Showing papers on "Fibonacci number published in 1992"

Phyllotaxis as a physical self-organized growth process

Abstract: A specific crystalline order, involving the Fibonacci series, had until now only been observed in plants (phyllotaxis). Here, these patterns are obtained both in a physics laboratory experiment and in a numerical simulation. They arise from self-organization in an iterative process. They are selected depending on only one parameter describing the successive appearance of new elements, and on initial conditions. The ordering is explained as due to the system's trend to avoid rational (periodic) organization, thus leading to a convergence towards the golden mean.

Electronic spectral and wavefunction properties of one-dimensional quasiperiodic systems: a scaling approach

Abstract: We review the results of the scaling and multifractal analyses for the spectra and wave-functions of the finite-difference Schrodinger equation: Here V is a function of period 1 and ω is irrational. For the Fibonacci model, V takes only two values (it is constant except for discontinuities) and the spectrum is purely singular continuous (critical wavefunctions). When V is a smooth function, the spectrum is purely absolutely continuous (extended wavefunctions) for λ small and purely dense point (localized wavefunctions) for λ large. For an intermediate λ, the spectrum is a mixture of absolutely continuous parts and dense point parts which are separated by a finite number of mobility edges. There is no singular continuous part. (An exception is the Harper model V (x) = cos (2πx), where the spectrum is always pure and the singular continuous one appears at λ = 2.)

Repetitions in the Fibonacci infinite word

Abstract: Soit φ le nombre d'or ; nous prouvons que le mot infini de Fibonacci ne contient pas la puissance fractionnaire d'exposant superieur a 2+φ, et nous prouvons qu'il contient des puissances d'exposant superieur a 2+φ-e, quel que soit le nombre reel e>0

Fibonacci numbers and Fermat's last theorem

Abstract: numbers. As applications we obtain a new formula for the Fibonacci quotient Fp−( 5 p )/p and a criterion for the relation p |F(p−1)/4 (if p ≡ 1 (mod 4)), where p 6= 5 is an odd prime. We also prove that the affirmative answer to Wall’s question implies the first case of FLT (Fermat’s last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.

Golden mean arithmetic in the fractal branching of diffusion-limited aggregates.

Abstract: We use the wavelet transform microscope to explore the intricate fractal geometry of Witten and Sander's diffusion-limited aggregates. We report on the discovery of Fibonacci sequences in the internal hierarchical structure of these clusters. We discuss the relevance of the golden mean arithmetic to the numerically well established statistical self-similarity of these aggregates.

Fibonacci sequence, golden ratio, and a network of resistors

Abstract: It is shown that the ratio of the effective resistance of an infinite network of identical resistors to the resistance of a constituent resistor is equal to the golden ratio.

The Linear Assignment Problem

Abstract: We present a broad survey of recent polynomial algorithms for the linear assignment problem. They all use essentially alternating trees and/or strongly feasible trees. Most of them employ Dijkstra’s shortest path algorithm directly or indirectly. When properly implemented, each has the same complexity: O(n 3) for dense graphs with simple data structures and O(n 2 log n + nm) for sparse graphs using Fibonacci Heaps.

Localization in a one-dimensional Thue-Morse chain.

Abstract: The mean resistance of a one-dimensional wire is calculated with the use of Landauer formula for three types of arrangements: the random, Thue-Morse, and Fibonacci chain for which the positions of the atoms and the scattering strengths are modulated according to the prescribed rules. Comparison of the obtained numerical results shows that for the position modulation, a Thue-Morse chain is more localized than a Fibonacci chain, while for the scattering strength modulation it is less localized. It is shown that the Thue-Morse chain can be switched from being localized to being extended when the ratio of the strength modulation to the position modulation is increased. A similar change occurs in the generalized Thue-Morse chain.

Electronic structure and localization behaviour in the GaAs/AlAs Fibonacci superlattice

Abstract: The authors study the electronic structure of the GaAs/AlAs Fibonacci superlattice using a semi-empirical sp3s* tight-binding method. They find that a self-similar energy spectrum can be seen in the band structure, although the energy spectrum depends strongly on the wavevector parallel to the layers. Furthermore, they find that a localization character is enhanced due to the band hybridization, producing a spiky density of states and a localization-like effect of the wavefunctions in the hybridized energy region. The reasons for the localization-like behaviour are discussed briefly.

Electro-optic effect and transmission spectrum in a Fibonacci optical superlattice

Abstract: A new type of Fibonacci optical superlattice is analysed. Its electro-optic effect is studied. The phase-matching concept is proposed for the first time for the Fibonacci optical superlattice. The transmission spectrum is nonself-similar owing to the dispersive effect of the refractive index. An extinction phenomenon exists provided that the thicknesses of the domains are properly selected.

Structural characterization of three-component Fibonacci Ta/Al multilayer films.

Abstract: A new class of quasiperiodic superlattice structures called three-component Fibonacci structures has been studied both theoretically and experimentally. These structures with the characteristic irrational intervals A, B, and C can be produced by the substitution rule A\ensuremath{\rightarrow}AC, C\ensuremath{\rightarrow}B, and B\ensuremath{\rightarrow}A. The projection method is applied to deal with the pattern and index of their diffraction spectrum. The analytical results are compared with the experimental one from three-component Fibonacci Ta/Al superlattices. The experimental results are in good agreement with the numerical calculations using the model for compositionally modulated multilayers. Some possible applications of these structures are discussed.

Decagonal quasicrystal tilings

Abstract: A description is given of three new distinct quasiperiodic tilings that exhibit tenfold symmetry. Related to each other by decomposition, these tilings are most easily generated from their respective Fibonacci pentagrid dual tilings. The latter, one of which is singular, are based on the property that there are three principle Fibonacci quasiperiodic sequences that possess mirror symmetry. The resulting tilings, while somewhat analogous to the Penrose tilings, are more complex in that they contain inequivalent tiles of the same shape. Associated with each of these decagonal tilings, which belong to different local isomorphism classes, are one twofold and two different fivefold tilings. Twelve inflations or deflations are necessary before each of the latter reoccurs with the same orientation.

Renormalization group of generalized Fibonacci lattices.

Abstract: We investigate the renormalization group of the generalized Fibonacci lattices associated with the aperiodic sequences as constructed by the inflation rule: {A,B} --> {A(n)B(m),A}, in which m and n are positive integers. The derived renormalization group consists of 2n(n+m-1)+1 basic renormalization-group transformations. By suitable combinations of these basic transformations, local Green's function and local density of states at any sites can be calculated for electrons on the generalized Fibonacci lattices. The off-diagonal model is employed, and the local electronic densities of states at several sites are numerically calculated for some generalized Fibonacci lattices.

The classification of 3-manifolds with spines related to Fibonacci groups

Abstract: We study the topological structure of closed connected orientable 3-manifolds which admit spines corresponding to the standard presentation of Fibonacci groups.

Surface phonons in periodic and Fibonacci Nb/Cu superlattices.

Abstract: Surface phonons in both periodic and Fibonacci Nb/Cu metallic superlattice systems have been studied by means of the Brillouin light-scattering technique. Measurement of the phase velocity of the surface Rayleigh wave enabled us to compare the elastic properties of these two types of structures with those calculated from the elastic constants of bulk Cu and Nb. The results obtained have shown that the relevant parameter for determining the elastic behavior of quasiperiodic Fibonacci superlattices is the average distance between Nb/Cu interfaces, rather than the quasiperiodicity, which is the goverbing parameter in x-ray and structural analysis

On the restoration of translational invariance in the critical quantum Ising model on a Fibonacci chain

Abstract: The spectra of quasiperiodically modulated quantum Ising chains are investigated. For moderate values of the modulation strength a unified approach, based on the lifting of the degeneracy between the left- and right-moving eigenmodes is proposed. The local scaling behaviour reflecting partial restoration of translational invariance is established. Both the conformally invariant scaling for the lower modes and the multi-fractal scaling for the higher ones are described in terms of the same quantity. A simple but accurate calculational scheme is constructed, where the rearrangement in the level spectra is due to the interaction between left- and right-moving modes of the periodic system as induced by the quasiperiodic modulation.

State transition in a family of generalized Fibonacci lattices

Abstract: We study the electronic properties of a family of generalized Fibonacci lattices associated with the sequences which are given by the inflation rule (A, B)→(ABn,A) withn>1. Analytical and numerical analyses reveal the existence of a state transition phenomenon, i.e. a transition from the local density of states with a smooth behavior in some energy regions to another one without any smooth part.

Gap-labeling properties of the energy spectrum for one-dimensional Fibonacci quasilattices.

Abstract: We study the gap-labeling properties of the energy spectrum for one-dimensional Fibonacci quasilattices. We have obtained the occupation probabilities on subbands of the hierarchical energy spectrum and the step heights of the integrated density of states. It is analytically proved that the step height is equal to {m\ensuremath{\tau}}, where the braces denote the fractional part, and m is an integer that can be used to label the corresponding energy gap. Numerical simulation confirms these results.

My collaboration with Julia Robinson

Abstract: The name of Julia Robinson cannot be separated from Hilbert’s tenth problem. This is one of the twenty-three problems stated by David Hilbert in 1900. The section of his famous address [4] devoted to the tenth problem is so short that it can be cited here in full: 10. DETERMINATION OF THE SOLVABILITY OF A DIOPHANTINE EQUATION Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

Electronic properties of the generalized Fibonacci lattices: energy spectrum and wavefunction

Abstract: The authors study the electronic properties of a one-dimensional diagonal tight-binding model with potentials (Vn) arranged in generalized Fibonacci (GF) sequences. Using the negative-eigenvalue theorem, they calculate the density of states (DOS). The DOS and the V dependence of energy spectra for silver-mean (SM) and copper-mean (CM) series clearly show distinctive features. The relation of the energy spectral feature to the geometry of the underlying lattices is emphasized. Various states of the CM lattice are examined in detail by means of wavefunctions, resistances and a multifractal analysis. Critical states are characterized both by scaling transformations and by multifractal behaviours. They find that states with a strongly localized wavefunction under a given system size exhibit additional wavepackets with increasing system size. This shows that the localized behaviour of allowed states is a feature of the finite size of the system, and implies the absence of strongly localized states in a system of infinite size.

Zeckendorf representations using negative fibonacci numbers

Abstract: It is well known that every positive integer can be represented uniquely as a sum of distinct, nonconsecutive Fibonacci numbers (see, e.g., Brown [1]. This representation is called the Zeckendorf representation of the positive integer. Other Zeckendorf-type representations where the Fibonacci numbers are not necessarily consecutive are possible. Brown [2] considers one where a maximal number of distinct Fibonacci numbers are used rather than a minimal number.

An electron diffraction and microscopy investigation of quasi-periodic Ta-Al superlattices

Abstract: Quasi-periodic (Fibonacci sequence) Ta-Al multilayer films were fabricated by magnetron sputtering, and studied by electron and X-ray diffraction. Eleven orders of electron diffraction satellite spots were obtained. Their positions and intensities were in good agreement with the data from X-ray diffraction, and both were in excellent agreement with the theoretical positions predicted by the projection method, k = 2πD −1(n + mτ). Transmission electron microscope studies of the thin film cross-sections showed the well-formed layered structures of Fibonacci sequence Ta-Al superlattices. The films have textures with Ta [110] and Al [111] in the growth direction, and coherent stacking in the quasi-periodic multilayers.