Classification of reductive real spherical pairs II. The semisimple case

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple (see arXiv:1609.00963) and generalizes the results of Brion and Mikityuk in the (complex) spherical case.Comment: Extended revised version. Section 6 and Appendix B are new. To appear in Transformation Groups. 40

Multiplizitätenfreie Supervektorräume

In dieser Arbeit werden Darstellungen reduktiver Gruppen mit multiplizitätenfreier supersymmetrischer Algebra klassifiziert.In this thesis representations of reductive groups with a multiplicity-free supersymmetric algebra are classified

Classification of reductive real spherical pairs:I. The simple case

This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum.Comment: 44 pages, 8 tables. Minor corrections in Table 8. Published versio

Hydraulic modeling of a compact stormwater treatment device applying concepts of dynamic similitude

The development of compact treatment devices (CTDs) with high removal efficiencies and low space requirements is a key objective of urban stormwater treatment. Thus, many devices utilize a combination of sedimentation and upward-flow filtration in a single system. Here, sedimentation is used before filtration, which makes it difficult to evaluate the individual treatment stages separately. This study determines the removal efficiency by sedimentation and the expected filter load in a specific compact treatment device designed for a catchment area of up to 10,000 m2. In contrast to a full-scale investigation, small-scale physical hydraulic modeling is applied as a new cost-saving alternative. To validate upscaling laws, tracer signals and particle-size-specific removal efficiencies are determined for two geometrically similar models at different length scales. Thereby, Reynolds number similarity produces similar flow patterns, while the similarity of Hazen numbers allows to upscale removal efficiencies. Upscaling to the full-scale reveals that the filter in the device is only partly loaded by particulate matter that consists mostly of particles ≤63 μm. Thus, sedimentation upstream of a filter is of relevant importance in CTDs. The proposed dimensionless relationship may be used for particles from different catchments and helps to size the device accordingly. HIGHLIGHTS A small-scale approach is presented to study the gravity-driven removal efficiency in a compact stormwater treatment device that combines sedimentation and filtration.; The Hazen number is successfully applied to scale the gravity-driven removal of particles from a small- to a full-scale model.; The filter stage of the device is only partially loaded with particulate matter, which mostly consists of particles ≤63 μm.

Design-oriented evaluation of the hydrodynamics in a full-scale combined filter-lamella separator for urban stormwater treatment

The development of compact treatment devices with high removal efficiencies and low space requirements is a key objective of urban stormwater treatment. Thus, many devices utilize a combination of sedimentation and upward flow filtration in a single system. This study, for the first time, evaluates the flow field inside a combined filter-lamella separator via computational fluid dynamics. Herein, three objectives are investigated: (i) the flow field for different structural configurations, (ii) the distribution of particulate matter along the filter bed and (iii) the dynamic clogging in discrete filter zones, which is addressed by a clogging model derived from literature data. The results indicate that a direct combination of a filtration stage with a lamella separator promotes a uniform flow distribution. The distribution of particulate matter along the filter bed varies with configuration and particle size. Clogging, induced by particles in the spectrum <63 $\mu$m, creates gradients of hydraulic conductivity along the filter bed. After treating about half of Germany's annual runoff-efficient precipitation at a rainfall intensity of 5 L/(s·ha), the filtration rates increase in the front of the filter bed by +10%. Thus, long-term operating behavior is sensitive to efficient filter utilization in compact treatment devices