Electronic Research Announcements

 2016 , Volume 23

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Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra
Pavel Galashin and Vladimir Zolotov
2016, 23: 1-7 doi: 10.3934/era.2016.23.001 +[Abstract](2300) +[PDF](305.0KB)
We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm for constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space.
Asymptotic Hilbert polynomial and a bound for Waldschmidt constants
Marcin Dumnicki, Łucja Farnik and Halszka Tutaj-Gasińska
2016, 23: 8-18 doi: 10.3934/era.2016.23.002 +[Abstract](2834) +[PDF](374.0KB)
In the paper we give a method to compute an upper bound for the Waldschmidt constants of a wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasińska, Adv. Math. 2014, [5]. Our bound is given by a root of a suitable derivative of a certain polynomial associated with the asymptotic Hilbert polynomial.
Nonexistence results for a fully nonlinear evolution inequality
Qianzhong Ou
2016, 23: 19-24 doi: 10.3934/era.2016.23.003 +[Abstract](2373) +[PDF](310.7KB)
In this paper, a Liouville type theorem is proved for some global fully nonlinear evolution inequality via suitable choices of test functions and the argument of integration by parts.
Pentagrams, inscribed polygons, and Prym varieties
Anton Izosimov
2016, 23: 25-40 doi: 10.3934/era.2016.23.004 +[Abstract](2433) +[PDF](376.8KB)
The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of these invariants, R. Schwartz and S. Tabachnikov have recently proved that for polygons inscribed in a conic section one has $E_k = O_k$ for all $k$. In this paper we give a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our main observation is that for inscribed polygons the corresponding monodromy satisfies a certain self-duality relation. From this we also deduce that the space of inscribed polygons with fixed values of the monodromy invariants is an open dense subset in the Prym variety (i.e., a half-dimensional torus in the Jacobian) of the spectral curve. As a byproduct, we also prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons.
Banach limit in convexity and geometric means for convex bodies
Liran Rotem
2016, 23: 41-51 doi: 10.3934/era.2016.23.005 +[Abstract](2492) +[PDF](324.9KB)
In this note we construct Banach limits on the class of sequences of convex bodies. Surprisingly, the construction uses the recently introduced geometric mean of convex bodies. In the opposite direction, we explain how Banach limits can be used to construct a new variant of the geometric mean that has some desirable properties.
Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$
Stephen Coughlan, Łukasz Gołębiowski, Grzegorz Kapustka and Michał Kapustka
2016, 23: 52-68 doi: 10.3934/era.2016.23.006 +[Abstract](3029) +[PDF](442.5KB)
We present a list of arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$ and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$ for which no mirror construction is known.

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