Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system possessing $N$-peakon weak solutions, for all $N\geq 1$, in the setting of an integral formulation which is used in analysis for studying local well-posedness, global existence, and wave breaking for non-peakon solutions. Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH equation do not reduce to conserved integrals (constants of motion) for $2$-peakon weak solutions. This perplexing situation is addressed here by finding an explicit conserved integral for $N$-peakon weak solutions for all $N\geq 2$. When $N$ is even, the conserved integral is shown to provide a Hamiltonian structure with the use of a natural Poisson bracket that arises from reduction of one of the Hamiltonian structures of the mCH equation. But when $N$ is odd, the Hamiltonian equations of motion arising from the conserved integral using this Poisson bracket are found to differ from the dynamical equations for the mCH $N$-peakon weak solutions. Moreover, the lack of conservation of the two Hamiltonians of the mCH equation when they are reduced to $2$-peakon weak solutions is shown to extend to $N$-peakon weak solutions for all $N\geq 2$. The connection between this loss of integrability structure and related work by Chang and Szmigielski on the Lax pair for the mCH equation is discussed.Comment: Minor errata in Eqns. (32) to (34) and Lemma 1 have been fixe

Stress Testing German Industry Sectors: Results from a Vine Copula Based Quantile Regression

Measuring interdependence between probabilities of default (PDs) in different industry sectors of an economy plays a crucial role in financial stress testing. Thereby, regression approaches may be employed to model the impact of stressed industry sectors as covariates on other response sectors. We identify vine copula based quantile regression as an eligible tool for conducting such stress tests as this method has good robustness properties, takes into account potential nonlinearities of conditional quantile functions and ensures that no quantile crossing effects occur. We illustrate its performance by a data set of sector specific PDs for the German economy. Empirical results are provided for a rough and a fine-grained industry sector classification scheme. Amongst others, we confirm that a stressed automobile industry has a severe impact on the German economy as a whole at different quantile levels whereas e.g., for a stressed financial sector the impact is rather moderate. Moreover, the vine copula based quantile regression approach is benchmarked against both classical linear quantile regression and expectile regression in order to illustrate its methodological effectiveness in the scenarios evaluated.Comment: 12 page

Multispektral-bildgestütztes Sortieren von Biopartikeln

Bioparticles surround us in many areas of daily life. They are used, for example, in plant cultivation, the food industry or for environmental applications. The characterization of these particles is a challenge due to their small size. The present work therefore deals with the development and investigation of novel multispectral imaging techniques for the characterization and sorting of bioparticles in microfluidic flow. For this purpose, innovative microfluidic structures were tested to place particles in the focal plane of the optical system by controlled fluid rotation. These structures build the basis for optical characterization and image-based sorting at the single particle level. Microalgae of the type Haematococcus pluvialis (HP) were used as model organism for the investigations as well as for the subsequent image-based sorting. These microalgae are cultivated in a two-step process, where they change their morphological as well as their spectral properties if they are exposed to external stress factors. Such characteristics make them interesting for multispectral imaging analysis. During the development and optimization of biotechnological cultivation processes, continuous monitoring of the spectral and morphological properties of the cells is an important quality pa-rameter. Due to the large number of cells analyzed at the single cell level per measurement, pre-cise information about the physiological state of the cells as well as the composition of the entire population is possible. With the novel universally applicable multispectral imaging platform, several thousand particles per minute could be captured and classified. By implementing real-time classification, the entire analysis process could be significantly accelerated and automated. The measured values of the multispectral imaging correlate with the measured values of the chemically extracted dyes in the particles