Coefficient estimates, Landau's theorem and Lipschitz-type spaces on planar harmonic mappings

In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss the coefficient estimates and Landau's Theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.Comment: 17 page

Instabilities and subharmonic resonances of subsonic heated round jets, volume 2

When a jet is perturbed by a periodic excitation of suitable frequency, a large-scale coherent structure develops and grows in amplitude as it propagates downstream. The structure eventually rolls up into vortices at some downstream location. The wavy flow associated with the roll-up of a coherent structure is approximated by a parallel mean flow and a small, spatially periodic, axisymmetric wave whose phase velocity and mode shape are given by classical (primary) stability theory. The periodic wave acts as a parametric excitation in the differential equations governing the secondary instability of a subharmonic disturbance. The (resonant) conditions for which the periodic flow can strongly destabilize a subharmonic disturbance are derived. When the resonant conditions are met, the periodic wave plays a catalytic role to enhance the growth rate of the subharmonic. The stability characteristics of the subharmonic disturbance, as a function of jet Mach number, jet heating, mode number and the amplitude of the periodic wave, are studied via a secondary instability analysis using two independent but complementary methods: (1) method of multiple scales, and (2) normal mode analysis. It is found that the growth rates of the subharmonic waves with azimuthal numbers beta = 0 and beta = 1 are enhanced strongly, but comparably, when the amplitude of the periodic wave is increased. Furthermore, compressibility at subsonic Mach numbers has a moderate stabilizing influence on the subharmonic instability modes. Heating suppresses moderately the subharmonic growth rate of an axisymmetric mode, and it reduces more significantly the corresponding growth rate for the first spinning mode. Calculations also indicate that while the presence of a finite-amplitude periodic wave enhances the growth rates of subharmonic instability modes, it minimally distorts the mode shapes of the subharmonic waves

Interaction and disorder effects in topological insulators = Wechselwirkungs- und Unordnungseffekte in topologischen Isolatoren

In my thesis, I examine interaction and disorder effects in topological insulators (TIs). I focus on surface states of 3D TIs of various symmetry classes. The dissertation contains two reviewing chapters on Anderson localization and TIs. Original scientific work is presented on semiclassical transport, on localization in systems with sublattice symmetry, on interaction effects in experimentally realized thin 3D TI slabs, the quantum Hall effect in the latter and the superconducting instability

Field theoretic formulation and empirical tracking of spatial processes

Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and statistical physics can be used to understand behaviour in complex, spatially-extended multi-body systems. On the other hand, computer vision and statistical analysis can be used to study 4D microscopy data to observe and understand real spatial processes in vivo. On the rst of these fronts, analytical models are developed for abstract processes, which can be simulated on graphs and lattices before considering real-world applications in elds such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes, techniques originating in quantum eld theory such as canonical quantisation and the renormalization group are applied to reaction-di usion processes by analogy. These techniques are combined in the study of critical phenomena or critical dynamics. At this level, one is often interested in the scaling behaviour; how the correlation functions scale for di erent dimensions in geometric space. This can lead to a better understanding of how macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching random walk on various graphs is studied. In the thesis, a distinctly abstract approach is emphasised in order to support an algorithmic approach to parts of the formalism. A model of self-organised criticality, the Abelian sandpile model, is also considered. By exploiting a bijection between recurrent con gurations and spanning trees, an e cient Monte Carlo algorithm is developed to simulate sandpile processes on large lattices. On the second front, two case studies are considered; migratory patterns of leukaemia cells and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics are used to study the spatial dynamics of di erent leukaemia cell lineages before and after a treatment. One key result is that we can discriminate between migratory patterns in response to treatment, classifying cell motility in terms of sup/super/di usive regimes. For the second case study, a novel algorithm is developed to processes a 4D light-sheet microscopy dataset. The combination of transient uorescent markers and a poorly localised specimen in the eld of view leads to a challenging tracking problem. A fuzzy registration-tracking algorithm is developed to track mitotic events so as to understand their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces