The present thesis intents on analysing the truncated matricial Hamburger power moment problem in the general (degenerate and non-degenerate) case. Initiated due to manifold lines of research, by this time, outnumbering results and thoughts have been established that are concerned with specific subproblems within this field. The resulting presence of such a diversity as well as an extensively considered topic si- multaneously involves advantageous as well as obstructive aspects: on the one hand, we adopt the favourable possibility to capitalise on essential available results that proved beneficial within subsequent research. Nevertheless, on the other hand, we are obliged to illustrate major preparatory work in order to illucidate the comprehension of the attaching examination. Moreover, treating the matricial cases of the respective problems requires meticulous technical demands, in particular, in view of the chosen explicit approach to solving the considered tasks. Consequently, the first part of this thesis is dedicated to furnishing the necessary basis arranging the prime results of this research paper. Compul- sary notation as well as objects are introduced and thoroughly explained. Furthermore, the required techniques in order to achieve the desired results are characterised and ex- haustively discussed. Concerning the respective findings, we are afforded the opportunity to seise presentations and results that are, by this time, elaborately studied. Being equipped with mandatory cognisance, the thematically bipartite second and pivo- tal part objectives to describe all the possible values of all the solution functions of the truncated matricial Hamburger power moment problem M P [R; (s j ) 2n j=0 , ≤]. Aming this, we realise a first paramount achievement epitomising one of the two parts of the main results: Capturing an established representation of the solution set R 0,q [Π + ; (s j ) 2n j=0 , ≤] of the assigned matricial Hamburger moment problem via operating a specific algorithm of Schur-type, we expand these findings. We formulate a parameterisation of the set R 0,q [Π + ; (s j ) 2n j=0 , ≤] which is compatible with establishing respective equivalence classes within a certain subset of Nevanlinna pairs and utilise specific systems of orthogonal polynomials in order to entrench novel representations. In conclusion, we receive a para- meterisation that is valid within the entire upper open complex half-plane Π + . The second of the two prime parts changes focus to analysing all possible values of the functions belonging to R 0,q [Π + ; (s j ) 2n j=0 , ≤] in an arbitrary point w ∈ Π + . We gain two decisive conclusions: We identify these respective values to exhaust particular matrix balls 2n K[(s j ) 2n j=0 , w] := {F (w) | F ∈ R 0,q [Π + ; (s j ) j=0 , ≤]} the parameters of which are feasable to being described by specific rational matrix-valued functions and, in this course, enhance formerly established analyses. Moreover, we compile an alternative representation of the semi-radii constructing the respective matrix balls which manifests supportive in further consideration. We seise the achieved parameterisation of the set K[(s j ) 2n j=0 , w] and examine the behaviour of the respective sequences of left and right semi-radii. We recognise that these sequences of semi-radii associated with the respective matrix balls in the general case admit a particular monotonic behaviour. Consequently, with increasing number of given data, the resulting matrix balls are identified as being nested. Moreover, a proper description of the limit case of an infinite number of prescribed moments is facilitated.:1. Brief Historic Embedding and Introduction 2. Part I: Initialising Compulsary Cognisance Arranging Principal Achievements 2.1. Notation and Preliminaries 2.2. Particular Classes of Holomorphic Matrix-Valued Functions 2.3. Nevanlinna Pairs 2.4. Block Hankel Matrices 2.5. A Schur-Type Algorithm for Sequences of Complex p × q Matrices 2.6. Specific Matrix Polynomials 3. Part II: Momentous Results and Exposition – Improved Parameterisations of the Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.1. An Essential Step to a Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.2. Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 3.3. Particular Matrix Polynomials 3.4. Description of the Solution Set of the Truncated Matricial Hamburger Moment Problem by a Certain System of Orthogonal Matrix Polynomials 4. Part III: Prime Results and Exposition – Novel Description Balls 4.1. Particular Rational Matrix-Valued Functions 4.2. Description of the Values of the Solutions 4.3. Monotony of the Semi-Radii and Limit Balls of the Weyl Matrix 5. Summary of Principal Achievements and Prospects A. Matrix Theory B. Integration Theory of Non-Negative Hermitian Measures

Im Mittelpunkt der vorliegenden Dissertation steht eine neue Behandlung der Matrixversion eines speziellen klassischen Interpolationsproblems, welches unter dem Namen Schurproblem bekannt ist. Eine im Inneren des Einheitskreises holomorphe Matrixfunktion, welche ausschließlich kontraktive Werte annimmt, nennen wir Schurfunktion. Das matrizielle Schurproblem besteht dann darin, die Menge aller Schurfunktionen mit vorgegebener Anfangsfolge der Taylorentwicklung im Nullpunkt zu beschreiben. Insbesondere ist jene Situation zu charakterisieren, in der die Lösungsmenge des Problems nichtleer ist. In den vergangenen 50 Jahren wurden sehr erfolgreich verschiedene Lösungsstrategien zur Behandlung des matriziellen Schurproblems entwickelt. Das Ziel der vorliegenden Arbeit ist es, einen zu den bisher etablierten Herangehensweisen alternativen Zugang zu präsentieren. Wir sind nämlich bestrebt, ein matrizielles Schurproblem mit einem weiteren klassischen matriziellen Interpolationsproblem, dem sogenannten Carathéodoryproblem, in Verbindung zu bringen. Unter einer Carathéodoryfunktion verstehen wir eine im Inneren des Einheitskreises holomorphe quadratische Matrixfunktion, welche ausschließlich Werte mit nichtnegativ hermiteschem Realteil annimmt. Das matrizielle Carathéodoryproblem besteht dann darin, die Menge aller Carathéodoryfunktionen mit vorgegebener Anfangsfolge der Taylorentwicklung im Nullpunkt zu beschreiben. Insbesondere ist jene Situation zu charakterisieren, in der die Lösungsmenge des Problems nichtleer ist. Unsere Strategie beruht auf der Idee, mit einem konkreten matriziellen Schurproblem in geeigneter Weise ein matrizielles Carathéodoryproblem derart zu assoziieren, dass sich die Lösungen des Schurproblems mit gewissen ausgezeichneten Lösungen des Carathéodoryproblems identifizieren lassen. Somit können wir aus der Darstellung der Lösungsmenge des Carathéodoryproblems, welche sich als Bild einer gebrochenlinearen Transformation von Matrizen realisieren lässt, eine entsprechende Beschreibung der Lösungsmenge des ursprünglichen Schurproblems herleiten. Unser Vorgehen gliedert sich in mehrere Schritte, welche jeweils verschiedene Dimensionen des matriziellen Schurproblems beleuchten: Zuerst wenden wir uns einer algebraischen Untersuchung sogenannter (endlicher oder unendlicher) matrizieller Schurfolgen zu, welche über Kontraktivitätseigenschaften spezieller Blockmatrizen definiert sind. Diese Folgen stellen sich später gerade als Taylorkoeffizientenfolgen von matrixwertigen Schurfunktionen heraus. Wir konstruieren zu einer Schurfolge zunächst eine nichtnegativ definite Folge, welche über die nichtnegative Hermitizität einer Blocktoeplitzmatrix definiert ist, und darauf aufbauend eine sogenannte matrizielle Carathéodoryfolge. Weiterhin werden Aussagen zur inneren Struktur matrizieller Schurfolgen hergeleitet, das Erweiterungsproblem für endliche matrizielle Schurfolgen gelöst und sogenannte zentrale Schurfolgen untersucht. Zweitens werden aus endlichen matriziellen Schurfolgen dann spezielle Matrixpolynome konstruiert. Besonderes Augenmerk legen wir hierbei auf den Zusammenhang dieser Matrixpolynome mit gewissen Matrixpolynomen, welche aus der zur Schurfolge assoziierten Carathéodoryfolge gebildet werden und bei der Beschreibung der Lösungsmenge des matriziellen Carathéodoryproblems in den Resolventenmatrizen der gebrochenlinearen Transformationen auftreten. Es stellt sich heraus, dass die aus einer Schurfolge konstruierten Matrixpolynome in Blockzerlegungen der zur assoziierten Carathéodoryfolge gehörenden Matrixpolynome auftreten. Im dritten Schritt wenden wir uns holomorphen Matrixfunktionen zu und assoziieren mittels einer matriziellen Verallgemeinerung des Lemmas von Schwarz zu einer matriziellen Schurfunktion eine matrizielle Carathéodoryfunktion. Wir befassen uns außerdem mit der Taylorkoeffizientenfolge einer Schurfunktion und können so die Verbindung zu den vorherigen Überlegungen herstellen. Weiterhin untersuchen wir sogenannte Typ-II-zentrale Schurfunktionen. Als Höhepunkt dieser Arbeit leiten wir eine vollständige Lösung des matriziellen Schurproblems her. Die Frage der Lösbarkeit können wir rasch unter Zuhilfenahme unserer bisherigen Untersuchungen beantworten. Daraufhin identifizieren wir Lösungen des matriziellen Schurproblems mit speziellen Lösungen eines durch unsere bisherigen Betrachtungen nahegelegten assoziierten matriziellen Carathéodoryproblems. Unter Heranziehung der allgemeinen Lösung des matriziellen Carathéodoryproblems erhalten wir somit die gewünschte Beschreibung der Lösungsmenge des matriziellen Schurproblems mittels gebrochenlinearer Transformationen von Matrizen, welche aus den zuvor untersuchten Matrixpolynomen gebildet werden. Weiterhin identifizieren wir eine Teilmenge des Definitionsbereiches der gebrochenlinearen Transformationen, auf der diese sogar bijektive Abbildungen auf die Lösungsmenge des matriziellen Schurproblems liefern.

In this PhD thesis we investigate the space R^inv(M): the space of riemannian metrics on a spin manifold M whose associated Dirac operator is invertible. In particular we are interest in the bond between the topology of R^inv(M) and the topology of the underlying manifold M. We conjecture that the homotopy type of R^inv(M) is invariant under spin cobordism.

Ziel dieser Arbeit ist die Erarbeitung von effizienten (semi-)automatischen Verfahren zur Verknüpfung von Wissensbasen. Eine Vielzahl von Lösungsklassen können zu diesem Zweck eingesetzt werden. In dieser Arbeit werden ausschließlich deklarative Ansätze erörtert. Deklarative Ansätze gehen davon aus, dass das direkte Errechnen von Mappings zwischen Mengen von Ressourcen in vielen Fällen nur schwer möglich ist oder eines nicht vertretbaren Aufwands bedarf. Diese Ansätze zielen daher darauf ab, eine Ähnlichkeitsfunktion sowie einen Schwellwert zu finden, die zur Approximation eines Mappings genutzt werden können. Zwei Herausforderungen gehen mit dieser Modellierung des Problems einher: (a) Effizienz sowie (b) Genauigkeit und Vollständigkeit. Lösungen zu beiden Herausforderungen sowie auf echten Daten basierende Anwendungen dieser Lösungen werden in der Arbeit vorgestellt.

Following the classical theory of Baire category results for sets of measure-preserving transformations, this work develops a theory for Baire category results for sets of measure-preserving extensions. First the case is considered where a measure space and a sub-algebra are fixed, and extensions are considered to be any measure-preserving transformations which leave this sub-algebra invariant. In the latter case, extensions of a fixed measure-preserving transformation are considered. In both cases, it is shown that the set of weakly mixing extensions form a dense, G-delta set

In this work we introduce some category-theoretical concepts and techniques to study probability distributions on metric spaces and ordered metric spaces. In Chapter 1 we give an overview of the concept of a probability monad, first defined by Giry. Probability monads can be interpreted as a categorical tool to talk about random elements of a space X. We can consider these random elements as formal convex combinations, or mixtures, of elements of X. Spaces where the convex combinations can be actually evaluated are called algebras of the probability monad. In Chapter 2 we define a probability monad on the category of complete metric spaces and 1-Lipschitz maps called the Kantorovich monad, extending a previous construction due to van Breugel. This monad assigns to each complete metric space X its Wasserstein space PX. It is well-known that finitely supported probability measures with rational coefficients, or empirical distributions of finite sequences, are dense in the Wasserstein space. This density property can be translated into categorical language as a colimit of a diagram involving certain powers of X. The monad structure of P, and in particular the integration map, is uniquely determined by this universal property. We prove that the algebras of the Kantorovich monad are exactly the closed convex subsets of Banach spaces. In Chapter 3 we extend the Kantorovich monad of Chapter 2 to metric spaces equipped with a partial order. The order is inherited by the Wasserstein space, and is called the stochastic order. Differently from most approaches in the literature, we define a compatibility condition of the order with the metric itself, rather then with the topology it induces. We call the spaces with this property L-ordered spaces. On L-ordered spaces, the stochastic order induced on the Wasserstein spaces satisfies itself a form of Kantorovich duality. The Kantorovich monad can be extended to the category of L-ordered metric spaces. We prove that its algebras are the closed convex subsets of ordered Banach spaces, i.e. Banach spaces equipped with a closed cone. The category of L-ordered metric spaces can be considered a 2-category, in which we can describe concave and convex maps categorically as the lax and oplax morphisms of algebras. In Chapter 4 we develop a new categorical formalism to describe operations evaluated partially. We prove that partial evaluations for the Kantorovich monad, or partial expectations, define a closed partial order on the Wasserstein space PA over every algebra A, and that the resulting ordered space is itself an algebra. We prove that, for the Kantorovich monad, these partial expectations correspond to conditional expectations in distribution. Finally, we study the relation between these partial evaluation orders and convex functions. We prove a general duality theorem extending the well-known duality between convex functions and conditional expectations to general ordered Banach spaces.

This PhD project is on the algorithmic aspects of synthesis planning and mass spectrometry; two separate chemical problems concerning the understanding of molecules and how these behave. Part I: In synthesis planning, the goal is to synthesize a target molecule from available starting materials, possibly optimizing costs such as price or environmental impact of the process. Current algorithmic approaches to synthesis planning are usually based on selecting a bond set and finding a single good plan among those induced by it. We demonstrate that synthesis planning can be phrased as a combinatorial optimization problem on hypergraphs, not necessarily using a pre-defined bond set. For this, individual synthesis plans are modeled as directed hyperpaths embedded in a hypergraph of reactions (HoR) representing the chemistry of interest. As a consequence, application of a know polynomial time algorithm to find the K shortest hyperpaths yields the K best synthesis plans for a given target molecule. To this end, classical quality measures are discussed. Having K good plans to choose from has several benefits: It makes the synthesis planning process much more robust when in later stages adding further chemical details, it allows one to combine several notions of cost, and it provides a way to deal with imprecise yield estimates. An empirical study of our method illustrates the limitations of what a chemist can expect is feasible to compute, as well as the practical value of our method for cases where yield estimates are imprecise or unknown. To illustrate the realism of the approach, synthesis plans from our abstraction level are compared with detailed chemical synthesis plans from the literature. For this, a synthesis plan for Wieland-Miescher ketone and a synthesis plan for lysergic acid are used. In addition, equivalence of our structural definition of a hyperpath and two definitions from the hypergraph literature is shown. Part II: Mass spectrometry is an analytic technique for characterizing molecules and molecular mixtures, by gaining knowledge of their structure and composition from the way they fragment. In a mass spectrometer, molecules or molecular mixtures are ionized and fragmented, and the abundances of the different fragment masses are measured, resulting in so-called mass spectra. We suggest a new road map improving the current state-of-the art in computational methods for mass spectrometry. The main focus is on increasing the chemical realism of the modeling of the fragmentation process. Two core ingredients for this are i) describing the individual fragmentation reactions via graph transformation rules and ii) expressing the dynamics of the system via reaction rates and quasi-equilibrium theory. Graph transformation rules are used both for specifying the possible core fragmentation reactions, and for characterizing the reaction sites when learning values for the rates. We believe that this model describes chemical mechanisms more accurately than previous ones, and that this can lead to both better spectrum prediction and more explanatory power. Our modeling of system dynamics also allows better separation of instrument dependent and instrument independent parameters of the model.

This thesis is split into two parts, the first one is concerned with some problems in stochastic homogenization and the second addresses a problem in singular SPDEs. In the part on stochastic homogenization we are interested in developing large-scale regularity theories for random linear elliptic operators by using estimates for the homogenization error to transfer regularity from the homogenized operator to the heterogeneous one at large scales. In the whole-space case this has been done by Gloria, Neukamm, and Otto through means of a homogenization-inspired Campanato iteration. Here we are specifically interested in boundary regularity and as a model setting we consider random linear elliptic operators on the half-space with either homogeneous Dirichlet or Neumann boundary data. In each case we obtain a large-scale regularity theory and the main technical difficulty turns out to be the construction of a sublinear homogenization corrector that is adapted to the boundary data. The case of Dirichlet boundary data is taken from a joint work with Julian Fischer. In an attempt to head towards a percolation setting, we have also included a chapter concerned with the large-scale behaviour of harmonic functions on a domain with random holes assuming that these are 'well-spaced'. In the second part of this thesis we would like to provide a pathwise solution theory for a singular quasilinear parabolic initial value problem with a periodic forcing. The difficulty here is that the roughness of the data limits the regularity the solution such that it is not possible to define the nonlinear terms in the equation. A well-posedness result, therefore, comes with two steps: 1) Giving meaning to the nonlinear terms and 2) Showing that with this meaning the equation has a solution operator with some continuity properties. The solution theory that we develop in this contribution is a perturbative result in the sense that we think of the solution of the initial value problem as a perturbation of the solution of an associated periodic problem, which has already been handled in a work by Otto and Weber. The analysis in this part relies entirely on estimates for the heat semigroup. The results in the second part of this thesis will be in an upcoming joint work with Felix Otto and Jonas Sauer.

It is a widely accepted assumption that fish predation controls structure and functioning of aquatic food webs. In the past, however, a large part of effort was concentrated on lakes and reservoirs. Thus, the knowledge about stream ecosystems is much more rudimentary than that for lakes in this respect. The aim of this thesis, therefore, was to describe and assess the effects of fish predation in natural stream ecosystems. For that purpose a reach scale field experiment was set up using an experimental stretch with benthivorous fish and a fishless reference stretch. A wide range of effects of the fish predators on their stream invertebrates prey was studied. To discriminate between lethal and sublethal predation effects, measuring the physiological status of the organisms seemed promising. However, before it was possible to decide whether or not environmental stress, such as predation, might affect the physiological status, the internal control as well as the seasonal and species-specific variability of the energy amount stored had to be assessed. Thus, the concentration and seasonal dynamics of the major energy storage components triglycerides and glycogen were measured in two species of mayflies (Rhithrogena semicolorata and Ephemera danica) with contrasting life cycle strategies. E. danica is a burrowing, semivoltine collector-gatherer, R. semicolorata is univoltine and scrapes periphyton from stones. Although triglycerides are the major energy reserve in both species throughout the whole larval development (> 84 % of total energy storage) their seasonal dynamic differed considerably. In R. semicolorata the triglyceride concentration declined during the last weeks prior to emergence in both sexes. The same pattern was found in female larvae of E. danica, but not in male E. danica. It is suggested that females use triglycerides in the last larval stages for egg maturation, which is completed in the last larval instar. In male E. danica the triglyceride concentrations remained high until emergence, presumably due to their high energy demands as adults for their swarming flights and mating. The difference in seasonal variation of triglycerides between E. danica and R. semicolorata shows the influence of environmental factors on the dynamics of storage components. E. danica lived in a very stable environment (within the substratum). Therefore the dynamic of energy storage components was optimised with respect to maximal reproduction. R. semicolorata on the other hand, suffered from hostile environmental factors such as predation or food limitation due to low periphyton biomass after leaf sprout and following light limitation in spring. Consequently, the concentration of storage components decreased during spring. One conclusion from this study was that the measurement of storage components might reveal sublethal predation effects. However, season and sex of the organisms are important factors as well and have to be considered in the sampling design. To analyse sublethal predation effects behavioural changes due to the presence of benthivorous fish were measured. Drift as a low-energy cost means of migration may enable stream invertebrates to leave risky habitats or may even be a direct escape reaction after a predator encounter. While the control of drift activity by predators has received considerable interest from many researchers, it remains still unclear whether predators reduce or increase drift activity. Drift activity of stream invertebrates was influenced significantly by the presence or absence the two benthivorous fish species gudgeon (Gobio gobio) and stone loach (Barbatula barbatula). Contrary to previous studies gudgeon and stone loach reduced invertebrate drift density and drift activity of Baetis rhodani rather than inducing higher night-time drift. Further, species composition of the invertebrate drift differed significantly between the two stretches. A further conclusion from this study is therefore that drift is not generally a mechanism of active escape from benthos-feeding fish, as previously assumed. In addition, the reduced drift activity in the fish stretch might result in a compensation of the consumptive losses due to fish predation. Thus, in this study design the effects of fish predation on invertebrate community might be underestimated. To detect predation effects on the food web structure the reactions of the grazing mayfly Rhithrogena semicolorata and the shredding amphipod Gammarus pulex to strong predation by benthivorous fish were compared. It has been hypothesised that shredders are generally less vulnerable to fish predation and therefore less likely to be predation-controlled than grazers, because the latter are visible to the predators during their feeding on stone surfaces, while shredders may hide between leaves during foraging. Biomass of G. pulex was significantly reduced in the fish stretch while that of R. semicolorata was not. Since approximately 91 % of the annual production of G. pulex but only 12 % of R. semicolorata production was consumed by benthivorous gudgeon, the observed difference of G. pulex biomass between the fish and reference reach is likely due to a lethal predation effect. However, no sublethal predation effects such as reduced concentration of storage components (triglycerides, glycogen) or reduced reproductive success were observed for both species. Hence, in contrast to the initial hypothesis, in the studied stream the shredder was top-down-controlled, while the grazer was not. It is concluded that top-down control depends on the ecological characteristics of a specific predator-prey pair rather than on trophic guild of the prey. To assess the predation effects on the life history of merolimnic insects and its consequences on fecundity the larval development and emergence of R. semicolorata was studied. It was possible to show lethal and sublethal effects of predation by benthivorous fish (Gobio gobio, Barbatula barbatula). Predation consequently resulted in changes of larval development and population fitness. The presence of two benthivorous fish species (gudgeon and stone loach) led to slower larval development and a delayed emergence. However, no differences in the adult size and fecundity between the fish reach and the reference were observed. Nevertheless, the longer time spent in the larval phase resulted in a higher mortality and therefore in a lower mean population fitness. The presence of gudgeon alone, however, did not seem to influence larval development, growth or time of emergence and consequently fecundity. Further, strong lethal impact of gudgeon could not be detected. Thus, the population fitness measured as the product of adult density and egg number was not reduced by gudgeon alone. It is assumed that the stronger lethal impact in the combined fish experiment is caused mainly by stone loach because the proportion of mayfly consumption by stone loach to mayfly production shortly before emergence was higher than the proportion related to gudgeon. Thus another conclusion is that 1) the impact of predation seems to differ for the fish species and 2) lethal effects have a stronger impact on the population survival than life history changes. Combining the results mentioned above leads to the assumption that predation by benthivorous fish has the potential to shape invertebrate communities and food webs in streams. It was possible to show reductions of benthic densities and mean population fitness. The strength of trophic interactions seemed to be specific for the single predator-prey pairs here. Finally, it can be stated that contrary to previous assumptions consumption of the fish predators seemed to be more important for the prey populations than sublethal predation effects.

We are interested in a nonlinear filtering problem motivated by an information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. We solve this problem by `change of measure method\\\'' and show the existence of the density of the unnormalized conditional distribution which is a solution to the Zakai equation. Zakai equation is a linear SPDE which, in general, cannot be solved analytically. We apply Galerkin method to solve it numerically and show the convergence of Galerkin approximation in mean square. Lastly, we design an adaptive Galerkin filter with a basis of Hermite polynomials and we present numerical examples to illustrate the effectiveness of the proposed method. The work is closely related to the paper Frey and Schmidt (2010).