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.     

带粗糙初始值向列型液晶流的适定性

考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的.     

一类带齐次核的奇异重积分算子的范数及其应用

设核函数K(u,v)具有对称性和齐次性,对如下定义的奇异重积分算子T:(Tf)(y)=∫R_+~n K(‖x‖α,‖y‖α)f(x)dx,y∈R_+~n,其中‖x‖α=(x_1~α+…+x_n~α)~1/α(α>0),研究了T的范数及其应用.     

关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,[x,y]~(p~m)=1,[x,[x,y]]=[y,[x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).     

广义超特殊p-群的自同构群Ⅲ

确定了广义超特殊p-群G的自同构群的结构.设|G|=p~(2n+m),|■G|=p~m,其中n≥1,m≥2,Aut_fG是AutG中平凡地作用在Frat G上的元素形成的正规子群,则(1)当G的幂指数是p~m时,(i)如果p是奇素数,那么AutG/AutfG≌Z_((p-1)p~(m-2)),并且AutfG/InnG≌Sp(2n,p)×Zp.(ii)如果p=2,那么AutG=Aut_fG(若m=2)或者AutG/AutfG≌Z_(2~(m-3))×Z_2(若m≥3),并且AutfG/InnG≌Sp(2n,2)×Z_2.(2)当G的幂指数是p~(m+1)时,(i)如果p是奇素数,那么AutG=〈θ〉■Aut_fG,其中θ的阶是(p-1)p~(m-1),且Aut_f G/Inn G≌K■Sp(2n-2,p),其中K是p~(2n-1)阶超特殊p-群.(ii)如果p=2,那么AutG=〈θ_1,θ_2〉■Aut_fG,其中〈θ_1,θ_2〉=〈θ_1〉×〈θ_2〉≌Z_(2~(m-2))×Z_2,并且Aut_fG/Inn G≌K×Sp(2n-2,2),其中K是2~(2n-1)阶初等Abel 2-群.特别地,当n=1时...     

限制零点个数的亚纯函数的正规定则

设k,n(≥k+1)是两个正整数,a(≠0),b是两个有穷复数,F为区域D内的一族亚纯函数.如果对于任意的f∈F,f的零点重级大于等于k+1,并且在D内满足f+a[L(f)]~n-b至多有n-k-1个判别的零点,那么F在D内正规·这里L(f)=f~((k))(z)+a_1f~((k-1))(z)+…+a_(k-1)f'(z)+a_kf(z),其中a_1(z),a_2(z),…,a_k(z)是区域D上的全纯函数.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

ON LINEAR AND NONLINEAR RIEMANN-HILBERT PROBLEMS FOR REGULAR FUNCTION WITH VALUES IN A CLIFFORD ALGEBRA

This paper deals with the boundary value problems for regular function with valuesin a Clifford algebra: ()W=O, x∈R~n\Г, w~+(x)=G(x)W~-(x)+λf(x, W~+(x), W~-(x)), x∈Г; W~-(∞)=0,where Г is a Liapunov surface in R~n the differential operator ()=()/()x_1+()/()x_2+…+()/()x_ne_n, W(x) =∑_A, ()_AW_A(x) are unknown functions with values in a Clifford algebra ()_n Undersome hypotheses, it is proved that the linear baundary value problem (where λf(x, W~+(x),W~-(x)) =g(x)) has a unique solution and the nonlinear boundary value problem has atleast one solution.     

拟线性次椭圆方程组在Morrey空间上的部分正则性

证明了拟线性次椭圆方程组-X_α~*(a_(ij)~(αβ)(x,u)X_βu~j)=-X_α~*f_i~α+g_i,i=1,2,…,N,x∈Ω的弱解广义梯度Xu在Morrey空间L_x~(p,λ)(Ω,R~(mN))(p2)上的部分正则性,其中光滑实向量场族X=(X_1,X_2,…,X_m)满足H(o|¨)rmander有限秩条件,X_α~*是X_α的共轭;而且主项系数a_(ij)~(αβ)(x,u)关于x一致VMO(Vanishing Mean Oscillation的缩写,消失平均震荡)间断,且关于u为一致连续.     

一类对称函数的Schur凸性及其应用

对x=(x_1,…,x_n)∈[0,1)~n∪(1,+∞o)~n,定义对称函数■其中r∈N,i_1,i_2,…,i_n为非负整数.研究了F_n(x,r)的Schur凸性、Schur乘性凸性和Schur调和凸性.作为应用,用控制理论建立了一些不等式,特别地,给出了高维空间的一些新的几何不等式.     

ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETERS OF TWO-SIDED TRUNCATION DISTRIBUTION FAMILIES

Consider the two-sided truncation distrbution families written in the formf(x,θ)dx=w(θ_1, θ_2)h(x)I_([θ_1,θ_2])(x)dx, where θ=(θ_1,θ_2).T(x)=(t_1(x), t_2(x))=(min(x_1,…,x_m), max(x_1, …,x_m))is a sufficient statistic and its marginal density is denoted by f(t)dμ~T. The prior distribution of θ belongs to the familyF={G:∫‖θ‖~2dG(θ)<∞}.In this paper, the author constructs the empirical Bayes estimator (EBE) of θ, φ_n (t), by using the kernel estimation of f(t). Under a quite general assumption imposed upon f(t) and h(x), it is shown that φ_n(t) is an asymptotically optimal EBE of θ.     

单位方体上沿曲面的振荡积分在Sobolev空间上的有界性

研究了欧氏空间R~2中单位方体Q~2=[0,1]~2上沿曲面(t,s,γ(t,s))的振荡奇异积分算子T_(α,β)f(u,v,x)=∫_(Q~2)f(u-t,v-s,x-γ(t,s))e~(it~(-β_1)s~(-β_2))t~(-1-α_1)s~(-1-α_2)dtds从Sobolev空间L_τ~p(R~(2+n))到L~p(R~(2+n))中的有界性,其中x∈R~n,(u,v)∈R~2,(t,s,γ(t,s))=(t,s,t~(P_1)s~(q_1),t~(p_2)s~(q_2),…,t~(p_n)s~(q_n))为R~(2+n)上一个曲面,且β_1α_1≥0,β_2α_20.这些结果推广和改进了R~3上的某些已知的结果.作为应用,得到了乘积空间上粗糖核奇异积分算子的Sobolev有界性.     

Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth p _i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable.     

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ $({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.     

Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.     

系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调

研究了系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调, 其中~$\mathbb{F}$ 是一个素特征的代数闭域且~$\frak{gl}(2,\mathbb{F})$ 是系数在~$\mathbb{F}$ 上的~$2\times 2$ 阶矩阵李代数. 计算出所有~$\frak{gl}(2,\mathbb{F})$ 到模李超代数~$W(m,3,\underline{1})$ 的子模的导子和内导子. 从而一维上同调~$\textrm{H}^{1}(\frak{gl}(2,\mathbb{F}),W(m,3,\underline{1}))$ 可以完全用矩阵的形式表示.     

多圆柱上μ-Bloch空间之间的点乘子

设μ和ν是[0,1]上的正规函数,刻画了C~n中以多圆柱D~n为支撑集的广义Bloch型空间β_μ到β_ν之间的点乘子.     

The Initial Value Problems and Nonlinear Boundary Value Problems for a Class of the Second order Quasilinear Parabolio Systems

In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below $\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$ are discussed.The boundary value conditions are $\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$ $\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$ Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved.     

ON THE EXPECTED SAMPLE SIZES OF SOME POWER ONE TESTS FOR NORMAL MEAN WITH UNKNOWN VARIANCE

Suppose that $\[{x_1},{x_2}, \cdots \]$ are i i d. random variables on a probability space $\[(\Omega ,F,P)\]$ and $\[{x_1}\]$ is normally distributed with mean $\[\theta \]$ and variance $\[{\sigma ^2}\]$, both of which are unknown. Given $\[{\theta _0}\]$ and $\[0 < \alpha < 1\]$, we propose a concrete stopping rule T w. r. e.the $\[\{ {x_n},n \ge 1\} \]$ such that $$\[{P_{\theta \sigma }}(T < \infty ) \le \alpha \begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta \le {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[{P_{\theta \sigma }}(T < \infty ) = 1\begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta > {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[\mathop {\lim }\limits_{\theta \downarrow {\theta _0}} {(\theta - {\theta _0})^2}{({\ln _2}\frac{1}{{\theta - {\theta _0}}})^{ - 1}}{E_{\theta \sigma }}T = 2{\sigma ^2}{P_{{\theta _0}\sigma }}(T = \infty )\]$$ where $\[{\ln _2}x = \ln (\ln x)\]$.