Math & Physics Seminar SS 2003

Aims and Scope, Organization
The seminar aims at getting to know the research fields of colleagues from the mathematics and physics department. Talks will be presented on an interdisciplinary level, i.e. emphasis is put on explaining the basic questions, ideas, and tools.
Language English
meetings Tuesday, 16.15 - 17.45, 32/409 (Physics building)
22. 4. 2003 Heinz-Jürgen Schmidt, Spin systems and graph theory,
Systems of interacting spins (classical or quantum) can often be described by a graph. Its vertices correspond to the spin sites and its edges to pairs of interacting spins. It can be shown that the eigenvalues of the adjacency matrix J of this graph yield some useful information about the energy levels of the spin system. A certain property of the spin system (or its graph) which we called "n-cyclicity" seems to be crucial for further investigations. We do not know whether this concept which is a special kind of "n-colorability" has already be considered in graph theory. Applications of this approach to the properties of recently synthesized magnetic molecules will be outlined.
29. 4. 2003 Jürgen Schnack, Matthias Exler, Peter Hage, The end is (not) in sight: exact diagonalization, Lanczos, and Density Matrix Renormalization Group technique,
Although of finite dimensionality the determination of eigenvalues of the Hamilton operator of magnetic systems is practically impossible for realistic systems, even for small magnetic molecules. Nevertheless, some eigenvalues and eigenstates, and thus some physical properties, can be obtained using quasi exact or approximate methods.
6. 5. 2003 Horst Malchow, Selforganization in Population Dynamics,
Many mechanisms of temporal, spatial and spatio-temporal pattern formation in population dynamics are not known yet. In mathematical models, oscillations, waves, fronts as well as diffusive and/or advective instabilities of uniform distributions have been found. Using a simple prey-predator model, different routes to deterministic spatio-temporal chaos, wave and front propagation, stationary spatial coexistence of steady states etc. will be presented.
13. 5. 2003 May-Britt Kallenrode, Transport of energetic charged particles in the inner heliosphere,
The transport of energetic charged particles in the inner heliosphere is determined by a number of physical processes which can be summarized into different forms of transport equations. In this talk I will introduce a few of these mathematical models and their numerical treatment. Special emphasis is put on the discrepancy between an optimal mathematical or numerical model and basic physical concepts, such as conservation of particle numbers and the fact that particles should not propagate faster than with the speed of light.
20. 5. 2003 Christian Remling, Spectral theory of Schrödinger operators,
Schrödinger operators describe the dynamics of quantum systems. In this talk, I want to give an impression of how physical questions lead to mathematical questions, and I will discuss several results and open problems concerning these issues.
27. 5. 2003 Carsten Hartmann, An averaging principle for fast degrees of freedom with long-term correlations,
The function of many biomolecules comes from their dynamical properties and their ability to make statistically rare switches between different conformations. The influence of the fast scales on the conformation dynamics is discussed: Investigating the averaging principle for stochastic diffusion equations with fast and slow degrees of freedom, it is demonstrated how it fails if the fast scales exhibit long-term correlations. However, asymptotic multiscale analysis reveals error indicators and allows for designing the conformational free energy landscape in order to cover these effects.
3. 6. 2003 Bernd Heber, Modulation of cosmic rays - energetic charged particles - in the heliosphere,
Our knowledge of how cosmic rays are modulated in the inner heliosphere has been dramatically enlarged as a result of measurements from several missions launched in the past ten years and the progress in understanding the transport parameters in Parkers cosmic ray transport equation. This transport equation differs from the one described by Prof. Kallenrode that drift effects are included and that distribution of these particles are nearly isotropic. Measurements by the Ulysses spacecraft ( have direct implications for our understanding of the latitudinal transport of cosmic rays in the heliosphere. In my contribution I will focus on the Ulysses results and their implication for the diffusion tensor.
17. 6. 2003 private seminar of the group Macroscopic Systems and Quantum Theory
24. 6. 2003 Ali Ben Amor, On the equivalence between trace and capacitary inequalities for the abstract space of Bessel potentials,
Starting from a strongly continuous contarctive Markovian semigroup, we consturct a family of Banach spaces, the so called abstract spaces of Bessel potentials. We shall prove that the imbedding of such spaces in some Lebesgue spaces is equivalent to capacitary inequalities. The result relies, among others, on establishing a strong-type capacitary inequality.

1. 7. 2003 Olaf Müller, Representation of 3d geometry with fractal compressed displacement maps,
Current consumer graphics hardware is capable of rendering highly detailed 3d geometry with thousands of polygons at interactive framerates. Increasing geometry complexity causes increasing demand for bandwith to transfer the geometry to and memory to store it on the users computer. Displacement mapping is a recent method for compressing such geometries efficiently. By compressing the displacement maps with fractal techniques the total storage volume could further be minimized and other advantages like level of detail could be preserved.
8. 7. 2003 Christian Remling, Spectral theory of Schrödinger operators,
Schrödinger operators describe the dynamics of quantum systems. In this talk, I want to give an impression of how physical questions lead to mathematical questions, and I will discuss several results and open problems concerning these issues.
15. 7. 2003 Felix Homann, Solitons, nonlinear Schrödinger equations, and Hamiltonian systems,
Solitary wave or soliton solutions of nonlinear Schrödinger equations (NLS) are of great interest in many branches of physics, especially in nonlinear optics. While the NLS with cubic nonlinearity has been identified as an integrable Hamiltonian system, which relates to the existence of soliton solutions, in the interesting case of a saturable (photorefractive) nonlinearity the system is no longer integrable but still Hamiltonian. This is the starting point of a (numerical) approximation scheme which aims to model the interaction of solitary waves in the NLS with different nonlinearities. The talk will cover basic aspects of solitons, Hamiltonian systems, the NLS as a Hamiltonian system and how the Hamiltonian nature of the NLS can be used to approximate soliton dynamics. The related calculational problems will be touched.
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