Universität Stuttgart
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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 Spectral theory of quantum graphs(2012) Demirel, Semra; Weidl, Timo (Prof. TeknD)We study some spectral problems for quantum graphs with a potential. On the one hand we analyze the quantitative dependence of bound states of the Schrödingeroperator on the potential. On the other hand we generalize certain basic identities from the one-dimensional scattering theory to quantum graphs. The first paper is concerned with the study of the discrete negative spectra of quantum graphs. We use the method of trace identities (sum rules) to derive inequalities of Lieb-Thirring, Payne-Polya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false. The second paper deals with the scattering problem for the Schrödinger equation on the half-line with the Robin boundary condition at the origin. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation for the perturbation determinant and to trace identities of Buslaev-Faddeev type. In the third paper we generalize results from the half-line case to the full graph case. More precisely, we consider the Schrödinger problem on a star shaped graph with n edges joined at a single vertex. A trace formula is derived for the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula. Besides these three articles this thesis also contains some further results. The method of sum rules is applied to the modified Schrödinger operator with variable coefficients to obtain a Lieb-Thirring type inequality with optimal constant. Furthermore, Lieb-Thirring inequalities are studied for star shaped graphs by using variational arguments and the method of symmetry decomposition of the corresponding Hilbert space. In several cases this leads to optimal constants in the inequalities.Item Open Access Schaustücke : Einblicke in wissenschaftliche Sammlungen der Universität Stuttgart(Stuttgart : Universität Stuttgart, 2022) Wiatrowski, Frank (Gestaltung, Fotograf); Engstler, Katja Stefanie (Gestaltung); Ceranski, Beate (Vorwort); Rambach, Christiane (Vorwort)Die wissenschaftlichen Sammlungen der Universität zeugen von einer langen Lehr- und Forschungstradition. In Fakultäten und Instituten, in der Universitätsbibliothek und im Universitätsarchiv sind vielfältige Sammlungen beheimatet, zum Teil mit ungewöhnlichen oder gar einzigartigen Objekten. Die Broschüre gibt erste Einblicke in diese vielfach versteckte Welt der universitären Sammlungen in Stuttgart.Item Open Access From simplicial groups to crossed squares(2022) Asiki, Natalia-MariaSimplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.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 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.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 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 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.