-down approach starting with an overview and displaying detailed reaction networks on demand. This exploration method is also applied to compare metabolic networks in different species and from different resources. As for the analysis of dynamic networks, we present a framework to capture changes in the connectivity as well as changes in the attributes associated with the network’s elements.

Mitochondria are organelles present in most eukaryotic cells. They generate most of the cells adenosine triphosphate (ATP) supply which make them essential for cell viability. It is assumed that they are derived from a proteobacterial ancestor as they retain their own, drastically small genome. The importance in studying mitochondrial genome evolution came from the discovery of a large number of human diseases that are caused by mitochondrial dysfunction (e.g., Parkinson and Alzheimer). Many of these diseases are a result of a mutation in one of the mitochondrial genes or a defective mitochondrial DNA (mtDNA) maintenance, mostly caused by genetic defects in proteins involved in mtDNA replication. In order to explore the diversity and understand the evolution of mitochondrial genomes (mitogenomes) in animals, multiple methods have been developed in this study to deal with two biological problems related to the mitochondrial genome evolution. A new method for identifying the mitochondrial origins of replication is presented. This method deals with the problem of determining the origins of replication, which despite many previous efforts has remained non-trivial even in the small genomes of animal mitochondria. The replication mechanism is of central interest to understand the evolution of mitochondrial genomes since it allows the duplication of the genetic information. The extensive work that has been done to study the replication of mitochondrial genomes has generated the assumption of the strand displacement model (SDM) also known as the standard model of replication that is known to leave the mitochondrial H-strand in a single stranded state exposing it to mutation and damage. Later on, other models of replication have been suggested such as the strand coupled bidirectional replication model, its refinement which assumes the bidirectional mode but with a unidirectional start, and the \"RNA incorporation throughout the lagging strand\" (RITOLS) model proposed as a refinement of the strand displacement model. Based on the observation that the GC-skew is correlated with the distance from the replication origins in the light of the strand displacement model of replication, a new computational method to infer the position of both the heavy strand and the light strand origins from nucleotide skew data has been developed. The method has been applied in a comprehensive survey of deuterostome mitochondria where conserved positions of the replication origins for the vast majority of vertebrates and cephalochordates have been inferred. Deviations from the consensus picture are presumably associated with genome rearrangements. Additionally, two methods for the identification of tRNA remolding events throughout Metazoa have been developed. Remolding changes the identity of a tRNA by a duplication and a point mutation(s) of the anticodon. This new tRNA takes the identity of another tRNA which is then lost. This can lead to artifacts in the annotation of mitogenomes and thus in studies of mitogenomic evolution. In this work, novel methods are developed to detect tRNA remolding in large-scale data sets. The first method represents an extension of the similarity-based approach to determine remolding candidates with high confidence. This approach uses an extended set of criteria based on both sequence and structural similarities of the tRNAs in conjunction with statistical tests. The second method is a novel phylogeny-based likelihood method which evaluates specific topologies of gene phylogenies of the two tRNA families relevant to a putative remolding event. Both methods have been applied to survey tRNA remolding throughout animal evolution. At least three novel remolding events are identified in addition to the ones previously mentioned in the literature. A detailed analysis of these remoldings showed that many of them are derived ancestral events.

Genre and domain influence an author\''s style of writing and therefore a text\''s characteristics. Natural language processing is prone to such variations in textual characteristics: it is said to be genre and domain dependent. This thesis investigates genre and domain dependencies in sentiment analysis. Its goal is to support the development of robust sentiment analysis approaches that work well and in a predictable manner under different conditions, i.e. for different genres and domains. Initially, we show that a prototypical approach to sentiment analysis -- viz. a supervised machine learning model based on word n-gram features -- performs differently on gold standards that originate from differing genres and domains, but performs similarly on gold standards that originate from resembling genres and domains. We show that these gold standards differ in certain textual characteristics, viz. their domain complexity. We find a strong linear relation between our approach\''s accuracy on a particular gold standard and its domain complexity, which we then use to estimate our approach\''s accuracy. Subsequently, we use certain textual characteristics -- viz. domain complexity, domain similarity, and readability -- in a variety of applications. Domain complexity and domain similarity measures are used to determine parameter settings in two tasks. Domain complexity guides us in model selection for in-domain polarity classification, viz. in decisions regarding word n-gram model order and word n-gram feature selection. Domain complexity and domain similarity guide us in domain adaptation. We propose a novel domain adaptation scheme and apply it to cross-domain polarity classification in semi- and unsupervised domain adaptation scenarios. Readability is used for feature engineering. We propose to adopt readability gradings, readability indicators as well as word and syntax distributions as features for subjectivity classification. Moreover, we generalize a framework for modeling and representing negation in machine learning-based sentiment analysis. This framework is applied to in-domain and cross-domain polarity classification. We investigate the relation between implicit and explicit negation modeling, the influence of negation scope detection methods, and the efficiency of the framework in different domains. Finally, we carry out a case study in which we transfer the core methods of our thesis -- viz. domain complexity-based accuracy estimation, domain complexity-based model selection, and negation modeling -- to a gold standard that originates from a genre and domain hitherto not used in this thesis.

Die Zielsetzung der Arbeit ist die Anwendung bestehender Ansätze aus dem Forschungsgebiet des Service Engineering sowie deren konzeptionelle Erweiterung, um eine methodische Verfahrensweise bei der Entwicklung und Gestaltung von Dienstleistungen in der Branche erneuerbare Energien zu unterstützen. Im Fokus der Arbeit stehen dabei industrienahe, komplexe und variantenreiche Dienstleistungen der Branche, für deren Beschreibung Methoden und Werkzeuge dargelegt werden. Hierbei wird ein möglichst ganzheitlicher Ansatz verfolgt, der die Beschreibung von EE-Dienstleistungen auf mehreren Ebenen ermöglicht. Im Vorfeld der Anwendung konkreter Ansätze wird erhoben, welche Potenziale bezüglich des Service Engineering in der Branche bestehen und welche Anforderungen sich aus diesen ergeben. Zur Erfüllung dieser Anforderungen werden die Konzepte eines Metamodells zur ganzheitlichen Beschreibung von Dienstleistungen auf den Ebenen Komponente, Produkt, Prozess und Ressource angewendet und spezifisch erweitert. Auf dieser Grundlage erfolgt die Implementierung verschiedener Softwareartefakte, die in einem Dienstleistungsframework verknüpft werden und eine integrierte Beschreibung IT-basierter Dienstleistungen ermöglichen. Mittels eines Anwendungsfalls wird die Integration der Beschreibung und Konfiguration von Dienstleistungen auf der businesslogischen Ebene mit der Orchestrierung und Ausführung auf der Ebene der Softwareservices veranschaulicht. Die Arbeit hat die Zielsetzung, zum einen wissenschaftlich und zum anderen praktisch zur innovativen Entwicklungen im Gebiet der EE-Dienstleistungen beizutragen und somit den Betrachtungsfokus auf die technologie-begleitenden Dienstleistungen zu richten.

We studied the subjectivity of word maps on SU(n) and the length of the shortest elements in the central series of free group of rank 2 with some applications to almost laws in compact groups.