36,778 research outputs found
Ramsey Properties of Permutations
The age of each countable homogeneous permutation forms a Ramsey class. Thus,
there are five countably infinite Ramsey classes of permutations.Comment: 10 pages, 3 figures; v2: updated info on related work + some other
minor enhancements (Dec 21, 2012
Complexity spectrum of some discrete dynamical systems
We first study birational mappings generated by the composition of the matrix
inversion and of a permutation of the entries of matrices. We
introduce a semi-numerical analysis which enables to compute the Arnold
complexities for all the possible birational transformations. These
complexities correspond to a spectrum of eighteen algebraic values. We then
drastically generalize these results, replacing permutations of the entries by
homogeneous polynomial transformations of the entries possibly depending on
many parameters. Again it is shown that the associated birational, or even
rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Latin bitrades derived from groups
A latin bitrade is a pair of partial latin squares which are disjoint, occupy
the same set of non-empty cells, and whose corresponding rows and columns
contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin
bitrade is equivalent to three derangements whose product is the identity and
whose cycles pairwise have at most one point in common. By letting a group act
on itself by right translation, we show how some latin bitrades may be derived
from groups without specifying an independent group action. Properties of latin
trades such as homogeneousness, minimality (via thinness) and orthogonality may
also be encoded succinctly within the group structure. We apply the
construction to some well-known groups, constructing previously unknown latin
bitrades. In particular, we show the existence of minimal, -homogeneous
latin trades for each odd . In some cases these are the smallest known
such examples.Comment: 23 page
Partitioning 3-homogeneous latin bitrades
A latin bitrade is a pair of partial latin
squares which defines the difference between two arbitrary latin squares
and
of the same order. A 3-homogeneous bitrade has
three entries in each row, three entries in each column, and each symbol
appears three times in . Cavenagh (2006) showed that any
3-homogeneous bitrade may be partitioned into three transversals. In this paper
we provide an independent proof of Cavenagh's result using geometric methods.
In doing so we provide a framework for studying bitrades as tessellations of
spherical, euclidean or hyperbolic space.Comment: 13 pages, 11 figures, fixed the figures. Geometriae Dedicata,
Accepted: 13 February 2008, Published online: 5 March 200
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