Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant ∆ = 7, 9, 13, 19, 31, 37, 43, 61, 67, 103, 109, 127, 157 . A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree ` and conductor f . Assume the GRH for ζK(s). If 38(`− 1)(log f) log...

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...

Statistical learning algorithms often rely on the Euclidean distance. In practice, non-Euclidean or non-metric dissimilarity measures may arise when contours, spectra or shapes are compared by edit distances or as a consequence of robust object matching [1,2]. It is an open issue whether such measures are advantageous for statistical learning or whether they should be constrained to obey the me...

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a \graph space". The robot can locate itself by the presence of distinctively labeled \landmark" nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about th...

We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve th...