关于无穷级半纯函数的填充圆

本文应用Ａｈｌｆｏｒｓ覆盖曲面理论，在一定条件下证明了无穷级半纯函数强性填充圆的存在性，从而部分地解决了李国平提出的一个猜想。     

具有无界度的无限圆填充的刚性(英文)

刚性是圆填充理论的一个重要的性质.已经知道,平面上无限的有界度的圆填充的刚性可以用环绕数的方法来证明.本文应用环绕数和指标的技术,结合有限覆盖定理证明了几乎填满整个黎曼球面具有相同复形的无限无界度圆填充对M(o|¨)bius变换来说是等价的,也就是,一个圆填充是另一个圆填充在M(o|¨)bius变换下的像.这给出了无限无界度圆填充的刚性的一种新的证明.     

John 圆,拟圆及反射

利用内距离λD及构造的关于区域D的边界(a)D的反射RD得到了John 圆与拟圆的必要条件.     

拟圆的一个充要条件

本文把 Ｇｅｈｒｉｎｇ, Ｆ Ｗ．和 Ｍａｒｔｉｏ, Ｏ．~［１］给出的拟圆的一个必要条件精确化为充要条件．     

拟圆的一个充要条件

本文把Gehring,F.W.和Martio,O.［1］给出的拟圆的一个必要条件精确化为充要条件.     

拟对称极小的齐次完全集

本文用质量分布原理,证明了由有界正整数序列定义的Hausdorff维数为1的齐次完全集是一维拟对称极小的.     

拟圆的一个充分必要条件

利用拟双曲度量研究了平面拟共形映照中的拟圆，得到了拟圆的一个充分必要条件。     

拟共不变集和拟圆的特征

证明了ＮＥＤ集在拟共形映照下的像仍是ＮＥＤ集，建立了关于拟圆的一个充分必要条件。     

齐次完全集的拟对称极小性

作为Cantor型集的推广,文志英和吴军引入了齐次完全集的概念,并基于齐次完全集的基本区间的长度以及基本区间之间的间隔的长度,得到了齐次完全集的Hausdorff维数.本文研究齐次完全集的拟对称极小性,证明在某些条件下Hausdorff维数为1的齐次完全集是1维拟对称极小的.     

Approximation of Quasisymmetries Using Circle Packings

The Universal Teichmüller Space has recently gained the attention of physicists as a setting for developing string theory. Two common models for this space are the set of quasicircles and the set of quasisymmetries. The link between these models lies in the fact that the Riemann maps from D and D ^* to the complementary domains of quasicircles induce quasisymmetric automorphisms of . We develop a means of approximating these quasisymmetries given their associated quasicircles. This provides a concrete method for switching from the quasicircle to the quasisymmetry model of the Universal Teichmüller Space. Our approach uses discrete analytic functions induced by circle packings to approximate the Riemann maps. Received May 19, 1999, and in revised form April 6, 2000. Online publication September 22, 2000.     

关于圆和椭圆逼近显示的最优算法(英文)

本文给出了一个逼近显示圆的新算法。该算法是通过相交多边形而不是内接多边形逼近圆。由于构造相交多边形时其面积等于圆面积 ,因此新算法是最优逼近。同时还推广到椭圆     

The Densest Packing of 19 Congruent Circles in a Circle

Dense packings of n congruent circles in a circle were given by Kravitz in 1967 for n = 2,..., 16. In 1969 Pirl found the optimal packings for n 10, he also conjectured the dense configurations for 11 n 19. In 1994, Melissen provided a proof for n = 11. In this paper we exhibit the densest packing of 19 congruent circles in a circle with the help of a technique developed by Bateman and Erdös.     

On Large Lattice Packings of Spheres

The packing density of large lattice packings of spheres in Euclidean E ^d measured by the parametric density depends on the parameter and on the shape of the convex hull P of the sphere centers; in particular on the isoperimetric coefficient of P and on the second term in the Ehrhart polynomial of the lattice polytope P. We show in E ^d , d 2, that flat or spherelike polytopes generate less dense packings, whereas polytopes with suitably chosen large facets generate dense packings. This indicates that large lattice packings in E ³ of high parametric density may be good models for real crystals.     

Non-rigidity of Spherical Inversive Distance Circle Packings

We give a counterexample of Bowers–Stephenson’s conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.     

Packings of the complete directed graph with m-circuits

A packing of the complete directed symmetric graph DKv with m-circuits, denoted by(v,m)-DCP, is defined to he a family of are-disjoint m-circuits of DK, such that any one arc of DKv occurs in at most one m circuit. The packing number P(v,m) is the maximum number of m-circults in such a packing. The packing problem is to determine the value P(v,m) for everyinteger v ≥ m. In this paper, the problem is reduced to the case m 6 ≤v≤2m-[(4m-3的平方极) 1/2],for any fixed even integer m≥4,In particular,the values of P（v,m） are completely determined for m=12,14 and 16.     

Periodicity and Circle Packings of the Hyperbolic Plane

We prove that given a fixed radius r, the set of isometry-invariant probability measures supported on 'periodic' radius r-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius r-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius r-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius.     

Curved Hexagonal Packings of Equal Disks in a Circle

For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and one of the 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called ``billiards' simulation algorithm. A curved hexagonal packing pattern is invariant under a rotation. For , the density (covering fraction) of curved hexagonal packings tends to . The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present the new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness' of the curved hexagonal pattern for k ≤ 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality. Received May 15, 1995, and in revised form March 5, 1996.     

Approximation of Distributions by Bounded Sets

Let P be a probability distribution on a locally compact separable metric space (S,d). We study the following problem of approximation of a distribution P by a set A from a given class $\mathcal{A}\subset2^{S}$ : $$W(A,P)\equiv\int_{S}\varphi(d(x,A))P(dx)\to\min_{A\in\mathcal{A}},$$ where φ is a nondecreasing function. A special case where $\mathcal{A}$ consists of unions of bounded sets, $\mathcal{A}=\{\bigcup_{i=1}^{k}A_{i}:\Delta(A_{i})\leq K,\ i=1,\ldots,k\}$ , is considered in detail. We give sufficient conditions for the existence of an optimal approximative set and for the convergence of the sequence of optimal sets A _n found for measures P _n which satisfy P _n ? P. Current article is a follow-up to Käärik and Pärna (Acta Appl. Math. 78, 175–183, 2003; Acta Comment. Univ. Tartu. 8, 101–112, 2004) where the case of parametric sets was studied.