946 research outputs found
Primitive points in rational polygons
Let be a star-shaped polygon in the plane, with rational
vertices, containing the origin. The number of primitive lattice points in the
dilate is asymptotically Area as
. We show that the error term is both and . Both
bounds extend (to the above class of polygons) known results for the isosceles
right triangle, which appear in the literature as bounds for the error term in
the summatory function for Euler's .Comment: 17 page
Triangulating metric surfaces
We prove that any length metric space homeomorphic to a surface may be
decomposed into non-overlapping convex triangles of arbitrarily small diameter.
This generalizes a previous result of Alexandrov--Zalgaller for surfaces of
bounded curvature.Comment: 21 pages, 6 figure
Partitioning a Polygon Into Small Pieces
We study the problem of partitioning a given simple polygon into a
minimum number of polygonal pieces, each of which has bounded size. We give
algorithms for seven notions of `bounded size,' namely that each piece has
bounded area, perimeter, straight-line diameter, geodesic diameter, or that
each piece must be contained in a unit disk, an axis-aligned unit square or an
arbitrarily rotated unit square.
A more general version of the area problem has already been studied. Here we
are, in addition to , given positive real values such that
the sum equals the area of . The goal is to partition
into exactly pieces such that the area of is .
Such a partition always exists, and an algorithm with running time has
previously been described, where is the number of corners of . We give
an algorithm with optimal running time . For polygons with holes, we
get running time .
For the other problems, it seems out of reach to compute optimal partitions
for simple polygons; for most of them, even in extremely restricted cases such
as when is a square. We therefore develop -approximation algorithms
for these problems, which means that the number of pieces in the produced
partition is at most a constant factor larger than the cardinality of a minimum
partition. Existing algorithms do not allow Steiner points, which means that
all corners of the produced pieces must also be corners of . This has the
disappointing consequence that a partition does often not exist, whereas our
algorithms always produce useful partitions. Furthermore, an optimal partition
without Steiner points may require pieces for polygons where a
partition consisting of just pieces exists when Steiner points are allowed.Comment: 32 pages, 24 figure
Partitioning Regular Polygons into Circular Pieces I: Convex Partitions
We explore an instance of the question of partitioning a polygon into pieces,
each of which is as ``circular'' as possible, in the sense of having an aspect
ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters
of the smallest circumscribing circle to the largest inscribed disk. The
problem is rich even for partitioning regular polygons into convex pieces, the
focus of this paper. We show that the optimal (most circular) partition for an
equilateral triangle has an infinite number of pieces, with the lower bound
approachable to any accuracy desired by a particular finite partition. For
pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already
optimal. The square presents an interesting intermediate case. Here the
one-piece partition is not optimal, but nor is the trivial lower bound
approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082
with several somewhat intricate partitions.Comment: 21 pages, 25 figure
Collaboration in sensor network research: an in-depth longitudinal analysis of assortative mixing patterns
Many investigations of scientific collaboration are based on statistical
analyses of large networks constructed from bibliographic repositories. These
investigations often rely on a wealth of bibliographic data, but very little or
no other information about the individuals in the network, and thus, fail to
illustrate the broader social and academic landscape in which collaboration
takes place. In this article, we perform an in-depth longitudinal analysis of a
relatively small network of scientific collaboration (N = 291) constructed from
the bibliographic record of a research center involved in the development and
application of sensor network and wireless technologies. We perform a
preliminary analysis of selected structural properties of the network,
computing its range, configuration and topology. We then support our
preliminary statistical analysis with an in-depth temporal investigation of the
assortative mixing of selected node characteristics, unveiling the researchers'
propensity to collaborate preferentially with others with a similar academic
profile. Our qualitative analysis of mixing patterns offers clues as to the
nature of the scientific community being modeled in relation to its
organizational, disciplinary, institutional, and international arrangements of
collaboration.Comment: Scientometrics (In press
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