Piercing Families of Convex Sets with the <Emphasis Type="Italic">d</Emphasis>-Intersection Property in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

In this paper, we consider finite families of convex sets in ℝ ^d such that every d or fewer sets of the family have a common point. For some families of this type, we give upper bounds on the size of a finite set intersecting all sets of the family. This research was supported by the Russian Foundation for the Basic Research Grants No. 03-01-00801 and 06-01-00648, and by the President of the Russian Federation Grant No. MK-5724.2006.1.     

Generating Random Vectors in (ℤ/<Emphasis Type="Italic">p</Emphasis>ℤ)<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> via an Affine Random Process

This paper considers some random processes of the form X _n+1=T X _n +B _n (mod p) where B _n and X _n are random variables over (ℤ/pℤ) ^d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B _n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)² steps suffice to make X _n close to uniformly distributed where X ₀ is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p ^b steps are needed for X _n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X ₀ is the zero vector.     

Distributions of Points in <Emphasis Type="Italic">d</Emphasis> Dimensions and Large <Emphasis Type="Italic">k</Emphasis>-Point Simplices

We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1] ^d , such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ _k,d (n), the supremum of this minimum volume over all distributions of n points in [0,1] ^d , we show that c _k,d ⋅(log n)^1/(d−k+1)/n ^k/(d−k+1)≤Δ _k,d (n)≤c _k,d ′/n ^k/d for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ _k,d (n)≤c _k,d ″/n ^{k/d+(k−1)/(2d(d−1))}, where c _k,d ,c _k,d ′,c _k,d ″>0 are constants. A preliminary version of this paper appeared in COCOON ’05.     

The Characteristic Intersection Property of Line-Free Choquet Simplices in <Emphasis Type="Italic">E</Emphasis><Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

   Abstract. Generalizing the characteristic intersection property of Choquet simplices, it is proved that for line-free convex bodies B ₁ and B ₂ in E ^d , the following conditions are equivalent: (i) there is a line-free convex body B ⊂ E ^d such that every nonempty intersection B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , is a homothetic copy of B , (ii) both B ₁ and B ₂ are Choquet simplices and the nonempty intersections B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , are homothetic copies of a Choquet simplex B . All such triplets B ₁ ,B ₂ ,B are described.     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

Path properties of <Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-valued Gaussian random fields

General limit theorems are established for l ^p -valued Gaussian random fields indexed by a multidimensional parameter, which contain both almost sure moduli of continuity and limits of large increments for the l ^p -valued Gaussian random fields under Ṅ explicit conditions. This work was supported by NSERC Canada grants at Carleton University and by KOSEF-R01-2005-000-10696-0     

A generalization of Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> by using atoms

Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have the same property as H^P（X）. Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let （X, δ, μ） be the normalized space of （X, d, μ） in the sense of Macias and Segovia. We also study the relations of our function spaces for （X, d, μ） and （X, δ,μ）.     

A randomized Delaunay triangulation heuristic for the Euclidean Steiner tree problem in <Emphasis Type="Italic">ℜ</Emphasis><Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

We present a heuristic for the Euclidean Steiner tree problem in ℜ ^d for d≥2. The algorithm utilizes the Delaunay triangulation to generate candidate Steiner points for insertion, the minimum spanning tree to identify the Steiner points to remove, and second-order cone programming to optimize the location of the remaining Steiner points. Unlike other ESTP heuristics relying upon Delaunay triangulation, we insert Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points. We govern this neighbor generation procedure with a local search framework that extends effectively into higher dimensions. We present computational results on benchmark test problems in ℜ ^d for 2≤d≤5.     

Some results on 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2 ^n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 ^n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2 ^n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4 ^m 2 ⁿ designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4 ^m 2 ⁿ designs under the clear effects criterion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10571093, 10671099 and 10771123), the Research Foundation for Doctor Programme (Grant No. 20050055038) and the Natural Science Foundation of Shandong Province of China (Grant No. Q2007A05). Zhang’s research was also supported by the Visiting Scholar Program at Chern Institute of Mathematics.     

Relative mean curvature configurations for surfaces in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>, <Emphasis Type="Italic">n</Emphasis> ≥ 5

We define the relative mean curvature directions on surfaces immersed in ℝⁿ, n ≥ 4, generalizing the concept of mean curvature directions for surfaces in 4-space studied by Mello. We obtain their differential equations and study their corresponding generic configurations. *Work partially supported by DGCYT grant no. MTM2004-03244 and Unimontes-BR. †Work partially supported by DGCYT grant no. MTM2004-03244. ‡Work partially supported by DGCYT grant no. BFM2003-0203.