New Uniform Parametric Error Bounds

A uniform parametric error bound is a uniform error estimate for feasible solutions of a family of parametric mathematical programming problems. It has been proven useful in exact penalty formulation for bilevel programming problems. In this paper, we derive new sufficient conditions for the existence of uniform parametric error bounds.     

On Uniform Global Error Bounds for Convex Inequalities

This paper studies the existence of a uniform global error bound when a convex inequality g 0, where g is a closed proper convex function, is perturbed. The perturbation neighborhoods are defined by small arbitrary perturbations of the epigraph of its conjugate function. Under certain conditions, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if g satisfies the Slater condition and the solution set is bounded or its recession function satisfies the Slater condition. The results are used to derive lower bounds on the distance to ill-posedness.     

Multivariate R-O Varying Measures Part I: Uniform Bounds

A finite Borel measure µ on R^d is called R-O varying withindex F if there exist a GL(R^d)-valued function f varying regularlywith index (–F), an increasing function k: (0, ) (0,) with k(t) and k(t + 1)/k(t) c 1 as t , and a -finitemeasure on R^d\0 such that R-O varying measures generalize regularly varying measures introducedby Meerschaert (see M. M. Meerschaert, ‘Regular variationin R^k’, Proc. Amer. Math. Soc. 102 (1988) 341–348)and have numerous applications in limit theorems for probabilitymeasures. For an R-O varying measure µ and – < let denote the tail- andtruncated moment functions of µ in the direction || =1. The purpose of this paper is to show that R-O variation ofa measure implies sharp bounds on the growth rate of the tail-and truncated moment functions depending on the real parts ofthe eigenvalues of the index F along a compact set of directions.Furthermore, bounds on the ratio of these functions for certainvalues of a and b are obtained. 1991 Mathematics Subject Classification:60B10, 28C15.     

Lie Symmetries and Weak Transversality

We discuss the notion of weak transversality and clarify its role in the context of Lie group theory with several examples. In particular, we present some new results concerning models of fluid dynamics.     

Bounds for the Concentration of Asymptotically Normal Statistics

Let X₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal bound is derived for the concentration function of an arbitrary real-valued statistic T = T (X ₁,...,X _n ) for which ET² < . Applications are given for Wilcoxon"s rank-sum statistic, U-statistics, Student"s statistic, the two-sample Student statistic and linear regression.     

Transversality in Variational Analysis

We discuss various aspects of newly developed extension of the classical transversality theory to variational analysis and optimization theory. In particular, we give interpretations in transversality terms of some key results (relating to subdifferential calculus, necessary optimality conditions and linear convergence of alternating projections) and prove a set-valued version of the Thom transversality theorem for semi-algebraic objects.     

一致正规结构与Reich的公开问题的解答

在具有一致正规结构且其范数是一致Gateaux可微的Banach空间中,研究了Reich提出的公开问题.在给渐近非扩张映象作更适当的假设下,对Reich的公开问题给出了一个肯定的答复.所得结果在下列方面推广与改进了张石生教授的最新结果:(ⅰ)去掉了张教授的较强条件“迭代参数列收敛到零”;(ⅱ)去掉了张教授的较强假设“渐近非扩张映象有不动点”;(ⅲ)也去掉了张教授的较强条件“Banach压缩映象原理生成的序列强收敛”.而且,这些结果也推广与改进了先前由Reich,Shioji,Takahashi,Ueda及Wittmann等多位作者得到的相应结果.     

Some New Transversality Properties

The theorems of Ceva and Menelaus are concerned with cyclic products of ratios of lengths of collinear segments of triangles or more general polygons. These segments have one endpoint at a vertex of the polygon and one at the intersection point of a side with a suitable line. To these classical results we have recently added a selftransversality theorem in which the suitable line is determined by two other vertices. Here we present additional transversality properties in which the suitable line is determined either by a vertex and the intersection point of two diagonals, or by the intersection points of two pairs of such diagonals. Unexpectedly it turns out that besides several infinite families of systematic cases there are also a few sporadic cases.     

均匀分布与正态分布的教学设计

通过针对一元和二元均匀分布、正态分布实施重点教学,就概率统计的概念和知识点,讨论它们之间的联系及对比,再把分散的内容进行归纳。教学过程注重讲解均匀、正态反映的随机性的区别及各自的应用背景,使学生对这些抽象的概念有一个较生动化的认识．     

Transversality of the Shapley value

A few applications of the Shapley value are described. The main choice criterion is to look at quite diversified fields, to appreciate how wide is the terrain that has been explored and colonized using this and related tools. The title is inspired by a tutorial that one of the authors planned to deliver at the 7th meeting on Game Theory and Practice (Montreal, 2007), but was unable to do it for personal reasons. Thanks to Georges Zaccour whose invitation sparked the present survey.     

随机算子的拓扑横截性

本文给出了横截的随机算子的定义并介绍了它的性质及其应用.     

Transversality condition for infinite-horizon problems

We present necessary conditions of optimality for an infinitehorizon optimal control problem. The transversality condition is derived with the help of stability theory and is formulated in terms of the Lyapunov exponents of solutions to the adjoint equation. A problem without an exponential factor in the integral functional is considered. Necessary and sufficient conditions of optimality are proved for linear quadratic problems with conelike control constraints.     

Computing Bounds on the Expected Maximum of Correlated Normal Variables

We compute upper and lower bounds on the expected maximum of correlated normal variables (up to a few hundred in number) with arbitrary means, variances, and correlations. Two types of bounding processes are used: perfectly dependent normal variables, and independent normal variables, both with arbitrary mean values. The expected maximum for the perfectly dependent variables can be evaluated in closed form; for the independent variables, a single numerical integration is required. Higher moments are also available. We use mathematical programming to find parameters for the processes, so they will give bounds on the expected maximum, rather than approximations of unknown accuracy. Our original application is to the maximum number of people on-line simultaneously during the day in an infinite-server queue with a time-varying arrival rate. The upper and lower bounds are tighter than previous bounds, and in many of our examples are within 5% or 10% of each other. We also demonstrate the bounds’ performance on some PERT models, AR/MA time series, Brownian motion, and product-form correlation matrices.     

Explicit Error Bounds for Periodic Splines of Odd Order on a Uniform Mesh

The dependence relationships connecting equal interval splinesand their derivatives are analysed to obtain the form of theerror term when the spline is replaced by a general function.The defining equations for periodic splines of odd order ona uniform mesh are then expressed in terms of a positive definitecirculant matrix A and attainable bounds determined for thecondition number of A and for the norm of A^-1. In conjunctionwith the error term associated with the dependence relationships,this enables explicit error bounds to be established for thederivatives at the knots of the spline function. Some subsidiary results in the paper also relate to B-splineson a uniform mesh.     

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros.