有限环上的齐次重量与Mbius函数

讨论了有限环上齐次重量、M(o|¨)bius函数和欧拉phi-函数等函数之间的关系.在有限主理想环上给出了这些函数的易于计算的刻画,对于整数剩余类环把它们还原成了经典的数论M(o|¨)bius函数和数论欧拉phi-函数.     

S~4中具有调和共形高斯映照的超曲面

设x:M~n→S~(n+1)是球面S~(n+1)中的一个定向超曲面,其共形高斯映照G=(H,Hx+en+.1):M~n→R_1S~(n+3)是M(o|¨)bius变换群下的一个不变量,其中H,e(n+1)+1分别是超曲面x的平均曲率和单位法向量场.本文研究了S~4中具有调和共形高斯映照的超曲面,分类了具有调和共形高斯映照和常M(o|¨)bius数量曲率的超曲面,给出了具有调和共形高斯映照但不是Willmore超曲面的例子.     

有限偏序集上格的M?bius函数和特征多项式

在一些有限格上研究了格的特征多项式,确定了它们的表示式,并且得到这些格的M(o|¨)bius函数.     

Sn+1中具有调和M(o)bius曲率张量的超曲面

设x:Mn→Sn 1是(n 1)维单位球面Sn 1中的无脐点的超曲面.Sn 1中超曲面x有两个基本的共形不变量:M(o)bius度量g和M(o)bius第二基本形式B.当超曲面维数大于3时,在相差一个M(o)bius变换下这两个不变量完全决定了超曲面.另外M(o)bius形式Ф也是一个重要的不变量,在一些分类定理中Ф=0条件的假定是必要的.本文考虑了Sn 1(n≥3)中具有消失M(o)bius形式Ф的超曲面:对具有调和曲率张量的超曲面进行分类,进而,在M(o)bius度量的意义下,对Einstein超曲面和具有常截面曲率的超曲面也进行了分类.     

M(o)bius立方体的拓扑结构性质

超立方体网络是目前在超级计算机处理器结构中应用得最广泛的拓扑结构,M(o)bius立方体是超立方体的一种变形,已经被证明它在某些方面具有优于超立方体的拓扑性质.本文指出了n维M(o)bius立方体递归结构的一些重要拓扑性质.     

Möbius立方体的拓扑结构性质

超立方体网络是目前在超级计算机处理器结构中应用得最广泛的拓扑结构,M(o)bius立方体是超立方体的一种变形,已经被证明它在某些方面具有优于超立方体的拓扑性质.本文指出了n维M(o)bius立方体递归结构的一些重要拓扑性质.     

关于具有常平均曲率和数量曲率超曲面的M(o)bius几何的一个注记

设x:M→Sn+1(n≥3)是n+1-维单位球中的无脐点超曲面,M(o)bius不变量(g),φ,A和B分别表示x的M(o)bius度量,M(o)bius形式,Blaschke形式和M(o)bius第二基本形式.本文证明了如果x的M(o)bius形式φ平行,并且A+λ(g)+μB=0,其中λ,μ分别是定义在M上的光滑函数,那么φ=0,由此及李海中、王长平(2003年)文献中的分类定理给出了Sn+1中具有平行的M(o)bius形式及满足A+λ(g)+μB=0的超曲面的分类.此结果推广了他们及张廷枋(2003年)文献中的结果.     

一类0<α<1带有无限时滞的中立型脉冲微分方程mild解的存在性（英文）

本文研究了一类0α1带有无限时滞的中立型脉冲微分方程mild解的存在性的问题.利用解算子的相关性质及M(o|¨)nch不动点理论的方法,获得了这类方程的mild解并予以证明,且得到了解的存在性的结果.     

单位球面上具有非负M(o)bius截面曲率的超曲面

Let x : M → Sn 1 be a hypersurface in the (n 1)-dimensional unit sphere Sn 1 without umbilic point. The M(o)bius invariants of x under the M(o)bius transformation group of Sn 1 are M(o)bius metric, M(o)bius form, M(o)bius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem: Let x: M → Sn 1 (n ≥ 2) be an umbilic free hypersurface in Sn 1with nonnegative M(o)bius sectional curvature and with vanishing M(o)bius form. Then x is locally M(o)bius equivalent to one of the following hypersurfaces: (i) the torus Sk(a) × Sn-k(√1-a2) with 1 ≤ k ≤ n - 1; (ii) the pre-image of the stereographic projection of the standard cylinder Sk × Rn-k (∩) Rn 1 with 1 ≤ k ≤ n - 1; (iii) the pre-image of the stereographic projection of the cone in Rn 1: ～x(u, v, t) = (tu, tv),where (u,v,t) ∈ Sk(a) × Sn-k-1( √1- a2) × R .     

划分格中的Mbius函数和秩生成函数

设S={1,2,…,n},P(n)是由S的所有划分组成的集合.对于π,σ∈P(n),如果π中的每个块包含在σ的一个块里,就定义π≤σ,那么P(n)作成一个格.如果M(n,k)是由S的所有k部划分组成的集合,而L(n,k)是由M(n,k)生成的格.在P(n)和L(n,k)中,给出M(o|¨)bius函数,并且确定了特征多项式和秩生成函数的表示式.     

The Möbius invariants for circle packings

Möbius invariants of general circle packings are defined in terms of cross ratios. The necessary and sufficient conditions of existence of circle packings are established by the techniques of Möbius invariants. It is shown that circle packings are uniquely determined, up to Möbius transformations, by their Möbius invariants. The rigidity of infinite circle packings with bounded degree is proved using the approach of Möbius invariants.     

A probabilistic proof of Thurston's conjecture on circle packings

In 1985 William Thurston conjectured that one could use circle packings to approximate conformal mappings. This was confirmed by Burt Rodin and Dennis Sullivan with a proof which relied on the hexagonal nature of the packings involved. This paper provides a probabilistic proof which accomodates more general combinatorics by analysing the dynamics of invididual circle packings. One can use reversible Markov processes to model the movement of curvature and hyperbolic area among the circles of a packing as it undergoes adjustement, much as one can use them to model the movement of current in an electrical circuit. Each circle packing has a Markov process intimately coupled to its geometry; the crucial local rigidity of the packing then appears as a a Harnack inequality for discrete harmonic functions of the process.     

An infinite family of subcubic graphs with unbounded packing chromatic number

Recently, Balogh et al. (2018) answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded in the class of graphs with maximum degree 3. In this note, we present an explicit infinite family of subcubic graphs with unbounded packing chromatic number.     

基于有界度抛物复形的解析函数边值问题

该文讨论使用Circle Packing 方法来考虑解析函数边值问题. 寻求满足给定边界条件的解析函数, 是许多理论和实际问题中应用极为广泛的重要问题. 该文使用有界度的Circle Packing来构造给定区域上满足一定边界条件的解析函数, 为此首先讨论了 Circle Packing 映射与经典多项式之间的关系, 并在此基础上证明离散序列对解析函数的收敛性. 这个结果扩展了Carter和Rodin以及Dubejko早期使用正则6-packing取得的结果.     

SUBMANIFOLDS IN S~(n+p) WITH PARALLEL MBIUS FORM

§ 1　IntroductionLet x:M→ Sn+p be an n-dimensional submanifold in the unit sphere without umbilicpoints.Let { ei} be a local orthonormal basis with respectto the firstfundamental form I=dx·dx with dual basis{ θi} .Let II =∑ijαhαijθiθjeαbe the second fundamental form of xand H =∑αHαeαbe the mean curvature vector of x,where we use the range of indices:1≤ i,j,k,l≤ n,　　 n + 1≤α,β,γ≤ n + p,Hα=1n∑ihαij,{ eα} is a local orthonormal basis forthe normal bundle ofx.We defi…     

拟对称的有界度圆填充逼近

拟对称集和拟圆周集是万有Teichm(u|¨)er空间中两个常用模型.对于任一个由K-拟圆周诱导的拟对称,应用有界度圆填充的方法,构造了其近似映射,并证明了这些近似映射一致收敛于该拟对称.     

Genus Distributions of M(o|¨)bius Ladders

The genus distribution of a graph is a polynomial whose coefficients are the partition of the number of embeddings with respect to the genera. In this paper, the genus distribution of Mobius ladders is provided which is an infinite class of 3-connected simple graphs.     

The asymptotic distribution of circles in the orbits of Kleinian groups

Let P\mathcal{P} be a locally finite circle packing in the plane ℂ invariant under a non-elementary Kleinian group Γ and with finitely many Γ-orbits. When Γ is geometrically finite, we construct an explicit Borel measure on ℂ which describes the asymptotic distribution of small circles in P\mathcal{P}, assuming that either the critical exponent of Γ is strictly bigger than 1 or P\mathcal{P} does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Γ. Our construction also works for P\mathcal{P} invariant under a geometrically infinite group Γ, provided Γ admits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of P\mathcal{P} is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.     

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature) $\times$ (center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system $M_ D$ consisting of those $4 \times 4$ real matrices $W$ with $W^T Q_{D} \bW = Q_{W}$ where $Q_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 - \frac{1}{2}(x_1 +x_2 +x_3 + x_4)^2$ and $Q_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. On the parameter space $M_ D$ the group $\mathop{\it Aut}(Q_D)$ acts on the left, and $\mathop{\it Aut}(Q_W)$ acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group $O(3, 1)$. The right action of $\mathop{\it Aut}(Q_W)$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^2$ while the left action of $\mathop{\it Aut}(Q_D)$ is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of $\mathop{\it Aut}(Q_D)$, which we call the Apollonian group. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in $\mathop{\it Aut}(Q_D)$, the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer $4 \times 4$ matrices. We show these groups are hyperbolic Coxeter groups.     

The full power of the half-power

We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiled.