08 Fakultät Mathematik und Physik
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Item Open Access General properties of ionic complex fluids(2016) Bier, Markus; Dietrich, Siegfried (Prof. Dr.)Item Open Access Electron correlations in the 2D multilayer organic metal k-(BEDT-TTF)2I3 in magnetic fields(2004) Balthes, Eduard; Schweitzer, Dieter (Prof. Dr.)This work presents quantum oscillation experiments in quasi-twodimensional multilayer organic metals. They show that low integer Landau level filling factors nare present in the two-dimensional organic metal k-(BEDT-TTF)2I3 and give strong indications for the existence of the fractional filling factor n=½ in this material. By this the work shows the presence of electron localisation and electron correlation in a bulk metallic two-dimensional system. These effects are found in the normal conducting state of the organic superconductor k-(BEDT-TTF)2I3. The revolutionary discovery of the integer as well as the fractional quantum Hall effect in two-dimensional semiconducting single-layer systems invoked, i.a., the questions, whether these effects may also be present in other types of conductors and, especially, whether they may also occur in bulk three-dimensional crystals. Strong efforts were made to produce bilayer two-dimensional semiconductors, to control their interlayer coupling as well as electron tunnelling, to increase step by step the number of involved layers with the aim to realise the quantised Hall effects in 'bulk' multilayer and, finally, in 'infinite-layer' systems. Furthermore strong efforts are made in semiconducting two-dimensional systems to realise carrier densities above 1011/cm2 with mobilities exceeding 107cm2/Vs. k-(BEDT-TTF)2I3 is a metallic compound with a very high electron density of 2*1019/cm2 and a very high carrier mobility reaching about 5*108cm2/Vs. From its structural principle this organic material represents a system of 105 coupled metallic multilayers, which can be synthesised in very high purity and can be produced as three-dimensional bulk single crystals. Despite of this, the material shows strongly two-dimensional electronic properties under certain experimental conditions, as found in the frame of this work. In contrast to the characteristic situation in semiconducting two-dimensional systems, where (correlated) electrons move on a single quantised orbit, the strongly correlated carriers in k-(BEDT-TTF)2I3 move on various quantised orbits with even very different filling factors. These are the main conditions under which the above mentioned filling factors are found in k-(BEDT-TTF)2I3. Besides these characteristics, the present two-dimensional organic metal holds a number of further peculiarities, which may represent a challenge for the understanding of possible fractional filling factors and quantum limit in a macroscopic multilayer crystal with two-dimensional electronic properties. In addition, the present work resumes experiments on the influence of low-dimensionality onto the electronic properties of a number of low-dimensional multilayer organic conductors.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 Physics of inhomogeneous nematic liquid crystals : colloidal dispersions and multiple scattering of light(1999) Stark, Holger; Trebin, Hans-Rainer (Prof. Dr.)The habilitation thesis deals with two interesting aspects of nematic liquid crystals with an inhomogeneous orientational order induced either by dispersed particles or by thermal director fluctuations. In the first part, the phenomenological description of the nematic phase and its topological defects are reviewed. The second part addresses the physics of nematic colloidal dispersions as a novel challenging type of soft matter. We first investigate the nematic environment of one particle with homeotropic boundary condition. Three possible structures are identified and discussed in detail; the dipole, the Saturn-ring and the surface-ring configuration. Secondly, we treat dipolar and quadrupolar two-particle interactions with the help of a phenomenological theory. Thirdly, we calculate the anisotropic Stokes drag of a particle in a nematic environment which determines its Brownian motion. We then turn our interest towards colloidal dispersions in complex geometries where we identify the dipolar configuration and study its formation. Finally, we demonstrate that surface-induced nematic order above the nematic-isotropic phase transition results in a strongly attractive but short-range two-particle interaction. Its strength can be controlled by temperature and thereby induce flocculation in an otherwise stabilized dispersion. In the third part we study multiple scattering of light from thermal fluctuations of the director. We use this scattering mechanism to test our generalized theory for the diffuse transport of light and its temporal correlations in random anisotropic media. Diffusing light constitutes a successful regime for accessing multiply scattered light. In diffusing-wave spectroscopy it is used to monitor the dynamics of turbid systems. We first provide a review of all the fascinating facets of multiply scattered light, and we introduce the basic theory of diffuse light transport in isotropic systems.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 Quantum systems with balanced gain and loss, signatures of branch points, and dissociation effects(2014) Cartarius, Holger; Wunner, Günter (Prof. Dr.)Gain and loss to the wave function of quantum mechanics can in a convenient way be modelled by effective non-Hermitian Hamiltonians. Imaginary contributions to the potential introduce source and drain terms for the probability amplitude. A special class of non-Hermitian Hamiltonians are those which possess a parity-time symmetry. In spite of their non-Hermiticity these Hamiltonians allow for real energy eigenvalues, i.e. the existence of stationary states in the presence of balanced gain and loss. This effect has been identified theoretically in a large number of quantum systems. Its existence has also been proved experimentally in coupled optical wave guides. The wave guides are, however, only optical analogues of quantum systems. In the first part of this thesis it is shown from the theoretical side that Bose-Einstein condensates in a double-well setup are an ideal candidate for a first experimental realisation of a genuine quantum system with parity-time symmetry. When particles are removed from one well and coherently injected into the other the external potential is parity-time symmetric. To investigate the system the underlying time-independent and time-dependent Gross-Pitaevskii equations are solved numerically. It turns out that a subtle interplay between the nonlinearity of the Gross-Pitaevskii equation and the gain-loss effect leads to a complicated dynamics of the condensate wave function. However, the most important result is the existence of stationary states that are sufficiently stable to be observable in an experiment. Two suggestions for experimental realisations are presented. They are based on the idea of embedding the non-Hermitian parity-time-symmetric system into a larger structure described by a Hermitian Hamiltonian. A further effect of non-Hermitian Hamiltonians are so-called exceptional points, at which two resonances coalesce such that both their eigenvalues and wave functions become identical. It is shown that an exceptional point can unambiguously be identified by a characteristic non-exponential decay of the resonances. With numerically exact calculations for the hydrogen atom in crossed electric and magnetic fields this effect is verified in an experimentally accessible quantum system. The second part of the thesis is devoted to semiclassical Gaussian approximations to the Boltzmann operator, which have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. A numerically cheap frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters is developed. It is applied to the cases of Ar3 and Ar6. For these clusters classical-like transitions in one step from a bounded moiety to free particles are found for increasing temperatures. Additionally, the structure of the Ar6 cluster is studied in the bound configuration and during the dissociation. Quantum effects, i.e. differences with the purely classical case, manifest themselves in the low-temperature behaviour of the mean energy and specific heat as well as in a slight shift of the transition temperature. A first-order correction to the semiclassical propagator is used to improve the results of the calculation for Ar3, and it is shown how the correction can be used to objectively assess the validity of the frozen Gaussian approximation.Item Open Access Atomistic simulation of shock waves : from simple crystals to complex quasicrystals(2005) Roth, Johannes Werner; Trebin, Hans-Rainer (Prof. Dr.)This habilitation thesis describes molecular dynamics simulations of solids. The impact of shock waves on a number of solids is studied: In the first part binary icosahedral quasicrystals and Laves crystals are treated, in the second part monatomic dodecagonal quasicrystals and body centered cubic crystals are dealt with. The third part contains studies of intermediate phases and solitons which show up in the body centered cubic crystals if shocked along a three-fold axis. In all cases three ranges of different behavior are observed: if the shock waves are weak, elastic deformations occur, in a medium range elastic and plastic waves or phase transitions are observed. If the shock waves are strong, the initial structures are completely destroyed. In this work we are concerned especially with the range of medium strong shock waves. For the binary crystal structure fragmentation occurs. The emerging crystallites are rotated with respect to each other and separated by boundary layers which are several atomic distances thick. The main difference between crystal and quasicrystal are phason-like defects which lead to a continuous transition between the range of weak and medium shock waves. For the monatomic crystal structures the Dzugutov potential has been applied to stabilize the structures. Here we find in the range of medium shock waves phase transitions from quasicrystals and approximants to the body centered cubic structure. Depending on the orientation and strength of the shock waves the transition takes places within a few atomic layers or spread out across many layers. In the quasicrystal and the approximants atomic flips are observed in the elastically compress region. Body centered cubic crystals possess an inherent instability along the three-fold axes. In many materials, this leads to a phase transition to the so-called omega-phase. In our case the omega-phase is stable only in a small range of compression, thus a forth- and back-transformation from body centered cubic to the omega-phase takes place, as long as the phase transition front moves slower than the speed of sound. If this is no longer the case, solitons shock up which contain in their interior. In summary several differences could be observed between crystals and quasicrystals. The results obtained for the Dzugutov potentials are comparable to the outcome of simulation of shock waves in iron with materials-specific interactions.