Improved bounds for the crossing numbers of K_m,n and K_n

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

Mechanical Properties of Three-layer Boards of Pine (Pinus radiata) and Pigue (Piptocoma discolor) Owned by the PISMADE S.A. Company

The present study proposes to determine the mechanical properties of three-layer boards of pine (Pinus radiata) and pigue (Piptocoma discolor) owned by the PISMADE S.A. Company. The tests were carried out under established technical procedures, with the use of international standards, at the Materials Resistance laboratory of the Faculty of Mechanics located in the Higher Polytechnic School of Chimborazo. 24 test pieces belonging to three-layer boards provided by the company were evaluated, each of them of different dimensions for bending and compression tests. It was possible to verify the different resistances that the boards of these species support; bending and compression values for both pine and pigue were obtained, appropriate for their use in the industry. The information generated deems this material rigid and appropriate for its efficient and durable use. Keywords: compression, flexion, mechanical properties, wood boards. Resumen El presente estudio propone determinar las propiedades mecánicas en tableros tricapa de pino (Pinus radiata) y pigue (Piptocoma discolor) propiedad de la empresa PISMADE S.A. Los ensayos se los realizó bajo procedimientos técnicos, con la utilización de normas internacionales, dentro del laboratorio de Resistencia de Materiales de la facultad de Mecánica ubicado en la Escuela Superior Politécnica de Chimborazo. Evaluándose 24 probetas pertenecientes a tableros alistonados tricapa, cada una de ellas de diferentes dimensiones para ensayos de flexión y compresión, proporcionados por la empresa. Se pudo comprobar las diferentes resistencias que soportan los tableros de estas especies, obteniendo valores de flexión y compresión tanto para pino y pigue apropiados para su utilización en la industria. La información generada considera a este material rígido y apropiado para su uso eficiente y duradero. Palabras clave: compresión, flexión, propiedades mecánicas, tableros de madera

Regular (2+1)-dimensional black holes within non-linear Electrodynamics

(2+1)-regular static black hole solutions with a nonlinear electric field are derived. The source to the Einstein equations is an energy momentum tensor of nonlinear electrodynamics, which satisfies the weak energy conditions and in the weak field limit becomes the (2+1)-Maxwell field tensor. The derived class of solutions is regular; the metric, curvature invariants and electric field are regular everywhere. The metric becomes, for a vanishing parameter, the (2+1)-static charged BTZ solution. A general procedure to derive solutions for the static BTZ (2+1)-spacetime, for any nonlinear Lagrangian depending on the electric field is formulated; for relevant electric fields one requires the fulfillment of the weak energy conditions.Comment: 5 pages, Latex, 2 figure

Stability properties of black holes in self-gravitating nonlinear electrodynamics

We analyze the dynamical stability of black hole solutions in self-gravitating nonlinear electrodynamics with respect to arbitrary linear fluctuations of the metric and the electromagnetic field. In particular, we derive simple conditions on the electromagnetic Lagrangian which imply linear stability in the domain of outer communication. We show that these conditions hold for several of the regular black hole solutions found by Ayon-Beato and Garcia.Comment: 15 pages, no figure

Accurate pre- and post-eruption orbital periods for the dwarf/classical nova V1017 Sgr

V1017 Sgr is a classical nova (from 1919) that displayed an earlier dwarf nova eruption (from 1901) and underwent two more dwarf nova events (in 1973 and 1991). Previous work on this bright system in quiescence (V = 13.5) has consisted only of a few isolated magnitudes, a few spectra and an ambiguous claim of an orbital period of 5.714 d based on nine radial velocities. To test this period, we have collected 2896 magnitudes (plus 53 in the literature) in the UBVRIJHKL bands from 1897-2016, making an essentially complete photometric history of this unique cataclysmic variable. We find that the light curve in all bands is dominated by the ellipsoidal modulations of a G giant companion star, with a post-eruption (after the 1919 nova event) orbital period of 5.786290 ± 0.000032 d. This is the longest period for any classical nova; the accretion must be powered by the nuclear evolution of the companion star and dwarf nova events occur only because the outer parts of the large disc are cool enough to be unstable. Further, we measure the pre-eruption orbital period (from 1907-1916). The orbital period has decreased by 273 ± 61 parts per million across the 1919 eruption, with the significance of the period change being at the 5.7σ confidence level. This is startling and mystifying for nova theory, because the three known period-change effects cannot account for a period decrease in V1017 Sgr, much less one of such a large size