13,099 research outputs found
Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan
The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the
generalized fan have been calculated exactly for arbitrary size as well as
arbitrary individual edge and node reliabilities, using transfer matrices of
dimension four at most. While the all-terminal reliabilities of these graphs
are identical, the special case of identical edge () and node ()
reliabilities shows that their two-terminal reliabilities are quite distinct,
as demonstrated by their generating functions and the locations of the zeros of
the reliability polynomials, which undergo structural transitions at
Integrable Combinatorics
We review various combinatorial problems with underlying classical or quantum
integrable structures. (Plenary talk given at the International Congress of
Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1
Yang-Baxter maps: dynamical point of view
A review of some recent results on the dynamical theory of the Yang-Baxter
maps (also known as set-theoretical solutions to the quantum Yang-Baxter
equation) is given. The central question is the integrability of the transfer
dynamics. The relations with matrix factorisations, matrix KdV solitons,
Poisson Lie groups, geometric crystals and tropical combinatorics are discussed
and demonstrated on several concrete examples.Comment: 24 pages. Extended version of lectures given at the meeting
"Combinatorial Aspect of Integrable Systems" (RIMS, Kyoto, July 2004
Fractals from genomes: exact solutions of a biology-inspired problem
This is a review of a set of recent papers with some new data added. After a
brief biological introduction a visualization scheme of the string composition
of long DNA sequences, in particular, of bacterial complete genomes, will be
described. This scheme leads to a class of self-similar and self-overlapping
fractals in the limit of infinitely long constotuent strings. The calculation
of their exact dimensions and the counting of true and redundant avoided
strings at different string lengths turn out to be one and the same problem. We
give exact solution of the problem using two independent methods: the
Goulden-Jackson cluster method in combinatorics and the method of formal
language theory.Comment: 24 pages, LaTeX, 5 PostScript figures (two in color), psfi
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