3,682 research outputs found

### Clusters of Cycles

A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar
2--co nnected finite or countable graph $G$ of girth $r$ and maximal
vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the
plane, denote it by $P(G)$, i.e. such that: (i) all interior vertices are of
degree $q$, (ii) all interior faces (denote their number by $p_r$) are
combinatorial $r$-gons and (implied by (i), (ii)) (iii) all vertices, edges and
interior faces form a cell-complex.
An example of $(r,q)$-polycycle is the skeleton of $(r^q)$, i.e. of the
$q$-valent partition of the sphere $S^2$, Euclidean plane $R^2$ or hyperbolic
plane $H^2$ by regular $r$-gons. Call {\it spheric} pairs
$(r,q)=(3,3),(3,4),(4,3),(3,5),(5,3)$; for those five pairs $P(r^q)$ is $(r^q)$
without the exterior face; otherwise $P(r^q)=(r^q)$.
We give here a compact survey of results on $(r,q)$-polycycles.Comment: 21. to in appear in Journal of Geometry and Physic

### Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension
d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the
largest diameter over all lattice (d, k)-polytopes. We develop a computational
framework to determine {\delta}(d, k) for small instances. We show that
{\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3,
4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and
is achieved, up to translation, by a Minkowski sum of lattice vectors

### Properties of parallelotopes equivalent to Voronoi's conjecture

A parallelotope is a polytope whose translation copies fill space without
gaps and intersections by interior points. Voronoi conjectured that each
parallelotope is an affine image of the Dirichlet domain of a lattice, which is
a Voronoi polytope. We give several properties of a parallelotope and prove
that each of them is equivalent to it is an affine image of a Voronoi polytope.Comment: 18 pages (submitted

### Berge Sorting

In 1966, Claude Berge proposed the following sorting problem. Given a string
of $n$ alternating white and black pegs on a one-dimensional board consisting
of an unlimited number of empty holes, rearrange the pegs into a string
consisting of $\lceil\frac{n}{2}\rceil$ white pegs followed immediately by
$\lfloor\frac{n}{2}\rfloor$ black pegs (or vice versa) using only moves which
take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the
alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ such {\em Berge
2-moves} for $n\geq 5$. Extending Berge's original problem, we consider the
same sorting problem using {\em Berge $k$-moves}, i.e., moves which take $k$
adjacent pegs to $k$ vacant adjacent holes. We prove that the alternating
string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge 3-moves for
$n\not\equiv 0\pmod{4}$ and in $\lceil\frac{n}{2}\rceil+1$ Berge 3-moves for
$n\equiv 0\pmod{4}$, for $n\geq 5$. In general, we conjecture that, for any $k$
and large enough $n$, the alternating string can be sorted in
$\lceil\frac{n}{2}\rceil$ Berge $k$-moves. This estimate is tight as
$\lceil\frac{n}{2}\rceil$ is a lower bound for the minimum number of required
Berge $k$-moves for $k\geq 2$ and $n\geq 5$.Comment: 10 pages, 2 figure

### Understanding Image Virality

Virality of online content on social networking websites is an important but
esoteric phenomenon often studied in fields like marketing, psychology and data
mining. In this paper we study viral images from a computer vision perspective.
We introduce three new image datasets from Reddit, and define a virality score
using Reddit metadata. We train classifiers with state-of-the-art image
features to predict virality of individual images, relative virality in pairs
of images, and the dominant topic of a viral image. We also compare machine
performance to human performance on these tasks. We find that computers perform
poorly with low level features, and high level information is critical for
predicting virality. We encode semantic information through relative
attributes. We identify the 5 key visual attributes that correlate with
virality. We create an attribute-based characterization of images that can
predict relative virality with 68.10% accuracy (SVM+Deep Relative Attributes)
-- better than humans at 60.12%. Finally, we study how human prediction of
image virality varies with different `contexts' in which the images are viewed,
such as the influence of neighbouring images, images recently viewed, as well
as the image title or caption. This work is a first step in understanding the
complex but important phenomenon of image virality. Our datasets and
annotations will be made publicly available.Comment: Pre-print, IEEE Conference on Computer Vision and Pattern Recognition
(CVPR), 201

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