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Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.


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Book ChapterDOI
01 Jan 1973
TL;DR: In this paper, a nonlinear problem in differential geometry is discussed: to characterize all smooth functions on a two-dimensional sphere which can be obtained in this manner from some riemannian metric.
Abstract: Publisher Summary This chapter discusses a nonlinear problem in differential geometry. Any metric ds 2 on a two-dimensional sphere S 2 determines a Gauss curvature function K satisfying the Gauss–Bonnet formula. This chapter discusses the converse question: to characterize all smooth functions on a two-dimensional sphere which can be obtained in this manner from some riemannian metric. To characterize all Gauss curvature functions belonging to metrics ds 2 which are conformally related to the standard metric ds 0 2 , so that ds 2 = λ ds 0 2 , where λ is a positive function on the sphere. This requires the determination of the single function λ in terms of the given function K .

190 citations

01 Sep 2012
TL;DR: In this article, the authors generalized the Wasserstein metric to reaction-diffusion systems with reversible mass-action kinetic and showed that this gradient structure can be generalized to systems including electrostatic interactions and correct energy balance via coupling to the heat equation.
Abstract: In recent years the theory of the Wasserstein metric has opened up new treatments of diffusion equations as gradient systems, where the free energy or entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction–diffusion systems with reversible mass-action kinetic. The metric is constructed using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential.The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in Albinus et al (2002 Nonlinearity 15 367–83), which stimulated this work, have a gradient structure.

190 citations

Journal ArticleDOI
TL;DR: This article proposes a new distance measure based on the biharmonic differential operator that has all the desired properties and provides a nice trade-off between nearly geodesic distances for small distances and global shape-awareness for large distances.
Abstract: Measuring distances between pairs of points on a 3D surface is a fundamental problem in computer graphics and geometric processing. For most applications, the important properties of a distance are that it is a metric, smooth, locally isotropic, globally “shape-aware,” isometry-invariant, insensitive to noise and small topology changes, parameter-free, and practical to compute on a discrete mesh. However, the basic methods currently popular in computer graphics (e.g., geodesic and diffusion distances) do not have these basic properties. In this article, we propose a new distance measure based on the biharmonic differential operator that has all the desired properties. This new surface distance is related to the diffusion and commute-time distances, but applies different (inverse squared) weighting to the eigenvalues of the Laplace-Beltrami operator, which provides a nice trade-off between nearly geodesic distances for small distances and global shape-awareness for large distances. The article provides theoretical and empirical analysis for a large number of meshes.

189 citations

Journal ArticleDOI
TL;DR: A new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties, and conditions are derived for these parameters that guarantee monotonic improvement in the single-step convergence rate.
Abstract: This part of the paper introduces some possible implementations of Self-Scaling Variable Metric algorithms based on the theory presented in Part I. These implementations are analyzed theoretically ...

189 citations

Proceedings ArticleDOI
01 Jan 1995
TL;DR: It is proved that the Elmore delay is an absolute upper bound on the 50% delay of an RC tree response and that this bound holds for input signals other than steps, and that the actual delay asymptotically approaches theElmore delay as the input signal rise time increases.
Abstract: The Elmore delay is an extremely popular delay metric, particularly for RC tree analysis. The widespread usage of this metric is mainly attributable to it being the most accurate delay measure that is a simple analytical function of the circuit parameters. The only drawbacks to this delay metric are the uncertainty as to whether it is an optimistic or a pessimistic estimate, and the restriction to step response delay estimation. In this paper, we prove that the Elmore delay is an absolute upper bound on the 50% delay of an RC tree response. Moreover, we prove that this bound holds for input signals other than steps, and that the actual delay asymptotically approaches the Elmore delay as the input signal rise time increases. A lower bound on the delay is also developed using the Elmore delay and the second moment of the impulse response. The utility of this bound is for understanding the accuracy and the limitations of the Elmore delay metric as we use it for design automation.

189 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202253
20213,191
20203,141
20192,843
20182,731
20172,341