Sufficient Conditions for Error Bounds and Applications

Our aim in this paper is to present sufficient conditions for error bounds in terms of Fréchet and limiting Fréchet subdifferentials in general Banach spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (x, y) C × D, g(x, y, u) = 0, where g takes values in an infinite-dimensional space and u plays the role of a parameter. This symmetric structure offers us the choice of imposing conditions either on C or D. We use these results to prove the nonemptiness and weak-star compactness of Fritz–John and Karush–Kuhn–Tucker multiplier sets, to establish the Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions.     

E—matrices and Several Necessary and Sufficient Conditions

AER-x.,Aiscalledanonnegative(nonPositive)matrixifau>O(a,jO(aiirtO),markedA>O(AO(i#j),auO,wecallp(A)=max{IA,l,lA,l'.-.,lAnl},(whereA,,A2,..',A"areeigenva1ueofA)asspectralradiusofA.LemmalIfAe7'.,thenareanonnegativesquarematrxBandapositivenumberksuchthatA=B-kI(whereIistheidentitymatrixofordern).ProofLetbti=…     

Necessary and Sufficient Conditions for Zero-stiffness Postbuckling

In the course of sensitivity analysis of the initial postbuckling behavior of elastic structures, a special case may occur which is referred to as zero-stiffness postbuckling. The purpose of this paper is to present sufficient and necessary conditions for this special case which is associated with a favorable form of transition from imperfection sensitivity to imperfection insensitivity in the course of sensitivity analysis of the initial postbuckling path. Koiter's initial postbuckling analysis is chosen as the vehicle for quantification of the initial postbuckling behavior. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

幂幺矩阵的充要条件

研究幂幺矩阵的充要条件,利用矩阵的秩和齐次线性方程组解空间的维数.将m=2时幂幺矩阵的充要条件推广到一般幂幺矩阵的充要条件,得出了幂幺矩阵可对角化的结果,并将幂幺矩阵的充要条件平行地推广到幂幺线性变换.     

Two Suficient and Necesary Conditions for Quasidisks

ＴｗｏＳｕｆｉｃｉｅｎｔａｎｄＮｅｃｅｓａｒｙＣｏｎｄｉｔｉｏｎｓｆｏｒＱｕａｓｉｄｉｓｋｓ＊）ＣｈｕＹｕｍｉｎｇ（褚玉明）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕｎａｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｃｈａｎｇｓｈａ，４１００８２）...     

多元二次函数极值的研究

本文用多元函数的二阶泰勒展式给出了多元二次函数极值存在的充分必要条件,同时证明极值也是最值;用线性代数知识刻画了多元二次函数无极值时的特性.将这些方法和结论融入教学,设计了若干思考题.     

Necessary and Sufficient Conditions for Efficiency Via Convexificators

Based on the extended Ljusternik Theorem by Jiménez-Novo, necessary conditions for weak Pareto minimum of multiobjective programming problems involving inequality, equality and set constraints in terms of convexificators are established. Under assumptions on generalized convexity, necessary conditions for weak Pareto minimum become sufficient conditions.     

Necessary and Sufficient Conditions for Minimax Fractional Programming

We establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions. Subsequently, we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality, strong duality, and strict converse duality theorems.     

幻方的几个充要条件

建立等重方阵的概念,并运用它阐述幻方、2次幻方、3次幻方及中心对称互补幻方的充要条件,根据这些结论构作出一些8阶、9阶2次中心对称互补幻方.     

The Minimax Equality; Sufficient and Necessary Conditions

The paper produces new versions of the minimax theorem based on original conditions. Moreover, we investigate not only the sufficiency, but also the necessity of such conditions. The proofs are very simple and preclude any topological technique.     

Competition between Most Rapid Approach Paths: Necessary and Sufficient Conditions

We revisit the optimality of the most rapid approach paths for problems of the calculus of variation in infinite horizon, which are scalar and linear w.r.t. the derivative. Our approach is based on the characterization of the value function in terms of the viscosity solutions of a Hamilton-Jacobi equation. We obtain a new necessary and sufficient condition when there is only one turnpike, but characterize also the optimality of several turnpikes in competition. In this last case, nonsmooth analysis is used. Finally, we illustrate the results on a fishery management problem, for which the growth function has a depensation and the price is variable.This work was done during the visit of the second author at the Laboratoire dAnalyse des Systèmes et de Biométrie, Montpellier, France. He is grateful to the Institut National de la Recherche Agronomique for the support.     

复正定矩阵的充要条件

本文研究Hermite正定矩阵、实正定矩阵的推广概念复正定矩阵,给出了复方阵复正定性的一些充分必要条件。     

Necessary and Sufficient Conditions for Solving Infinite-Dimensional Linear Inequalities

The existence of a feasible solution to a system of infinite-dimensional linear inequalities is characterized by a topological generalization of the Farkas Condition. If this result is specialized to a finite-dimensional vector space with finite positive cone, then a geometric proof of the classic Minkowski-Farkas Lemma is obtained. A dual version leads to an infinite-dimensional extension of the Theorem of the Alternative.     

Necessary and Sufficient Conditions of Doubly Weighted Hardy-Littlewood-Sobolev Inequality

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we consider whole ranges of p and q, i.e., 0 p ≤∞ and 0 q ≤∞.     

Equivalent Conditions for Local Error Bounds

In this paper we present two classes of equivalent conditions for local error bounds in finite dimensional spaces. We formulate conditions of the first class by using subderivatives, subdifferentials and strong slopes for nearby points outside the referenced set, and show that these conditions actually characterize a uniform version of the local error bound property. We demonstrate this uniformity for the max function of a finite collection of smooth functions, and as a consequence we show that quasinormality constraint qualifications guarantee the existence of local error bounds. We further present the second class of equivalent conditions for local error bounds by using the various limits defined on the boundary of the referenced set. In presenting these conditions, we exploit the variational geometry of the referenced set in a systematic way and unify some existing results in the literature.     

行半正定矩阵的充要条件及其性质

文章给出行半正定矩阵的概念,并对它进行讨论,获得行半正定矩阵的一些充要条件及其性质。     

The Nemitskii Operator in Holder Spaces: Some Necessary and Sufficient Conditions

The Nemitskii operator in the Hölder spaces C^0, () with an open bounded subset of Rⁿ, is studied; necessary and sufficientconditions are given for boundedness, uniform continuity, anduniformly continuous differentiability on bounded sets.     

跳过程随机可比的充要条件

本文研究跳过程的随机可比性·基于文献［２］和［３］,证明两个跳过程随机可比的充要条件是其ｑ对可比·同时,改进了［６］中唯一性问题的结果     

Necessary and Sufficient Conditions for Viability for Nonlinear Evolution Inclusions

We consider the viability problem for nonlinear evolutions inclusions of the form u′(t)?∈?Au(t)?+?F(u(t)), where A is an m-dissipative (possible nonlinear and multi-valued) operator acting in a Banach space X, K?X is a nonempty, locally closed set and $F:K{\user1{ \rightsquigarrow }}X$ is with nonempty, convex, closed and bounded values. We define the concept of A-quasi-tangent set to K at a given point ξ?∈?K and we prove a necessary condition for C ⁰-viability expressed in terms of this new tangency concept. We next show that, under various natural extra-assumptions, the necessary condition is also sufficient. We extend the results to the quasi-autonomous case, we deal with the existence of noncontinuable or even global C ⁰-solutions and, as applications, we deduce a comparison result and a sufficient condition for null controllability.