scholarly journals Number of bounded distance equivalence classes in hulls of repetitive Delone sets

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber ◽  
Lorenzo Sadun
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Bounded Distance ◽  
Local Complexity ◽  
Delone Set ◽  
Delone Sets ◽  
Uniformly Bounded

<p style='text-indent:20px;'>Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.</p>

scholarly journals A dichotomy for bounded displacement equivalence of Delone sets

2021 ◽  
pp. 1-18
Author(s):  
YOTAM SMILANSKY ◽  
YAAR SOLOMON
Keyword(s):  
Compact Space ◽  
Equivalence Classes ◽  
Natural Topology ◽  
Fell Topology ◽  
Local Complexity ◽  
Substitution Tilings ◽  
Local Topology ◽  
Delone Sets

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

scholarly journals Bipolar Fuzzy Relations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur
Keyword(s):  
Equivalence Class ◽  
Level Set ◽  
Equivalence Relation ◽  
Level Sets ◽  
Equivalence Classes ◽  
Fuzzy Relation ◽  
Fuzzy Relations ◽  
Transitive Relation ◽  
Fuzzy Partition

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.

On an Equivalence Relation for Free Mappings Embeddeable in a Flow

2003 ◽  
Vol 13 (07) ◽  
pp. 1911-1915 ◽  
Author(s):  
Z. Leśniak
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes

We consider an equivalence relation for a given free mapping f of the plane. Under the assumption that f is embeddable in a flow {ft : t ∈ R} we describe the structure of equivalence classes of the relation. Finally, we prove that f restricted to each equivalence class is a Sperner homeomorphism.

scholarly journals Tower systems for linearly repetitive Delone sets

2010 ◽  
Vol 31 (6) ◽  
pp. 1595-1618 ◽  
Author(s):  
JOSÉ ALISTE-PRIETO ◽  
DANIEL CORONEL
Keyword(s):  
Rate Of Convergence ◽  
Transition Matrices ◽  
Symbolic Systems ◽  
Delone Set ◽  
Nested Sequence ◽  
Delone Sets ◽  
Uniformly Bounded ◽  
Tower System

AbstractIn this paper we study linearly repetitive Delone sets and prove, following the work of Bellissard, Benedetti and Gambaudo, that the hull of a linearly repetitive Delone set admits a properly nested sequence of box decompositions (tower system) with strictly positive and uniformly bounded (in size and norm) transition matrices. This generalizes a result of Durand for linearly recurrent symbolic systems. Furthermore, we apply this result to give a new proof of a classic estimation of Lagarias and Pleasants on the rate of convergence of patch frequencies.

scholarly journals Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

2021 ◽  
Vol 59 ◽  
pp. 1-29
Author(s):  
Petr Ambroz ◽  
Zuzana Masakova ◽  
Edita Pelantova
Keyword(s):  
Finite Alphabet ◽  
Incidence Matrices ◽  
Project Method ◽  
Infinite Word ◽  
Bounded Distance ◽  
Local Complexity ◽  
Self Similar ◽  
Numeration Systems ◽  
Average Lattice ◽  
Delone Sets

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

scholarly journals Local Complexity of Delone Sets and Crystallinity

2002 ◽  
Vol 45 (4) ◽  
pp. 634-652 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Peter A. B. Pleasants
Keyword(s):  
Growth Rates ◽  
Linear Growth ◽  
Closed Ball ◽  
Covering Radius ◽  
Ideal Crystal ◽  
Minimum Radius ◽  
Local Complexity ◽  
Delone Set ◽  
Delone Sets

AbstractThis paper characterizes when a Delone set X in is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X, let NX(T) count the number of translation-inequivalent patches of radius T in X and let MX(T) be the minimum radius such that every closed ball of radius MX(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to X being an ideal crystal.Explicitly, for NX(T), if R is the covering radius of X then either NX(T) is bounded or NX(T) ≥ T/2R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions.For MX(T), either MX(T) is bounded or MX(T) ≥ T/3 for all T > 0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has MX(T) ≥ c(n)T for all T > 0, for a certain constant c(n) which depends on the dimension n of X and is > 1/3 when n > 1.

On the expressibility hierarchy of Magidor-Malitz quantifiers

1983 ◽  
Vol 48 (3) ◽  
pp. 542-557 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

AbstractWe prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for PC classes.Let M ⊨ Qnx1 … xnφ(x1 … xn) mean that there is an uncountable subset A of ∣M∣ such that for every a1 …, an ∈ A, M ⊨ φ[a1, …, an].Theorem 1.1 (Shelah) (♢ℵ1). For every n ∈ ωthe classKn+1 = {‹A, R› ∣ ‹A, R› ⊨ ¬ Qn+1x1 … xn+1R(x1, …, xn+1)} is not an ℵ0-PC-class in the logic ℒn, obtained by closing first order logic underQ1, …, Qn. I.e. for no countable ℒn-theory T, isKn+1the class of reducts of the models of T.Theorem 1.2 (Rubin) (♢ℵ1). Let M ⊨ QE x yφ(x, y) mean that there is A ⊆ ∣M∣ such thatEA, φ = {‹a, b› ∣ a, b ∈ A and M ⊨ φ[a, b]) is an equivalence relation on A with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let KE = {‹A, R› ∣ ‹A, R› ⊨ ¬ QExyR(x, y)}. Then KE is not an ℵ0-PC-class in the logic gotten by closing first order logic under the set of quantifiers {Qn ∣ n ∈ ω) which were defined in Theorem 1.1.

scholarly journals Trees and amenable equivalence relations

1990 ◽  
Vol 10 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Scot Adams
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Measure Space ◽  
Polynomial Growth ◽  
Finite Measure ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Borel Equivalence Relation

AbstractLet R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.

A generalization of Sierpiński's paradoxical decompositions: Coloring semialgebraic grids

2012 ◽  
Vol 77 (4) ◽  
pp. 1165-1183
Author(s):  
James H. Schmerl
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
First Order ◽  
Coordinate Axis ◽  
Paradoxical Decompositions

AbstractA structure is an n-grid if each Ei, is an equivalence relation on A and whenever X and Y are equivalence classes of, respectively, distinct Ei, and Ej, then X ∩ Y is finite. A coloring χ: A → n is acceptable if whenever X is an equivalence class of Ei, then {x ∈ X: χ(x) = i} is finite. If B is any set, then the n-cube Bn = (Bn; E0, …, En−1) is considered as an n-grid, where the equivalence classes of Ei are the lines parallel to the i-th coordinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpihski [17], proved that ℝn has an acceptable coloring iff 2ℵ0 ≤ ℵn−2. The main result is: if is a semialgebraic (i.e., first-order definable in the field of reals) n-grid, then the following are equivalent: (1) if embeds all finite n-cubes, then 2ℵ0 ≤ ℵn−2: (2) if embeds ℝn, then 2ℵ0 ≤ ℵn−2; (3) has an acceptable coloring.

Some applications of coarse inner model theory

1997 ◽  
Vol 62 (2) ◽  
pp. 337-365 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Class ◽  
Set Theory ◽  
Model Theory ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Inner Model ◽  
Descriptive Set

AbstractThe Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .

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