On Tilings of Quadrants and Rectangles and Rectangular Pattern

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.

Chair Tilings非周期艺术图案的生成

在Chairtilings结构的基础上构造出新的函数来生成彩色非周期图案,其原胞具有丰富的图形结构,为计算机实现非周期拼砌提供了新方法·

Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases

Let Tn be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T+n be Tn with an extra 2 x 2 square. We investigate signed tilings of rectangles by Tn and T+n . We show that a rectangle has a signed tiling by Tn 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+n, 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.

Tilings与谱对偶性质的证明

Tilings与谱分别在几何和分析中起着重要的作用,有许多猜测涉及到它们之间的联系。二者之间没有直接的共轭关系,在二者较强条件下,已给出了Tilings与谱的一些特征性质;现利用不等式逼近相应恒等式的方法,证明了其中几个重要的基本定理。

Tilings与谱特征性质的证明

Tilings与谱分别在几何和分析中起着重要的作用,有许多猜测涉及到它们之间的联系。文献1中就二者在较强条件下,给出了Tilings与谱的一些特征性质。本文利用不等式逼近相应恒等式的方法,证明了其中几个重要的基本定理。

Tiling a Plane with Semi-Regular Equilateral Polygons with 2m-Sides

In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of the same type. Tiling a plane with semi-regular polygons depends not only on the type of a semi-regular polygon, but also on its interior angles that join at a node. In relation to the interior angles, semi-regular equilateral polygons with the same or different interior angles can be joined in the nodes. Here, we shall first consider tiling a plane with semi-regular equilateral polygons with 2m-sides. The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of , whereare the non-negative integers which are not equal to zero at the same time, and are the interior angles of a semi-regular equilateral polygon from the characteristic angle. It is shown that of all semi-regular equilateral polygons with 2m-sides, a plane can be tiled only with the semi-regular equilateral quadrilaterals and semi-regular equilateral hexagons. Then, the problem of tiling a plane with semi-regular equilateral quadrilaterals is analyzed in detail, and then the one with semi-regular equilateral hexagons. For these semi-regular polygons, all possible solutions of the corresponding Diophantine equations were analyzed and all nodes were determined, and then the problem for different values of characteristic elements was observed. For some of the observed cases of tiling a plane with these semi-regular polygons, some graphical presentations of tiling constructions are also given.

一种Tiling Array信号识别流程的并行优化

针对一种非常有效的信号识别算法——滑窗法(sliding window,SW),在其嵌入的串行信号识别流程基础上提出并行处理方案,并基于MPI(message passing interface)平台实现并行优化版本。这在基因芯片研究领域是个崭新的思路。最后,根据公共数据Affymetrix(2005) Tiling Array,展示SW并行流程的运行性能。实验结果表明,信号识别流程的并行化对大规模的数据运算非常有效且十分必要。

正规Tiling的邻居系

该文借助Tiling的拓扑图和边邻居图,给出了正规Tiling中两个Tiles拥有相同的邻居系的一个充分必要条件,并利用此条件证明了正规Tiling中不可能存在三个不同的Tiles具有相同的邻居系,从而在一定程度上回答了Gr_(u|¨)nbaum的一个公开问题.

Regular tilings的L(d,1)-标号着色(英文)

L(d,1)- 标号着色是L(2,1)- 标号着色的推广,这一图的点着色问题来自于无线电波中的频道分配问题,要求图中相邻顶点所着的颜色相差至少d,距离为2的顶点所着颜色必须不相同.由于d=0,1,2时regulartilings的L(d,1) 标号着色数已由Calamoneri和Petreschi给出,本文研究d≥3时所有3种regulartilings的L(d,1) 标号着色,给出它们的L(d,1) 标号着色数.结合Calamoneri和Petres chi的结果,对所有非负整数d,regulartilings的L(d,1) 标号着色数已完全确定.

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

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.

密度方法在谱与tilings关系中的一个应用

谱对与tiling对存在某些确定的联系,两者在小波理论、离散Fourier分析与三角逼近理论中有着直接的应用.本文将从谱与tilings的基本性质出发,利用密度方法,得到谱与tilings关系中集合的勒贝格测度的一些估计,为进一步研究此类问题奠定基础.这里的研究是基于谱与tilings的基本性质,与共轭Fuglede猜想密切相关.

基于遗传算法的Tiling覆盖策略天文卫星任务规划

天文卫星机遇目标任务规划是一个复杂的多目标优化问题。针对Tiling覆盖策略的机遇目标任务规划要求及其约束条件进行抽象,建立任务规划问题模型,在规划模型基础上设计基于遗传算法的多目标优化任务规划算法TPA,并通过实例数据验证了不同参数条件下的求解。在解决Tiling覆盖策略的天文卫星机遇目标多目标任务规划问题时,所提方法能够在保证算法收敛性的同时兼顾优先级和规划路径,满足规划需求。

线性变换下的谱与Tiling的一些性质

令Ω=In,(Ω,Λ)为谱对当且仅当(Ω,Λ)为Tiling对。在任意可逆的线性变换下,相对应的谱对、Tiling对等价性质保持不变。Lagarias、Reeds与Wang利用密度法已证成立,现给出为另一种方法证明,即利用不等式逼近相应的恒等式。

Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner Bases

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.

整数自仿Tiling的Tile数字集(英文)

关于由一个扩张矩阵A∈Mn(Z)和数字集D={d1,d2,...,dm}■Zn生成的整数自仿Tiling,已经有很多研究结果。其中一个重要的问题是判定一个数字集在什么条件下能生成一个Tile。在一维情况下,已知结果有:标准数字集,乘积形式数字集,弱乘积形式数字集都是Tile数字集。在本文中,我们把弱乘积形式的概念推广到高维,并证明它们都是Tile数字集。

Substitutions of vertex configuration of Ammann–Beenker tiling in framework of Ammann lines

The Ammann–Beenker tiling is a typical model for two-dimensional octagonal quasicrystals. The geometric properties of local configurations are the key to understanding its formation mechanism. We study the configuration correlations in the framework of Ammann lines, giving an in-depth inspection of this eightfold symmetric structure. When both the vertex type and the orientation are taken into account, strict confinements of neighboring vertices are found. These correlations reveal the structural properties of the quasilattice and also provide substitution rules of vertex along an Ammann line.

Tiling Rectangles with Gaps by Ribbon Right Trominoes

We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.

Every Tiling of the First Quadrant by Ribbon <i>L n</i>-Ominoes Follows the Rectangular Pattern

Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by two ribbon L-shaped n-ominoes. An application of our result is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for an infinite family of sets of tiles: , n even, has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an square by n/2 vertical rectangles, with a tiling by n/2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L-shaped n-ominoes. We show that none of these results are valid for any odd n. The rectangular pattern of a tiling of the first quadrant persists if we add an extra tile to , n even. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. We also show that our main result implies that a skewed L-shaped n-omino, n even, is not a replicating tile of order k2 for any odd k.

Revisiting a Tiling Hierarchy (II)

In a recent paper, we revisited Golomb’s hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles. In this classification, for several levels in Golomb’s hierarchy, more types appear. We showed that there is no general relationship among tiling capabilities for types corresponding to same level. Then we found the relationships from Golomb’s hierarchy that remain valid in this setup and found those that fail. As a consequence we discovered two alternative tiling hierarchies. The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones. We show that almost all of them fail. If one refines the hierarchy for tile sets that tile rectangles and for deficient regions then most of the implications of tiling capabilities can be recovered.

A Tiling Lemma and Its Application to the Ratio Test for Convergence of Series

We prove that any collection which tiles the positive integers must contain one of two types of sub-collections. We then use this result to prove a variation of the Ratio Test for convergence of series. This version of the Ratio Test shows the convergence of certain series for which the Root Test (which is known to be more powerful than the conventional Ratio Test) fails. This version of the Ratio Test is also used to prove a version of the Banach Contraction Principle for self-maps of a complete metric space.