The Huber theorem for non-compact conformally flat manifolds

It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Received: April 14, 2000; revised version: March 20, 2001     

约束装箱问题的混合遗传算法求解

本将最佳适应法和遗传算法相结合，提出了一种新的启发式混合遗传算法对具有时间约束的装箱问题进行求解，给出了具体的算法步骤，试算结果表明基于启发式算法的混合遗传算法适合于求解各种约束条件下的大规模装箱问题。     

Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with specified asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.     

Recent developments in polymeric charge transfer complexes

The recent literature on polymeric charge transfer complexes is reviewed with emphasis on the author's own work. After a definition of the area and a survey of investigations on the spatial arrangement of donor and acceptor sites in the solid state, a variety of applications of these complexes is presented. Electrically conductive polymers are excluded. These applications are: compatibilization of polymer blends, liquid crystalline supramolecular organization, new developments in photo-conductivity, electroluminescence, nonlinear optical properties, photorefractivity and reversible optical storage.     

On the performance of peeling algorithms

The peeling of a d-dimensional set of points is usually performed with successive calls to a convex hull algorithm; the optimal worst-case convex hull algorithm, known to have an O(n^˙ Log (n)) execution time, may give an O(n^˙n^˙ Log (n)) to peel all the set; an O(n^˙n) convex hull algorithm, m being the number of extremal points, is shown to peel every set with an O(n-n) time, and proved to be optimal; an implementation of this algorithm is given for planar sets and spatial sets, but the latter give only an approximate O(n^˙n) performance.     

TERNARY OPERATIONS AS PRIMITIVE NOTIONS FOR PLANE GEOMETRY II

We proved in the first part [1] that plane geometry over Pythagorean fields is axiomatizable by quantifier-free axioms in a language with three individual constants, one binary and three ternary operation symbols. In this paper we prove that two of these operation symbols are superfluous.     

Polynomial-time approximation schemes for packing and piercing fat objects

We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.     

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with discontinuous boundary conditions, which include Dirichlet boundary conditions on one part of the boundary and Neumann boundary conditions on another part of the boundary. We study the heat kernel asymptotics of Zaremba boundary value problem. The construction of the asymptotic solution of the heat equation is described in detail and the heat kernel is computed explicitly in the leading approximation. Some of the first nontrivial coefficients of the heat kernel asymptotic expansion are computed explicitly. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tail probabilities of the limiting null distributions of the Anderson–Stephens statistics

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) z_i, i=1,…,n, Anderson and Stephens (Biometrika 59 (1972) 613–621) proposed testing procedures based on the statistics S_max=max_u S(u) and S_min=min_u S(u), where u is a unit vector and nS(u) is the sum of squares of u′z_i's. In this paper, we also consider another test statistic S_range=S_max−S_min. We provide formulas for the P-values of S_max, S_min, S_range by approximating tail probabilities of the limiting null distributions by means of the tube method, an integral-geometric approach for evaluating tail probability of the maximum of a Gaussian random field. Monte Carlo simulations for examining the accuracy of the approximation and for the power comparison of the statistics are given.     

A Representation of a Family of Secret Sharing Matroids

Deciding whether a matroid is secret sharing or not is a well-known open problem. In Ng and Walker [6] it was shown that a matroid decomposes into uniform matroids under strong connectivity. The question then becomes as follows: when is a matroid m with N uniform components secret sharing? When N = 1, m corresponds to a uniform matroid and hence is secret sharing. In this paper we show, by constructing a representation using projective geometry, that all connected matroids with two uniform components are secret sharing