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This thesis reports on investigations on the structure and reactivity of dipeptide-alkali metal complexes, a series of ruthenium bearing catalysts, dysprosium based single molecule magnets and organometallic di-cobalt complexes. A variety of experimental and theoretical methods was used dependent on the problem: collision induced dissociation, hydrogen/deuterium exchange reactions, gas phase reactions with \(D_2\), infrared multiple-photon dissociation and the determination of minimum energy structures, IR absorption spectra, transition states and electronic transitions based on density functional theory.
A case study was carried out to explore the influence of alkali metal ions on the gas phase structure of the dipeptide Carnosine. CID experiments on protonated Carnosine and its alkali metal complexes in an ion trap resulted in different fragment pathways dependent on the size of the alkali metal. The complexation of small ions (\(Li^+\) and \(Na^+\)) promoted the cleavage of bonds in the molecules backbone under CID, while \(Rb^+\)- and \(Cs^+\)-Carnosine complexes underwent the exclusive loss of the alkali metal. CID breakdown curves reflected the different binding behavior of the alkali ions to Carnosine. Gas phase H/D exchange reactions with \(D_2O\) resulted in the exchange of several protons of the protonated dipeptide, while its alkali metal complexes underwent no exchange reactions. DFT derived energetical minimum isomers exhibited only charge solvated tridentate structures, whereas salt bridge as well as charge solvated binding motives are reported in literature on complexes of alkali metal ions and oligopeptides. This study was published in a similar version as a paper in Zeitschrift für Physikalische Chemie.
A combination of the four dipeptides Carnosine, Anserine, GlyHis and HisGly with alkali metal ions was investigated with the help of CID, IR-MPD spectroscopy and H/D exchange reactions with \(ND_3\). The aim of the survey was to elucidate the influence of the methyl-group at the histidine ring, of the peptide sequence and chain length on the binding motives of the alkali ions. The experimental results were compared to DFT derived minimum energetical isomers. A moderate accordance was found for DFT predicted IR absorptions to IR-MPD spectra. A systematic nomenclature was developed reflecting all binding motives of the four dipeptides to alkali ions. Carnosine complexes all alkali metal ions in an uniform motive. DFT derived energetical minimum isomers of the three other dipeptides showed strong conformational changes with increasing size of the alkali ion. The most favored binding motive of all peptides was the tridentate complexation of the alkali ion by a carboxylic and an amidic oxygen atom, while the electron donating nitrogen atom either belongs to the Histidine ring or the amine group. The ability to form hydrogen bonds in a certain binding motive is essential for the preference of the Histidine or amine nitrogen atom as an electron donor. The charge solvated binding motive is the most common within all found isomers. Several structures exhibited hydrogen bonded protons. Those can be interpretated as intermediates between the charge solvated and the salt bridge binding motive. CID breakdown curves of the cationic complexes of the dipeptides with \(K^+\), \(Rb^+\) and \(Cs^+\) resulted in a fair agreement of \(E^{50\%}_{com}\) values with DFT derived Gibbs free binding energies. CID led to multiple fragments of the \(Li^+\) and \(Na^+\) dipeptide complexes and to an insufficient correlation between the \(E^{50\%}_{com}\) values and metal-dipeptide free binding enthalpies. Gas phase H/D exchange reactions of the protonated dipeptides with \(ND_3\) resulted in the exchange of all labile protons with comparable relative partial rate constants. The assumption of coexisting single and double exchange reactions per single collision led to an enhancement in quality of the pseudo first order kinetic fits of the experimental derived data. The \(Li^+\), \(Na^+\) and \(K^+\) complexes of the dipeptides exhibited a reduction in the number of exchanged protons, significantly lower rate constants for H/D exchange and only single exchange reactions.
The complexation of the doubly charged cationic transition metal \(Zn^{2+}\) by deprotonated Carnosine led to crucial conformational changes with respect to the alkali metal complexes. Former DFT calculations on the gas phase structure of \([Carn-H,Zn^{II}]^+\) were now compared to IR-MPD spectra. IR-MPD spectra exhibited several of the DFT predicted IR absorptions while the overall agreement in the position of bands is only partially satisfactory. The complex \([Carn-H,Zn^{II}]^+\) was furthermore used in order to study the band dependent enhancement of fragmentation efficiency by application of a resonant 2-color IR-MPD pump/probe scheme. In literature, it is assumed that the slopes of linear fits to the log-log scale of experimental data (fragmentation efficiency vs. laser pulse energy) correlate to the number of photons needed for fragmentation. No reasonable number of photons for the fragmentation of the molecule was derived with this approach. However, it could be shown that the number of photons of the pump laser needed for fragmentation is reduced by the use of a second IR color. The change of the delay between the pump and probe laser pulse had an influence on the shape of the absorption bands. Irradiation with the probe laser pulse before the pump laser caused a heating of the molecule which resulted in a broadening of bands. No broadening was observed when the probe laser was applied simultaneously or after the pump laser. CID and IR-MPD fragmentation channels differed in their relative abundance. Furthermore, relative abundancies of fragments were specific to the excited vibrational motions. This study provides essential approaches for the further study of the mechanism of resonant 2-color IR-MPD spectroscopy.
Several ruthenium catalysts for transfer hydrogenation reactions were synthesized by L. Ghoochany (research group W. Thiel, TU Kaiserlautern). CID measurements on isotopic labeled species led to the following conclusion about the activation process of the catalyst: a nitrogen-ruthenium bond is broken, the pyrimidine ring of the substituted 2-R-4-(2-pyridinyl)pyrimidine ligand rotates about 160° and a carbon-ruthenium bond is formed under subsequent loss of a HCl (or DCl) molecule. The mass spectrometers CID amplitude was calibrated with a set of “thermometer ions”. CID breakdown curves were used for determination of \(E^{50\%}_{com}\) values of three differently substituted catalysts. Finally, activation energies were estimated by means of the calibration. The resulting activation energies showed a qualitative correlation to DFT derived activation energies. These results are part of a manuscript which was submitted to Chemistry – A European Journal and is currently in the review process. Further studies on this series of transition metal complexes included CID on ligand exchanged species, 1- and 2-color IR-MPD spectroscopy, gas phase reactions with \(D_2\) and DFT based modeling of the reaction coordinate of the \(D_2\) insertion. The exchange of the anionic chlorido ligand in solution led to three complexes with different fragmentation thresholds. CID derived activation amplitudes correspond well to the order predicted by the hard/soft acids/bases (HSAB) concept. 1-color IR-MPD experiments on two complexes showed only a few bands. Resonant 2-color IR-MPD increased the overall fragmentation efficiency and uncovered several dark bands. DFT derived IR absorption spectra correlate well to IR-MPD spectra while some bands are still not observable. Gas phase reactions with \(D_2\) showed an increase of the mass of the activated complex of +4 m/z. This was interpreted in terms of an incorporation of a \(D_2\) molecule under heterolytical cleavage of the \(D_2\) molecule and can be compared to a back reaction of the activation. The reaction coordinate of the \(D_2\) incorporation was modeled with DFT at the B3LYP/cc-pVTZ level of theory and different activation energies were derived dependent on the substituent. Reactions of three differently substituted complexes with \(D_2\) resulted in different relative partial rate constants. The comparison to rate constants derived from transition state theory showed a qualitative but not quantitative correlation to the experimental results. This study contributes to our ongoing work on the assignment and isolation of reaction intermediates in the gas phase.
A series of dysprosium based complexes was synthesized by A. Bhunia (research group P. W. Roesky, KIT) and studied within the collaborative research center SFB/TRR 88 “3MET”. We contributed to this work with ESI-MS, CID and experiments on H/D exchange reactions with \(ND_3\) in the gas phase. Those complexes consist of a central triple-charged dysprosium cation and two identical salen-type ligands which allow for a complexation of up to two transition metals. The monometallic dysprosium complex shows single molecule magnet (SMM) behavior in SQUID measurements, while the incorporation of two double-charged manganese cations leads to ferromagnetic behavior. The interaction of terminal amine groups with the manganese ions caused a hinderance of the exchange H/D exchange reaction with \(ND_3\) in the gas phase. Alternatively, the terminal amine groups of the monometallic dysprosium complex allow for the bond of two \(Ni^{2+}(tren)\) complexes. ESI-MS studies showed anionic as well as cationic complexes due to deprotonation or protonation in solution. CID studies led to fragmentation schemes which correlate quite well to the predicted structures of the complexes. These results are part of two publications in Inorganic Chemistry and Dalton Transactions. Further studies on this series of mono-, di- and trimetallic complexes are reported in this thesis. H/D exchange reactions with \(D_2O\) in solution yielded in an exchange of all labile protons for the cationic complexes. Anionic complexes underwent a partial or a complete exchange of labile protons. A comparison of 1- and 2-color IR-MPD spectra of anionic and cationic complexes as well as H/D exchanged species allowed for the assignment of vibrational bands. Furthermore, preferred protonation sites were derived by comparing the results of IR-MPD experiments and H/D exchange reactions in solution and in the gas phase. This study contributes to our ongoing work on the determination of magnetic properties of isolated ions in the gas phase at the Helmholtz-Zentrum Berlin.
The complex \([(^4CpCo)_2(\mu-C_2Ph_2)]\) (\(^4Cp\) = tetraisopropyl-cyclopentadiene) was synthesized by J. Becker (research group H. Sitzmann, TU Kaiserslautern). The cationic complex and several reaction products were characterized by ESI-MS. Some of the experimental data contributed to the diploma thesis of J. Becker. The cationic reaction products and the complex itself were subject of IR spectroscopic characterization. IR-MPD efficiency changed crucially with modification of the complex, yielding \([(^4CpCo)_2(\mu-C_2Ph_2)X]^+ (X=H, (H+CH_3CN), Cl, O)\). The contribution of various fragmentation channels to the overall fragmentation efficiency was studied in detail. An increase of photon flux resulted in a saturation of preferred \(C_2Ph_2\) loss, additional alkyl fragments out of the \(^4Cp\) rings arising. Several absorption bands were found in the mid- and near-IR region. A model system from literature was used to identify seemingly levels of DFT theory by reference to X-ray crystal structure data. The B3LYP and the B97D functional with cc-pVDZ and Stuttgart 1997 ECP basis sets were identified for calculations of the complex \([(^4CpCo)_2(\mu-C_2Ph_2)]^+\) and of its reaction products. An elongation of the Co-Co bond distance was observed for the cationic reaction products with \(Cl^-\) and \(O^{2-}\). Calculations with B3LYP and B97D resulted in different electronic ground states. We did not obtain a good agreement of calculated vibrational modes and recorded IR-MPD spectra. DFT predicted more absorption bands than observed, especially those corresponding to aliphatic symmetric \(CH_n (n=2, 3)\) and aromatic CH stretch motions. Future 2-color IR-MPD experiments might resolve currently prevailing discrepancies. TD-DFT calculations yielded several electronic transitions that do not correspond to the IR-MPD spectra. The chosen levels of theory for DFT and TD-DFT calculations does not seem to be appropriate. IR-MPD spectra have to be remeasured in order to normalize the spectra to photon flux. Furthermore, a different strategy has to be developed for ab initio calculations on the complexes under study.
A combination of various methods applied to isolated ions in the gas phase and in solution allowed for the study of their structure, binding energies and reactivity. 1- and 2-color IR-MPD spectroscopy combined with DFT predicted absorption spectra of different isomers enabled an assignment of vibrational bands and binding motives of the molecules. The derived results are important for further studies on the binding behavior of peptides and the reaction behavior of metal complexes.

Factorization of multivariate polynomials is a cornerstone of many applications in computer algebra. To compute it, one uses an algorithm by Zassenhaus who used it in 1969 to factorize univariate polynomials over \(\mathbb{Z}\). Later Musser generalized it to the multivariate case. Subsequently, the algorithm was refined and improved.
In this work every step of the algorithm is described as well as the problems that arise in these steps.
In doing so, we restrict to the coefficient domains \(\mathbb{F}_{q}\), \(\mathbb{Z}\), and \(\mathbb{Q}(\alpha)\) while focussing on a fast implementation. The author has implemented almost all algorithms mentioned in this work in the C++ library factory which is part of the computer algebra system Singular.
Besides, a new bound on the coefficients of a factor of a multivariate polynomial over \(\mathbb{Q}(\alpha)\) is proven which does not require \(\alpha\) to be an algebraic integer. This bound is used to compute Hensel lifting and recombination of factors in a modular fashion. Furthermore, several sub-steps are improved.
Finally, an overview on the capability of the implementation is given which includes benchmark examples as well as random generated input which is supposed to give an impression of the average performance.

Most innovation in the automotive industry is driven by embedded systems. They make usage of dynamic adaption to environmental changes or component/subsystem failures for remaining safe. Following this evolution, fault tree analysis techniques have been extended with concept for dynamic adaptation but resulting techniques like state event fault tree analysis, are not widely used in practice.
In this report we present the results of a controlled experiment that analyze these two techniques (State Events Fault Trees and Faul trees combined with markov chains) with regard to their applicability and efficiency in modeling dynamic behavior of dynamic embedded systems.
The experiment was conducted with students of the TU Kaiserslautern to modeli different safety aspects of an ambient assisted living system.
The main results of the experiment show that SEFTs where more easy and effective to use.

Most of the evolution in ambient assisted living is due to embedded
systems that dynamically adapt themself to react to environmental
changes or component/subsystem failures to maintain a certain level of
safety. Following this evolution fault tree analysis techniques have been
extended with concept for dynamic adaptation but resulting techniques
such as dynamic fault trees or state event fault trees analysis are not
widely used as expected.
In this report we describe a controlled experiment to analyze these two
techniques with regard to their applicability and efficiency in modeling
dynamic behavior of ambient assisted living systems.
Results of the experiment show that Dynamic Fault Trees are easier and more effective
to use, although they produce better results (models) with State Events Fault Trees.

The application behind the subject of this thesis are multiscale simulations on highly heterogeneous particle-reinforced composites with large jumps in their material coefficients. Such simulations are used, e.g., for the prediction of elastic properties. As the underlying microstructures have very complex geometries, a discretization by means of finite elements typically involves very fine resolved meshes. The latter results in discretized linear systems of more than \(10^8\) unknowns which need to be solved efficiently. However, the variation of the material coefficients even on very small scales reveals the failure of most available methods when solving the arising linear systems. While for scalar elliptic problems of multiscale character, robust domain decomposition methods are developed, their extension and application to 3D elasticity problems needs to be further established.
The focus of the thesis lies in the development and analysis of robust overlapping domain decomposition methods for multiscale problems in linear elasticity. The method combines corrections on local subdomains with a global correction on a coarser grid. As the robustness of the overall method is mainly determined by how well small scale features of the solution can be captured on the coarser grid levels, robust multiscale coarsening strategies need to be developed which properly transfer information between fine and coarse grids.
We carry out a detailed and novel analysis of two-level overlapping domain decomposition methods for the elasticity problems. The study also provides a concept for the construction of multiscale coarsening strategies to robustly solve the discretized linear systems, i.e. with iteration numbers independent of variations in the Young's modulus and the Poisson ratio of the underlying composite. The theory also captures anisotropic elasticity problems and allows applications to multi-phase elastic materials with non-isotropic constituents in two and three spatial dimensions.
Moreover, we develop and construct new multiscale coarsening strategies and show why they should be preferred over standard ones on several model problems. In a parallel implementation (MPI) of the developed methods, we present applications to real composites and robustly solve discretized systems of more than \(200\) million unknowns.

This thesis deals with generalized inverses, multivariate polynomial interpolation and approximation of scattered data. Moreover, it covers the lifting scheme, which basically links the aforementioned topics. For instance, determining filters for the lifting scheme is connected to multivariate polynomial interpolation. More precisely, sets of interpolation sites are required that can be interpolated by a unique polynomial of a certain degree. In this thesis a new class of such sets is introduced and elements from this class are used to construct new and computationally more efficient filters for the lifting scheme.
Furthermore, a method to approximate multidimensional scattered data is introduced which is based on the lifting scheme. A major task in this method is to solve an ordinary linear least squares problem which possesses a special structure. Exploiting this structure yields better approximations and therefore this particular least squares problem is analyzed in detail. This leads to a characterization of special generalized inverses with partially prescribed image spaces.

Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients.
We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods.
The work is divided into three parts.
In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients.
The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations.

This thesis is separated into three main parts: Development of Gaussian and White Noise Analysis, Hamiltonian Path Integrals as White Noise Distributions, Numerical methods for polymers driven by fractional Brownian motion.
Throughout this thesis the Donsker's delta function plays a key role. We investigate this generalized function also in Chapter 2. Moreover we show by giving a counterexample, that the general definition for complex kernels is not true.
In Chapter 3 we take a closer look to generalized Gauss kernels and generalize these concepts to the case of vector-valued White Noise. These results are the basis for Hamiltonian path integrals of quadratic type. The core result of this chapter gives conditions under which pointwise products of generalized Gauss kernels and certain Hida distributions have a mathematical rigorous meaning as distributions in the Hida space.
In Chapter 4 we discuss operators which are related to applications for Feynman Integrals as differential operators, scaling, translation and projection. We show the relation of these operators to differential operators, which leads to the well-known notion of so called convolution operators. We generalize the central homomorphy theorem to regular generalized functions.
We generalize the concept of complex scaling to scaling with bounded operators and discuss the relation to generalized Radon-Nikodym derivatives. With the help of this we consider products of generalized functions in chapter 5. We show that the projection operator from the Wick formula for products with Donsker's deltais not closable on the square-integrable functions..
In Chapter 5 we discuss products of generalized functions. Moreover the Wick formula is revisited. We investigate under which conditions and on which spaces the Wick formula can be generalized to. At the end of the chapter we consider the products of Donsker's delta function with a generalized function with help of a measure transformation. Here also problems as measurability are concerned.
In Chapter 6 we characterize Hamiltonian path integrands for the free particle, the harmonic oscillator and the charged particle in a constant magnetic field as Hida distributions. This is done in terms of the T-transform and with the help of the results from chapter 3. For the free particle and the harmonic oscillator we also investigate the momentum space propagators. At the same time, the $T$-transform of the constructed Feynman integrands provides us with their generating functional. In Chapter 7, we can show that the generalized expectation (generating functional at zero) gives the Greens function to the corresponding Schrödinger equation.
Moreover, with help of the generating functional we can show that the canonical commutation relations for the free particle and the harmonic oscillator in phase space are fulfilled. This confirms on a mathematical rigorous level the heuristics developed by Feynman and Hibbs.
In Chapter 8 we give an outlook, how the scaling approach which is successfully applied in the Feynman integral setting can be transferred to the phase space setting. We give a mathematical rigorous meaning to an analogue construction to the scaled Feynman-Kac kernel. It is open if the expression solves the Schrödinger equation. At least for quadratic potentials we can get the right physics.
In the last chapter, we focus on the numerical analysis of polymer chains driven by fractional Brownian motion. Instead of complicated lattice algorithms, our discretization is based on the correlation matrix. Using fBm one can achieve a long-range dependence of the interaction of the monomers inside a polymer chain. Here a Metropolis algorithm is used to create the paths of a polymer driven by fBm taking the excluded volume effect in account.

The Bus Evacuation Problem (BEP) is a vehicle routing problem that arises in emergency planning. It models the evacuation of a region from a set of collection points to a set of capacitated shelters with the help of buses, minimizing the time needed to bring the last person out of the endangered region.
In this work, we describe multiple approaches for finding both lower and upper bounds for the BEP, and apply them in a branch and bound framework. Several node pruning techniques and branching rules are discussed. In computational experiments, we show that solution times of our approach are significantly improved compared to a commercial integer programming solver.

This thesis is concerned with a phase field model for brittle fracture.
The high potential of phase field modeling in computational fracture mechanics lies in the generality of the approach and the straightforward numerical implementation, combined with a good accuracy of the results in the sense of continuum fracture mechanics.
However, despite the convenient numerical application of phase field fracture models, a detailed understanding of the physical properties is crucial for a correct interpretation of the numerical results. Therefore, the driving mechanisms of crack propagation and nucleation in the proposed phase field fracture model are explored by a thorough numerical and analytical investigation in this work.