5,964 research outputs found
A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2
Using Singer polygons, we construct locally finite affine buildings of types
~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This
construction produces very explicit descriptions of these buildings as well as
very short presentations of the lattices. All but one of the ~C_2-buildings are
necessarily exotic. To the knowledge of the author, these are the first
presentations of lattices in buildings of type ~C_2. Integral and rational
group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions
In this paper we develop an integer-affine classification of
three-dimensional multistory completely empty convex marked pyramids. We apply
it to obtain the complete lists of compact two-dimensional faces of
multidimensional continued fractions lying in planes with integer distances to
the origin equal 2, 3, 4 ... The faces are considered up to the action of the
group of integer-linear transformations. In conclusion we formulate some actual
unsolved problems associated with the generalizations for n-dimensional faces
and more complicated face configurations.Comment: Minor change
Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras
We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6
Topological Data Analysis with Bregman Divergences
Given a finite set in a metric space, the topological analysis generalizes
hierarchical clustering using a 1-parameter family of homology groups to
quantify connectivity in all dimensions. The connectivity is compactly
described by the persistence diagram. One limitation of the current framework
is the reliance on metric distances, whereas in many practical applications
objects are compared by non-metric dissimilarity measures. Examples are the
Kullback-Leibler divergence, which is commonly used for comparing text and
images, and the Itakura-Saito divergence, popular for speech and sound. These
are two members of the broad family of dissimilarities called Bregman
divergences.
We show that the framework of topological data analysis can be extended to
general Bregman divergences, widening the scope of possible applications. In
particular, we prove that appropriately generalized Cech and Delaunay (alpha)
complexes capture the correct homotopy type, namely that of the corresponding
union of Bregman balls. Consequently, their filtrations give the correct
persistence diagram, namely the one generated by the uniformly growing Bregman
balls. Moreover, we show that unlike the metric setting, the filtration of
Vietoris-Rips complexes may fail to approximate the persistence diagram. We
propose algorithms to compute the thus generalized Cech, Vietoris-Rips and
Delaunay complexes and experimentally test their efficiency. Lastly, we explain
their surprisingly good performance by making a connection with discrete Morse
theory
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