Geometric Hyperplanes of Lie Incidence Geometries

A geometric hyperplane of a point--line geometry is a proper subspace which meets each line non-trivially. If H is a hyperplane of a projective space P, and the point line geometry has an embedding in P , then the pullback from H is a geometric hyperplane of . We show that all geometric hyperplanes arise in this way for polar spaces of typeD _n , the Grassmann space of lines, and the exceptional geometry E _1,6 . The actual geometric hyperplanes are studied in several cases.     

The Supersolvable Order of Hyperplanes of an Arrangement

This paper mainly gives a sufficient and necessary condition for an order of hyperplanes of a graphic arrangement being supersolvable. In addition, we give the relations between the set of supersolvable orders of hyperplanes and the set of quadratic orders of hyperplanes for a supersolvable arrangement.     

Results on Uniqueness Problem for Meromorphic Mappings Sharing Moving Hyperplanes in General Position Under More General and Weak Conditions

The aim of the paper is to deal with the algebraic dependence and uniqueness problem for meromorphic mappings by using the new second main theorem with different weights involved the truncated counting functions, and some interesting uniqueness results are obtained under more general and weak conditions where the moving hyperplanes in general position are partly shared by mappings from Cⁿ into P^N(C), which can be seen as the improvements of previous well-known results.     

基于Bregman距离函数的可靠性分析

针对概率结构可靠性问题,引入Bregman距离函数,建立了基于同伦算法(HM)的可靠性分析模型.利用极限状态方程,将可靠性指标求解转化为一个非线性约束优化问题.结合同伦思想的基本理论和Bregman距离函数,构造同伦方程组,采用路径跟踪算法对该方程组进行求解.通过相应的数值算例探讨了不同函数形式以及不同程度非线性问题的可靠性计算,并与其他方法计算结果进行了对比,分析结果表明该模型能够有效求解概率结构可靠性问题.     

The Distribution of Robust Distances

Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chi-squared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw's minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large samples, the chi-squared approximation to the distances of the sample data from the MCD center with respect to the MCD shape is poor. We provide an improved F approximation that gives accurate outlier rejection points for various sample sizes.     

On the Minimum Distances of Non-Binary Cyclic Codes

We deal with the minimum distances of q-ary cyclic codes of length q ^m - 1 generated by products of two distinct minimal polynomials, give a necessary and sufficient condition for the case that the minimum distance is two, show that the minimum distance is at most three if q > 3, and consider also the case q = 3.     

链状正则图的平均距离

本文构造了一类链状正则图G_k∶δ,求出了它们的平均距离D(G_k.δ),并得到关系式上式等号成立当且仅当δ=4f且k=0.这个估计式指出了施容华猜想[1]D(G)≤n/(δ 1)不成立. 文中进一步证明了这一类链状正则图有最大的直径,所以可以作出猜想: 若G是n阶连通图,则D(G)<(n 1)/(δ 1),其中δ是图G的最小度。     

Affine-Invariant Distances, Envelopes and Symmetry Sets

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.     

Distances to spaces of measurable and integrable functions

Given a complete probability space and a Banach space X we establish formulas to compute the distance from a function to the spaces of strongly measurable functions and Bochner integrable functions. We study the relationship between these distances and use them to prove some quantitative counterparts of Pettis’ measurability theorem. We also give several examples showing that some of our estimates are sharp.     

Computability of Minimizers and Separating Hyperplanes

We prove in recursive analysis an existence theorem for computable minimizers of convex computable continuous real-valued functions, and a computable separation theorem for convex sets in ?^m. Mathematics Subject Classification: 03F60, 52A40.     

On the Distances between Latin Squares and the Smallest Defining Set Size

In this note, we show that for each Latin square L of order , there exists a Latin square of order n such that L and differ in at most cells. Equivalently, each Latin square of order n contains a Latin trade of size at most . We also show that the size of the smallest defining set in a Latin square is .     

Two Characterizations of Smooth Norms

In an arbitrary Minkowski space M, n2, let there be given an arbitrary finite set P of weighted points whose affine hull is n-dimensional. We show that the unit ball B of M has smooth boundary if and only if each median hyperplane (minimizing the sum of weighted distances with respect to P) is spanned by n affinely independent points from P. Moreover, B has the same property if and only if every center hyperplane (minimizing the maximal weighted distance with respect to P) has the same maximal distance to at least n+1 affinely independent points from P.     

About Distances between Points on the Plane

Mathematical Notes -     

Measurable Sets With Excluded Distances

For a set of distances D = {d ₁,..., d _k } a set A is called D-avoiding if no pair of points of A is at distance d _i for some i. We show that the density of A is exponentially small in k provided the ratios d ₁/d ₂, d ₂/d ₃, …, d _k-1/d _k are all small enough. This resolves a question of Székely, and generalizes a theorem of Furstenberg–Katznelson–Weiss, Falconer–Marstrand, and Bourgain. Several more results on D-avoiding sets are presented. Received: January 2007, Revision: February 2008, Accepted: February 2008     

Unit Distances and Diameters in Euclidean Spaces

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d. As a corollary, we determine the exact maximum number of unit distances for all even d≥6 and the exact maximum number of diameters for all d≥4 and all n sufficiently large depending on d. This material is based upon work supported by the South African National Research Foundation.     

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

Negacyclic codes of length 2s over the Galois ring GR(2a,m) are linearly ordered under set-theoretic inclusion,i.e.,they are the ideals <(x + 1)i>,0 ≤ i ≤ 2sa,of the chain ring GR(2a,m)[x]/.This structure is used to obtain the symbol-pair distances of all such negacyclic codes.Among others,for the special case when the alphabet is the finite field F2m (i.e.,a =1),the symbol-pair distance distribution of constacyclic codes over F2m verifies the Singleton bound for such symbol-pair codes,and provides all maximum distance separable symbol-pair constacyclic codes of length 2s over F2m.     

剔除中位数偏差数据法用于招投评标的探讨

提出剔除与中位数偏超过一定值的数据,然后求平均数的方法,与“去掉一个最高分和去掉一个最低分求平均数”相比,该方法有效去掉极端数据的影响,用于招标评标,更具科学性和合理性,文中给出了具体的实例分析。     

向量到子空间的距离及其应用

给出了向量到有限维子空间距离的定义及求法 ,并推广到向量到可数无限维子空间距离 .采用两种方法求距离并比较了它们的运算量 .揭示了 Cholesky分解法与 Schmidt正交化方法的内在联系 .最后利用向量到子空间距离给出了矛盾方程组最小二乘解的求法     

基于直觉模糊距离的群决策专家意见聚合分析

提出了一种基于直觉模糊距离来聚合专家个人意见为一个优化的群体意见一致度的新方法.首先,根据直觉模糊集的几何意义,定义了两个直觉模糊集之间的距离;然后,利用直觉模糊距离来聚合专家两两之间对备选方案意见的一致度,综合考虑每位专家的相对重要权重,得到专家群体对备选方案意见的综合一致度;最后通过一个具体实例来说明这种方法的具体应用及计算过程.