On the existence of bi-pyramidal central configurations of the n + 2-body problem with an n-gon base

Abstract. In this paper we prove the existence of central con gurations of the n + 2{body problem where n equal masses are located at the vertices of a regular n{gon and the remaining 2 masses, which are not necessarily equal, are located on the straight line orthogonal to the plane containing the n{gon passing through its center. Here this kind of central con gurations is called bi{pyramidal central con gurations. In particular, we prove that if the masses mn+1 and mn+2 and their positions satisfy convenient relations, then the con guration is central. We give explicitly those relations.

Food Indicators and Their Relationship with 10 to 12 Year-olds' Subjective Well-Being

This study aimed to test subjective indicators designed to analyze the role food plays in children’s lives, explore children’s personal well-being, and evaluate the relationship between these two phenomena. It was conducted on 371 children aged 10 to 12 by means of a selfadministered questionnaire. Results showed a marked interest in food on the part of children, who consider taste and health the most important indicators when it comes to eating. They demonstrated a high level of personal well-being, measured using Cummins & Lau’s adapted version of the Personal Well- Being Index–School Children (PWI-SC) (2005), overall life satisfaction (OLS) and satisfaction with various life domains (friends, family, sports, food and body). Regression models were conducted to explain satisfaction with food, taking as independent variables the interest children have in food, the importance they give to different reasons for eating, scores from the PWI-SC, OLS and satisfaction with various life domains. In the final model, it was found that OLS, health indicators, satisfaction with health from the PWI-SC and satisfaction with your body contribute to explaining satisfaction with food. The results obtained suggest that satisfaction with food is a relevant indicator in the exploration of children’s subjective well-being, calling into question the widespread belief that these aspects are of exclusive interest to adults. They also seem to reinforce the importance of including food indicators in any study aimed at exploring the well-being of the 10 to 12 year-old population.

Central configurations of nested rotated regular tetrahedra

In this paper we prove that there are only two different classes of central configura- tions with convenient masses located at the vertices of two nested regular tetrahedra: either when one of the tetrahedra is a homothecy of the other one, or when one of the tetrahedra is a homothecy followed by a rotation of Euler angles = = 0 and = of the other one. We also analyze the central configurations with convenient masses located at the vertices of three nested regular tetrahedra when one them is a homothecy of the other one, and the third one is a homothecy followed by a rotation of Euler angles = = 0 and = of the other two. In all these cases we have assumed that the masses on each tetrahedron are equal but masses on different tetrahedra could be different.

Central configurations of nested regular polyhedra for the spatial 2n–body problem

We consider 2n masses located at the vertices of two nested regular polyhedra with the same number of vertices. Assuming that the masses in each polyhedron are equal, we prove that for each ratio of the masses of the inner and the outer polyhedron there exists a unique ratio of the length of the edges of the inner and the outer polyhedron such that the configuration is central.

Central configurations of three nested regular polyhedra for the spatial 3n–body problem

Three regular polyhedra are called nested if they have the same number of vertices n, the same center and the positions of the vertices of the inner polyhedron ri, the ones of the medium polyhedron Ri and the ones of the outer polyhedron Ri satisfy the relation Ri = ri and Ri = Rri for some scale factors R > > 1 and for all i = 1, . . . , n. We consider 3n masses located at the vertices of three nested regular polyhedra. We assume that the masses of the inner polyhedron are equal to m1, the masses of the medium one are equal to m2, and the masses of the outer one are equal to m3. We prove that if the ratios of the masses m2/m1 and m3/m1 and the scale factors and R satisfy two convenient relations, then this configuration is central for the 3n–body problem. Moreover there is some numerical evidence that, first, fixed two values of the ratios m2/m1 and m3/m1, the 3n–body problem has a unique central configuration of this type; and second that the number of nested regular polyhedra with the same number of vertices forming a central configuration for convenient masses and sizes is arbitrary.

Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 in R3

In this paper we will find a continuous of periodic orbits passing near infinity for a class of polynomial vector fields in R3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane and that possess a “generalized heteroclinic loop” formed by two singular points e+ and e− at infinity and their invariant manifolds � and . � is an invariant manifold of dimension 1 formed by an orbit going from e− to e+, � is contained in R3 and is transversal to . is an invariant manifold of dimension 2 at infinity. In fact, is the 2–dimensional sphere at infinity in the Poincar´e compactification minus the singular points e+ and e−. The main tool for proving the existence of such periodic orbits is the construction of a Poincar´e map along the generalized heteroclinic loop together with the symmetry with respect to .

Symmetric periodic orbits near heteroclinic loops at infinity for a class of polynomial vector fields

For polynomial vector fields in R3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

Equilibrium points and central configurations for the Lennard-Jones 2-and 3-body problems

Abstract. In this paper we study the relative equilibria and their stability for a system of three point particles moving under the action of a Lennard{Jones potential. A central con guration is a special position of the particles where the position and acceleration vectors of each particle are proportional, and the constant of proportionality is the same for all particles. Since the Lennard{Jones potential depends only on the mutual distances among the particles, it is invariant under rotations. In a rotating frame the orbits coming from central con gurations become equilibrium points, the relative equilibria. Due to the form of the potential, the relative equilibria depend on the size of the system, that is, depend strongly of the momentum of inertia I. In this work we characterize the relative equilibria, we nd the bifurcation values of I for which the number of relative equilibria is changing, we also analyze the stability of the relative equilibria.

Symmetric periodic orbits near a heteroclinic loop in R3 formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper we consider C1 vector fields X in R3 having a “generalized heteroclinic loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere S2 and a diameter connecting the north with the south pole. The north pole is an attractor on S2 and a repeller on . The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S2. We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient Poincar´e map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3 satisfying this dynamics.

Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

In this paper we consider vector fields in R3 that are invariant under a suitable symmetry and that posses a “generalized heteroclinic loop” L formed by two singular points (e+ and e −) and their invariant manifolds: one of dimension 2 (a sphere minus the points e+ and e −) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e −). In particular, we analyze the dynamics of the vector field near the heteroclinic loop L by means of a convenient Poincar´e map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in R3, and the second one is the charged rhomboidal four body problem.

The role of Cysteine 6.47 in class A GPCRs

Abstract Background: The CWxP motif of transmembrane helix 6 (x: any residue) is highly conserved in class A GPCRs. Within this motif, W6.48 is a big star in the theory of the global “toggle switch” because of its key role in the activation mechanism of GPCRs upon ligand binding. With all footlights focused on W6.48, the reason why the preceding residue, C6.47, is largely conserved is still unknown. The present study is aimed to fill up this lack of knowledge by characterizing the role of C6.47 of the CWxP motif. Results: A complete analysis of available crystal structures has been made alongside with molecular dynamics simulations of model peptides to explore a possible structural role for C6.47. Conclusions: We conclude that C6.47 does not modulate the conformation of the TM6 proline kink and propose that C6.47 participates in the rearrangement of the TM6 and TM7 interface accompanying activation.

La puesta en práctica de la interseccionalidad política: el caso de las políticas LGTB en Cataluña

En el marco de la discusión académica está emergiendo con fuerza la necesidad de abordar la interacción entre distintas desigualdades en el diseño e implementación de políticas públicas, siendo el enfoque interseccional una propuesta teórica muy interesante. Se detecta, sin embargo, que hacen falta más análisis empíricos que permitan dar cuenta de los avances y obstáculos con que se encuentran las políticas públicas para implementar la interseccionalidad entre diversas desigualdades. En este artículo analizamos las recientes políticas de igualdad LGTB, centrándonos en el caso catalán, para ver cómo adoptan la relación entre desigualdades, y detectar cuáles son los factores clave que interfieren en la implementación de un modelo interseccional.

La RSC en las memorias contables de las empresas del mercado continuo español

Un analisis de la informacion aportada por algunas de companias admitidas a cotizacion en Bolsa en 2008, permite concluir que aun son demasiadas las empresas que siguen sin presentar informacion medioambiental en la memoria de las cuentas anuales, segun 10 requerido por el PGC07. Y menos las que presentan provisiones y contingencias medioambientales.