The number of planar central configurations for the 4-body problem is finite when 3 mass positions are fixed

In the n{body problem a central con guration is formed when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. Lindstrom showed for n = 3 and for n > 4 that if n ? 1 masses are located at xed points in the plane, then there are only a nite number of ways to position the remaining nth mass in such a way that they de ne a central con guration. Lindstrom leaves open the case n = 4. In this paper we prove the case n = 4 using as variables the mutual distances between the particles.

Increasing stair climbing in a train station: effects of contextual variables and visibility

Accumulation of physical activity during daily living is a current public health target that is influenced by the layout of the built environment. This study reports how the layout of the environment may influence responsiveness to an intervention. Pedestrian choices (n = 41 717) between stairs and the adjacent escalators were monitored for seven weeks in a train station (Birmingham, UK). After a 3.5 week baseline period, a stair riser banner intervention to increase stair climbing was installed on two staircases adjacent to escalators and monitoring continued for a further 3.5 weeks. Logistic regression analyses revealed that the visibility of the intervention, defined as the area of visibility in the horizontal plane opposite to the direction of travel (termed the isovist) had a major effect on success of the intervention. Only the largest isovist produced an increase in stair climbing (isovist=77.6 m2, OR = 1.10, CIs 1.02-1.19; isovist=40.7 m2, OR = 0.98, CIs 0.91-1.06; isovist=53.2 m2, OR = 1.00, CIs 0.95-1.06). Additionally, stair climbing was more common during the morning rush hour (OR = 1.56, CIs 1.80-2.59) and at higher levels of pedestrian traffic volume (OR = 1.92, CIs 1.68-2.21). The layout of the intervention site can influence responsiveness to point-of-choice interventions. Changes to the design of train stations may maximize the choice of the stairs at the expense of the escalator by pedestrians leaving the station.

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.

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.