Thue-Morse Sequence Research Articles

Factor-balanced S-adic languages

A set of words, also called a language, is letter-balanced if the number of occurrences of each letter only depends on the length of the word, up to a constant. Similarly, a language is factor-balanced if the difference of the number of occurrences of any given factor in words of the same length is bounded. The most prominent example of a letter-balanced but not factor-balanced language is given by the Thue–Morse sequence. We establish connections between the two notions, in particular for languages given by substitutions and, more generally, by sequences of substitutions. We show that the two notions essentially coincide when the sequence of substitutions is proper. For the class of Thue–Morse–Sturmian languages, we give a full characterisation of factor-balancedness.

Linear independence of series related to the Thue–Morse sequence along powers

Abstract The Thue–Morse sequence $\{t(n)\}_{n\geqslant 0}$ is the indicator function of the parity of the number of ones in the binary expansion of nonnegative integers n, where $t(n)=1$ (resp. $=0$ ) if the binary expansion of n has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E. Miyanohara by showing that, for a fixed Pisot or Salem number $\beta>\sqrt {\varphi }=1.272019\ldots $ , the set of the numbers $$\begin{align*}1,\quad \sum_{n\geqslant1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant1}\frac{t(n^2)}{\beta^{n}},\quad \dots, \quad \sum_{n\geqslant1}\frac{t(n^k)}{\beta^{n}},\quad \dots \end{align*}$$ is linearly independent over the field $\mathbb {Q}(\beta )$ , where $\varphi :=(1+\sqrt {5})/2$ is the golden ratio. Our result yields that for any integer $k\geqslant 1$ and for any $a_1,a_2,\ldots ,a_k\in \mathbb {Q}(\beta )$ , not all zero, the sequence { $a_1t(n)+a_2t(n^2)+\cdots +a_kt(n^k)\}_{n\geqslant 1}$ cannot be eventually periodic.

The Thue–Morse continued fractions in characteristic 2 are algebraic

Let $a, b$ be distinct, non-constant polynomials in $\mathbb F_2 [z]$. Let $\xi_{a, b}$ be the power series in $\mathbb F_2((z^{-1}))$ whose sequence of partial quotients is the Thue–Morse sequence over $\{a, b\}$. We establish that $\xi_{a,b}$ is algebra

Pseudorandom Binary Sequences: Quality Measures and Number-Theoretic Constructions

In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order k, expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

Deleting Terms of the Divergent p-Series and Reciprocals of Primes Series Using the Thue-Morse Sequence

Deleting Terms of the Divergent p-Series and Reciprocals of Primes Series Using the Thue-Morse Sequence

Research on degree correlations of the Thue–Morse hierarchical network

In this paper, we construct a hierarchical network generated from the Thue–Morse sequence. First, we deduce that the degree distribution of the Thue–Morse hierarchical network follows an exponential distribution. Then, we compute the degree correlations by calculating the average degree of all neighbors of nodes which have degree k, denoted [Formula: see text]. By getting an accurate range of [Formula: see text], we show that [Formula: see text] and the Thue–Morse hierarchical network is assortative. Finally, we conclude that the hierarchical networks generated from substitution sequences still have the same properties as deterministic hierarchical networks.

Degree distributions and motif profiles of Thue–Morse complex network

In this paper, we study both the horizontal visibility graph (HVG) and the limited penetrable horizontal visibility graph (LPHVG) mapped from the Thue–Morse sequence. Firstly, we map the series to complex networks by using visibility graph algorithms. Then, we obtain the analytical degree distribution of the Thue–Morse HVG through an iterative method. And we also derive the degree distribution of the Thue–Morse LPHVG with a penetrable distance of ρ=1. Finally, we investigate the profile of sequential 4-node motifs for the Thue–Morse HVG. The relative frequencies of the 4-node motifs for the Thue–Morse HVG converge to a constant vector (12,16,16,16,0,0). The numerical simulations coincide excellently with theoretical results of the degree distributions and motif profiles.

Tunable bandstop filtering specialities in superconducting Thue–Morse photonic multilayers

We theoretically investigate wide and tunable bandstop filters based on one-dimensional superconducting photonic multilayers. These multilayers are composed of superconductors (Nb) and semiconductors (GaAs), arranged in accordance with the Thue–Morse sequence. The wide transmission bandgap of the multilayers can be utilized as a bandstop filter, with greater accuracy achieved by larger sequence number and thinner semiconductor. Our results demonstrate that both the bandwidth and center frequency decrease as the ambient temperature and semiconductor thickness increase. Especially, the stopband disappears when the temperature exceeds the critical temperature (Tc) of Nb. On the other hand, increasing hydrostatic pressure and superconductor thickness leads to monotonous increases in both the bandwidth and center frequency. Moreover, for thin superconductors or high temperatures close to Tc, the quality factor is high and tunable in a wide range. Our work presents a flexible solution for optical bandstop filters, which may also find potential applications in optical switching and modulating.

Values of Binary Partition Function Represented by a Sum of Three Squares

Let m be a positive integer and b_{m}(n) be the number of partitions of a non-negative integer n with parts being powers of 2, where each part can take m colors. We show that if m=2^{k}-1, then the natural density of n such that b_{m}(n) cannot be represented as a sum of three squares exists, and equals 1/12 for k=1, 2 and 1/6 for kge 3. In particular, for m=1 the equation b_{1}(n)=x^2+y^2+z^2 has a solution in integers if and only if n is not of the form 2^{2k+2}(8s+2t_{s}+3)+i for i=0, 1 and k, s are non-negative integers, and where t_{n} is the nth term in the Prouhet–Thue–Morse sequence. A similar characterization is obtained for the solutions in n of the equation b_{2^k-1}(n)=x^2+y^2+z^2.

Algebraic Automatic Continued Fractions in Characteristic 2

Abstract We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first family $\mathcal {P}$, and its sum modulo $2$, the Thue–Morse sequence, belongs to the second family $\mathcal {G}$. The family $\mathcal {G}$ contains all the iterated sums of sequences from the $\mathcal {P}$ and more.

An arithmetical property of the real numbers generated by Thue–Morse sequence along squares

An arithmetical property of the real numbers generated by Thue–Morse sequence along squares

Two-block substitutions and morphic words

We consider in general two-block substitutions and their fixed points. We prove that some of them have a simple structure: their fixed points are morphic sequences. Others are intrinsically more complex, such as the Kolakoski sequence. We prove this for the Thue-Morse sequence in base 3/2.

The absorption properties of one-dimensional spherical photonic crystals based on plasma material

The first theoretical treatment of the electromagnetic properties of one-dimensional plasma spherical photonic crystals (1D PSPCs) in the non-magnetized state is carried out to obtain the expressions for the electric and magnetic fields. The utilization of the transfer matrix method provides a new idea for the exploration of the 1D PSPCs, based on which the expressions for the absorptance of the 1D PSPCs are derived. A structure of quasi-periodic 1D PSPCs arranged according to the Thue-Morse sequence is proposed to acquire the ultra-wide absorption bands (ABs) at large incidence angles by adding an air layer and setting the plasma frequency in a gradient. And the factors affecting the ABs are meticulously analyzed. It is figured out that the primary factors affecting ABs are topology and plasma frequency, both of which have vigoroso influences on the bandwidths and positions of ABs. However, the initial radius and the collision frequency prevailingly affect the amplitude of the absorptance. The unexceptionable geometric properties of the 1D PSPCs make them sensitive to parameter changes and can be applied in the field of absorbers and radomes, which have great application prospects and development space.

Vanadium oxide metal-insulator phase transition in different types of one-dimensional photonic microcavities

The optical properties of vanadium dioxide (VO2) can be tuned via metal-insulator transition. In this work, different types of one-dimensional photonic structure-based microcavities that embed vanadium dioxide have been studied in the spectral range between 900 nm and 2000 nm. In particular, VO2 has been sandwiched between: i) two photonic crystals made of SiO2 and ZrO2; ii) two aperiodic structures made of SiO2 and ZrO2 that follow the Thue-Morse sequence; iii) two disordered photonic structures, made of SiO2 and ZrO2 in which the disorder is introduced either by a random sequence of the two materials or by a random variation of the thicknesses of the layers; iv) two four material-based photonic crystals made of SiO2, Al2O3, Y2O3, and ZrO2. The ordered structures i and iv show, respectively, one and two intense transmission valleys with defect modes, while the aperiodic and disordered structures ii and iii show a manifold of transmission valleys due to their complex layered configurations. The metal-insulator transition of VO2, controlled by temperature, results in a modulation of the optical properties of the microcavities.

Correlations of the Thue–Morse sequence

The pair correlations of the Thue–Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue–Morse sequence with general real weights are effectively determined by a single value of the balanced 2-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue–Morse sequence vanish, and that, for any even n, the n-point correlations of the balanced Thue–Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue–Morse system. We finish by showing how the correlations of the Thue–Morse system with general real weights can be derived from the balanced 2-point correlations.

Magneto-Optical Faraday Effect in Quasicrystalline and Aperiodic Microresonator Structures

We theoretically and numerically investigate magnetophotonic microresonators formed by a magnetic layer sandwiched between two reflective multilayers with different layer arrangements. Quasicrystals with the Fibonacci layer sequence and aperiodic structures with the Thue–Morse sequence are all compared to the conventional photonic crystal Bragg microresonators. The magneto-optical spectral properties of such magnetophotonic structures are completely different from each other and from a uniform magnetic film. In multilayered structures of various order types, microresonator modes are excited. The feature of multilayered structures with arrangements different from a periodic one is that they support the excitation of the multiple microresonator modes in a limited visible and near-infrared spectral range. The wavelengths of the two microresonator modes in a regular photonic crystal differ by more than one octave. This feature of the quasi-crystalline and aperiodic microresonators is important for applications in devices based on the Faraday effect.

Non-reciprocal spatial and quasi-reciprocal angular Goos-Hänchen shifts around double CPA-LPs in PT-symmetric Thue-Morse photonic crystals.

We theoretically investigate the Goos-Hänchen (GH) shifts of reflected light beams in Thue-Morse photonic crystals. The systems are constituted by two Thue-Morse dielectrics multilayers and satisfy parity-time (PT) symmetry. Double coherent perfect absorption laser points (CPA-LPs) are achieved in the parameter space composed of the incident angle and the gain-loss factor. Dramatic changes in the phase of reflection coefficient induce giant positive and negative spatial GH shifts at the CPA-LPs, while great angular GH shifts exist around the exceptional points (EPs). The spatial GH shifts present non-reciprocity for the forward and backward incident light waves near the double CPA-LPs, while the angular GH shifts are quasi-reciprocal. Increasing the Thue-Morse sequence number, these characteristics are approved around multiple CPA-LPs as well. Our work could pave the way to explore high-accuracy optical sensors.

Möbius orthogonality of the Thue–Morse sequence along Piatetski-Shapiro numbers

Möbius orthogonality of the Thue–Morse sequence along Piatetski-Shapiro numbers

Non-injectivity of infinite interval exchange transformations and generalized Thue-Morse sequences

In this paper we study the non-injectivity arising in infinite interval exchange transformations. In particular, we build and analyze an infinite family of infinite interval exchanges semi-conjugated to generalized Thue-Morse subshifts, whose non-injectivity occurs at a characterizable finite set of points.

Analysis of interface mode localization in disordered photonic crystal structure

A disordered photonic crystal (D-PhC) structure is analyzed to study the interface mode localization characteristics. The design comprises a bilayer-disordered PhC structure, where layers are arranged in Thue–Morse sequence (TMS). The impact of local symmetric substructures on eigenstates coupling is also considered over a wider wavelength range. The mode hybridization study is carried out for varying refractive index contrast values of TMS structures at an operating wavelength of 550, 632.8, and 750 nm, respectively. The dispersion analysis confirms the localization of bulk guided, and edge-guided modes for different incidence angles at the structural local resonators. Further, increasing the RI contrast value leads to generation of hybrid interface modes of very high electric field intensity. Thus, showing its potential applications in both sensing and light guiding applications. Moreover, because of the higher surface electric field intensity this structure can also be used for fluorescence-based detection and surface-enhanced Raman spectroscopy as well.