Good covers are algorithmically unrecognizable

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are algorithmically unrecognizable. More precisely, the intersection pattern of a good cover can be stored in a simplicial complex called nerve which records which subfamilies of the good cover intersect. A simplicial complex is topologically d-representable if it is isomorphic to the nerve of a good cover in R^d. We prove that it is algorithmically undecidable whether a given simplicial complex is topologically d-representable for any fixed d \geq 5. The result remains also valid if we replace good covers with acyclic covers or with covers by open d-balls. As an auxiliary result we prove that if a simplicial complex is PL embeddable into R^d, then it is topologically d-representable. We also supply this result with showing that if a "sufficiently fine" subdivision of a k-dimensional complex is d-representable and k \leq (2d-3)/3, then the complex is PL embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in version

A Note on Non-compact Cauchy surface

It is shown that if a space-time has non-compact Cauchy surface, then its topological, differentiable, and causal structure are completely determined by a class of compact subsets of its Cauchy surface. Since causal structure determines its topological, differentiable, and conformal structure of space-time, this gives a natural way to encode the corresponding structures into its Cauchy surface

The Gabor wave front set of compactly supported distributions

We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set

Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria

In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak $L^p$ norms in the space variables and log improvement in the time variable.Comment: 14 pages, to appea

Global wellposedness for a certain class of large initial data for the 3D Navier-Stokes Equations

In this article, we consider a special class of initial data to the 3D Navier-Stokes equations on the torus, in which there is a certain degree of orthogonality in the components of the initial data. We showed that, under such conditions, the Navier-Stokes equations are globally wellposed. We also showed that there exists large initial data, in the sense of the critical norm $B^{-1}_{\infty,\infty}$ that satisfies the conditions that we considered.Comment: 13 pages, updated references for v

Cosmological spacetimes not covered by a constant mean curvature slicing

We show that there exist maximal globally hyperbolic solutions of the Einstein-dust equations which admit a constant mean curvature Cauchy surface, but are not covered by a constant mean curvature foliation.Comment: 11 page

About Starobinsky inflation

It is believed that soon after the Planck era, space time should have a semi-classical nature. According to this, the escape from General Relativity theory is unavoidable. Two geometric counter-terms are needed to regularize the divergences which come from the expected value. These counter-terms are responsible for a higher derivative metric gravitation. Starobinsky idea was that these higher derivatives could mimic a cosmological constant. In this work it is considered numerical solutions for general Bianchi I anisotropic space-times in this higher derivative theory. The approach is ``experimental'' in the sense that there is no attempt to an analytical investigation of the results. It is shown that for zero cosmological constant $\Lambda=0$, there are sets of initial conditions which form basins of attraction that asymptote Minkowski space. The complement of this set of initial conditions form basins which are attracted to some singular solutions. It is also shown, for a cosmological constant $\Lambda> 0$ that there are basins of attraction to a specific de Sitter solution. This result is consistent with Starobinsky's initial idea. The complement of this set also forms basins that are attracted to some type of singular solution. Because the singularity is characterized by curvature scalars, it must be stressed that the basin structure obtained is a topological invariant, i.e., coordinate independent.Comment: Version accepted for publication in PRD. More references added, a few modifications and minor correction

Production of medium-mass neutron-rich nuclei in reactions induced by 136Xe projectiles at 1 A GeV on a beryllium target

Production cross sections of medium-mass neutron-rich nuclei obtained in the fragmentation of 136Xe projectiles at 1 A GeV have been measured with the FRagment Separator (FRS) at GSI. 125Pd was identified for the first time. The measured cross sections are compared to 238U fission yields and model calculations in order to determine the optimum reaction mechanism to extend the limits of the chart of the nuclides around the r-process waiting point at N=82.Comment: 9 pages, 6 figure