We prove that with probability tending to 1, a 1-relator group with at least 3 generators and the relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.

Modern power systems are a very complex dynamical system spreading over a large geographical area. They indicate very complex phenomena in some parts as well as in the whole interconnected power system. In large and complex systems, there are coherent groups of generators. A group of generators in one area swings against a group of generators in another area. In this paper, using the Huang's Em...

The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schrödinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU (1, 1) and SU (2) coherent states are the unique states which minim...

In this work we study a kind of coherence condition on FIGmodules, which generalizes the usual notion of finite generation. We prove that a module is coherent, in the appropriate sense, if and only if its generators, as well as its torsion, appears in only finitely many degrees. Using this technical result, we prove that the category of coherent FIG-modules is abelian, independent of any assump...

The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schrödinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU (1, 1) and SU (2) coherent states are the unique states which minim...

The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schrödinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU(1, 1) and SU(2) coherent states are the unique states which minimiz...

Blood flow data from direct measurements (4D flow MRI) or numerical simulations opens new possibilities for the understanding of the development of cardiac diseases. However, before this new data can be used in clinical studies or for diagnosis, it is important to develop a notion of the characteristics of typical flow structures. To support this process we developed a novel blood flow clusteri...

One of the ubiquitous features of real-life turbulent flows is the existence and persistence of coherent vortices. Here we show that such coherent vortices can be extracted as clusters of Lagrangian trajectories. We carry out the clustering on a weighted graph, with the weights measuring pairwise distances of fluid trajectories in the extended phase space of positions and time. We then extract ...

Roughly speaking, clustering evolving networks aims at detecting structurally dense subgroups in networks that evolve over time. This implies that the subgroups we seek for also evolve, which results in many additional tasks compared to clustering static networks. We discuss these additional tasks and difficulties resulting thereof and present an overview on current approaches to solve these pr...