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1.
Benoit Loridant 《Mathematica Slovaca》2008,58(2):241-251
If A is a 2 × 2 expanding matrix with integral coefficients, and
⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ +
is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive.
It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable.
To a quadratic polynomial x
2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.
This research was supported by the Austrian Science Fundation (FWF), projects S9610 and S9612, that are part of the Austrian
National Research Network “Analytic Combinatorics and Probabilistic Number theory”. 相似文献
2.
Let U
λ be the union of two unit intervals with gap λ. We show that U
λ is a self-similar set satisfying the open set condition if and only if U
λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can
be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations
of the set of all such λ’s, as well as the corresponding tiling patterns, are given.
The first author was partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation
and NSFC (10571100). The second author was partially supported by NSFC (70371074) and NFSC (10571104). 相似文献
3.
M. N. Kolountzakis 《Discrete and Computational Geometry》2000,23(4):537-553
We consider polygons with the following ``pairing property': for each edge of the polygon there is precisely one other edge
parallel to it. We study the problem of when such a polygon K tiles multiply the plane when translated at the locations Λ , where Λ is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier analysis. As a first application of our approach we establish a necessary
and sufficient condition for K to tile with a given lattice Λ . (This was first found by Bolle for the case of convex polygons—notice that all convex polygons that tile, necessarily have
the pairing property and, therefore, our theorems apply to them.) Our main result is a proof that a large class of such polygons
tile multiply only quasi-periodically, which for us means that Λ must be a finite union of translated two-dimensional lattices in the plane. For the particular case of convex polygons we
show that all convex polygons which are not parallelograms tile multiply only quasi-periodically, if at all.
Received February 24, 1999, and in revised form August 26, 1999, and October 9, 1999. 相似文献
4.
Jian-qin Zhou 《应用数学学报(英文版)》2008,24(2):185-194
Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L(l, h, x, y), Aguil5 et al. define a discrete iteration L(p) = L(l + 2p, h + 2p, x + p, y + p), p = 0, 1, 2,..., over L-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by L(l, h, x,y), which is said to be a procreating k-tight tile if L(p)(p = 0, 1, 2, ~ ~ ~ ) are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L(l, h, x, y), 0 ≤ y - x ≤ 2k + 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches. 相似文献
5.
A measurable set Q ⊂
R
n
is a wavelet set for an expansive matrix A if F
−1
(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in
R
n
associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are,
up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with
the matrix B = A
t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes. 相似文献
6.
Abstract. Let T be a self-affine tile that is generated by an expanding integral matrix A and a digit set D . It is known that many properties of T are invariant under the Z -similarity of the matrix A . In [LW1] Lagarias and Wang showed that if A is a 2 × 2 expanding matrix with |det(A)| = 2 , then the Z -similar class is uniquely determined by the characteristic polynomial of A . This is not true if |det(A)| > 2. In this paper we give complete classifications of the Z -similar classes for the cases |det(A)| =3, 4, 5 . We then make use of the classification for |det(A)| =3 to consider the digit set D of the tile and show that μ(T) >0 if and only if D is a standard digit set. This reinforces the conjecture in [LW3] on this. 相似文献
7.
Aristotle contended that (regular) tetrahedra tile space, an opinion that remained widespread until it was observed that non-overlapping
tetrahedra cannot subtend a solid angle of 4π around a point if this point lies on a tetrahedron edge. From this 15th century argument, we can deduce that tetrahedra do
not tile space but, more than 500 years later, we are unaware of any known non-trivial upper bound to the packing density
of tetrahedra. In this article, we calculate such a bound. To this end, we show the existence, in any packing of regular tetrahedra,
of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is not fully covered by the packing. The bound
on the amount of space that is not covered in each sphere is obtained in a recursive way by building on the solid angle argument.
The argument can be readily modified to apply to other polyhedra. The resulting lower bound on the fraction of empty space
in a packing of regular tetrahedra is 2.6…×10−25 and reaches 1.4…×10−12 for regular octahedra. 相似文献
8.
Richard Kenyon 《Inventiones Mathematicae》1992,107(1):637-651
Summary Aperturbation of a tiling of a region inR
n
is a set of isometries, one applied to each tile, so that the images of the tiles tile the same region.We show that a locally finite tiling of an open region inR
2 with tiles which are closures of their interiors isrigid in the following sense: any sufficiently small perturbation of the tiling must have only earthquake-type discontinuities, that is, the discontinuity set consists of straight lines and arcs of circles, and the perturbation near such a curve shifts points along the direction of that curve.We give an example to show that this type of rigidity does not hold inR
n
, forn>2.Using rigidity in the plane we show that any tiling problem with a finite number of tile shapes (which are topological disks) is equivalent to a polygonal tiling problem, i.e. there is a set of polygonal shapes with equivalent tiling combinatorics.Oblatum 19-III-1991 相似文献
9.
For a convex polygonP withn sides, a ‘partitioning’ ofP inton−2 nonoverlapping triangles each of whose vertices is a vertex ofP is called a triangulation or tiling, and each triangle is a tile. Each tile has a given cost associated with it which may
differ one from another. This paper considers the problem of finding a tiling ofP such that the sum of the costs of the tiles used is a minimum, and explores the curiosity that (an abstract formulation of)
it can be cast as a linear program. Further the special structure of the linear program permits a recursive O(n
3) algorithm.
Research and reproduction of this report were partially supported by the National Science Foundation Grants MCS-8119774, MCS-7926009
and ECS-8012974; Department of Energy Contract DE-AM03-76SF00326, PA# DE-AT03-76ER72018; Office of Naval Research Contract
N00014-75-C-0267.
Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and donot necessarily reflect the views of the above sponsors. 相似文献
10.
E. Rémila 《Discrete and Computational Geometry》1998,20(2):189-204
We first give a new presentation of an algorithm, from W. Thurston, to tile polygons with ``calissons' (i.e., lozenges formed
from two cells of the triangular lattice Λ ). Afterward, we use a similar method to get a linear algorithm to tile polygons with m -leaning bars (parallelograms of length m formed from 2m cells of Λ ) and equilateral triangles (whose sides have length m ) and we produce a quadratic algorithm to tile polygons with m -leaning bars.
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Received July 3, 1996, and in revised form May 14, 1997. 相似文献
11.
Self-Affine Sets and Graph-Directed Systems 总被引:1,自引:0,他引:1
Abstract. A self-affine set in R
n
is a compact set T with A(T)= ∪
d∈ D
(T+d) where A is an expanding n× n matrix with integer entries and D
={d
1
, d
2
,···, d
N
} ⊂ Z
n
is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the
case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in
wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether
such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T
o
≠ ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer. 相似文献
12.
D. Fletcher 《Discrete and Computational Geometry》2011,46(2):394-403
We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging
edge-to-edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles, and we apply
our method to present via these rules a single prototile that can only tile ℝ3 aperiodically, and a pair of square tiles that can only tile ℝ2 aperiodically. 相似文献
13.
Tai-Man Tang 《Journal of Mathematical Sciences》2007,144(5):4504-4510
Wildly embedded tiles in ℝ3 with spherical boundary are discussed. The construction of the topologically complicated, crumpled cube tiles is reviewed.
We construct an infinite family of wildly embedded, cellular tiles with Fox-Artin-type wild points. Finally, a condition on
the set of wild points on a cellular tile is given to show that certain wild cells cannot be tiles. Several observations are
recorded for further investigations.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 203–211, 2005. 相似文献
14.
María del Carmen Moure 《Discrete and Computational Geometry》2009,42(4):722-739
Dekking (Adv. Math. 44:78–104, 1982; J. Comb. Theory Ser. A 32:315–320, 1982) provided an important method to compute the boundaries of lattice rep-tiles as a ‘recurrent set’ on a free group of a finite
alphabet. That is, those tilings are generated by lattice translations of a single tile, and there is an expanding linear
map that carries tiles to unions of tiles. The boundary of the tile is identified with a sequence of words in the alphabet
obtained from an expanding endomorphism (substitution) on the alphabet. In this paper, Dekking’s construction is generalized
to address tilings with more than one tile, and to have the elements of the tilings be generated by both translation and rotations.
Examples that fall within the scope of our main result include self-replicating multi-tiles, self-replicating tiles for crystallographic
tilings and aperiodic tilings. 相似文献
15.
R. Adler T. Nowicki G. Świrszcz C. Tresser S. Winograd 《Indagationes Mathematicae》2018,29(3):831-841
This is a companion paper to Adleret al. (in press, 2015). There, we proved the existence of an absorbing invariant tile for the Error Diffusion dynamics on an acute simplex when the input is constant and “ergodic” and we discuss the geometry of this tile. Under the same assumptions we prove here that said invariant tile (a fundamental set of the lattice generated by the vertices of the simplex) which is a finite union of polytopes have the property that any union of the intersections of the tile with the Voronoï regions of the vertices is a tile for a different, explicitly defined lattice. 相似文献
16.
本文研究了数字集具有严格乘积形式的自相似tile成为框架集的问题.利用Zak变换,证明了自相似tile是框架集的充要条件是其数字集D={0,1,2…,N-1}. 相似文献
17.
Balint Farkas Mate Matolcsi Peter Mora 《Journal of Fourier Analysis and Applications》2006,12(5):483-494
Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde’s Spectral Set
Conjecture in both directions. Namely, in
Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a nonspectral tile.
In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias
and Wang [14]. In particular, we prove here that the USC and the "tile → spectral" direction of Fuglede’s conjecture are equivalent
in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó [13] for the existence
of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples
of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas [2], and obtain
nonspectral tiles in
. Fuglede’s conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case
is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1]. 相似文献
18.
Rong Bao GU Tai Xiang SUN Ting Ting ZHENG 《数学学报(英文版)》2005,21(4):873-880
Let G be a graph (i.e., a finite one-dimensional polyhedron) and f : G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f. 相似文献
19.
Denis S. Krotov 《Designs, Codes and Cryptography》2011,61(3):315-329
A vertex coloring of a graph is called “perfect” if for any two colors a and b, the number of the color-b neighbors of a color-a vertex x does not depend on the choice of x, that is, depends only on a and b (the corresponding partition of the vertex set is known as “equitable”). A set of vertices is called “completely regular”
if the coloring according to the distance from this set is perfect. By the “weight distribution” of some coloring with respect
to some set we mean the information about the number of vertices of every color at every distance from the set. We study the
weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (in particular,
with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the
color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions.
Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating
the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very
simple formula for the weight enumerator of an arbitrary perfect coloring. 相似文献
20.
If π is a set of primes, a finite group G is block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there exists a prime p ∈ π such that α and β lie in different Brauer p-blocks. A group G is block separated if it is separated by the set of prime divisors of |G|. Given a set π with n different primes, we construct an example of a solvable π-group G which is block separated but it is not separated by every proper subset of π.
Received: 22 December 2004 相似文献