Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant

An upper bound on the k-modem illumination problem

A variation on the classical polygon illumination problem was introduced in [Aichholzer et. al. EuroCG'09]. In this variant light sources are replaced by wireless devices called k-modems, which can penetrate a fixed number k, of "walls". A point in the interior of a polygon is "illuminated" by a k-modem if the line segment joining them intersects at most k edges of the polygon. It is easy to construct polygons of n vertices where the number of k-modems required to illuminate all interior points is Omega(n/k). However, no non-trivial upper bound is known. In this paper we prove that the number of k-modems required to illuminate any polygon of n vertices is at most O(n/k). For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of 6n/k + 1. Moreover, we present an O(n log n) time algorithm to achieve this bound.Comment: 9 pages, 4 figure

Efeito da L-lisina na alimentação in ovo de embriões avícolas

O presente estudo teve como objetivo avaliar os efeitos da alimentação in ovo utilizando L-lisina sobre a eclodibilidade, relação pinto/ovo, desenvolvimento do trato gastrointestinal (do nascimento aos sete dias pós-eclosão) e desempenho dos pintos de um a sete dias. Foram utilizados 350 ovos férteis da linhagem Rhode Island Red (matrizes com 47 semanas). O delineamento experimental foi o completamente casualizado com os tratamentos constituídos por dois controles e cinco níveis crescentes de L-lisina com 50 repetições (ovos) cada. Os dados coletados foram submetidos a regressão polinomial a 5%. Houve efeito significativo (p<0,05) sobre a eclodibilidade, mortalidade embrionária intermediária, mortalidade intermediária tardia e ovos bicados, com uma gradual queda na eclodibilidade a partir do aumento dos níveis de L-lisina utilizados. Foram observadas diferenças significativas (p<0,05) sobre o peso do coração e do pâncreas de pintos com um dia; e no consumo de ração, percentual de ganho de peso e comprimento do ceco dos pintos aos sete dias. Os resultados deste estudo indicam que a alimentação in ovo utilizando L-lisina afeta diretamente a eclodibilidade dos ovos inoculados. Usando 0,5% e 1,0% de L-lisina, houve um efeito positivo no peso do pinto, do coração, do pâncreas e sobre o desempenho dos pintinhos de um a 7 dias, sem afetar negativamente a relação pintinho/ovo e o trato gastrointestinal

Some results on the laplacian spectra of Token graphs

This version of the contribution has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/978-3-030-83823-2_11. Use of this Accepted Version is subject to the publisher's Accepted Manuscript terms of use http://www.spingernature.com/gp/open-research/policies/accepted-manuscript-terms.We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this work, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that 1=h=k=n2 , the Laplacian spectrum of Fh(G) is contained in the Laplacian spectrum of Fk(G) . Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement G¯¯¯¯ . This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs.The research of C. Dalf´o and M. A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The research of C. Dalf´o has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The research of C. Huemer was supported by PID2019-104129GBI00/ AEI/ 10.13039/501100011033 and Gen. Cat. DGR 2017SGR1336. F. J. Zaragoza Mart´ınez acknowledges the support of the National Council of Science and Technology (Conacyt) and its National System of Researchers (SNI). This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 734922.Peer ReviewedPostprint (author's final draft