Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual

We fix three natural numbers k, n, N, such that n+k+1 = N, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of N hyperplanes in a k-dimensional affine space, the other is an arrangement of N hyperplanes in an n-dimensional affine space. We assign weights α ₁, . . . , α _N to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of k-dimensional hypergeometric integrals and the second is a matrix of n-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler’s gamma function multiplied by a product of exponentials of the weights. Supported in part by NSF grant DMS-0244579.     

涉及移动目标的亚纯映射唯一性定理

本文首先证明了一个新的从C~n到P~N(C)的亚纯映射第二基本定理,其中涉及到带有不同权重的截断型计算函数;其次利用这个新的第二基本定理,考虑了退化的亚纯映射在分担处于一般位置的移动超平面下的唯一性问题,并在较弱的条件下获得了一个唯一性结果,改进了已有的一些经典结果.     

Detecting Planar Patches in an Unorganised Set of Points in Space

Given n points in 3D, sampled from k original planes (with sampling errors), a new probabilistic method for detecting coplanar subsets of points in O(k ⁶) steps is introduced. The planes are reconstructed with small probability of error. The algorithm reduces the problem of reconstruction to the problem of clustering in R ³ and thereby produces effective results. The algorithm is significantly faster than other known algorithms in most cases.     

Chambers of arrangements of hyperplanes and Arrow's impossibility theorem

Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H⁺ and the other half-space H⁻. Let B={+,−}. For H∈A, define a map by (if C⊆H⁺) and (if C⊆H⁻). Define . Let Chm=Ch×Ch×?×Ch (m times). Then the maps induce the maps . We will study the admissible maps which are compatible with every . Suppose |A|?3 and m?2. Then we will show that A is indecomposable if and only if every admissible map is a projection to a component. When A is a braid arrangement, which is indecomposable, this result is equivalent to Arrow's impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.     

On Meromorphically Normal Families of Meromorphic Mappings of Several Complex Variables into P(C)

We shall prove some meromorphic normality criteria for families of meromorphic mappings of several complex variables into P^N(C), the complex N-dimensional projective space, related to Nochka's Picard type theorems. Some related results (e.g., extension theorems, improved normality criteria, and quasi-normality criteria) will be obtained also. The technique in this paper mainly depends on Stoll's normality criteria for families of non-negative divisors on a domain of Cⁿ.     

Spherical uniformity and some characterizations of the Cauchy distribution

Given a fixed line L (in Rⁿ) and a uniform distribution of points (c) on the unit sphere, L(t_c), the point of intersection of L and the hyperplane P · c = 0, leads to a mapping X_n : Rⁿ → R, which is shown to have a Cauchy distribution.     

Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

For a compact set K\subset R ^d with nonempty interior, the Markov constants M _n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M _n (K) = n ² /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B ^d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] ^d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.     

The diffeomorphic types of the complements of arrangements in ??3 II

For any arrangement of hyperplanes in ℂℙ³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ℂℙ³. This work was partially supported by NSA grant and NSF grant     

Parameter spaces of separating hyperplanes

Every pair of relatively disjoint polytopes is dual to the parameter space of all their separating hyperplanes, which is also a polytope. For a polytope whose interior is disjoint from the relative interior of another polytope, the parameter space of all separating hyperplanes is a polytope of the same dimension. One face of this parameter space parametrizes the separating hyperplanes that also simultaneously support both polytopes. A separating hyperplane corresponds to a vertex of this face if and only if no other hyperplanes support the polytopes at the same intersection points. If all the vertices of the polytopes have all their coordinates in an ordered field, then the same results and their proofs hold with the same ordered field.     

Arrangements and Ranking Patterns

In the unidimensional unfolding model, given m objects in general position on the real line, there arise 1 + m(m − 1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. Change of the position of these m objects results in change of the ranking pattern. In this paper we use arrangement theory to determine the number of ranking patterns theoretically for all m and numerically for m ≤ 8. We also consider the probability of the occurrence of each ranking pattern when the objects are randomly chosen. Received March 5, 2005