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The main topic of this treatise is the solution of two problems from the general theory of linear partial differential equations with constant coefficients. While surjectivity criteria for linear partial differential operators in spaces of smooth functions over an open subset of euclidean space and distributions were proved by B. Malgrange and L. Hörmander in 1955, respectively 1962, concrete evaluation of these criteria is still a highly non-trivial task. In particular, it is well-known that surjectivity in the space of smooth functions over an open subset of euclidean space does not automatically imply surjectivity in the space of distributions. Though, examples for this fact all live in three or higher dimensions. In 1966, F. Trèves conjectured that in the two dimensional setting surjectivity of a linear partial differential operator on the smooth functions indeed implies surjectivity on the space of distributions. An affirmative solution to this problem is presented in this treatise. The second main result solves the so-called problem of (distributional) parameter dependence for solutions of linear partial differential equations with constant coefficients posed by J. Bonet and P. Domanski in 2006. It is shown that, in dimensions three or higher, this problem in general has a negative solution even for hypoelliptic operators. Moreover, it is proved that the two dimensional case is again an exception, because in this setting the problem of parameter dependence always has a positive solution.
As an interface between an individual and its environment, the skin is a major site of direct exposure to exogenous substances. Once absorbed, these substances may interact with different biomolecules within the skin. The aryl hydrocarbon receptor (AhR) signaling pathway is one mechanism whereby the skin responds to exposures, predominantly through the induction or upregulation of metabolizing enzymes. One known physiological role of the AhR in many tissues is its involvement in the control of cell cycle progression. In skin, almost nothing is known about this physiological function. Moreover, the question whether frequently used naturally occurring phenolic derivatives like eugenol and isoeugenol impact on the AhR within the skin has rarely been studied so far. Eugenol and isoeugenol are due to their odour referred to as fragrances. The ubiquitous distribution of eugenol and isoeugenol results in an almost unavoidable contact with these substances in our daily lives. Despite this fact, their molecular mechanisms of action in skin are poorly understood. There is evidence supporting the hypothesis that these substances may impact on the AhR. On the one hand, eugenol is shown to induce cytochrome P450 1A1 (CYP1A1), a well-known target gene of the AhR. On the other hand, their known anti-proliferative properties might also be mediated by the AhR, based on its physiological function. In order to proof this hypothesis, it was investigated whether eugenol and isoeugenol impact on the AhR signaling pathway in skin cells. Results revealed that eugenol as well as isoeugenol impact on the AhR signaling pathway in skin cells. Both substances caused the translocation of the AhR into the nucleus, induced the expression of the well-known AhR target genes CYP1A1 and AhR repressor (AhRR) and exhibited impact on cell cycle progression. Both substances caused an AhR-dependent cell cycle arrest in skin cells, modulated protein levels of several cell cycle regulatory proteins, inhibited DNA synthesis and thereby reduced cell numbers. The comparison of wildtype cells to AhR knockdown cells revealed an influence of the AhR on cell cycle progression in skin cells in the absence of exogenous ligands. AhR knockdown cells exhibited a slower progression through the cell cycle caused by an accumulation of cells in the G0/G1 phase of the cell cycle and a decreased DNA synthesis rate. Modulation of cell cycle regulatory proteins involved in the transition from the G0/G1 to the S phase of the cell cycle was altered in AhR knockdown cells as well. To conclude, eugenol as well as isoeugenol were able to impact on the AhR signaling pathway in skin cells. Their molecular mechanisms of action are similar to those of classical AhR ligands, although their structural characteristics strongly differ from that of these ligands. In the absence of exogenous ligands the AhR promotes cell cycle progression in many tissues and this knowledge could be expanded on skin-derived cells within the scope of this thesis.
In this thesis, we mainly investigate geometric properties of optimal codebooks for random elements $X$ in a seperable Banach space $E$. Here, for a natural number $ N $ and a random element $X$ , an $N$-optimal codebook is an $ N $-subset in the underlying Banach space $E$ which gives a best approximation to $ X $ in an average sense. We focus on two types of geometric properties: The global growth behaviour (growing in $N$) for a sequence of $N$-optimal codebooks is described by the maximal (quantization) radius and a so-called quantization ball. For many distributions, such as central-symmetric distributions on $R^d$ as well as Gaussian distributions on general Banach spaces, we are able to estimate the asymptotics of the quantization radius as well as the quantization ball. Furthermore, we investigate local properties of optimal codebooks, in particular the local quantization error and the weights of the Voronoi cells induced by an optimal codebook. In the finite-dimensional setting, we are able to proof for many interesting distributions classical conjectures on the asymptotic behaviour of those properties. Finally, we propose a method to construct sequences of asymptotically optimal codebooks for random elements in infinite dimensional Banach spaces and apply this method to construct codebooks for stochastic processes, such as fractional Brownian Motions.
In recent years, the study of dynamical systems has developed into a central research area in mathematics. Actually, in combination with keywords such as "chaos" or "butterfly effect", parts of this theory have been incorporated in other scientific fields, e.g. in physics, biology, meteorology and economics. In general, a discrete dynamical system is given by a set X and a self-map f of X. The set X can be interpreted as the state space of the system and the function f describes the temporal development of the system. If the system is in state x ∈ X at time zero, its state at time n ∈ N is denoted by f^n(x), where f^n stands for the n-th iterate of the map f. Typically, one is interested in the long-time behaviour of the dynamical system, i.e. in the behaviour of the sequence (f^n(x)) for an arbitrary initial state x ∈ X as the time n increases. On the one hand, it is possible that there exist certain states x ∈ X such that the system behaves stably, which means that f^n(x) approaches a state of equilibrium for n→∞. On the other hand, it might be the case that the system runs unstably for some initial states x ∈ X so that the sequence (f^n(x)) somehow shows chaotic behaviour. In case of a non-linear entire function f, the complex plane always decomposes into two disjoint parts, the Fatou set F_f of f and the Julia set J_f of f. These two sets are defined in such a way that the sequence of iterates (f^n) behaves quite "wildly" or "chaotically" on J_f whereas, on the other hand, the behaviour of (f^n) on F_f is rather "nice" and well-understood. However, this nice behaviour of the iterates on the Fatou set can "change dramatically" if we compose the iterates from the left with just one other suitable holomorphic function, i.e. if we consider sequences of the form (g∘f^n) on D, where D is an open subset of F_f with f(D)⊂ D and g is holomorphic on D. The general aim of this work is to study the long-time behaviour of such modified sequences. In particular, we will prove the existence of holomorphic functions g on D having the property that the behaviour of the sequence of compositions (g∘f^n) on the set D becomes quite similarly chaotic as the behaviour of the sequence (f^n) on the Julia set of f. With this approach, we immerse ourselves into the theory of universal families and hypercyclic operators, which itself has developed into an own branch of research. In general, for topological spaces X, Y and a family {T_i: i ∈ I} of continuous functions T_i:X→Y, an element x ∈ X is called universal for the family {T_i: i ∈ I} if the set {T_i(x): i ∈ I} is dense in Y. In case that X is a topological vector space and T is a continuous linear operator on X, a vector x ∈ X is called hypercyclic for T if it is universal for the family {T^n: n ∈ N}. Thus, roughly speaking, universality and hypercyclicity can be described via the following two aspects: There exists a single object which allows us, via simple analytical operations, to approximate every element of a whole class of objects. In the above situation, i.e. for a non-linear entire function f and an open subset D of F_f with f(D)⊂ D, we endow the space H(D) of holomorphic functions on D with the topology of locally uniform convergence and we consider the map C_f:H(D)→H(D), C_f(g):=g∘f|_D, which is called the composition operator with symbol f. The transform C_f is a continuous linear operator on the Fréchet space H(D). In order to show that the above-mentioned "nice" behaviour of the sequence of iterates (f^n) on the set D ⊂ F_f can "change dramatically" if we compose the iterates from the left with another suitable holomorphic function, our aim consists in finding functions g ∈ H(D) which are hypercyclic for C_f. Indeed, for each hypercyclic function g for C_f, the set of compositions {g∘f^n|_D: n ∈ N} is dense in H(D) so that the sequence of compositions (g∘f^n|_D) is kind of "maximally divergent" " meaning that each function in H(D) can be approximated locally uniformly on D via subsequences of (g∘f^n|_D). This kind of behaviour stands in sharp contrast to the fact that the sequence of iterates (f^n) itself converges, behaves like a rotation or shows some "wandering behaviour" on each component of F_f. To put it in a nutshell, this work combines the theory of non-linear complex dynamics in the complex plane with the theory of dynamics of continuous linear operators on spaces of holomorphic functions. As far as the author knows, this approach has not been investigated before.
The German Mittelstand is closely linked to the success of the German economy. Mittelstand firms, thereof numerous Hidden Champions, significantly contribute to Germany’s economic performance, innovation, and export strength. However, the advancing digitalization poses complex challenges for Mittelstand firms. To benefit from the manifold opportunities offered by digital technologies and to defend or even expand existing market positions, Mittelstand firms must transform themselves and their business models. This dissertation uses quantitative methods and contributes to a deeper understanding of the distinct needs and influencing factors of the digital transformation of Mittelstand firms. The results of the empirical analyses of a unique database of 525 mid-sized German manufacturing firms, comprising both firm-related information and survey data, show that organizational capabilities and characteristics significantly influence the digital transformation of Mittelstand firms. The results support the assumption that dynamic capabilities promote the digital transformation of such firms and underline the important role of ownership structure, especially regarding family influence, for the digital transformation of the business model and the pursuit of growth goals with digitalization. In addition to the digital transformation of German Mittelstand firms, this dissertation examines the economic success and regional impact of Hidden Champions and hence, contributes to a better understanding of the Hidden Champion phenomenon. Using quantitative methods, it can be empirically proven that Hidden Champions outperform other mid-sized firms in financial terms and promote regional development. Consequently, the results of this dissertation provide valuable research contributions and offer various practical implications for firm managers and owners as well as policy makers.
Given a compact set K in R^d, the theory of extension operators examines the question, under which conditions on K, the linear and continuous restriction operators r_n:E^n(R^d)→E^n(K),f↦(∂^α f|_K)_{|α|≤n}, n in N_0 and r:E(R^d)→E(K),f↦(∂^α f|_K)_{α in N_0^d}, have a linear and continuous right inverse. This inverse is called extension operator and this problem is known as Whitney's extension problem, named after Hassler Whitney. In this context, E^n(K) respectively E(K) denote spaces of Whitney jets of order n respectively of infinite order. With E^n(R^d) and E(R^d), we denote the spaces of n-times respectively infinitely often continuously partially differentiable functions on R^d. Whitney already solved the question for finite order completely. He showed that it is always possible to construct a linear and continuous right inverse E_n for r_n. This work is concerned with the question of how the existence of a linear and continuous right inverse of r, fulfilling certain continuity estimates, can be characterized by properties of K. On E(K), we introduce a full real scale of generalized Whitney seminorms (|·|_{s,K})_{s≥0}, where |·|_{s,K} coincides with the classical Whitney seminorms for s in N_0. We equip also E(R^d) with a family (|·|_{s,L})_{s≥0} of those seminorms, where L shall be a a compact set with K in L-°. This family of seminorms on E(R^d) suffices to characterize the continuity properties of an extension operator E, since we can without loss of generality assume that E(E(K)) in D^s(L).
In Chapter 2, we introduce basic concepts and summarize the classical results of Whitney and Stein.
In Chapter 3, we modify the classical construction of Whitney's operators E_n and show that |E_n(·)|_{s,L}≤C|·|_{s,K} for s in[n,n+1).
In Chapter 4, we generalize a result of Frerick, Jordá and Wengenroth and show that LMI(1) for K implies the existence of an extension operator E without loss of derivatives, i.e. we have it fulfils |E(·)|_{s,L}≤C|·|_{s,K} for all s≥0. We show that a large class of self similar sets, which includes the Cantor set and the Sierpinski triangle, admits an extensions operator without loss of derivatives.
In Chapter 5 we generalize a result of Frerick, Jordá and Wengenroth and show that WLMI(r) for r≥1 implies the existence of a tame linear extension operator E having a homogeneous loss of derivatives, such that |E(·)|_{s,L}≤C|·|_{(r+ε)s,K} for all s≥0 and all ε>0.
In the last chapter we characterize the existence of an extension operator having an arbitrary loss of derivatives by the existence of measures on K.
Variational inequality problems constitute a common basis to investigate the theory and algorithms for many problems in mathematical physics, in economy as well as in natural and technical sciences. They appear in a variety of mathematical applications like convex programming, game theory and economic equilibrium problems, but also in fluid mechanics, physics of solid bodies and others. Many variational inequalities arising from applications are ill-posed. This means, for example, that the solution is not unique, or that small deviations in the data can cause large deviations in the solution. In such a situation, standard solution methods converge very slowly or even fail. In this case, so-called regularization methods are the methods of choice. They have the advantage that an ill-posed original problem is replaced by a sequence of well-posed auxiliary problems, which have better properties (like, e.g., a unique solution and a better conditionality). Moreover, a suitable choice of the regularization term can lead to unconstrained auxiliary problems that are even equivalent to optimization problems. The development and improvement of such methods are a focus of current research, in which we take part with this thesis. We suggest and investigate a logarithmic-quadratic proximal auxiliary problem (LQPAP) method that combines the advantages of the well-known proximal-point algorithm and the so-called auxiliary problem principle. Its exploration and convergence analysis is one of the main results in this work. The LQPAP method continues the recent developments of regularization methods. It includes different techniques presented in literature to improve the numerical stability: The logarithmic-quadratic distance function constitutes an interior point effect which allows to treat the auxiliary problems as unconstrained ones. Furthermore, outer operator approximations are considered. This simplifies the numerical solution of variational inequalities with multi-valued operators since, for example, bundle-techniques can be applied. With respect to the numerical practicability, inexact solutions of the auxiliary problems are allowed using a summable-error criterion that is easy to implement. As a further advantage of the logarithmic-quadratic distance we verify that it is self-concordant (in the sense of Nesterov/Nemirovskii). This motivates to apply the Newton method for the solution of the auxiliary problems. In the numerical part of the thesis the LQPAP method is applied to linearly constrained, differentiable and nondifferentiable convex optimization problems, as well as to nonsymmetric variational inequalities with co-coercive operators. It can often be observed that the sequence of iterates reaches the boundary of the feasible set before being close to an optimal solution. Against this background, we present the strategy of under-relaxation, which robustifies the LQPAP method. Furthermore, we compare the results with an appropriate method based on Bregman distances (BrPAP method). For differentiable, convex optimization problems we describe the implementation of the Newton method to solve the auxiliary problems and carry out different numerical experiments. For example, an adaptive choice of the initial regularization parameter and a combination of an Armijo and a self-concordance step size are evaluated. Test examples for nonsymmetric variational inequalities are hardly available in literature. Therefore, we present a geometric and an analytic approach to generate test examples with known solution(s). To solve the auxiliary problems in the case of nondifferentiable, convex optimization problems we apply the well-known bundle technique. The implementation is described in detail and the involved functions and sequences of parameters are discussed. As far as possible, our analysis is substantiated by new theoretical results. Furthermore, it is explained in detail how the bundle auxiliary problems are solved with a primal-dual interior point method. Such investigations have by now only been published for Bregman distances. The LQPAP bundle method is again applied to several test examples from literature. Thus, this thesis builds a bridge between theoretical and numerical investigations of solution methods for variational inequalities.
Why they rebel peacefully: On the violence-reducing effects of a positive attitude towards democracy
Under the impression of Europe’s drift into Nazism and Stalinism in the first half of the 20th century, social psychological research has focused strongly on dangers inherent in people’s attachment to a political system. The dissertation at hand contributes to a more differentiated perspective by examining violence-reducing aspects of political system attachment in four consecutive steps: First, it highlights attachment to a social group as a resource for violence prevention on an intergroup level. The results suggest that group attachment fosters self-control, a well-known protective factor against violence. Second, it demonstrates violence-reducing influences of attachment on a societal level. The findings indicate that attachment to a democracy facilitate peaceful and prevent violent protest tendencies. Third, it introduces the concept of political loyalty, defined as a positive attitude towards democracy, in order to clarify the different approaches of political system attachment. A set of three studies show the reliability and validity of a newly developed political loyalty questionnaire that distinguishes between affective and cognitive aspects. Finally, the dissertation differentiates former findings with regard to protest tendencies using the concept of political loyalty. A set of two experiments show that affective rather than cognitive aspects of political loyalty instigate peaceful protest tendencies and prevent violent ones. Implications of this dissertation for political engagement and peacebuilding as well as avenues for future research are discussed.
The development of our society contributed to increased occurrence of emerging substances (pesticides, pharmaceuticals, personal care products, etc.) in wastewater. Because of their potential hazard on ecosystems and humans, Wastewater Treatment Plants (WWTPs) need to adapt to better remove these compounds. Technology or policy development should however comply with sustainable development, e.g. based on Life Cycle Assessment (LCA) metrics. Nevertheless, the reliability or consistency of LCA results can sometimes be debatable. The main objective of this work was to explore how LCA can better support the implementation of innovative wastewater treatment options, in particular including removal benefits. The method was applied to support solutions for pharmaceuticals elimination from wastewater, regarding: (i) UV technology design, (ii) choice of advanced technology and (iii) centralized or decentralized treatment policy. The assessment approach followed by previous authors based on net impacts calculation seemed very promising to consider both environmental effects induced by treatment plant operation and environmental benefits obtained from pollutants removal. It was therefore applied to compare UV configuration types. LCA outcomes were consistent with degradation kinetics analysis. For the comparison of advanced technologies and policy scenarios, the common practice (net impacts based on EDIP method) was compared to other assessments, to better consider elimination benefits. First, USEtox consensus was applied for the avoided (eco)toxicity impacts, in combination with the recent method ReCiPe for generated impacts. Then, an eco-efficiency indicator (EFI) was developed to weigh the treatment efforts (generated impacts based on EDIP and ReCiPe methods) by the average removal efficiency (overcoming (eco)toxicity uncertainty issues). In total, the four types of comparative assessment showed the same trends: (i) ozonation and activated carbon perform better than UV irradiation, and (ii) no clear advantage distinguished between policy scenarios. It cannot be however concluded that advanced treatment of pharmaceuticals is not necessary because other criteria should be considered (risk assessment, bacterial resistance, etc.) and large uncertainties were embedded in calculations. Indeed, a significant part of this work was dedicated to the discussion of uncertainty and limitations of the LCA outcomes. At the inventory level, it was difficult to model technology operation at development stage. For impact assessment, the newly developed characterization factors for pharmaceuticals (eco)toxicity showed large uncertainties, mainly due to the lack of data and quality for toxicity tests. The use of information made available under REACH framework to develop CFs for detergent ingredients tried to cope with this issue but the benefits were limited due to the mismatch of information between REACH and USEtox method. The highlighted uncertainties were treated with sensitivity analyses to understand their effects on LCA results. This research work finally presents perspectives on the use of transparently generated data (technology inventory and (eco)toxicity factors) and further development of EFI indicator. Also, an accent is made on increasing the reliability of LCA outcomes, in particular through the implementation of advanced techniques for uncertainty management. To conclude, innovative technology/product development (e.g. based on circular economy approach) needs the involvement of all types of actors and the support from sustainability metrics.