Comments on numerical solution of boundary value problems of the Laplace equation and calculation of eigenvalues by the grid method

The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners

More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons

A recent precise formulation of the hoop conjecture in four spacetime dimensions is that the Birkhoff invariant $\beta$ (the least maximal length of any sweepout or foliation by circles) of an apparent horizon of energy $E$ and area $A$ should satisfy $\beta \le 4 \pi E$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies $\ell^2 \le \pi A$. We have tested these conjectures on the horizons of all four-charged rotating black hole solutions of ungauged supergravity theories and find that they always hold. They continue to hold in the the presence of a negative cosmological constant, and for multi-charged rotating solutions in gauged supergravity. Surprisingly, they also hold for the Ernst-Wild static black holes immersed in a magnetic field, which are asymptotic to the Melvin solution. In five spacetime dimensions we define $\beta$ as the least maximal area of all sweepouts of the horizon by two-dimensional tori, and find in all cases examined that $\beta(g) \le \frac{16 \pi}{3} E$, which we conjecture holds quiet generally for apparent horizons. In even spacetime dimensions $D=2N+2$, we find that for sweepouts by the product $S^1 \times S^{D-4}$, $\beta$ is bounded from above by a certain dimension-dependent multiple of the energy $E$. We also find that $\ell^{D-2}$ is bounded from above by a certain dimension-dependent multiple of the horizon area $A$. Finally, we show that $\ell^{D-3}$ is bounded from above by a certain dimension-dependent multiple of the energy, for all Kerr-AdS black holes.Comment: 25 page

Ten-decimal tables of the logarithms of complex numbers and for the transformation from Cartesian to polar coordinates

Ten-Decimal Tables of the Logarithms of Complex Numbers and for the Transformation from Cartesian to Polar Coordinates contains Tables of mathematical functions up to ten-decimal value. These tables are compiled in the Department for Approximate Computations of the Institute of Exact Mechanics and Computational Methods of the U.S.S.R. Academy of Sciences. The computations are carried out by this department in conjunction with the Computational-Experimental Laboratory of the Institute.This book will be of value to mathematicians and researchers