A comparison of directed formulations for the capacitated minimal spanning tree problem

In this paper, we compare linear integer programming directed formulations for the capacitated minimal spanning tree (CMST) problem. This problem is directly related to network design and consists of finding the cheapest way to link a set of terminals at different locations with a central node (a computer site, for instance). The present work focuses on a single commodity flow formulation presented in [3,4]. This formulation includes two sets of variables and, hence, it also must include coupling constraints between the two sets of variables. These coupling constraints can be defined in several ways. The main result of this work establishes that when the strongest form of the coupling constraints is used in the flow formulation (as in [4]), the equivalent formulation using only theX _ij variables satisfies the so-called multistar constraints. These constraints are shown to be stronger than the weakest form of the generalized subtour breaking constraints and, hence, our result improves on the characterization given in [5] for the flow formulation presented in [4]. For certain parameters, the multistar constraints induce facets of the non-directed CMST polytope (see [2]). We also show that it is not possible to compare the linear relaxations of a formulation that involves the multistar constraints with one that involves the generalized subtour breaking constraints.     

Arrangements of hyperplanes with property D

We denote by the complement of the complexification of a real arrangement of hyperplanes. It is known that there is a certain technical property, called property D, on real arrangements of hyperplanes such that: if a real arrangement of hyperplanes is simplicial then has property D, and if has property D then is aK(, 1) space. Our main goal is to prove that: if has property D then is simplicial. We also prove that a quasi-simplicial arrangement is always simplicial.     

Self-consistent solutions of the semiclassical Einstein equations with cosmological constant

The general solution to the semiclassical backreaction equation is found for conformally invariant free quantum fields in spatially flat homogeneous and isotropic spacetime with Cosmological constant and with no classical source when the ratio of the renormalisation parameters/=9/4. It contains a two-parameter family of bouncing solutions that avoid the singularity. There are several one-parameter families which do not have particle horizons. The stability of these solutions is investigated and it is found that they are stable when and have different signs. However, when both parameters have the same sign the set of stable solutions is restricted by the condition 0 < < 1/9. In both cases these solutions have a final de Sitter stage.     

Weyl equation in some anisotropic stiff fluid universes

The Weyl equation (massless Dirac equation) is studied in a family of exact solutions of the Einstein equations whose material content is a perfect fluid with stiff equation of state (p=) and which are of Bianchi type I. The field equation is solved exactly for some members of the family.