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- Wissenssoziologie, Karl Mannheim, Paul Boghossian, science wars, Relationismus, Denkstile, Objektivierungsweisen, Wirklichkeits- und Wahrheitsverständnisse, Seinsgebundenheit, perspektivische Vorstrukturierung, Inkommensurabilität, Debatten (1)
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Sie haben eine spannende politische Diskussion in der Klasse. Das Gros Ihrer Schüler*innen ist wach, interessiert und engagiert. Alles läuft prima. Doch dann passiert's: Einer oder eine von ihnen stellt – absichtlich oder unreflektiert – eine extremistische oder verschwörungstheoretische Aussage in den Raum. Und nun?
Debatten führen nicht immer zu einem Konsens. Selbst die Vorlage von Beweisen bewirkt nicht immer eine Überzeugung der Gegenseite. Dies zeigt sich nicht nur in der Geschichte der Wissenschaften (vgl. Ludwik Fleck, Bruno Latour), sondern auch in der in unterschiedlichen Disziplinen geführten zeitgenössischen Debatte unter dem Label ‚science wars‘ zwischen einem Realismus und Konstruktivismus beziehungsweise Relativismus. Unterschiede in ihren Legitimierungen zeigen systematisch verschiedene Wirklichkeits- und Wahrheitsverständnisse, die sich aus den vom Seinsstandort der Perspektive abhängigen Grundannahmen konstituieren. Über einen wissenssoziologischen Zugriff wird es möglich die (sozio-)strukturlogische Konstitution von Perspektivität zu analysieren, die eine epistemologisch vorstrukturierte Revolvierung untereinander inkommensurabler Beiträge in der Debatte aufdeckt, was als Erklärung für ungelöste Debatten in Wissenschaft, Politik und Alltag überhaupt fungieren kann.
Die vorliegende Arbeit orientiert sich in ihrem Vorgehen an dem von Paul Boghossian veröffentlichten Werk ‚Angst vor der Wahrheit‘ als zeitgenössischen Vertreter eines Neuen Realismus. Hierbei werden zum einen den direkten Bezügen von Boghossian die Aussagen der kritisierten Perspektiven (v.a. Latour und Goodman) gegenübergestellt, als auch zum anderen weitere Spielarten eines Konstruktivismus (kognitionstheoretischer Konstruktivismus nach Maturana und Varela, soziologischer Konstruktivismus nach Berger und Luckmann, Wissenschaftssoziologie am Beispiel von Bloor und Latour, die Systemtheorie von Luhmann sowie postkonstruktivistische Positionen) in den Dimensionen ‚Wissensverständnis‘, ‚Subjektrelevanz‘ und ‚Einstellung zu einer naturalistischen Grundlage‘ vorgestellt. Es wird eine systematische und beidseitige Fehlinterpretation in der Debatte zwischen Realismus und Konstruktivismus sichtbar. Diese wird auf die Seinsgebundenheit von Perspektiven nach dem Verständnis einer mannheimschen Wissenssoziologie zurückgeführt. Anhand einer Rekonstruktion der Erkenntnistheorie des frühen Mannheims (1922: ‚Strukturanalyse der Erkenntnistheorie‘) wird die (sozio-)strukturlogische Konstitution erkenntnistheoretischer Elemente von Grundwissenschaften herausgearbeitet, wodurch denkstilgemäße Objektivierungen (und damit Wahrheitsverständnisse) unterschieden werden können. Diese Unterschiede erklären nicht nur die Inkommensurabilität von heterogenen Perspektiven in Debatten, sondern zeigen auf, dass das Aufeinandertreffen der Debattierenden vorstrukturiert sind. Der Ablauf einer Debatte ist soziostrukturell determiniert. Abschließend wird in der vorliegenden Arbeit diskutiert, inwiefern der verfahrenen Situation einer Debatte entgegengewirkt werden kann und auf welche Weise eine wissenssoziologische Analyse zu einem gegenseitigen Verständnis zwischen debattierenden Parteien beitragen kann.
Trotz des Rückgangs der Einwohner*innenzahl, kommunaler wohnungspolitischer Maßnahmen und der Pandemie steigen die Göttinger Mieten weiterhin. Besonders Menschen mit geringen Einkommen haben nach wie vor große Probleme, eine bezahlbare Wohnung in Göttingen zu finden. In diesem Wohnraumatlas zeigen wir die Entwicklung der Angebotsmieten auf. Zudem verdeutlichen wir, dass der Mietwohnungsmarkt in Teilmärkte segmentiert ist, für deren Identifizierung wir Ansätze liefern. Damit wollen wir stadtpolitisch Aktiven und anderen Interessierten Materialien an die Hand geben, um die Wohnungspolitik der Stadt einordnen zu können.
While humans find it easy to process visual information from the real world, machines struggle with this task due to the unstructured and complex nature of the information. Computer vision (CV) is the approach of artificial intelligence that attempts to automatically analyze, interpret, and extract such information. Recent CV approaches mainly use deep learning (DL) due to its very high accuracy. DL extracts useful features from unstructured images in a training dataset to use them for specific real-world tasks. However, DL requires a large number of parameters, computational power, and meaningful training data, which can be noisy, sparse, and incomplete for specific domains. Furthermore, DL tends to learn correlations from the training data that do not occur in reality, making DNNs poorly generalizable and error-prone.
Therefore, the field of visual transfer learning is seeking methods that are less dependent on training data and are thus more applicable in the constantly changing world. One idea is to enrich DL with prior knowledge. Knowledge graphs (KG) serve as a powerful tool for this purpose because they can formalize and organize prior knowledge based on an underlying ontological schema. They contain symbolic operations such as logic, rules, and reasoning, and can be created, adapted, and interpreted by domain experts. Due to the abstraction potential of symbols, KGs provide good prerequisites for generalizing their knowledge. To take advantage of the generalization properties of KG and the ability of DL to learn from large-scale unstructured data, attempts have long been made to combine explicit graph and implicit vector representations. However, with the recent development of knowledge graph embedding methods, where a graph is transferred into a vector space, new perspectives for a combination in vector space are opening up.
In this work, we attempt to combine prior knowledge from a KG with DL to improve visual transfer learning using the following steps: First, we explore the potential benefits of using prior knowledge encoded in a KG for DL-based visual transfer learning. Second, we investigate approaches that already combine KG and DL and create a categorization based on their general idea of knowledge integration. Third, we propose a novel method for the specific category of using the knowledge graph as a trainer, where a DNN is trained to adapt to a representation given by prior knowledge of a KG. Fourth, we extend the proposed method by extracting relevant context in the form of a subgraph of the KG to investigate the relationship between prior knowledge and performance on a specific CV task. In summary, this work provides deep insights into the combination of KG and DL, with the goal of making DL approaches more generalizable, more efficient, and more interpretable through prior knowledge.
Startups are essential agents for the evolution of economies and the creative destruction of established market conditions for the benefit of a more effective and efficient economy. Their significance is manifested in their drive for innovation and technological advancements, their creation of new jobs, their contribution to economic growth, and their impact on increased competition and increased market efficiency. By reason of their attributes of newness and smallness, startups often experience a limitation in accessing external financial resources. Extant research on entrepreneurial finance examines the capital structure of startups, various funding tools, financing environments in certain regions, and investor selection criteria among other topics. My dissertation contributes to this research area by examining the becoming increasingly important funding instrument of venture debt. Prior research on venture debt only investigated the business model of venture debt, the concept of venture debt, the selection criteria of venture debt providers, and the role of patents in the venture debt provider’s selection process. Based on qualitative and quantitative methods, the dissertation outlines the emergence of venture debt in Europe as well as the impact of venture debt on startups to open up a better understanding of venture debt.
The results of the qualitative studies indicate that venture debt was formed based on a ‘Kirznerian’ entrepreneurial opportunity and venture debt impacts startups positive and negative in their development via different impact mechanisms.
Based on these results, the dissertation analyzes the empirical impact of venture debt on a startup’s ability to acquire additional financial resources as well as the role of the reputation of venture debt providers. The results suggest that venture debt increases the likelihood of acquiring additional financial resources via subsequent funding rounds and trade sales. In addition, a higher venture debt provider reputation increases the likelihood of acquiring additional financial resources via IPOs.
THE NONLOCAL NEUMANN PROBLEM
(2023)
Instead of presuming only local interaction, we assume nonlocal interactions. By doing so, mass
at a point in space does not only interact with an arbitrarily small neighborhood surrounding it,
but it can also interact with mass somewhere far, far away. Thus, mass jumping from one point to
another is also a possibility we can consider in our models. So, if we consider a region in space, this
region interacts in a local model at most with its closure. While in a nonlocal model this region may
interact with the whole space. Therefore, in the formulation of nonlocal boundary value problems
the enforcement of boundary conditions on the topological boundary may not suffice. Furthermore,
choosing the complement as nonlocal boundary may work for Dirichlet boundary conditions, but
in the case of Neumann boundary conditions this may lead to an overfitted model.
In this thesis, we introduce a nonlocal boundary and study the well-posedness of a nonlocal Neu-
mann problem. We present sufficient assumptions which guarantee the existence of a weak solution.
As in a local model our weak formulation is derived from an integration by parts formula. However,
we also study a different weak formulation where the nonlocal boundary conditions are incorporated
into the nonlocal diffusion-convection operator.
After studying the well-posedness of our nonlocal Neumann problem, we consider some applications
of this problem. For example, we take a look at a system of coupled Neumann problems and analyze
the difference between a local coupled Neumann problems and a nonlocal one. Furthermore, we let
our Neumann problem be the state equation of an optimal control problem which we then study. We
also add a time component to our Neumann problem and analyze this nonlocal parabolic evolution
equation.
As mentioned before, in a local model mass at a point in space only interacts with an arbitrarily
small neighborhood surrounding it. We analyze what happens if we consider a family of nonlocal
models where the interaction shrinks so that, in limit, mass at a point in space only interacts with
an arbitrarily small neighborhood surrounding it.
People are increasingly concerned about how meat affects the environment, human health, and animal welfare, yet eating and enjoying meat remains a norm. Unsurprisingly, many people are ambivalent about meat—evaluating it as both positive and negative. Here, we propose that meat-related conflict is multidimensional and depends on people’s dietary group: Omnivores’ felt ambivalence relates to multiple negative associations that oppose a predominantly positive attitude towards meat, and veg*ans’ ambivalence relates to various positive associations that oppose a predominantly negative attitude. A qualitative study (N = 235; German) revealed that omnivores and veg*ans experience meat-related ambivalence due to associations with animals, sociability, sustainability, health, and sensory experiences. To quantify felt ambivalence in these domains, we developed the Meat Ambivalence Questionnaire (MAQ). We validated the MAQ in four pre-registered studies using self-report and behavioral data (N = 3,485; German, UK, representative US). Both omnivores and veg*ans reported meat-related ambivalence, but with differences across domains and their consequences for meat consumption. Specifically, ambivalence was associated with less meat consumption in omnivores (especially sensory-/animal-based ambivalence) and more meat consumption in veg*ans (especially sensory-/socially-based ambivalence). Network analyses shed further light on the nomological net of the MAQ while controlling for a comprehensive set of determinants of meat consumption. By introducing the MAQ, we hope to provide researchers with a tool to better understand how ambivalence accompanies behavior change and maintenance.
The benefits of prosocial power motivation in leadership: Action orientation fosters a win-win
(2023)
Power motivation is considered a key component of successful leadership. Based on its dualistic nature, the need for power (nPower) can be expressed in a dominant or a prosocial manner. Whereas dominant motivation is associated with antisocial behaviors, prosocial motivation is characterized by more benevolent actions (e.g., helping, guiding). Prosocial enactment of the power motive has been linked to a wide range of beneficial outcomes, yet less has been investigated what determines a prosocial enactment of the power motive. According to Personality Systems Interactions (PSI) theory, action orientation (i.e., the ability to self-regulate affect) promotes prosocial enactment of the implicit power motive and initial findings within student samples verify this assumption. In the present study, we verified the role of action orientation as an antecedent for prosocial power enactment in a leadership sample (N = 383). Additionally, we found that leaders personally benefited from a prosocial enactment strategy. Results show that action orientation through prosocial power motivation leads to reduced power-related anxiety and, in turn, to greater leader well-being. The integration of motivation and self-regulation research reveals why leaders enact their power motive in a certain way and helps to understand how to establish a win-win situation for both followers and leaders.
Striving for sustainable development by combating climate change and creating a more social world is one of the most pressing issues of our time. Growing legal requirements and customer expectations require also Mittelstand firms to address sustainability issues such as climate change. This dissertation contributes to a better understanding of sustainability in the Mittelstand context by examining different Mittelstand actors and the three dimensions of sustainability - social, economic, and environmental sustainability - in four quantitative studies. The first two studies focus on the social relevance and economic performance of hidden champions, a niche market leading subgroup of Mittelstand firms. At the regional level, the impact of 1,645 hidden champions located in Germany on various dimensions of regional development is examined. A higher concentration of hidden champions has a positive effect on regional employment, median income, and patents. At the firm level, analyses of a panel dataset of 4,677 German manufacturing firms, including 617 hidden champions, show that the latter have a higher return on assets than other Mittelstand firms. The following two chapters deal with environmental strategies and thus contribute to the exploration of the environmental dimension of sustainability. First, the consideration of climate aspects in investment decisions is compared using survey data from 468 European venture capital and private equity investors. While private equity firms respond to external stakeholders and portfolio performance and pursue an active ownership strategy, venture capital firms are motivated by product differentiation and make impact investments. Finally, based on survey data from 443 medium-sized manufacturing firms in Germany, 54% of which are family-owned, the impact of stakeholder pressures on their decarbonization strategies is analyzed. A distinction is made between symbolic (compensation of CO₂-emissions) and substantive decarbonization strategies (reduction of CO₂-emissions). Stakeholder pressures lead to a proactive pursuit of decarbonization strategies, with internal and external stakeholders varying in their influence on symbolic and substantial decarbonization strategies, and the relationship influenced by family ownership.
COVID-19 was a harsh reminder that diseases are an aspect of human existence and mortality. It was also a live experiment in the formation and alteration of disease-related attitudes. Not only are these attitudes relevant to an individual’s self-protective behavior, but they also seem to be associated with social and political attitudes more broadly. One of these attitudes is Social Darwinism, which holds that a pandemic benefits society by enabling nature “to weed out the weak”. In two countries (N = 300, N = 533), we introduce and provide evidence for the reliability, validity, and usefulness of the Disease-Related Social Darwinism (DRSD) Short Scale measuring this concept. Results indicate that DRSD is meaningful related to other central political attitudes like Social Dominance Orientation, Authoritarianism and neoliberalism. Importantly, the scale significantly predicted people’s protective behavior during the Pandemic over and above general social Darwinism. Moreover, it significantly predicted conservative attitudes, even after controlling for Social Dominance Orientation.
The COVID-19 pandemic has affected schooling worldwide. In many places, schools closed for weeks or months, only part of the student body could be educated at any one time, or students were taught online. Previous research discloses the relevance of schooling for the development of cognitive abilities. We therefore compared the intelligence test performance of 424 German secondary school students in Grades 7 to 9 (42% female) tested after the first six months of the COVID-19 pandemic (i.e., 2020 sample) to the results of two highly comparable student samples tested in 2002 (n = 1506) and 2012 (n = 197). The results revealed substantially and significantly lower intelligence test scores in the 2020 sample than in both the 2002 and 2012 samples. We retested the 2020 sample after another full school year of COVID-19-affected schooling in 2021. We found mean-level changes of typical magnitude, with no signs of catching up to previous cohorts or further declines in cognitive performance. Perceived stress during the pandemic did not affect changes in intelligence test results between the two measurements.
Non-probability sampling is a topic of growing relevance, especially due to its occurrence in the context of new emerging data sources like web surveys and Big Data.
This thesis addresses statistical challenges arising from non-probability samples, where unknown or uncontrolled sampling mechanisms raise concerns in terms of data quality and representativity.
Various methods to quantify and reduce the potential selectivity and biases of non-probability samples in estimation and inference are discussed. The thesis introduces new forms of prediction and weighting methods, namely
a) semi-parametric artificial neural networks (ANNs) that integrate B-spline layers with optimal knot positioning in the general structure and fitting procedure of artificial neural networks, and
b) calibrated semi-parametric ANNs that determine weights for non-probability samples by integrating an ANN as response model with calibration constraints for totals, covariances and correlations.
Custom-made computational implementations are developed for fitting (calibrated) semi-parametric ANNs by means of stochastic gradient descent, BFGS and sequential quadratic programming algorithms.
The performance of all the discussed methods is evaluated and compared for a bandwidth of non-probability sampling scenarios in a Monte Carlo simulation study as well as an application to a real non-probability sample, the WageIndicator web survey.
Potentials and limitations of the different methods for dealing with the challenges of non-probability sampling under various circumstances are highlighted. It is shown that the best strategy for using non-probability samples heavily depends on the particular selection mechanism, research interest and available auxiliary information.
Nevertheless, the findings show that existing as well as newly proposed methods can be used to ease or even fully counterbalance the issues of non-probability samples and highlight the conditions under which this is possible.
Emotionen gelten als Spiegelbild unserer persönlichen Bedürfnislage. Insbesondere in Konflikt- oder Mediationsgesprächen ist es demnach wichtig, nicht nur über den Moment zu sprechen, an dem ein Streit entstanden ist, sondern auch Bedürfnisse und Gefühle aufzudecken, die unser Handeln, Denken und Fühlen beeinflusst haben. Die folgenden Materialien zeigen, wie man als Lehrkraft Emotionen und Streit mit Grundschulkindern behandeln kann.
Family firms play a crucial role in the DACH region (Germany, Austria, Switzerland). They are characterized by a long tradition, a strong connection to the region, and a well-established network. However, family firms also face challenges, especially in finding a suitable successor. Wealthy entrepreneurial families are increasingly opting to establish Single Family Offices (SFOs) as a solution to this challenge. An SFO takes on the management and protection of family wealth. Its goal is to secure and grow the wealth over generations. In Germany alone, there are an estimated 350 to 450 SFOs, with 70% of them being established after the year 2000. However, research on SFOs is still in its early stages, particularly regarding the role of SFOs as firm owners. This dissertation delves into an exploration of SFOs through four quantitative empirical studies. The first study provides a descriptive overview of 216 SFOs from the DACH-region. Findings reveal that SFOs exhibit a preference for investing in established companies and real estate. Notably, only about a third of SFOs engage in investments in start-ups. Moreover, SFOs as a group are heterogeneous. Categorizing them into three groups based on their relationship with the entrepreneurial family and the original family firm reveals significant differences in their asset allocation strategies. Subsequent studies in this dissertation leverage a hand-collected sample of 173 SFO-owned firms from the DACH region, meticulously matched with 684 family-owned firms from the same region. The second study focusing on financial performance indicates that SFO-owned firms tend to exhibit comparatively poorer financial performance than family-owned firms. However, when members of the SFO-owning family hold positions on the supervisory or executive board of the firm, there's a notable improvement. The third study, concerning cash holdings, reveals that SFO-owned firms maintain a higher cash holding ratio compared to family-owned firms. Notably, this effect is magnified when the SFO has divested its initial family firms. Lastly, the fourth study regarding capital structure highlights that SFO-owned firms tend to display a higher long-term debt ratio than family-owned firms. This suggests that SFO-owned firms operate within a trade-off theory framework, like private equity-owned firms. Furthermore, this effect is stronger for SFOs that sold their original family firm. The outcomes of this research are poised to provide entrepreneurial families with a practical guide for effectively managing and leveraging SFOs as a strategic long-term instrument for succession and investment planning.
Coastal erosion describes the displacement of land caused by destructive sea waves,
currents or tides. Due to the global climate change and associated phenomena such as
melting polar ice caps and changing current patterns of the oceans, which result in rising
sea levels or increased current velocities, the need for countermeasures is continuously
increasing. Today, major efforts have been made to mitigate these effects using groins,
breakwaters and various other structures.
This thesis will find a novel approach to address this problem by applying shape optimization
on the obstacles. Due to this reason, results of this thesis always contain the
following three distinct aspects:
The selected wave propagation model, i.e. the modeling of wave propagation towards
the coastline, using various wave formulations, ranging from steady to unsteady descriptions,
described from the Lagrangian or Eulerian viewpoint with all its specialties. More
precisely, in the Eulerian setting is first a steady Helmholtz equation in the form of a
scattering problem investigated and followed subsequently by shallow water equations,
in classical form, equipped with porosity, sediment portability and further subtleties.
Secondly, in a Lagrangian framework the Lagrangian shallow water equations form the
center of interest.
The chosen discretization, i.e. dependent on the nature and peculiarity of the constraining
partial differential equation, we choose between finite elements in conjunction
with a continuous Galerkin and discontinuous Galerkin method for investigations in the
Eulerian description. In addition, the Lagrangian viewpoint offers itself for mesh-free,
particle-based discretizations, where smoothed particle hydrodynamics are used.
The method for shape optimization w.r.t. the obstacle’s shape over an appropriate
cost function, constrained by the solution of the selected wave-propagation model. In
this sense, we rely on a differentiate-then-discretize approach for free-form shape optimization
in the Eulerian set-up, and reverse the order in Lagrangian computations.
Anmerkung: Es handelt sich um die 1. Auflage der Dissertation.
2. überarbeitete Auflage siehe:
"https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2166".
Ausgangspunkt der politisch-ikonographischen Untersuchung, in deren Zentrum zwei Staatsporträts König Maximilians II. von Bayern stehen, ist die Beobachtung, dass diese beiden Bildnisse grundsätzlich unterschiedliche Inszenierungsformen wählen. Das erste von Max Hailer gefertigte Werk zeigt Maximilian II. im vollen bayerischen Krönungsornat und greift eine tradierte Darstellungsweise im Staatsporträt auf. Es entstand zwei Jahre nach Maximilians II. Thronbesteigung und damit nach den revolutionären Unruhen der Jahre 1848/49 im Jahr 1850. Das zweite wurde von Joseph Bernhardt 1857 bis 1858 gemalt und im Jahr 1858 zum zehnjährigen Thronjubiläum des Monarchen erstmals präsentiert. Die Inszenierung ändert sich im zweiten Bildnis: Das bayerische Krönungsornat ist der Generalsuniform gewichen, ebenso weitere Details, die sich noch in der ersten Darstellung finden: Draperie und Wappen fehlen, der übliche bayerisch-königliche Thronsessel ist durch einen anderen ersetzt. In den Hintergrund gedrängt ist die Verfassung, immerhin seit 1818 staatliche Rechtsgrundlage des bayerischen Königreichs. Die beiden Staatsporträts Maximilians II. leiten offensichtlich von den Herrscherbildnissen im vollen bayerischen Krönungsornat seines Großvaters Maximilian I. und Vaters Ludwig I. über zu einer solchen in Uniform mit Krönungsmantel wie sie sich bei Napoleon III. und Friedrich Wilhelm IV. finden und wie sie sein Sohn Ludwig II. weiterführte. Es stellt sich somit die Frage, welche Faktoren zu diesem prägnanten Wandel in der Inszenierung Maximilians II. als König von Bayern führten. Die Arbeit geht der These nach, dass beide Darstellungen grundlegend auf eine reaktionäre, gegen die Revolution 1848/49 gerichtete Politik ausgelegt sind, wobei dieser reaktionäre Charakter in Maximilians II. Bildnis von 1858 noch eine Steigerung im Vergleich zu derjenigen von 1850 erfährt. Zudem wandelt sich die innenpolitisch-historische Ausrichtung des ersten Porträts bei der zweiten Darstellung des bayerischen Monarchen in eine außenpolitisch-progressive. Die Legitimation Maximilians II. begründet sich nicht mehr, wie bei ersterem, in der Geschichte und der Herrschaft der Wittelsbacher, sondern in seinen eigenen Errungenschaften und seiner eigenen Herrschaft. Dieser Wechsel der politischen Bildaussage fußt sowohl auf den politischen Veränderungen und Entwicklungen innerhalb und außerhalb Bayerns als auch auf der Entwicklung des Staatsporträts in der Mitte des 19. Jahrhunderts. Nach nur zehn Jahren wird so eine veränderte Botschaft über Maximilians II. Position und Machtanspruch ausgesendet.
Modern decision making in the digital age is highly driven by the massive amount of
data collected from different technologies and thus affects both individuals as well as
economic businesses. The benefit of using these data and turning them into knowledge
requires appropriate statistical models that describe the underlying observations well.
Imposing a certain parametric statistical model goes along with the need of finding
optimal parameters such that the model describes the data best. This often results in
challenging mathematical optimization problems with respect to the model’s parameters
which potentially involve covariance matrices. Positive definiteness of covariance matrices
is required for many advanced statistical models and these constraints must be imposed
for standard Euclidean nonlinear optimization methods which often results in a high
computational effort. As Riemannian optimization techniques proved efficient to handle
difficult matrix-valued geometric constraints, we consider optimization over the manifold
of positive definite matrices to estimate parameters of statistical models. The statistical
models treated in this thesis assume that the underlying data sets used for parameter
fitting have a clustering structure which results in complex optimization problems. This
motivates to use the intrinsic geometric structure of the parameter space. In this thesis,
we analyze the appropriateness of Riemannian optimization over the manifold of positive
definite matrices on two advanced statistical models. We establish important problem-
specific Riemannian characteristics of the two problems and demonstrate the importance
of exploiting the Riemannian geometry of covariance matrices based on numerical studies.
Even though proper research on Cauchy transforms has been done, there are still a lot of open questions. For example, in the case of representation theorems, i.e. the question when a function can be represented as a Cauchy transform, there is 'still no completely satisfactory answer' ([9], p. 84). There are characterizations for measures on the circle as presented in the monograph [7] and for general compactly supported measures on the complex plane as presented in [27]. However, there seems to exist no systematic treatise of the Cauchy transform as an operator on $L_p$ spaces and weighted $L_p$ spaces on the real axis.
This is the point where this thesis draws on and we are interested in developing several characterizations for the representability of a function by Cauchy transforms of $L_p$ functions. Moreover, we will attack the issue of integrability of Cauchy transforms of functions and measures, a topic which is only partly explored (see [43]). We will develop different approaches involving Fourier transforms and potential theory and investigate into sufficient conditions and characterizations.
For our purposes, we shall need some notation and the concept of Hardy spaces which will be part of the preliminary Chapter 1. Moreover, we introduce Fourier transforms and their complex analogue, namely Fourier-Laplace transforms. This will be of extraordinary usage due to the close connection of Cauchy and Fourier(-Laplace) transforms.
In the second chapter we shall begin our research with a discussion of the Cauchy transformation on the classical (unweighted) $L_p$ spaces. Therefore, we start with the boundary behavior of Cauchy transforms including an adapted version of the Sokhotski-Plemelj formula. This result will turn out helpful for the determination of the image of the Cauchy transformation under $L_p(\R)$ for $p\in(1,\infty).$ The cases $p=1$ and $p=\infty$ are playing special roles here which justifies a treatise in separate sections. For $p=1$ we will involve the real Hardy space $H_{1}(\R)$ whereas the case $p=\infty$ shall be attacked by an approach incorporating intersections of Hardy spaces and certain subspaces of $L_{\infty}(\R).$
The third chapter prepares ourselves for the study of the Cauchy transformation on subspaces of $L_{p}(\R).$ We shall give a short overview of the basic facts about Cauchy transforms of measures and then proceed to Cauchy transforms of functions with support in a closed set $X\subset\R.$ Our goal is to build up the main theory on which we can fall back in the subsequent chapters.
The fourth chapter deals with Cauchy transforms of functions and measures supported by an unbounded interval which is not the entire real axis. For convenience we restrict ourselves to the interval $[0,\infty).$ Bringing once again the Fourier-Laplace transform into play, we deduce complex characterizations for the Cauchy transforms of functions in $L_{2}(0,\infty).$ Moreover, we analyze the behavior of Cauchy transform on several half-planes and shall use these results for a fairly general geometric characterization. In the second section of this chapter, we focus on Cauchy transforms of measures with support in $[0,\infty).$ In this context, we shall derive a reconstruction formula for these Cauchy transforms holding under pretty general conditions as well as results on the behaviur on the left half-plane. We close this chapter by rather technical real-type conditions and characterizations for Cauchy transforms of functions in $L_p(0,\infty)$ basing on an approach in [82].
The most common case of Cauchy transforms, those of compactly supported functions or measures, is the subject of Chapter 5. After complex and geometric characterizations originating from similar ideas as in the fourth chapter, we adapt a functional-analytic approach in [27] to special measures, namely those with densities to a given complex measure $\mu.$ The chapter is closed with a study of the Cauchy transformation on weighted $L_p$ spaces. Here, we choose an ansatz through the finite Hilbert transform on $(-1,1).$
The sixth chapter is devoted to the issue of integrability of Cauchy transforms. Since this topic has no comprehensive treatise in literature yet, we start with an introduction of weighted Bergman spaces and general results on the interaction of the Cauchy transformation in these spaces. Afterwards, we combine the theory of Zen spaces with Cauchy transforms by using once again their connection with Fourier transforms. Here, we shall encounter general Paley-Wiener theorems of the recent past. Lastly, we attack the issue of integrability of Cauchy transforms by means of potential theory. Therefore, we derive a Fourier integral formula for the logarithmic energy in one and multiple dimensions and give applications to Fourier and hence Cauchy transforms.
Two appendices are annexed to this thesis. The first one covers important definitions and results from measure theory with a special focus on complex measures. The second appendix contains Cauchy transforms of frequently used measures and functions with detailed calculations.