Pair Of Metrics Research Articles

Compatible and Almost Compatible Pseudo-Riemannian Metrics

In the present paper, the notions of compatible and almost compatible Riemannian and pseudo-Riemannian metrics are introduced. These notions are motivated by the theory of compatible Poisson structures of hydrodynamic type (local and nonlocal) and generalize the notion of flat pencils of metrics, which plays an important role in the theory of integrable systems of hydrodynamic type and Dubrovin's theory of Frobenius manifolds. Compatible metrics generate compatible Poisson structures of hydrodynamic type (these structures are local for flat metrics and nonlocal if the metrics are not flat). For the “nonsingular” case in which the eigenvalues of a pair of metrics are distinct, we obtain a complete explicit description of compatible and almost compatible metrics.

Measuring reusability of legacy software systems

There is an increasing need to reuse existing legacy software systems within the context of a modern software architecture. There are two ways to achieve this, either by re-engineering or by wrapping. In this paper a pair of metrics are discussed for deciding whether a program is reusable at all, and if so, whether to convert it or to wrap it. Managers should be aware of both alternatives and be able to decide which alternative is best suited for them. The metrics proposed in this paper are intended to support this decision making. The metrics have emerged from the experience of the author in conducting re-engineering and wrapping projects in the last ten years. Copyright © 1998 John Wiley & Sons, Ltd.

Convergence of approximate solutions to conservation laws

Publisher Summary This chapter discusses a few results on the theoretical side of conservation laws concerning the convergence of approximate solutions. The general setting is a system of n conservation laws in one space dimension. In the context of conservation laws, the maximum norm and the total variation norm provide a natural pair of metrics in which to investigate stability; the maximum norm serving as a measure of the amplitude of the solution and the total variation norm as a measure of the gradient of the solution. Their role is indicated by the theorem concerning the stability and convergence of Glimm's random choice method applied to the Cauchy problem with initial data having small total variation.