Non-periodic Sequences Research Articles

Frustrated rotations in nematic monolayers.

Tabe and Yokoyama found recently that the optical axis in a chiral monolayer of a ferronematic rotates when water evaporates from the bath: the chiral molecules act as propellers. When the axis is blocked at the lateral walls of the trough, the accumulated rotation inside creates huge splays and bends. We discuss the relaxation of these tensions, assuming that a single dust particle nucleates disclination pairs. For the simplest geometry, we then predict a long delay time followed by a non-periodic sequence of "bursts". These ideas are checked by numerical simulations.

Electronic transmission in quasiperiodic serial stub structures

We present exact results on the electronic transmission through quantum stub waveguidesarranged in a Fibonacci quasiperiodic pattern. Discretizing the Schrödinger equation, wemap the problem into an equivalent tight binding form and study the transmissionspectrum using the transfer matrix method. We emphasize the effect of local positionalcorrelations in a Fibonacci quantum stub array that may lead to resonant eigenstates.Using the real space renormalization group ideas we unravel various local clusters of stubsresponsible for resonance. Extended eigenstates have been shown to exist and we find that,under some special circumstances, the electronic charge density exhibits a totally periodiccharacter in such a non-periodic sequence. Our method is completely generaland can be applied to any arbitrary sequence of stubs: periodic, quasiperiodicor random. This may lead to a possible experimental verification of the role ofpositional correlations in the transport behaviour of a class of mesoscopic devices.

Single-electron transistor based on modulation-doped SiGe heterostructures

We report the characterization of a single-electron transistor based on bended wires fabricated on modulation-doped SiGe two-dimensional electron gas. Electrical measurements show a diamond-shaped stability plot and a nonperiodic sequence of conductance peaks. The device behavior suggests the presence of disorder-induced multiple islands along the wire. Conductance oscillations remain well pronounced above liquid helium temperature.

Analytical transmission electron microscopy of oscillatory zoned grandite garnets

Oscillatory zoned grandite garnets from skarns of the Oslo rift region (southern Norway) have been studied using analytical transmission electron microscopy (ATEM). Our results show lamellae with a width varying from about 10 nm to more than 500 nm in both periodic and non-periodic sequences, which represent oscillatory zoning at a very fine scale. The compositions of adjacent lamellae alternate bimodally from Xand=0.55–0.75 to Xand=0.95–1.0. High resolution (HRTEM) images show that the compositional interfaces are sharp and coherent. Oscillatory zoning at the nm-scale puts constrains on the origin of oscillatory zoning in grandites and underlines the dynamic nature of the crystal growth process. The influence of growth rate is discussed against the background of local growth dynamics and external forcing at different length scales. This study emphasises the importance of adequate spatial resolution in analysing chemical variations in grandite garnets and shows that commonly applied methods (e.g. optical microscopy, backscattered electron imaging and electron or ion microprobe) are not sufficient to detect 'real' zonation patterns. Finally, proposed hypotheses on the origin of birefringence of grandite garnets with intermediate compositions are reviewed and discussed on the basis of the observed microstructure.

Recursive procedures for measuring disorder in non-periodic sequences and lattices

Recursive procedures are developed for calculating the conditional probabilities and kth-order entropy for non-periodic sequences and lattices. The entropies are used to provide a measure of disorder and this paper generalizes previous work in two ways: the use of three or more different atoms in the chain; providing a measure for disorder in a lattice constructed from non-periodic sequences. In addition, a quantitative comparison of the disorder in the Fibonacci and Thue-Morse sequences with three different types of atom is discussed.

Transmission of neutrons through non-periodic magnetic lattices

We consider the problem of comparing the suitability of various magnetic lattices for the spin polarization of neutrons. To do this, we use a discrete model to construct measures of transmission through one-dimensional lattices. We examine the model for non-periodic lattices that are generated from sequences and compare them with a periodic lattice. The periodic lattice gives rise to broader regions of spin polarization so that it should be easier to produce polarized beams, whereas the non-periodic sequences produce narrow regions more suitable for fine control. The most appropriate measure to illustrate these effects is seen to be that which simply measures the difference between spin+ and spin− transmission.

無周期な乱数系列の生成法の提案

無周期な乱数系列の生成法の提案

A RECYCLED CHARACTERIZATION OF KNEADING SEQUENCES

For any infinite sequence E on two symbols one can define two sequences of positive integers S(E) (the splitting times) and T(E) (the cosplitting times), which each describe the self-replicative structure of E. If E is the kneading sequence of a unimodal map, it is known that S(E) and T(E) carry a lot of information on the dynamics, and that they are disjoint. We show the reverse implication: A nonperiodic sequence E is the kneading sequence of some unimodal map if the sequences S(E) and T(E) are disjoint.

Descendants of Primitive Substitutions

Let s = (A, τ) be a primitive substitution. To each decomposition of the form τ (h) = uhv we associate a primitive substitution D[(h,u)](s) defined on the set of return words to h . The substitution D[(h,u)](s) is called a descendant of s and its associated dynamical system is the induced system (Xh, Th) on the cylinder determined by h . We show that \( {\cal D}(s) \) , the set of all descendants of s , is finite for each primitive substitution s . We consider this to be a symbolic counterpart to a theorem of Boshernitzan and Carroll which states that an interval exchange transformation defined over a quadratic field has only finitely many descendants. If s fixes a nonperiodic sequence, then \( {\cal D}(s) \) contains a recognizable substitution. Under certain conditions the set \( \Omega (s) = \bigcap_{s^\prime \in {\cal D}(s)}{\cal D}(s^\prime ) \) is nonempty.

Filters and filter banks for periodic signals, the Zak transform, and fast wavelet decomposition

We present a new approach to filtering and reconstruction of periodic signals. The tool that proves to handle these tasks very efficiently is the discrete Zak transform. The discrete Zak transform can be viewed as the discrete Fourier transform performed on the signal blocks. It also can be considered the polyphase representation of periodic signals. Fast filtering-decimation-interpolation-reconstruction algorithms are developed in the Zak transform domain both for the undersampling and critical sampling cases. The technique of finding the optimal biorthogonal filter banks, i.e., those that would provide the best reconstruction even in the undersampling case, is presented. An algorithm for orthogonalization of nonorthogonal filters is developed. The condition for perfect reconstruction for the periodic signals is derived. The generalizations are made for the nonperiodic sequences, and several ways to apply the developed technique to the nonperiodic sequences are considered. Finally, the developed technique is applied to recursive filter banks and the discrete wavelet decomposition.

A Generalization of Cobham's Theorem

If a nonperiodic sequence X is the image by a morphism of a fixed point of both a primitive substitution σ and a primitive substitution τ , then the dominant eigenvalues of the matrices of σ and τ are multiplicatively dependent. This is the way we propose to generalize Cobham's theorem.

Intertwinings of periodic sequences

A study of nonperiodic sequences {w j} jϵ Z which are intertwinings of two periodic sequences with maximal overlap. Expressions in terms of Beatty sequences {⌋ci + d ⌋} iϵ Z are given in case (i) both periodic sequences have the same periodic cycle and in case(ii) the periodic sequences have coprime periods. The extreme sequences in case (i) occur in a paper of Morse and Hedlund [9] and in case (ii) in a paper by Coven and Hedlund [3].

Concatenated Reed-Solomon/convolutional coding for data transmission in CDMA-based cellular systems

A robust error control scheme for data transmission in CDMA-based cellular systems is proposed which employs outer Reed-Solomon codes concatenated with inner convolutional codes. The performance of this scheme is analyzed assuming nonperiodic random spreading sequences and a Rake receiver with perfect knowledge of the channel. In particular, a simple model for the memoryless inner coding channel that encompasses the effects of multiple access interference, self-noise and thermal noise is first derived. Using new tight upper bounds on bit- and symbol-error probabilities of convolutional codes over Nakagami, Rayleigh, and Rician fading multipath channels, the performance of the concatenated coding scheme is then evaluated. The Reed-Solomon/convolutional coding scheme has been adopted by the European RACE Project Code Division Testbed (CODIT) and implemented in an experimental testbed. The code design methodology, which has been used to specify the 9.6-, 64-, and 128-kbit/s data traffic channels of the CODIT testbed, is presented and the single-cell CDMA capacity is computed.

Electron energy spectra of one-dimensional nondiagonal nonperiodic lattices.

We use degenerate perturbation theory to study the electronic energy spectra of one-dimensional two-component nondiagonal nonperiodic lattices, where the nearest-neighbor hopping integral ${\mathit{t}}_{\mathit{n}}$ takes two values T (strong bond) and t (weak bond) arranged in a nonperiodic sequence. To second order in the small parameter t/T, we obtain the density of states of the golden-mean (Fibonacci), silver-mean, and copper-mean lattices. The quantitative predictions for the density of states of these lattices are in good agreement with the numerical results obtained by exact diagonalization. The branching properties of energy spectra of these lattices are discussed. \textcopyright{} 1996 The American Physical Society.

Periodic and nonperiodic interstratification in the chlorite-biotite series

A systematic TEM investigation of interstratified chlorite-biotite crystals showed that the crystals are composed of domains of periodically interstratified chlorite-biotite, non-periodically interstratified chlorite-biotite, biotite, and chlorite. The interstratified chlorite-biotite occurs as a vein filling and was apparently crystallized from a hydrothermal solution. The complex structure of the interstratified chlorite-biotite presumably results from a nonlinear growth phenomenon occurring under a nonequilibrium state. A simple nonlinear dynamics model derived from Duffing`s equation was constructed with an additional chemical potential that accounts for the variation of structural configuration of tetrahedral sheets or 2:1 layers in chlorite and biotite, a simple periodic fluctuation of hydrothermal fluid composition, and a simple damping force for two-dimensional lattice misfit on (001) resulting from the intergrowth of different types of layers with different structural configurations and other dissipation effects. Solutions to the equations of the model show that periodic interstratification, nonperiodic interstratification, and domains of the two end-member components (biotite, chlorite) can be formed during crystallization under various conditions. The nonperiodic sequences of biotite and chlorite layers along the c axis in the interstratified crystals produced by this model are chaotic rather than random. The calculations suggest that both periodic and nonperiodic interstratifications can result from periodic externalmore » force, e.g., compositional fluctuation of the fluid.« less

Crystallization of thermotropic liquid crystalline HBA/HNA copolymers

The crystallization behaviour of random copolymers based on p-hydroxybenzoic acid (HBA) and 2-hydroxy-6-naphthoic acid (HNA) is reviewed. First a summary is given of what is known about the structure and thermal transitions of the constituent homopolymers. Next the crystallization behaviour of HBA/HNA copolymers is described starting with the glass transition, after which the two crystallization processes that have been found in these copolyesters are treated. The crystallization of the HBA/HNA copolymers will be compared to the crystallization of similar random copolymers that are not liquid crystalline. Two models have been proposed to describe the solid state of the HBA/HNA copolymers: the non-periodic layer (NPL) model and the paracrystalline lattice (PCL) model. In the NPL model, ordered domains are formed by the lateral register of similar but non-periodic sequences of neighbouring chains, whereas in the PCL model ordered domains are formed by the presence of conformational correlations between adjacent monomers without explicit sequence matching. The two models are compared in view of the experimental observations about the structure and the morphology of the crystallized material. This review ends with some conclusions.

Predictive Poincaré control: A control theory for chaotic systems.

One of the most interesting features of chaotic systems is the large number of unstable orbits embedded in a chaotic attractor. In this work, we propose a global chaos-control technique called predictive Poincar\'e control (PPC) that permits stabilization of a predefined solution, using only small control pulses. We prove this result for a large class of n-dimensional chaotic systems. The predefined solution can be a periodic or nonperiodic oscillation, expressed by a periodic or nonperiodic symbolic sequence [S. Hayes, C. Grebogi, and E. Ott, Phys. Rev. Lett. 70, 3031 (1993)]. We apply the general PPC scheme to the well known Lorenz model and study its robustness with respect to parasitic effects.

Measurement of disorder in three-component nonperiodic sequences

The information theoretic entropy provides a useful measure of the amount of disorder in a sequence of two or more symbols (components). The first-order and second-order conditional entropies H 1 and H 2 are calculated for several nonperiodic deterministic sequences, namely, the general three-component Fibonacci and the three-component Thue-Morse sequences. The sequences are arranged according to the amount of disorder measured by H 1. The results are compared with the corresponding two-component nonperiodic sequences.

Recursive procedures for measuring disorder in non-periodic sequences

Recursive procedures are found for calculating the kth-order entropy of non-periodic sequences. These entropies are used as measures of disorder of the sequences and we treat: the Fibonacci sequence and generalizations; the Thue-Morse sequence together with generalizations and the period-doubling sequence. A discussion of the relative ordering of the Thue-Morse and the Fibonacci sequences is given and it is found too simplistic to use any one measure for comparison.

Complexity in a spatially uniform continuum fault model

Recently, Rice [1993] pointed out that, up to now, the self‐organizing models which have produced complex nonperiodic sequences of events have all been sensitive to the spatial discretization used, and thus did not have a well defined continuum limit. He went on the suggest that spatial nonuniformity or “inherent discreteness” may be a necessary ingredient in allowing the complexity to develop in these systems. In this paper, I present a counterexample to this suggestion: a spatially uniform model with a well defined continuum limit is shown to give rise to complex nonperiodic sequences. The complexity arises in the deterministic model from inertial dynamics with a velocity‐weakening frictional instability, with the instability being stabilized at short lengthscales by a viscous term. The numerical results are shown to be independent of the spatial discretization for discretizations small compared to the viscous lengthscale. Furthermore, the qualitative features of the complexity produced are seen to be invariant with respect to two very different types of small scale cutoffs, implying a universality of the results with respect to the details of the small scale cutoff.