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Optimal Control of Partial Integro-Differential Equations and Analysis of the Gaussian Kernel
(2018)
An important field of applied mathematics is the simulation of complex financial, mechanical, chemical, physical or medical processes with mathematical models. In addition to the pure modeling of the processes, the simultaneous optimization of an objective function by changing the model parameters is often the actual goal. Models in fields such as finance, biology or medicine benefit from this optimization step.
While many processes can be modeled using an ordinary differential equation (ODE), partial differential equations (PDEs) are needed to optimize heat conduction and flow characteristics, spreading of tumor cells in tissue as well as option prices. A partial integro-differential equation (PIDE) is a parital differential equation involving an integral operator, e.g., the convolution of the unknown function with a given kernel function. PIDEs occur for example in models that simulate adhesive forces between cells or option prices with jumps.
In each of the two parts of this thesis, a certain PIDE is the main object of interest. In the first part, we study a semilinear PIDE-constrained optimal control problem with the aim to derive necessary optimality conditions. In the second, we analyze a linear PIDE that includes the convolution of the unknown function with the Gaussian kernel.
In this thesis, we present a new approach for estimating the effects of wind turbines for a local bat population. We build an individual based model (IBM) which simulates the movement behaviour of every single bat of the population with its own preferences, foraging behaviour and other species characteristics. This behaviour is normalized by a Monte-Carlo simulation which gives us the average behaviour of the population. The result is an occurrence map of the considered habitat which tells us how often the bat and therefore the considered bat population frequent every region of this habitat. Hence, it is possible to estimate the crossing rate of the position of an existing or potential wind turbine. We compare this individual based approach with a partial differential equation based method. This second approach produces a lower computational effort but, unfortunately, we lose information about the movement trajectories at the same time. Additionally, the PDE based model only gives us a density profile. Hence, we lose the information how often each bat crosses special points in the habitat in one night. In a next step we predict the average number of fatalities for each wind turbine in the habitat, depending on the type of the wind turbine and the behaviour of the considered bat species. This gives us the extra mortality caused by the wind turbines for the local population. This value is used for a population model and finally we can calculate whether the population still grows or if there already is a decline in population size which leads to the extinction of the population. Using the combination of all these models, we are able to evaluate the conflict of wind turbines and bats and to predict the result of this conflict. Furthermore, it is possible to find better positions for wind turbines such that the local bat population has a better chance to survive. Since bats tend to move in swarm formations under certain circumstances, we introduce swarm simulation using partial integro-differential equations. Thereby, we have a closer look at existence and uniqueness properties of solutions.
Background and rationale: Changing working conditions demand adaptation, resulting in higher stress levels in employees. In consequence, decreased productivity, increasing rates of sick leave, and cases of early retirement result in higher direct, indirect, and intangible costs. Aims of the Research Project: The aim of the study was to test the usefulness of a novel translational diagnostic tool, Neuropattern, for early detection, prevention, and personalized treatment of stress-related disorders. The trial was designed as a pilot study with a wait list control group. Materials and Methods: In this study, 70 employees of the Forestry Department Rhineland-Palatinate, Germany, were enrolled. Subjects were block-randomized according to the functional group of their career field, and either underwent Neuropattern diagnostics immediately, or after a waiting period of three months. After the diagnostic assessment, their physicians received the Neuropattern Medical Report, including the diagnostic results and treatment recommendations. Participants were informed by the Neuropattern Patient Report, and were eligible to an individualized Neuropattern Online Counseling account. Results: The application of Neuropattern diagnostics significantly improved mental health and health-related behavior, reduced perceived stress, emotional exhaustion, overcommitment and possibly, presenteeism. Additionally, Neuropattern sensitively detected functional changes in stress physiology at an early stage, thus allowing timely personalized interventions to prevent and treat stress pathology. Conclusion: The present study encouraged the application of Neuropattern diagnostics to early intervention in non-clinical populations. However, further research is required to determine the best operating conditions.
Industrial companies mainly aim for increasing their profit. That is why they intend to reduce production costs without sacrificing the quality. Furthermore, in the context of the 2020 energy targets, energy efficiency plays a crucial role. Mathematical modeling, simulation and optimization tools can contribute to the achievement of these industrial and environmental goals. For the process of white wine fermentation, there exists a huge potential for saving energy. In this thesis mathematical modeling, simulation and optimization tools are customized to the needs of this biochemical process and applied to it. Two different models are derived that represent the process as it can be observed in real experiments. One model takes the growth, division and death behavior of the single yeast cell into account. This is modeled by a partial integro-differential equation and additional multiple ordinary integro-differential equations showing the development of the other substrates involved. The other model, described by ordinary differential equations, represents the growth and death behavior of the yeast concentration and development of the other substrates involved. The more detailed model is investigated analytically and numerically. Thereby existence and uniqueness of solutions are studied and the process is simulated. These investigations initiate a discussion regarding the value of the additional benefit of this model compared to the simpler one. For optimization, the process is described by the less detailed model. The process is identified by a parameter and state estimation problem. The energy and quality targets are formulated in the objective function of an optimal control or model predictive control problem controlling the fermentation temperature. This means that cooling during the process of wine fermentation is controlled. Parameter and state estimation with nonlinear economic model predictive control is applied in two experiments. For the first experiment, the optimization problems are solved by multiple shooting with a backward differentiation formula method for the discretization of the problem and a sequential quadratic programming method with a line search strategy and a Broyden-Fletcher-Goldfarb-Shanno update for the solution of the constrained nonlinear optimization problems. Different rounding strategies are applied to the resulting post-fermentation control profile. Furthermore, a quality assurance test is performed. The outcomes of this experiment are remarkable energy savings and tasty wine. For the next experiment, some modifications are made, and the optimization problems are solved by using direct transcription via orthogonal collocation on finite elements for the discretization and an interior-point filter line-search method for the solution of the constrained nonlinear optimization problems. The second experiment verifies the results of the first experiment. This means that by the use of this novel control strategy energy conservation is ensured and production costs are reduced. From now on tasty white wine can be produced at a lower price and with a clearer conscience at the same time.
This thesis considers the general task of computing a partition of a set of given objects such that each set of the partition has a cardinality of at least a fixed number k. Among such kinds of partitions, which we call k-clusters, the objective is to find the k-cluster which minimises a certain cost derived from a given pairwise difference between objects which end up the same set. As a first step, this thesis introduces a general problem, denoted by (||.||,f)-k-cluster, which models the task to find a k-cluster of minimum cost given by an objective function computed with respect to specific choices for the cost functions f and ||.||. In particular this thesis considers three different choices for f and also three different choices for ||.|| which results in a total of nine different variants of the general problem. Especially with the idea to use the concept of parameterised approximation, we first investigate the role of the lower bound on the cluster cardinalities and find that k is not a suitable parameter, due to remaining NP-hardness even for the restriction to the constant 3. The reductions presented to show this hardness yield the even stronger result which states that polynomial time approximations with some constant performance ratio for any of the nine variants of (||.||,f)-k-cluster require a restriction to instances for which the pairwise distance on the objects satisfies the triangle inequality. For this restriction to what we informally refer to as metric instances, constant-factor approximation algorithms for eight of the nine variants of (||.||,f)-k-cluster are presented. While two of these algorithms yield the provably best approximation ratio (assuming P!=NP), others can only guarantee a performance which depends on the lower bound k. With the positive effect of the triangle inequality and applications to facility location in mind, we discuss the further restriction to the setting where the given objects are points in the Euclidean metric space. Considering the effect of computational hardness caused by high dimensionality of the input for other related problems (curse of dimensionality) we check if this is also the source of intractability for (||.||,f)-k-cluster. Remaining NP-hardness for restriction to small constant dimensionality however disproves this theory. We then use parameterisation to develop approximation algorithms for (||.||,f)-k-cluster without restriction to metric instances. In particular, we discuss structural parameters which reflect how much the given input differs from a metric. This idea results in parameterised approximation algorithms with parameters such as the number of conflicts (our name for pairs of objects for which the triangle inequality is violated) or the number of conflict vertices (objects involved in a conflict). The performance ratios of these parameterised approximations are in most cases identical to those of the approximations for metric instances. This shows that for most variants of (||.||,f)-k-cluster efficient and reasonable solutions are also possible for non-metric instances.
At any given moment, our senses are assaulted with a flood of information from the environment around us. We need to pick our way through all this information in order to be able to effectively respond to that what is relevant to us. In most cases we are usually able to select information relevant to our intentions from what is not relevant. However, what happens to the information that is not relevant to us? Is this irrelevant information completely ignored so that it does not affect our actions? The literature suggests that even though we mayrnignore an irrelevant stimulus, it may still interfere with our actions. One of the ways in which irrelevant stimuli can affect actions is by retrieving a response with which it was associated. An irrelevant stimulus that is presented in close temporal contiguity with a relevant stimulus can be associated with the response made to the relevant stimulus " an observation termed distractor-response binding (Rothermund, Wentura, & De Houwer, 2005). The studies presented in this work take a closer look at such distractor-response bindings, and therncircumstances in which they occur. Specifically, the study reported in chapter 6 examined whether only an exact repetition of the distractor can retrieve the response with which it was associated, or whether even similar distractors may cause retrieval. The results suggested that even repeating a similar distractor caused retrieval, albeit less than an exact repetition. In chapter 7, the existence of bindings between a distractor and a response were tested beyond arnperceptual level, to see whether they exist at an (abstract) conceptual level. Similar to perceptual repetition, distractor-based retrieval of the response was observed for the repetition of concepts. The study reported in chapter 8 of this work examined the influence of attention on the feature-response binding of irrelevant features. The results pointed towards a stronger binding effects when attention was directed towards the irrelevant feature compared to whenrnit was not. The study in chapter 9 presented here looked at the processes underlying distractor-based retrieval and distractor inhibition. The data suggest that motor processes underlie distractor-based retrieval and cognitive process underlie distractor inhibition. Finally, the findings of all four studies are also discussed in the context of learning.
This thesis is focused on improving the knowledge on a group of threatened species, the European cave salamanders (genus Hydromantes). There are three main sections gathering studies dealing with different topics: Ecology (first part), Life traits (second part) and Monitoring methodologies (third part). First part starts with the study of the response of Hydromantes to the variation of climatic conditions, analysing 15 different localities throughout a full year (CHAPTER I; published in PEERJ in August 2015). After that, the focus moves on identify which is the operative temperature that these salamander experience, including how their body respond to variation of environmental temperature. This study was conducted using one of the most advanced tool, an infrared thermocamera, which gave the opportunity to perform detailed observation on salamanders body (CHAPTER II; published in JOURNAL OF THERMAL BIOLOGY in June 2016). In the next chapter we use the previous results to analyse the ecological niche of all eight Hydromantes species. The study mostly underlines the mismatch between macro- and microscale analysis of ecological niche, showing a weak conservatism of ecological niches within the evolution of species (CHAPTER III; unpublished manuscript). We then focus only on hybrids, which occur within the natural distribution of mainland species. Here, we analyse if the ecological niche of hybrids shows divergences from those of parental species, thus evaluating the power of hybrids adaptation (CHAPTER IV; unpublished manuscript). Considering that hybrids may represent a potential threat for parental species (in terms of genetic erosion and competition), we produced the first ecological study on an allochthonous mixed population of Hydromantes, analysing population structure, ecological requirements and diet. The interest on this particular population mostly comes by the fact that its members are coming from all three mainland Hydromantes species, and thus it may represent a potential source of new hybrids (CHAPTER V; accepted in AMPHIBIA-REPTILIA in October 2017). The focus than moves on how bioclimatic parameters affect species within their distributional range. Using as model species the microendemic H. flavus, we analyse the relationship between environmental suitability and local abundance of the species, also focusing on all intermediate dynamics which provide useful information on spatial variation of individual fitness (CHAPTER VI; submitted to SCIENTIFIC REPORTS in November 2017). The first part ends with an analysis of the interaction between Hydromantes and Batracobdella algira leeches, the only known ectoparasite for European cave salamanders. Considering that the effect of leeches on their hosts is potentially detrimental, we investigated if these ectoparasites may represent a further threat for Hydromantes (CHAPTER VII; submitted to INTERNATIONAL JOURNAL FOR PARASITOLOGY: PARASITES AND WILDLIFE in November 2017). The second part is related to the reproduction of Hydromantes. In the first study we perform analyses on the breeding behaviour of several females belonging to a single population, identifying differences and similarities occurring in cohorting females (CHAPTER VIII; published in NORTH-WESTERN JOURNAL OF ZOOLOGY in December 2015). In the second study we gather information from all Hydromantes species, analysing size and development of breeding females, and identifying a relationship between breeding time and climatic conditions (CHAPTER IX; submitted to SALAMANDRA in June 2017). In the last part of this thesis, we analyse two potential methods for monitoring Hydromantes populations. In the first study we evaluate the efficiency of the marking method involving Alpha tags (CHAPTER X; published in SALAMANDRA in October 2017). In the second study we focus on evaluating N-mixtures models as a methodology for estimating abundance in wild populations (CHAPTER XI; submitted to BIODIVERSITY & CONSERVATION in October 2017).
Early life adversity (ELA) poses a high risk for developing major health problems in adulthood including cardiovascular and infectious diseases and mental illness. However, the fact that ELA-associated disorders first become manifest many years after exposure raises questions about the mechanisms underlying their etiology. This thesis focuses on the impact of ELA on startle reflexivity, physiological stress reactivity and immunology in adulthood.
The first experiment investigated the impact of parental divorce on affective processing. A special block design of the affective startle modulation paradigm revealed blunted startle responsiveness during presentation of aversive stimuli in participants with experience of parental divorce. Nurture context potentiated startle in these participants suggesting that visual cues of childhood-related content activates protective behavioral responses. The findings provide evidence for the view that parental divorce leads to altered processing of affective context information in early adulthood.
A second investigation was conducted to examine the link between aging of the immune system and long-term consequences of ELA. In a cohort of healthy young adults, who were institutionalized early in life and subsequently adopted, higher levels of T cell senescence were observed compared to parent-reared controls. Furthermore, the results suggest that ELA increases the risk of cytomegalovirus infection in early childhood, thereby mediating the effect of ELA on T cell-specific immunosenescence.
The third study addresses the effect of ELA on stress reactivity. An extended version of the Cold Pressor Test combined with a cognitive challenging task revealed blunted endocrine response in adults with a history of adoption while cardiovascular stress reactivity was similar to control participants. This pattern of response separation may best be explained by selective enhancement of central feedback-sensitivity to glucocorticoids resulting from ELA, in spite of preserved cardiovascular/autonomic stress reactivity.
Stiftungsunternehmen sind Unternehmen, die sich ganz oder teilweise im Eigentum einer gemeinnützigen oder privaten Stiftung befinden. Die Anzahl an Stiftungsunternehmen in Deutschland ist in den letzten Jahren deutlich gestiegen. Bekannte deutsche Unternehmen wie Aldi, Bosch, Bertelsmann, LIDL oder Würth befinden sich im Eigentum von Stiftungen. Einige von ihnen, wie beispielsweise Fresenius, ZF Friedrichshafen oder Zeiss, sind sogar an der Börse notiert. Die Mehrzahl der Stiftungsunternehmen entsteht dadurch, dass Unternehmensgründer oder Unternehmerfamilien ihr Unternehmen in eine Stiftung einbringen, anstatt es zu vererben oder zu verkaufen.
Die Motive hierfür sind vielfältig und können familiäre Gründe (z. B. Kinderlosigkeit, Vermeidung von Familienstreit), unternehmensbezogene Gründe (z. B. Möglichkeit der langfristigen Planung durch stabile Eigentümerstruktur) und steuerliche Gründe (Vermeidung oder Reduzierung der Erbschaftssteuer) haben oder sind durch die Person des Gründers motiviert (Möglichkeit, das Unternehmen auch nach dem eigenen Tod über die Stiftung noch weiterhin zu prägen). Aufgrund der Tatsache, dass Stiftungsunternehmen zumeist aus Familienunternehmen hervorgehen, wird in der Forschung häufig nicht zwischen Familien- und Stiftungsunternehmen differenziert. Aus diesem Grund werden in dieser Dissertation zu Beginn anhand des Drei-Kreis-Modells für Familienunternehmen die Unterschiede zwischen Stiftungs- und Familienunternehmen dargestellt. Die Ergebnisse zeigen, dass nur eine sehr geringe Anzahl von Stiftungsunternehmen eine große Ähnlichkeit zu klassischen Familienunternehmen aufweist. Die meisten Stiftungsunternehmen unterscheiden sich zum Teil sehr stark von Familienunternehmen. Diese Ergebnisse verdeutlichen, dass Stiftungsunternehmen als separates Forschungsfeld betrachtet werden sollten.
Da innerhalb der Gruppe der Stiftungsunternehmen ebenfalls eine starke Heterogenität herrscht, werden im Anschluss Performanceunterschiede innerhalb der Gruppe der Stiftungsunternehmen untersucht. Hierzu wurden die Daten von 142 deutschen Stiftungsunternehmen für die Jahre 2006-2016 erhoben und mittels einer lineareren Regression ausgewertet. Die Ergebnisse zeigen, dass zwischen den verschiedenen Typen signifikante Unterschiede herrschen. Unternehmen, die von einer gemeinnützigen Stiftung gehalten werden, weisen eine signifikant schlechtere Performance auf, als Unternehmen die eine private Stiftung als Shareholder haben.
Im nächsten Schritt wird die Gruppe der börsennotierten Stiftungsunternehmen untersucht. Mittels einer Ereignisstudie wird getestet, wie sich die Stiftung als Eigentümer eines börsennotierten Unternehmens auf den Shareholder Value auswirkt. Die Ergebnisse zeigen, dass eine Anteilsverringerung einer Stiftung einen positiven Einfluss auf den Shareholder Value hat. Stiftungen werden vom Kapitalmarkt dementsprechend negativ bewertet. Aufgrund der divergierenden Ziele von Stiftung und Unternehmen birgt die Verbindung zwischen Stiftung und Unternehmen potentielle Konflikte und Herausforderungen für die beteiligten Personen. Mittels eines qualitativen explorativen Ansatzes, wird basierend auf Interviews, ein Modell entwickelt, welches die potentiellen Konflikte in Stiftungsunternehmen anhand des Beispiels der Doppelstiftung aufzeigt.
Im letzten Schritt werden Handlungsempfehlungen in Form eines Entwurfs für einen Corporate Governance Kodex erarbeitet, die (potentiellen) Stifterinnen und Stiftern helfen sollen, mögliche Konflikte entweder zu vermeiden oder bereits bestehende Probleme zu lösen.
Die Ergebnisse dieser Dissertation sind relevant für Theorie und Praxis. Aus theoretischer Sicht liegt der Wert dieser Untersuchungen darin, dass Forscher künftig besser zwischen Stiftungs- und Familienunternehmen unterscheiden können. Zudem bringt diese Arbeit den aktuellen Forschungsstand zum Thema Stiftungsunternehmen weiter. Außerdem bietet diese Dissertation insbesondere potentiellen Stiftern einen Überblick über die verschiedenen Ausgestaltungsmöglichkeiten und die Vor- und Nachteile, die diese Konstruktionen mit sich bringen. Die Handlungsempfehlungen ermöglichen es Stiftern, vorab potentielle Gefahren erkennen zu können und diese zu umgehen.
A basic assumption of standard small area models is that the statistic of interest can be modelled through a linear mixed model with common model parameters for all areas in the study. The model can then be used to stabilize estimation. In some applications, however, there may be different subgroups of areas, with specific relationships between the response variable and auxiliary information. In this case, using a distinct model for each subgroup would be more appropriate than employing one model for all observations. If no suitable natural clustering variable exists, finite mixture regression models may represent a solution that „lets the data decide“ how to partition areas into subgroups. In this framework, a set of two or more different models is specified, and the estimation of subgroup-specific model parameters is performed simultaneously to estimating subgroup identity, or the probability of subgroup identity, for each area. Finite mixture models thus offer a fexible approach to accounting for unobserved heterogeneity. Therefore, in this thesis, finite mixtures of small area models are proposed to account for the existence of latent subgroups of areas in small area estimation. More specifically, it is assumed that the statistic of interest is appropriately modelled by a mixture of K linear mixed models. Both mixtures of standard unit-level and standard area-level models are considered as special cases. The estimation of mixing proportions, area-specific probabilities of subgroup identity and the K sets of model parameters via the EM algorithm for mixtures of mixed models is described. Eventually, a finite mixture small area estimator is formulated as a weighted mean of predictions from model 1 to K, with weights given by the area-specific probabilities of subgroup identity.