08 Fakultät Mathematik und Physik
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Item Open Access Contributions to the integral representation theory of groups(2004) Hertweck, Martin; Kimmerle, Wolfgang (apl. Prof. Dr.)This thesis contributes to the integral representation theory of groups. Topics treated include: the integral isomorphism problem --- if the group rings ZG and ZH are isomorphic, are the finite groups G and H isomorphic?, the Zassenhaus conjecture concerning automorphisms of integral group rings --- can each augmentation preserving automorphism of ZG be written as the product of an automorphism of G and a central automorphism?, and the normalizer problem --- in the unit group of ZG, is G only normalized by the obvious units? It is well known that these topics are closely related. Though counterexamples are known to each of these questions, our knowledge about such problems is still rather incomplete. A semilocal analysis of the known counterexample to the integral isomorphism problem is performed, which leads to new insight into the structure of the underlying groups. At the same time, this gives strong evidence for the existence of non-isomorphic groups of odd order having isomorphic semilocal group rings. It is shown how in the "semilocal case", counterexamples to the Zassenhaus conjecture can be produced with relatively minor effort. More importantly, it is shown for the first time that there is no local-global principle for automorphisms: An automorphism of a semilocal group ring (corresponding to an invertible bimodule M) need not give rise to a global automorphism (none of the modules in the genus of M is free from one side). In another part of this thesis, the normalizer problem for infinite groups is discussed. Research begun by Mazur is continued, and extensions of results of Jespers, Juriaans, de Miranda und Rogerio are obtained: By reduction to the finite group case, the normalizer problem is answered in the affirmative for certain classes of groups. The hypercenter of the unit group of RG, where G is a periodic group and R a G-adapted ring, is investigated too. If the hypercenter is not equal to the center, then G is a so called Q*-group, and then the hypercenter is described explicitly. The description in the R=Z case was obtained independently by Li and Parmenter, using different methods. The approach given here emphazises the connection to the normalizer problem and has a group-theoretical flavor. Moreover, it is shown that the second center of the unit group of ZG coincides with the finite conjugacy center. By way of contrast, the thesis ends with a little observation, intended to raise hopes that significant applications of integral representation theory to finite group theory will be found some day. In search of a proof of Glauberman's Z_p-star-Theorem (for odd p) which is independent from the classification, the following detail is noticed: If x is an element of order 3 in a finite group G which does not commute with any of its distinct conjugates, then chi(x), for any irreducible character chi of G, is an integral muliple of a root of unity.Item Open Access On the elementary theory of Heller triangulated categories(2013) Künzer, Matthias; König, Steffen (Prof. Dr.)Verdier's formalism of triangulated categories works with triangles, which fit into octahedra. These triangles enjoy a morphism prolongation property, but those octahedra do not. We establish a formalism of n-triangles such that the 2-triangles coincide with Verdier's triangles, such that the 3-triangles are particular Verdier octahedra, and such that n-triangles appear for all n. Now morphism prolongation holds for all n. Following Heller, we let the n-triangles be governed by an isotransformation between two shift functors on the stable category of n-pretriangles.Item Open Access Applications of Cartan and tractor calculus to conformal and CR-geometry(2007) Leitner, Felipe; Kühnel, Wolfgang (Prof.)The main object of this Habilitationsschrift is the geometric study of solutions of overdetermined conformally invariant differential equations via the use of Cartan and tractor calculus. This study fits into the broader research field of conformal and parabolic invariant theory. Parts of our investigations take special attention to conformal Lorentzian and spin geometry, which provides a link to the theories of modern physics. The present text originated from a collection of research articles and other works of the author, which emerged since the year 2003. In order to make the text basically self contained with uniform notations and conventions I decided to prefix an extended introductory chapter. An English and German summary are included as well.Item Open Access Concentrated patterns in biological systems(2003) Winter, Matthias; Mielke, Alexander (Prof.)We study pattern formation for reaction-diffusion systems of mathematical biology in the case of the Gierer-Meinhardt system. In this thesis we show that there is a critical growth rate of the inhibitor such that the position of boundary spikes is given by a linear combination of the boundary curvature and a Green function. There are two main results. The first one concerns the existence of boundary spikes for the activator. It says that the solutions are such that in the neighborhood of a boundary point for which the linear combination mentioned above possesses a nondegenerate critical point in tangential direction there is a spike (i.e. a peak whose spatial extension contracts but which after rescaling has a limit profile). Outside this boundary point the solutions are constant in first approximation. The proof uses Liapunov-Schmidt reduction, fixed point theorems and asymptotic analysis. The second main result concerns stability and says that the stability of this boundary spike depends on the parameters of the system. We assume that the linear combination from above possesses a nondegenerate local maximum at that boundary point. Then the stability depends on the size of a time relaxation constant. The proof studies small eigenvalues (i.e. they converge to zero) using asymptotic analysis. These small eigenvalue are connected with the second tangential derivatives of this linear combination. Large eigenvalues are explored using nonlocal eigenvalue problems.Item Open Access Asymptotische Entwicklungen des Robbins-Monro-Prozesses(1998) Dippon, Jürgen; Walk, Harro (Prof. Dr.)Zur Schätzung der Nullstelle x einer unbekannten Regressionsfunktion, deren Funktionswert f(X(n)) an der Stelle X(n) nur mit einem zufälligen Fehler V(n) mittels Y(n)=f(X(n))-V(n) beobachtet werden kann, schlugen Robbins und Monro (1951) die Rekursion X(n+1)=X(n)-a/n Y(n) vor. In der vorliegenden Arbeit werden Edgeworth-Entwicklungen des Robbins-Monro-Prozesses vorgestellt, welche eine Approximation der Verteilungsfunktion von sqrt(n)(X(n)-x) mit Resttermen der Ordnung o(1/sqrt(n)) und o(1/n) ermöglichen. Ausgehend von einer Idee von Walk zur linearen Approximation des Robbins-Monro-Prozesses wird die Rekursion in eine Summe von Multilinearformen in den Beobachtungsfehlern V(n) aufgelöst. Die Gültigkeit dieser Darstellungen wird in Kapitel 1 für quasi- und sublineare Regressionsfunktionen nachgewiesen. In Kapitel 2 werden die Entwicklungen der ersten vier Kumulanten der Darstellungsformen ermittelt. Dadurch ist die Form der Edgeworth-Entwicklung bereits festgelegt. Die dort gefundene asymptotische Entwicklung der Verzerrung könnte auch für weitere stochastische Approximationsverfahren von Interesse sein, da sie eine Korrektur des rekursiven Schätzers erlaubt. Zum Nachweis der Gültigkeit der Edgeworth-Entwicklungen der Darstellungsformen werden in Kapitel 3 die Methode der charakteristischen Funktionen und das Smoothing Lemma von Esseen verwendet. Der Beweis baut auf Ideen von Helmers, Callaert, Janssen, Veraverbeke, Bickel, Goetze und van Zwet auf, die Edgeworth-Entwicklungen für L- und U-Statistiken untersucht haben. In Kapitel 4 werden diese Ergebnisse auf den Robbins-Monro-Prozess angewendet. Damit kann die Überdeckungswahrscheinlichkeit von Konfidenzintervallen für x mit einem Restterm der Ordnung O(1/n) angegeben werden. Weitere Folgerungen betreffen Cornish-Fisher-Entwicklungen der Quantilfunktion und eine Edgeworth-Korrektur der Verteilungsfunktion.Item Open Access Mathematical modeling of coupled free flow and porous medium systems(2016) Rybak, Iryna; Rohde, Christian (Prof. Dr.)Different classes of physical systems with common interfaces arise in a variety of environmental and industrial problems. Striking examples originate from terrestrial-atmospheric contact zones, surface water-groundwater interaction, filters and fuel cells, where a free fluid system is in contact with a porous medium. Flow and transport processes in these systems evolve on multiple length and time scales, contributing to the complexity of these systems both from the modeling and the numerical side. An additional contributing factor to this complexity is the existence of multiple classes of entities, which include phases, interfaces between phases, and common curves that form at the boundary between three phases. Classical coupling approaches and traditional porous medium model formulations lead to reliable results in limited cases, however, applications require more realistic settings. The focus of the thesis is on derivation of mathematical models for multiphase multi-component porous medium systems that take into account lower dimensional entities, formulation of coupling conditions at the sharp interface and the transition region between porous medium and free flow systems, computation of effective parameters for the macroscale models, development and analysis of efficient numerical algorithms for coupled problems, and numerical simulation of applications.