Multiple Coverings of the Plane with Triangles

We prove that any 43-fold covering of the plane with translates of a triangle can be decomposed into two coverings.     

Mathematical modelling and analytical solution for workpiece temperature in grinding

This paper deals with modelling the workpiece temperature field produced during the grinding process. The proposed model is given in terms of a two-dimensional boundary-value problem where the interdependence among the grinding wheel, the workpiece and the coolant is described by two variable functions in the boundary condition. An explicit integral form solution is constructed using the Laplace and Fourier transforms and the Green’s function method.     

Comment on “The Free Will Theorem”

In a recent paper Conway and Kochen, Found. Phys. 36, 2006, claim to have established that theories of the Ghirardi-Rimini-Weber (RW) type, i.e., of spontaneous wave function collapse, cannot be made relativistic. On the other hand, relativistic GRW-type theories have already been presented, in my recent paper, J. Stat. Phys. 125, 2006, and by Dowker and Henson, J. Stat. Phys. 115, 2004. Here, I elucidate why these are not excluded by the arguments of Conway and Kochen.      

How many black holes fit on the head of a pin?

The Bekenstein–Hawking entropy of certain black holes can be computed microscopically in string theory by mapping the elusive problem of counting microstates of a strongly gravitating black hole to the tractable problem of counting microstates of a weakly coupled D-brane system, which has no event horizon, and indeed comfortably fits on the head of a pin. We show here that, contrary to widely held beliefs, the entropy of spherically symmetric black holes can easily be dwarfed by that of stationary multi-black-hole “molecules” of the same total charge and energy. Thus, the corresponding pin-sized D-brane systems do not even approximately count the microstates of a single black hole, but rather those of a zoo of entropically dominant multicentered configurations. Fourth Award in the 2007 Essay Competition of the Gravity Research Foundation.     

Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems

The algebro-geometric approach for integrability of semi-Hamiltonian hydrodynamic type systems is presented. The class of symmetric hydrodynamic type systems is defined and the calculation of the associated Riemann surfaces is greatly simplified for this class. Many interesting and physically motivated examples are investigated.     

On the Zero-Energy-Limit Solution for the Modified Gross–Pitaevskii Equation

The modified Gross–Pitaevskii equation was derived and solved to obtain the 1D solution in the zero-energy limit. This stationary solution could account for the dominated contributions due to the kinetic effect as well as the chemical potential in inhomogeneous Bose gases.