122 research outputs found
Lattice Properties of Oriented Exchange Graphs and Torsion Classes
The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent
quivers whose edges correspond to mutations. When the quiver admits a
nondegenerate Jacobi-finite potential, the exchange graph admits a natural
acyclic orientation called the oriented exchange graph, as shown by Br\"ustle
and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the
poset of functorially finite torsion classes of a certain finite dimensional
algebra. We prove that lattices of torsion classes are semidistributive
lattices, and we use this result to conclude that oriented exchange graphs with
finitely many elements are semidistributive lattices. Furthermore, if the
quiver is mutation-equivalent to a type A Dynkin quiver or is an oriented
cycle, then the oriented exchange graph is a lattice quotient of a lattice of
biclosed subcategories of modules over the cluster-tilted algebra, generalizing
Reading's Cambrian lattices in type A. We also apply our results to address a
conjecture of Br\"ustle, Dupont, and P\'erotin on the lengths of maximal green
sequences.Comment: Changes to abstract and introduction; in v3, minor changes
throughout, added Lemma 7.3; in v4, abstract slightly changed, final version;
in v5, Lemma 7.3 from v4 removed because of an error in its proof. We give a
new proof of Lemma 7.4, which cited Lemma 7.
Root system chip-firing II: Central-firing
Jim Propp recently proposed a labeled version of chip-firing on a line and
conjectured that this process is confluent from some initial configurations.
This was proved by Hopkins-McConville-Propp. We reinterpret Propp's labeled
chip-firing moves in terms of root systems: a "central-firing" move consists of
replacing a weight by for any positive root
that is orthogonal to . We show that central-firing is always
confluent from any initial weight after modding out by the Weyl group, giving a
generalization of unlabeled chip-firing on a line to other types. For
simply-laced root systems we describe this unlabeled chip-firing as a number
game on the Dynkin diagram. We also offer a conjectural classification of when
central-firing is confluent from the origin or a fundamental weight.Comment: 30 pages, 6 figures, 1 table; v2, v3: minor revision
Root system chip-firing I: Interval-firing
Jim Propp recently introduced a variant of chip-firing on a line where the
chips are given distinct integer labels. Hopkins, McConville, and Propp showed
that this process is confluent from some (but not all) initial configurations
of chips. We recast their set-up in terms of root systems: labeled chip-firing
can be seen as a root-firing process which allows the moves for whenever
, where is the set of
positive roots of a root system of Type A and is a weight of this
root system. We are thus motivated to study the exact same root-firing process
for an arbitrary root system. Actually, this central root-firing process is the
subject of a sequel to this paper. In the present paper, we instead study the
interval root-firing processes determined by for
whenever or , for any . We prove that these interval-firing processes are always confluent,
from any initial weight. We also show that there is a natural way to
consistently label the stable points of these interval-firing processes across
all values of so that the number of weights with given stabilization is a
polynomial in . We conjecture that these Ehrhart-like polynomials have
nonnegative integer coefficients.Comment: 54 pages, 12 figures, 2 tables; v2: major revisions to improve
exposition; v3: to appear in Mathematische Zeitschrift (Math. Z.
Oriented Flip Graphs and Noncrossing Tree Partitions
International audienceGiven a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras
PerceptNet:A Human Visual System Inspired Neural Network for Estimating Perceptual Distance
Traditionally, the vision community has devised algorithms to estimate the
distance between an original image and images that have been subject to
perturbations. Inspiration was usually taken from the human visual perceptual
system and how the system processes different perturbations in order to
replicate to what extent it determines our ability to judge image quality.
While recent works have presented deep neural networks trained to predict human
perceptual quality, very few borrow any intuitions from the human visual
system. To address this, we present PerceptNet, a convolutional neural network
where the architecture has been chosen to reflect the structure and various
stages in the human visual system. We evaluate PerceptNet on various
traditional perception datasets and note strong performance on a number of them
as compared with traditional image quality metrics. We also show that including
a nonlinearity inspired by the human visual system in classical deep neural
networks architectures can increase their ability to judge perceptual
similarity. Compared to similar deep learning methods, the performance is
similar, although our network has a number of parameters that is several orders
of magnitude less
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