121 research outputs found
Planar Rosa : a family of quasiperiodic substitution discrete plane tilings with -fold rotational symmetry
We present Planar Rosa, a family of rhombus tilings with a -fold
rotational symmetry that are generated by a primitive substitution and that are
also discrete plane tilings, meaning that they are obtained as a projection of
a higher dimensional discrete plane. The discrete plane condition is a relaxed
version of the cut-and-project condition. We also prove that the Sub Rosa
substitution tilings with -fold rotational symmetry defined by Kari and
Rissanen do not satisfy even the weaker discrete plane condition. We prove
these results for all even . This completes our previously published
results for odd values of
Two Counting Problems in Geometric Triangulations and Pseudoline Arrangements
The purpose of this dissertation is to study two problems in combinatorial geometry in regard to obtaining better bounds on the number of geometric objects of interest: (i) monotone paths in geometric triangulations and (ii) pseudoline arrangements.
\medskip(i) A directed path in a graph is monotone in direction of if every edge in the path has a positive inner product with . A path is monotone if it is monotone in some direction. Monotone paths are studied in optimization problems, specially in classical simplex algorithm in linear programming. We prove that the (maximum) number of monotone paths in a geometric triangulation of points in the plane is . This improves an earlier upper bound of ; the current best lower bound is (Dumitrescu~\etal, 2016).
\medskip (ii) Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, for instance in the study of sorting networks. Let be the number of nonisomorphic arrangements of pseudolines and let . The problem of estimating was posed by Knuth in 1992. Knuth conjectured that and also derived the first upper and lower bounds: and . The upper bound underwent several improvements, (Felsner, 1997), and (Felsner and Valtr, 2011), for large . Here we show that for some constant . In particular, for large . This improves the previous best lower bound, , due to Felsner and Valtr (2011). Our arguments are elementary and geometric in nature. Further, our constructions are likely to spur new developments and improved lower bounds for related problems, such as in topological graph drawings.
\medskip Developing efficient algorithms and computer search were key to verifying the validity of both results
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
(-1)-enumeration of plane partitions with complementation symmetry
We compute the weighted enumeration of plane partitions contained in a given
box with complementation symmetry where adding one half of an orbit of cubes
and removing the other half of the orbit changes the weight by -1 as proposed
by Kuperberg. We use nonintersecting lattice path families to accomplish this
for transpose-complementary, cyclically symmetric transpose-complementary and
totally symmetric self-complementary plane partitions. For symmetric
transpose-complementary and self-complementary plane partitions we get partial
results. We also describe Kuperberg's proof for the case of cyclically
symmetric self-complementary plane partitions.Comment: 41 pages, AmS-LaTeX, uses TeXDraw; reference adde
Master index
Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química
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