# American Institute of Mathematical Sciences

ISSN:
1935-9179

eISSN:
1935-9179

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## Electronic Research Announcements

January 2016 , Volume 23

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2016, 23: 1-7 doi: 10.3934/era.2016.23.001 +[Abstract](605) +[PDF](305.0KB)
Abstract:
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.
2016, 23: 8-18 doi: 10.3934/era.2016.23.002 +[Abstract](746) +[PDF](374.0KB)
Abstract:
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.
2016, 23: 19-24 doi: 10.3934/era.2016.23.003 +[Abstract](672) +[PDF](310.7KB)
Abstract:
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.
2016, 23: 25-40 doi: 10.3934/era.2016.23.004 +[Abstract](616) +[PDF](376.8KB)
Abstract:
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.
2016, 23: 41-51 doi: 10.3934/era.2016.23.005 +[Abstract](734) +[PDF](324.9KB)
Abstract:
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.
2016, 23: 52-68 doi: 10.3934/era.2016.23.006 +[Abstract](719) +[PDF](442.5KB)
Abstract:
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.

2016  Impact Factor: 0.483