A special family of stacked central configurations: Lagrange plus Euler in one

This is a post-peer-review, pre-copyedit version of an article published in Journal of dynamics and differential equations. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10884-018-9647-1We show the existence of a family of stacked central configurations in the planar five-body problem with a special property. Three bodies m1, m2 and m3, ordered from left to right, are collinear and form an Euler central configuration, and the other two bodies m4 and m5, together with m2 are at the vertices of an equilateral triangle and form a Lagrange central configuration.Peer ReviewedPostprint (author's final draft

Symmetry, bifurcation and stacking of the central configurations of the planar 1+4 body problem

In this work we are interested in the central configurations of the planar 1+4 body problem where the satellites have different infinitesimal masses and two of them are diametrically opposite in a circle. We can think this problem as a stacked central configuration too. We show that the configuration are necessarily symmetric and the other sattelites has the same mass. Moreover we proved that the number of central configuration in this case is in general one, two or three and in the special case where the satellites diametrically opposite have the same mass we proved that the number of central configuration is one or two saying the exact value of the ratio of the masses that provides this bifurcation.Comment: 9 pages, 2 figures. arXiv admin note: text overlap with arXiv:1103.627

Central configurations in the five-body problem : Rhombus plus one

We show the existence of central configurations in the planar five-body problem where four bodies are located at the vertices of a rhombus, called rhombus plus one central configurations. Concretely we prove analytically their existence when one diagonal is nearly equal to the sides of the rhombus and when the two diagonals are either equal or nearly equal. In addition, we prove that given a rhombus plus one configuration, the corresponding vector of positive masses that makes the configuration central, if exists, is unique

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, m1,m2 and m3, are collinear and ordered from left to right, while the other two, m4 and m5, are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with m1=m3, there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89-107, 2010), of stacked five-body central configuration where the segments m4m5 and m1m3 do not intersect

Physico-chemical analysis of Albian (Lower Cretaceous) amber from San Just (Spain) : implications for palaeoenvironmental and palaeoecological studies

Amber from a Lower Cretaceous outcrop at San Just, located in the Eastern Iberian Peninsula (Escucha Formation, Maestrat Basin), was investigated to evaluate its physico-chemical properties. Thermogravimetric (TG) and Differential Thermogravimetric (DTG) analyses, infra-red spectroscopy, elemental and C-isotope analyses were performed. Physico-chemical differences between the internal light nuclei and the peripheral darker portions of San Just amber can be attributed to processes of diagenetic alteration that preferentially took place in the external amber border colonized by microorganisms (fungi or bacteria) when the resin was still liquid or slightly polymerized. δ13Camber values of different pieces of the same sample, from the nucleus to the external part, are remarkably homogeneous, as are δ13Camber values of the darker peripheral portions and lighter inner parts of the same samples. Hence, neither invasive microorganisms, nor diagenetic alteration changed the bulk isotopic composition of the amber. δ13C values of different amber samples range from -21.1‰ to -24‰, as expected for C3 plant-derived material. C-isotope analysis, coupled to palaeobotanical, TG and DTG data and infra-red spectra, suggests that San Just amber was exuded by only one conifer species, belonging to either the Cheirolepidiaceae or Aracauriaceae, coniferous families probably living under stable palaeoenvironmental and palaeoecological conditions

Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry

Convex central configurations of the 4-body problem with two pairs of equal masses

Agraïments: The first and third authors are partially supported by FAPEMIG grant APQ-001082/14. The third author is partially supported by CNPq grant 472321/2013-7 and by FAPEMIG grant PPM-00516-15. The second and third autors are supported by CAPES CSF-PVE grant 88881.030454/2013-01.MacMillan and Bartky in 1932 proved that there is a unique isosceles trapezoid central configuration of the 4--body problem when two pairs of equal masses are located at adjacent vertices. After this result the following conjecture was well known between people working on central configurations: The isosceles trapezoid is the unique convex central configuration of the planar 4--body problem when two pairs of equal masses are located at adjacent vertices. We prove this conjecture

Home language, school language and children's literacy attainments: A systematic review of evidence from low‐ and middle‐income countries

The general consensus in the field is that when the home language is different from the language of instruction in school then children's literacy attainments could slow down. In this 26-year review of the literature on children's literacy attainments in low- to middle-income countries, 40 correlational, ethnographic and intervention studies provide the data. We test the ‘home language advantage’ hypothesis where we expect children who speak the same language at home and school to show better literacy learning. We also examine other attributes in the home language and literacy environment (HLLE). Among the multivariate studies, trends differ across countries, age and grade levels, and child measures. Rather than a universal home language advantage, the evidence shows that home language advantage is context-sensitive. The correlational and ethnographic evidence point to a multiple risk factors model of home and school language disconnection; and the ethnographic and intervention studies provide complementary evidence of both feelings of unease, disempowerment and wish to help among family members, and increased confidence following guided support. Possible underlying mechanisms are examined through parallel synthesis of evidence from multiple research methods on three HLLE dimensions—books-at-home, home tutoring and adult literacy practices. The data partially corroborate findings from high-income countries (e.g. home environments impact literacy development, responsive parenting is present across families) but also bring focus on context-specific realities. Neither low-income nor low-print environments are uniform constraints because communities differ and some homes use available resources more efficiently than others