In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization...

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on (R, g), d ≥ 3, when metric g is non-trapping and approaches the Euclidean metric like 〈x〉 with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metric...

We obtain KSS, Strichartz and certain weighted Strichartz estimates for the wave equation on (R, g), d ≥ 3, when metric g is non-trapping and approaches the Euclidean metric like 〈x〉 with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metri...

Metric spaces are among the most important widely studied topics in mathematics. In recent years, Mathematicians began to investigate using other metrics different from Euclidean metric. These metrics also find their place computer age in addition to their importance in geometry. In this paper, we consider the plane geometry with the generalized absolute value metric and define trigonometric fu...

Abstract The space of Euclidean cone metrics on centrically symmetric octahedra with fixed angles ?i < 2 ? , total surface area 1, has a natural hyperbolic metric, and is locally isometric to 3-space. metric completion the ideal tetrahedron whose dihedral are half cone-deficits ? .

Distance-based methods in machine learning and pattern recognition have to rely on a metric distance between points in the input space. Instead of specifying a metric a priori, we seek to learn the metric from data via kernel methods and multidimensional scaling (MDS) techniques. Under the classification setting, we define discriminant kernels on the joint space of input and output spaces and p...

1.1. Differential equations of surfaces Let F be a smooth orientable surface in 3-dimensional Euclidean space. The Euclidean metric induces a metric Ω on this surface, which in turn generates the complex structure of a Riemann surface R. Under such a parametrization, which is called conformal, the surface F is given by an immersion F = (F1, F2, F3) : R → R, and the metric is conformal: Ω = e dz...

This paper presents a novel approach for video-based face recognition. We define a metric based on an average 2 L Euclidean distance between two videos as the classifier. This metric makes use of Earth Mover’s Distance (EMD) as the underlying similarity measurement between videos. Earth Mover’s Distance is a recently proposed metric for geometric pattern matching and it reflects the average gro...

2 Euclidean structures 3 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Weighted triangulations . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Thurston triangulations . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Duality triangulations . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Equivalence of metric triangulations . . . . . . . . . . . . ...