When natural phenomena and data-based relations are driven by dynamics which are not purely local, they cannot be described satisfactorily by partial differential equations. As a consequence, mathematical models governed by nonlocal operators are of interest. This thesis is concerned with nonlocal operators of the form $\mathcal{L}u(x) = PV \int_{\mathbb{R}^d} (u(x)-u(y)) K(x,dy), x \in \mathbb{R}^d$, which are determined through a family of Borel measures $K=(K(x, \cdot))_{x \in \mathbb{R}^d}$ on $\mathbb{R}^d$ and which act on the vector space of Borel measurable functions $u: \mathbb{R}^d \rightarrow \mathbb{R}$. For a large class of families $K$, namely those where $K$ is a symmetric transition kernel satisfying a specific non-degeneracy condition, a variational theory for nonlocal equations of the type $\mathcal{L}u=f$ is established which builds upon gadgets from both measure theory and classical analysis. While measure theory is used to provide a nonlocal integration by parts formula that allows to set up a reasonable variational formulation of the above equation in dependency of the particular boundary condition (Dirichlet, Robin, Neumann) considered, Hilbert space theory and fixed-point approaches are utilized to develop sufficient conditions for the existence of variational solutions. This theory is then applied to two specific realizations of $\mathcal{L}$ of interest before a weak maximum principle is established which is finally used to study overlapping domain decomposition methods for the nonlocal and homogeneous Dirichlet problem.

Knapp 90 Jahre nach Erscheinen des Buchs von Paul Graindor zu den „Bustes et Statues-Portraits d'Egypte Romaine“ widmet sich mit der vorliegenden Dissertation erstmals wieder eine monographische Studie der marmornen Bildnisplastik der römischen Provinz Aegyptus von ihrer Gründung im Jahr 30 v. Chr. bis zum Ende des 3. Jhs. n. Chr. Basierend auf einer umfassenden Zusammenstellung bekannter, aber auch bislang unpublizierter Portraits sowie einer Neudokumentation zahlreicher Objekte gelingt erstmalig eine belastbare chronologische und typologische Auswertung dieser Bildnisse. Zwar bilden dabei die Darstellungen aus weißem Marmor die zentrale und auch quantitativ bei weitem größte Materialgruppe, doch es finden auch Bildnisse aus anderen Werkstoffen wie Bronze, Kalkstein, Gips oder Alabaster Berücksichtigung. Da die Provinz aufgrund geringer eigener Marmorvorkommen fast ausschließlich auf Importe angewiesen war, sind die Marmorbildnisse ein exzellentes Forschungsobjekt, um nicht nur den Handel von Marmor nach Ägypten und seine Distribution und Weiterverarbeitung in der Provinz zu untersuchen, sondern auch damit verbundene handwerkliche Besonderheiten, wie die häufig zu beobachtenden Ergänzungen mit Stuck- oder Steinelementen. Darüber hinaus werden auch Überlegungen zur Semantik des Materials sowie der Herkunft und dem Selbstverständnis der dargestellten Personen angestellt.

Case-Based Reasoning (CBR) is a symbolic Artificial Intelligence (AI) approach that has been successfully applied across various domains, including medical diagnosis, product configuration, and customer support, to solve problems based on experiential knowledge and analogy. A key aspect of CBR is its problem-solving procedure, where new solutions are created by referencing similar experiences, which makes CBR explainable and effective even with small amounts of data. However, one of the most significant challenges in CBR lies in defining and computing meaningful similarities between new and past problems, which heavily relies on domain-specific knowledge. This knowledge, typically only available through human experts, must be manually acquired, leading to what is commonly known as the knowledge-acquisition bottleneck. One way to mitigate the knowledge-acquisition bottleneck is through a hybrid approach that combines the symbolic reasoning strengths of CBR with the learning capabilities of Deep Learning (DL), a sub-symbolic AI method. DL, which utilizes deep neural networks, has gained immense popularity due to its ability to automatically learn from raw data to solve complex AI problems such as object detection, question answering, and machine translation. While DL minimizes manual knowledge acquisition by automatically training models from data, it comes with its own limitations, such as requiring large datasets, and being difficult to explain, often functioning as a "black box". By bringing together the symbolic nature of CBR and the data-driven learning abilities of DL, a neuro-symbolic, hybrid AI approach can potentially overcome the limitations of both methods, resulting in systems that are both explainable and capable of learning from data. The focus of this thesis is on integrating DL into the core task of similarity assessment within CBR, specifically in the domain of process management. Processes are fundamental to numerous industries and sectors, with process management techniques, particularly Business Process Management (BPM), being widely applied to optimize organizational workflows. Process-Oriented Case-Based Reasoning (POCBR) extends traditional CBR to handle procedural data, enabling applications such as adaptive manufacturing, where past processes are analyzed to find alternative solutions when problems arise. However, applying CBR to process management introduces additional complexity, as procedural cases are typically represented as semantically annotated graphs, increasing the knowledge-acquisition effort for both case modeling and similarity assessment. The key contributions of this thesis are as follows: It presents a method for preparing procedural cases, represented as semantic graphs, to be used as input for neural networks. Handling such complex, structured data represents a significant challenge, particularly given the scarcity of available process data in most organizations. To overcome the issue of data scarcity, the thesis proposes data augmentation techniques to artificially expand the process datasets, enabling more effective training of DL models. Moreover, it explores several deep learning architectures and training setups for learning similarity measures between procedural cases in POCBR applications. This includes the use of experience-based Hyperparameter Optimization (HPO) methods to fine-tune the deep learning models. Additionally, the thesis addresses the computational challenges posed by graph-based similarity assessments in CBR. The traditional method of determining similarity through subgraph isomorphism checks, which compare nodes and edges across graphs, is computationally expensive. To alleviate this issue, the hybrid approach seeks to use DL models to approximate these similarity calculations more efficiently, thus reducing the computational complexity involved in graph matching. The experimental evaluations of the corresponding contributions provide consistent results that indicate the benefits of using DL-based similarity measures and case retrieval methods in POCBR applications. The comparison with existing methods, e.g., based on subgraph isomorphism, shows several advantages but also some disadvantages of the compared methods. In summary, the methods and contributions outlined in this work enable more efficient and robust applications of hybrid CBR and DL in process management applications.

The goal of this work is to compare operators that are defined on probably varying Hilbert spaces. Distance concepts for operators as well as convergence concepts for such operators are explained and examined. For distance concepts we present three main notions. All have in common that they use space-linking operators that connect the spaces. At first, we look at unitary maps and compare the unitary orbits of the operators. Then, we consider isometric embeddings, which is based on a concept of Joachim Weidmann. Then we look at contractions but with more norm equations in comparison. The latter idea is based on a concept of Olaf Post called quasi-unitary equivalence. Our main result is that the unitary and isometric distances are equal provided the operators are both self-adjoint and have 0 in their essential spectra. In the third chapter, we focus specifically on the investigation of these distance terms for compact operators or operators in p-Schatten classes. In this case, the interpretation of the spectra as null sequences allows further distance investigation. Chapter four deals mainly with convergence terms of operators on varying Hilbert spaces. The analyses in this work deal exclusively with concepts of norm resolvent convergence. The main conclusion of the chapter is that the generalisation for norm resolvent convergence of Joachim Weidmann and the generalisation of Olaf Post, called quasi-unitary equivalence, are equivalent to each other. In addition, we specify error bounds and deal with the convergence speed of both concepts. Two important implications of these convergence notions are that the approximation is spectrally exact, i.e., the spectra converge suitably, and that the convergence is transferred to the functional calculus of the bounded functions vanishing at infinity.

Globalization significantly transforms labor markets. Advances in production technologies, transportation, and political integration reshape how and where goods and services are produced. Local economic conditions and diverse policy responses create varying speeds of change, affecting regions' attractiveness for living and working -- and promoting mobility. Competition for talent necessitates a deep understanding of why individuals choose specific destinations, how to ensure their effective labor market integration, and what workplace factors affect workers' well-being. This thesis focuses on two crucial aspects of labor market change -- Migration and workplace technological change. It contributes to our understanding of the determinants of labor mobility, the factors facilitating migrant integration, and the role of workplace automation for worker well-being. Chapter 2 investigates the relationship between minimum wages (MWs) and regional worker mobility in the EU. EU citizens are free to work anywhere in the common market, which allows them to take advantage of the significant variation in MWs across the EU. However, although MWs are set at the national level, it is also their local relevance that varies substantially -- depending on factors such as the share of affected workers or the extent to which they shift local compensation levels. These variations may attract workers from elsewhere, from within a country or from abroad. Analyzing regional variations in the Kaitz index, a measure of local MW impact, reveals that higher MWs can significantly increase inflows of low-skilled EU workers, particularly in central Europe. Chapter 3 examines the inequality in returns to skills experienced by immigrants, focusing on the role of linguistic proximity between migrants' origin and destination countries. Harmonized individual-level data from nine linguistically diverse migrant-hosting economies allows for an analysis of the wage gaps faced by immigrants from various origins, implicitly indicating how well they and their skills are integrated into the local labor markets. The analysis reveals that greater linguistic distance is associated with a higher wage penalty for highly skilled immigrants and a lower position in the wage distribution for those without tertiary education. Chapter 4 investigates an institutional factor potentially relevant for the integration of immigrants -- the labor market impact of Confucius Institutes (CIs), Chinese government-sponsored institutions that promote Chinese language and culture abroad. CIs have been found to foster trade and cultural exchange, indicating their potential relevance in shaping attitudes and trust of natives towards China and Chinese individuals. Examining the relationship between local CI presence and the wages of Chinese immigrants in local labor markets of the United States, the analysis reveals that CIs associate with significantly reduced wages for nearby residing Chinese immigrants. An event study demonstrates that the mere announcement of a new CI negatively impacts local wages for Chinese immigrants, independent of the CI's actual opening. Chapter 5 explores how working in automatable jobs affects life satisfaction in Germany. Following earlier literature, we classify occupations by potential for automation, and define the top third of occupations in this metric as \textit{automatable jobs}. We find workers in highly automatable jobs reporting a lower life satisfaction. Moreover, we detect a non-linearity, where workers in moderately automatable jobs (the second third of the distribution) experience a positive association with life satisfaction. Overall, the negative relationship of automation is most pronounced among younger and blue-collar workers, irrespective of the non-linearity.