7,720 research outputs found

    On First-Order Definable Colorings

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    We address the problem of characterizing HH-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

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    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

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    Not much is known about the dynamics outside the support of the maximal entropy measure μ\mu for holomorphic endomorphisms of CPk\mathbb{CP}^k. In this article we study the structure of the dynamics on the Julia set, which is typically larger than Supp(μ)Supp(\mu). The Julia set is the support of the so-called Green current TT, so it admits a natural filtration by the supports of the exterior powers of TT. For 1≤q≤k1\leq q \leq k, let Jq=Supp(Tq)J_q= Supp(T^q). We show that for a generic point of Jq∖Jq+1J_q\setminus J_{q+1} there are at least (k−q)(k-q) "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

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    Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on Rd\mathbb{R}^d 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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