Metrics on complex manifolds

In this note we discuss various canonical metrics on complex manifolds. A generalization of the Bergman metric is proposed and the relations of metrics on moduli spaces are commented. At last, we review some existence theorems of solutions to the Strominger system.     

Nonsingularity in second-order cone programming via the smoothing metric projector

Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker p...     

Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L ^p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L ^p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.      

Minty’s lemma and vector variational-like inequalities

In this paper, we consider two vector versions of Minty’s Lemma and obtain existence theorems for three kinds of vector variational-like inequalities. The results presented in this paper are extension and improvement of the corresponding results of other authors. J. Zafarani was partially supported by the Center of Excellence for Mathematics (University of Isfahan).     

Derivation of quantum Chernoff metric with perturbation expansion method

We investigate a measure of distinguishability defined by the quantum Chernoff bound, which naturally induces the quantum Chernoff metric over a manifold of quantum states. Based on a quantum statistical model, we alternatively derive this metric by means of perturbation expansion. Moreover, we show that the quantum Chernoff metric coincides with the infinitesimal form of the quantum Hellinger distance, and reduces to the variant version of the quantum Fisher information for the single-parameter case. We also give the exact form of the quantum Chernoff metric for a qubit system containing a single parameter.     

一种无参考型模糊图像质量评价方法

图像的边缘信息是人眼观察和识别物体的重要特征,根据模糊图像相对于清晰图像其边缘特征发生较大变化的特点,提出了一种基于边缘锐度的无参考模糊图像质量评价方法。首先,通过文中所示方法寻找图像中的所有阶跃边缘;其次,根据一些原则选择合适的部分边缘;最终,计算这些合适边缘的锐利程度作为图像的模糊度评价依据。实验结果表明,该方法相比于全参考模型SSIM能够更好地评价高斯模糊、离焦模糊等模糊类型图像,与主观评价结构相关性强,更符合人眼视觉系统特性,并且易于实现。     

On the classification of projectively flat Finsler metrics with constant flag curvature

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature K. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when K=0$\mathbf{K}=0$, K=−1$\mathbf{K}=-1$ and K=1$\mathbf{K}=1$ are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature K=1$\mathbf{K}=1$ determined by a Minkowski norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetric Finsler metrics much easier than before.     

Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension

Motivated by the definition of combinatorial scalar curvature given by Cooper and Rivin, we introduce a new combinatorial scalar curvature. Then we define the discrete quasi-Einstein metric, which is a combinatorial analogue of the constant scalar curvature metric in smooth case. We find that discrete quasi-Einstein metric is critical point of both the combinatorial Yamabe functional and the quadratic energy functional we defined on triangulated 3-manifolds. We introduce combinatorial curvature flows, including a new type of combinatorial Yamabe flow, to study the discrete quasi-Einstein metrics and prove that the flows produce solutions converging to discrete quasi-Einstein metrics if the initial normalized quadratic energy is small enough. As a corollary, we prove that nonsingular solution of the combinatorial Yamabe flow with nonpositive initial curvatures converges to discrete quasi-Einstein metric. The proof relies on a careful analysis of the discrete dual-Laplacian, which we interpret as the Jacobian matrix of curvature map.     

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.     

Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

For a general subcritical second-order elliptic operator P   in a domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ (or noncompact manifold), we construct Hardy-weight W which is optimal   in the following sense. The operator P−λW$P-\lambda W$ is subcritical in Ω   for all λ<1$\lambda <1$, null-critical in Ω   for λ=1$\lambda =1$, and supercritical near any neighborhood of infinity in Ω   for any λ>1$\lambda >1$. Moreover, if P   is symmetric and W>0$W>0$, then the spectrum and the essential spectrum of W⁻¹P$W^{-1}P$ are equal to [1,∞)$[1,\infty )$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu=0$Pu=0$, the existence of which depends on the subcriticality of P in Ω.