We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. Similar relations for metrics of arbitrary signature on a manifold are conside...

Applying a theorem due to Belopol’ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on R endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we c...

k-Nearest Neighbours (k-NN) is a well understood and widely-used approach to classification and regression problems. In many cases, such applications of k-NN employ the standard Euclidean distance metric for the determination of the set of nearest neighbours to a particular test data sample. This paper investigates the use of a data-driven evolutionary approach, named KTree, for the automatic c...

We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has a (essentially, unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.

This paper examines the problem of database organization and retrieval based on computing metric pairwise distances. A low-dimensional Euclidean approximation of a high-dimensional metric space is not efficient, while search in a high-dimensional Euclidean space suffers from the “curse of dimensionality”. Thus, techniques designed for searching metric spaces must be used. We evaluate several su...

This work 1 studies music classification using the 1-Nearest Neighbor rule comparing the Euclidean distance metric to an information distance metric based on Kolmogorov Complexity. The reason for this comparison is two-fold. First, to understand the music classification task and how similarity measures play a role. Secondly, to contribute to the knowledge regarding effective similarity measures...

The planar fixed-radius near neighbor problem can be stated as follows: Preprocess a set P of N points in the plane so that all points of P lying within some fixed radius r of a new point can be listed effectively. This problem has many practical applications in domains as varied as molecular graphics, statistics, air traffic control or data transmission [3]. Although a great deal of work has b...

The Delaunay triangulation characterizes a natural neighbour relation amongst points distributed in a Euclidean space. In this survey we examine extensions of the Delaunay paradigm that have been used to define triangle meshes for representing smooth surfaces embedded in three dimensional Euclidean space. Progress in this area has stemmed primarily from work done in surface reconstruction and s...

We address the problem of computing Riemannian normal coordinates on real, compact Stiefel manifold orthonormal frames. The are based so-called exponential and associated logarithm map enable one to transfer almost any computational procedure realm manifold. To compute is solve (local) geodesic endpoint problem. Instead restricting consideration geodesics with respect a single selected metric, ...