Folds

Dekking, Michel; Poorten, Alf van der

doi:10.1007/bf03023552

Cited by 59 publications

(55 citation statements)

References 12 publications

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“…As the main result of this section, we prove that these sequences modulo 2 are all 2-automatic in the sense of Salon [12], [13]. As said in the introduction, automatic sequences have been widely and deeply studied in the recent years, as a general reference, one can read the survey by Dekking, Mendes France and van der Poorten [8] or the survey by Allouche [1]. For the two-dimensional automatic sequences, see [12], [13].…”

Section: Automaticity Propertiesmentioning

confidence: 95%

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Hankel determinants of the Thue-Morse sequence

Allouche¹,

Peyrière²,

Zhou³

et al. 1998

Annales De L’institut Fourier

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Section: Automaticity Propertiesmentioning

confidence: 95%

“…formal power series over GF2(X) -and also in physics in relation to quasicrystals (see, for instance, [I], [7], [8], [II], [16], [17], [18]). …”

mentioning

confidence: 99%

Hankel determinants of the Thue-Morse sequence

Allouche¹,

Peyrière²,

Zhou³

et al. 1998

Annales De L’institut Fourier

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“…Numerous papers, including the survey [18], are devoted to paperfolding sequences. In this Section we consider folded continued fractions and we prove that they are always transcendental.…”

Section: Folded Continued Fractionsmentioning

confidence: 99%

“…-Let f = (f n ) n 0 be the paperfolding sequence on the alphabet {a, b} associated with the sequence (e n ) n 0 in {+1, −1} Z 0 of folding instructions. We first notice that f is not eventually periodic since no paperfolding sequence is eventually periodic (see [18]). …”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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Continued fractions and transcendental numbers

Adamczewski¹,

Bugeaud²,

Davison³

2006

Ann. inst. Fourier

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Noiselets

Coifman

Geshwind

Meyer

2001

Applied and Computational Harmonic Analysis

179

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Noiselets are functions which are noise-like in the sense that they are totally uncompressible by orthogonal wavelet packet methods. We describe a library of such functions and demonstrate a few of their noise-like properties. © 2001 Academic Press INTRODUCTIONAs the reader undoubtedly knows, various effective algorithms exist for using wavelets and wavelet packets to process data, for example, for compression or noise removal. In these algorithms, analysis of data is achieved because one is able to find rapid decay in the distribution of values of the data, when it is transformed into wavelet or wavelet packet bases.In practice one finds that the few large values in the transformed data describe the interesting part of the data, and the vast majority of values, which are small, represent a noise term. See, for example, [6].The performance of these algorithms is impressive and might lull one into the belief that analysis of any "interesting" structure can be carried out via wavelet packet analysis. Of course this cannot be so, and this paper gives constructions of large families of functions which give worst case behavior for orthogonal wavelet packet compression schemes.Noiselets are functions which give worst case behavior for the aforementioned type of orthogonal wavelet packet analysis. In particular, this paper gives explicit examples of (complex-valued) noiselets for which all Haar-Walsh wavelet packet coefficients have exactly the same absolute value. So, in some sense, noiselets are "noise-like," and in particular, noiselets are totally uncompressible by orthogonal wavelet packet methods.Although noiselets are noise-like in the sense of being spread in time and frequency, there are patterns lurking in them. Certain families of noiselets arise as bases for the spaces of the Haar multiresolution analysis. These bases are computationally good in the same way that wavelet packets are; they come with fast algorithms for forward and inverse transforms and there are trees of bases with the structure needed to support the best-basis algorithm. These good properties of noiselets are no coincidence. Noiselets are constructed via a multiscale iteration in exactly the same way as wavelet packets, but with a twist. So in some sense noiselets have the structure of wavelet packets. Because of this fast computational structure, the possibility exists that noiselets will be valuable tools for certain applications, rather than simply representing counterexamples.Another source of pattern within noiselets is that one finds within their construction certain classical fractal generating mechanisms. In fact, a whole class of noiselets are nothing but the distributional derivatives of the classical paper folding curves (see [4] for an introduction to paper folding). Hence noiselets provide a counterexample to the philosophical view of analysis with which this note began. Indeed, one sees that certain interesting multiscale mechanisms can produce well-organized data which are nonetheless invisible to our standard analysis tools...

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Folds

Cited by 59 publications

References 12 publications

Hankel determinants of the Thue-Morse sequence

Hankel determinants of the Thue-Morse sequence

Continued fractions and transcendental numbers

Noiselets

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