A GENERALIZED TRANSVERSALITY IN GLOBAL ANALYSIS

E. Zeidler in [10] mentions that transversality is certainly one of the most important concepts of modern mathematics, which provided an answer to the question: when is the preimage of a manifold a manifold. This is the celebrated transversality theorem, which has been applied widely to differential topology and dynamic system in [1,2] and [3].     

On the Global Visibility of the Singularity in Quasi-Spherical Collapse

We analyze here the issue of local versus global visibility of a singularity that forms in gravitational collapse of a dust cloud, which has important implications for the weak and strong versions of the cosmic censorship hypothesis. We find conditions for when a singularity will be only locally naked, rather than being globally visible, thus preserving the weak censorship hypothesis. The conditions for the formation of a black hole or a naked singularity in the Szekeres quasi-spherical collapse models are worked out. The causal behaviour of the singularity curve is studied by examining the outgoing radial null geodesics, and the final outcome of collapse is related to the nature of the regular initial data specified on an initial hypersurface from which the collapse evolves. An interesting feature that emerges is that the singularity in Szekeres spacetimes can be directionally naked.     

The Structure of the Extreme Schwarzschild-de Sitter Space-time

The extreme Schwarzschild-de Sitter space-timeis a spherically symmetric solution of Einstein'sequations with a cosmological constant Lambda and massparameter m > 0 which is characterized by thecondition that 9m² = 1. The globalstructure of this space-time is here analyzed in detail.Conformal and embedding diagrams are constructed, andsynchronous coordinates which are suitable for a discussion of the cosmic no-hair conjecture arepresented. The permitted geodesic motions are alsoanalyzed. By a careful investigation of the geodesicsand the equations of geodesic deviation, it is shown that specific families of observers escape fromfalling into the singularity and approach nonsingularasymptotic regions which are represented by specialpoints in the complete conformal diagram.The redshift of signals emitted by particles whichfall into the singularity, as detected by those observers which escape, is also calculated.     

Direct local and global approximation theorems for some linear positive operators

In this paper we establish direct local and global approximation theorems for Baskakov type operators and Szasz - Mirakjan type operators, respectively.