3 research outputs found
Algorithms for Permutation Groups and Cayley Networks
110 pagesBases, subgroup towers and strong generating sets (SGSs) have played a key role
in the development of algorithms for permutation groups. We analyze the computational
complexity of several problems involving bases and SGSs, and we use subgroup towers
and SGSs to construct dense networks with practical routing schemes.
Given generators for G ≤ Sym(n), we prove that the problem of computing a
minimum base for G is NP-hard. In fact, the problem is NP-hard for cyclic groups and
elementary abelian groups. However for abelian groups with orbits of size less than 8, a
polynomial time algorithm is presented for computing minimum bases. For arbitrary permutation groups a greedy algorithm for approximating minimum
bases is investigated. We prove that if G ≤ Sym(n) with a minimum base of size k, then
the greedy algorithm produces a base of size Ω (k log log n)
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Classical and Quantum Discrete Dynamical Systems
We study deterministic and quantum dynamics from a constructive "finite"
point of view, since the introduction of a continuum, or other actual
infinities in physics poses serious conceptual and technical difficulties,
without any need for these concepts to physics as an empirical science. For a
consistent description of the symmetries of dynamical systems at different
times and the symmetries of the various parts of such systems, we introduce
discrete analogs of the gauge connections. Gauge structures are particularly
important to describe the quantum behavior. We show that quantum behavior is
the result of a fundamental inability to trace the identity of
indistinguishable objects in the process of evolution. Information is available
only on invariant statements and values, relating to such objects. Using
mathematical arguments of a general nature we can show that any quantum
dynamics can be reduced to a sequence of permutations. Quantum interferences
occur in the invariant subspaces of permutation representations of symmetry
groups of dynamical systems. The observable values can be expressed in terms of
permutation invariants. We also show that for the description of quantum
phenomena, instead of a nonconstructive number system --- the field of complex
numbers, it is enough to use cyclotomic fields --- the minimal extentions of
natural numbers suitable for quantum mechanics. Finite groups of symmetries
play a central role in this article. The interest in such groups has an
additional motivation in physics. Numerous experiments and observations in
particle physics point to an important role of finite groups of relatively low
orders in a number of fundamental processes.Comment: author's English translation added; 74 pages in English, 80 pages in
Russian; English translation in Phys. Part. Nucl. ISSN 1063-7796 (2013) 44,
No. 1, pp 47-91 was done without author's control; V3: Section 4 revised; V4:
some correction