1,752 research outputs found
Some properties of a Rudin-Shapiro-like sequence
We introduce the sequence defined by , where denotes the number of inversions (i.e.,
occurrences of 10 as a scattered subsequence) in the binary representation of
n. We show that this sequence has many similarities to the classical
Rudin-Shapiro sequence. In particular, if S(N) denotes the N-th partial sum of
the sequence , we show that ,
where G is a certain function that oscillates periodically between
and .Comment: 21 pages, 6 figure
Spectrum of a Rudin-Shapiro-like sequence
We show that a recently proposed Rudin-Shapiro-like sequence, with balanced weights, has purely singular continuous diffraction spectrum, in contrast to the well-known Rudin-Shapiro sequence whose diffraction is absolutely continuous. This answers a question that had been raised about this new sequence
Surface Magnetization and Critical Behavior of Aperiodic Ising Quantum Chains
We consider semi-infinite two-dimensional layered Ising models in the extreme
anisotropic limit with an aperiodic modulation of the couplings. Using
substitution rules to generate the aperiodic sequences, we derive functional
equations for the surface magnetization. These equations are solved by
iteration and the surface magnetic exponent can be determined exactly. The
method is applied to three specific aperiodic sequences, which represent
different types of perturbation, according to a relevance-irrelevance
criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant
perturbation, the surface magnetization vanishes with a square root
singularity, like in the homogeneous lattice. For the period-doubling sequence,
the perturbation is marginal and the surface magnetic exponent varies
continuously with the modulation amplitude. Finally, the Rudin-Shapiro
sequence, which corresponds to the relevant case, displays an anomalous surface
critical behavior which is analyzed via scaling considerations: Depending on
the value of the modulation, the surface magnetization either vanishes with an
essential singularity or remains finite at the bulk critical point, i.e., the
surface phase transition is of first order.Comment: 8 pages, 7 eps-figures, uses RevTex and epsf, minor correction
Can Kinematic Diffraction Distinguish Order from Disorder?
Diffraction methods are at the heart of structure determination of solids.
While Bragg-like scattering (pure point diffraction) is a characteristic
feature of crystals and quasicrystals, it is not straightforward to interpret
continuous diffraction intensities, which are generally linked to the presence
of disorder. However, based on simple model systems, we demonstrate that it may
be impossible to draw conclusions on the degree of order in the system from its
diffraction image. In particular, we construct a family of one-dimensional
binary systems which cover the entire entropy range but still share the same
purely diffuse diffraction spectrum.Comment: 5 pages, 1 figure; two typos in the recursion relations for the
autocorrelation coefficients were correcte
The Mahler measure of the Rudin-Shapiro polynomials
Littlewood polynomials are polynomials with each of their coefficients in
{-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable
flatness property on the unit circle of the complex plane is given by the
Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure
and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of
the complex plane have the same size. It is also shown that the Mahler measure
and the maximum norm of the Rudin-Shapiro polynomials have the same size even
on not too small subarcs of the unit circle of the complex plane. Not even
nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials
have been known before
Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence
A formal inverse of a given automatic sequence (the sequence of coefficients
of the composition inverse of its associated formal power series) is also
automatic. The comparison of properties of the original sequence and its formal
inverse is an interesting problem. Such an analysis has been done before for
the Thue{Morse sequence. In this paper, we describe arithmetic properties of
formal inverses of the generalized Thue-Morse sequences and formal inverses of
two modifications of the Rudin{Shapiro sequence. In each case, we give the
recurrence relations and the automaton, then we analyze the lengths of strings
of consecutive identical letters as well as the frequencies of letters. We also
compare the obtained results with the original sequences.Comment: 20 page
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