## Doctoral Thesis

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#### Faculty / Organisational entity

In conventional radio communication systems, the system design generally starts from the transmitter (Tx), i.e. the signal processing algorithm in the transmitter is a priori selected, and then the signal processing algorithm in the receiver is a posteriori determined to obtain the corresponding data estimate. Therefore, in these conventional communication systems, the transmitter can be considered the master and the receiver can be considered the slave. Consequently, such systems can be termed transmitter (Tx) oriented. In the case of Tx orientation, the a priori selected transmitter algorithm can be chosen with a view to arrive at particularly simple transmitter implementations. This advantage has to be countervailed by a higher implementation complexity of the a posteriori determined receiver algorithm. Opposed to the conventional scheme of Tx orientation, the design of communication systems can alternatively start from the receiver (Rx). Then, the signal processing algorithm in the receiver is a priori determined, and the transmitter algorithm results a posteriori. Such an unconventional approach to system design can be termed receiver (Rx) oriented. In the case of Rx orientation, the receiver algorithm can be a priori selected in such a way that the receiver complexity is minimum, and the a posteriori determined transmitter has to tolerate more implementation complexity. In practical communication systems the implementation complexity corresponds to the weight, volume, cost etc of the equipment. Therefore, the complexity is an important aspect which should be taken into account, when building practical communication systems. In mobile radio communication systems, the complexity of the mobile terminals (MTs) should be as low as possible, whereas more complicated implementations can be tolerated in the base station (BS). Having in mind the above mentioned complexity features of the rationales Tx orientation and Rx orientation, this means that in the uplink (UL), i.e. in the radio link from the MT to the BS, the quasi natural choice would be Tx orientation, which leads to low cost transmitters at the MTs, whereas in the downlink (DL), i.e. in the radio link from the BS to the MTs, the rationale Rx orientation would be the favorite alternative, because this results in simple receivers at the MTs. Mobile radio downlinks with the rationale Rx orientation are considered in the thesis. Modern mobile radio communication systems are cellular systems, in which both the intracell and intercell interferences exist. These interferences are the limiting factors for the performance of mobile radio systems. The intracell interference can be eliminated or at least reduced by joint signal processing with consideration of all the signals in the considered cell. However such joint signal processing is not feasible for the elimination of intercell interference in practical systems. Knowing that the detrimental effect of intercell interference grows with its average energy, the transmit energy radiated from the transmitter should be as low as possible to keep the intercell interference low. Low transmit energy is required also with respect to the growing electro-phobia of the public. The transmit energy reduction for multi-user mobile radio downlinks by the rationale Rx orientation is dealt with in the thesis. Among the questions still open in this research area, two questions of major importance are considered here. MIMO is an important feature with respect to the transmit power reduction of mobile radio systems. Therefore, first questionconcerns the linear Rx oriented transmission schemes combined with MIMO antenna structures. The investigations of the MIMO benefit on the linear Rx oriented transmission schemes are studied in the thesis. Utilization of unconventional multiply connected quantization schemes at the receiver has also great potential to reduce the transmit energy. Therefore, the second question considers the designing of non-linear Rx oriented transmission schemes combined with multiply connected quantization schemes.

Channel estimation is of great importance in many wireless communication systems, since it influences the overall performance of a system significantly. Especially in multi-user and/or multi-antenna systems, i.e. generally in multi-branch systems, the requirements on channel estimation are very high, since the training signals or so called pilots that are used for channel estimation suffer from multiple access interference. Recently, in the context with such systems more and more attention is paid to concepts for joint channel estimation (JCE) which have the capability to eliminate the multiple access interference and also the interference between the channel coefficients. The performance of JCE can be evaluated in noise limited systems by the SNR degradation and in interference limited systems by the variation coefficient. Theoretical analysis carried out in this thesis verifies that both performance criteria are closely related to the patterns of the pilots used for JCE, no matter the signals are represented in the time domain or in the frequency domain. Optimum pilots like disjoint pilots, Walsh code based pilots or CAZAC code based pilots, whose constructions are described in this thesis, do not show any SNR degradation when being applied to multi-branch systems. It is shown that optimum pilots constructed in the time domain become optimum pilots in the frequency domain after a discrete Fourier transformation. Correspondingly, optimum pilots in the frequency domain become optimum pilots in the time domain after an inverse discrete Fourier transformation. However, even for optimum pilots different variation coefficients are obtained in interference limited systems. Furthermore, especially for OFDM-based transmission schemes the peak-to-average power ratio (PAPR) of the transmit signal is an important decision criteria for choosing the most suitable pilots. CAZAC code based pilots are the only pilots among the regarded pilot constructions that result in a PAPR of 0 dB for the transmit signal that origins in the transmitted pilots. When summarizing the analysis regarding the SNR degradation, the variation coefficient and the PAPR with respect to one single service area and considering the impact due to interference from other adjacent service areas that occur due to a certain choice of the pilots, one can conclude that CAZAC codes are the most suitable pilots for the application in JCE of multi-carrier multi-branch systems, especially in the case if CAZAC codes that origin in different mother codes are assigned to different adjacent service areas. The theoretical results of the thesis are verified by simulation results. The choice of the parameters for the frequency domain or time domain JCE is oriented towards the evaluated implementation complexity. According to the chosen parameterization of the regarded OFDM-based and FMT-based systems it is shown that a frequency domain JCE is the best choice for OFDM and a time domain JCE is the best choice for FMT applying CAZAC codes as pilots. The results of this thesis can be used as a basis for further theoretical research and also for future JCE implementation in wireless systems.

The scientific and industrial interest devoted to polymer/layered silicate
nanocomposites due to their outstanding properties and novel applications resulted
in numerous studies in the last decade. They cover mostly thermoplastic- and
thermoset-based systems. Recently, studies in rubber/layered silicate
nanocomposites were started, as well. It was presented how complex maybe the
nanocomposite formation for the related systems. Therefore the rules governing their
structure-property relationships have to be clarified. In this Thesis, the related
aspects were addressed.
For the investigations several ethylene propylene diene rubbers (EPDM) of polar and
non-polar origin were selected, as well as, the more polar hydrogenated acrylonitrile
butadiene rubber (HNBR). The polarity was found to be beneficial on the
nanocomposite formation as it assisted to the intercalation of the polymer chains
within the clay galleries. This favored the development of exfoliated structures.
Finding an appropriate processing procedure, i.e. compounding in a kneader instead
of on an open mill, the mechanical performance of the nanocomposites was
significantly improved. The complexity of the nanocomposite formation in
rubber/organoclay system was demonstrated. The deintercalation of the organoclay
observed, was traced to the vulcanization system used. It was evidenced by an
indirect way that during sulfur curing, the primary amine clay intercalant leaves the
silicate surface and migrates in the rubber matrix. This was explained by its
participation in the sulfur-rich Zn-complexes created. Thus, by using quaternary
amine clay intercalants (as it was presented for EPDM or HNBR compounds) the
deintercalation was eliminated. The organoclay intercalation/deintercalation detected
for the primary amine clay intercalants, were controlled by means of peroxide curing
(as it was presented for HNBR compounds), where the vulcanization mechanism
differs from that of the sulfur curing.
The current analysis showed that by selecting the appropriate organoclay type the
properties of the nanocomposites can be tailored. This occurs via generating different
nanostructures (i.e. exfoliated, intercalated or deintercalated). In all cases, the
rubber/organoclay nanocomposites exhibited better performance than vulcanizates
with traditional fillers, like silica or unmodified (pristine) layered silicates.The mechanical and gas permeation behavior of the respective nanocomposites
were modelled. It was shown that models (e.g. Guth’s or Nielsen’s equations)
developed for “traditional” vulcanizates can be used when specific aspects are taken
into consideration. These involve characteristics related to the platy structure of the
silicates, i.e. their aspect ratio after compounding (appearance of platelet stacks), or
their orientation in the rubber matrix (order parameter).

Sterisch anspruchsvolle Cyclopentadienyl-Liganden wurden zur Stabilisierung neuer Mono(cyclopentadienyl) Verbindungen der schweren Erdalkalimetalle eingesetzt und deren Funktionalisierbarkeit dieser Spezies wurde exemplarisch durch die Synthese neutraler Tripeldecker-Sandwichkomplexe demonstriert. Die dabei ausgebildeten Molekülstrukturen lassen sich mittels DFT-Rechnungen zuverlässig vorhersagen. In diesem Zusammenhang wurde ebenfalls der Cyclononatetraenyl-Ligand, dessen Komplexeigenschaften bisher nur unzureichend untersucht wurden, eingesetzt. Im Rahmen dieser Arbeit gelang die Synthese des Bis(cyclononatetraenyl)bariums, Ba(C9H9)2, und dessen spektroskopische Charakterisierung. DFT-Rechnungen sagen für diesen Komplex eine Metallocenstruktur mit nahezu parallelen Ringen und einem Ba-Ring Abstand von 2.37 Å voraus. Durch den Einsatz des Tetraisopropylcyclopentadienyl (4Cp) und Tri(tert.-butyl)cyclopentadienyl (Cp’)-Liganden gelang die Synthese von Bis- und Monocyclopentadienyl-Verbindungen der frühen und späten Lanthanoide. Besonders interessant in diesem Zusammenhang ist die erfolgreiche Darstellung des Azido-Clusters, [Na(dme)3]2[4Cp6Yb6(N3)14] (4Cp= (Me2CH)4C5H), der die unterschiedlichen Koordinationsmöglichkeiten des Azido-Liganden in einem einzigen Komplex vereint. Vergleichbare Komplexe waren in der Organolanthanoidchemie bisher unbekannt. Durch Substitution am Cyclopentadienyl-System lassen sich dessen elektronische und sterische Eigenschaften signifikant verändern. Die Auswirkungen dieser Effekte können sehr eindrucksvoll an Manganocen-Komplexen demonstriert werden, in denen sich der low- und high-spin Zustand energetisch nur sehr wenig unterscheiden. Der elektronische Grundzustand einer Reihe unterschiedlich substituierter Manganocen-Komplexe wurde mittels Festkörpermagnetismus, ESR, Röntgenstrukturanalyse, EXAFS und variabler Temperatur UV-Vis Spektroskopie bestimmt, und mit dem Substitutionsmuster am Cyclopentadienyl-System korreliert. Spin-Gleichgewichte ließen sich für [(Me3C)C5H4]2Mn, [(Me3C)2C5H3]2Mn und [(Me3C)(Me3Si)C5H3]2Mn nachweisen. Theoretische Rechnungen postulieren, dass Cerocen, Ce(C8H8)2, ein Beispiel für Moleküle mit gemischt-konfiguriertem Grundzustand sei, der durch 80 % [(Ce)f1e2u(cot)e2u3] und 20 % [(Ce)f0e2u(cot)e2u4] beschreiben werden könne. Obwohl dieses Molekül bereits seit 1976 bekannt ist, ist dessen elektronische Struktur bis heute sehr umstritten. Im Rahmen dieser Arbeit wurden neue Synthesekonzepte für diese Verbindung entwickelt und die elektronische Struktur mittels magnetischer Messungen im Festkörper, EXAFS und XANES Studien untersucht. Die dabei erhaltenen Daten sind in sehr guter Übereinstimmung mit den theoretischen Rechnungen und belegen die Bedeutung eines gemischt-konfigurierten Grundzustandes bei der Bindung in Organometallkomplexen der f-Block Metalle. Während in Cerocen nur ein temperaturunabhängiger Paramagnetismus (TIP) beobachtet werden kann, findet man eine starke Temperaturabhängigkeit der magnetischen Suszeptibilität in Ytterbium Systemen des Typs Cp’2Yb(bipy’) [Cp´ und bipy´ sind substituierte Cyclopentadienyl- oder 4,4’-substituierter 2,2’-Bipyridyl-Liganden]. Temperaturabhängige XANES-Experimenten belegen, dass auch in diesen Systemen ein gemischt-konfigurierter Grundzustand vorliegt, der durch [(Yb)f14(bipy)b1()0] und [(Yb)f13(bipy)b1()1] beschreiben werden kann. Der relative Anteil beider Wellenfunktionen zum Grundzustand wird durch Substitution am 2,2’-Bipyridyl- oder Cyclopentadienyl-System signifikant beeinflusst. Modelle, mit denen sich dieses Verhalten qualitativ beschreiben lässt, wurden im Rahmen dieser Arbeit entwickelt. Ein kinetisch stabilisiertes, adduktfreies Titanocen wurde unter Verwendung des Di(tert.-butyl)cyclopentadienyl Liganden hergestellt und dessen Reaktivität gegenüber kleinen Molekülen, z.B. CO, N2 und H2 untersucht. Im Rahmen der Reaktivitätsstudien wurden ebenfalls 2,2’-Bipyridyl Addukte an das Cp’2Ti Fragment synthetisiert und deren magnetische Eigenschaften erforscht. Durch Variationen am 2,2’-Bipyridyl System lässt sich das Singlet-Triplet Splitting in diesem System gezielt steuern.

This thesis investigates the constrained form of the spherical Minimax location problem and the spherical Weber location problem. Specifically, we consider the problem of locating a new facility on the surface of the unit sphere in the presence of convex spherical polygonal restricted regions and forbidden regions such that the maximum weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized and the sum of the weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized. It is assumed that a forbidden region is an area on the surface of the unit sphere where travel and facility location are not permitted and that distance is measured using the great circle arc distance. We represent a polynomial time algorithm for the spherical Minimax location problem for the special case where all the existing facilities are located on the surface of a hemisphere. Further, we have developed algorithms for spherical Weber location problem using barrier distance on a hemisphere as well as on the unit sphere.

In modern textile manufacturing industries, the function of human eyes to detect disturbances in the production processes which yield defective products is switched to cameras. The camera images are analyzed with various methods to detect these disturbances automatically. There are, however, still problems with in particular semi-regular textures which are typical for weaving patterns. We study three parts of that problem of automatic texture analysis: image smoothing, texture synthesis and defect detection. In image smoothing, we develop a two dimensional kernel smoothing method with locally and directionally adaptive bandwidths allowing correlation in the errors. Two approaches are used in synthesising texture. The first is based on constructing a generalized Ising energy function in the Markov Random Field setup, and for the second, we use two-dimensional periodic bootstrap methods for semi-regular texture synthesis. We treat defect detection as multihypothesis testing problem with the null hypothesis representing the absence of defects and the other hypotheses representing various types of defects. We develop a test based on a nonparametric regression setup, and we use the bootstrap for approximating the distribution of our test statistic.

An autoregressive-ARCH model with possible exogeneous variables is treated. We estimate the conditional volatility of the model by applying feedforward networks to the residuals and prove consistency and asymptotic normality for the estimates under the rate of feedforward networks complexity. Recurrent neural networks estimates of GARCH and value-at-risk is studied. We prove consistency and asymptotic normality for the recurrent neural networks ARMA estimator under the rate of recurrent networks complexity. We also overcome the estimation problem in stochastic variance models in discrete time by feedforward networks and the introduction of a new distributions on the innovations. We use the method to calculate market risk such as expected shortfall and Value-at risk. We tested this distribution together with other new distributions on the GARCH family models against other common distributions on the financial market such as Normal Inverse Gaussian, normal and the Student's t- distributions. As an application of the models, some German stocks are studied and the different approaches are compared together with the most common method of GARCH(1,1) fit.

This thesis deals with the development of thermoplastic polyolefin elastomers using recycled polyolefins and ground tyre rubber (GTR). The disposal of worn tyres and their economic recycling mean a great challenge nowadays. Material recycling is a preferred way in Europa owing to legislative actions and ecological arguments. This first step with worn tyres is already done in this direc-tion as GTR is available in different fractions in guaranteed quality. As the traditional applications of GTR are saturated, there is a great demand for new, value-added products containing GTR. So, the objective of this work was to convert GTR by reac-tive blending with polyolefins into thermoplastic elastomers (TPE) of suitable me-chanical and rheological properties. It has been established that bituminous reclamation of GTR prior to extrusion melt compounding with polyolefins is a promising way of TPE production. By this way the sol-content (acetone soluble fraction) of the GTR increases and the GTR particles can be better incorporated in the corresponding polyolefin matrix. The adhesion be-tween GTR and matrix is given by molecular intermingling in the resulting interphase. GTR particles of various production and mean particle size were involved in this study. As polyolefins recycled low-density polyethylene (LDPE), recycled high-density polyethylene (HDPE) and polypropylene (PP) were selected. First, the opti-mum conditions for the GTR reclamation in bitumen were established (160 °C < T < 180 °C; time ca. 4 hours). Polyolefin based TPEs were produced after GTR reclamation in extrusion compounding. Their mechanical (tensile behaviour, set properties), thermal (dynamic-mechanical thermal analysis, differential scanning calorimetry) and rheological properties (both in low- and high-shear rates ) were determined. The PE-based blends contained an ethylene/propylene/diene (EPDM) rubber as compatibilizer and their composition was as follows: PE/EPDM/GTR:bitumen = 50/25/25:25. The selected TPEs met the most important criterion, i.e. elongation at break > 100 %; compression set < 50%. The LDPE-based TPE (TPE(LDPE)) showed better me-chanical performance compared to the TPE(HDPE). This was assigned to the higher crystallinity of the HDPE. The PP-based blends of the compositions PP/(GTR-bitumen) 50/50 and 25/75, whereby the ratio of GTR/bitumen was 60/40, outperformed those containing non-reclaimed GTR. The related blends showed also a better compatibility with a PP-based commercial thermoplastic dynamic vulcanizate (TDV). Surprisingly, the mean particle size of the GTR, varied between < 0.2 and 0.4-0.7 mm, had a small effect on the mechanical properties, however somewhat larger for the rheological behaviour of the TPEs produced.

Metallocenes containing diarylethene type photochromic switches are synthesized, characterized and tested in polyolefin catalysts. Propylene polymerizations using unbridged bis(2,3-dibenzo[b]thiophen-3-yl)cyclopenta[b]thien-3-yl)zirconium dichloride/MAO (80) treated with 254nm UV irradiation produced bimodal polymer distributions by GPC. This was due to an increase in the low molecular weight fractions when the closed form of the catalyst/photoswitch was made. Comparison with similarly structured catalyst without photoisomerization properties did not produce bimodal polymer under identical conditions. Propylene polymerizations made with dimethylsilyl[(1,5-dimethyl-3-phenylcyclopenta[b]thien-6-yl)][(2,3-dibenzothien-3-yl)cyclopenta[b]thien-6-yl)]zirconium dichloride/MAO (86) with 254nm UV irradiation caused a 3 fold increase in the polymer molecular weight. Polymers made with ethylene and ethylene/hexene using (80) after UV irradiation did not show differences in measured polymer properties. Polymerizations with ethylene/ hexene mixtures using (86) had increased activity and co-monomer (hexene) incorporation with UV irradiation.

It is considered an analytical model of defaultable bond portfolio in terms of its face value process. The face value process dynamically evolves with time and incorporates changes caused by recovery payment on default followed by purchasing of new bonds. The further studies involve properties, distribution and control of the face value process.

The aim of the thesis is the numerical investigation of saturated, stationary, incompressible Newtonian flow in porous media when inertia is not negligible. We focus our attention to the Navier-Stokes system with two pressures derived by two-scale homogenization. The thesis is subdivided into five Chapters. After the introductory remarks on porous media, filtration laws and upscaling methods, the first chapter is closed by stating the basic terminology and mathematical fundamentals. In Chapter 2, we start by formulating the Navier-Stokes equations on a periodic porous medium. By two-scale expansions of the velocity and pressure, we formally derive the Navier-Stokes system with two pressures. For the sake of completeness, known existence and uniqueness results are repeated and a convergence proof is given. Finally, we consider Stokes and Navier-Stokes systems with two pressures with respect to their relation to Darcy's law. Chapter 3 and Chapter 4 are devoted to the numerical solution of the nonlinear two pressure system. Therefore, we follow two approaches. The first approach which is developed in Chapter 3 is based on a splitting of the Navier-Stokes system with two pressures into micro and macro problems. The splitting is achieved by Taylor expanding the permeability function or by discretely computing the permeability function. The problems to be solved are a series of Stokes and Navier-Stokes problems on the periodicity cell. The Stokes problems are solved by an Uzawa conjugate gradient method. The Navier-Stokes equations are linearized by a least-squares conjugate gradient method, which leads to the solution of a sequence of Stokes problems. The macro problem consists of solving a nonlinear uniformly elliptic equation of second order. The least-squares linearization is applied to the macro problem leading to a sequence of Poisson problems. All equations will be discretized by finite elements. Numerical results are presented at the end of Chapter 3. The second approach presented in Chapter 4 relies on the variational formulation in a certain Hilbert space setting of the Navier-Stokes system with two pressures. The nonlinear problem is again linearized by the least-squares conjugate gradient method. We obtain a sequence of Stokes systems with two pressures. For the latter systems, we propose a fast solution method which relies on pre-computing Stokes systems on the periodicity cell for finite element basis functions acting as right hand sides. Finally, numerical results are discussed. In Chapter 5 we are concerned with modeling and simulation of the pressing section of a paper machine. We state a two-dimensional model of a press nip which takes into account elasticity and flow phenomena. Nonlinear filtration laws are incorporated into the flow model. We present a numerical solution algorithm and the chapter is closed by a numerical investigation of the model with special focus on inertia effects.

The present thesis deals with a novel approach to increase the resource usage in digital communications. In digital communication systems, each information bearing data symbol is associated to a waveform which is transmitted over a physical medium. The time or frequency separations among the waveforms associated to the information data have always been chosen to avoid or limit the interference among them. By doing so, n the presence of a distortionless ideal channel, a single receive waveform is affected as little as possible by the presence of the other waveforms. The conditions necessary to meet the absence of any interference among the waveforms are well known and consist of a relationship between the minimum time separation among the waveforms and their bandwidth occupation or, equivalently, the minimum frequency separation and their time occupation. These conditions are referred to as Nyquist assumptions. The key idea of this work is to relax the Nyquist assumptions and to transmit with a time and/or frequency separation between the waveforms smaller than the minimum required to avoid interference. The reduction of the time and/or frequency separation generates not only an increment of the resource usage, but also a degradation in the quality of the received data. Therefore, to maintain a certain quality in the received signal, we have to increase the amount of transmitted power. We investigate the trade-off between the increment of the resource usage and the correspondent performance degradation in three different cases. The first case is the single carrier case in which all waveforms have the same spectrum, but have different temporal locations. The second one is the multi carrier case in which each waveform has its distinct spectrum and occupies all the available time. Finally, the hybrid case when each waveform has its unique time and frequency location. These different cases are framed within the general system modelling developed in the thesis so that they can be easily compared. We evaluate the potential of the key idea of the thesis by choosing a set of four possible waveforms with different characteristics. By doing so, we study the influence of the waveform characteristics in the three system configurations. We propose an interpretation of the results by modifying the well-known Shannon capacity formula and by explicitly expressing its dependency on the increment of resource usage and on the performance degradation. The results are very promising. We show that both in the case of a single carrier system with a time limited waveform and in the case of a multi-carrier system with a frequency limited waveform, the reduction of the time or frequency separation, respectively, has a positive effect on the channel capacity. The latter, depending on the actual SNR, can double or increase even more significantly.

Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations. In the thesis, we have developed a framework, unifying many important algebras in the classes of \(G\)- and \(GR\)-algebras and studied their ring-theoretic properties. Let \(A\) be a \(G\)-algebra in \(n\) variables. We establish necessary and sufficient conditions for \(A\) to have a Poincar'e-Birkhoff-Witt (PBW) basis. Further on, we show that besides the existence of a PBW basis, \(A\) shares some other properties with the commutative polynomial ring \(\mathbb{K}[x_1,\ldots,x_n]\). In particular, \(A\) is a Noetherian integral domain of Gel'fand-Kirillov dimension \(n\). Both Krull and global homological dimension of \(A\) are bounded by \(n\); we provide examples of \(G\)-algebras where these inequalities are strict. Finally, we prove that \(A\) is Auslander-regular and a Cohen-Macaulay algebra. In order to perform symbolic computations with modules over \(GR\)-algebras, we generalize Gröbner bases theory, develop and respectively enhance new and existing algorithms. We unite the most fundamental algorithms in a suite of applications, called "Gröbner basics" in the literature. Furthermore, we discuss algorithms appearing in the non-commutative case only, among others two-sided Gröbner bases for bimodules, annihilators of left modules and operations with opposite algebras. An important role in Representation Theory is played by various subalgebras, like the center and the Gel'fand-Zetlin subalgebra. We discuss their properties and their relations to Gröbner bases, and briefly comment some aspects of their computation. We proceed with these subalgebras in the chapter devoted to the algorithmic study of morphisms between \(GR\)-algebras. We provide new results and algorithms for computing the preimage of a left ideal under a morphism of \(GR\)-algebras and show both merits and limitations of several methods that we propose. We use this technique for the computation of the kernel of a morphism, decomposition of a module into central characters and algebraic dependence of pairwise commuting elements. We give an algorithm for computing the set of one-dimensional representations of a \(G\)-algebra \(A\), and prove, moreover, that if the set of finite dimensional representations of \(A\) over a ground field \(K\) is not empty, then the homological dimension of \(A\) equals \(n\). All the algorithms are implemented in a kernel extension Plural of the computer algebra system Singular. We discuss the efficiency of computations and provide a comparison with other computer algebra systems. We propose a collection of benchmarks for testing the performance of algorithms; the comparison of timings shows that our implementation outperforms all of the modern systems with the combination of both broad functionality and fast implementation. In the thesis, there are many new non-trivial examples, and also the solutions to various problems, arising in different fields of mathematics. All of them were obtained with the developed theory and the implementation in Plural, most of them are treated computationally in this thesis for the first time.

The thesis is focused on modelling and simulation of a Joint Transmission and Detection Integrated Network (JOINT), a novel air interface concept for B3G mobile radio systems. Besides the utilization of the OFDM transmission technique, which is a promising candidate for future mobile radio systems, and of the duplexing scheme time division duplexing (TDD), the subdivision of the geographical domain to be supported by mobile radio communications into service areas (SAs) is a highlighted concept of JOINT. A SA consists of neighboring sub-areas, which correspond to the cells of conventional cellular systems. The signals in a SA are jointly processed in a Central Unit (CU) in each SA. The CU performs joint channel estimation (JCE) and joint detection (JD) in the form of the receive-zero-forcing (RxZF) Filter for the uplink (UL) transmission and joint transmission (JT) in the form of the transmit-zero-forcing (TxZF) Filter for the downlink (DL) transmission. By these algorithms intra-SA multiple access interference (MAI) can be eliminated within the limits of the used model so that unbiased data estimates are obtained, and most of the computational effort is moved from mobile terminals (MTs) to the CU so that the MTs can do with low complexity. A simulation chain of JOINT has been established in the software MLDesigner by the author based on time discrete equivalent lowpass modelling. In this simulation chain, all key functionalities of JOINT are implemented. The simulation chain is designed for link level investigations. A number of channel models are implemented both for the single-SA scenario and the multiple-SA scenario so that the system performance of JOINT can be comprehensively studied. It is shown that in JOINT a duality or a symmetry of the MAI elimination in the UL and in the DL exists. Therefore, the typical noise enhancement going along with the MAI elimination by JD and JT, respectively, is the same in both links. In the simulations also the impact of channel estimation errors on the system performance is studied. In the multiple-SA scenario, due to the existence of the inter-SA MAI, which cannot be suppressed by the algorithms of JD and JT, the system performance in terms of the average bit error rate (BER) and the BER statistics degrades. A collection of simulation results show the potential of JOINT with respect to the improvement of the system performance and the enhancement of the spectrum e±ciency as compared to conventional cellular systems.

Since its invention by Sir Allistair Pilkington in 1952, the float glass process has been used to manufacture long thin flat sheets of glass. Today, float glass is very popular due to its high quality and relatively low production costs. When producing thinner glass the main concern is to retain its optical quality, which can be deteriorated during the manufacturing process. The most important stage of this process is the floating part, hence is considered to be responsible for the loss in the optical quality. A series of investigations performed on the finite products showed the existence of many short wave patterns, which strongly affect the optical quality of the glass. Our work is concerned with finding the mechanism for wave development, taking into account all possible factors. In this thesis, we model the floating part of the process by an theoretical study of the stability of two superposed fluids confined between two infinite plates and subjected to a large horizontal temperature gradient. Our approach is to take into account the mixed convection effects (viscous shear and buoyancy), neglecting on the other hand the thermo-capillarity effects due to the length of our domain and the presence of a small stabilizing vertical temperature gradient. Both fluids are treated as Newtonian with constant viscosity. They are immiscible, incompressible, have very different properties and have a free surface between them. The lower fluid is a liquid metal with a very small kinematic viscosity, whereas the upper fluid is less dense. The two fluids move with different velocities: the speed of the upper fluid is imposed, whereas the lower fluid moves as a result of buoyancy effects. We examine the problem by means of small perturbation analysis, and obtain a system of two Orr-Sommerfeld equations coupled with two energy equations, and general interface and boundary conditions. We solve the system analytically in the long- and short- wave limit, by using asymptotic expansions with respect to the wave number. Moreover, we write the system in the form of a general eigenvalue problem and we solve the system numerically by using Chebyshev spectral methods for fluid dynamics. The results (both analytical and numerical) show the existence of the small-amplitude travelling waves, which move with constant velocity for wave numbers in the intermediate range. We show that the stability of the system is ensured in the long wave limit, a fact which is in agreement with the real float glass process. We analyze the stability for a wide range of wave numbers, Reynolds, Weber and Grashof number, and explain the physical implications on the dynamics of the problem. The consequences of the linear stability results are discussed. In reality in the float glass process, the temperature strongly influences the viscosity of both molten metal and hot glass, which will have direct consequences on the stability of the system. We investigate the linear stability of two superposed fluids with temperature dependent viscosities by considering a different model for the viscosity dependence of each fluid. Although, the temperature-viscosity relationships for glass and metal are more complex than those used in our computations, our intention is to emphasize the effects of this dependence on the stability of the system. It is known from the literature that in the case of one fluid, the heat, which causes viscosity to decrease along the domain, usually destabilizes the flow. For the two superposed fluids problem we investigate this behaviour and discuss the consequences of the linear stability in this new case.

Over the last decades, mathematical modeling has reached nearly all fields of natural science. The abstraction and reduction to a mathematical model has proven to be a powerful tool to gain a deeper insight into physical and technical processes. The increasing computing power has made numerical simulations available for many industrial applications. In recent years, mathematicians and engineers have turned there attention to model solid materials. New challenges have been found in the simulation of solids and fluid-structure interactions. In this context, it is indispensable to study the dynamics of elastic solids. Elasticity is a main feature of solid bodies while demanding a great deal of the numerical treatment. There exists a multitude of commercial tools to simulate the behavior of elastic solids. Anyhow, the majority of these software packages consider quasi-stationary problems. In the present work, we are interested in highly dynamical problems, e.g. the rotation of a solid. The applicability to free-boundary problems is a further emphasis of our considerations. In the last years, meshless or particle methods have attracted more and more attention. In many fields of numerical simulation these methods are on a par with classical methods or superior to them. In this work, we present the Finite Pointset Method (FPM) which uses a moving least squares particle approximation operator. The application of this method to various industrial problems at the Fraunhofer ITWM has shown that FPM is particularly suitable for highly dynamical problems with free surfaces and strongly changing geometries. Thereby, FPM offers exactly the features that we require for the analysis of the dynamics of solid bodies. In the present work, we provide a numerical scheme capable to simulate the behavior of elastic solids. We present the system of partial differential equations describing the dynamics of elastic solids and show its hyperbolic character. In particular, we focus our attention to the constitutive law for the stress tensor and provide evolution equations for the deviatoric part of the stress tensor in order to circumvent limitations of the classical Hooke's law. Furthermore, we present the basic principle of the Finite Pointset Method. In particular, we provide the concept of upwinding in a given direction as a key ingredient for stabilizing hyperbolic systems. The main part of this work describes the design of a numerical scheme based on FPM and an operator splitting to take the different processes within a solid body into account. Each resulting subsystem is treated separately in an adequate way. Hereby, we introduce the notion of system-inherent directions and dimensional upwinding. Finally, a coupling strategy for the subsystems and results are presented. We close this work with some final conclusions and an outlook on future work.

The HMG-CoA reductase inhibitors SIM, LOV, ATV, PRA, FV and NKS were investigated for their effects on human SkMCs. We were able to demonstrate that statins can induce oxidative stress (ROS formation, GSH-depletion, TBARS), apoptosis (, caspase-3 activity, nuclear morphology) and necrosis (LDH-leakage) in hSkMCs. After incubation with statins, the sequence of cellular events starts by the increased formation of ROS (30 min) followed by caspase-3 activation (2-4 hours) and necrosis (LDH-leakage) and formation of condensed and fragmented nuclei after 24-72 hours. It was shown that, antioxidants (NAC, DTT, TPGS, M-2 and M-3) and the HMG-CoA reductase downstream metabolites (MVA, F, FPP, GG and GGPP) protected against statin-induced ROS formation, caspase-3 activation and partially from necrosis. The caspase-3 inhibitor Ac-DEVD-CHO rescues cells partially from necrosis. These results suggest that the statin-induced necrosis is HMG-CoA dependent and occurs secondary to apoptosis, which by decrease of ATP is driven into necrosis. The increase of ATP observed at low concentrations and early time points suggest an increased glycolytic activity. This was confirmed by increased PDK-4 gene expression and increased PFK2/F-2,6-BPase expression both activator of glycolysis. Glycolysis was also confirmed for some statins by increased cellular lactate concentations. The consequence of PDK-4 mediated pyruvate dehydrogenase inactivation is the metabolic switching from fatty acid to amino acid from proteins as energy source. The oxidative stress hypothesis was further supported by the induction of the FOXO3A transcription factor, which is involved in regulating MnSOD-2 expression in the mitochondrium. The mechanism by which statins produce ROS is still not resolved. There is an indirect evidence from our experiments as well as from the literature, that immediately after the statin treatment, intracellular Ca2+ is mobilized due to HMG-CoA reductase inhibition, which after mitochondrial uptake could lead to increased ROS formation.

In the first part of this work, called Simple node singularity, are computed matrix factorizations of all isomorphism classes, up to shiftings, of rank one and two, graded, indecomposable maximal Cohen--Macaulay (shortly MCM) modules over the affine cone of the simple node singularity. The subsection 2.2 contains a description of all rank two graded MCM R-modules with stable sheafification on the projective cone of R, by their matrix factorizations. It is given also a general description of such modules, of any rank, over a projective curve of arithmetic genus 1, using their matrix factorizations. The non-locally free rank two MCM modules are computed using an alghorithm presented in the Introduction of this work, that gives a matrix factorization of any extension of two MCM modules over a hypersurface. In the second part, called Fermat surface, are classified all graded, rank two, MCM modules over the affine cone of the Fermat surface. For the classification of the orientable rank two graded MCM R-modules, is used a description of the orientable modules (over normal rings) with the help of codimension two Gorenstein ideals, realized by Herzog and Kühl. It is proven (in section 4), that they have skew symmetric matrix factorizations (over any normal hypersurface ring). For the classification of the non-orientable rank two MCM R-modules, we use a similar idea as in the case of the orientable ones, only that the ideal is not any more Gorenstein.

In this thesis we have discussed the problem of decomposing an integer matrix \(A\) into a weighted sum \(A=\sum_{k \in {\mathcal K}} \alpha_k Y^k\) of 0-1 matrices with the strict consecutive ones property. We have developed algorithms to find decompositions which minimize the decomposition time \(\sum_{k \in {\mathcal K}} \alpha_k\) and the decomposition cardinality \(|\{ k \in {\mathcal K}: \alpha_k > 0\}|\). In the absence of additional constraints on the 0-1 matrices \(Y^k\) we have given an algorithm that finds the minimal decomposition time in \({\mathcal O}(NM)\) time. For the case that the matrices \(Y^k\) are restricted to shape matrices -- a restriction which is important in the application of our results in radiotherapy -- we have given an \({\mathcal O}(NM^2)\) algorithm. This is achieved by solving an integer programming formulation of the problem by a very efficient combinatorial algorithm. In addition, we have shown that the problem of minimizing decomposition cardinality is strongly NP-hard, even for matrices with one row (and thus for the unconstrained as well as the shape matrix decomposition). Our greedy heuristics are based on the results for the decomposition time problem and produce better results than previously published algorithms.

In the thesis the task of channel estimation in beyond 3G service area based mobile radio air interfaces is considered. A system concept named Joint Transmission and Detection Integrated Network (JOINT) forms the target platform for the investigations. A single service area of JOINT is considered, in which a number of mobile terminals is supported by a number of radio access points, which are connected to a central unit responsible for the signal processing. The modulation scheme of JOINT is OFDM. Pilot-aided channel estimation is considered, which has to be performed only in the uplink of JOINT, because the duplexing scheme TDD is applied. In this way, the complexity of the mobile terminals is reduced, because they do not need a channel estimator. Based on the signals received by the access points, the central unit estimates the channel transfer functions jointly for all mobile terminals. This is done by resorting to the a priori knowledge of the radiated pilot signals and by applying the technique of joint channel estimation, which is developed in the thesis. The quality of the gained estimates is judged by the degradation of their signal-to-noise ratio as compared to the signal-to-noise ratio of the respective estimates gained in the case of a single mobile terminal radiating its pilots. In the case of single-element receive antennas at the access points, said degradation depends solely on the structure of the applied pilots. In the thesis it is shown how by a proper design of the pilots the SNR degradation can be minimized. Besides using appropriate pilots, the performance of joint channel estimation can be further improved by the inclusion of additional a-priori information in the estimation process. An example of such additional information would be the knowledge of the directional properties of the radio channels. This knowledge can be gained if multi-element antennas are applied at the access points. Further, a-priori channel state information in the form of the power delay profiles of the radio channels can be included in the estimation process by the application of the minimum mean square error estimation principle for joint channel estimation. After having intensively studied the problem of joint channel estimation in JOINT, the thesis rounds itself by considering the impact of the unavoidable channel estimation errors on the performance of data estimation in JOINT. For the case of small channel estimation errors occurring due to the presence of noise at the access points, the performance of joint detection in the uplink and of joint transmission in the downlink of JOINT are investigated based on simulations. For the uplink, which utilizes joint detection, it is shown to which degree the bit error probability increases due to channel estimation errors. For the downlink, which utilizes joint transmission, channel estimation errors lead to an increase of the required transmit power, which can be quantified by the simulation results.