Computational Group Theory

2011 ◽  
pp. 2113-2161
Author(s):  
Bettina Eick ◽  
Gerhard Hiß ◽  
Derek Holt ◽  
Eamonn O’Brien
Keyword(s):  
Group Theory ◽  
Computational Group Theory

Computational group theory

1986 ◽  
Vol 11 (1) ◽  
pp. 93-94
Author(s):  
F.W. Roush
Keyword(s):  
Group Theory ◽  
Computational Group Theory

scholarly journals Complete characterization of the stability of cluster synchronization in complex dynamical networks

2016 ◽  
Vol 2 (4) ◽  
pp. e1501737 ◽  
Author(s):  
Francesco Sorrentino ◽  
Louis M. Pecora ◽  
Aaron M. Hagerstrom ◽  
Thomas E. Murphy ◽  
Rajarshi Roy
Keyword(s):  
Group Theory ◽  
Complete Characterization ◽  
Power Grids ◽  
Cluster Synchronization ◽  
General Technique ◽  
Dynamical Networks ◽  
Computational Group Theory ◽  
Node Network ◽  
Engineered Systems ◽  
The Stability

Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based on the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. Understanding how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an optoelectronic experiment on a five-node network that confirms the synchronization patterns predicted by the theory.

scholarly journals Nice Efficient Presentions for all Small Simple Groups and their Covers

2004 ◽  
Vol 7 ◽  
pp. 266-283 ◽  
Author(s):  
Colin M. Campbell ◽  
George Havas ◽  
Colin Ramsay ◽  
Edmund F. Robertson
Keyword(s):  
Group Theory ◽  
Power Relations ◽  
Simple Groups ◽  
Computational Group Theory ◽  
The Status ◽  
Coset Enumeration ◽  
Covering Groups ◽  
Computational Properties ◽  
Selection Of ◽  
Better Than

AbstractPrior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

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