Showing papers on "Fibonacci number published in 1989"

Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

Abstract: It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H mn=δ m, n+1+δ m, n−1+δ m, n μ[(n+1)α]−[nα]) where α=(√5−1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzeroμ, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.

Multifractal wave functions on a Fibonacci lattice

Abstract: Analyse des fonctions d'onde sur un reseau de Fibonacci (a partir) du point de vue multifractal. Obtention, exactement, d'une fonction d'entropie [f(α)] representant la distribution de densite de probabilite d'une particule pour cet etat au centre du spectre. Presentation de calculs numeriques pour les autres etats. Une analyse des proprietes d'echelle de taille finie montre que les fonctions d'onde critiques ordonnees sont multifractales

The golden section in the measurement theory

Abstract: The main ideas and results of the algorithmic measurement theory, i.e. the constructive trend in the mathematic measurement theory are discussed. The theory in question dates back to the problem of choosing the best system of scale weights (Fibonacci, thirteenth century). The theory uses the asymmetry principle of measurement and is connected with the brilliant mathematical achievements, viz, the golden section, the Fibonacci numbers and Pascal's triangle. New methods of number representation, i.e. the Fibonacci and the golden ratio codes have been developed. They are used in computer engineering.

Electronic properties of the tight-binding Fibonacci Hamiltonian

Abstract: The analysis of the electronic properties of the tight-binding Fibonacci Hamiltonian is carried out using dynamical systems techniques. Two classes of Fibonacci sequences are considered, corresponding to the cases when there are two types of building blocks and also when there are three types of building blocks. The recursion relations for the traces of the transfer matrices are determined and studied for various extensions of the Fibonacci case with the golden mean. Some differences are obtained between the various types of second-order Fibonacci sequences and are most likely due to long-range order. Applications of this work to the transmission of light through a multilayered medium and the electrical resistance of a one-dimensional quasicrystal, as determined by the Landauer formula, are presented.

Phyllotaxis or the properties of spiral lattices. - II. Packing of circles along logarithmic spirals

Abstract: Phyllotaxis can be identified with the study of spiral lattices which are useful as models for many botanical structures (arrangements of the inner florets of a daisy, of the scales of a pineapple...). We consider a geometrical idealization of such networks : a lattice of tangent circles aligned along a logarithmic spiral. using conditions for close-packing of such circles, we show that the parastichy numbers belong to a generalized Fibonacci sequence. Moreover, if « regular » parastichy transitions only occur in the lattice, the divergence tends to a noble number. On the contrary a rational number is reached after an infinite sequence of singular transitions.

Ultrasonic spectrum in Fibonacci acoustic superlattices.

Abstract: A new type of superlattice, a Fibonacci acoustic superlattice, is presented. Its ultrasonic spectrum, which is excited by the piezoelectric effect, has been studied theoretically and experimentally. The results are in good agreement with each other.

Uniqueness in Finite Measurement

Abstract: This article surveys recent investigations of real sequences (d 1,..., d n ) which arise in the theory of measurement from considerations of uniqueness for numerical representations of qualitative relations on finite sets. The sequences we discuss arise from measurement problems which include measurement of subjective probability, extensive measurement, difference measurement, and additive conjoint measurement. The measurement problems lead to sequences with fascinating combinatorial and number-theoretic properties.

Attractors of some volume-nonpreserving Fibonacci trace maps

Abstract: On montre qu'un role important est joue par certains pseudo-invariants dans le comportement des attracteurs. Sur la base de la presence de ces attracteurs, on illustre la coexistence de spectres d'excitation reguliers et de type cantor

Dynamic structure factor for the Fibonacci-chain quasicrystal

Abstract: We have performed an exact real-space renormalization-group calculation of the complete wave-vector- and frequency-dependent-response function, S(q,\ensuremath{\omega}), for the one-dimensional Fibonacci-chain quasicrystal with ``Goldstone'' dynamics as in the case of phonons or magnons We present surface plots, which are highly structured, of the full response S(q,\ensuremath{\omega}), for different values of the coupling ratio In addition, we have obtained a hierarchy of dispersion curves of \ensuremath{\omega} versus q which contain features related to the gap structure of the excitation spectrum The limiting case of equal couplings, while still nontrivial because of geometric effects, is analytically tractable and gives a useful test of the renormalization-group treatment, of which it is a special case, and when extended using degenerate perturbation theory it provides an interpretation of the general situation

Energy spectra for one-dimensional quasiperiodic potentials: bandwidth, scaling, mapping and relation with local isomorphism

Abstract: Energy spectra for one-dimensional tight-binding models with two types of quasiperiodic potentials are studied, for which the incommensurability is characterised by quadratic irrationals. One is the step potential model, for which the structure is a generalised Fibonacci chain. For special structures, scaling properties of the spectrum are found numerically; a critical index delta for the total bandwidth is determined. After deriving recursion relations for the trace of transfer matrices, it is shown that generalised Fibonacci chains have the same energy spectrum if and only if they are locally isomorphic. The other potential is sinusoidal, for which the critical index delta is determined at the critical point.

Renormalization group for a quasiperiodic Schrödinger operator

Abstract: The real-space renormalization group for a generalized Fibonacci Hamiltonian is constructed. The spectrum is shown to have the hierarchical structure of a zero-measure Cantor set guided by the continuous fraction representation of the incommmensurate frequency ϖ of the problem. The fractal properties of the spectrum are discussed Nous construisons un groupe de renormalisation dans l'espace reel pour hamiltonien de Fibonacci generalise. Nous montrons que le spectre a la structure hierarchique d'un ensemble de Cantor de mesure nulle lie a la representation en fraction continue de la frequence incommensurable du probleme. Nous discutons des proprietes fractales du spectre

Nature of the Quasi-Energy Spectrum of Quasi-Periodically Driven N-Level Quantum Systems

Abstract: Extending recent work on two-level systems by Luck, Orland and Smilansky, we show that the Fourier transform of the time evolution operator of N-level systems, driven by a quasi-periodic external force generated from a Fibonacci sequence is not, in general, a denumerable set of ?-functions, implying that the dynamics is not quasi-periodic. The possibility of a mixed discrete and continuous spectrum for N???3 is pointed out.

Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs

Abstract: Fibonacci numbers are studied with respect to the topological theory of benzenoid hydrocarbons. These numbers are identified as the number of Kekule structures of nonbranched all-benzenoid hydrocarbons, the number of matchings of paths, the number of independent sets of vertices of paths, the number of nonattacking rooks of certain rook boards, as well as the number of Clar structures of certain benzenoid hydrocarbons. Fibonacci numbers were also identified as the number of conjugated circuits of certain benzenoid hydrocarbons and thus they were also related to the structure-resonance model. Maximal independent sets of caterpillar trees are also shown to be Fibonacci numbers.

Fractal nature of the electronic structure of a Penrose tiling lattice in a magnetic field

Abstract: The one-electron energy spectrum of a Penrose tiling lattice in a magnetic field is studied with a tight-binding Hamiltonian. We show the following remarkable results characteristic of a Penrose lattice. (1) The density of states in a magnetic field has a central peak with zero width at zero energy. It is shown that the zero-energy states correspond to the ring states in which wavefunction has nonvanishing amplitudes only at the sites circling the origin. (2) The magnetic field dependence of the energy spectrum shows a butterfly shape caused by Landau quantization. (3) The magnetic field dependence of the energy spectrum also shows a fractal nature. In particular it is characterized by two periods whose ratio is equal to the golden mean \((1+\sqrt{5})/2\), and two periods comprising a Fibonacci sequence. We have clarified the origin of this fractal behavior of the energy spectrum analytically.

Magnetic levels in quasiperiodic superlattices.

Abstract: With numerical calculations it is shown that in a Fibonacci superlattice, in a magnetic field parallel to the layers, the self-similarity in the length scale is reproduced in the energy-level structure. In particular, the cyclotron-orbit-center dispersion of the energy levels shows a field-dependent structure which is the same at values of the magnetic field, which are related by integer powers of \ensuremath{\tau}, the golden mean [(\ensuremath{\surd}5+1)/2]. The conditions for which this effect can be observed are discussed.

Reflection of soft x-rays and extreme ultraviolet from a metallic Fibonacci quasi-superlattice

Abstract: The reflectivity of TE soft x-rays and extreme ultraviolet from a metallic Fibonacci quasi-superlattice is investigated Based on the hydrodynamic model and by using the transfer matrix method we obtained a formal expression of the reflectivity for the case of a metallic multilayer in the form of a Fibonacci sequence The numerical results for reflectivities are shown for a set of multilayers with finite generation number Besides the interesting self-similar pattern of the reflectance peaks, the numbers of which are arranged just corresponding the Fibonacci sequence, we find that the stronger reflectance peak larger height and width moves to a higher-frequency region compared with the usual standard superlattice, which stimulates interest in the study and making of soft x-rays and extreme ultraviolet reflectors