5,645 research outputs found
Drawing Area-Proportional Euler Diagrams Representing Up To Three Sets
Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where non-convex curves are necessary, our method draws an appropriate diagram using non-convex polygons. Thus, we are now always able to automatically visualize data for up to three sets
Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree
Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is
a spectral sequence starting at the Khovanov homology of the link and
converging to the Heegaard Floer homology of its branched double cover. The aim
of this paper is to explicitly calculate this spectral sequence in terms of
bordered Floer homology. There are two primary ingredients in this computation:
an explicit calculation of bimodules associated to Dehn twists, and a general
pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on
computing the bimodules; this part focuses on the pairing theorem for polygons,
in order to prove that the spectral sequence constructed in the previous part
agrees with the one constructed by Ozsv\'ath and Szab\'o.Comment: 85 pages, 19 figures, v3: Version to appear in Journal of Topolog
New constructions of two slim dense near hexagons
We provide a geometrical construction of the slim dense near hexagon with
parameters . Using this construction, we construct
the rank 3 symplectic dual polar space which is the slim dense near
hexagon with parameters . Both the near hexagons are
constructed from two copies of a generalized quadrangle with parameters (2,2)
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads
This is the first of two papers in which we prove that a cell model of the
moduli space of curves with marked points and tangent vectors at the marked
points acts on the Hochschild co--chains of a Frobenius algebra. We also prove
that a there is dg--PROP action of a version of Sullivan Chord diagrams which
acts on the normalized Hochschild co-chains of a Frobenius algebra. These
actions lift to operadic correlation functions on the co--cycles. In
particular, the PROP action gives an action on the homology of a loop space of
a compact simply--connected manifold.
In this first part, we set up the topological operads/PROPs and their cell
models. The main theorems of this part are that there is a cell model operad
for the moduli space of genus curves with punctures and a tangent
vector at each of these punctures and that there exists a CW complex whose
chains are isomorphic to a certain type of Sullivan Chord diagrams and that
they form a PROP. Furthermore there exist weak versions of these structures on
the topological level which all lie inside an all encompassing cyclic
(rational) operad.Comment: 50 pages, 7 figures. Newer version has minor changes. Some material
shifted. Typos and small things correcte
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