4,827 research outputs found
On the Plants Leaves Boundary, "Jupe \`a Godets" and Conformal Embeddings
The stable profile of the boundary of a plant's leaf fluctuating in the
direction transversal to the leaf's surface is described in the framework of a
model called a "surface \`a godets". It is shown that the information on the
profile is encoded in the Jacobian of a conformal mapping (the coefficient of
deformation) corresponding to an isometric embedding of a uniform Cayley tree
into the 3D Euclidean space. The geometric characteristics of the leaf's
boundary (like the perimeter and the height) are calculated. In addition a
symbolic language allowing to investigate statistical properties of a "surface
\`a godets" with annealed random defects of curvature of density is
developed. It is found that at the surface exhibits a phase transition
with critical exponent from the exponentially growing to the flat
structure.Comment: 17 pages (revtex), 8 eps-figures, to appear in Journal of Physics
Ricci-flat graphs with girth at least five
A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges.
Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau,
Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009].
In this paper, we classified all Ricci-flat connected graphs with girth at
least five: they are the infinite path, cycle (), the
dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We
also construct many Ricci-flat graphs with girth 3 or 4 by using the root
systems of simple Lie algebras.Comment: 14 pages, 15 figure
Chiral extensions of chiral polytopes
Given a chiral d-polytope K with regular facets, we describe a construction
for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is
finite whenever K is finite. We provide explicit examples of chiral 4-polytopes
constructed in this way from chiral toroidal maps.Comment: 21 pages. [v2] includes several minor revisions for clarit
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
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