Smart Geodesic Tracing in GRworkbench

We have developed a new tool for numerical work in General Relativity: GRworkbench. We discuss how GRworkbench's implementation of a numerically-amenable analogue to Differential Geometry facilitates the development of robust and chart-independent numerical algorithms. We consider, as an example, geodesic tracing on two charts covering the exterior Schwarzschild space-time.     

A rigid subspace of the real line whose square is a homogeneous subspace of the plane

Working in ZFC, we give an example as indicated in the title.

     

Forces along Equidistant Particle Paths

Two particles on the sphere leave the equator moving due south and travel at a constant and equal speed along a geodesic colliding at the south pole. An observer who is unaware of the curvature of the space will conclude that there is an attractive force acting between the particles. On the other hand, if particles travel at the same speed (initially parallel) along geodesics in the hyperbolic plane, then the particle paths diverge. Imagine two particles in the hyperbolic plane that are bound together at a constant distance with their center of mass traveling along a geodesic path at a constant velocity, then the force due to the curvature of the space acts to break the bond and increases as a quadratic function of the velocity. We consider this problem for the sphere and the hyperbolic plane and we give the exact formula for the apparent force between the particles. This revised version was published online in July 2006 with corrections to the Cover Date.     

Riemannian Manifolds in Which Certain Curvature Operator Has Constant Eigenvalues along Each Circle

Riemannian manifolds for which a natural curvature operator has constant eigenvalues on circles are studied. A local classification in dimensions two and three is given. In the 3-dimensional case one gets all locally symmetric spaces and all Riemannian manifolds with the constant principal Ricci curvatures r ₁ = r ₂ = 0, r ₃= 0 , which are not locally homogeneous, in general.     

Parametric and Non Homogeneous Semi-Markov Process for HIV Control

In AIDS control, physicians have a growing need to use pragmatically useful and interpretable tools in their daily medical taking care of patients. Semi-Markov process seems to be well adapted to model the evolution of HIV-1 infected patients. In this study, we introduce and define a non homogeneous semi-Markov (NHSM) model in continuous time. Then the problem of finding the equations that describe the biological evolution of patient is studied and the interval transition probabilities are computed. A parametric approach is used and the maximum likelihood estimators of the process are given. A Monte Carlo algorithm is presented for realizing non homogeneous semi-Markov trajectories. As results, interval transition probabilities are computed for distinct times and follow-up has an impact on the evolution of patients.      

Fluoride effect on the palladium–phenanthroline catalyzed carbonylation of nitroarenes to carbamates

Fluorides promote the palladium–phenanthroline catalyzed carbonylation of nitroarenes to carbamates. The effect is more evident on the rate of the reaction at short reaction times, but a positive effect on selectivity is also observed under certain conditions. The effect is observed even under conditions under which chloride inhibits the reaction. Tetraethylammonium is a better countercation than sodium. Copyright © 2007 John Wiley & Sons, Ltd.     

Some Remarks on the Geodesic Completeness of Compact Nonpositively Curved Spaces

We study the following question: does a compact nonpositively curved space have a totally geodesic core?