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 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...

Consider an n-point metric space M = (V, δ), and a transmission range assignment r : V → R that maps each point v ∈ V to the disk of radius r(v) around it. The symmetric disk graph (henceforth, SDG) that corresponds to M and r is the undirected graph over V whose edge set includes an edge (u, v) if both r(u) and r(v) are no smaller than δ(u, v). SDGs are often used to model wireless communicati...

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying th...

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α ≥ 1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings int...

In order to achieve the wide range of the robotic application it is necessary to provide iterative motions among points of the goals. For instance, in the industry mobile robots can replace any components between a storehouse and an assembly department. Ammunition replacement is widely used in military services. Working place is possible in ports, airports, waste site and etc. Mobile agents can...

A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. We characterize metric gauges that are locally branched Euclidean and discuss an obstruction to removing the branching. Our characterization is a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to pro...

The present experiments investigated two characteristics of subjects' multidimensional representations: their dimensional organization and metric structure, for both analyzable and integral stimuli. In Experiment 1, subjects judged the dissimilarity between all pairs of stimuli differing in brightness and size (analyzable stimuli), while in Experiment 2, subjects made dissimilarity judgments fo...