Let C be a plane convex body. For arbitrary points , denote by , n a b E  ab the Euclidean length of the line-segment . Let be a longest chord of C parallel to the line-segment . The relative distance between the points and is the ratio of the Euclidean distance between and b to the half of the Euclidean distance between and . In this note we prove the triangle inequality in with the relative ...

Beacon attraction is a movement system whereby a robot (modeled as a point in 2D) moves in a free space so as to always locally minimize its Euclidean distance to an activated beacon (also a point). This results in the robot moving directly towards the beacon when it can, and otherwise sliding along the edge of an obstacle. When a robot can reach the activated beacon by this method, we say that...

For modelling non-stationary spatial random elds Z = fZ(x) : x 2 R n ; n 2g a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x; y) = varZ(x) ? Z(y)], (x; y) 2 R n R n , depends only on the Euclidean distance in the deformed space through a stationary and isotropic variogram. We prove uniqueness of this model in two diierent cases: (i) is s...

This study investigated French /y/ and /u/ produced by one native speaker (NS) and seven Japanesespeaking learners; four of them received lessons on /u/ with ultrasound visual feedback of the tongue shape and position. Acoustic measurements of 10 repetitions of six words (coronal context) are compared to those of isolated /y/ and /u/, and of the non-words /tyty/ and /tutu/, both before and afte...

Let C be a convex body. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. The aim of the paper is to find upper bounds for the minimum of the relative lengths of the sides of convex hexagons and

We study solution concepts for economic games that are resistant to local deviations. Strategy spaces are l subsets of R and local deviations are small in the Euclidean metric. We define local Nash equilibrium and local evolutionarily stable strategy, and present applications to Walrasian outcomes in Cournot games and separating outcomes in screening models.  2001 Elsevier Science B.V. All rig...

This paper derives the distribution of the deviation distance to visit an alternative fuel station. Distance is measured as the Euclidean distance on a continuous plane. The distribution explicitly considers the vehicle range and whether the round trip between origin and destination can be made. Three cases are examined: fuel is available at both origin and destination, fuel is available at eit...

Given a real vector space V equipped with an Euclidean metric, (after rescaling) any p-form φ ∈ V p V ∗ defines a calibration on V . This note identifies an exterior differential system whose integral submanifolds are precisely the critical submanifolds of the calibration. In particular, calibrated submanifolds are necessarily integral submanifolds of the system. The result is extended to calib...

Let (X, d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9 logn n2 . This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound. We discus related questions and consider in part...

An intuitive method for distributing N points on a manifold A ⊂ Rd is to consider minimal s-energy arrangements of points that interact through a power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R 0 . Under what conditions will these points be “uniformly” distributed on A for large N? In this talk I will present recent results characterizing asymptotic properties ...