We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2...

In the k-supplier problem, we are given a set of clients C and set of facilities F located in a metric (C∪F, d), along with a bound k. The goal is to open a subset of k facilities so as to minimize the maximum distance of a client to an open facility, i.e., minS⊆F :|S|=k maxv∈C d(v, S), where d(v, S) = minu∈S d(v, u) is the minimum distance of client v to any facility in S. We present a 1 + √ 3...

Introduction. This paper is concerned with several disconnected developments in distance geometry. §1 deals with the congruent imbedding of metric spaces in euclidean or Hubert spaces. By showing that the validity of the Pythagorean theorem insures the essentially euclidean character of the metric, the basic role this theorem plays in euclidean geometry is seen to be fully justified. In §2 the ...

Quantifying the dissimilarity (or distance) between two sequences is essential to the study of action potential (spike) trains in neuroscience and genetic sequences in molecular biology. In neuroscience, traditional methods for sequence comparisons rely on techniques appropriate for multivariate data, which typically assume that the space of sequences is intrinsically Euclidean. More recently, ...

We consider the Schr\"odinger operator $$ P=h^2\Delta_g +V on $\Bbb{R}^n$ equipped with a metric $g$ that is Euclidean outside compact set. The real-valued potential $V$ assumed to be compactly supported and smooth except at {\it conormal singularities} of order $-1-\alpha$ along hypersurface $Y$. For $\alpha>2$ (or even $\alpha>1$ if classical flow unique), we show $E_0$ non-trapping energy fo...

In this work, we define a new metric of the distance and depth of penetration between two convex polyhedral bodies. The metric is computed by means of a linear program with three variables and m+n constraints, where m and n are the number of facets of the two polyhedral bodies. As a result, this metric can be computed with O(n+m) algorithmic complexity, superior to the best algorithms known for...

Existing deep embedding methods in vision tasks are capable of learning a compact Euclidean space from images, where Euclidean distances correspond to a similarity metric. To make learning more effective and efficient, hard sample mining is usually employed, with samples identified through computing the Euclidean feature distance. However, the global Euclidean distance cannot faithfully charact...

Teramoto et al. [TAKD06] defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from S, a bounded subset of R. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion o...

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structur...