Abstract: The atomic surfaces modelling technique has been used to solve the structure of the basic Ni-rich Al–Co–Ni decagonal phase. Formula Al70.6Co6.7Ni22.7, space group P\overline{10}, five-dimensional unit-cell parameters: d1 = d4 = 4.752 (3) A, d2 = d3 = 3.360 (2) A, d5 = 8.1710 (2) A; α12 = α34 = 69.295°, α13 = α24 = 45°, α14 = 41.410°, α23 = αi5 = 90° (i = 1–4), V = 291.2 (7) A5; Dx = 3.887 Mg m−3. Refinement based on |F|; 2767 unique reflections (|F| > 0), 749 parameters, R = 0.17, wR = 0.06. Describing the structure of quasicrystals embedded in n-dimensional superspace in principle takes advantage of n-dimensional periodicity to select the minimal set of degrees of freedom for the structure. The method of modelling of the atomic surfaces yielded the first fully detailed structure solution of this phase. Comparison with numerous former, less accurate models confirms several features already derived, but adds a new essential insight of the structure and its complexity. The atoms fill the space forming recurrent structure motifs, which we will (generically) refer to as clusters. However, no unique cluster exists, although differences are small. Each cluster shows a high degree of structural disorder. This gives rise to a large configurational entropy, as much as expected in a phase which is stable at high temperature. On the other side, the cluster spatial arrangement is perfectly quasiperiodic. These considerations, corroborated by analysis of the structural relationship with neighbouring periodic phases, strongly suggest the existence of a non-local, long-range interaction term in the total energy which may be essential to the stability.

Abstract: We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t) . We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t , then the attractor is a continuous image of a torus. Moreover the global attractor attracts all the solutions of the NS system with exponential rate, that is, the attractor is exponential. We also consider the 2D Navier–Stokes system with rapidly oscillating external force g(x,t,t/e) , which has the average as e → 0+. We assume that the function g(x,t,z) has a bounded primitive with respect to z and the averaged NS system has a small Grashof number that provides a simple structure of the averaged global attractor. Then we prove that the distance from the global attractor of the original NS system to the attractor of the averaged NS system is less than a small power of e .

Abstract: We report on confined two-dimensional (2D) dipole clusters formed by small ferromagnetic particles floating at the liquid-air interface and confined by nonuniform external magnetic field. The particles self assemble into hexagonally ordered clusters whose lattice constant can be magnetically tuned. We study the area S, the energy E, the chemical potential $\ensuremath{\mu},$ and the lattice constant a, of 2D clusters as functions of particle number N for $Nl130.$ We develop a continuum approximation which accounts fairly well for the smooth part of $\ensuremath{\mu}(N),S(N),$ and $a(N)$ dependences. In addition to these dependences, we observe quasiperiodic fluctuations with dips at ``magic'' numbers corresponding to particularly symmetric particle configurations. We demonstrate that these fluctuations are related to the cluster symmetry and to the cluster center of mass position.

Abstract: A method based on the idea of a discontinuity mapping is derived for predicting the characteristics of system attractors that occur following a grazing intersection of a two-frequency, quasiperiodic oscillation with a two-dimensional impact surface in a three-dimensional state space. Within certain restrictions, the correction to the non-impacting flow afforded by the discontinuity mapping is computable using quantities determined solely by the non-impacting flow and the properties of the impact surface and the associated impact mapping in the immediate vicinity of the initial grazing contact. A model example is discussed to illustrate the quantitative predictive power of the discontinuity-mapping approach even relatively far away in parameter space from the original grazing intersection. Finally, constraints on the applicability of the methodology are described in detail with suggestions for suitable modifications.

Abstract: To investigate the influence of electronic interaction on the metal-insulator transition (MIT), we consider the Aubry-Andre (or Harper) model which describes a quasiperiodic one-dimensional quantum system of non-interacting electrons and exhibits an MIT. For a two-particle system, we study the effect of a Hubbard interaction on the transition by means of the transfer-matrix method and finite-size scaling. In agreement with previous studies we find that the interaction localizes some states in the otherwise metallic phase of the system. Nevertheless, the MIT remains unaffected by the interaction. For a long-range interaction, many more states become localized for sufficiently large interaction strength and the MIT appears to shift towards smaller quasiperiodic potential strength.

Abstract: Quasi-phase-matching optical parametric and cascaded parametric processes in a two-component quasiperiodic superlattice were studied in theory and experiment. This letter demonstrates how to obtain red at 666 nm and blue at 443 nm simultaneously from the superlattice using a 532 nm laser as a pump through these two processes mentioned above. The result confirms that some nonlinear frequency conversion processes occurring in a high-dimension χ(2) nonlinear photonic crystal may be efficiently achieved in such a one-dimension quasiperiodic optical superlattice.

Abstract: The influence of disorder on ultracold atomic Bose gases in quasiperiodic optical lattices is discussed in the framework of the one-dimensional Bose-Hubbard model. It is shown that simple periodic modulations of the well depths generate a rich phase diagram consisting of superfluid, Mott insulator, Bose-glass and Anderson localized phases. The detailed evolution of mean occupation numbers and number fluctuations as function of modulation amplitude and interaction strength is discussed. Finally, the signatures of the different phases, especially of the Bose-glass phase, in matter-wave interference experiments are investigated.

Abstract: We introduce an algorithm for applying a cross-wavelet transform to analysis of quasiperiodic variations in a time-series, and introduce significance tests for the technique. We apply a continuous wavelet transform and the cross-wavelet algorithm to the Pearson-Readhead VLBI survey sources using data obtained from the University of Michigan 26-m parabloid at observing frequencies of 14.5, 8.0, and 4.8 GHz. Thirty of the sixty-two sources were chosen to have sufficient data for analysis, having at least 100 data points for a given time-series. Of these thirty sources, a little more than half exhibited evidence for quasiperiodic behavior in at least one observing frequency, with a mean characteristic period of 2.4 yr and standard deviation of 1.3 yr. We find that out of the thirty sources, there were about four time scales for every ten time series, and about half of those sources showing quasiperiodic behavior repeated the behavior in at least one other observing frequency.

Abstract: The overlayer covering a (110) face of a Nb crystal annealed at 1500--2000 K in UHV has been studied by Auger electron and photoemission spectroscopies, low-energy electron diffraction (LEED) and scanning tunneling microscopy (STM). This layer, which results from the surface segregation of oxygen dissolved in Nb bulk, corresponds to a thin niobium oxide with a ${\mathrm{NbO}}_{x\ensuremath{\approx}1}$ stoichiometry as shown by photoemission with synchrotron radiation. Both LEED and STM investigations show the complex structure of the oxide overlayer with two orientations rotated by 109\ifmmode^\circ\else\textdegree\fi{}. LEED diagrams reveal the epitaxial relationship between lattices of the oxide overlayer and the metal. From STM observations, each domain in the oxide layer consists of a quasiperiodic arrangement of strictly parallel sticks. Analysis of all the results shows that each stick characterizes a small NbO crystal with a typical $3.5\ifmmode\times\else\texttimes\fi{}1.4{\mathrm{nm}}^{2}$ size. Therefore, the oxide layer can be described as a side to side arrangement of these NbO nanocrystals (fcc) on Nb(110) (bcc). Two kinds of epitaxial relationship between these two lattices are found: (i) the relative arrangement of NbO nanocrystals is determined by the underlying Nb(110) lattice; (ii) each nanocrystal develops an epitaxy relationship with the metal surface since the NbO lattice presents one (111) plane parallel to Nb(110) and one NbO 〈110〉 direction is parallel to one Nb 〈111〉 direction (Kurdjumov-Sachs-type alignment). The origin of this structure is discussed in terms of NbO/Nb misfit.

Abstract: We study the energy transport properties of one-dimensional nonlinear aperiodic lattice models in this paper. It is found that, compared with their periodic and disordered counterparts, the quasiperiodic and fractal models exhibit critical macroscopic behavior in the low-temperature region, while in the high-temperature region all the models show the same property of energy transport. The relationship between the observed macroscopic behaviors and the localization theory is discussed.

Abstract: Bloch-like surface waves associated with a quasiperiodic structure are observed for the first time in a classic wave propagation experiment which consists of pulse propagation with a shallow fluid covering a quasiperiodically drilled bottom. We show that a transversal pulse propagates as a plane wave with quasiperiodic modulation, displaying the characteristic undulatory propagation in this quasiperiodic systems and reinforcing the idea that analogous concepts to Bloch functions can be applied to quasicrystals under certain circumstances.

Abstract: The zero-temperature phase diagram of a quasiperiodic anisotropic $\mathrm{XY}$ quantum spin chain in a transverse magnetic field is studied In the isotropic limit, the system exhibits an infinite number of quantum phase transitions with a devil's-staircase-like magnetization curve Anisotropy decreases the number of transitions A mechanism is proposed for the successive transitions based on the ordering of finite clusters of spins along the chain

Abstract: Results of the strip projection method have established that formation of antiphase boundaries (APBs) is the initial stage of a periodic-to-quasiperiodic transformation in superlattices. The Al5Ti3−to-quasiperiodic superlattice transformation, observed in Al-rich γ-TiAl intermetallics, is a classic example of such a transition. Energy of the relevant APBs would, therefore, be expected to play an important role in the occurrence of this transition. In the two-dimensional Monte Carlo simulations carried out in this work, change in the APB energy, accomplished by suitably altering the pair interaction energies, was found to explain the diffraction effects observed in the alloys with various compositions depicting this transition. The results of the simulation bring out salient features of the atomic arrangements in imperfectly ordered quasiperiodic and near-quasiperiodic structures encountered in this transition.

Abstract: We study the transport of electrons in a Fibonacci magnetic superlattice produced on a two-dimensional electron gas modulated by parallel magnetic-field stripes arranged in a Fibonacci sequence. Both the transmission coefficient and conductance exhibit self similarity and the six-circle property. The presence of extended states yields a finite conductivity at infinite length, that may be detected as an abrupt change in the conductance as the Fermi energy is varied, much as a metal-insulator transition. This is a unique feature of transport in this kind of structure, arising from its inherent two-dimensional nature.

Abstract: We give a rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. More precisely, the system considered is a quantum two-level system in a time-dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewise-constant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function.

Abstract: We report the observation of an anomalous ${}^{27}\mathrm{Al}\ensuremath{-}\mathrm{NMR}$ response of a single grain ${\mathrm{Al}}_{72.4}{\mathrm{Pd}}_{20.5}{\mathrm{Mn}}_{7.1}$ icosahedral quasicrystal at low temperatures. In an external magnetic field of 6 T and upon decreasing temperature, we observe a sharp 100% increase of the resonance linewidth at 2.5 K. No further changes of the linewidth are observed down to 0.05 K. The linewidth enhancement is accompanied by a small but distinct increase of the spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}.$ These anomalies are absent in external fields of 2.5 T and below. Our observations indicate unusual variations in the stability of isolated magnetic moments in a quasiperiodic metallic environment.

Abstract: Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schrodinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states.

Abstract: The solution to the problem of detection of subsequences-fragments in a quasiperiodic sequence is presented. The case is analyzed where (1) the quasiperiodic sequence includes only identical subsequences- fragments; (2) the serial numbers of the first members (instants of time of beginning) of subsequences-frag- ments are determinate (not random) but unknown quantities; (3) the number of subsequences-fragments in the quasiperiodic sequence is unknown; (4) the quasiperiodic sequence is corrupted by the Gaussian uncorrelated noise with known variance; and (5) the boundaries of the interval of observation of the corrupted sequence do not split the first and the last subsequences-fragments of the uncorrupted quasiperiodic sequence that is hidden from observation into two parts. It is established that this problem is a specific problem of testing hypotheses of the mean of a random Gaussian vector. The efficient a posteriori computational algorithm for solving this problem is substantiated. The recurrent formulas of step-by-step discrete optimization are obtained. They ensure a decision is made on the basis of the maximum-likelihood criterion. The estimates of time and space complexity of the algorithm are given. They are related to the parameters of the problem. The results of numer- ical simulation are presented.

Abstract: 0 Introduction 1 One-Dimensional Small Divisor Problems (On Holomorphic Germs and Circle Diffeomorphisms) 1.1 Linearization of the quadratic polynomial. Size of Siegel disks 1.2 Herman rings. Differentiable conjugacy of diffeomorphisms of the circle 1.3 Gevrey classes 2 Finite-Dimensional Small Divisor Problems 2.1 Linearization of germs of holomorphic diffeomorphisms of \((\mathbb{C}^n, 0)\) 2.2 Elliptic fixed points and KAM theory 2.3 \(\mathbb{Z}^k\)-actions 2.4 Diffeomorphisms of compact manifolds 3 KAM Theory and Hamiltonian Systems 3.1 Twist maps 3.2 Euler-Lagrange flows 3.3 n-body problem 4 Linear Quasiperiodic Skew-Products, Spectral Theory and Hamiltonian Partial Differential Equations 4.1 Reducibility of skew-products 4.2 Spectral theory and integrated density of states 4.3 Nonlinear Hamiltonian PDEs References

Abstract: We study quasiperiodic solutions of the localized induction approximation in terms of the elliptic functions of Weierstrass. They describe the Kida-class motion of a thin vortex filament in an incompressible inviscid fluid. Our solution includes various filament shapes such as the vortex ring, the helicoidal filament, the plane sinusoidal filament, and the Hasimoto type-1 soliton filament.

Abstract: The magnetostatic excitation in antiferromagnetic superlattices (antiferromagnetic/nonmagnetic layered structure) grown following the Fibonacci sequence has been studied. The dispersion relations of the magnetostatic spin wave spectra and the precession amplitudes of the total magnetization in each layer are numerically obtained. The eigenfrequency spectra are divided into two branches, ${\ensuremath{\omega}}^{\ensuremath{-}}$ and ${\ensuremath{\omega}}^{+}.$ For each branch, the distribution of eigenfrequency spectra exhibits triadic Cantor-set subband structures with self-similar features. The eigenfrequency spectra distribution strongly depends on the in-plane wave vector and the thickness of antiferromagnetic and nonmagnetic layers. For most of the eigenfrequencies, especially in the triadic regions, the profiles of precession amplitudes of total magnetization in the quasiperiodic system are critical and self-similar. For the eigenfrequencies near the edges of bands, the profiles of precession amplitudes of total magnetization are extended with a sine modulation. Besides the critical and extended states, a few states at the edges of the subbands are still quasilocalized. The corresponding profiles of precession amplitudes of total magnetization either decay or oscillate with exponential attenuation from the surface into the film.

Abstract: 27Al NMR measurements of the transverse magnetization decay in a decagonal quasicrystal Al 7 2 . 6 Ni 1 0 . 5 Co 1 6 . 9 detected at temperatures between 300 and 4 K slow, low-activation-energy, diffusive atomic jumps that exhibit properties compatible with the elementary excitations of the quasiperiodic lattices in the form of phason jumps. However, identical atomic motion was observed also in "vacancy-ordered" bcc crystalline Al 5 0 Cu 3 5 Ni 1 5 , indicating that this motion is not quasicrystals specific, but a feature of close-packed atomic structures either periodic or quasiperiodic-that contain structural vacancies.

Abstract: A simple system is studied, involving a single nondispersive breaking wave and its interaction with two dispersive modes through a resonant triad. The dynamics of this system are shown to be quite rich, through a combined theoretical and numerical analysis. A sharply defined traveling wave with a corner seems to attract almost all initial data with enough energy, provided the nondispersive wave is unstable to the other two when standing alone. In other cases, the solution converges to quasiperiodic final states, unless extra symmetries force the solution to converge to simpler configurations.