08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Stability of compact symmetric spaces(2022) Semmelmann, Uwe; Weingart, GregorIn this article, we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.Item Open Access Gendo-Frobenius algebras and comultiplication(2022) Yırtıcı, ÇiğdemGendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.Item Open Access On a stochastic Camassa-Holm type equation with higher order nonlinearities(2020) Rohde, Christian; Tang, HaoThe subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.Item Open Access General interface problems. 1(1994) Nicaise, Serge; Sändig, Anna-MargareteWe study transmission problems for elliptic operators of order 2m with general boundary and interface conditions, introducing new covering conditions. This allows to prove solvability, regularity and asymptotics of solutions in weighted Sobolev spaces. We give some numerical examples for the location of the singular exponents.Item Open Access An approximation of solutions to heat equations defined by generalized measure theoretic Laplacians(2020) Ehnes, Tim; Hambly, BenWe consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure μ, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.Item Open Access Analysis of target data-dependent greedy kernel algorithms : convergence rates for f-, f· P- and f/P-greedy(2022) Wenzel, Tizian; Santin, Gabriele; Haasdonk, BernardData-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.Item Open Access Ordinal patterns in clusters of subsequent extremes of regularly varying time series(2020) Oesting, Marco; Schnurr, AlexanderIn this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine.Item Open Access On quiver Grassmannians and orbit closures for gen-finite modules(2021) Pressland, Matthew; Sauter, JuliaWe show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.Item Open Access Revisiting and generalizing the dual iteration for static and robust output‐feedback synthesis(2021) Holicki, Tobias; Scherer, Carsten W.The dual iteration was introduced in a conference paper in 1997 by Iwasaki as an iterative and heuristic procedure for the challenging and non‐convex design of static output‐feedback controllers. We recall in detail its essential ingredients and go beyond the work of Iwasaki by demonstrating that the framework of linear fractional representations allows for a seamless extension of the dual iteration to output‐feedback designs of practical relevance, such as the design of robust or robust gain‐scheduled controllers. In the paper of Iwasaki, the dual iteration is solely based on, and motivated by algebraic manipulations resulting from the elimination lemma. We provide a novel control theoretic interpretation of the individual steps, which paves the way for further generalizations of the powerful scheme to situations where the elimination lemma is not applicable. As an illustration, we extend the dual iteration to a design of static output‐feedback controllers with multiple objectives. We demonstrate the approach with numerous numerical examples inspired from the literature.Item Open Access Lossless transformations and excess risk bounds in statistical inference(2023) Györfi, László; Linder, Tamás; Walk, HarroWe study the excess minimum risk in statistical inference, defined as the difference between the minimum expected loss when estimating a random variable from an observed feature vector and the minimum expected loss when estimating the same random variable from a transformation (statistic) of the feature vector. After characterizing lossless transformations, i.e., transformations for which the excess risk is zero for all loss functions, we construct a partitioning test statistic for the hypothesis that a given transformation is lossless, and we show that for i.i.d. data the test is strongly consistent. More generally, we develop information-theoretic upper bounds on the excess risk that uniformly hold over fairly general classes of loss functions. Based on these bounds, we introduce the notion of a δ -lossless transformation and give sufficient conditions for a given transformation to be universally δ -lossless. Applications to classification, nonparametric regression, portfolio strategies, information bottlenecks, and deep learning are also surveyed.