Hyperbolic formulations and numerical relativity II: Asymptotically constrained systems of the Einstein equations

We study asymptotically constrained systems for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. First, we examine the previously proposed "$\lambda$-system", which introduces artificial flows to constraint surfaces based on the symmetric hyperbolic formulation. We show that this system works as expected for the wave propagation problem in the Maxwell system and in general relativity using Ashtekar's connection formulation. Second, we propose a new mechanism to control the stability, which we call the ``adjusted system". This is simply obtained by adding constraint terms in the dynamical equations and adjusting its multipliers. We explain why a particular choice of multiplier reduces the numerical errors from non-positive or pure-imaginary eigenvalues of the adjusted constraint propagation equations. This ``adjusted system" is also tested in the Maxwell system and in the Ashtekar's system. This mechanism affects more than the system's symmetric hyperbolicity.Comment: 16 pages, RevTeX, 9 eps figures, added Appendix B and minor changes, to appear in Class. Quant. Gra

Bankruptcy Treatment of Intellectual Property Assets: An Economic Analysis

With the rise of intellectual property in the modern economy, bankruptcy treatment of intellectual property assets has taken on ever greater importance. The law in this area must balance different approaches to asset management. Viewing the world from an ex ante perspective, intellectual property laws seek to foster investment in research and development. Freedom of contract plays a central role in maximizing the potential value of intellectual property by encouraging a robust licensing market to exploit the value of intellectual creativity. By contrast, the bankruptcy system generally views asset management from an ex post standpoint, focusing narrowly on how to maximize the value of a failing or failed enterprise. Thus, bankruptcy law affords trustees and debtors substantial leeway to rescind contracts and reorder the affairs of the failed entity. This article examines the rather complex rules governing the treatment of intellectual property assets in bankruptcy and suggests various reforms that could better promote economic efficiency