Universität Stuttgart
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Item Open Access Dominant dimensions of finite dimensional algebras(2012) Abrar, Muhammad; König, Steffen (Prof. Dr. rer. nat.)We study the dominant dimensions of three classes of finite dimensional algebras, namely hereditary algebras, quotient algebras of trees and serial algebras. We see that a branching vertex plays a key role to establish that the dominant dimension (dom.dim) of hereditary algebras (quivers) is at most one. We define arms of a tree and split trees into two classes: trees without arms and trees with arms. Like hereditary algebras, it turns out that the dominant dimension of the quotient algebras of trees can not exceed one. For serial algebras A associated to linearly oriented quiver with n vertices, we give lower and upper bounds of dom.dimA, and show that the bounds are optimal. It is also shown that some of the algebras A satisfy the conditions in the higher dimensional version of the Auslander's correspondence. Further we consider serial algebras corresponding to one-oriented-cycle quiver Q with n vertices, and give optimal bounds for a special subclass of these algebras. We conjecture that for any non self-injective quotient algebra A of Q dom.dimA is at most 2n-3, where the number of vertices n is bigger than 2.. Finally, we construct few examples of algebras having large (finite) dominant dimensions.Item Open Access Adaptive higher order discontinuous Galerkin methods for porous-media multi-phase flow with strong heterogeneities(2018) Kane, Birane; Siebert, Kunibert (Prof. Dr.)In this thesis, we develop, analyze, and implement adaptive discontinuous Galerkin (DG) finite element solvers for the efficient simulation of porous-media flow problems. We consider 2d and 3d incompressible, immiscible, two-phase flow in a possibly strongly heterogeneous and anisotropic porous medium. Discontinuous capillarypressure functions and gravity effects are taken into account. The system is written in terms of a phase-pressure/phase-saturation formulation. First and second order Adams-Moulton time discretization methods are combined with various interior penalty DG discretizations in space, such as the symmetric interior penalty Galerkin (SIPG), the nonsymmetric interior penalty Galerkin (NIPG) and the incomplete interior penalty Galerkin (IIPG). These fully implicit space time discretizations lead to fully coupled nonlinear systems requiring to build a Jacobian matrix at each time step and in each iteration of a Newton-Raphson method. We provide a stability estimate of the saturation and the pressure with respect to initial and boundary data. We also derive a-priori error estimates with respect to the L2(H1) norm for the pressure and the L∞(L2)∩L2(H1) norm for the saturation. Moving on to adaptivity, we implement different strategies allowing for a simultaneous variation of the element sizes, the local polynomial degrees and the time step size. These approaches allow to increase the local polynomial degree when the solution is estimated to be smooth and refine locally the mesh otherwise. They also grant more flexibility with respect to the time step size without impeding the convergence of the method. The aforementioned adaptive algorithms are applied in series of homogeneous, heterogeneous and anisotropic test cases. To our knowledge, this is the first time the concept of local hp-adaptivity is incorporated in the study of 2d and 3d incompressible, immiscible, two-phase flow problems. Delving into the issue of efficient linear solvers for the fully-coupled fully-implicit formulations, we implement a constrained pressure residual (CPR) two-stage preconditioner that exploits the algebraic properties of the Jacobian matrices of the systems. Furthermore, we provide an open-source DG two-phase flow simulator, based on the software framework DUNE, accompanied by a set of programs including instructions on how to compile and run them.Item Open Access Über Öffnungen und Neigungen von Unterräumen eines Banachraumes(1971) Gurarij, V. I.; Wagenknecht, Monika (Übersetzerin)In der vorliegenden Arbeit wird eine Reihe von Charakteristiken der wechselseitigen Anordnung von Unterräumen eines Banachraumes untersucht. Es werden einige Anwendungen in der Geometrie von Banachräumen angeführt.Item Open Access Fibrations of spheres by great spheres over division algebras and their differentiability(1990) Grundhöfer, Theo; Hähl, HermannThe authors answer the natural question: When is the fibration of S {2n-1} by great (n-1)-spheres determined by a division algebra differentiable? They show that this happens only for the classical Hopf fibrations, which are determined by the classical division algebras R, C, H and O.Item Open Access Über eine Klasse geschlossener Flächen(1969) Aleksandrov, A. D.; Wagenknecht, Monika (Übersetzerin)Wir betrachten die geschlossenen zweimal stetig differenzierbaren Flächen, die wir T-Flächen nennen, und die folgende Eigenschaften besitzen: 1) Eine T-Fläche besitzt Gebiete positiver und negativer Gaußscher Krümmung, die voneinander durch stückweise glatte Kurven getrennt sind; 2) die Gaußsche Krümmung verschwindet nur auf diesen Kurven (wir können annehmen, daß die Gaußsche Krümmung auf einer Menge von inneren Punkten der Gebiete, in denen sie das Zeichen nicht wechselt, verschwindet, deren Häufungspunkte eine nirgends-dichte Menge auf den Grenzkurven bilden); 3) die Totalkrümmung der Gebiete positiver Krümmung beträgt 4pi. Die Torusfläche stellt das einfachste Beispiel einer T-Fläche dar. Es existieren T-Flächen von beliebigem Geschlecht p größer gleich 1. Satz I. Auf jeder T-Fläche bilden die Gebiete positiver Krümmung ein zusammenhängendes Stück einer geschlossenen konvexen Fläche. Die Grenzkurven (die dieses einzige Gebiet mit positiver Krümmung von den Gebieten mit negativer Krümmung trennen) sind geschlossene konvexe Kurven, deren jede in einer Tangentialebene an die Fläche liegt. Satz II. Sind zwei analytische T-Flächen isometrisch, so sind sie entweder kongruent oder symmetrisch. Z.B. erlaubt die Torusfläche keine nichttrivialen isometrischen Abbildungen. Satz III. Jede analytische T-Fläche ist starr.Item Open Access Long wave approximation over and beyond the natural time scale(2024) Hofbauer, Sarah; Schneider, Guido (Prof. Dr.)Item Open Access Generalized polygons with doubly transitive ovoids(2013) Krinn, Boris; Stroppel, Markus (apl. Prof. Dr. rer. nat.)This thesis studies finite generalized quadrangles and hexagons which contain an ovoid. An ovoid is a set of mutually opposite points of maximum size. The first objective of the present work is to show that a generalized quadrangle or hexagon is a classical polygon if it contains an ovoid in the case of quadrangles or an ovoid-spread pairing in the case of hexagons, such that a group of given isomorphism type acts on the polygon in such a way that it leaves the ovoid invariant. The groups in use here are Suzuki and Ree groups. The second part of the thesis is devoted to the problem of determining all groups which can act on a generalized quadrangle or generalized hexagon in such a way that they operate doubly transitively on an ovoid of this polygon. It will turn out that these groups are essentially the known examples of groups acting on a classical or semi-classical ovoid of a classical polygon. In the first two chapters, meant as an introduction to the problem, constructions of the relevant generalized polygons are given and known results concerning the existence question of ovoids in known generalized quadrangles and hexagons are collected. In the following chapter, the known polarity of the symplectic quadrangle is presented and it is shown that the symplectic quadrangle can be reconstructed from the action of the Suzuki group on the absolute elements of this polarity. Then we show that any generalized quadrangle which contains an ovoid, such that a Suzuki group acts on the ovoid, is isomorphic to the symplectic quadrangle. In Chapter 4, analogous to the approach used for symplectic quadrangles, the known polarity of the split Cayley hexagon is described and it is shown that the split Cayley hexagon can be reconstructed from the action of the Ree group on the absolute points of the polarity. Then we show that any generalized hexagon that contains an ovoid-spread-pairing, on which a Ree group acts, is isomorphic to the classical split-Cayley hexagon. Both of these results are achieved without the use of classification results. The last three chapters are devoted to the problem of determining all groups which can act doubly transitively on an ovoid of a generalized quadrangle or generalized hexagon. This chapter uses the classification of finite simple groups (via the classification of finite doubly transitive groups). For hexagons the result is that only unitary groups and Ree groups are possible. This result, together with the one obtained in the previous chapter and a theorem by Joris De Kaey provides that the generalized hexagon is classical and the ovoid is classical if the ovoid belongs to an ovoid-spread-pairing. The result for quadrangles is less smooth. A further restriction on the order of the quadrangle is needed, namely that the number of points per line and the number of lines per point coincide and that this number is a prime power. This was not necessary in the case of the hexagons. With this additional assumption, we show that only orthogonal groups or Suzuki groups can act on these ovoids.Item Open Access Sketched stable planes(2003) Wich, Anke; Stroppel, Markus (Prof. Dr.)Standard objects in classical (topological) geometry are the real affine and hyperbolic planes. Both of them can be seen as (open) subplanes of the real projective plane (endowed with the standard topology) and thus share a common theory. This may serve as a brief illustration of the importance of the notion of embeddability. One particularly nice class of topological planes are the so called stable planes - in fact, the above examples are stable planes; as well as the projective planes over the real and complex numbers, Hamilton quaternions and Cayley octaves, the so called classical planes. Moreover, every open subplane of a stable plane again is a stable plane. Consequently, one way of understanding a given stable plane is trying to embed it into one of more profound acquaintanceship, preferredly one of the classical planes. An elegant way of constructing stable planes uses stable partitions of Lie groups. Planes of that type can be treated more efficiently studying these groups along with certain stabilisers, the so called sketches, rather than the original geometries. This method has so far yielded results in several cases where intrinsic methods had not been gratifying. Maier in his dissertation gives a classification of all 4-dimensional connected Lie groups which allow for a stable partition. Only one of them, the Frobenius group Gamma - the semidirect product of the real numbers and the 3-dimensional Heisenberg group - had not been expected, and it hosts an infinite number of stable partitions. Our objective is whether or not the resulting stable planes are embeddable into an already well known plane. Using sketches, it can be proved that none of these planes is embeddable into the classical complex projective plane. As an interesting counterpoint, those planes - hostile as they are towards being embedded into classical planes - do contain an abundance of both, affine and non-affine 2-dimensional classical subplanes. The full automorphism group of such a plane does not contain a certain selection of classical groups. Some conclusions can be drawn as to how soluble this automorphism group is : either it is soluble or it contains one copy of a subgroup with Lie algebra sl(2,R). The normaliser Gamma in the full automorphism group turns out to be soluble, after all. On a more general basis, the interplay of being a sketched geometry and a stable plane is studied : Is there any particular reason why all the examples of sketched stable planes so far have been point homogeneous geometries? And indeed, any line homegeneous sketched stable plane is necessarily flag homogeneous.Item Open Access Rigorous compilation for near-term quantum computers(2024) Brandhofer, Sebastian; Polian, Ilia (Prof.)Quantum computing promises an exponential speedup for computational problems in material sciences, cryptography and drug design that are infeasible to resolve by traditional classical systems. As quantum computing technology matures, larger and more complex quantum states can be prepared on a quantum computer, enabling the resolution of larger problem instances, e.g. breaking larger cryptographic keys or modelling larger molecules accurately for the exploration of novel drugs. Near-term quantum computers, however, are characterized by large error rates, a relatively low number of qubits and a low connectivity between qubits. These characteristics impose strict requirements on the structure of quantum computations that must be incorporated by compilation methods targeting near-term quantum computers in order to ensure compatibility and yield highly accurate results. Rigorous compilation methods have been explored for addressing these requirements as they exactly explore the solution space and thus yield a quantum computation that is optimal with respect to the incorporated requirements. However, previous rigorous compilation methods demonstrate limited applicability and typically focus on one aspect of the imposed requirements, i.e. reducing the duration or the number of swap gates in a quantum computation. In this work, opportunities for improving near-term quantum computations through compilation are explored first. These compilation opportunities are included in rigorous compilation methods to investigate each aspect of the imposed requirements, i.e. the number of qubits, connectivity of qubits, duration and incurred errors. The developed rigorous compilation methods are then evaluated with respect to their ability to enable quantum computations that are otherwise not accessible with near-term quantum technology. Experimental results demonstrate the ability of the developed rigorous compilation methods to extend the computational reach of near-term quantum computers by generating quantum computations with a reduced requirement on the number and connectivity of qubits as well as reducing the duration and incurred errors of performed quantum computations. Furthermore, the developed rigorous compilation methods extend their applicability to quantum circuit partitioning, qubit reuse and the translation between quantum computations generated for distinct quantum technologies. Specifically, a developed rigorous compilation method exploiting the structure of a quantum computation to reuse qubits at runtime yielded a reduction in the required number of qubits of up to 5x and result error by up to 33%. The developed quantum circuit partitioning method optimally distributes a quantum computation to distinct separate partitions, reducing the required number of qubits by 40% and the cost of partitioning by 41% on average. Furthermore, a rigorous compilation method was developed for quantum computers based on neutral atoms that combines swap gate insertions and topology changes to reduce the impact of limited qubit connectivity on the quantum computation duration by up to 58% and on the result fidelity by up to 29%. Finally, the developed quantum circuit adaptation method enables to translate between distinct quantum technologies while considering heterogeneous computational primitives with distinct characteristics to reduce the idle time of qubits by up to 87% and the result fidelity by up to 40%.Item Open Access Über eine Schauderbasis im Raum stetiger Funktionen mit kompaktem metrischem Definitionsbereich(1969) Vacher, F. S.; Wagenknecht, Monika (Übersetzerin)Im vorliegenden Artikel wird die Existenz einer Basis im Raum C(Q) stetiger Funktionen bewiesen, die auf einem beliebigen Kompaktum Q definiert sind; für einige Räume C(Q), insbesondere für den Raum C(K), wo K der Hilbert-Würfel ist, wird die Konstruktion einer Basis explizit angegeben.