## 15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY

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#### Keywords

- Gyroscopic (1)
- Hypergraph (1)
- Invariante (1)
- Mustererkennung (1)
- Raumplanung - Mittelzentren - Kleinstädte (1)
- Tensorfeld (1)
- hypergraph (1)
- invariant (1)
- moment (1)
- point cloud (1)

#### Faculty / Organisational entity

In the recent years small towns have experienced several negative developments. Especially in rural areas there are demographic problems and in the sector of retailing. Reforms in state administration resulted in the reduction of county administration seats. Also professional health care services are being reduced.
The thesis explores the effectiveness of three middle order centres (which in two cases are formed by more than one town) to fulfil their respective functions for their regions (complementary regions).
The spatial dominance of these towns in the sectors of jobs and services (retailing, secondary education – up to college level – health and entertainment) is surveyed.
The analysis is done with statistical material already collected by various institutions and by several own surveys. Interviews were done with experts.
Thus each middle order center and its complementary region is evaluated.
Haslach/Hausach/Wolfach performs best, albeit their demographic development is not dynamic. Bad Krozingen/Staufen has some shortcomings in its performance by the most dynamic demographic development of the three entities. Titisee-Neustadt's performance can be ranked second.
In a final chapter further research topics are listed.

This PhD-Thesis deals with the calculation and application of a new class of invariants, that can be used to recognize patterns in tensor fields (i.e. scalar fields, vector fields und matrix fields), and by the composition of scalar fields with delta-functions also to point-clouds.
In the first chapter an overview over already existing invariants is given.
In the second chapter the general definition of the new invariants is given:
starting with a tensor field a set of moment tensor is created via folding in tensor-product manner with different orders of the tensor product of the positional vector. From these, rotational invariant values are calculated via contraction of tensor products. An algorithm to get a complete and independent set of invariants from a given moment tensor set is described. Furthermore methods to make these sets of invariants invariant against translation, rotation, scaling, and affine transformation.
In the third chapter, a method to optimize the calculation of these sets of invariants is described: every invariant can be modeled as undirected graph comprising multiple sub-graphs representing partially contracted tensor products of the moment tensors.
The composition of the sets of invariants is optimized by a clever choice of the decomposition into sub-graphs, all paths creating a hyper-graph of sub-graphs where each node describes a composition step. Finally, C++-source-code is created, which optimized using the symmetry of the different tensors and tensor-products, and a comparison of the effort to other calculation methods of invariants is given.
The fourth chapter describes the application of the invariants to object recognition in point-clouds from 3D-scans. To do this, the invariants of sub-sets of point-clouds are stored for every known object. Afterwards, invariants are calculated from an unknown point-cloud and tried to find them in the database to assign it to one of the known objects. Benchmarks using three 3D-object databases are made testing time and recognition rate.

On Gyroscopic Stabilization
(2012)

This thesis deals with systems of the form
\(
M\ddot x+D\dot x+Kx=0\;, \; x \in \mathbb R^n\;,
\)
with a positive definite mass matrix \(M\), a symmetric damping matrix \(D\) and a positive definite stiffness
matrix \(K\).
If the equilibrium in the system is unstable, a small disturbance is enough to set the system in motion again. The motion of the system sustains itself, an effect which is called self-excitation or self-induced vibration. The reason behind this effect is the presence of negative damping, which results for example from dry friction.
Negative damping implies that the damping matrix \(D\) is indefinite or negative definite. Throughout our work, we assume \(D\) to be indefinite, and that the system possesses both stable and unstable modes and thus is unstable.
It is now the idea of gyroscopic stabilization to mix the modes of a system with indefinite damping such
that the system is stabilized without introducing further
dissipation. This is done by adding gyroscopic forces \(G\dot x\) with a suitable
skew-symmetric matrix \(G\) to the left-hand side. We call \(G=-G^T\in\mathbb R^{n\times n}\) a gyroscopic stabilizer for
the unstable system, if
\(
M\ddot x+(D+ G)\dot x+Kx=0
\)
is asymptotically stable. We show the existence of \(G\) in space dimensions three and four.