| In this paper we demonstrate that two common metrics, symmetric set diierence, and Eu-clidian distance, have normalized forms which are nevertheless metrics. The rst of these jA4Bj=jABj is easily established and generalizes to measure spaces. The second applies to vectors in R n and is given by kX?Y k=(kXk+kY k). That this is a metric is more dii-cult to demonstrate and is true for Euclidian ...

The main results are summarized by Figure 1. They demonstrate the resiliency of the isomorphism constructed in [Nai99] between weighted cohomology and a variant of weighted L2 cohomology. Our attention is restricted from generic locally symmetric spaces to spaces whose ends are hyperbolic, diffeomorphic to (0,∞) × (S1)n−1, and carry exponentially warped product metrics. For weighting functions ...

in this paper, we propose a new definition of intuitionistic fuzzyquasi-metric and pseudo-metric spaces based on intuitionistic fuzzy points. weprove some properties of intuitionistic fuzzy quasi- metric and pseudo-metricspaces, and show that every intuitionistic fuzzy pseudo-metric space is intuitionisticfuzzy regular and intuitionistic fuzzy completely normal and henceintuitionistic fuzzy nor...

We show that any rectilinear polygonal subdivision in the plane can be converted into a “guillotine” subdivision whose length is at most twice that of the original subdivision. “Guillotine” subdivisions have a simple recursive structure that allows one to search for “optimal” such subdivisions in polynomial time, using dynamic programming. In particular, a consequence of our main theorem is a v...

is precompact in the C1,α topology. Here R denotes the Riemann curvature tensor, vol the volume and diam the diameter of (M,g). Thus, for any sequence of metrics gi on n-manifolds Mi satisfying (1.1), there is a subsequence, also called gi, and diffeomorphisms φi : M∞ → Mi such that the metrics φi gi converge in the C 1,α topology to a limit metric g∞ onM∞, for any α < 1. In particular, there a...

Many algorithms for reducing the number of triangles in a surface model have been proposed, but to date there has been little theoretical analysis of the approximations they produce. Previously we described an algorithm that simplifies polygonal models using a quadric error metric. This method is fast and produces high quality approximations in practice. Here we provide some theory to explain w...

In a recent paper, Lee and Moharrami [13] construct a family of metrics (Xn, dn) with |Xn| = N , such that (Xn, √ dn) embeds into L2 with constant distortion, but embedding (Xn, dn) into a metric of negative type requires distortion Ω̃(log 1N). In this paper, we build on their analysis, and improve their result by showing an Ω̃(log 1N) lower bound for embedding (Xn, dn) into a metric of negative ...

We investigate the Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest Σ3 = W (SU(3)) (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. Complex projective space CP(2) that is a supers...

In this work we present a CGAL (Computational Geometry Algorithm Library) implementation of the line segment Voronoi diagram under the L∞ metric, building on top of the existing line segment Voronoi diagram under the Euclidean (L2) metric. CGAL is an open-source collection of geometric algorithms implemented in C++, used in both academia and industry. We also discuss a possible application of t...

We consider gradient descent equations for energy functionals of the type S(u) = 1 2 〈u(x), A(x)u(x)〉L2 + ∫ Ω V (x, u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for S where the gradient is an element of the Sobolev spac...