48,403 research outputs found
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
There is a growing body of results in the theory of discrete point sets and
tiling systems giving conditions under which such systems are pure point
diffractive. Here we look at the opposite direction: what can we infer about a
discrete point set or tiling, defined through a primitive substitution system,
given that it is pure point diffractive? Our basic objects are Delone multisets
and tilings, which are self-replicating under a primitive substitution system
of affine mappings with a common expansive map . Our first result gives a
partial answer to a question of Lagarias and Wang: we characterize repetitive
substitution Delone multisets that can be represented by substitution tilings
using a concept of "legal cluster". This allows us to move freely between both
types of objects. Our main result is that for lattice substitution multiset
systems (in arbitrary dimensions) being a regular model set is not only
sufficient for having pure point spectrum--a known fact--but is also necessary.
This completes a circle of equivalences relating pure point dynamical and
diffraction spectra, modular coincidence, and model sets for lattice
substitution systems begun by the first two authors of this paper.Comment: 36 page
Controlled Fuzzy Parallel Rewriting
We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of cells) in which developmental errors may occur. In essence this model is the fuzzy analogue of the derivation-controlled iteration grammar. Under minor assumptions on the family of control languages and on the family of fuzzy languages in the underlying iteration grammar, we show (i) regular control does not provide additional generating power to the model, (ii) the number of fuzzy substitutions in the underlying iteration grammar can be reduced to two, and (iii) the resulting family of fuzzy languages possesses strong closure properties, viz. it is a full hyper-AFFL, i.e., a hyper-algebraically closed full Abstract Family of Fuzzy Languages
Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces
Model sets (or cut and project sets) provide a familiar and commonly used
method of constructing and studying nonperiodic point sets. Here we extend this
method to situations where the internal spaces are no longer Euclidean, but
instead spaces with p-adic topologies or even with mixed Euclidean/p-adic
topologies.
We show that a number of well known tilings precisely fit this form,
including the chair tiling and the Robinson square tilings. Thus the scope of
the cut and project formalism is considerably larger than is usually supposed.
Applying the powerful consequences of model sets we derive the diffractive
nature of these tilings.Comment: 11 pages, 2 figures; dedicated to Peter Kramer on the occasion of his
65th birthda
A Glimpse at Mathematical Diffraction Theory
Mathematical diffraction theory is concerned with the analysis of the
diffraction measure of a translation bounded complex measure . It
emerges as the Fourier transform of the autocorrelation measure of .
The mathematically rigorous approach has produced a number of interesting
results in the context of perfect and random systems, some of which are
summarized here.Comment: 6 pages; Invited talk at QTS2, Krakow, July 2001; World Scientific
proceedings LaTeX styl
Tree Regular Model Checking for Lattice-Based Automata
Tree Regular Model Checking (TRMC) is the name of a family of techniques for
analyzing infinite-state systems in which states are represented by terms, and
sets of states by Tree Automata (TA). The central problem in TRMC is to decide
whether a set of bad states is reachable. The problem of computing a TA
representing (an over- approximation of) the set of reachable states is
undecidable, but efficient solutions based on completion or iteration of tree
transducers exist. Unfortunately, the TRMC framework is unable to efficiently
capture both the complex structure of a system and of some of its features. As
an example, for JAVA programs, the structure of a term is mainly exploited to
capture the structure of a state of the system. On the counter part, integers
of the java programs have to be encoded with Peano numbers, which means that
any algebraic operation is potentially represented by thousands of applications
of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an
extended version of tree automata whose leaves are equipped with lattices. LTAs
allow us to represent possibly infinite sets of interpreted terms. Such terms
are capable to represent complex domains and related operations in an efficient
manner. We also extend classical Boolean operations to LTAs. Finally, as a
major contribution, we introduce a new completion-based algorithm for computing
the possibly infinite set of reachable interpreted terms in a finite amount of
time.Comment: Technical repor
Kinematic Diffraction from a Mathematical Viewpoint
Mathematical diffraction theory is concerned with the analysis of the
diffraction image of a given structure and the corresponding inverse problem of
structure determination. In recent years, the understanding of systems with
continuous and mixed spectra has improved considerably. Simultaneously, their
relevance has grown in practice as well. In this context, the phenomenon of
homometry shows various unexpected new facets. This is particularly so for
systems with stochastic components. After the introduction to the mathematical
tools, we briefly discuss pure point spectra, based on the Poisson summation
formula for lattice Dirac combs. This provides an elegant approach to the
diffraction formulas of infinite crystals and quasicrystals. We continue by
considering classic deterministic examples with singular or absolutely
continuous diffraction spectra. In particular, we recall an isospectral family
of structures with continuously varying entropy. We close with a summary of
more recent results on the diffraction of dynamical systems of algebraic or
stochastic origin.Comment: 30 pages, invited revie
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