4,583 research outputs found
Classification of 3-dimensional integrable scalar discrete equations
We classify all integrable 3-dimensional scalar discrete quasilinear
equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An
equation Q=0 is called integrable if it may be consistently imposed on all
3-dimensional elementary faces of the 4-dimensional lattice.
Under the natural requirement of invariance of the equation under the action
of the complete group of symmetries of the cube we prove that the only
nontrivial (non-linearizable) integrable equation from this class is the
well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2
has been made.) (Version 3: A few small corrections: one more reference added,
the main statement stated more explicitly.)Comment: 20 p. LaTeX + 1 EPS figur
Enumeration of Nonsingular Buekenhout Unitals
The only known enumeration of Buekenhout unitals occurs in the Desarguesian plane .  In this paper we develop general techniques for enumerating the nonsingular Buekenhoutunitals embedded in any two-dimensional translation plane, and apply these techniques to obtain such an enumeration in the regular nearfield planes, the odd-order Hall planes, and the flag-transitive affine planes.  We also provide some computer data for small-order André planes of index two and give partial results toward an enumeration in this case
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
- …