13,305 research outputs found
A topological invariant of line arrangements
We define a new topological invariant of line arrangements in the complex
projective plane. This invariant is a root of unity defined under some
combinatorial restrictions for arrangements endowed with some special torsion
character on the fundamental group of their complements. It is derived from the
peripheral structure on the group induced by the inclusion map of the boundary
of a tubular neigborhood in the exterior of the arrangement. By similarity with
knot theory, it can be viewed as an analogue of linking numbers. This is an
orientation-preserving invariant for ordered arrangements. We give an explicit
method to compute the invariant from the equations of the arrangement, by using
wiring diagrams introduced by Arvola, that encode the braid monodromy.
Moreover, this invariant is a crucial ingredient to compute the depth of a
character satisfying some resonant conditions, and complete the existent
methods by Libgober and the first author. Finally, we compute the invariant for
extended MacLane arrangements with an additional line and observe that it takes
different values for the deformation classes.Comment: 19 pages, 5 figure
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Moderate Growth Time Series for Dynamic Combinatorics Modelisation
Here, we present a family of time series with a simple growth constraint.
This family can be the basis of a model to apply to emerging computation in
business and micro-economy where global functions can be expressed from local
rules. We explicit a double statistics on these series which allows to
establish a one-to-one correspondence between three other ballot-like
strunctures
A solution to one of Knuth's permutation problems
We answer a problem posed recently by Knuth: an n-dimensional box, with edges
lying on the positive coordinate axes and generic edge lengths W_1 < W_2 < ...
< W_n, is dissected into n! pieces along the planes x_i = x_j. We describe
which pieces have the same volume, and show that there are C_n distinct
volumes, where C_n denotes the nth Catalan number.Comment: 4 pages, 2 figures
The propagator for the step potential using the path decomposition expansion
We present a direct path integral derivation of the propagator in the
presence of a step potential. The derivation makes use of the Path
Decomposition Expansion (PDX), and also of the definition of the propagator as
a limit of lattice paths.Comment: To appear in DICE 2008 conference proceeding
Fewest repetitions in infinite binary words
A square is the concatenation of a nonempty word with itself. A word has
period p if its letters at distance p match. The exponent of a nonempty word is
the quotient of its length over its smallest period.
In this article we give a proof of the fact that there exists an infinite
binary word which contains finitely many squares and simultaneously avoids
words of exponent larger than 7/3. Our infinite word contains 12 squares, which
is the smallest possible number of squares to get the property, and 2 factors
of exponent 7/3. These are the only factors of exponent larger than 2. The
value 7/3 introduces what we call the finite-repetition threshold of the binary
alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive
threshold
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