Inspired by work of Hubery [Hub] and Fomin, Shapiro and Thurston [FST06] related to cluster algebras, we construct a bijection between certain curves on a cylinder and the string modules over a path algebra of type Ãn. We show that under this bijection irreducible maps and the Auslander-Reiten translation have a geometric interpretation. Furthermore we prove that the dimension of extension groups can be expressed in terms of intersection numbers. Finally we explain the connection to cluster algebras and apply our results to describe the exchange graph in type Ãn. / Angeregt durch Arbeiten zu Cluster-Algebren von Hubery [Hub] und Fomin, Shapiro und Thurston [FST06] konstruieren wir eine Bijektion zwischen gewissen Kurven auf einem Zylinder und den Fadenmoduln über einer Wege-Algebra vom Typ Ãn. Wir zeigen, daß unter dieser Bijektion sowohl irreduzible Abbildungen als auch die Auslander-Reiten-Verschiebung eine geometrische Interpretation haben. Weiterhin beweisen wir, daß sich die Dimension der Erweiterungsgruppen mittels Anzahlen von Schnittpunkten ausdrücken läßt. Schließlich erklären wir die Verbindung zu Cluster-Algebren und verwenden unsere Ergebnisse um den Austauschgraph im Typ Ãn zu beschreiben.

Title: Modules over Gorenstein rings Author: David Pospíšil Department: Department of Algebra Supervisor: Prof. RNDr. Jan Trlifaj, DSc. Supervisor's e-mail address: trlifaj@karlin.mff.cuni.cz Abstract: The dissertation collects my actual contributions to the clas- sification of (co)tilting modules and classes over Gorenstein rings. Com- pared with the original intent we get a more general result in classification of (co)tilting classes namely for general commutative noetherian rings (see the third paper in this dissertation). The dissertation consists of an introduction and three papers with coauthors. The first paper (published in Contemp. Math.) contains a classification of all (co)tilting modules and classes over 1- Gorenstein commutative rings. The second paper (published in J. Algebra) contains a classification of all tilting classes over regular rings of Krull dimen- sion 2 and also a classification of all tilting modules in the local case. Finally the third paper (preprint) contains a classification of all (co)tilting classes and also torsion pairs over general commutative noetherian rings. All these classi- fications are in terms of subsets of the spectrum of the ring and by associated prime ideals of modules. Keywords: (co)tilting module, (co)tilting class, torsion pair, Gorenstein ring, regular ring,...

The thesis deals with the issue of measuring the height of the center of gravity of SUV´s. The first part of thesis is devoted to the issue of center of gravity and the influence of the height of the center of gravity on driving stability. Furthermore, individual methods of measuring the height of the center of gravity in passenger cars are discussed. The analytic part of the thesis is devoted to experimental measurement of the height of the center of gravity of selected SUV´s at different vehicle load, by the method of tilting the vehicle on the axle. The results of the measurements showed that the height of the center of gravity for vehicles with a capacity for five people is in the range from 606 to 697 mm. The lowest value belongs to the vehicle Toyota RAV4 Hybrid, when loaded by the driver. The highest value was found for the vehicle Kia Sportage, when loaded by three passengers. The Landover Defender, which is vehicle, that has a capacity for nine people, was found to have a height of center of gravity ranging from 705 to 874 mm. The lowest value was found when loaded by the driver, the highest value when loaded by nine passengers. Only for the Nissan Qashqai was the growing dependence between the load increase and the height position of the vehicle's center of gravity confirmed. A similar trend was found for the Toyota RAV4 Hybrid, Jeep Compass and Landover Defender. The determined values can be used in the analysis of traffic accidents of SUV’s.

The aim of this thesis is the design of the rotary tilting table controlled the fourth and fifth axis for CNC centrum.První part includes the search for pivotally tilting tables. The second part includes the choice of technical paremetrů pivotally tilting table for selected CNC center, design of possible options , selection of appropriate options, design of the selected option.

Inspired by work of Hubery [Hub] and Fomin, Shapiro and Thurston [FST06] related to cluster algebras, we construct a bijection between certain curves on a cylinder and the string modules over a path algebra of type Ãn. We show that under this bijection irreducible maps and the Auslander-Reiten translation have a geometric interpretation. Furthermore we prove that the dimension of extension groups can be expressed in terms of intersection numbers. Finally we explain the connection to cluster algebras and apply our results to describe the exchange graph in type Ãn. / Angeregt durch Arbeiten zu Cluster-Algebren von Hubery [Hub] und Fomin, Shapiro und Thurston [FST06] konstruieren wir eine Bijektion zwischen gewissen Kurven auf einem Zylinder und den Fadenmoduln über einer Wege-Algebra vom Typ Ãn. Wir zeigen, daß unter dieser Bijektion sowohl irreduzible Abbildungen als auch die Auslander-Reiten-Verschiebung eine geometrische Interpretation haben. Weiterhin beweisen wir, daß sich die Dimension der Erweiterungsgruppen mittels Anzahlen von Schnittpunkten ausdrücken läßt. Schließlich erklären wir die Verbindung zu Cluster-Algebren und verwenden unsere Ergebnisse um den Austauschgraph im Typ Ãn zu beschreiben.

In radiation therapy treatment planning, recent works have used machine learning based on historically delivered plans to automate the process of producing clinically acceptable plans. Compared to traditional approaches such as repeated weighted-sum optimization or multicriteria optimization (MCO), automated planning methods have, in general, the benefits of low computational times and minimal user interaction, but on the other hand lack the flexibility associated with general-purpose frameworks such as MCO. Machine learning approaches can be especially sensitive to deviations in their dose prediction due to certain properties of the optimization functions usually used for dose mimicking and, moreover, suffer from the fact that there exists no general causality between prediction accuracy and optimized plan quality.In this thesis, we present a means of unifying ideas from machine learning planning methods with the well-established MCO framework. More precisely, given prior knowledge in the form of either a previously optimized plan or a set of historically delivered clinical plans, we are able to automatically generate Pareto optimal plans spanning a dose region corresponding to plans which are achievable as well as clinically acceptable. For the former case, this is achieved by introducing dose--volume constraints; for the latter case, this is achieved by fitting a weighted-data Gaussian mixture model on pre-defined dose statistics using the expectation--maximization algorithm, modifying it with exponential tilting and using specially developed optimization functions to take into account prediction uncertainties.Numerical results for conceptual demonstration are obtained for a prostate cancer case with treatment delivered by a volumetric-modulated arc therapy technique, where it is shown that the methods developed in the thesis are successful in automatically generating Pareto optimal plans of satisfactory quality and diversity, while excluding clinically irrelevant dose regions. For the case of using historical plans as prior knowledge, the computational times are significantly shorter than those typical of conventional MCO. / Inom strålterapiplanering har den senaste forskningen använt maskininlärning baserat på historiskt levererade planer för att automatisera den process i vilken kliniskt acceptabla planer produceras. Jämfört med traditionella angreppssätt, såsom upprepad optimering av en viktad målfunktion eller flermålsoptimering (MCO), har automatiska planeringsmetoder generellt sett fördelarna av lägre beräkningstider och minimal användarinteraktion, men saknar däremot flexibiliteten hos allmänna ramverk som exempelvis MCO. Maskininlärningsmetoder kan vara speciellt känsliga för avvikelser i dosprediktionssteget på grund av särskilda egenskaper hos de optimeringsfunktioner som vanligtvis används för att återskapa dosfördelningar, och lider dessutom av problemet att det inte finns något allmängiltigt orsakssamband mellan prediktionsnoggrannhet och kvalitet hos optimerad plan. I detta arbete presenterar vi ett sätt att förena idéer från maskininlärningsbaserade planeringsmetoder med det väletablerade MCO-ramverket. Mer precist kan vi, givet förkunskaper i form av antingen en tidigare optimerad plan eller en uppsättning av historiskt levererade kliniska planer, automatiskt generera Paretooptimala planer som täcker en dosregion motsvarande uppnåeliga såväl som kliniskt acceptabla planer. I det förra fallet görs detta genom att introducera dos--volym-bivillkor; i det senare fallet görs detta genom att anpassa en gaussisk blandningsmodell med viktade data med förväntning--maximering-algoritmen, modifiera den med exponentiell lutning och sedan använda speciellt utvecklade optimeringsfunktioner för att ta hänsyn till prediktionsosäkerheter.Numeriska resultat för konceptuell demonstration erhålls för ett fall av prostatacancer varvid behandlingen levererades med volymetriskt modulerad bågterapi, där det visas att metoderna utvecklade i detta arbete är framgångsrika i att automatiskt generera Paretooptimala planer med tillfredsställande kvalitet och variation medan kliniskt irrelevanta dosregioner utesluts. I fallet då historiska planer används som förkunskap är beräkningstiderna markant kortare än för konventionell MCO.

Construction sites are among the most common areas to experience soil erosion and sediment transport due to the mandatory foundation tasks such as excavation and land grubbing. Thus, temporary sediment barriers are installed along the perimeter to prevent sediment transport from the site. Erosion and sediment transport control measures may include, but not limited to, physical and chemical processes such as the use of a silt fence and polyacrylamide product. Runoff from construction sites and other impervious surfaces are routinely discharged into ponds for treatment before being released into a receiving water body. Stormwater harvesting from a pond for irrigation of adjacent lands is promoted as one approach to reducing pond discharge while supplementing valuable potable water used for irrigation. The reduction of pond discharge reduces the mass of pollutants in the discharge. In the dissertation, presented is the investigation of the effectiveness of temporary sediment barriers and then, development of a modeling approach to a stormwater harvesting pond to provide a comprehensive stormwater management pollution reduction assessment tool. The first part of the research presents the investigation of the performance efficiencies of silt fence fabrics in turbidity and sediment concentration removal, and the determination of flowthrough-rate on simulated construction sites in real time. Two silt fence fabrics, (1) woven and the other (2) nonwoven were subjected to material index property tests and a series of field-scale tests with different rainfall intensities and events for different embankment slopes on a tilting test-bed. Collected influent and effluent samples were analyzed for sediment concentration and turbidity, and the flow-through-rate for each fabric was evaluated. Test results revealed that the woven and nonwoven silt fence achieved 11 and 56 percent average turbidity reduction iv efficiency, respectively. Each fabric also achieved 20 and 56 percent average sediment concentration removal efficiency, respectively. Fabric flow-through-rates were functions of the rainfall intensity and embankment slope. The nonwoven fabric exhibited higher flow-throughrates than the woven fabric in both field-scale and laboratory tests. In the second part of the study, a Stormwater Harvesting and Assessment for Reduction of Pollution (SHARP) model was developed to predict operation of wet pond used for stormwater harvesting. The model integrates the interaction of surface water and groundwater in a catchment area. The SHARP model was calibrated and validated with actual pond water elevation data from a stormwater pond at Miramar Lakes, Miramar, Florida. Model evaluation showed adequate prediction of pond water elevation with root mean square error between 0.07 and 0.12 m; mean absolute error was between 0.018 and 0.07 m; and relative index of agreement was between 0.74 and 0.98 for both calibration and validation periods. The SHARP model is capable of assessing harvesting safe-yield and discharge from a pond, including the prediction of the percentage of runoff into a harvesting pond that is not discharged. The combination of silt fence and/or polyacrylamide PAM before stormwater harvesting pond in a treatment train for the reduction of pollutants from construction sites has the potential of significantly exceeding a performance standard of 85 percent reduction typically required by local authorities. In fact, the stringent requirement of equaling pre- and post-development pollutant loading is highly achievable by the treatment train approach. The significant contribution from the integration of the SHARP model to the treatment train is that real-time assessment of pollutant loading reduction by volume can be planned and controlled to achieve target performance standards.

Mit dem Grundwasserwiederanstieg in Braunkohleabraumkippen werden die aus der Pyritverwitterung resultierenden Stoffausträge an Sulfat-, Eisen-, Schwermetall- und H+-Ionen gelöst. Im Rahmen dieser Dissertation wurde eine Methodik entwickelt, mit deren Hilfe Problembereiche ausgehalten und somit Maßnahmen im aktiven mitteldeutschen Tagebaubetrieb ergriffen werden können, um die zukünftige Beeinflussung der umgebenden Grund- und Oberflächenwasserkörper zu minimieren. An Vorfeldsedimenten konnten in Feld-eluaten, Stoßbeprobungen und Verwitterungsversuchen geochemische Eigenschaften ermittelt werden. In resultierenden Pufferungsversuchen aus karbonathaltigen Geschiebe-mergeln und Hauptaciditätsträgern konnte eine langfristige Minderung der Aciditätswirkung nachgewiesen werden und ein adaptierter Regelkippenaufbau mit laminaren, alterierenden Sichtaufbau begründet werden. In umliegenden Altkippengrundwässern sind Pufferung und Sulfatreduktion als Wiederfestlegungsprozesse der AMD-Problemstoffe belegt worden. / For lignite mining extensive overburden masses have to be moved. Due to the ventilation of the overburden by atmospheric oxygen, there is a weathering of mostly tertiary sulfides. The rebound of groundwater in future tippings dissolves sulfate, iron, heavy metal and H+ ions, resulting from the pyrite weathering. The partial mobilization of overburden sulfides are opposed to hydrogeochemical buffer reactions e.g. the buffering by carbonates as the first step of buffering. Therefore, there are the questions to the mining operators of the measures that can be taken to minimize the geochemical influence of the surrounding ground and surface water bodies. Object of this PhD-thesis is to lead a methodology that helps to characterize the future tilting substrates to find technological and strategic measures for minimizing the acid water formation in the active open pit operation. In Field eluates and weathering tests in the laboratory and in the field, sediments from dry drill holes in the forefront of open pits “Schleenhain” and “Peres” it could be shown that the geological facies formation of sediments has a decisive influence on geochemical characteristics. As the main acidifying sediment the tertiary aquifer number 2 (lying part) and number 3 can be identified with their high sulfur contents. With increasing time of oxygen exposure sulfate, iron, heavy metal and H + ions released massively. Furthermore, it appears that carbonate buffer essentially are available as glacial till only in cohesive Quaternary. With the resulting buffering experiments from glacial till and the most acidic aquifer sediments a long-term retention of iron, heavy metal and H + - ion and a reduction of sulfate release can be shown, if there is a share of at least 40% glacial till to the lying aquifer number 2 sediments or 20% glacial till to the aquifer number 3. The groundwater quality monitoring of unstructured resaturated old dumps near to the active open pits is comparable to a field test. In addition to weathering zones with high levels of pollutants in the presence of carbonates, buffering processes and sulfate reduction with precipitation of problematic substances in secondary mineral phases can be detected. Blending the research results of geological and geochemical data, an important, in principle selectively recoverable, buffer potential already exists. The determined mixing ratio from 80-60 mass-% acidic sediments to 20-40 mass-% buffering sediment from the buffering experiments can be realized in tilting. In the open pit “Schleenhain” the missing buffering material can be compensated by mass offset from the open pit “Peres”. With the use of the already existing equipment, it is possible to establish a laminar, alternately tipping body with good geochemical and geotechnical conditions.

Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the exchange graph. This allows us to provide a good global description of the exchange graphs of arbitrary mutation-infinite quivers. In particular we show that the exchange graph of an acyclic quiver is a tree if (and in fact only if) any two vertices are connected by at least two arrows. Furthermore we give classification results for the simplicial complexes and thereby obtain a partial positive answer to Happel's question. Another consequence of our findings is a confirmation of Unger's conjecture about the infinite number of components of the tilting exchange graph in all but finitely many cases. Finally we generalise and conceptualise our results by introducing what we call "polynomial quivers", stating several conjectures about "polynomial quiver mutation", and giving proofs in special cases.

L'objet de cette thèse est de proposer un cadre complet, du haut niveau au bas niveau, de génération de trajectoires pour un groupe de systèmes dynamiques indépendants. Ce cadre, basé sur la résolution de l'équation de Burgers pour la génération de trajectoires, est appliqué à un modèle original de drone trirotor et utilise la platitude des deux systèmes différentiels considérés. La première partie du manuscrit est consacrée à la génération de trajectoires. Celle-ci est effectuée en créant formellement, par le biais de la platitude du système considéré, des solutions à l'équation de la chaleur. Ces solutions sont transformées en solution de l'équation de Burgers par la transformation de Hopf-Cole pour correspondre aux formations voulues. Elles sont optimisées pour répondre à des contraintes spécifiques. Plusieurs exemples de trajectoires sont donnés.La deuxième partie est consacrée au suivi autonome de trajectoire par un drone trirotor. Ce drone est totalement actionné et un contrôleur en boucle fermée non-linéaire est proposé. Celui-ci est testé en suivant, en roulant, des trajectoires au sol et en vol. Un modèle est présenté et une démarche pour le contrôle est proposée pour transporter une charge pendulaire. / This thesis is dedicated to the creation of a complete framework, from high-level to low-level, of trajectory generation for a group of independent dynamical systems. This framework, based for the trajectory generation, on the resolution of Burgers equation, is applied to a novel model of trirotor UAV and uses the flatness of the two levels of dynamical systems.The first part of this thesis is dedicated to the generation of trajectories. Formal solutions to the heat equation are created using the differential flatness of this equation. These solutions are transformed into solutions to Burgers' equation through Hopf-Cole transformation to match the desired formations. They are optimized to match specific requirements. Several examples of trajectories are given.The second part is dedicated to the autonomous trajectory tracking by a trirotor UAV. This UAV is totally actuated and a nonlinear closed-loop controller is suggested. This controller is tested on the ground and in flight by tracking, rolling or flying, a trajectory. A model is presented and a control approach is suggested to transport a pendulum load.