@phdthesis{Fruehwirth2025, author = {Fr{\"u}hwirth, Lorenz}, title = {The Asymptotic Behavior of Birkhoff- and Lacunary Sums}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-15677}, school = {Universit{\"a}t Passau}, pages = {109 Seiten}, year = {2025}, abstract = {This doctoral thesis consists of three independently published research articles on the asymptoic behaviour of Lacunary- and Birkhoff sums. The former are sums formed by periodic functions and exponentially growing sequences of natural numbers. The corresponding summands often exhibit behavior typical of independent and identically distributed random variables. The methods used are of an analytical and probabilistic nature. The Birkhoff sums considered in this work are generated by the Kronecker sequence and by discontinuous functions. The methods employed are from the field of metric number theory, specifically classical results from continued fraction theory are utilized.}, language = {en} } @phdthesis{Hasenpflug2025, author = {Hasenpflug, Mareike}, title = {Slice sampling on Riemannian manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-15903}, school = {Universit{\"a}t Passau}, pages = {iii, 118 Seiten}, year = {2025}, abstract = {This thesis is concerned with hybrid slice samplers for approximate sampling of distributions on Riemannian manifolds. First for distributions on the Euclidean unit sphere, and then for distributions on general Riemannian manifolds we introduce a geodesic-based hybrid slice sampler, called geodesic slice sampler. Under mild regularity assumptions, we establish reversibility with respect to the target distribution for this sampler and positive semi-definiteness of the corresponding operator. Moreover, on compact Riemannian manifolds we show uniform ergodicity with explicit constants for the geodesic slice sampler if the target distribution has a bounded density with respect to the Riemannian measure. As an important building block of this sampler, we provide an explicit expression for the shrinkage procedure proposed in (Neal, 2003) in terms of a Markov kernel. We establish that this kernel is reversible with respect to the uniform distribution on the target set and that its corresponding operator is positive semi-definite. Beyond the geodesic slice sampler, we apply these results also to elliptical slice sampling (Murray, Adams, MacKay, 2010) to obtain a proof for its reversibility with respect to the target distribution and positive semi-definiteness of the corresponding operator.}, language = {en} } @article{GishbolinerGlockSgueglia2025, author = {Gishboliner, Lior and Glock, Stefan and Sgueglia, Amedeo}, title = {Tight Hamilton cycles with high discrepancy}, series = {Combinatorics, Probability and Computing (1469-2163)}, volume = {34 (2025)}, journal = {Combinatorics, Probability and Computing (1469-2163)}, number = {4}, publisher = {Cambridge University Press}, address = {Cambridge}, issn = {1469-2163}, doi = {10.1017/S0963548325000057}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19166}, pages = {565 -- 584}, year = {2025}, abstract = {In this paper, we study discrepancy questions for spanning subgraphs of k-uniform hypergraphs. Our main result is that, for any integers k ≥ 3 and r ≥ 2, any r-colouring of the edges of a k-uniform n-vertex hypergraph G with minimum (k-1)-degree δ(G) ≥ (1/2+o(1))n contains a tight Hamilton cycle with high discrepancy, that is, with at least n/r +� (n) edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Tur{\´a}n-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.}, language = {en} } @article{Hofstadler2025, author = {Hofstadler, Julian}, title = {Optimal convergence rates of MCMC integration for functions with unbounded second moment}, series = {Journal of Applied Probability}, volume = {62 (2025)}, journal = {Journal of Applied Probability}, number = {3}, publisher = {Cambridge University Press}, address = {Cambridge}, doi = {10.1017/jpr.2024.108}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19310}, pages = {1069 -- 1075}, year = {2025}, abstract = {We study the Markov chain Monte Carlo estimator for numerical integration for func- tions that do not need to be square integrable with respect to the invariant distribution. For chains with a spectral gap we show that the absolute mean error for L^p functions, with p ∈ (1, 2), decreases like n^(1/p)-1 , which is known to be the optimal rate. This improves currently known results where an additional parameter δ > 0 appears and the convergence is of order n^((1+δ)/p)-1 .}, language = {en} } @phdthesis{Henle2025, author = {Henle, Mona}, title = {Multi-Leader Congestion Games with an Adversary}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19683}, school = {Universit{\"a}t Passau}, pages = {112 Seiten}, year = {2025}, abstract = {In this thesis, we introduced a congestion game with multiple leaders and a single follower (adversary) which is motivated by security applications with congestion effects. Our objective was to understand the result and the impact of selfish acting individuals in these games. In this regard, we analyzed the existence, the computation and the quality of (approximate) pure Nash equilibria. First, we observed that an exact pure Nash equilibrium always exists in the resulting strategic game among the leaders if the resource cost coefficients are identical and the underlying congestion game is a matroid congestion game. If one of these two conditions is not fulfilled, the existence of PNE is not ensured anymore in general. Consequently, we focused on approximate equilibria. For the case of symmetric singleton strategies, one of our main result established that K ≈ 1.1974, the unique solution of a cubic polynomial equation, is the smallest possible factor such that the existence of a K-approximate equilibrium is guaranteed for all instances of the game. To this end, we presented an efficient algorithm which computes a K-approximate PNE. Furthermore, we showed that the factor K is tight by providing an instance where no α-approximate PNE with α < K exists. However, for a specific symmetric singleton instance there might be a better α-approximate PNE, i.e., with α < K. A given instance could even admit an exact PNE. We provided therefore a polynomial time procedure that computes a best approximate PNE of a given instance. In particular, this procedure can verify the existence of an exact PNE in a given instance efficiently and, if it exists, can also determine the corresponding load vector. Finally, for symmetric singleton instances with two resources, we compared the total cost of a best (cheapest) and worst (most expensive) PNE to the total cost of an optimal outcome, termed by the price of stability and the price of anarchy, respectively. In particular, we verified that the PoS and the PoA are 4/3.}, subject = {Spieltheorie}, language = {en} } @phdthesis{Schiermeier2025, author = {Schiermeier, Kathrin}, title = {Multidimensional Wavelets and Neural Networks}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19742}, school = {Universit{\"a}t Passau}, pages = {xx, 168 Seiten}, year = {2025}, abstract = {The construction of scaling functions and wavelets in multiple dimensions and for arbitrary scaling matrices is a challenging task entailing some complexities. Existing approaches mainly focus on the two-dimensional case using dyadic or quincunx sampling. This thesis aims to develop a method to construct multidimensional scaling and wavelet filters yielding orthogonal scaling functions and wavelets under the usage of convolutional neural networks. We start by recalling substantial fundamentals of ideals, modules, Fourier analysis, filterbanks and multiresolution analyses, where the mentioned concepts are already considered in an arbitrary dimensional setting to prepare the proof of the main result. There, we show the connection between multivariate scaling functions and multidimensional filters possessing certain properties. This enables us to construct scaling functions and corresponding wavelets by discrete filter design. Exploiting the link between the discrete wavelet decomposition, filterbanks and neural networks, we utilize the latter to do so. Being the main difficulty of this process, we especially focus on the Cohen criterion, which concerns the zeros of the Fourier transform of the scaling filter in modulus representing a multivariate trigonometric polynomial. After transferring the Bernstein inequality for univariate trigonomic polynomials to multiple dimensions, we present a method to derive a finite set of inequality constraints implying that the Cohen criterion holds true for a given multivariate cosine sum. Afterwards, we introduce neural networks and TensorFlow as the main tools to execute the described approach, formulate the described objective as an optimization problem and present some smaller numerical experiments and their results. A second objective of this thesis is the construction of filters possessing a unimodular modulation vector and therefore the ability to be completed to a perfect reconstruction filterbank. Both - the construction and the filterbank completion - can also be considered in a neural network framework as we will detail in the last section of this thesis alongside with the presentation of corresponding numerical experiments. In the context of filterbank completion, a further observation which allows to complete any given interpolatory filter to a perfect reconstruction filterbank in a very intuitive and simple way is presented. Furthermore, we explain that any given unimodular filter can be rendered interpolatory through prefiltering.}, language = {en} }