205 research outputs found
Multivariate matrix-exponential distributions
We review what is currently known about one-dimensional distributions on
the non-negative reals with rational Laplace transform, also known as
matrix-exponential distributions. In particular we discuss a flow
interpreation which enables one to mimic certain probabilisticly
inspired arguments which are known from the theory of phase-type distributions.
We then move on to present ongoing research for higher dimensions.
We discuss a characterization result, some closure properties, and
a number of examples. Finally we present open problems and future
perspectives
A Reciprocity Theorem for Monomer-Dimer Coverings
The problem of counting monomer-dimer coverings of a lattice is a
longstanding problem in statistical mechanics. It has only been exactly solved
for the special case of dimer coverings in two dimensions. In earlier work,
Stanley proved a reciprocity principle governing the number of dimer
coverings of an by rectangular grid (also known as perfect matchings),
where is fixed and is allowed to vary. As reinterpreted by Propp,
Stanley's result concerns the unique way of extending to so
that the resulting bi-infinite sequence, for , satisfies a
linear recurrence relation with constant coefficients. In particular, Stanley
shows that is always an integer satisfying the relation where unless 2(mod 4) and
is odd, in which case . Furthermore, Propp's method is
applicable to higher-dimensional cases. This paper discusses similar
investigations of the numbers , of monomer-dimer coverings, or
equivalently (not necessarily perfect) matchings of an by rectangular
grid. We show that for each fixed there is a unique way of extending
to so that the resulting bi-infinite sequence, for , satisfies a linear recurrence relation with constant coefficients. We
show that , a priori a rational number, is always an integer, using a
generalization of the combinatorial model offered by Propp. Lastly, we give a
new statement of reciprocity in terms of multivariate generating functions from
which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete
Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes
The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
Combinatorics, Probability and Computing
The main theme of this workshop was the use of probabilistic
methods in combinatorics and theoretical computer science. Although
these methods have been around for decades, they are being refined all
the time: they are getting more and more sophisticated and powerful.
Another theme was the study of random combinatorial structures,
either for their own sake, or to tackle extremal questions. The workshop
also emphasized connections between probabilistic combinatorics and
discrete probability
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