In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-S...In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.展开更多
Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set...Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set T=T(M,D)defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty.If D is of the form D={0,v,...,(|det M|-1)v},we say that T has a collinear digit set.The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets.In particular,we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball.Moreover,we show that in this case,T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling{T+z:z∈Z^(3)}induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.展开更多
We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 ...We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.展开更多
Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equa...Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equals to the order of the associate symmetry group. We apply the result to the study of the Levy Dragon.展开更多
为了探究Vector 3D Tiles格式在三维矢量地物表达方面的效果和性能,研究了一套Vector 3D Tiles格式生产和表达工具,用于在Cesium平台上对矢量数据进行可视化展示。主要工作包括两部分:一是将数据从传统的矢量格式转换为Vector 3D Tiles...为了探究Vector 3D Tiles格式在三维矢量地物表达方面的效果和性能,研究了一套Vector 3D Tiles格式生产和表达工具,用于在Cesium平台上对矢量数据进行可视化展示。主要工作包括两部分:一是将数据从传统的矢量格式转换为Vector 3D Tiles格式,二是在Cesium平台上展示转换后的Vector 3D Tiles数据。为了验证方法可行性,采用广州市地下管线数据开展了实验,对Shapefile、GeoJSON二维矢量格式进行处理,生成Vector 3D Tiles格式后,在Cesium平台上进行三维可视化展示。通过不同格式数据的加载效率和呈现效果比较,证明了矢量切片数据比原始矢量格式加载更快、渲染更平滑。在此基础上,对矢量切片数据基于自定义三维样式的渲染能力进行了验证。展开更多
文摘In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.
基金supported by a grant funded by the Austrian Science Fund and the Russian Science Foundation(Grant No.I 5554)supported by National Natural Science Foundation of China(Grant No.12101566)。
文摘Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set T=T(M,D)defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty.If D is of the form D={0,v,...,(|det M|-1)v},we say that T has a collinear digit set.The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets.In particular,we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball.Moreover,we show that in this case,T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling{T+z:z∈Z^(3)}induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.
基金supported by Japanese Ministry of Education,Culture,Sports,Science and Technology,Grantin Aid for Fundamental Research (Grant No.21540010)the Japanese Society for the Promotion of Science (Grant No.08F08714)
文摘We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.
基金Supported by the National Natural Science Foundation of China(No.10631040)
文摘Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equals to the order of the associate symmetry group. We apply the result to the study of the Levy Dragon.
文摘为了探究Vector 3D Tiles格式在三维矢量地物表达方面的效果和性能,研究了一套Vector 3D Tiles格式生产和表达工具,用于在Cesium平台上对矢量数据进行可视化展示。主要工作包括两部分:一是将数据从传统的矢量格式转换为Vector 3D Tiles格式,二是在Cesium平台上展示转换后的Vector 3D Tiles数据。为了验证方法可行性,采用广州市地下管线数据开展了实验,对Shapefile、GeoJSON二维矢量格式进行处理,生成Vector 3D Tiles格式后,在Cesium平台上进行三维可视化展示。通过不同格式数据的加载效率和呈现效果比较,证明了矢量切片数据比原始矢量格式加载更快、渲染更平滑。在此基础上,对矢量切片数据基于自定义三维样式的渲染能力进行了验证。
