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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.
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.
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.
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.
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.