Kalinin N.,Apartado Postal |
Kalinin N.,National Research University Higher School of Economics
Discrete and Computational Geometry | Year: 2017
Suppose that there exists a hypersurface with the Newton polytope (Formula presented.), which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of (Formula presented.) to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of (Formula presented.) from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points (Formula presented.) with prescribed multiplicities (Formula presented.). Suppose that the minimal lattice width (Formula presented.) of the Newton polygon (Formula presented.) of the curve C is at least (Formula presented.). Using tropical floor diagrams (a certain degeneration of (Formula presented.) on a horizontal line) we prove that (Formula presented.)In the case (Formula presented.) this estimate becomes (Formula presented.). That rewrites as (Formula presented.) for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary (Formula presented.)) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not a priori clear what is a collection of generic points in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory. © 2017 Springer Science+Business Media New York
Castro-Garay P.,University of Sonora |
Manzanares-Martinez J.,Apartado Postal |
Corella-Madueno A.,University of Sonora |
Rosas-Burgos A.,University of Sonora |
And 3 more authors.
Proceedings of SPIE - The International Society for Optical Engineering | Year: 2015
We have found experimentally the transmittance of normal incident circularly polarized light due to new chiral mixture that was distorted by electric field. The chiral mixture was achieved by mixtures of two nematic liquid crystals (5OCB and 5CB) and S-1-bromo-2-methylbutane. We have found a regime of circular Bragg diffraction for certain values of concentrations and thickness. Optical diffraction phenomenon have received particular attention in research for optical and electro-optical applications, such as low -voltage modulators, reflective phase gratings and smart reflectors. © 2015 SPIE.