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Filtering, Approximation and Portfolio Optimization for Shot-Noise Models and the Heston Model
(2012)

We consider a continuous time market model in which stock returns satisfy a stochastic differential equation with stochastic drift, e.g. following an Ornstein-Uhlenbeck process. The driving noise of the stock returns consists not only of Brownian motion but also of a jump part (shot noise or compound Poisson process). The investor's objective is to maximize expected utility of terminal wealth under partial information which means that the investor only observes stock prices but does not observe the drift process. Since the drift of the stock prices is unobservable, it has to be estimated using filtering techniques. E.g., if the drift follows an Ornstein-Uhlenbeck process and without
jump part, Kalman filtering can be applied and optimal strategies can be computed explicitly. Also in other cases, like for an underlying
Markov chain, finite-dimensional filters exist. But for certain jump processes (e.g. shot noise) or certain nonlinear drift dynamics explicit computations, based on discrete observations, are no longer possible or existence of finite dimensional filters is no longer valid. The same
computational difficulties apply to the optimal strategy since it depends on the filter. In this case the model may be approximated by
a model where the filter is known and can be computed. E.g., we use statistical linearization for non-linear drift processes, finite-state-Markov chain approximations for the drift process and/or diffusion approximations for small jumps in the noise term.
In the approximating models, filters and optimal strategies can often be computed explicitly. We analyze and compare different approximation methods, in particular in view of performance of the corresponding utility maximizing strategies.

In this study, two outstanding subgroups of organic-inorganic hybrid materials have been investigated. The first part covers the design, synthesis, characterization and application of seven novel Metal Organic Frameworks (MOFs) containing functionalized biphenyl dicarboxylates as linkers. In the second part, the surface modification of the metal oxides ZrO2, TiO2 and Al2O3 using phosphonate derivates is reported.
Firstly three functionalized MOF structures; ZnBrBPDC, ZnNO2BPDC and ZnNH2BPDC were synthesised using 4,4´-biphenyldicarboxylic acid derivatives with different functional groups (-Br, -NO2, -NH2) Powder X-ray diffraction (PXRD) measurements indicated that the synthesised MOFs posses the interpenetrated IRMOF-9 structure with a cubic topology, which was also confirmed with single crystal X-ray measurements. The chemical structure of the MOF materials was further proved by solid state NMR and IR measurements. N2 adsorption measurements showed Type I isotherms for all three structures with large surface areas. TGA measurements of the evacuated samples were in good agreement with the elemental analysis data. The results proved that their thermal stability is between 325 °C - 450 °C.
Adsorption properties of these MOF structures were tested using light alkanes (CH4, C2H6, C3H8, and n-C4H10) at three different temperatures. For all adsorbents, the maximum uptakes were observed at 273 K. When the temperature was increased, the amount of the adsorbed gas decreased. All three MOFs showed strong affinities for n-butane. The lowest uptakes were observed for CH4.
The effect of functional groups on the IRMOF series was also examined by synthesizing amide functionalized biphenyl linkers. For this purpose, four different linkers containing amides with different alkyl chains (C1-C4) were synthesized and used for the synthesis of four new MOF structures ZnAcBPDC, ZnPrBPDC, ZnBuBPDC and ZnPeBPDC.
PXRD measurements of ZnAcBPDC indicated that the structure contains two different phases. PXRD patterns of ZnPrBPDC, ZnBuBPDC and ZnPeBPDC revealed non-interpenetrated structures which were further proved by single crystal X-ray measurements. The chemical structure of the MOF materials was further confirmed by X-ray spectoscopy, solid state NMR and IR measurements.
N2 adsorption measurements of the MOF structures were carried out using different activation methods. For all four MOFs, Type I isotherms were obtained. ZnAcBPDC showed the highest BET surface area. ZnAcBPDC and ZnBuBPDC were tested for their alkane, alkene and CO2 adsorption capacities.
In the second part of the work, the surface modification of three different metal oxides, ZrO2, TiO2 and Al2O3 was performed. For this purpose firstly three different fluorescent phosphonate derivatives containing thiophene units were synthesized from their halo derivatives in a four step synthesis and then used as coupling molecules for the surface modification. Nine different surfaces were obtained (38@TiO2, 39@TiO2, 40@TiO2, 38@Al2O3, 39@Al2O3, 40@Al2O3, 38@ZrO2, 39@ZrO2, 40@ZrO2).
All three modified metal oxide surfaces were characterized using elemental analysis, solid state NMR and IR spectroscopy. The BET surface areas of the materials were determined by N2 adsorption measurements. TGA was used to determine the stability of the surfaces. Maximum loadings were obtained for ZrO2 surfaces.
Due to the strong luminescence of the coupling molecules, the modified surfaces were checked for their light emission. All ZrO2 and Al2O3 surfaces showed fluorescence with exception of 40@Al2O3. On the other hand, for the modified TiO2 surfaces, no fluorescence could be observed.

Image restoration and enhancement methods that respect important features such as edges play a fundamental role in digital image processing. In the last decades a large
variety of methods have been proposed. Nevertheless, the correct restoration and
preservation of, e.g., sharp corners, crossings or texture in images is still a challenge, in particular in the presence of severe distortions. Moreover, in the context of image denoising many methods are designed for the removal of additive Gaussian noise and their adaptation for other types of noise occurring in practice requires usually additional efforts.
The aim of this thesis is to contribute to these topics and to develop and analyze new
methods for restoring images corrupted by different types of noise:
First, we present variational models and diffusion methods which are particularly well
suited for the restoration of sharp corners and X junctions in images corrupted by
strong additive Gaussian noise. For their deduction we present and analyze different
tensor based methods for locally estimating orientations in images and show how to
successfully incorporate the obtained information in the denoising process. The advantageous
properties of the obtained methods are shown theoretically as well as by
numerical experiments. Moreover, the potential of the proposed methods is demonstrated
for applications beyond image denoising.
Afterwards, we focus on variational methods for the restoration of images corrupted
by Poisson and multiplicative Gamma noise. Here, different methods from the literature
are compared and the surprising equivalence between a standard model for
the removal of Poisson noise and a recently introduced approach for multiplicative
Gamma noise is proven. Since this Poisson model has not been considered for multiplicative
Gamma noise before, we investigate its properties further for more general
regularizers including also nonlocal ones. Moreover, an efficient algorithm for solving
the involved minimization problems is proposed, which can also handle an additional
linear transformation of the data. The good performance of this algorithm is demonstrated
experimentally and different examples with images corrupted by Poisson and
multiplicative Gamma noise are presented.
In the final part of this thesis new nonlocal filters for images corrupted by multiplicative
noise are presented. These filters are deduced in a weighted maximum likelihood
estimation framework and for the definition of the involved weights a new similarity measure for the comparison of data corrupted by multiplicative noise is applied. The
advantageous properties of the new measure are demonstrated theoretically and by
numerical examples. Besides, denoising results for images corrupted by multiplicative
Gamma and Rayleigh noise show the very good performance of the new filters.

In this thesis we consider the problem of maximizing the growth rate with proportional and fixed costs in a framework with one bond and one stock, which is modeled as a jump diffusion with compound Poisson jumps. Following the approach from [1], we prove that in this framework it is optimal for an investor to follow a CB-strategy. The boundaries depend only on the parameters of the underlying stock and bond. Now it is natural to ask for the investor who follows a CB-strategy which is given by the stopping times \((\tau_i)_{i\in\mathbb N}\) and impulses \((\eta_i)_{i\in\mathbb N}\) how often he has to rebalance. In other words we want to obtain the limit of the inter trading times
\[
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n(\tau_{i+1}-\tau_{i}).
\]
We are able to obtain this limit which is given by the expected first exit time of the risky fraction process from some interval under the invariant measure of the Markov chain \((\eta_i)_{i\in\mathbb N}\) using the Ergodic Theorem from von Neumann and Birkhoff. In general, it is difficult to obtain the expectation of the first exit time for the process with jumps. Because of the jump part, when the process crosses the boundaries of the interval an overshoot may occur which makes it difficult to obtain the distribution. Nevertheless we can obtain the first exit time if the process has only negative jumps using scale functions. The main difficulty of this approach is that the scale functions are known only up to their Laplace transforms. In [2] and [3] the closed-form expression for the scale function of the Levy process with phase-type distributed jumps is obtained. Phase-type distributions build a rich class of positive-valued distributions: the exponential, hyperexponential, Erlang, hyper-Erlang and Coxian distributions. Since the scale function is given as a function in a closed form we can differentiate to obtain the expected first exit time using the fluctuation identities explicitly.
[1] Irle, A. and Sass,J.: Optimal portfolio policies under fixed and proportional transaction costs, Advances in Applied Probability 38, 916-942.
[2] Egami, M., Yamazaki, K.: On scale functions of spectrally negative Levy processes with phase-type jumps, working paper, July 3.
[3]Egami, M., Yamazaki, K.: Precautionary measures for credit risk management in jump models, working paper, June 17.

This thesis deals with the relationship between no-arbitrage and (strictly) consistent price processes for a financial market with proportional transaction costs
in a discrete time model. The exact mathematical statement behind this relationship is formulated in the so-called Fundamental Theorem of Asset Pricing (FTAP). Among the many proofs of the FTAP without transaction costs there
is also an economic intuitive utility-based approach. It relies on the economic
intuitive fact that the investor can maximize his expected utility from terminal
wealth. This approach is rather constructive since the equivalent martingale measure is then given by the marginal utility evaluated at the optimal terminal payoff.
However, in the presence of proportional transaction costs such a utility-based approach for the existence of consistent price processes is missing in the literature. So far, rather deep methods from functional analysis or from the theory of random sets have been used to show the FTAP under proportional transaction costs.
For the sake of existence of a utility-maximizing payoff we first concentrate on a generic single-period model with only one risky asset. The marignal utility evaluated at the optimal terminal payoff yields the first component of a
consistent price process. The second component is given by the bid-ask prices
depending on the investors optimal action. Even more is true: nearby this consistent price process there are many strictly consistent price processes. Their exact structure allows us to apply this utility-maximizing argument in a multi-period model. In a backwards induction we adapt the given bid-ask prices in such a way so that the strictly consistent price processes found from maximizing utility can be extended to terminal time. In addition possible arbitrage opportunities of the 2nd kind vanish which can present for the original bid-ask process. The notion of arbitrage opportunities of the 2nd kind has been so
far investigated only in models with strict costs in every state. In our model
transaction costs need not be present in every state.
For a model with finitely many risky assets a similar idea is applicable. However, in the single-period case we need to develop new methods compared
to the single-period case with only one risky asset. There are mainly two reasons
for that. Firstly, it is not at all obvious how to get a consistent price process
from the utility-maximizing payoff, since the consistent price process has to be
found for all assets simultaneously. Secondly, we need to show directly that the
so-called vector space property for null payoffs implies the robust no-arbitrage condition. Once this step is accomplished we can à priori use prices with a
smaller spread than the original ones so that the consistent price process found
from the utility-maximizing payoff is strictly consistent for the original prices.
To make the results applicable for the multi-period case we assume that the prices are given by compact and convex random sets. Then the multi-period case is similar to the case with only one risky asset but more demanding with regard to technical questions.

The goal of this work is to develop a simulation-based algorithm, allowing the prediction
of the effective mechanical properties of textiles on the basis of their microstructure
and corresponding properties of fibers. This method can be used for optimization of the
microstructure, in order to obtain a better stiffness or strength of the corresponding fiber
material later on. An additional aspect of the thesis is that we want to take into account the microcontacts
between fibers of the textile. One more aspect of the thesis is the accounting for the thickness of thin fibers in the
textile. An introduction of an additional asymptotics with respect to a small parameter,
the relation between the thickness and the representative length of the fibers, allows a
reduction of local contact problems between fibers to 1-dimensional problems, which
reduces numerical computations significantly.
A fiber composite material with periodic microstructure and multiple frictional microcontacts
between fibers is studied. The textile is modeled by introducing small geometrical
parameters: the periodicity of the microstructure and the characteristic
diameter of fibers. The contact linear elasticity problem is considered. A two-scale
approach is used for obtaining the effective mechanical properties.
The algorithm using asymptotic two-scale homogenization for computation of the
effective mechanical properties of textiles with periodic rod or fiber microstructure
is proposed. The algorithm is based on the consequent passing to the asymptotics
with respect to the in-plane period and the characteristic diameter of fibers. This
allows to come to the equivalent homogenized problem and to reduce the dimension
of the auxiliary problems. Further numerical simulations of the cell problems give
the effective material properties of the textile.
The homogenization of the boundary conditions on the vanishing out-of-plane interface
of a textile or fiber structured layer has been studied. Introducing additional
auxiliary functions into the formal asymptotic expansion for a heterogeneous
plate, the corresponding auxiliary and homogenized problems for a nonhomogeneous
Neumann boundary condition were deduced. It is incorporated into the right hand
side of the homogenized problem via effective out-of-plane moduli.
FiberFEM, a C++ finite element code for solving contact elasticity problems, is
developed. The code is based on the implementation of the algorithm for the contact
between fibers, proposed in the thesis.
Numerical examples of homogenization of geotexiles and wovens are obtained in the
work by implementation of the developed algorithm. The effective material moduli
are computed numerically using the finite element solutions of the auxiliary contact
problems obtained by FiberFEM.

The discrete nature of the dispersed phase (swarm of droplet) in stirred and pulsed liquid-liquid extraction columns makes its mathematical modelling of such complex system a tedious task. The dispersed phase is considered as a population of droplets distributed randomly with respect to their internal properties (such as: droplet size and solute concentration) at a specific location in space. Hence, the population balance equation has been emerged as a mathematical tool to model and describe such complex behaviour. However, the resulting model is too complicated. Accordingly, the analytical solution of such a mathematical model does not exist except for particular cases. Therefore, numerical solutions are resorted to in general. This is due to the inherent nonlinearities in the convective and diffusive terms as well as the appearance of many integrals in the source term. However, modelling and simulation of liquid extraction columns is not an easy task because of the discrete nature of the dispersed phase, which consist of population of droplets. The natural frame work for taking this into account is the population balance approach.
In part of this doctoral thesis work, a rigours mathematical model based on the bivariate population balance frame work (the base of LLECMOD ‘‘Liquid-Liquid Extraction Column Module’’) for the steady state and dynamic simulation of pulsed (sieve plate and packed) liquid-liquid extraction columns is developed. The model simulates the coupled hydrodynamic and mass transfer for pulsed (packed and sieve plate) extraction columns. The model is programmed using visual digital FORTRAN and then integrated into the LLECMOD program. Within LLECMOD the user can simulate different types of extraction columns including stirred and pulsed ones. The basis of LLECMOD depends on stable robust numerical algorithms based on an extended version of a fixed pivot technique after Attarakih et al., 2003 (to take into account interphase solute transfer) and advanced computational fluid dynamics numerical methods. Experimental validated correlations are used for the estimation of the droplet terminal velocity in extraction columns based on single and swarm droplet experiments in laboratory scale devices. Additionally, recent published correlations for turbulent energy dissipation, droplet breakage and coalescence frequencies are discussed as been used in this version of LLECMOD. Moreover, coalescence model from literature derived from a stochastical description have been modified to fit the deterministic population model. As a case study, LLECMOD is used here to simulate the steady state performance of pulsed extraction columns under different operating conditions, which include pulsation intensity and volumetric flow rates are simulated. The effect of pulsation intensity (on the holdup, mean droplet diameter and solute concentration) is found to have more profound effect on systems of high interfacial tension. On the hand, the variation of volumetric flow rates have substantial effect on the holdup, mean droplet diameter and solute concentration profiles for chemical systems with low interfacial tension. Two chemical test systems recommended by the European Federation of Chemical Engineering (water-acetone (solute)-n-butyl acetate and water-acetone (solute)-toluene) and an industrial test system are used in the simulation. Model predictions are successfully validated against steady state and transient experimental data, where good agreements are achieved. The simulated results (holdup, mean droplet diameter and mass transfer profiles) compared to the experimental data show that LLECMOD is a powerful simulation tool, which can efficiently predict the dynamic and steady state performance of pulsed extraction columns.
In other part of this doctoral thesis work, the steady state performance of extraction columns is studied taking into account the effect of dispersed phase inlet condition (light or heavy phase is dispersed) and the direction of mass transfer (from continuous to dispersed phase and vice versa) using the population balance framework. LLECMOD, a program that uses multivariate population balance models, is extended to take into account the direction of mass transfer and the dispersed phase inlet. As a case study, LLECMOD is used to simulate pilot plant RDC columns where the steady state mean flow properties (dispersed phase hold up and droplet mean diameter) and the solute concentration profiles are compared to the available experimental data. Three chemical systems were used: sulpholane–benzene–n-heptane, water–acetone–toluene and water–acetone–n-butyl acetate. The dispersed phase inlet and the direction of mass transfer as well as the chemical system physical properties are found to have profound effect on the steady state performance of the RDC column. For example, the mean droplet diameter is found to persist invariant when the heavy phase is dispersed and the extractor efficiency is higher when the direction of mass transfer is from the continuous to the dispersed phase. For the purpose of experimental validation, it is found that LLECMOD predictions are in good agreement with the available experimental data concerning the dispersed phase hold up, mean droplet diameter and solute concentration profiles in both phases.
In a further part of this doctoral thesis, a mathematical model is developed for liquid extraction columns based on the multivariate population balance equation (PBE) and the primary secondary particle method (PSPM) introduced by Attarakih, 2010 (US Patent Application: 0100106467). It is extended to include the momentum balance for the dispersed phase. The advantage of momentum balance is to eliminate the need for often conflicting correlations used in estimating the terminal velocity of single and swarm of droplets. The resulting mathematical model is complex due to the integral nature of the population balance equation. To reduce the complexity of this model, while maintaining most of the information drawn from the continuous population balance equation, the concept of the PSPM is used. Based on the multivariate population balance equation and the PSPM a mathematical model is developed for any liquid extraction column. The secondary particle could be envisaged as a fluid particle carrying information about the distribution as it is evolved in space and time, in the meanwhile the primary particles carry the mean properties of the population such as total droplet concentration; mean droplet diameter dispersed phase hold up and so on. This information reflects the particle-particle interactions (breakage and coalescence) and transport (convection and diffusion). The developed model is discretized in space using a first-order upwind method, while semi-implicit first-order scheme in time is used to simulate a pilot plant RDC extraction column. Here the effect of the number of primary particles (classes) on the final predicted solution is investigated. Numerical results show that the solution converge fast even as the number of primary particle is increased. The terminal droplet velocity of the individual primary particle is found the most sensitive to the number of primary particles. Other mean population properties like the droplet mean diameter, mean hold up and the concentration profiles are also found to converge along the column height by increasing the number of primary particles. The predicted steady state profiles (droplet diameter, holdup and the concentration profiles) along a pilot RDC extraction column are compared to the experimental data where good agreement is achieved.
In addition to this a robust rigorous mathematical model based on the bivariate population balance equation is developed to predict the steady state and dynamic behaviour of the interacting hydrodynamics and mass transfer in Kühni extraction columns. The developed model is extended to include the momentum balance for the calculation of the droplet velocity. The effects of step changes in the important input variables (such as volumetric flow rates, rotational speed, inlet solute concentrations etc.) on the output variables (dispersed phase holdup, mean droplet diameter and the concentration profiles) are investigated.
The last topic of this doctoral thesis is developed to transient problems. The unsteady state analysis reveals the fact that the largest time constant (slowest response) is due to the mass transfer. On the contrary, the hydrodynamic response of the dispersed phase holdup is very fast when compared to the mass transfer due to the relative fast motion of the dispersed droplets with respect to the continuous phase. The dynamic behaviour of the dispersed and continuous phases shows a lag time that increases away from the feed points of both phases. Moreover, the solute concentration response shows a highly nonlinear behaviour due to both positive and negative step changes in the input variables. The simulation results are in good agreement with the experimental ones and show the usefulness of the model.

Mechanisms underlying the biological effects of coffee and its constituents are incompletely understood. Many effects have been attributed solely to caffeine, neglecting that coffee is a mixture of many chemical substances. Some authors suggest that the main mechanism of action of caffeine is to antagonize adenosine receptors (AR); a second effect is the inhibition of phosphodiesterases with the subsequent accumulation of cAMP and an intensification of the effects of catecholamines. Although the inhibition of phosphodiesterases may contribute to the actions of caffeine, there is growing evidence that most pharmacological effects of this xanthine result from antagonism of AR.
One of the main objectives of this work was to investigate whether substances other than caffeine in coffee may influence the homeostasis of intracellular cyclic nucleotides in vitro and in vivo. The influence of selected coffee compounds, extracts and brews on key elements involved in the adenosine receptor-mediated signaling pathway have been investigated.
A further aim of this work was also to determine if coffee or some coffee constituents may have a stimulatory effect on the cellular heme oxygenase activity (HO-activity). Two coffee extracts, a slightly (AB1) and an intensively roasted coffee (AB2), were studied along with selected individual compounds. Caffeine and low substituted pyrazines showed no effect on the HO-activity, while NMP, pyrazines with a greater substitution pattern such as Tetramethylpyrazine (TMP) and 2-Ethyl-3,5(6)-dimethylpyrazine (2-E-3,5-DMP) and both coffee extracts significantly induced the HO-activity in liver hepatocellular carcinoma (HepG2), intestinal colo-rectal adenocarcinoma (Caco-2) and in some instances in monocytic leukemia (MM6) cells.
It was found that caffeine, theophylline, coffee extracts from conventional or functional coffees, pyrazines (2,3-DE-6-MP, 2-Isobutyl-3-methoxyP), 5-CQA and caffeic acid all significantly inhibited the basal cytoplasmatic PDE activity in lysates of lung tumour xenograft cells (LXFL529L) and human platelets. To a somewhat lesser extent, PDE inhibition was also found in experiments performed with paraxanthine and other pyrazines (2-E-3,5-DMP, TMP and 2-E-5-MP). Thus the degree of roasting has a considerable impact on constituents of influence for PDE activity. Caffeine, coffee polyphenols and some pyrazines and further, as yet unknown roasting products appear to represent the main modulating constituents.
In two coffee intervention studies, a short-term (8 weeks) and a long-term study (24 weeks), comprising 8 and 84 healthy volunteers respectively, we examined extracellular key elements of the adenosine pathway including plasma adenosine levels and adenosine deaminase activity. Additionally, we studied the intracellular cAMP concentration and the PDE activity in platelets as surrogate biomarkers of adipocytes.
Results of in vitro experiments had suggested that the concentrations of caffeine and coffee extracts required to obtain a half maximal inhibition were in the upper range of physiological conditions. Yet, it was demonstrated for the first time in vivo that moderate consumption of coffee can modulate the activity of platelet phosphodiesterases in humans in long and short term. In both studies, the first exposure to coffee showed a strong inhibition (p<0.001) of the PDE activity in the platelet lysates of the participants while the second coffee phase showed no or a slight effect when compared with the first coffee intervention.
In both studies a significant increase (p<0.001) in intraplatelet cAMP concentrations during the wash-out phase (after the first coffee phase) was observed. This response could be due to inhibition of the PDE activity in the previously phase extending in to the wash out phase. However, the behavior of cAMP in the following study phases cannot be easily explained. It may be hypothesized that this effect is attributable to adaptive effects to allow PDE inhibition. One possibility is the modulation of the expression of membrane-bound adenosine receptors in platelet precursors, which still have a nucleus. This may potentially influence adenylate cyclase activity in mature platelets. For the observed effects, in addition to caffeine other ingredients of coffee appear to play a role. The findings suggest that monitoring of cAMP homeostasis in platelets is not a useful surrogate biomarker for effects in other tissues.
Neither the activity of adenosine deaminase nor the adenosine concentrations in plasma were markedly modulated by the coffee consumption in both trials. This may reflect the fact that adenosine is subject to quick and effective enzymatic turnover by phosphorylation (adenosine kinase) or deamination (adenosine deaminase) allowing keep its concentration within a well balanced homeostasis. However, it is also well known, that considerable variability exists in the responses to coffee drinking. In part, such variability is due to caffeine tolerance, but there is also evidence for a genetic background.
Altogether the data reported here provide further evidence for the perception that coffee consumption is associated with beneficial health effects demonstrated for the cAMP enhancement in platelets, known to counteract platelet aggregation. The effects observed for the influence of cellular heme oxygenase (HO) are in line with the well documented antioxidative activity of coffee and its constituents.

Dealing with information in modern times involves users to cope with hundreds of thousands of documents, such as articles, emails, Web pages, or News feeds.
Above all information sources, the World Wide Web presents information seekers with great challenges.
It offers more text in natural language than one is capable to read.
The key idea for this research intends to provide users with adaptable filtering techniques, supporting them in filtering out the specific information items they need.
Its realization focuses on developing an Information Extraction system,
which adapts to a domain of concern, by interpreting the contained formalized knowledge.
Utilizing the Resource Description Framework (RDF), which is the Semantic Web's formal language for exchanging information,
allows extending information extractors to incorporate the given domain knowledge.
Because of this, formal information items from the RDF source can be recognized in the text.
The application of RDF allows a further investigation of operations on recognized information items, such as disambiguating and rating the relevance of these.
Switching between different RDF sources allows changing the application scope of the Information Extraction system from one domain of concern to another.
An RDF-based Information Extraction system can be triggered to extract specific kinds of information entities by providing it with formal RDF queries in terms of the SPARQL query language.
Representing extracted information in RDF extends the coverage of the Semantic Web's information degree and provides a formal view on a text from the perspective of the RDF source.
In detail, this work presents the extension of existing Information Extraction approaches by incorporating the graph-based nature of RDF.
Hereby, the pre-processing of RDF sources allows extracting statistical information models dedicated to support specific information extractors.
These information extractors refine standard extraction tasks, such as the Named Entity Recognition, by using the information provided by the pre-processed models.
The post-processing of extracted information items enables representing these results in RDF format or lists, which can now be ranked or filtered by relevance.
Post-processing also comprises the enrichment of originating natural language text sources with extracted information items by using annotations in RDFa format.
The results of this research extend the state-of-the-art of the Semantic Web.
This work contributes approaches for computing customizable and adaptable RDF views on the natural language content of Web pages.
Finally, due to the formal nature of RDF, machines can interpret these views allowing developers to process the contained information in a variety of applications.

This thesis is devoted to furthering the tropical intersection theory as well as to applying the
developed theory to gain new insights about tropical moduli spaces.
We use piecewise polynomials to define tropical cocycles that generalise the notion of tropical Cartier divisors to higher codimensions, introduce an intersection product of cocycles with tropical cycles and use the connection to toric geometry to prove a Poincaré duality for certain cases. Our
main application of this Poincaré duality is the construction of intersection-theoretic fibres under a
large class of tropical morphisms.
We construct an intersection product of cycles on matroid varieties which are a natural
generalisation of tropicalisations of classical linear spaces and the local blocks of smooth tropical
varieties. The key ingredient is the ability to express a matroid variety contained in another matroid variety by a piecewise polynomial that is given in terms of the rank functions of the corresponding
matroids. In particular, this enables us to intersect cycles on the moduli spaces of n-marked abstract
rational curves. We also construct a pull-back of cycles along morphisms of smooth varieties, relate
pull-backs to tropical modifications and show that every cycle on a matroid variety is rationally
equivalent to its recession cycle and can be cut out by a cocycle.
Finally, we define families of smooth rational tropical curves over smooth varieties and construct a tropical fibre product in order to show that every morphism of a smooth variety to the moduli space of abstract rational tropical curves induces a family of curves over the domain of the morphism.
This leads to an alternative, inductive way of constructing moduli spaces of rational curves.