Inequalities for Stationary Poisson Cuboid Processes

A cuboid is a rectangular parallelepipedon. By the notion “stationary Poisson cuboid process” we understand a stationary Poisson hyperplane process which divides the Euclidean space ?^d into cuboids. It is equivalent to speak of a stationary Poisson cuboid tessellation. The distributions of volume and total edge length of the typical cuboid and the origin-cuboid of a stationary Poisson cuboid process are considered. It is shown that these distributions become minimal, in the sense of a specific order relation, in the case of quasi-isotropy. A possible connection to a more general problem, treated in [6], is also discussed.     

关于换位子范数的若干不等式

当f是[0,∞)上的非负算子的单调函数时,得到了对任何非交换Banach函数空间范数都有‖f (A)X - Xf (B)‖ 54‖f ( AX - XB )‖,其中A,B是τ-可测正算子, X是收缩算子.     

Inequalities from Poisson brackets

We introduce the notion of tropicalization for Poisson structures on ${\mathbb{R}}^n$ with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to ${\mathbb{C}}^n$ viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus.As an example, we consider the canonical Poisson bracket on the dual Poisson–Lie group $G^1$ for $G=U\left(n\right)$ in the cluster coordinates of Fomin–Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand–Zeitlin completely integrable system of Guillemin–Sternberg and Flaschka–Ratiu.     

关于Mate-Nevai型非线性积分不等式

主要目的是将Mate-Nevai型积分不等式推广到非线性情形,得到了两个具有广泛意义的Mate-Nevai型非线性积分不等式,利用所得结论讨论某些积分方程解的有界性.     

On Some Noncoercive Variational Inequalities

We study existence and regularity of solutions of noncoercive variational inequalities.     

三阶可微函数的若干不等式

通过建立与Bullen不等式有关的积分恒等式,利用Halder不等式、Grüss不等式、Chebychev不等式,给出三阶可微函数的一些不等式.     

几个算子不等式的最佳常数

该文给出了单位圆上解析函数中φ(z),|φ(z)<1,对于 Hilbert空间 H上任一真压缩算子 A,算子φ(A)的范数估计,得出 sup b~2(φ)=2, sup b~2(φ)=1. b(φ)=sup(||φ(A)||(1-||A||~2) ||φ(A)||~2)~(1/2),b(φ)=sup{||φ(A)||(1-||A||~2) inf ||φ(A)x||~2}~(1/2),这里A取遍H上一切真压缩算子,并且若φ(z)=sum from σ=0 to n(0/σ),|φ(z)|,|φ(z)|<1,则||φ(A)|| ||φ(A)||≤2n,其中φ(z)=z~nφ(z~(-1)),且2是与n,φ(z)及A无关常数中的最小者.     

解线性不等式的神经网络

本文提出两个解线性不等式的Hopfiedl-Tank型的神经网络。第一个网络模拟同时松弛投影方法，第二个网络是二次规划方法。当线性不等式的解集非空时，这两个方法都给出该线性不等式的解。同时我们还给出了这两个网络的数值模拟。     

凸函数的Hadamard不等式的若干推广

本文获得两个定理 ,它们均是不等式f a +b2 1b -a∫baf (x) dx f (a) +f (b)2(其中 f是 [a,b]上的连续凸函数 )的推广 .     

On Some Nonlinear Diophantine Inequalities with Primes

Mathematical Notes -     

On Certain Inequalities for Some Analytic Functions and Differential Subordinations

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关于新几何体Γ_(-p)K的几个不等式

本文研究了新几何体Γ_(-p)K与L_p-质心体和L_p-John椭球有关的几个有趣不等式,包括L_p-Busemann-Petty质心不等式的一种隔离,同时探讨了新几何体Γ_(-p)K的单调性．     

Some Inequalities for Tree Martingales

In this paper we study tree martingales and proved that if 1≤α,β〈∞,1≤p〈∞ then for every predictable tree martingale f=（ft,t∞T）and E[σ^（P）（f）]〈∞,E[S^（P）（f）]〈∞,it holds that ‖（St^（p）（f）,t∈T）‖M^α∞≤Cαβ‖f‖p^αβ,‖（σt^（p）（f）,t∈T）‖M^α,β‖f‖P^αβ,where Cαβ depends only on α and β.     

Some Inequalities for Two Simplices

In this paper, we establish some inequalities for two n-dimensional simplices in the n-dimensional Euclidean space E ⁿ .     

关于对数和指数的几个不等式

研究了关于对数和指数的两个函数:gα(x)=((ln(1+x))/x)~α及h_β(x)=((e~x-1)/x)~β.得到当x0时,g_α(x)+h_β(x)2及g_α(x)h_β(x)1这两个不等式成立的充分必要条件.     

含有n个无关变元的离散不等式

对一些基本的不等式进行了推广,给出了含有n个无关变元的非线性的离散不等式.所得结果推广了已有的一些结论.     

On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.