In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by ...In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by using self-similar tilings for the acceptable dilations on the Heisenberg group.展开更多
Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investiga...Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.展开更多
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is...We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.展开更多
The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by p...The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.展开更多
We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an od...We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4? is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4? is twice the number of tilings by dominoes of a (2m+1)×(2m+1)?deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set?T+<sub style="margin-left:-6px;">4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4? and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4? and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.展开更多
In this paper, we deduce Wiener number of some connected subgraphs in tilings (4, 4, 4, 4) and (4, 6, 12), which are in Archimedean tilings. And compute their average distance.
Edge-to-edge tilings of the sphere by congruent a^(2)bc-quadrilaterals are classified as 3 classes:(1)A 1-parameter family of quadrilateral subdivisions of the octahedron with24 tiles,and a flip modification for one s...Edge-to-edge tilings of the sphere by congruent a^(2)bc-quadrilaterals are classified as 3 classes:(1)A 1-parameter family of quadrilateral subdivisions of the octahedron with24 tiles,and a flip modification for one special parameter;(2)a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n≥3;(3)a 3-layer earth map tiling with 8n tiles for each n≥2,and two flip modifications for each odd n.The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings.展开更多
In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by ...In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by ε^m,m ≥ 2,m ∈ N,F^k,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.展开更多
Based on the substitution rule and symmetry, we propose a method to generate an octagonal quasilattice consisting ofsquare and rhombus tiles. Local configurations and Ammann lines are used to guide the growth of the t...Based on the substitution rule and symmetry, we propose a method to generate an octagonal quasilattice consisting ofsquare and rhombus tiles. Local configurations and Ammann lines are used to guide the growth of the tiles in a quasiperiodicorder. The structure obtained is a perfect eight-fold symmetric quasilattice, which is confirmed by the radial distributionfunction and the diffraction pattern.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
基金Sponsored by the NSFC (10871003, 10701008, 10726064)the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007001040)
文摘In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by using self-similar tilings for the acceptable dilations on the Heisenberg group.
文摘Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.
文摘We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
文摘The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.
文摘We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4? is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4? is twice the number of tilings by dominoes of a (2m+1)×(2m+1)?deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set?T+<sub style="margin-left:-6px;">4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4? and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4? and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.
文摘In this paper, we deduce Wiener number of some connected subgraphs in tilings (4, 4, 4, 4) and (4, 6, 12), which are in Archimedean tilings. And compute their average distance.
基金supported by the Key Projects of Zhejiang Natural Science Foundation(No.LZ22A010003)ZJNU Shuang-Long Distinguished Professorship Fund(No.YS304319159)。
文摘Edge-to-edge tilings of the sphere by congruent a^(2)bc-quadrilaterals are classified as 3 classes:(1)A 1-parameter family of quadrilateral subdivisions of the octahedron with24 tiles,and a flip modification for one special parameter;(2)a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n≥3;(3)a 3-layer earth map tiling with 8n tiles for each n≥2,and two flip modifications for each odd n.The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings.
基金Supported by FEDER funds through COMPETE Operational Programme Factors of Competitiveness(Programa Operacional Factores de Competitividade)Supported by FSE+3 种基金Supported by Portuguese funds through the Center for Researchand Development in Mathematics and Applications(University of Aveiro)the Portuguese Foundation for Science and Technology(FCT Fundao para a Ciência e a Tecnologia)project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690supported partially by an NSERC Canada Discovery Grant
文摘In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by ε^m,m ≥ 2,m ∈ N,F^k,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.
基金the National Natural Science Foun-dation of China(Grant No.11674102)。
文摘Based on the substitution rule and symmetry, we propose a method to generate an octagonal quasilattice consisting ofsquare and rhombus tiles. Local configurations and Ammann lines are used to guide the growth of the tiles in a quasiperiodicorder. The structure obtained is a perfect eight-fold symmetric quasilattice, which is confirmed by the radial distributionfunction and the diffraction pattern.
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。
