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152
Weighted Dirac combs with pure point diffraction
, 2004
"... A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a ..."
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Cited by 76 (37 self)
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A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.
Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
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Cited by 56 (21 self)
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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Cited by 40 (5 self)
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
Surprises in Diffuse Scattering
 Z. Kristallogr
, 2000
"... . Diffuse scattering is usually associated with some disorder in the analyzed material. Different kinds of disorder may produce different diffuse scattering  or not. In this letter, we demonstrate some aspects of the variety of diffuse scattering that occurs even in very simple examples, and how u ..."
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Cited by 22 (16 self)
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. Diffuse scattering is usually associated with some disorder in the analyzed material. Different kinds of disorder may produce different diffuse scattering  or not. In this letter, we demonstrate some aspects of the variety of diffuse scattering that occurs even in very simple examples, and how unawareness may lead astray. 1 Introduction For a long time, the diffuse part in diffraction spectra played only a minor role in crystallography. This was mainly due to experimental restrictions but also to a lack of theoretical studies. In this letter, we investigate some simple models with disorder, both deterministic and random, where the diffuse background of the diffraction must not be neglected in the structural analysis. We first start with an introduction to the language of mathematical diffraction theory, which is necessary to get a full understanding of the spectrum. The following two examples have exactly the same diffraction pattern albeit their disorder is of completely differen...
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 17 (4 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Palindrome Complexity Bounds For Primitive Substitution Sequences
, 2000
"... We consider onesided infinite words generated via iteration by primitive substitutions on finite alphabets and provide bounds on the palindrome complexity function as well as uniform bounds on the frequencies of palindromes in such words. As an application of these bounds, we prove that the str ..."
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Cited by 15 (7 self)
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We consider onesided infinite words generated via iteration by primitive substitutions on finite alphabets and provide bounds on the palindrome complexity function as well as uniform bounds on the frequencies of palindromes in such words. As an application of these bounds, we prove that the strongly palindromic sequences in a primitive substitution dynamical system form a set of measure zero.
Entropy rates and finitestate dimension
 THEORETICAL COMPUTER SCIENCE
, 2005
"... The effective fractal dimensions at the polynomialspace level and above can all be equivalently defined as the Centropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspacedimension is equivalent to the PSPACEentropy rate. At lower levels of c ..."
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Cited by 14 (1 self)
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The effective fractal dimensions at the polynomialspace level and above can all be equivalently defined as the Centropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspacedimension is equivalent to the PSPACEentropy rate. At lower levels of complexity the equivalence proofs break down. In the polynomialtime case, the Pentropy rate is a lower bound on the pdimension. Equality seems unlikely, but separating the Pentropy rate from pdimension would require proving P != NP. We show that at the finitestate level, the opposite of the polynomialtime case happens: the REGentropy rate is an upper bound on the finitestate dimension. We also use the finitestate genericity of AmbosSpies and Busse (2003) to separate finitestate dimension from the REGentropy rate. However, we point out that a blockentropy rate characterization of finitestate dimension follows from the work of Ziv and Lempel (1978) on finitestate compressibility and the compressibility characterization of finitestate dimension by Dai, Lathrop, Lutz, and Mayordomo (2004). As applications of the REGentropy rate upper bound and the blockentropy rate characterization, we prove that every regular language has finitestate dimension 0 and that normality is equivalent to finitestate dimension 1.