锥体积泛函与仿射不等式(英文)

本文研究了凸多胞形的锥体积泛函.利用投影体以及Lutwak、杨和张最近所建立的仿射等周不等式,得到了刻划平行四边形特征的一个崭新不等式和用锥体积泛函以及投影体的体积所表达的关于配极体体积的严格下界.     

Extending a Partially Ordered Set: Links with its Lattice of Ideals

A well-known result of Bonnet and Pouzet bijectively links the set of linear extensions of a partial order P with the set of maximal chains of its lattice of ideals I(P). We extend this result by showing that there is a one-to-one correspondence between the set of all extensions of P and the set of all sublattices of I(P) which are chain-maximal in the sense that every chain which is maximal (for inclusion) in the sublattice is also maximal in the lattice.We prove that the absence of order S as a convex suborder of P is equivalent to the absence of I(S) as a convex suborder of I(P). Let S be a set of partial orders and let us call S-convex-free any order that does not contain any order of S as a convex suborder. We deduce from the previous results that there is a one-to-one correspondence between the set of S-convex-free extensions of P and the set of I(S)-convex-free chain-maximal sublattices of I(P). This can be applied to some classical classes of orders (total orders and in the finite case, weak orders, interval orders, N-free orders). In the particular case of total orders this gives as a corollary the result of Bonnet and Pouzet.     

New Combinations of Convex Sets

The family of convex sets in a (finite dimensional) real vector space admits several unary and binary operations – dilatation, intersection, convex hull, vector sum – which preserve convexity. These generalize to convex functions, where there are in fact further operations of this kind. Some of the latter may be regarded as combinations of two such operations, acting on complementary subspaces. In this paper, a general theory of such mixed operations is introduced, and some of its consequences developed.     

The Median Stabilization Degree of a Median Algebra

The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. We determine the value of msd of a graphic n-cube Qn and we derive an estimation of msd for the natural median operator of Rn which is sharp up to one or two units. Interestingly, msd of Qn and of Rn grows like log1.5n. Finally, we characterize median algebras and median graphs of msd 1 in terms of forbidden subspaces.     

Equality in Distribution in a Convex Ordering Family

Let F and G be the respective distributions of nonnegative random variables X and Y satisfying the convex ordering. We investigate the class of functions h for which the equality E[h(X)] = E[h(Y)] guarantees F = G. It leads to extensions of some existing results and at the same time offers a somewhat simpler proof.     

Banach空间上凸泛函的某些对偶性质

本语文对Ｂａｎａｃｈ空间上的凸泛函建立了若干对偶性质,通过对偶的手段,建立了非常简洁的凸泛函极小化序列弱收敛的特征定理。     

A note on quadratic forms

We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C ₁, C ₂, C ₃ be three symmetric matrices in ℜ ^n×n , if max{x ^T C ₁ x,x ^T C ₂ x,x ^T C ₃ x}≥0 for all x∈ℜ ⁿ , it is proved that there exist t _i ≥0 (i=1,2,3) such that ∑ _i=1 ³ t _i =1 and ∑ _i=1 ³ t _i C _i has at most one negative eigenvalue. Received February 18, 1997 / Revised version received October 1, 1997? Published online June 11, 1999     

线性约束0-1 二次规划的一个定界技术

本文给出确定线性约束0－1二次规划问题最优值下界的方法,该方法结合McBride和Yormark的思想和总体优化中定下界的方法,证明了所定的界较McBride和Yormark的要好．求解线性约束0－1二次规划问题的分支定界算法可以利用本文的定界技术．     

Smooth representation of a parametric polyhedral convex set with application to sensitivity in optimization

We show in this paper that if a polyhedral convex set is defined by a parametric linear system with smooth entries, then it possesses local smooth representation almost everywhere. This result is then applied to study the differentiability of the solutions and the marginal functions of several classes of parametric optimization problems.