Universität Stuttgart
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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 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 General mathematical model for the period chirp in interference lithography(2023) Bienert, Florian; Graf, Thomas; Abdou Ahmed, MarwanItem 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 A new estimate for the Ginzburg-Landau approximation on the real axis(1994) Schneider, GuidoModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. For scalar parabolic equations for which instability occurs at nonzero wavelength, we show that the associated Ginzburg-Landau equation dominates the dynamics of the nonlinear problem locally, at least over a long timescale. We develop a method which is simpler than previous ones and allows initial conditions of lower regularity. It involves a careful handling of the critical modes in the Fourier-transformed problem and an estimate of Gronwall's type. As an example, we treat the Kuramoto-Shivashinsky equation. Moreover, the method enables us to handle vector-valued problems.Item Open Access Topologische Ovale(1981) Buchanan, Thomas; Hähl, Hermann; Löwen, RainerDer Ausgangspunkt für die hier dargestellten Untersuchungen war die Frage, ob es lokalkompakte topologische Laguerreebenen gibt, in denen die topologische Dimension des Punktraums größer ist als vier. Im Zusammenhang damit galt es zu klären, ob es in kompakten topologischen projektiven Ebenen einer Dimension größer als vier Ovale geben kann, die in topologischer Hinsicht gutartig sind. In dieser Arbeit werden wir beide Fragen verneinend entscheiden (wobei wir allerdings die betrachteten Ebenen als endlichdimensional voraussetzen); dabei genügt als topologische Voraussetzung an die Ovale ihre Abgeschlossenheit im Punktraum.