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A matrix A is called completely positive if there exists an entrywise nonnegative matrix B such that A = BB^T. These matrices can be used to obtain convex reformulations of for example nonconvex quadratic or combinatorial problems. One of the main problems with completely positive matrices is checking whether a given matrix is completely positive. This is known to be NP-hard in general. rnrnFor a given matrix completely positive matrix A, it is nontrivial to find a cp-factorization A=BB^T with nonnegative B since this factorization would provide a certificate for the matrix to be completely positive. But this factorization is not only important for the membership to the completely positive cone, it can also be used to recover the solution of the underlying quadratic or combinatorial problem. In addition, it is not a priori known how many columns are necessary to generate a cp-factorization for the given matrix. The minimal possible number of columns is called the cp-rank of A and so far it is still an open question how to derive the cp-rank for a given matrix. Some facts on completely positive matrices and the cp-rank will be given in Chapter 2. Moreover, in Chapter 6, we will see a factorization algorithm, which, for a given completely positive matrix A and a suitable starting point, computes the nonnegative factorization A=BB^T. The algorithm therefore returns a certificate for the matrix to be completely positive. As introduced in Chapter 3, the fundamental idea of the factorization algorithm is to start from an initial square factorization which is not necessarily entrywise nonnegative, and extend this factorization to a matrix for which the number of columns is greater than or equal to the cp-rank of A. Then it is the goal to transform this generated factorization into a cp-factorization. This problem can be formulated as a nonconvex feasibility problem, as shown in Section 4.1, and solved by a method which is based on alternating projections, as proven in Chapter 6. On the topic of alternating projections, a survey will be given in Chapter 5. Here we will see how to apply this technique to several types of sets like subspaces, convex sets, manifolds and semialgebraic sets. Furthermore, we will see some known facts on the convergence rate for alternating projections between these types of sets. Considering more than two sets yields the so called cyclic projections approach. Here some known facts for subspaces and convex sets will be shown. Moreover, we will see a new convergence result on cyclic projections among a sequence of manifolds in Section 5.4. In the context of cp-factorizations, a local convergence result for the introduced algorithm will be given. This result is based on the known convergence for alternating projections between semialgebraic sets. To obtain cp-facrorizations with this first method, it is necessary to solve a second order cone problem in every projection step, which is very costly. Therefore, in Section 6.2, we will see an additional heuristic extension, which improves the numerical performance of the algorithm. Extensive numerical tests in Chapter 7 will show that the factorization method is very fast in most instances. In addition, we will see how to derive a certificate for the matrix to be an element of the interior of the completely positive cone. As a further application, this method can be extended to find a symmetric nonnegative matrix factorization, where we consider an additional low-rank constraint. Here again, the method to derive factorizations for completely positive matrices can be used, albeit with some further adjustments, introduced in Section 8.1. Moreover, we will see that even for the general case of deriving a nonnegative matrix factorization for a given rectangular matrix A, the key aspects of the completely positive factorization approach can be used. To this end, it becomes necessary to extend the idea of finding a completely positive factorization such that it can be used for rectangular matrices. This yields an applicable algorithm for nonnegative matrix factorization in Section 8.2. Numerical results for this approach will suggest that the presented algorithms and techniques to obtain completely positive matrix factorizations can be extended to general nonnegative factorization problems.
Die Dissertation beschäftigt sich mit einer neuartigen Art von Branch-and-Bound Algorithmen, deren Unterschied zu klassischen Branch-and-Bound Algorithmen darin besteht, dass
das Branching durch die Addition von nicht-negativen Straftermen zur Zielfunktion erfolgt
anstatt durch das Hinzufügen weiterer Nebenbedingungen. Die Arbeit zeigt die theoretische Korrektheit des Algorithmusprinzips für verschiedene allgemeine Klassen von Problemen und evaluiert die Methode für verschiedene konkrete Problemklassen. Für diese Problemklassen, genauer Monotone und Nicht-Monotone Gemischtganzzahlige Lineare Komplementaritätsprobleme und Gemischtganzzahlige Lineare Probleme, präsentiert die Arbeit
verschiedene problemspezifische Verbesserungsmöglichkeiten und evaluiert diese numerisch.
Weiterhin vergleicht die Arbeit die neue Methode mit verschiedenen Benchmark-Methoden
mit größtenteils guten Ergebnissen und gibt einen Ausblick auf weitere Anwendungsgebiete
und zu beantwortende Forschungsfragen.
Two areas were selected to represent major process regimes of Mediterranean rangelands. In the County of Lagads (Greece), situated east of the city of Thessaloniki, livestock grazing with sheep and goats is a major factor of the rural economy. In suitable areas, it is complemented by agricultural use. The region of Ayora (Spain) is located west of the city of Valencia. It is one of regions most affected by fires in Spain. First of all, long time series of satellite data were compiled for both regions on the basis of Landsat sensors, which cover the time until 1976 (Ayora) and 1984 (Lagadas) with one image per year. Using a rigorous processing scheme, the data were geometrically and radiometrically corrected Specific attention was given to an exact sensor calibration, the radiometric intercalibration of Landsat-TM and "MSS. Proportional cover of photosynthetically active vegetation was identified as a suitable quantitative indicator for assessing the state of rangelands. Using Spectral Mixture Analysis (SMA) it was inferred for all data sets. The extensive data base procured this way enabled to map fire events in the Ayora area based on sequential diachronic sets and provide fire dates, perimeter as well as fire recurrence for each pixel. The increasing fire frequency in the past decades is in large parts attributed to the accelerated abandonment of the area that leads to an encroachment of shrublands and the accumulation of combustible biomass. On the basis of the fire mapping results, a spatial and temporal stratification of the data set allowed to asses plant recovery dynamics on the landscape level through linear trend analysis. The long history of fire events in the Mediterranean frequently leads to processes of auto-succession. Following an initial dominance of herbaceous vegetation this commonly leads to similar plant communities as the ones present before the fire. On a temporal axis, this results in typical exponential post-fire trajectories which could also be shown in this study. The analysis of driving factors for post-fire dynamics confirmed the importance of aspect and slope. Locations with lower amounts of solar irradiation and favourable water supply yielded faster recovery rates and higher post-fire vegetation cover levels. In most cases, the vegetation cover levels observed before the fire were not reached within the post-fire observation period. In the area of Lagadas, linear trend analysis and additional statistical parameters were used to infer a degradation index. This could be used to illustrate a complex pattern of stability, regeneration and degradation of vegetation cover. These different processes and states are found in close proximity and are clearly determined by topography and elevation. Following a sequence of analyses, it was found that in particular steep, narrow valleys show positive trends, while negative trends are more abundant on plain or gently undulating areas. Considering the local grazing regime, this spatial differentiation was related to the accessibility of specific locations. Subsequently, animal numbers on community level were used to calculate efficient stocking rates and assess the temporal development of their relation with vegetation cover. This calculation of temporal trajectories illustrated that only some communities show the expected negative relation. To the contrary, a positive relation or even changing relation patterns are observed. This signifies recent concentration and intensification processes in the grazing scheme, as a result of which animals are kept in sheds, where additional feedstuffs are provided. In these cases, free roaming of livestock animals is often confined to some hours every day, which explains the spatial preference of easily accessible areas by the shepherds. Beyond these temporal trends, it was analysed whether the grazing pattern is equally reflected in a spatial trend. Making use of available geospatial information layers, the efforts required to reach each location was expressed as a cost. Then, cost zones could be defined and woody vegetation cover as a grazing indicator could be inferred for the different zones. Animal sheds were employed as starting features for this piospheric analysis, which could be mapped from very high spatial resolution Quickbird image data. The result was a clearly structured gradient showing increasing woody vegetation cover with increasing cost distance. On the basis of these two pilot studies, the elements of a monitoring and interpretation framework identified at the beginning of the work were evaluated and a formal interpretation scheme was presented.
Abenteuer Sprache: Spanisch und Französisch im Kontakt mit amerikanischen Ureinwohnersprachen
(2006)
Sprache - ein Abenteuer?! Spanisch und Französisch - beide romanischen Sprachen sind in Amerika mit Ureinwohnersprachen in Kontakt getreten. In der vorliegenden Dissertation wird der Sprachkontakt zwischen Spanisch und den Maya-Sprachen (Maya yucateco) mit dem Kontakt zwischen Französisch und den Algonkin-Sprachen (Montagnais) vergleichend untersucht. Besondere Schwerpunkte bilden dabei die ersten Kontakte, die Fixierung der Ureinwohnersprachen durch die Europäer und die bis heute in Yucatán und Québec spürbaren linguistischen Konsequenzen dieser europäisch-indigenen Sprachdurchdringung. Obwohl der Einfluss der genannten Ureinwohnersprachen auf die beiden romanischen Sprachen insgesamt gesehen für relativ gering gehalten wird, stellt er einen wichtigen und interessanten Aspekt bei der Sprachbetrachtung dar. Ohne das Werk der europäischen Missionare hätte die Reichhaltigkeit der indigenen Sprachen nicht bis zum heutigen Tage lebendig gehalten werden können. Wir verdanken ihnen die ersten "sprachwissenschaftlichen Abhandlungen, mit denen sie einen beachtlichen Beitrag für die Linguistik geleistet haben, auch wenn dieser von Zeit zu Zeit sehr kontrovers diskutiert wird. Der vermutete Überhang an Indigenismen im heutigen amerikanischen Spanisch kann im Anschluss an meine Untersuchungen bestätigt werden. Während im Französischen Québecs die Entlehnungen aus dem Montagnais vorwiegend im Bereich der Flora und Fauna anzutreffen sind, hat sich im yukatekischen Spanisch im Raum Mérida eine Art "Regionalcode" herausgebildet, der etliche Einflüsse aus dem Maya yucateco enthält. Die Reise der beiden Sprachen über den Ozean, die Kontaktaufnahme ihrer Sprecher, die Arbeit der Missionare und die Ergebnisse des Sprachkontakts sind in jeder Hinsicht so abenteuerlich wie eine Reise in die untersuchten Sprachgebiete!
Die vorliegende Arbeit entstand im Rahmen des EU INTERREG NWE IVB Projektes "ForeStClim - Transnational Forestry Management Strategies in Response to Regional Climate Change Impacts". Zum Zweck der Verbesserung des Prozessverständnisses von Abflussprozessen in Wäldern sowie zur Validierung und Weiterentwicklung eines GIS-basierten Tools (GIS-DRP) zur Erstellung von Abflussprozesskarten wurden auf 25 Test-Plots in vier Einzugsgebieten in Rheinland-Pfalz und dem Großherzogtum Luxemburg boden-hydrologische Untersuchungen durchgeführt. Auf Grundlage dieser Untersuchungen konnten große intraspezifische Unterschiede im Abflussverhalten von Waldstandorten erhoben werden. Die Differenzen werden dabei hauptsächlich durch das Substrat, die bodenphysikalischen Eigenschaften, die Nutzung bzw. deren Intensität und die Vorfeuchte bedingt. Es wurde nachgewiesen, dass Wälder generell hohe Infiltrationsraten aufweisen und verzögerte Zwischenabflussprozesse begünstigen. Durch einen prinzipiell naturnahen Waldbau und etwaige Meliorationsmaßnahmen auf Niederertragsstandorten bestehen zudem Möglichkeiten positiv auf die Wasserretention und das Wasserspeichervermögen eines Forstbestandes einzuwirken. Die mittels GIS-DRP erstellten Abflussprozesskarten der vier Testgebiete wurden durch die Ergebnisse der Geländeuntersuchungen sowie der Abflussprozesskartierung nach SCHERRER (2006) validiert. Hierdurch wurden für die Abflussgenerierung wichtige Parameter ermittelt und Optimierungsansätze erarbeitet, welche anschließend in GIS-DRP implementiert werden konnten. Verschlämmungsprozesse auf Ackerflächen können nun durch das modifizierte GIS-DRP-Werkzeug identifiziert werden. Zudem war es möglich, Extrem-Ereignis basierte Abflussprozesskarten zu etablieren, die Hot Spots der Abflussgenerierung identifizieren können. Die Einführung des Abflussprozesses "dSSF" (tiefer Zwischenabfluss) wurde durch eine neue Klassifizierung des geologischen Ausgangssubstrates erreicht. Forstwirten und Entscheidungsträgern im Waldmanagement wird somit die Möglichkeit geboten, Expertenwissen in ihre Planungen einfließen zu lassen. Hierdurch kann zum einen positiv auf den Landschaftswasserhaushalt eingewirkt werden, da gezielt auf Flächen nachteiliger Abflussbildung geeignete Maßnahmen des dezentralen Hochwasserschutzes angewandt werden können. Zum anderen werden Potentiale für bestmögliche Waldwachstumsvoraussetzungen in einem Landschaftsraum aufgezeigt. Der nachhaltigen Nutzung von Wäldern wird somit auch im Kontext des Klimawandels Rechnung getragen.
The following dissertation contains three studies examining academic boredom development in five high-track German secondary schools (AVG-project data; Study 1: N = 1,432; Study 2: N = 1,861; Study 3: N = 1,428). The investigation period spanned 3.5 years, with four waves of measurement from grades 5 to 8 (T1: 5th grade, after transition to secondary school; T2: 5th grade, after mid-term evaluations; T3: 6th grade, after mid-term evaluations; T4: 8th grade, after mid-term evaluations). All three studies featured cross-sectional and longitudinal analyses, separating, and comparing the subject domains of mathematics and German.
Study 1 provided an investigation of academic boredom’s factorial structure alongside correlational and reciprocal relations of different forms of boredom and academic self-concept. Analyses included reciprocal effects models and latent correlation analyses. Results indicated separability of boredom intensity, boredom due to underchallenge and boredom due to overchallenge, as separate, correlated factors. Evidence for reciprocal relations between boredom and academic self-concept was limited.
Study 2 examined the effectiveness and efficacy of full-time ability grouping for as a boredom intervention directed at the intellectually gifted. Analyses included propensity score matching, and latent growth curve modelling. Results pointed to limited effectiveness and efficacy for full-time ability grouping regarding boredom reduction.
Study 3 explored gender differences in academic boredom development, mediated by academic interest, academic self-concept, and previous academic achievement. Analyses included measurement invariance testing, and multiple-indicator-multi-cause-models. Results showed one-sided gender differences, with boys reporting less favorable boredom development compared to girls, even beyond the inclusion of relevant mediators.
Findings from all three studies were embedded into the theoretical framework of control-value theory (Pekrun, 2006; 2019; Pekrun et al., 2023). Limitations, directions for future research, and practical implications were acknowledged and discussed.
Overall, this dissertation yielded important insights into boredom’s conceptual complexity. This concerned factorial structure, developmental trajectories, interrelations to other learning variables, individual differences, and domain specificities.
Keywords: Academic boredom, boredom intensity, boredom due to underchallenge, boredom due to overchallenge, ability grouping, gender differences, longitudinal data analysis, control-value theory
Energy transport networks are one of the most important infrastructures for the planned energy transition. They form the interface between energy producers and consumers and their features make them good candidates for the tools that mathematical optimization can offer. Nevertheless, the operation of energy networks comes with two major challenges. First, the nonconvexity of the equations that model the physics in the network render the resulting problems extremely hard to solve for large-scale networks. Second, the uncertainty associated to the behavior of the different agents involved, the production of energy, and the consumption of energy make the resulting problems hard to solve if a representative description of uncertainty is to be considered.
In this cumulative dissertation we study adaptive refinement algorithms designed to cope with the nonconvexity and stochasticity of equations arising in energy networks. Adaptive refinement algorithms approximate the original problem by sequentially refining the model of a simpler optimization problem. More specifically, in this thesis, the focus of the adaptive algorithm is on adapting the discretization and description of a set of constraints.
In the first part of this thesis, we propose a generalization of the different adaptive refinement ideas that we study. We sequentially describe model catalogs, error measures, marking strategies, and switching strategies that are used to set up the adaptive refinement algorithm. Afterward, the effect of the adaptive refinement algorithm on two energy network applications is studied. The first application treats the stationary operation of district heating networks. Here, the strength of adaptive refinement algorithms for approximating the ordinary differential equation that describes the transport of energy is highlighted. We introduce the resulting nonlinear problem, consider network expansion, and obtain realistic controls by applying the adaptive refinement algorithm. The second application concerns quantile-constrained optimization problems and highlights the ability of the adaptive refinement algorithm to cope with large scenario sets via clustering. We introduce the resulting mixed-integer linear problem, discuss generic solution techniques, make the link with the generalized framework, and measure the impact of the proposed solution techniques.
The second part of this thesis assembles the papers that inspired the contents of the first part of this thesis. Hence, they describe in detail the topics that are covered and will be referenced throughout the first part.
Bei der Preisberechnung von Finanzderivaten bieten sogenannte Jump-diffusion-Modelle mit lokaler Volatilität viele Vorteile. Aus mathematischer Sicht jedoch sind sie sehr aufwendig, da die zugehörigen Modellpreise mittels einer partiellen Integro-Differentialgleichung (PIDG) berechnet werden. Wir beschäftigen uns mit der Kalibrierung der Parameter eines solchen Modells. In einem kleinste-Quadrate-Ansatz werden hierzu Marktpreise von europäischen Standardoptionen mit den Modellpreisen verglichen, was zu einem Problem optimaler Steuerung führt. Ein wesentlicher Teil dieser Arbeit beschäftigt sich mit der Lösung der PIDG aus theoretischer und vor allem aus numerischer Sicht. Die durch ein implizites Zeitdiskretisierungsverfahren entstandenen, dicht besetzten Gleichungssysteme werden mit einem präkonditionierten GMRES-Verfahren gelöst, was zu beinahe linearem Aufwand bezüglich Orts- und Zeitdiskretisierung führt. Trotz dieser effizienten Lösungsmethode sind Funktionsauswertungen der kleinste-Quadrate-Zielfunktion immer noch teuer, so dass im Hauptteil der Arbeit Modelle reduzierter Ordnung basierend auf Proper Orthogonal Decomposition Anwendung finden. Lokale a priori Fehlerabschätzungen für die reduzierte Differentialgleichung sowie für die reduzierte Zielfunktion, kombiniert mit einem Trust-Region-Ansatz zur Globalisierung liefern einen effizienten Algorithmus, der die Rechenzeit deutlich verkürzt. Das Hauptresultat der Arbeit ist ein Konvergenzbeweis für diesen Algorithmus für eine weite Klasse von Optimierungsproblemen, in die auch das betrachtete Kalibrierungsproblem fällt.
Water-deficit stress, usually shortened to water- or drought stress, is one of the most critical abiotic stressors limiting plant growth, crop yield and quality concerning food production. Today, agriculture consumes about 80-90% of the global freshwater used by humans and about two thirds are used for crop irrigation. An increasing world population and a predicted rise of 1.0-2.5-°C in the annual mean global temperature as a result of climate change will further increase the demand of water in agriculture. Therefore, one of the most challenging tasks of our generation is to reduce the amount water used per unit yield to satisfy the second UN Sustainable Development Goal and to ensure global food security. Precision agriculture offers new farming methods with the goal to improve the efficiency of crop production by a sustainable use of resources. Plant responses to water stress are complex and co-occur with other environmental stresses under natural conditions. In general, water stress causes plant physiological and biochemical changes that depend on the severity and the duration of the actual plant water deficit. Stomatal closure is one of the first responses to plant water stress causing a decrease in plant transpiration and thus an increase in plant temperature. Prolonged or severe water stress leads to irreversible damage to the photosynthetic machinery and is associated with decreasing chlorophyll content and leaf structural changes (e.g., leaf rolling). Since a crop can already be irreversibly damaged by only mild water deficit, a pre-visual detection of water stress symptoms is essential to avoid yield loss. Remote sensing offers a non-destructive and spatio-temporal method for measuring numerous physiological, biochemical and structural crop characteristics at different scales and thus is one of the key technologies used in precision agriculture. With respect to the detection of plant responses to water stress, the current state-of-the-art hyperspectral remote sensing imaging techniques are based on measurements of thermal infrared emission (TIR; 8-14 -µm), visible, near- and shortwave infrared reflectance (VNIR/SWIR; 0.4-2.5 -µm), and sun-induced fluorescence (SIF; 0.69 and 0.76 -µm). It is, however, still unclear how sensitive these techniques are with respect to water stress detection. Therefore, the overall aim of this dissertation was to provide a comparative assessment of remotely sensed measures from the TIR, SIF, and VNIR/SWIR domains for their ability to detect plant responses to water stress at ground- and airborne level. The main findings of this thesis are: (i) temperature-based indices (e.g., CWSI) were most sensitive for the detection of plant water stress in comparison to reflectance-based VNIR/SWIR indices (e.g., PRI) and SIF at both, ground- and airborne level, (ii) for the first time, spectral emissivity as measured by the new hyperspectral TIR instrument could be used to detect plant water stress at ground level. Based on these findings it can be stated that hyperspectral TIR remote sensing offers great potential for the detection of plant responses to water stress at ground- and airborne level based on both TIR key variables, surface temperature and spectral emissivity. However, the large-scale application of water stress detection based on hyperspectral TIR measures in precision agriculture will be challenged by several problems: (i) missing thresholds of temperature-based indices (e.g., CWSI) for the application in irrigation scheduling, (ii) lack of current TIR satellite missions with suitable spectral and spatial resolution, (iii) lack of appropriate data processing schemes (including atmosphere correction and temperature emissivity separation) for hyperspectral TIR remote sensing at airborne- and satellite level.
Death is perceived as a severe threat to the self. Although it is certain that everyone has to die, people usually don't think about the finiteness of their life. Everything reminding of death is ignored, rationalized and death-related thoughts and fears are pushed out of mind (TMT; Pyszczynski et al., 1999). However, people differ in their ability to regulate negative affect and to access their self-system (Kuhl, 2001). As death is assumed to arouse existential fears, the ability to regulate such fears is particularly important, higher self-access could be relevant in defending central personal values. This thesis aimed at showing existential fears under mortality salience and effects of self-regulation of affect under mortality salience. In two studies (Chapter 2) implicit negative affect under mortality salience was demonstrated. An additional study (Chapter 3) shows the effects of self-regulation on implicit negative affect, whereas four studies in Chapter 4 displayed differences in self-access under mortality salience depending on people- ability of self-regulating negative affect.