A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

The relative isoperimetric inequality inside an open, convex cone $\mathcal{C}$ states that, at fixed volume, $B_{r} \cap\mathcal{C}$ minimizes the perimeter inside $\mathcal{C}$ . Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov’s proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal{C}$ . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167–211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.     

Veljan-Korchmaros型不等式的稳定性

关于Euclidean空间En(n≥2)中单形的几何不等式,由于支撑函数或径向函数的表达式很难找到,因此一般很难用Hausdorff度量或径向度量来度量两个单形的偏差,使得涉及单形的几何不等式的稳定性的研究比较困难.利用单形棱长在确定单形时起决定性作用这一事实,引进了两个单形偏正度量的概念,从而较好地解决了单形偏正度量的问题,并建立了著名的Veljan-Korchmaros 不等式的稳定性版本.作为推论,还导出了一系列Veljan-Korchmaros型不等式的稳定性版本.     

Stability for Parametric Vector Quasi-Equilibrium Problems With Variable Cones

This article is devoted to the study of stability conditions for a class of quasi-equilibrium problems with variable cones in normed spaces. We introduce concepts of upper and lower semicontinuity of vector-valued mappings involving variable cones and their properties, we also propose a key hypothesis. Employing this hypothesis and such concepts, we investigate sufficient/necessary conditions of the Hausdorff semicontinuity/continuity for solution mappings to such problems. We also discuss characterizations for the hypothesis which do not contain information regarding solution sets. As an application, we consider the special case of traffic network problems. Our results are new or improve the existing ones.     

对偶Aleksandrov-Fenchel不等式的稳定性

多个几何体(主要是凸体(convex boby)和星体(star body))相似“偏差”的一个度量方法被引进,在此度量下,利用R~n中H(?)lder不等式的一个加强获得了另一类对偶Aleksandrov-Fenchel型不等式的稳定性版本(stability version)。     

Veljan-Korchmaros不等式的稳定性

关于Euclidean空间E~n(n■2)中单形的几何不等式,由于支撑函数或径向函数的表达式很难找到,因此一般很难用Hausdorff度量或径向度量来度量两个单形的"偏差",使得涉及单形的几何不等式的稳定性的研究一直为空白.本文利用单形棱长在确定单形时起决定性作用这一事实,引进了两个单形"偏正"度量的概念,从而较好地解决了单形偏正度量的问题,由此建立了著名的Veljan-Korchmaros不等式的稳定性版本,作为推论,导出了单形中著名的Weitzenb■ck不等式和Euler不等式的稳定性版本,最后提出了几个有待解决的问题.     

A Universal Inequality for Stability of Coarse Lipschitz Embeddings

Let X and Y be two pointed metric spaces.In this article,we give a generalization of the Cheng-Dong-Zhang theorem for coarse Lipschitz embeddings as follows:If f:X→Y is a standard coarse Lipschitz embedding,then for each x^*∈Lip₀(X) there exist α,γ> 0 depending only on f and Q_x*G Lip₀(Y) with ‖Q_x*‖_Lip ≤α‖x^*‖_Lip such that ■,for all x∈X.Coarse stability for a pair of metric spaces is studied.This can be consi...     

Global Stability Results for the Weak Vector Variational Inequality

In this paper, we consider the global stability of solutions of a Weak Vector Variational Inequality in a finite-dimensional Euclidean space. Upper semi-continuity of the solution set mapping is established. And by a scalarization method, we derive a sufficient condition that guarantees the lower semi-continuity of the solution set mapping for the Weak Vector Variational Inequality     

Stability of Quasimonotone Variational Inequality Under Sign-Continuity

Whenever the data of a Stampacchia variational inequality, that is, the set-valued operator and/or the constraint map, are subject to perturbations, then the solution set becomes a solution map, and the study of the stability of this solution map concerns its regularity. An important literature exists on this topic, and classical assumptions, for monotone or quasimonotone set-valued operators, are some upper or lower semicontinuity. In this paper, we limit ourselves to perturbations on the constraint map, and it is proved that regularity results for the solution maps can be obtained under some very weak regularity hypothesis on the set-valued operator, namely the lower or upper sign-continuity.     

Quantitative Stability of the Brunn-Minkowski Inequality for Sets of Equal Volume

The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume:If |A| =|B| ＞ 0 and |A + B|1/n =(2 + δ)|A|1/n for some small δ,then,up to a translation,both A and B are close (in terms of δ) to a convex set K.Although this result was already proved by the authors in a previous paper,the present paper provides a more elementary proof that the authors believe has its own interest.Also,the result here provides a stronger estimate for the stability exponent than the previous result of the authors.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

拟平移不变拓扑锥与局部β-凸空间的共轭锥

[1]中提出的局部β-凸分析问题从本质上来说是一种非线性凸分析问题 .为了刻画和研究局部β-凸空间 X的共轭锥 X*β ,本文在抽象凸锥上引进具有拟平移不变性质的拓扑结构 ,第一部分重点研究局部生成拓扑锥与赋范拓扑锥 .第二部分将这两种拓扑锥的一般理论应用于局部 β - 凸空间的共轭锥 X*β 的研究 ,得到 (X*β,U| A)与 (X*β ,‖‖ )的局部生成性与完备性定理等 .     

Wallis不等式的新改进

对Wallis不等式做出了新的改进.与目前掌握的文献相比,所得结果精度更高,且容易估计误差.同时还导出了(2n(2-n)1!)!!!的一个渐近展开式.     

Cones for the Moulton planes

We present a two-parameter family of affinely connected surfaces which admit the cylinder group as a collineation group of their geodesics. The Moulton Planes in the radial model of Betten, the circular cone, as well as the real affine plane, are part of this family. The Moulton Planes occur in this family in the same way as the real affine plane is contained in a family of cones with decreasing steepness.      

混合投影体的极与混合相交体的Aleksandrov-Fenchel 不等式的稳定性

引进了多个几何体(主要是凸体(Convex body)和星体(Star body)) 相似``偏差'的一个度量方法, 从而推广了已有的相似``偏差'度量方法.并在此度量下,利用Rⁿ 中Hölder不等式的一个加强获得了文献[1]建立的混合投影体的极的Aleksandrov-Fenchel不等式和文献[2]建立的混合相交体的Aleksandrov-Fenchel不等式的稳定性版本.     

On the Stability of the p-Affine Isoperimetric Inequality

Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in ?² in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in ?² such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.     

基于积分不等式法的区间变时滞不确定系统鲁棒稳定性研究

研究了一类区间变时滞不确定系统鲁棒稳定性问题.在不确定性为泛数有界的情况下,通过构造包含简单积分项和多重积分项的Lyapunov-Krasovskii泛函,结合新的积分不等式,在未忽略有用项的前提下,利用其更紧的界定条件来处理交叉项,从而建立了基于线性矩阵不等式(LMI)形式的时滞相关稳定性新判据.方法不涉及自由权矩阵技术和任何模型变抱,减少了理论和计算上的复杂性.数值算例表明,所提出的判据是有效的,和一些已有文献相比具有更低的保守性.     

二锥形分离杯之间血液流动稳定性的窄间隙理论

本文在获得血液分离器锥形分离杯内(其中一杯静止,另一杯以ω等角速旋转)血液流动的边界摄动解的基础上[1],采用窄间隙稳定性理论,证明了带轴向流的二锥形分离杯(其中一杯静止,另一杯以ω等角速旋转)之间旋转密度分层血液流动的稳定性.     

Areas and Volumes for Null Cones

Motivated by recent work of Choquet-Bruhat et al. (Class Quantum Gravity 26(135011), 22, 2009), we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop–Gromov relative volume monotonicity theorem and Günther’s volume comparison theorem. We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above.     

Stability of Linear Inequality Systems in a Parametric Setting

In this paper, we propose a parametric approach to the stability theory for the solution set of a semi-infinite linear inequality system in the n-dimensional Euclidean space . The main feature of this approach is that the coefficient perturbations are modeled through the so-called mapping of parametrized systems, which assigns to each parameter, ranging in a metric space, a subset of . Each vector of this image set provides the coefficients of an inequality in and the whole image set defines the inequality system associated with the parameter. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. The paper is focused mainly on the structural stability of the feasible set mapping, providing a characterization of the Berge lower-semicontinuity property. The role played by the strong Slater qualification is analyzed in detail.This research has been partially supported by Grant PB98-0975 from DGES (Spain), Grant BFM2002-04114-C02 (01–02) from MCYT (Spain), FEDER (European Union), and Bancaja-UMH (Spain).