Universität Stuttgart
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Item Open Access Reconstruction of μXRCT data sets using the ASTRA toolbox(2020) Voland, PaulItem Open Access Descriptions of some double Burnside rings(2017) Krauß, NoraThe double Burnside R-algebra B_R(G,G) of a finite group G with coefficients in a commutative ring R has been introduced by S. Bouc. It is R-linearly generated by finite (G,G)-bisets, modulo a relation identifying disjoint union and sum. Its multiplication is induced by the tensor product. It contains the bifree double Burnside R-algebra B_R^Delta(G,G) generated by bifree finite (G,G)-bisets. Let S_n denote the symmetric group on n letters. For R in {Q, Z, Z_(2), F_2, Z_(3), F_3}, we calculate B_R(S_3,S_3) and B_R^Delta(S_4,S_4).Item Open Access Basic representation theory of crossed modules(2018) Truong, MonikaA group corresponds to a topological space with one nontrivial homotopy group. A crossed module corresponds to a topological space with two nontrivial homotopy groups. In classical group theory, Cayley's Theorem constructs for every group G an injective group morphism to the symmetric group S_G. For a crossed module V, we have a similar statement. For every category C, we have the symmetric crossed module S_C. For every crossed module V, we construct an injective crossed module morphism to the symmetric crossed module S_VCat. Suppose given an R-linear category M. On the one hand, we obtain the invertible monoidal category Aut_R(M) by means of category theory. On the other hand, we have the symmetric crossed module S_M as in the Cayley context. In S_M, we have the crossed submodule Aut^CM_R(M) containing only the R-linear elements of S_M. We consider the corresponding invertible monoidal category (Aut^CM_R(M))Cat. We show that there exists a monoidal isofunctor Real_M : (Aut^CM_R(M))Cat -~-> Aut_R(M). This means that starting with M, we obtain essentially the same object via crossed module theory as via category theory. A representation of a group G on an R-module N is given by a group morphism G -> Aut_R(N). Analogously, a representation of a crossed module V on an R-linear category M is given by a crossed module morphism V -> Aut^CM_R(M). We begin to study the representation theory of crossed modules.Item Open Access The Generalized Minimum Manhattan Network Problem(2015) Schnizler, MichaelIn this thesis we consider the Generalized Minimum Manhattan Network Problem: given a set containing n pairs of points in R2 or Rd, the goal is to find a rectilinear network of minimal length which contains a path of minimal length (a so-called Manhattan path) between the two points of each pair. We restrict our search to a discrete subspace and show that under specific conditions an optimal solution can be found in polynomial time using a dynamic program. The conditions concern the intersection graph of the bounding boxes of the pairs. Its maximum degree as well as the treewidth must be bounded by two constants which are independent of n. Finally, we present a simple greedy algorithm for practical purposes.Item Open Access Preconditioning techniques for coupled Stokes Darcy systems(2020) Schmalfuß, JennyItem Open Access The minimal projective resolution of Z_(2) over Z_(2)S_4(2017) Nitsche, SebastianWe consider the trivial module Z_(2) over the group ring Z_(2)S_4. Using the Wedderburn image of Z_(2)S_4, a minimal projective resolution of Z_(2) over Z_(2)S_4 with regular behavior is constructed. We give a closed formula for the projective terms and the differentials. The minimal projective resolution is used to calculate the cohomology groups of S_4 over Z_(2). In 1974, Thomas gave a description of the cohomology ring of S_4 as a factor ring of a polynomial ring. As far as we were able to compare both using Magma, our calculation is in accordance with his result.Item Open Access Universelle Konstruktionen für Relationen(2024) Bechtel, JonasIn der Arbeit wird die Kategorie der Mengen und Relationen betrachtet, deren Objekte Mengen und deren Morphismen Relationen sind. Gegeben seien zwei Relationen, welche als Ursprungsmenge die gleiche Menge aufweisen. Für diese Relationen können je nach Situation die Eigenschaften linkstotal, rechtstotal, linkseindeutig und rechtseindeutig gefordert sein. Für die sich hieraus ergebenden Ausgangssituationen wurde untersucht, wann stets ein Pushout konstruiert werden kann und für welche Situationen es Gegenbeispiele zur Existenz eines Pushouts gibt. Existiert ein Pushout, so ist in der Arbeit auch die Pushout-Konstruktion angegeben. Des Weiteren wurde die Faktorisierung einer Relation über ihren Graphen betrachtet und hierfür eine universelle Konstruktion angegeben.Item Open Access Simulating stochastic processes with variational quantum circuits(2022) Fink, DanielSimulating future outcomes based on past observations is a key task in predictive modeling and has found application in many areas ranging from neuroscience to the modeling of financial markets. The classical provably optimal models for stationary stochastic processes are so-called ϵ-machines, which have the structure of a unifilar hidden Markov model and offer a minimal set of internal states. However, these models are not optimal in the quantum setting, i.e., when the models have access to quantum devices. The methods proposed so far for quantum predictive models rely either on the knowledge of an ϵ-machine, or on learning a classical representation thereof, which is memory inefficient since it requires exponentially many resources in the Markov order. Meanwhile, variational quantum algorithms (VQAs) are a promising approach for using near-term quantum devices to tackle problems arising from many different areas in science and technology. Within this work, we propose a VQA for learning quantum predictive models directly from data on a quantum computer. The learning algorithm is inspired by recent developments in the area of implicit generative modeling, where a kernel-based two-sample-test, called maximum mean discrepancy (MMD), is used as a cost function. A major challenge of learning predictive models is to ensure that arbitrarily many time steps can be simulated accurately. For this purpose, we propose a quantum post-processing step that yields a regularization term for the cost function and penalizes models with a large set of internal states. As a proof of concept, we apply the algorithm to a stationary stochastic process and show that the regularization leads to a small set of internal states and a constantly good simulation performance over multiple future time steps, measured in the Kullback-Leibler divergence and the total variation distance.Item Open Access The resolution equivalence for n-complexes(2022) Klein, VeronikaAn injective resolution equivalence is constructed which generalizes the construction from classical homological algebra to n-complexes. The construction proceeds by showing that a functor in the converse direction is an equivalence. The injective resolution functor is defined to be its inverse.Item Open Access On the Bisson-Tsemo model category of graphs(2022) Hess, JannikGezeigt wird, auf Grundlage von Bisson und Tsemo, dass die Kategorie der Graphen eine Modellkategorie darstellt. Es wird ein hinreichendes Kriterium dafür gezeigt, dass ein Graphmorphismus ein Quasiisomorphismus ist. Des Weiteren werden einige Beispiele und Gegenbeispiele konstruiert.