Integral estimates of conformal metrics

In this article we show that there exists a rational number , depending only on the dimension of the manifold such that the th-power of the conformal factor is bounded in norm in terms of volume bound and the square norm bound of the scalar curvature of the conformal metrics. Some applications are also given.

     

Homogenization of two-phase metrics and applications

We consider two-phase metrics of the form ϕ(x, ξ) ≔ , where α,β are fixed positive constants and B _α, B _β are disjoint Borel sets whose union is ℝ^N, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying

Examples of fuzzy metrics and applications

In this paper we present new examples of fuzzy metrics in the sense of George and Veeramani. The examples have been classified attending to their construction and most of the well-known fuzzy metrics are particular cases of those given here. In particular, novel fuzzy metrics, by means of fuzzy and classical metrics and certain special types of functions, are introduced. We also give an extension theorem for two fuzzy metrics that agree in its nonempty intersection. Finally, we give an application of this type of fuzzy metrics to color image processing. We propose a fuzzy metric that simultaneously takes into account two different distance criteria between color image pixels and we use this fuzzy metric to filter noisy images, obtaining promising results. This application is also illustrative of how fuzzy metrics can be used in other engineering problems.     

On some applications of Gauduchon metrics

Geometriae Dedicata - We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with the implications of algebro-geometric nature...     

Integral formulas for compact spacelike hypersurfaces in de sitter space and their applications to Goddard's conjecture

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS ₁ ⁿ⁺¹ (1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result. Li Haizhong is supported by NNSFC No.19701017 and Basic Science Research Foundation of Tsinghua University and Chen Weihua is supported by NNSFC No. 19571005     

Geodesic metrics on fractals and applications to heat kernel estimates

It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of pr...     

Integral closure of a cubic extension and applications

In this paper, we compute the integral closure of a cubic extension over a Noetherian unique factorization domain. We also present some applications to triple coverings and to rank two reflexive sheaves over an algebraic variety.

     

Derivatives with respect to metrics and applications: subgradient marching algorithm

This paper introduces a subgradient descent algorithm to compute a Riemannian metric that minimizes an energy involving geodesic distances. The heart of the method is the Subgradient Marching Algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N ² log(N)) operations on a discrete grid of N points. It performs a front propagation that computes a subgradient of a discrete geodesic distance. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distances.     

Construction of larger Riemannian metrics with bounded sectional curvatures and applications

For any (not necessarily complete) Riemannian manifold, we constructa larger Riemannian metric which is complete and with boundedsectional curvatures. As an application, log-Sobolev inequalitiesare established on arbitrary Riemannian manifolds with referencemeasures having smooth and strictly positive densities. In particular,a conjecture of the second-named author (see [J. Math. Anal.Appl. 300 (2004) 426–435]) is solved.     

Geometry of the cotangent bundle with Sasakian metrics and its applications

The main aim of this paper is to study paraholomorpic Sasakian metric and Killing vector field with respect to the Sasakian metric in the cotangent bundle.     

Sturdy Harmonic Functions and their Integral Representations

Sturdy harmonic functions constitute all but the least tractable of the positive harmonic functions in potential-theoretic settings. They are the uniform limits on compact sets of positive, bounded harmonic functions and are also produced by a simple integral representation on the boundary of a natural compactification of the space on which they are defined. The boundary of that compactification is metrizable, and more regular for the Dirichlet problem, in general, than is the Martin boundary if that boundary is even defined in the setting.     

Means and their applications

General considerations about means and expectations are preceded by a short historical overview. The existing approaches to the definition of means are classified into three groups: approximational, functional, and axiomatic. In some particular cases all of them are equivalent. The problem of meaningfulness of means is discussed for ordinal data and for some important cases of metric data. A survey of the main areas of applications: decision theory, group decision, insurance, economical equity and inequality is also provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.     

Integral representations for polynomial projections and their kernels

In this paper, we discuss the conditions on polynomial projections that allow special integral representations for their kernels. These representations are based on a new biorthonormality criterion. As an application, we obtain new integral representations for the Legendre and Gegenbauer kernels.     

Correlation inequalities and their applications

Correlation inequalities and their applications to the question of the presence or absence of phase transitions is classical lattice systems are considered. A major part of the known inequalities is obtained as a corollary of a single general theorem.Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 16, pp. 5–37, 1979.