13,127 research outputs found
On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
We study a non-trivial extreme case of the orchard problem for
pseudolines and we provide a complete classification of pseudoline arrangements
having triple points and double points. We have also classified those
that can be realized with straight lines. They include new examples different
from the known example of B\"or\"oczky. Since Melchior's inequality also holds
for arrangements of pseudolines, we are able to deduce that some combinatorial
point-line configurations cannot be realized using pseudolines. In particular,
this gives a negative answer to one of Gr\"unbaum's problems. We formulate some
open problems which involve our new examples of line arrangements.Comment: 5 figures, 11 pages, to appear in Periodica Mathematica Hungaric
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Metastable states influence on the magnetic behavior of the triangular lattice: Application to the spin-chain compound Ca3Co2O6
It is known that the spin-chain compound Ca3Co2O6 exhibits very interesting
plateaus in the magnetization as a function of the magnetic field at low
temperatures. The origin of them is still controversial. In this paper we study
the thermal behavior of this compound with a single-flip Monte Carlo simulation
on a triangular lattice and demonstrate the decisive influence of metastable
states in the splitting of the ferrimagnetic 1/3 plateau below 10 K. We
consider the [Co2O6]n chains as giant magnetic moments described by large Ising
spins on planar clusters with open boundary conditions. With this simple
frozen-moment model we obtain stepped magnetization curves which agree quite
well with the experimental results for different sweeping rates. We describe
particularly the out-of-equilibrium states that split the low-temperature 1/3
plateau into three steps. They relax thermally to the 1/3 plateau, which has
long-range order at the equilibrium. Such states are further analyzed with
snapshots unveiling a domain-wall structure that is responsible for the
observed behavior of the 1/3 plateau. A comparison is also given of the exact
results in small triangular clusters with our Monte Carlo results, providing
further support for our thermal description of this compound.Comment: 8 pages, 11 figures, submitted to PR
M-curves of degree 9 with deep nests
The first part of Hilbert's sixteenth problem deals with the classification
of the isotopy types realizable by real plane algebraic curves of given degree
. For , one restricts the study to the case of the -curves. For
, the classification is still wide open. We say that an -curve of
degree 9 has a deep nest if it has a nest of depth 3. In the present paper, we
prohibit 10 isotopy types with deep nests and no outer ovals.Comment: 16 pages, 11 figures v.4 minimal correction
Convex-Arc Drawings of Pseudolines
A weak pseudoline arrangement is a topological generalization of a line
arrangement, consisting of curves topologically equivalent to lines that cross
each other at most once. We consider arrangements that are outerplanar---each
crossing is incident to an unbounded face---and simple---each crossing point is
the crossing of only two curves. We show that these arrangements can be
represented by chords of a circle, by convex polygonal chains with only two
bends, or by hyperbolic lines. Simple but non-outerplanar arrangements
(non-weak) can be represented by convex polygonal chains or convex smooth
curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was
made as a poster at the 21st International Symposium on Graph Drawing,
Bordeaux, France, September 2013, and published in Lecture Notes in Computer
Science 8242, Springer, 2013, pp. 522--52
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