In many practical situations, we are faced with a necessity to combine sophisticated mathematical knowledge about the analyzed systems with informal expert knowledge. To make this combination natural, it is desirable to reformulate the abstract mathematical knowledge in understandable intuitive terms. In this paper, we show how this can be done for an abstract metric. 1 For Data Fusion, It Is D...

We discuss a number of distance functions encountered in the theory of computation, including metrics, ultra-metrics, quasi-metrics, generalized ultrametrics, partial metrics, d-ultra-metrics, and generalized metrics. We consider their properties, associated fixed-point theorems, and some general applications they have within the theory of computation. We consider in detail the applications of ...

We study the effectiveness of metrics for MultiRobot Motion-Planning (MRMP) when using RRT-style sampling-based planners. These metrics play the crucial role of determining the nearest neighbors of configurations and in that they regulate the connectivity of the underlying roadmaps produced by the planners and other properties like the quality of solution paths. After screening over a dozen dif...

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler-Einstein metrics, and are automatically Kähler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existen...

We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.

the general relatively isotropic mean landsberg metrics contain the general relativelyisotropic landsberg metrics. a class of finsler metrics is given, in which the mentioned two conceptsare equivalent. in this paper, an interpretation of general relatively isotropic mean landsberg metrics isfound by using c-conformal transformations. some necessary conditions for a general relativelyisotropic ...

Lipschitz condition is a natural notion of function regularity in this context, and the norm dual to the mixed Lipschitz space is a natural distance between measures. In this paper, we consider the tensor product of spaces equipped with tree metrics and give effective formulas for the mixed Lipschitz norm and its dual. We also show that these norms behave well when approximating an arbitrary me...