7,722 research outputs found
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
Bounding differences in Jager Pairs
Symmetrical subdivisions in the space of Jager Pairs for continued
fractions-like expansions will provide us with bounds on their difference.
Results will also apply to the classical regular and backwards continued
fractions expansions, which are realized as special cases
Fatou directions along the Julia set for endomorphisms of CP^k
Not much is known about the dynamics outside the support of the maximal
entropy measure for holomorphic endomorphisms of . In this
article we study the structure of the dynamics on the Julia set, which is
typically larger than . The Julia set is the support of the
so-called Green current , so it admits a natural filtration by the supports
of the exterior powers of . For , let . We
show that for a generic point of there are at least
"Fatou directions" in the tangent space. We also give estimates for the
rate of expansion in directions transverse to the Fatou directions.Comment: Final, shorter version, to appear in J. Math. Pures App
Geometry of Log-Concave Density Estimation
Shape-constrained density estimation is an important topic in mathematical
statistics. We focus on densities on that are log-concave, and
we study geometric properties of the maximum likelihood estimator (MLE) for
weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the
optimal log-concave density is piecewise linear and supported on a regular
subdivision of the samples. This defines a map from the space of weights to the
set of regular subdivisions of the samples, i.e. the face poset of their
secondary polytope. We prove that this map is surjective. In fact, every
regular subdivision arises in the MLE for some set of weights with positive
probability, but coarser subdivisions appear to be more likely to arise than
finer ones. To quantify these results, we introduce a continuous version of the
secondary polytope, whose dual we name the Samworth body. This article
establishes a new link between geometric combinatorics and nonparametric
statistics, and it suggests numerous open problems.Comment: 22 pages, 3 figure
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