24 research outputs found
Sorting Jordan sequences in linear time
For a Jordan curve C in the plane, let x_{1},x_{2},...,x_{n} be the abscissas of the intersection points of C with the x-axis, listed in the order the points occur on C. We call x_{1},x_{2},...,x_{n} a Jordan sequence. In this paper we describe an O(n)-time algorithm for recognizing and sorting Jordan sequences. The problem of sorting such sequences arises in computational geometry and computational geography. Our algorithm is based on a reduction of the recognition and sorting problem to a list-splitting problem. To solve the list-splitting problem we use level linked search trees
Meanders: Exact Asymptotics
We conjecture that meanders are governed by the gravitational version of a
c=-4 two-dimensional conformal field theory, allowing for exact predictions for
the meander configuration exponent \alpha=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and
the semi-meander exponent {\bar\alpha}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This
result follows from an interpretation of meanders as pairs of fully packed
loops on a random surface, described by two c=-2 free fields. The above values
agree with recent numerical estimates. We generalize these results to a score
of meandric numbers with various geometries and arbitrary loop fugacities.Comment: new version with note added in proo
Meanders: A Direct Enumeration Approach
We study the statistics of semi-meanders, i.e. configurations of a set of
roads crossing a river through n bridges, and possibly winding around its
source, as a toy model for compact folding of polymers. By analyzing the
results of a direct enumeration up to n=29, we perform on the one hand a large
n extrapolation and on the other hand we reformulate the available data into a
large q expansion, where q is a weight attached to each road. We predict a
transition at q=2 between a low-q regime with irrelevant winding, and a large-q
regime with relevant winding.Comment: uses harvmac (l), epsf, 16 figs included, uuencoded, tar compresse
A linear time algorithm to remove winding of a simple polygon
AbstractIn this paper, we present a linear time algorithm to remove winding of a simple polygon P with respect to a given point q inside P. The algorithm removes winding by locating a subset of Jordan sequence that is in the proper order and uses only one stack
Exact Meander Asymptotics: a Numerical Check
This note addresses the meander enumeration problem: "Count all topologically
inequivalent configurations of a closed planar non self-intersecting curve
crossing a line through a given number of points". We review a description of
meanders introduced recently in terms of the coupling to gravity of a
two-flavored fully-packed loop model. The subsequent analytic predictions for
various meandric configuration exponents are checked against exact enumeration,
using a transfer matrix method, with an excellent agreement.Comment: 48 pages, 24 figures, tex, harvmac, eps
SU(N) Meander Determinants
We propose a generalization of meanders, i.e., configurations of
non-selfintersecting loops crossing a line through a given number of points, to
SU(N). This uses the reformulation of meanders as pairs of reduced elements of
the Temperley-Lieb algebra, a SU(2)-related quotient of the Hecke algebra, with
a natural generalization to SU(N). We also derive explicit formulas for SU(N)
meander determinants, defined as the Gram determinants of the corresponding
bases of the Hecke algebra.Comment: TeX using harvmac.tex and epsf.tex, 60 pages (l-mode), 5 figure
Meanders and the Temperley-Lieb algebra
The statistics of meanders is studied in connection with the Temperley-Lieb
algebra. Each (multi-component) meander corresponds to a pair of reduced
elements of the algebra. The assignment of a weight per connected component
of meander translates into a bilinear form on the algebra, with a Gram matrix
encoding the fine structure of meander numbers. Here, we calculate the
associated Gram determinant as a function of , and make use of the
orthogonalization process to derive alternative expressions for meander numbers
as sums over correlated random walks.Comment: 85p, uuencoded, uses harvmac (l mode) and epsf, 88 figure