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In this thesis we study a variant of the quadrature problem for stochastic differential equations (SDEs), namely the approximation of expectations \(\mathrm{E}(f(X))\), where \(X = (X(t))_{t \in [0,1]}\) is the solution of an SDE and \(f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R}\) is a functional, mapping each realization of \(X\) into the real numbers. The distinctive feature in this work is that we consider randomized (Monte Carlo) algorithms with random bits as their only source of randomness, whereas the algorithms commonly studied in the literature are allowed to sample from the uniform distribution on the unit interval, i.e., they do have access to random numbers from \([0,1]\).
By assumption, all further operations like, e.g., arithmetic operations, evaluations of elementary functions, and oracle calls to evaluate \(f\) are considered within the real number model of computation, i.e., they are carried out exactly.
In the following, we provide a detailed description of the quadrature problem, namely we are interested in the approximation of
\begin{align*}
S(f) = \mathrm{E}(f(X))
\end{align*}
for \(X\) being the \(r\)-dimensional solution of an autonomous SDE of the form
\begin{align*}
\mathrm{d}X(t) = a(X(t)) \, \mathrm{d}t + b(X(t)) \, \mathrm{d}W(t), \quad t \in [0,1],
\end{align*}
with deterministic initial value
\begin{align*}
X(0) = x_0 \in \mathbb{R}^r,
\end{align*}
and driven by a \(d\)-dimensional standard Brownian motion \(W\). Furthermore, the drift coefficient \(a \colon \mathbb{R}^r \to \mathbb{R}^r\) and the diffusion coefficient \(b \colon \mathbb{R}^r \to \mathbb{R}^{r \times d}\) are assumed to be globally Lipschitz continuous.
For the function classes
\begin{align*}
F_{\infty} = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{\sup}\bigr\}
\end{align*}
and
\begin{align*}
F_p = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{L_p}\bigr\}, \quad 1 \leq p < \infty.
\end{align*}
we have established the following.
\[\]
\(\textit{Theorem 1.}\)
There exists a random bit multilevel Monte Carlo (MLMC) algorithm \(M\) using
\[
L = L(\varepsilon,F) = \begin{cases}\lceil{\log_2(\varepsilon^{-2}}\rceil, &\text{if} \ F = F_p,\\
\lceil{\log_2(\varepsilon^{-2} + \log_2(\log_2(\varepsilon^{-1}))}\rceil, &\text{if} \ F = F_\infty
\end{cases}
\]
and replication numbers
\[
N_\ell = N_\ell(\varepsilon,F) = \begin{cases}
\lceil{(L+1) \cdot 2^{-\ell} \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F = F_p,\\
\lceil{(L+1) \cdot 2^{-\ell} \cdot \max(\ell,1) \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F=f_\infty
\end{cases}
\]
for \(\ell = 0,\ldots,L\), for which exists a positive constant \(c\) such that
\begin{align*}
\mathrm{error}(M,F) = \sup_{f \in F} \bigl(\mathrm{E}(S(f) - M(f))^2\bigr)^{1/2} \leq c \cdot \varepsilon
\end{align*}
and
\begin{align*}
\mathrm{cost}(M,F) = \sup_{f \in F} \mathrm{E}(\mathrm{cost}(M,f)) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases}
(\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\
(\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty
\end{cases}
\end{align*}
for every \(\varepsilon \in {]0,1/2[}\).
\[\]
Hence, in terms of the \(\varepsilon\)-complexity
\begin{align*}
\mathrm{comp}(\varepsilon,F) = \inf\bigl\{\mathrm{cost}(M,F) \colon M \ \text{is a random bit MC algorithm}, \mathrm{error}(M,F) \leq \varepsilon\bigr\}
\end{align*}
we have established the upper bound
\begin{align*}
\mathrm{comp}(\varepsilon,F) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases}
(\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\
(\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty
\end{cases}
\end{align*}
for some positive constant \(c\). That is, we have shown the same weak asymptotic upper bound as in the case of random numbers from \([0,1]\). Hence, in this sense, random bits are almost as powerful as random numbers for our computational problem.
Moreover, we present numerical results for a non-analyzed adaptive random bit MLMC Euler algorithm, in the particular cases of the Brownian motion, the geometric Brownian motion, the Ornstein-Uhlenbeck SDE and the Cox-Ingersoll-Ross SDE. We also provide a numerical comparison to the corresponding adaptive random number MLMC Euler method.
A key challenge in the analysis of the algorithm in Theorem 1 is the approximation of probability distributions by means of random bits. A problem very closely related to the quantization problem, i.e., the optimal approximation of a given probability measure (on a separable Hilbert space) by means of a probability measure with finite support size.
Though we have shown that the random bit approximation of the standard normal distribution is 'harder' than the corresponding quantization problem (lower weak rate of convergence), we have been able to establish the same weak rate of convergence as for the corresponding quantization problem in the case of the distribution of a Brownian bridge on \(L_2([0,1])\), the distribution of the solution of a scalar SDE on \(L_2([0,1])\), and the distribution of a centered Gaussian random element in a separable Hilbert space.

Activity recognition has continued to be a large field in computer science over the last two decades. Research questions from 15 years ago have led to solutions that today support our daily lives. Specifically, the success of smartphones or more recent developments of other smart devices (e.g., smart-watches) is rooted in applications that leverage on activity analysis and location tracking (fitness applications and maps). Today we can track our physical health and fitness and support our physical needs by merely owning (and using) a smart-phone. Still, the quality of our lives does not solely rely on fitness and physical health but also more increasingly on our mental well-being. Since we have learned how practical and easy it is to have a lot of functions, including health support on just one device, it would be specifically helpful if we could also use the smart-phone to support our mental and cognitive health if need be.
The ultimate goal of this work is to use sensor-assisted location and motion analysis to support various aspects of medically valid cognitive assessments.
In this regard, this thesis builds on Hypothesis 3: Sensors in our ubiquitous environment can collect information about our cognitive state, and it is possible to extract that information. In addition, these data can be used to derive complex cognitive states and to predict possible pathological changes in humans. After all, not only is it possible to determine the cognitive state through sensors but also to assist people in difficult situations through these sensors.
Thus, in the first part, this thesis focuses on the detection of mental state and state changes.
The primary purpose is to evaluate possible starting points for sensor systems in order to enable a clinically accurate assessment of mental states. These assessments must work on the condition that a developed system must be able to function within the given limits of a real clinical environment.
Despite the limitations and challenges of real-life deployments, it was possible to develop methods for determining the cognitive state and well-being of the residents. The analysis of the location data provides a correct classification of cognitive state with an average accuracy of 70% to 90%.
Methods to determine the state of bipolar patients provide an accuracy of 70-80\% for the detection of different cognitive states (total seven classes) using single sensors and 76% for merging data from different sensors. Methods for detecting the occurrence of state changes, a highlight of this work, even achieved a precision and recall of 95%.
The comparison of these results with currently used standard methods in psychiatric care even shows a clear advantage of the sensor-based method. The accuracy of the sensor-based analysis is 60% higher than the accuracy of the currently used methods.
The second part of this thesis introduces methods to support people’s actions in stressful situations on the one hand and analyzes the interaction between people during high-pressure activities on the other.
A simple, acceleration based, smartwatch instant feedback application was used to help laypeople to learn to perform CPR (cardiopulmonary resuscitation) in an emergency on the fly.
The evaluation of this application in a study with 43 laypersons showed an instant improvement in the CPR performance of 50%. An investigation of whether training with such an instant feedback device can support improved learning and lead to more permanent effects for gaining skills was able to confirm this theory.
Last but not least, with the main interest shifting from the individual to a group of people at the end of this work, the question: how can we determine the interaction between individuals within a group of people? was answered by developing a methodology to detect un-voiced collaboration in random ad-hoc groups. An evaluation with data retrieved from video footage provides an accuracy of up to more than 95%, and even with artificially introduced errors rates of 20%, still an accuracy of 70% precision, and 90% recall can be achieved.
All scenarios in this thesis address different practical issues of today’s health care. The methods developed are based on real-life datasets and real-world studies.

Biological clocks exist across all life forms and serve to coordinate organismal physiology with periodic environmental changes. The underlying mechanism of these clocks is predominantly based on cellular transcription-translation feedback loops in which clock proteins mediate the periodic expression of numerous genes. However, recent studies point to the existence of a conserved timekeeping mechanism independent of cellular transcription and translation, but based on cellular metabolism. These metabolic clocks were concluded based upon the observation of circadian and ultradian oscillations in the level of hyperoxidized peroxiredoxin proteins. Peroxiredoxins are enzymes found almost ubiquitously throughout life. Originally identified as H2O2 scavengers, recent studies show that peroxiredoxins can transfer oxidation to, and thereby regulate, a wide range of cellular proteins. Thus, it is conceivable that peroxiredoxins, using H2O2 as the primary signaling molecule, have the potential to integrate and coordinate much of cellular physiology and behavior with metabolic changes. Nonetheless, it remained unclear if peroxiredoxins are passive reporters of metabolic clock activity or active determinants of cellular timekeeping. Budding yeast possess an ultradian metabolic clock termed the Yeast Metabolic Cycle (YMC). The most obvious feature of the YMC is a high amplitude oscillation in oxygen consumption. Like circadian clocks, the YMC temporally compartmentalizes cellular processes (e.g. metabolism) and coordinates cellular programs such as gene expression and cell division. The YMC also exhibits oscillations in the level of hyperoxidized peroxiredoxin proteins.
In this study, I used the YMC clock model to investigate the role of peroxiredoxins in cellular timekeeping, as well as the coordination of cell division with the metabolic clock. I observed that cytosolic 2-Cys peroxiredoxins are essential for robust metabolic clock function. I provide direct evidence for oscillations in cytosolic H2O2 levels, as well as cyclical changes in oxidation state of a peroxiredoxin and a model peroxiredoxin target protein during the YMC. I noted two distinct metabolic states during the YMC: low oxygen consumption (LOC) and high oxygen consumption (HOC). I demonstrate that thiol-disulfide oxidation and reduction are necessary for switching between LOC and HOC. Specifically, a thiol reductant promotes switching to HOC, whilst a thiol oxidant prevents switching to HOC, forcing cells to remain in LOC. Transient peroxiredoxin inactivation triggered rapid and premature switching from LOC to HOC. Furthermore, I show that cell division is normally synchronized with the YMC and that deletion of typical 2-Cys peroxiredoxins leads to complete uncoupling of cell division from metabolic cycling. Moreover, metabolic oscillations are crucial for regulating cell cycle entry and exit. Intriguingly, switching to HOC is crucial for initiating cell cycle entry whilst switching to LOC is crucial for cell cycle completion and exit. Consequently, forcing cells to remain in HOC by application of a thiol reductant leads to multiple rounds of cell cycle entry despite failure to complete the preceding cell cycle. On the other hand, forcing cells to remain in LOC by treating with a thiol oxidant prevents initiation of cell cycle entry.
In conclusion, I propose that peroxiredoxins – by controlling metabolic cycles, which are in turn crucial for regulating the progression through cell cycle – play a central role in the coordination of cellular metabolism with cell division. This proposition, thus, positions peroxiredoxins as active players in the cellular timekeeping mechanism.

Diversification is one of the main pillars of investment strategies. The prominent 1/N portfolio, which puts equal weight on each asset is, apart from its simplicity, a method which is hard to outperform in realistic settings, as many studies have shown. However, depending on the number of considered assets, this method can lead to very large portfolios. On the other hand, optimization methods like the mean-variance portfolio suffer from estimation errors, which often destroy the theoretical benefits. We investigate the performance of the equal weight portfolio when using fewer assets. For this we explore different naive portfolios, from selecting the best Sharpe ratio assets to exploiting knowledge about correlation structures using clustering methods. The clustering techniques separate the possible assets into non-overlapping clusters and the assets within a cluster are ordered by their Sharpe ratio. Then the best asset of each portfolio is chosen to be a member of the new portfolio with equal weights, the cluster portfolio. We show that this portfolio inherits the advantages of the 1/N portfolio and can even outperform it empirically. For this we use real data and several simulation models. We prove these findings from a statistical point of view using the framework by DeMiguel, Garlappi and Uppal (2009). Moreover, we show the superiority regarding the Sharpe ratio in a setting, where in each cluster the assets are comonotonic. In addition, we recommend the consideration of a diversification-risk ratio to evaluate the performance of different portfolios.

In an overall effort to contribute to the steadily expanding EO literature, this cumulative dissertation aims to help the literature to advance with greater clarity, comprehensive modeling, and more robust research designs. To achieve this, the first paper of this dissertation focuses on the consistency and coherence in variable choices and modeling considerations by conducting a systematic quantitative review of the EO-performance literature. Drawing on the plethora of previous EO studies, the second paper employs a comprehensive meta-analytic structural equation modeling approach (MASEM) to explore the potential for unique component-level relationships among EO’s three core dimensions in antecedent to outcome relationships. The third paper draws on these component-level insights and performs a finer-grained replication of the seminal MASEM of Rosenbusch, Rauch, and Bausch (2013) that proposes EO as a full mediator between the task environment and firm performance. The fourth and final paper of this cumulative dissertation illustrates exigent endogeneity concerns inherent in observational EO-performance research and provides guidance on how researchers can move towards establishing causal relationships.

Problems, Chances and Limitations of Facilitating Self-Directed Learning at a German Gymnasium
(2020)

Self-directed learning is becoming more important than ever. In a rapidly changing world, learners must be ready to face new obstacles. Self-directed learning gives the learners the chance to adapt to these social contextual changes. But facilitating self-directed learning in formal settings seems to be a risky task and venture. To accomplish its facilitation, many limits must be overcome.
In this thesis, lessons at a German school called a Gymnasium – the type of school where learners can get the highest school level degree – were observed in order to find out in how far elements of self-directed learning can be found in the observed lessons. For the comparison, the process elements of Knowles’ book “Self-Directed Learning: A Guide for Learners and Teachers” from 1975 were adapted to the observations of the lessons.
A central part of the observations and interviews of the teachers was to find out which limitations in the facilitation of self-directed learning can be found in terms of the institutional framework and the attitude of the teachers. The results of the observations highly differentiated. Whereas in many of the observed scientific lessons, many elements of self-directed learning were found, the lessons in social studies were teacher-directed. Also, a different attitude between the teachers was found in terms of the support for self-directed learning.
Importantly, the thesis includes the scientific critic of self-directed learning instead of excluding it and proposes the facilitation of Grow’s “Self-Directed-Learning Model” (1991) where the level of the learner’s self-directed learning is supposed to progress during school. This thesis is relevant for educators, curriculum developers, teachers and policymakers to help them identify the difficulties and chances to facilitate SDL in formal settings.

Although today’s bipeds are capable of demonstrating impressive locomotion skills, in many aspects, there’s still a big gap compared to the capabilities observed in humans. Partially, this is due to the deployed control paradigms that are mostly based on analytical approaches. The analytical nature of those approaches entails strong model dependencies – regarding the robotic platform as well as the environment – which makes them prone to unknown disturbances. Recently, an increasing number of biologically-inspired control approaches have been presented from which a human-like bipedal gait emerges. Although the control structures only rely on proprioceptive sensory information, the smoothness of the motions and the robustness against external disturbances is impressive. Due to the lack of suitable robotic platforms, until today the controllers have been mostly applied to
simulations.
Therefore, as the first step towards a suitable platform, this thesis presents the Compliant Robotic Leg (CARL) that features mono- as well as biarticular actuation. The design is driven by a set of core-requirements that is primarily derived from the biologically-inspired behavior-based bipedal locomotion control (B4LC) and complemented by further functional aspects from biomechanical research. Throughout the design process, CARL is understood as a unified dynamic system that emerges from the interplay of the mechanics, the electronics, and the control. Thus, having an explicit control approach and the respective gait in mind, the influence of each subsystem on the characteristics of the overall system is considered
carefully.
The result is a planar robotic leg whose three joints are driven by five highly integrated linear SEAs– three mono- and two biarticular actuators – with minimized reflected inertia. The SEAs are encapsulated by FPGA-based embedded nodes that are designed to meet the hard application requirements while enabling the deployment of a full-featured robotic framework. CARL’s foot is implemented using a COTS prosthetic foot; the sensor information is obtained from the deformation of its main structure. Both subsystems are integrated into a leg structure that matches the proportions of a human with a size of 1.7 m.
The functionality of the subsystems, as well as the overall system, is validated experimentally. In particular, the final experiment demonstrates a coordinated walking motion and thereby confirms that CARL can produce the desired behavior – a natural looking, human-like gait is emerging from the interplay of the behavior-based walking control and the mechatronic system. CARL is robust regarding impacts, the redundant actuation system can render the desired joint torques/impedances, and the foot system supports the walking structurally while it provides the necessary sensory information. Considering that there is no movement of the upper trunk, the angle and torque profiles are comparable to the ones found in humans.

With the technological advancement in the field of robotics, it is now quite practical to acknowledge the actuality of social robots being a part of human's daily life in the next decades. Concerning HRI, the basic expectations from a social robot are to perceive words, emotions, and behaviours, in order to draw several conclusions and adapt its behaviour to realize natural HRI. Henceforth, assessment of human personality traits is essential to bring a sense of appeal and acceptance towards the robot during interaction.
Knowledge of human personality is highly relevant as far as natural and efficient HRI is concerned. The idea is taken from human behaviourism, with humans behaving differently based on the personality trait of the communicating partners. This thesis contributes to the development of personality trait assessment system for intelligent human-robot interaction.
The personality trait assessment system is organized in three separate levels. The first level, known as perceptual level, is responsible for enabling the robot to perceive, recognize and understand human actions in the surrounding environment in order to make sense of the situation. Using psychological concepts and theories, several percepts have been extracted. A study has been conducted to validate the significance of these percepts towards personality traits.
The second level, known as affective level, helps the robot to connect the knowledge acquired in the first level to make higher order evaluations such as assessment of human personality traits. The affective system of the robot is responsible for analysing human personality traits. To the best of our knowledge, this thesis is the first work in the field of human-robot interaction that presents an automatic assessment of human personality traits in real-time using visual information. Using psychology and cognitive studies, many theories has been studied. Two theories have been been used to build the personality trait assessment system: Big Five personality traits assessment and temperament framework for personality traits assessment.
By using the information from the perceptual and affective level, the last level, known as behavioural level, enables the robot to synthesize an appropriate behaviour adapted to human personality traits. Multiple experiments have been conducted with different scenarios. It has been shown that the robot, ROBIN, assesses personality traits correctly during interaction and uses the similarity-attraction principle to behave with similar personality type. For example, if the person is found out to be extrovert, the robot also behaves like an extrovert. However, it also uses the complementary attraction theory to adapt its behaviour and complement the personality of the interaction partner. For example, if the person is found out to be self-centred, the robot behaves like an agreeable in order to flourish human-robot interaction.

Synapses are connections between different nerve cells that form an essential link in neural signal transmission. It is generally distinguished between electrical and chemical synapses, where chemical synapses are more common in the human brain and are also the type we deal with in this work.
In chemical synapses, small container-like objects called vesicles fill with neurotransmitter and expel them from the cell during synaptic transmission. This process is vital for communication between neurons. However, to the best of our knowledge no mathematical models that take different filling states of the vesicles into account have been developed before this thesis was written.
In this thesis we propose a novel mathematical model for modeling synaptic transmission at chemical synapses which includes the description of vesicles of different filling states. The model consists of a transport equation (for the vesicle growth process) plus three ordinary differential equations (ODEs) and focuses on the presynapse and synaptic cleft.
The well-posedness is proved in detail for this partial differential equation (PDE) system. We also propose a few different variations and related models. In particular, an ODE system is derived and a delay differential equation (DDE) system is formulated. We then use nonlinear optimization methods for data fitting to test some of the models on data made available to us by the Animal Physiology group at TU Kaiserslautern.

The Directive 97/23/EC of the European Parliament and of the Council of 29 May 1997 on the approximation of the laws of the Member States concerning pressure equipment (European Commision, 1997) is the basis of the legal framework for protection of pressure equipment within the European Union. Codes and standards are useful to comply with the legal and regulatory responsibilities stipulated in PED Directive regarding the protection of pressure equipment against overpressure, sizing, and selection safety relief devices.
Rupture disk devices are primary relief devices to protect vessels, pipe, and equipment against overpressure. A rupture disk bursts once the so-called burst pressure is reached in the protected system, thereby discharging flow and preventing further increase in pressure. Currently, rupture disks are sized with standards and codes assuming the worst-case scenario at burst pressure. There is however no standardized procedure for sizing rupture disks with two-phase flow and there lacks suited test-facilities, test-sections, and reliable experimental data for model validation. Sizing rupture disk vent-line systems with current characteristic numbers comes with significant uncertainties, especially for high-velocity compressible flows (Schmidt, 2015).
Zero-Emission and Green Safety are current trends for organizations that seek to attain innovative protection concepts beyond regulatory compliance. A procedure to size a rupture disk vent-line should accurately determine the discharge rate and pressure-drop across a rupture disk, from the point of rupture disk activation to the point when the system depressurizes fully. This procedure is critical for further safety considerations, such as for modeling the dispersion of toxic gases released during emergency-relief and calculating the emissions to the environment with time.
Over-dimensioning is one measure taken today to mitigate uncertainties encountered while sizing with current methods. This is not always an option, as over-dimensioning the rupture disk vent-line system leads to unnecessary financial costs. It may also cause malfunction of the collecting systems downstream when the fluids discharged are more than the design limits. Emissions to the environment are thereby potentially higher than necessary, causing excessive harm to the environment. Under-dimensioning, on the other hand, may lead to hazardous incidents with loss of human life and equipment. This work has therefore focused on the investigation of the mass flow rate and pressure-drop through rupture disk devices with compressible gas and two-phase flow.
The experimental focus was in the design, construction, and commissioning of a high-capacity, high-pressure industry-scale test facility for testing small- to large-diameter rupture disks and other fittings with gas flow. The resulting test facility is suited to test safety devices and pipe fittings at near realistic flow conditions at pressures up to 150 bar. This work also presents the design of a pilot plant for testing rupture disks with air/water two-phase flow. These test facilities open-up new frontiers for capacity testing because they have precise and state-of-the-art measurement and instrumentation. Experimental results from these facilities deliver reliable experimental data to validate proposed sizing procedures for rupture disk devices.
The theoretical focus was on the development of a reliable rupture disk sizing procedure for compressible gas and two-phase flow. This required phenomenological studies of flow through rupture disks with both experiments and CFD studies. Better suited rupture disk characteristic numbers and model parameters for determining the mass flow rate and pressure-drop across rupture disks are identified. The proposed sizing procedure with compressible gas and two-phase flow predicts the dischargeable mass flow rate and pressure-drop across a rupture disk within ±4 % of measured value. Experimental validation has been undertaken with different types of rupture disks. The procedure is suited for determine the mass flow rate and pressure-drop through rupture disk seamlessly, from the point of rupture disk activation (worst-case scenario) to the point when the system fully depressurizes beyond regulatory compliance.