Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of $n$-dimensional Descartes configurations, which consist of $n+2$ mutually touching spheres. We work in the space $M_D^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those $(n+2) \times (n+2)$ real matrices $W$ with $W^T Q_{D,n} W = Q_{W,n}$ where $Q_{D,n} = x_1^2 + \cdots + x_{n+2}^2 - ({1}/{n})(x_1 +\cdots + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + \cdots + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. On the parameter space $M_D^n$ of augmented curvature-center matrices, the group ${\it Aut}(Q_{D,n})$ acts on the left and ${\it Aut}(Q_{W,n})$ acts on the right. Both these groups are isomorphic to the $(n+2)$-dimensional Lorentz group $O(n+1,1)$, and give two different "geometric" actions. The right action of ${\it Aut}(Q_{W,n})$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^n$ while the left action of ${\it Aut}(Q_{D,n})$ is defined only on the parameter space $M_D^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in ${\it Aut}(Q_{D,n})$, with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational).     

Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group. It is observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature x centers of all circles are integral. We show that (up to scale) there are exactly eight different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group O(3, 1).     

Apollonian circle packings: number theory

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: . Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple.     

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...     

树的导出匹配覆盖

The induced matching cover number of a graph G without isolated vertices,denoted by imc(G),is the minimum integer k such that G has k induced matchings M1,M2,…,Mk such that,M1∪M2 ∪…∪Mk covers V(G).This paper shows if G is a nontrivial tree,then imc(G) ∈ {△*0(G),△*0(G) + 1,△*0(G)+2},where △*0(G) = max{d0(u) + d0(v) :u,v ∈ V(G),uv ∈ E(G)}.     

旗传递点本原且基柱为${\rm PSL}_n(q)$的$(v,k,4)$-对称设计

若$\cal D$为一个非平凡旗传递点本原对称$(v,k,4)$设计, 其基柱为${\rm PSL}_n(q)$且$G\leq {\rm Aut}(\cal D)$. 那么, $\cal D$ 必为$2$-$(15,8,4)$设计且${\rm Soc}(G)={\rm PSL}_2(9)$.     

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 $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

交比和Poincare度量在平面拟共形映射下的偏差

研究(1)若f是 R²到 R²上的k -拟共形映射, 则对任意x₁,x₂,x₃,x_4∈R²有16^{\frac1k-1}(|(x₁,x_2, x₃,x₄)|+1)^{\frac1k}&;\leq&; \left|\left(f(x_1), f(x_2),f(x_3),f(x_4)\right)\right|+1\\&; \leq&; 16^{k-1}\left(|(x_1,x_2,x_3,x_4)|+1\right)^{k}; \end{eqnarray*}(2)若f是R²到R²上的k -拟共形映射, D是R²中的任一真子域,则对任意x₁,x₂∈D有\begin{eqnarray*}\frac1k\lambda_D(x_1,x_2)+4(\frac1k-1)\log2&;\leq&; \lambda_{f(D)} (f(x_1),f(x_2))\\&;\leq &;k\lambda_D(x_1,x_2)+4(k-1)\log2.\end{eqnarray*}了交比和Poincar\'e度量在平面拟共形映射下的偏差估计, 得到了如下两个结果.     

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)).     

UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION

Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.