On substitution tilings and Delone sets without finite local complexity

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.

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

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-59-2021-1-29 ◽

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.

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Local Complexity of Delone Sets and Crystallinity

Canadian Mathematical Bulletin ◽

10.4153/cmb-2002-058-0 ◽

2002 ◽

Vol 45 (4) ◽

pp. 634-652 ◽

Cited By ~ 12

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.

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Delone Sets with Finite Local Complexity: Linear Repetitivity Versus Positivity of Weights

Discrete & Computational Geometry ◽

10.1007/s00454-012-9455-z ◽

2012 ◽

Vol 49 (2) ◽

pp. 335-347 ◽

Cited By ~ 2

Author(s):

Adnene Besbes ◽

Michael Boshernitzan ◽

Daniel Lenz

Keyword(s):

Local Complexity ◽

Delone Sets

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Equicontinuous Delone Dynamical Systems

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-090-3 ◽

2013 ◽

Vol 65 (1) ◽

pp. 149-170 ◽

Cited By ~ 6

Author(s):

Johannes Kellendonk ◽

Daniel Lenz

Keyword(s):

Dynamical Systems ◽

The Other ◽

Almost Periodic ◽

Other Hand ◽

Local Complexity ◽

Dirac Comb ◽

Delone Set ◽

Delone Sets

AbstractWe characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with finite local complexity, the only equicontinuous systems are then shown to be the crystallographic ones. On the other hand, within the class without finite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems that are not crystallographic. Our results solve the problem posed by Lagarias as to whether a Delone set whose Dirac comb is strongly almost periodic must be crystallographic.

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

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021157 ◽

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>

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Highly symmetric random substitution tilings

Zeitschrift für Kristallographie ◽

10.1524/zkri.2008.1048 ◽

2008 ◽

Vol 223 (11-12) ◽

Cited By ~ 1

Author(s):

Juan Garcia Escudero

Keyword(s):

Substitution Tilings ◽

Random Substitution

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COMPLEXITY MAPS REVEAL CLUSTERS IN NEURONAL ARBORIZATIONS

Fractals ◽

10.1142/s0218348x94000363 ◽

1994 ◽

Vol 02 (02) ◽

pp. 297-301

Author(s):

B. DUBUC ◽

S. W. ZUCKER ◽

M. P. STRYKER

Keyword(s):

Computer Vision ◽

Fractal Dimension ◽

Tree Growth ◽

Local Structure ◽

Dendritic Tree ◽

Local Information ◽

Growth Patterns ◽

Rate Of Growth ◽

Local Complexity ◽

Formal Definitions

A central issue in characterizing neuronal growth patterns is whether their arbors form clusters. Formal definitions of clusters have been elusive, although intuitively they appear to be related to the complexity of branching. Standard notions of complexity have been developed for point sets, but neurons are specialized "curve-like" objects. Thus we consider the problem of characterizing the local complexity of a "curve-like" measurable set. We propose an index of complexity suitable for defining clusters in such objects, together with an algorithm that produces a complexity map which gives, at each point on the set, precisely this index of complexity. Our index is closely related to the classical notions of fractal dimension, since it consists in determining the rate of growth of the area of a dilated set at a given scale, but it differs in two significant ways. First, the dilation is done normal to the local structure of the set, instead of being done isotropically. Second, the rate of growth of the area of this new set, which we named "normal complexity", is taken at a fixed (given) scale instead instead of around zero. The results will be key in choosing the appropriate representation when integrating local information in low level computer vision. As an application, they lead to the quantification of axonal and dendritic tree growth in neurons.

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ET0L SYSTEMS FOR COMPOSITE DODECAGONAL QUASICRYSTAL PATTERNS

International Journal of Modern Physics B ◽

10.1142/s0217979201004654 ◽

2001 ◽

Vol 15 (08) ◽

pp. 1165-1175 ◽

Cited By ~ 9

Author(s):

JUAN GARCÍA ESCUDERO

Keyword(s):

Pisot Number ◽

Substitution Tilings ◽

Inflation Factor ◽

L Systems ◽

Inflation Rules

Two types of deterministic substitution tilings with 12-fold symmetry and a Pisot number as inflation factor are generated and described in terms of bracketed L-systems. Composition of the inflation rules allows to construct other types of dodecagonal patterns which can be described with the help of ET0L-systems and may be used in order to derive nondeterministic models of quasicrystal structures.

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NONLINEAR GATED EXPERTS FOR TIME SERIES: DISCOVERING REGIMES AND AVOIDING OVERFITTING

International Journal of Neural Systems ◽

10.1142/s0129065795000251 ◽

1995 ◽

Vol 06 (04) ◽

pp. 373-399 ◽

Cited By ~ 161

Author(s):

ANDREAS S. WEIGEND ◽

MORGAN MANGEAS ◽

ASHOK N. SRIVASTAVA

Keyword(s):

Time Series ◽

Real World ◽

Markov Models ◽

Multilayer Perceptrons ◽

Time Step ◽

Local Complexity ◽

Previous State ◽

Segmentation Task ◽

Gating Network ◽

Update Rules

In the analysis and prediction of real-world systems, two of the key problems are nonstationarity (often in the form of switching between regimes), and overfitting (particularly serious for noisy processes). This article addresses these problems using gated experts, consisting of a (nonlinear) gating network, and several (also nonlinear) competing experts. Each expert learns to predict the conditional mean, and each expert adapts its width to match the noise level in its regime. The gating network learns to predict the probability of each expert, given the input. This article focuses on the case where the gating network bases its decision on information from the inputs. This can be contrasted to hidden Markov models where the decision is based on the previous state(s) (i.e. on the output of the gating network at the previous time step), as well as to averaging over several predictors. In contrast, gated experts soft-partition the input space, only learning to model their region. This article discusses the underlying statistical assumptions, derives the weight update rules, and compares the performance of gated experts to standard methods on three time series: (1) a computer-generated series, obtained by randomly switching between two nonlinear processes; (2) a time series from the Santa Fe Time Series Competition (the light intensity of a laser in chaotic state); and (3) the daily electricity demand of France, a real-world multivariate problem with structure on several time scales. The main results are: (1) the gating network correctly discovers the different regimes of the process; (2) the widths associated with each expert are important for the segmentation task (and they can be used to characterize the sub-processes); and (3) there is less overfitting compared to single networks (homogeneous multilayer perceptrons), since the experts learn to match their variances to the (local) noise levels. This can be viewed as matching the local complexity of the model to the local complexity of the data.

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