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The harmonic Faber operator
(2018)
P. K. Suetin points out in the beginning of his monograph "Faber Polynomials and Faber Series" that Faber polynomials play an important role in modern approximation theory of a complex variable as they are used in representing analytic functions in simply connected domains, and many theorems on approximation of analytic functions are proved with their help [50]. In 1903, the Faber polynomials were firstly discovered by G. Faber. It was Faber's aim to find a generalisation of Taylor series of holomorphic functions in the open unit disc D in the following way. As any holomorphic function in D has a Taylor series representation f(z)=\sum_{\nu=0}^{\infty}a_{\nu}z^{\nu} (z\in\D) converging locally uniformly inside D, for a simply connected domain G, Faber wanted to determine a system of polynomials (Q_n) such that each function f being holomorphic in G can be expanded into a series
f=\sum_{\nu=0}^{\infty}b_{\nu}Q_{\nu} converging locally uniformly inside G. Having this goal in mind, Faber considered simply connected domains bounded by an analytic Jordan curve. He constructed a system of polynomials (F_n) with this property. These polynomials F_n were named after him as Faber polynomials. In the preface of [50], a detailed summary of results concerning Faber polynomials and results obtained by the aid of them is given. An important application of Faber polynomials is e.g. the transfer of known assertions concerning polynomial approximation of functions belonging to the disc algebra to results of the approximation of functions being continuous on a compact continuum K which contains at least two points and has a connected complement and being holomorphic in the interior of K. In this field, the Faber operator denoted by T turns out to be a powerful tool (for an introduction, see e.g. D. Gaier's monograph). It
assigns a polynomial of degree at most n given in the monomial basis \sum_{\nu=0}^{n}a_{\nu}z^{\nu} with a polynomial of degree at most n given in the basis of Faber polynomials \sum_{\nu=0}^{n}a_{\nu}F_{\nu}. If the Faber operator is continuous with respect to the uniform norms, it has a unique continuous extension to an operator mapping the disc algebra onto the space of functions being continuous on the whole compact continuum and holomorphic in its interior. For all f being element of the disc algebra and all polynomials P, via the obvious estimate for the uniform norms ||T(f)-T(P)||<= ||T|| ||f-P||, it can be seen that the original task of approximating F=T(f) by polynomials is reduced to the polynomial approximation of the function f. Therefore, the question arises under which conditions the Faber operator is continuous and surjective. A fundamental result in this regard was established by J. M. Anderson and J. Clunie who showed that if the compact continuum is bounded by a rectifiable Jordan curve with bounded boundary rotation and free from cusps, then the Faber operator with respect to the uniform norms is a topological isomorphism. Now, let f be a harmonic function in D. Similar as above, we find that f has a uniquely determined representation f=\sum_{\nu=-\infty}^{\infty}a_{\nu}p_{\nu}
converging locally uniformly inside D where p_{n}(z)=z^{n} for n\in\N_{0} and p_{-n}(z)=\overline{z}^{n} for n\in\N}. One may ask whether there is an analogue for harmonic functions on simply connected domains G. Indeed, for a domain G bounded by an analytic Jordan curve, the conjecture that each function f being harmonic in G has a uniquely determined representation f=\sum_{\nu= \infty}^{\infty}b_{\nu}F_{\nu} where F_{-n}(z)=\overline{F_{n}(z\)} for n\inN, converging locally uniformly inside G, holds true. Let now K be a compact continuum containing at least two points and having a connected complement. A main component of this thesis will be the examination of the harmonic Faber operator mapping a harmonic polynomial given in the basis of the harmonic monomials \sum_{\nu=-n}^{n}a_{\nu}p_{\nu} to a harmonic polynomial given as \sum_{\nu=-n}^{n}a_{\nu}F_{\nu}.
If this operator, which is based on an idea of J. Müller, is continuous with respect to the uniform norms, it has a unique continuous extension to an operator mapping the functions being continuous on \partial\D onto the continuous functions on K being
harmonic in the interior of K. Harmonic Faber polynomials and the harmonic Faber operator will be the objects accompanying us throughout
our whole discussion. After having given an overview about notations and certain tools we will use in our consideration in the first chapter, we begin our studies with an introduction to the Faber operator and the harmonic Faber operator. We start modestly and consider domains bounded by an analytic Jordan curve. In Section 2, as a first result, we will show that, for such a domain G, the harmonic Faber operator has a unique continuous extension to an operator mapping the space of the harmonic functions in D onto the space
of the harmonic functions in G, and moreover, the harmonic Faber
operator is an isomorphism with respect to the topologies of locally
uniform convergence. In the further sections of this chapter, we illumine the behaviour of the (harmonic) Faber operator on certain function spaces. In the third chapter, we leave the situation of compact continua bounded by an analytic Jordan curve. Instead we consider closures of domains bounded by Jordan curves having a Dini continuous curvature. With the aid of the concept of compact operators and the Fredholm alternative, we are able to show that the harmonic Faber operator is a topological isomorphism. Since, in particular, the main result of the third chapter holds true for closures K of domains bounded by analytic Jordan curves, we can make use of it to obtain new results concerning the approximation of functions being continuous on K and harmonic in the interior of K by harmonic polynomials. To do so, we develop techniques applied by L. Frerick and J. Müller in [11] and adjust them to our setting. So, we can transfer results about the classic Faber operator to the harmonic Faber operator. In the last chapter, we will use the theory of harmonic Faber polynomials
to solve certain Dirichlet problems in the complex plane. We pursue
two different approaches: First, with a similar philosophy as in [50],
we develop a procedure to compute the coefficients of a series \sum_{\nu=-\infty}^{\infty}c_{\nu}F_{\nu} converging uniformly to the solution of a given Dirichlet problem. Later, we will point out how semi-infinite programming with harmonic Faber polynomials as ansatz functions can be used to get an approximate solution of a given Dirichlet problem. We cover both approaches first from a theoretical point of view before we have a focus on the numerical implementation of concrete examples. As application of the numerical computations, we considerably obtain visualisations of the concerned Dirichlet problems rounding out our discussion about the harmonic Faber polynomials and the harmonic Faber operator.
Quadratische Optimierungsprobleme (QP) haben ein breites Anwendungsgebiet, wie beispielsweise kombinatorische Probleme einschließlich des maximalen Cliquenroblems. Motzkin und Straus [25] zeigten die Äquivalenz zwischen dem maximalen Cliquenproblem und dem standard quadratischen Problem. Auch mathematische Statistik ist ein weiteres Anwendungsgebiet von (QP), sowie eine Vielzahl von ökonomischen Modellen basieren auf (QP), z.B. das quadratische Rucksackproblem. In [5] Bomze et al. haben das standard quadratische Optimierungsproblem (StQP) in ein Copositive-Problem umformuliert. Im Folgenden wurden Algorithmen zur Lösung dieses copositiviten Problems von Bomze und de Klerk in [6] und Dür und Bundfuss in [9] entwickelt. Während die Implementierung dieser Algorithmen einige vielversprechende numerische Ergebnisse hervorbrachten, konnten die Autoren nur die copositive Neuformulierung des (StQP)s lösen. In [11] präsentierte Burer eine vollständig positive Umformulierung für allgemeine (QP)s, sogar mit binären Nebenbedingungen. Leider konnte er keine Methode zur Lösung für ein solches vollständig positives Problem präsentieren, noch wurde eine copositive Formulierung vorgeschlagen, auf die man die oben erwähnten Algorithmen modifizieren und anwenden könnte, um diese zu lösen. Diese Arbeit wird einen neuen endlichen Algorithmus zur Lösung eines standard quadratischen Optimierungsproblems aufstellen. Desweiteren werden in dieser Thesis copositve Darstellungen für ungleichungsbeschränkte sowie gleichungsbeschränkte quadratische Optimierungsprobleme vorgestellt. Für den ersten Ansatz wurde eine vollständig positive Umformulierung des (QP) entwickelt. Die copositive Umformulierung konnte durch Betrachtung des dualen Problems des vollständig positiven Problems erhalten werden. Ein direkterer Ansatz wurde gemacht, indem das Lagrange-Duale eines äquivalenten quadratischen Optimierungsproblems betrachtet wurde, das durch eine semidefinite quadratische Nebenbedingung beschränkt wurde. In diesem Zusammenhang werden Bedingungen für starke Dualität vorgeschlagen.
Um 1500 bildeten die Regensburger Juden eine der letzten Judengemeinden, die noch in einer Reichsstadt des Heiligen Römischen Reichs lebten. Der Regensburger Stadtrat ging mit immer weitreichenderen Restriktionen gegen ihre hergebrachten Rechte vor. Versuche, zu einer verbindlichen Regelung über die gegenseitigen Rechte und Pflichten zu kommen, führten zu keinem Ergebnis. Das Innsbrucker Regiment wurde schließlich damit beauftragt, die beiden Parteien im Wege eines Gerichtsverfahrens zu einer einvernehmlichen Lösung zu bewegen.
Der Innsbrucker Prozess gehört zu den bemerkenswertesten Aspekten der Geschichte der Juden in dieser Zeit. Über seine Hintergründe, Entwicklungen und Akteure liegt nun erstmals eine quellenfundierte Detailstudie vor. Sie zeigt, mit welcher Entschlossenheit die Regensburger Judengemeinde ihre Rechte verteidigte. Selbst als der Stadtrat Ende Februar 1519 die Juden gewaltsam aus der Stadt schaffen ließ, als Judenviertel, Synagoge und jüdischer Friedhof zerstört waren, gab das noch laufende Verfahren in Innsbruck der Judengemeinde berechtigte Hoffnung auf eine erfolgreiche Gegenwehr. Tatsächlich konnten weder Kaiser Karl V. noch die Reichsstadt Regensburg das Unrecht der Vertreibung ungeahndet ad acta legen.
Das Working Paper beleuchtet den Themenbereich‚Demografie und Migration und arbeitet Herausforderungen für die Raumentwicklung der Großregion ab. Insbesondere legt es einen Fokus auf die grenzüberschreitende Wohnmobilität an den Grenzen des Großherzogtums; die Bevölkerungsalterung und die Sicherung der Daseinsvorsorge im Gesundheitsbereich in ländlichen Gebieten.
The forward effect of testing refers to the finding that retrieval practice of previously studied information increases retention of subsequently studied other information. It has recently been hypothesized that the forward effect (partly) reflects the result of a reset-of-encoding (ROE) process. The proposal is that encoding efficacy decreases with an increase in study material, but testing of previously studied information resets the encoding process and makes the encoding of the subsequently studied information as effective as the encoding of the previously studied information. The goal of the present study was to verify the ROE hypothesis on an item level basis. An experiment is reported that examined the effects of testing in comparison to restudy on items’ serial position curves. Participants studied three lists of items in each condition. In the testing condition, participants were tested immediately on non-target lists 1 and 2, whereas in the restudy condition, they restudied lists 1 and 2. In both conditions, participants were tested immediately on target list 3. Influences of condition and items’ serial learning position on list 3 recall were analyzed. The results showed the forward effect of testing and furthermore that this effect varies with items’ serial list position. Early target list items at list primacy positions showed a larger enhancement effect than middle and late target list items at non-primacy positions. The results are consistent with the ROE hypothesis on an item level basis. The generalizability of the ROE hypothesis across different experimental tasks, like the list-method directed-forgetting task, is discussed.
Die patienten-fokussierte Psychotherapieforschung hat das Ziel, den Erfolg von Psychotherapie durch die kontinuierliche Messung und Rückmeldung von Prozessvariablen zu verbessern. Es konnte bereits gezeigt werden, dass nicht nur Patienten-spezifische Charakterisitika, wie die Symptomreduktion, sondern auch dyadische Merkmale, wie die therapeutische Beziehung, indikativ sind. Ein vielversprechender neuer Ansatz bzgl. der Messung dyadischer Charakteristika ist nonverbale Synchronie, die definiert ist als Bewegungskoordination zwischen Interaktionspartnern. Nonverbale Synchronie kann inzwischen objektiv und automatisch in Therapievidoes gemessen werden, was die Methodik frei von Biases wie selektiver Wahrnehmung oder sozialer Erwünschtheit macht. Frühe Studien aus der Sozial- und Entwicklungspsychologie konnten Zusammenhänge mit sozialer Bindung und Sympathie finden. Erste Studien aus der Psychotherapieforschung weisen auf Zusammenhänge zwischen nonverbaler Synchronie und der Therapiebeziehung sowie dem Therapieerfolg hin und geben erste Hinweise darauf, dass nonverbale Synchronie eine zusätzliche Informationsquelle für dyadische Aspekte sein kann, mit der man zukünftig frühzeitig Therapieerfolge vorhersagen könnte. Die vorliegende Arbeit beinhaltet drei Studien zu nonverbaler Synchronie in der ambulanten Psychotherapie und Zusammenhängen mit therapeutischen Prozessen. In Studie 1 wurde nonverbale Synchronie in einer diagnose-heterogenen Stichprobe von N=143 Patienten zu Therapiebeginn gemessen. Mittels Mehrebenenanalysen konnte die Validität der Messmethodik bestätigt werden. Des weiteren wurden Zusammenhänge mit bestimmten Artes des Therapieerfolgs gefunden: Patienten, die unverändert die Therapie abbrachen zeigten das niedrigste Level an Synchronie, während Patienten, die unverändert die Therapie zu Ende führten das höchste Level hatte und Patienten mit einer reliablen Symptomreduktion ein mittleres Level an nonverbaler Synchrony aufwiesen (auch unter Kontrolle der Therapiebeziehung). In Studie 2 wurden nonverbale Synchronie und die Bewegungsmenge zu Therapiebeginn und zum Therapieende erfasst und in zwei Stichproben von Patienten mit Depression (N=68) und Patienten mit Angststörungen (N=25) verglichen. Mehrebenenanalysen zeigten weniger Bewegungsmenge und Synchronie bei Dyaden mit depressiven Patienten, wobei sich beide Gruppen zum Therapieende nicht mehr in der nonverbalen Synchronie unterschieden. In Studie 3 wurde nonverbale Synchronie in einer Stichprobe von N=111 Patienten mit Sozialer Phobie zu vier Zeitpunkten im Therapieverlauf gemessen (N=346 Videos). Mehrebenenanalysen zeigten einen kontinuierlich sinkenden Verlauf der Synchronie und einen Moderationseffekt auf den Zusammenhang zwischen frühen Verbesserungen und dem Therapieerfolg.
Theoretischer Hintergrund: Essstörungen sind schwere psychische Störungen, welche aufgrund ihrer Komplexität, der hohen Mortalitätsrate sowie häufiger Chronifizierungen zu den Herausforderungen für Therapie und Forschung zählen. Die Herzratenvariabilität, als Indikator autonomer Regulation, scheint insbesondere bei Anorexie-Patientinnen zu Gunsten einer höheren parasympathischen Aktivität verschoben. Dieser Befund lässt sich anhand des Model Of Neurovisceral Integration erklären: Gemäß dieses Modells stellt eine erhöhte Herzratenvariabilität einen Hinweis für erfolgreiche Selbstregulation dar. Letztere scheint für restriktives Essverhalten essentiell, während sie bei impulsiven Verhaltensweisen wie Essanfälle und Erb-rechen reduziert sein sollte. Die bisherige Studienlage zur Herzratenvariabilität bei Essstörungen ist aufgrund der begrenzten Anzahl der Studien, der geringen Stichprobengrößen und Nicht- Berücksichtigung sinnvoller Drittvariablen jedoch noch inkonsistent und oftmals widersprüchlich. Neben der physiologischen Komponente werden in der Essstörungssymptomatik Veränderungen im kognitiven und emotionalen Erleben beschrieben. Zur Untersuchung beider Konstrukte erweisen sich Methoden des Ecological Momentary Assessment als aufschlussreich, da hierbei das Verhalten im Alltag der Patienten erhoben wird. Die bisherige Literatur zeigte bislang eine gute Anwendbarkeit der Methodik bei Essstörungspatienten, wobei die Anzahl der Studien gering ist. So fehlen bislang Studien, welche Emotionen und Kognitionen in Bezug zu Mahlzeiten und Sättigungsempfindungen setzen, obgleich solche Zusammenhänge in der kognitiven Verhaltenstherapie als zentral angesehen werden. Methode: Zu Beginn einer stationären psychosomatischen Behandlung wurden bei N=51 Probandinnen (Anorexia Nervosa: 19, Bulimia Nervosa: 15, gesunde Kontrollgruppe: 17) zeit- und frequenzanalytische Parameter der Herzratenvariabilität unter Berücksichtigung des Alters und des BMI in einer standardisierten fünfminütigen Laboruntersuchung untersucht. Am selben Tag fand außerdem eine stündliche Erhebung von Essverhalten, essstörungsspezifischen Kognitionen und negativen Emotionen mittels Smartphone statt. Am Ende der Behandlung wurde die Untersuchung wiederholt. Allgemein lineare Modelle wurden ebenso wie Mehrebenenmodelle zur statistischen Überprüfung der Hypothesen eingesetzt. Ergebnisse: Anorexie-Patientinnen zeigten tendenziell eine höhere parasympathische Aktivität als gesunde Probandinnen. Im Vergleich zu den beiden anderen Gruppen wiesen Bulimie-Patientinnen die niedrigste HRV auf. Antidepressiva führten zu einer Verringerung der HRV, genauso wie bei Anorexie-Patientinnen die Krankheitsdauer. Zusammenhänge mit erlebten Essanfällen konnten nicht festgestellt werden. Im Therapieverlauf zeigte sich, dass sich bei Anorexie-Patientinnen die HRV nach erfolgreicher Gewichtszunahme signifikant verringerte. Des Weiteren zeigten Essstörungspatientinnen höhere Ausprägungen in essstörungsspezifischen Kognitionen und negativen Emotionen während des Messtages. Mahlzeiten führten zu einer Verschlechterung der Stimmung, insbesondere bei restriktiven Anorexie-Patientinnen. Das Sättigungsempfinden einer Mahlzeit hatte einen signifikanten Einfluss auf die Bewertung dieser bei der klinischen Stichprobe, nicht jedoch bei gesunden Probandinnen. Am Ende der psychosomatischen Behandlung zeigte sich eine deutliche Verbesserung der essstörungsspezifischen Kognitionen und Mahlzeit-Bewertungen. Mahlzeiten hatten überdies einen geringeren Einfluss auf die Stimmung als zu Behandlungsbeginn. Diskussion: Die Auffälligkeiten im psychischen und physiologischen Bereich bei Essstörungspatientinnen sind Ausdruck eines vielschichtigen Krankheitsbildes, welches jedoch durch intensive Therapieangebote veränderbar ist. Das Hinzuziehen sinnvoller Drittvariablen erscheint bei Untersuchungen zur Herzratenvariabilität bei Essstörungspatienten essentiell. Darüber hin-aus zeigt die vorliegende Studie erstmals Zusammenhänge zwischen Mahlzeiten, Sättigungsempfinden und Essstörungssymptomatik mittels Ecological Momentary Assessment. Diese Methodik bietet einen inkrementellen Nutzen in der Erhebung verhaltensnaher Therapieerfolge. Resultierende Therapieansätze und Implikationen der Studie werden aufgezeigt.
The implicit power motive is one of the most researched motives in motivational psychology—at least in adults. Children have rarely been subject to investigation and there are virtually no results on behavioral and affective correlates of the implicit power motive in children. As behavior and affect are important components of conceptual validation, the empirical data in this dissertation focused on identifying three correlates, namely resource control behavior (study 1), power stress (study 2), and persuasive behavior (study 3). In each study, the implicit power motive was measured via the Picture Story Exercise, using an adapted version for children. Children across samples were between 4 and 11 years old.
Results from study 1 and 2 showed that children’s power-related behavior corresponded with evidence from adult samples: children with a high implicit power motive secure attractive resources and show negative reactions to a thwarted attempt to exert influence. Study 3 contradicted existing evidence with adults in that children’s persuasive behavior was not associated with nonverbal, but with verbal strategies of persuasion. Despite this inconsistency, these results are, together with the validation of a child-friendly Picture Story Exercise version, an important step into further investigating and confirming the concept of the implicit power motive and how to measure it in children.
Production of biomass feedstock for methanation in Europe has focused on silages of maize and cereals. As ecological awareness has increased in the last several years, more attention is being focused on perennial energy crops (PECs). Studies of specific PECs have shown that their cultivation may enhance agrobiodiversity and increase soil organic carbon stocks while simultaneously providing valuable feedstock for methanation. This study was designed to compare soil quality indicators under annual energy crops (AECs), PECs and permanent grassland (PGL) on the landscape level in south-western Germany. At a total 25 study sites, covering a wide range of parent materials, the cropping systems were found adjacent to each other. Stands were commercially managed, and PECs included different species such as the Cup Plant, Tall Wheatgrass, Giant Knotweed, Miscanthus, Virginia Mallow and Reed Canary Grass. Soil sampling was carried out for the upper 20 cm of soil. Several soil quality indicators, including soil organic carbon (Corg), soil microbial biomass (Cmic), and aggregate stability, showed that PECs were intermediate between AEC and PGL systems. At landscape level, mean Corg content for (on average) 6.1-year-old stands of PEC was 22.37 (±7.53) g kg1, compared to 19.23 (±8.08) and 32.08 (±10.11) for AEC and PGL. Cmic contents were higher in PECs (356 ± 241 lgCg1) compared to AECs (291 ± 145) but significantly lower than under PGL (753 ± 417). The aggregate stability increased by almost 65% in PECs compared to AEC but was still 57% lower than in PGL. Indicator differences among cropping systems were more pronounced when inherent differences in the parent material were accounted for in the comparisons. Overall, these results suggest that the cultivation of PECs has positive effects on soil quality indicators. Thus, PECs may offer potential to make the production of biomass feedstock more sustainable.
The dissertation deals with methods to improve design-based and model-assisted estimation techniques for surveys in a finite population framework. The focus is on the development of the statistical methodology as well as their implementation by means of tailor-made numerical optimization strategies. In that regard, the developed methods aim at computing statistics for several potentially conflicting variables of interest at aggregated and disaggregated levels of the population on the basis of one single survey. The work can be divided into two main research questions, which are briefly explained in the following sections.
First, an optimal multivariate allocation method is developed taking into account several stratification levels. This approach results in a multi-objective optimization problem due to the simultaneous consideration of several variables of interest. In preparation for the numerical solution, several scalarization and standardization techniques are presented, which represent the different preferences of potential users. In addition, it is shown that by solving the problem scalarized with a weighted sum for all combinations of weights, the entire Pareto frontier of the original problem can be generated. By exploiting the special structure of the problem, the scalarized problems can be efficiently solved by a semismooth Newton method. In order to apply this numerical method to other scalarization techniques as well, an alternative approach is suggested, which traces the problem back to the weighted sum case. To address regional estimation quality requirements at multiple stratification levels, the potential use of upper bounds for regional variances is integrated into the method. In addition to restrictions on regional estimates, the method enables the consideration of box-constraints for the stratum-specific sample sizes, allowing minimum and maximum stratum-specific sampling fractions to be defined.
In addition to the allocation method, a generalized calibration method is developed, which is supposed to achieve coherent and efficient estimates at different stratification levels. The developed calibration method takes into account a very large number of benchmarks at different stratification levels, which may be obtained from different sources such as registers, paradata or other surveys using different estimation techniques. In order to incorporate the heterogeneous quality and the multitude of benchmarks, a relaxation of selected benchmarks is proposed. In that regard, predefined tolerances are assigned to problematic benchmarks at low aggregation levels in order to avoid an exact fulfillment. In addition, the generalized calibration method allows the use of box-constraints for the correction weights in order to avoid an extremely high variation of the weights. Furthermore, a variance estimation by means of a rescaling bootstrap is presented.
Both developed methods are analyzed and compared with existing methods in extensive simulation studies on the basis of a realistic synthetic data set of all households in Germany. Due to the similar requirements and objectives, both methods can be successively applied to a single survey in order to combine their efficiency advantages. In addition, both methods can be solved in a time-efficient manner using very comparable optimization approaches. These are based on transformations of the optimality conditions. The dimension of the resulting system of equations is ultimately independent of the dimension of the original problem, which enables the application even for very large problem instances.
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.
Possibilités et opportunités d’une organisation démocratique du système scolaire luxembourgeois
(2018)
« Aujourd’hui, le socialisme et la démocratie ne sont plus de simples questions réservées aux partis politiques, ce sont des questions vitales. L’école et les enseignants devront s’en saisir (...) et donner au peuple de demain, par un enseignement solide, mais surtout par une éducation formatrice, les moyens de remplir sa mission principale : régner ensemble. » – ein luxemburger Schulfreund, 1920
“Sozialismus und Demokratie sind heute nicht mehr bloße Parteifragen, es sind Lebensfragen. Schule und Lehrer kommen an ihnen nicht vorbei (…), durch gediegenen Unterricht, vor allem aber durch charakterbildende Erziehung das Volk von morgen zu seiner großen Aufgabe des Mitherrschens zu befähigen.” – ein luxemburger Schulfreund, 1920
Fostering positive and realistic self-concepts of individuals is a major goal in education worldwide (Trautwein & Möller, 2016). Individuals spend most of their childhood and adolescence in school. Thus, schools are important contexts for individuals to develop positive self-perceptions such as self-concepts. In order to enhance positive self-concepts in educational settings and in general, it is indispensable to have a comprehensive knowledge about the development and structure of self-concepts and their determinants. To date, extensive empirical and theoretical work on antecedents and change processes of self-concept has been conducted. However, several research gaps still exist, and several of these are the focus of the present dissertation. Specifically, these research gaps encompass (a) the development of multiple self-concepts from multiple perspectives regarding stability and change, (b) the direction of longitudinal interplay between self-concept facets over the entire time period from childhood to late adolescence, and (c) the evidence that a recently developed structural model of academic self-concept (nested Marsh/Shavelson model [Brunner et al., 2010]) fits the data in elementary school students, (d) the investigation of structural changes in academic self-concept profile formation within this model, (e) the investigation of dimensional comparison processes as determinants of academic self-concept profile formation in elementary school students within the internal/external frame of reference model (I/E model; Marsh, 1986), (f) the test of moderating variables for dimensional comparison processes in elementary school, (g) the test of the key assumptions of the I/E model that effects of dimensional comparisons depend to a large degree on the existence of achievement differences between subjects, and (h) the generalizability of the findings regarding the I/E model over different statistical analytic methods. Thus, the aim of the present dissertation is to contribute to close these gaps with three studies. Thereby, data from German students enrolled in elementary school to secondary school education were gathered in three projects comprising the developmental time span from childhood to adolescence (ages 6 to 20). Three vital self-concept areas in childhood and adolescence were in-vestigated: general self-concept (i.e., self-esteem), academic self-concepts (general, math, reading, writing, native language), and social self-concepts (of acceptance and assertion). In all studies, data were analyzed within a latent variable framework. Findings are discussed with respect to the research aims of acquiring more comprehensive knowledge on the structure and development of significant self-concept in childhood and adolescence and their determinants. In addition, theoretical and practical implications derived from the findings of the present studies are outlined. Strengths and limitations of the present dissertation are discussed. Finally, an outlook for future research on self-concepts is given.
Kartenschätze aus Italien
(2018)
Die Entdeckungen der Neuzeit sowie verbesserte Druckverfahren führten ab dem 16. Jahrhundert zu einem enormen Aufschwung der Kartographie. Gerade in Italien entstanden blühende kartographische Zentren mit exzellentem Ruf, die innerhalb kurzer Zeit große Fortschritte hinsichtlich Genauigkeit und Übersichtlichkeit machten. Aus dem der Universitätsbibliothek Trier vermachten Nachlass des Kartensammlers Fritz Hellwig werden drei repräsentative Beispiele vorgestellt.
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.
The economic growth theory analyses which factors affect economic growth and tries to analyze how it can last. A popular neoclassical growth model is the Ramsey-Cass-Koopmans model, which aims to determine how much of its income a nation or an economy should save in order to maximize its welfare. In this thesis, we present and analyze an extended capital accumulation equation of a spatial version of the Ramsey model, balancing diffusive and agglomerative effects. We model the capital mobility in space via a nonlocal diffusion operator which allows for jumps of the capital stock from one location to an other. Moreover, this operator smooths out heterogeneities in the factor distributions slower, which generated a more realistic behavior of capital flows. In addition to that, we introduce an endogenous productivity-production operator which depends on time and on the capital distribution in space. This operator models the technological progress of the economy. The resulting mathematical model is an optimal control problem under a semilinear parabolic integro-differential equation with initial and volume constraints, which are a nonlocal analog to local boundary conditions, and box-constraints on the state and the control variables. In this thesis, we consider this problem on a bounded and unbounded spatial domain, in both cases with a finite time horizon. We derive existence results of weak solutions for the capital accumulation equations in both settings and we proof the existence of a Ramsey equilibrium in the unbounded case. Moreover, we solve the optimal control problem numerically and discuss the results in the economic context.
Ziel der Dissertation ist es, den Hochwasserschutz und das Management extremer Hoch-wasser für das Einzugsgebiet der Isar zu verbessern mit Hinblick darauf, wie sich vorhandene und neu zu schaffende Retentionsräume mit optimaler Wirkung für das gesamte Flusssystem einsetzen lassen. Dafür sind Kenntnisse über extreme Ereignisse und deren Auswirkung auf die betrachteten Einzugsgebiete notwendig. Großskalige Niederschläge in Mitteleuropa werden überwiegend durch Vb-artige Zugbahnen ausgelöst. Die Relevanz für Bayern zeigt die Auswertung des neuesten Kataloges der Vb-Zugbahnen für den Zeitraum 1959 bis 2015. In den Monaten April bis Oktober haben Vb-Zugbahnen zu ca. 30 % der beobachten Hochwasser beigetragen. Im Sommer führt sogar jedes zweite Vb-Tief zu Hochwasser. Im Donaueinzugsgebiet können 50 % der 20 größten Hochwasser direkt auf Vb-Zugbahnen zurückgeführt werden, weitere 25 % durch ähnliche Zugbahnen oder auf eine Vb aktiven Phase. Über die Hälfe der größten Hochwasser traten dabei in Bezug zu einer Serie von Vb-Tiefs auf. 60 % der Vb-Zugbahnen sind Teil einer Serie von Vb-Tiefs. Aus wiederkehrenden Niederschlägen persistenter Zugbahnen resultieren mehrgipflige Hochwasserwellen, die insbesondere für Rückhalteräume betrachtet werden müssen (DIN 19700). Die Detailuntersuchung erfolgt unter besonderer Beachtung der Untersuchungen zu den Vb-Zugbahnen. Das Isareinzugsgebiet mit 8900 km-² besitzt mit den Seen im Voralpenland große natürliche Retentionsräume und mit dem Sylvensteinspeicher im alpinen Einzugsgebiet den größten staatlichen Speicher Bayerns. Für die Wirkungsanalyse von gekoppelten Hoch-wasserrückhalteräumen in komplexen Einzugsgebieten müssen Ganglinien mit einem Nie-derschlag-Abfluss-Modell generiert werden, die den Wellenablauf des Hochwassers im ge-samten Einzugsgebiet repräsentieren. Die Dissertation analysiert, wie sich der Einsatz ver-schiedener Verfahren zur Vorgabe der Eingangsniederschläge auswirkt. Dabei liegt der Schwerpunkt der Untersuchung auf dem Niederschlagsverlauf. Es wird ein Verfahren zur Ableitung von Ganglinien aus standardisierten beobachteten Niederschlagsverläufen entwi-ckelt. Die Hochwasserganglinien, generiert aus synthetischen Niederschlagsverläufen der Bemessung, werden am Beispiel des Sylvensteinspeichers mit den drei größten abgelaufe-nen Hochwasserereignissen verglichen und diskutiert, ob mit dem neuen Verfahren die Cha-rakteristik der beobachten Hochwasser besser wiedergeben wird. Der Fokus liegt dabei auf der Wellenüberlagerung. Es kann für das ganze Gebiet gezeigt werden, dass die mit der neuen Methode standardisierten beobachteten Niederschlagsverläufe besser geeignet sind, die Wellenüberlagerung wiederzugeben, da zeitliche Unterschiede durch die Staueffekte an den Alpen berücksichtigt werden, wie sie bei Vb-Zugbahn geprägten Niederschlägen entste-hen. Es kann daher bei ähnlichen Fragestellungen empfohlen werden, diese Methode in der Praxis als Variante hinzuzuziehen, um die natürlichen Prozesse repräsentativer zu beschrei-ben. Für die Simulation mit dem N-A-Modell LARSIM werden die Unsicherheiten durch Varianten-rechnungen gezeigt. Es hat sich herausgestellt, dass nicht nur der Niederschlagsverlauf und die Vorbedingungen des Ereignisses eine große Auswirkung auf die Kalibrierung der Ab-flussbeiwerte im N-A-Modell haben, sondern auch das gewählte Flood-Routing-Verfahren und die Gerinnerauheit. Schließlich wird die Bewertung der potenziellen Standorte durchgeführt. Es wird berechnet, wo das Hochwasser zurückgehalten werden muss, um sowohl eine lokale Reduktion des Hochwasserscheitels, als auch gleichzeitig eine möglichst große Schutzwirkung für das Ge-samtsystem zu ermöglichen. Priorisiert werden Rückhaltestandorte, die praktisch umsetzbar sind und den größten Nutzen haben. Die Untersuchung einer Doppelwelle, die durch eine Serie von Vb-Zugbahnen entstehen kann, zeigt, wie die Einschätzung potenzieller Standorte verändern kann. Der alpine und zum Teil der voralpine Raum reagieren mit kurzen steilen Ganglinien und sind gegenüber Doppelwellen weniger sensitiv, weil kaum Wellenüberlagerung entsteht. Für den Sylvensteinspeicher, der im alpinen Raum liegt, können daher kurze Niederschlagspausen für eine schnelle Entlastung des Speicherraumes genutzt werden. Un-terhalb von Seen mit einem großen Retentionsvermögen erzeugen Doppelwellen aufgrund der langen Retentionsäste durch die Wellenüberlagerung deutlich höhere Abflüsse als Ein-zelwellen. Rückhalt an der oberen Isar ist unter diesen Kriterien am optimalsten. Empfohlene Maßnahmen - ohne Bauaufwand - konnten bereits umgesetzt werden und verbessern den Hochwasserschutz und das Hochwassermanagement an der Isar. Die Auswertungen zeigen, dass in den Monaten April, Mai, September und Oktober die Hochwasserereignisse in Folge von Vb-Zugbahnen im Zuge der Klimaveränderung häufiger und in den Sommermonaten extremer werden könnten.
Salivary alpha-amylase (sAA) influences the perception of taste and texture, features both relevant in acquiring food liking and, with time, food preference. However, no studies have yet investigated the relationship between basal activity levels of sAA and food preference. We collected saliva from 57 volunteers (63% women) who we assessed in terms of their preference for different food items. These items were grouped into four categories according to their nutritional properties: high in starch, high in sugar, high glycaemic index, and high glycaemic load. Anthropometric markers of cardiovascular risk were also calculated. Our findings suggest that sAA influences food
preference and body composition in women. Regression analysis showed that basal sAA activity is inversely associated with subjective but not self-reported behavioural preference for foods high in sugar. Additionally, sAA and subjective preference are associated with anthropometric markers of cardiovascular risk. We believe that this pilot study points to this enzyme as an interesting candidate to consider among the physiological factors that modulate eating behaviour.
We will consider discrete dynamical systems (X,T) which consist of a state space X and a linear operator T acting on X. Given a state x in X at time zero, its state at time n is determined by the n-th iteration T^n(x). We are interested in the long-term behaviour of this system, that means we want to know how the sequence (T^n (x))_(n in N) behaves for increasing n and x in X. In the first chapter, we will sum up the relevant definitions and results of linear dynamics. In particular, in topological dynamics the notions of hypercyclic, frequently hypercyclic and mixing operators will be presented. In the setting of measurable dynamics, the most important definitions will be those of weakly and strongly mixing operators. If U is an open set in the (extended) complex plane containing 0, we can define the Taylor shift operator on the space H(U) of functions f holomorphic in U as Tf(z) = (f(z)- f(0))/z if z is not equal to 0 and otherwise Tf(0) = f'(0). In the second chapter, we will start examining the Taylor shift on H(U) endowed with the topology of locally uniform convergence. Depending on the choice of U, we will study whether or not the Taylor shift is weakly or strongly mixing in the Gaussian sense. Next, we will consider Banach spaces of functions holomorphic on the unit disc D. The first section of this chapter will sum up the basic properties of Bergman and Hardy spaces in order to analyse the dynamical behaviour of the Taylor shift on these Banach spaces in the next part. In the third section, we study the space of Cauchy transforms of complex Borel measures on the unit circle first endowed with the quotient norm of the total variation and then with a weak-* topology. While the Taylor shift is not even hypercyclic in the first case, we show that it is mixing for the latter case. In Chapter 4, we will first introduce Bergman spaces A^p(U) for general open sets and provide approximation results which will be needed in the next chapter where we examine the Taylor shift on these spaces on its dynamical properties. In particular, for 1<=p<2 we will find sufficient conditions for the Taylor shift to be weakly mixing or strongly mixing in the Gaussian sense. For p>=2, we consider specific Cauchy transforms in order to determine open sets U such that the Taylor shift is mixing on A^p(U). In both sections, we will illustrate the results with appropriate examples. Finally, we apply our results to universal Taylor series. The results of Chapter 5 about the Taylor shift allow us to consider the behaviour of the partial sums of the Taylor expansion of functions in general Bergman spaces outside its disc of convergence.
Die vorliegende Arbeit befasst sich mit einer komplexen Fragestellung: Wie geschieht der dynamische Umbau der sprachlichen Strukturen unter der Wirkung der innersprachlichen und außersprachlichen Parameter. Im Fokus der Forschung steht der Mechanismus des Werdens der Sprachstruktur, der hier als ein einziger Modus des Daseins der Sprache betrachtet wird. Als Material der Untersuchung dient die Operationalisierung der Bestandteile der verbalen Wortbildungsprozesse in der deutschen Sprache. Die Auswahl des verbalen Teils des Vokabulars ist dadurch bedingt, dass diese Wortart ein Zentralelement ist, das die ganze Sprachmaterie konsolidiert. Als einer der Schlüsselparameter gilt dabei der Frequenzfaktor, der bisher keinen einheitlichen Status in der Sprachtheorie bekommen hat. Die Suche nach dem Ursprung der Macht dieses Faktors führt unumgänglich über die Grenzen des Sprachsystems hinaus. Die Beobachtungen über das Verhalten des Frequenzfaktors in den Prozessen und Strukturen unterschiedlichster Natur lassen behaupten, dass wir es hier mit einem sehr komplexen Phänomen zu tun haben, das ein Bestandteil des allgemeinen kognitiven Anpassungsmechanismus des Menschen zur Umwelt ist. Als solcher ist er auch ein unveräußerlicher Aspekt der Semiose, des Sprachzeichens.
The changing views on the evolutionary relationships of extant Salamandridae (Amphibia: Urodela)
(2018)
The phylogenetic relationships among members of the family Salamandridae have been repeatedly investigated over the last 90 years, with changing character and taxon sampling. We review the changing composition and the phylogenetic position of salamandrid genera and species groups and add a new phylogeny based exclusively on sequences of nuclear genes. Salamandrina often changed its position depending on the characters used. It was included several times in a clade together with the primitive newts (Echinotriton, Pleurodeles, Tylototriton) due to their seemingly ancestral morphology. The latter were often inferred as a monophyletic clade. Respective monophyly was almost consistently established in all molecular studies for true salamanders (Chioglossa, Lyciasalamandra, Mertensiella, Salamandra), modern Asian newts (Cynops, Laotriton, Pachytriton, Paramesotriton) and modern New World newts (Notophthalmus, Taricha). Reciprocal non-monophyly has been established through molecular studies for the European mountain newts (Calotriton, Euproctus) and the modern European newts (Ichthyosaura, Lissotriton, Neurergus, Ommatotriton, Triturus) since Calotriton was identified as the sister lineage of Triturus. In pre-molecular studies, their respective monophyly had almost always been assumed, mainly because a complex courtship behaviour shared by their respective members. Our nuclear tree is nearly identical to a mito-genomic tree, with all but one node being highly supported. The major difference concerns the position of Calotriton, which is no longer nested within the modern European newts. This has implications for the evolution of courtship behaviour of European newts. Within modern European newts, Ichthyosaura and Lissotriton changed their position compared to the mito-genomic tree. Previous molecular trees based on seemingly large nuclear data sets, but analysed together with mitochondrial data, did not reveal monophyly of modern European newts since taxon sampling and nuclear gene coverage was too poor to obtain conclusive results. We therefore conclude that mitochondrial and nuclear data should be analysed on their own.
Sample surveys are a widely used and cost effective tool to gain information about a population under consideration. Nowadays, there is an increasing demand not only for information on the population level but also on the level of subpopulations. For some of these subpopulations of interest, however, very small subsample sizes might occur such that the application of traditional estimation methods is not expedient. In order to provide reliable information also for those so called small areas, small area estimation (SAE) methods combine auxiliary information and the sample data via a statistical model.
The present thesis deals, among other aspects, with the development of highly flexible and close to reality small area models. For this purpose, the penalized spline method is adequately modified which allows to determine the model parameters via the solution of an unconstrained optimization problem. Due to this optimization framework, the incorporation of shape constraints into the modeling process is achieved in terms of additional linear inequality constraints on the optimization problem. This results in small area estimators that allow for both the utilization of the penalized spline method as a highly flexible modeling technique and the incorporation of arbitrary shape constraints on the underlying P-spline function.
In order to incorporate multiple covariates, a tensor product approach is employed to extend the penalized spline method to multiple input variables. This leads to high-dimensional optimization problems for which naive solution algorithms yield an unjustifiable complexity in terms of runtime and in terms of memory requirements. By exploiting the underlying tensor nature, the present thesis provides adequate computationally efficient solution algorithms for the considered optimization problems and the related memory efficient, i.e. matrix-free, implementations. The crucial point thereby is the (repetitive) application of a matrix-free conjugated gradient method, whose runtime is drastically reduced by a matrx-free multigrid preconditioner.
In the context of accelerated global socio-environmental change, the Water-Energy-Food Nexus has received increasing attention within science and international politics by promoting integrated resource governance. This study explores the scientific nexus debates from a discourse analytical perspective to reveal knowledge and power relations as well as geographical settings of nexus research. We also investigate approaches to socio-nature relations that influence nexus research and subsequent political implications. Our findings suggest that the leading nexus discourse is dominated by natural scientific perspectives and a neo-Malthusian framing of environmental challenges. Accordingly, the promoted cross-sectoral nexus approach to resource governance emphasizes efficiency, security, future sustainability, and poverty reduction. Water, energy, and food are conceived as global trade goods that require close monitoring, management and control, to be achieved via quantitative assessments and technological interventions. Within the less visible discourse, social scientific perspectives engage with the social, political, and normative elements of the Water-Energy-Food Nexus. These perspectives criticize the dominant nexus representation for itsmanagerial, neoliberal, and utilitarian approach to resource governance. The managerial framing is critiqued for masking power relations and social inequalities, while alternative framings acknowledge the political nature of resource governance and socio-nature relations. The spatial dimensions of the nexus debate are also discussed. Notably, the nexus is largely shaped by western knowledge, yet applied mainly in specific regions of the Global South. In order for the nexus to achieve integrative solutions for sustainability, the debate needs to overcome its current discursive and spatial separations. To this end, we need to engage more closely with alternative nexus discourses, embrace epistemic pluralism and encourage multi-perspective debates about the socio-nature relations we actually intend to promote.
In the present study a non-motion-stabilized scanning Doppler lidar was operated on board of RV Polarstern in the Arctic (June 2014) and Antarctic (December 2015– January 2016). This is the first time that such a system measured on an icebreaker in the Antarctic. A method for a motion correction of the data in the post-processing is presented.
The wind calculation is based on vertical azimuth display (VAD) scans with eight directions that pass a quality control. Additionally a method for an empirical signal-tonoise ratio (SNR) threshold is presented, which can be calculated for individual measurement set-ups. Lidar wind profiles are compared to total of about 120 radiosonde profiles and also to wind measurements of the ship.
The performance of the lidar measurements in comparison with radio soundings generally shows small root mean square deviation (bias) for wind speed of around 1ms-1(0.1ms-1) and for wind direction of around 10 (1). The post-processing of the non-motion-stabilized data shows comparably high quality to studies with motion-stabilized systems.
Two case studies show that a flexible change in SNR threshold can be beneficial for special situations. Further the studies reveal that short-lived low-level jets in the atmospheric boundary layer can be captured by lidar measurements with a high temporal resolution in contrast to routine radio soundings. The present study shows that a non-motionstabilized Doppler lidar can be operated successfully on an
icebreaker. It presents a processing chain including quality control tests and error quantification, which is useful for further measurement campaigns.
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.
Surveys are commonly tailored to produce estimates of aggregate statistics with a desired level of precision. This may lead to very small sample sizes for subpopulations of interest, defined geographically or by content, which are not incorporated into the survey design. We refer to subpopulations where the sample size is too small to provide direct estimates with adequate precision as small areas or small domains. Despite the small sample sizes, reliable small area estimates are needed for economic and political decision making. Hence, model-based estimation techniques are used which increase the effective sample size by borrowing strength from other areas to provide accurate information for small areas. The paragraph above introduced small area estimation as a field of survey statistics where two conflicting philosophies of statistical inference meet: the design-based and the model-based approach. While the first approach is well suited for the precise estimation of aggregate statistics, the latter approach furnishes reliable small area estimates. In most applications, estimates for both large and small domains based on the same sample are needed. This poses a challenge to the survey planner, as the sampling design has to reflect different and potentially conflicting requirements simultaneously. In order to enable efficient design-based estimates for large domains, the sampling design should incorporate information related to the variables of interest. This may be achieved using stratification or sampling with unequal probabilities. Many model-based small area techniques require an ignorable sampling design such that after conditioning on the covariates the variable of interest does not contain further information about the sample membership. If this condition is not fulfilled, biased model-based estimates may result, as the model which holds for the sample is different from the one valid for the population. Hence, an optimisation of the sampling design without investigating the implications for model-based approaches will not be sufficient. Analogously, disregarding the design altogether and focussing only on the model is prone to failure as well. Instead, a profound knowledge of the interplay between the sample design and statistical modelling is a prerequisite for implementing an effective small area estimation strategy. In this work, we concentrate on two approaches to address this conflict. Our first approach takes the sampling design as given and can be used after the sample has been collected. It amounts to incorporate the survey design into the small area model to avoid biases stemming from informative sampling. Thus, once a model is validated for the sample, we know that it holds for the population as well. We derive such a procedure under a lognormal mixed model, which is a popular choice when the support of the dependent variable is limited to positive values. Besides, we propose a three pillar strategy to select the additional variable accounting for the design, based on a graphical examination of the relationship, a comparison of the predictive accuracy of the choices and a check regarding the normality assumptions.rnrnOur second approach to deal with the conflict is based on the notion that the design should allow applying a wide variety of analyses using the sample data. Thus, if the use of model-based estimation strategies can be anticipated before the sample is drawn, this should be reflected in the design. The same applies for the estimation of national statistics using design-based approaches. Therefore, we propose to construct the design such that the sampling mechanism is non-informative but allows for precise design-based estimates at an aggregate level.