Groups Complexity Cryptology
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- Actions of the Braid Group, and New Algebraic Proofs of Results of Dehornoy and Larue by Bacardit, Lluís and Dicks, Warren
- Strong law of large numbers on graphs and groups by Mosina, Natalia and Ushakov, Alexander
- The Zieschang–McCool method for generating algebraic mapping-class groups by Bacardit, Lluís and Dicks, Warren
- Cyclic rewriting and conjugacy problems by Diekert, Volker/ Duncan, Andrew and Myasnikov, Alexei G.
- Non-associative key establishment for left distributive systems by Kalka, Arkadius and Teicher, Mina
Strong law of large numbers on graphs and groups
1Department of Mathematics, Engineering, and Computer Science, CUNY/LAGCC, Long Island City, NY, USA.
2Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ, USA.
Citation Information: Groups – Complexity – Cryptology. Volume 3, Issue 1, Pages 67–103, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: 10.1515/gcc.2011.004, February 2011
- Published Online:
We consider (graph-)group-valued random element ξ, discuss the properties of a mean-set 𝔼(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.
Keywords.: Probability measures on graphs and groups; average; expectation; mean-set; strong law of large numbers; Chebyshev inequality; Chernoff bound; configuration of mean-sets; free group; shift search problem