Showing papers on "Fibonacci number published in 1988"

Three classes of one-dimensional, two-tile Penrose tilings and the Fibonacci Kronig-Penney model as a generic case

Abstract: We generalize the Fibonacci Penrose tiling to three classes of one-dimensional, two-tile Penrose tilings which can be obtained geometrically as well as recursively. From a numerical study of their spectral properties, we conclude that the Fibonacci case has the generic features of all three classes. As a model of epitaxial quasiperiodic superlattices we consider a Fibonacci Kronig-Penney model and give a physical picture relating structural to spectral properties.

Exact decimation approach to the Green’s functions of the Fibonacci-chain quasicrystal

Abstract: Using real-space renormalization-group techniques, an exact decimation transformation has been derived for the parameters characterizing the phonon or tight-binding properties of a one-dimensional quasicrystal, the Fibonacci chain, for the discrete rescaling factor \ensuremath{\tau} (the golden mean). A Green's function formalism is presented which, when combined with the rescaling transformation, provides an a priori exact determination of the Green's functions ${G}_{\mathrm{ij}}$. Iteration of the transformation provides numerical results for the density of states \ensuremath{\rho}(E) [or \ensuremath{\rho}(${\ensuremath{\omega}}^{2}$)], and its scaling properties near the upper and lower edges of any gap have been related to the cyclic behavior of our renormalization-group mapping.

Dynamics of an electron in quasiperiodic systems. I: Fibonacci model

Abstract: Dynamics of an electron in quasiperiodic systems is studied numerically. Calculations are carried out for the one-dimensional tight-binding model with diagonal or off-diagonal modulation obeying the Fibonacci sequence. The width of the wavepacket of an electron put on a single site at time t =0 exhibits such an overall time evolution as \(\sqrt{\langle\varDelta x^{2}\rangle}\sim t^{\alpha}\) (0<α<1). The dynamical index α decreases continuously with increasing the modulation strength. This anomalous power-law diffusion is successfully interpreted in terms of renormalization group arguments in both the strong and weak modulation limits.

On the response of a two-level quantum system to a class of time-dependent quasiperiodic perturbations

Abstract: We study analytically the response of a two-level quantum system to a certain class of time-dependent quasiperiodic perturbations generated by a Fibonacci sequence. We show that the quasi-energy spectrum (Fourier transform of the evolution operator) generically is not a denumerable sum of delta functions. Hence the response is not quasiperiodic. Several numerical investigations (Poincare sections, polarization fluctuation, etc.) suggest an intermediate kind of behavior between quasiperiodic and chaotic.

Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasiperiodic superlattices.

Abstract: A theoretical study of the electronic properties of GaAs/${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Al}}_{\mathrm{x}}$As quasiperiodic Fibonacci superlattices reveals the striking differences to be expected in the band structure, wave-function localization, and optical transitions of superlattices having quasiperiodic modulation either in the barrier width or in the barrier height. We show how these differences are related to the value of the Fibonacci invariant. The results of photoluminescence excitation spectroscopy experiments are also presented.

Theory of the Raman response in Fibonacci superlattices.

Abstract: Une theorie microscopique du spectre Raman polarise des chaines de Fibonacci est developpee et appliquee aux heterostructures GaAs-AlAs. L'accord avec l'experience est satisfaisant. Le traitement fait dans l'espace reel sur un systeme non periodique au sens strict du terme possede neanmoins des proprietes associees a l'invariance par translation des cristaux

Structural and electronic properties of nonperiodic superlattices

Abstract: A brief review of the structural and electronic properties of Fibonacci, Thue-Morse, and random superlattices (SLs) is given Quasiperiodic Fibonacci SLs and 1-D analogs of quasicrystals; their wave behavior is characterized by critical hierarchy of gaps The interest in random SLs focuses on the problem of Anderson localization Thue-Morse structures are based on an automatic sequence that is not random, periodic, or quasi-periodic The structural properties of these SLs and results of X-ray and Raman scattering (RS) experiments on GaAs-Al/sub 1-x/Ga/sub x/As and other semiconductor multilayers are discussed >

Jacobsthal and pell curves

Abstract: In an earlier paper [2], a study was made of Fibonacci and Lucas curves in the plane, and their Laser-printed graphs were exhibited. These graphs were drawn from the equations of the curves, rather than from the tabulated lists of values of the Cartesian coordinates x and y9 which also served a purpose of their own. It is desirable to extend the work in [2] and so produce a more complete theory. Here, we present basic information about the corresponding curves associated with (i) Pell and Pell-Lucas numbers, and (11) Jacobsthal and Jacobs thaiLucas numbers, in our nomenclature. Curves associated with (i) will carry the generic name of Pell curves while those connected with (ii) will be designated Jacobsthal curves, There seems to be no theory related to (i) and (ii) which corresponds to the result of Halsey [1] for Fibonacci numbers. To avoid unnecessary duplication in our discussion, we will consider the numbers in (i) and (ii) (as well as the Fibonacci and Lucas numbers) to be special instances of a general sequence {wn} whose relevant properties will be investigated. Thus, the Pell and Jacobsthal curves, as well as the Fibonacci and Lucas curves, may be thought of as members of a family of curves which we shall designate as w-curves. The two Pell curves and the two Jacobsthal curves resemble the Fibonacci and Lucas curvess so we will not reproduce them here* Instead, the reader is invited to compare them in the minds eye with the curves exhibited in [2].

Nonlinear dynamics of localization in a class of one-dimensional quasicrystals.

Abstract: We consider the diagonalization of quasiperiodic operators on generalizations of the Fibonacci lattice defined recursively by $(A,B)\ensuremath{\rightarrow}({A}^{n}B,A)$. The inflation symmetry of these lattices induces a three-dimensional nonlinear dynamical map on the traces of associated transfer matrices. We find the invariant manifolds for these trace maps to be twisted and pinched versions of the Fibonacci manifold. We investigate the effect of these pinches on the spectrum of a tight-binding Hamiltonian and consider the limit of weak incommensurability: $n\ensuremath{\rightarrow}\ensuremath{\infty}$.

Optical multistability in a nonlinear Fibonacci multilayer

Abstract: The transmission properties of a multilayered medium consisting of $N$ nonlinear slabs are studied. A general characteristic matrix formalism is applied to obtain the power dependence of the transmission coefficient. As an application, a nonlinear Fibonacci multilayer with as many as 55 and 233 nonlinear slabs is considered. The nonlinear quasiperiodic system is shown to offer a wide variety of bistable and multistable operations.

Angular dependence of phonon transmission through a Fibonacci superlattice

Abstract: Phonon imaging is employed to examine the propagation of acoustic phonons through a Fibonacci superlattice. Ballistic transmission of phonons with \ensuremath{ u}g850 GHz through 750 superlattice interfaces is detected. In addition, sharp variations in the phonon intensity with propagation angle are observed. These measurements are consistent with Monte Carlo simulations presented in this paper. Distinct stop bands are expected theoretically, and the angular dependence of these structures is remarkably similar to those predicted for a periodic superlattice.

Tables of Fibonacci and Lucas factorizations

Abstract: We list the known prime factors of the Fibonacci numbers Fn for n < 999 and Lucas numbers Ln for n < 500. We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject.

Quasiperiodic dynamics for a generalized third-order Fibonacci series.

Abstract: A simple quasiperiodic arrangement made up of three building blocks is introduced in terms of a third-order Fibonacci series. We obtain a dynamical system for the series of transfer matrices for the one-dimensional Schr\"odinger equation and derive the reduced dynamical map of the traces. This map has some unique features compared with the second-order Fibonacci series.

X-Ray Diffraction Patterns of Configurational Fibonacci Lattices

Abstract: Four types of the configurational Fibonacci lattices have been grown by molecular beam epitaxy. The golden mean τ in a present Fibonacci lattice is not associated with the spacing between the alternating layer but is associated with the species of atomic layers or the hexagonal-cubic packing sequence. Many satellite peaks are observed having a self-similarity feature when the instrumental resolution of the X-ray diffraction is extremely high. Special attention was paid for the ABC stacking Fibonacci sequence to compare with the structure of stacking fault observed in a metal alloy such as MgCuAl.

A probabilistic primality test based on the properties of certain generalized Lucas numbers

Abstract: After defining a class of generalized Fibonacci numbers and Lucas numbers, we characterize the Fibonacci pseudoprimes of the mth kind.In virtue of the apparent paucity of the composite numbers which are Fibonacci pseudoprimes of the mth kind for distinct values of the integral parameter m, a method, which we believe to be new, for finding large probable primes is proposed. An efficient computational algorithm is outlined.

Discrete mathematics with algorithms

Abstract: Partial table of contents: SETS AND ALGORITHMS: AN INTRODUCTION. Binary Arithmetic and the Magic Trick Revisited. Algorithms. Set Theory and the Magic Trick. Set Cardinality and Counting. ARITHMETIC. Exponentiation: A First Look. Three Inductive Proofs. How Good Is Fast Exponentiation? The ''Big Oh'' Notation. ARITHMETIC OF SETS. Binomial Coefficients. Permutations. The Binomial Theorem. NUMBER THEORY. Greatest Common Divisors. The Euclidean Algorithm. Fibonacci Numbers. Congruences and Equivalence Relations. An Application: Public Key Encryption Schemes. GRAPH THEORY. Building the LAN. Graphs. Trees and the LAN. Graphical Highlights. Index.

On a Problem of Diophantus

Abstract: Long ago Diophantus of Alexandria [4] noted that the numbers 1/16, 33/16, 68/16, and 105/16 all have the property that the product of any two increased by 1 is the square of a rational number. Much later, Fermat [5] notes that the product of any two of 1, 3, 8, and 120 increased by 1 is the square of an integer. In 1969, Davenport and Baker [3] showed that if the integers 1, 3, 8 and c have this property then c must be 120. From this it follows that there does not exist an integer d dif f erent from 1, 3, 5, and 120 such that the five numbers 1, 3, 8, 120, and d have the same property. In 1977, Hoggatt and Bergum [7] noted that 1 = F2, 3 = F4, 8 = F6 and 120 = 4•2 •3 •5 = 4F1F3F5 where Fn is the nth Fibonacci number, and were led to guess that the numbers F2n, F2n+2, F2n+4, and 4F2n+1F2n+2F2n+3 possessed this same property for every n ≧ 1. Moreover, since 1, 2, and 5 have the property that the product of any two decreased by 1 is the square of an integer and 1 = F1, 2 = F3, and 5 = F5, they guessed that there must exist an integer y such that the product of any two of 1, 2, 5, and y decreased by 1 must be a perfect square. More generally, they guessed that for every n ≧0 there must exist an integer yn such that the product of any two of F2n+1, F2n+3, F2n+5, and yn decreased by 1 must be a perfect square. Their guesses were only partly correct; however, they were able to prove the following theorems.

On the energy spectra of one-dimensional quasi-periodic systems

Abstract: The Cantor set character of the electronic energy spectrum of a one-dimensional quasi-crystals in the form of a Fibonacci sequence is analysed under the tight-binding approximation by means of a renormalisation procedure. The electronic structure of quasi-periodic superlattices is also studied with the conclusion that there scarcely exists any notable difference from periodic superlattices.

Universality class of a Fibonacci Ising model

Abstract: The specific heat of a certain ferromagnetic Fibonacci Ising model is shown to have a logarithmic singularity.

Fibonacci Numbers of the Forms PX2 ± 1, PX3 ± 1, Where P is Prime

Abstract: Let m denote a non-negative integer, Fm the mth Fibonacci number, p a prime. Fibonacci numbers of the forms x2, 2x2, x2+1, x2-1, px2, x3, 2x3, px3, p2x3, x3 ±1 have been studied by J. H. E. Cohn [1], R. Finkelstein [3], H. C. Williams [9], the author [7], [8], A. Petho [61, H. London & R. Finkelstein [5], and J. C. Lasarias & D. P. Weisser [4]. In this article, we find all solutions to each of the four equations: $$ {F_m} = p{x^2} + 1 $$ (A) $$ {F_m} = p{x^2} - 1 $$ (B) $$ {F_m} = p{x^3} + 1 $$ (C) $$ {F_m} = p{x^3} - 1 $$ (D)

Localization-delocalization transition for one-dimensional alloy potentials

Abstract: One-dimensional alloy potentials are characterized by their complexity σ which is a generalized Shannon entropy. For the potentials having σ > 1 localization of eigenstates is established. Evidence is given for localization-delocalization transition at σ = 1. Critical properties of Fibonacci potentials are explained.

Primes Having an Incomplete System of Residues for a Class of Second-Order Recurrences

Abstract: Shah [4] and Bruckner [1] showed that if p is a prime and p > 7, then the Fibonacci sequence {Fn} has an incomplete system of residues modulo p. Shah established this result for the cases in which p = 1, 9, 11, or 19 modulo 20, while Bruckner proved the result true for the re ma ini n g c ases in which p = 3 or 7 modulo 10. Burr [2] extended these results by dete rmining all the positive integers m for which the Fibonacci sequence has an incomplete system of residues modulo m.

Crystallography of spiral lattices

Abstract: Spiral lattices, beautifully illustrated in composite flowers such as daisies and discussed in a recent letter by Bursill, Peng and Fan, have concentric crystalline grains separated by quasicrystalline grain boundaries. They are generated by a simple algorithm parametrized by one golden (or noble) angle. The topology of the patterns and their crystallographic properties (floret shape, lattice planes) are set by Voronoi construction. Golden or noble angle, and thus Fibonacci or Lucas numbers of visible spirals in each grain, correspond to maximal uniformity of the pattern so that their ubiquity in nature is the consequence of a variational principle. Existing results are reviewed here, which answer some questions raised by Bursill et al.

More on the Problem of Diophantus

Abstract: Recently, a number of articles have appeared in the literature which deal with finding a set of four numbers such that the product of any two different members when increased by n is a perfect square, [1] to [3] and [6] to [9].

Asveld’s Polynomials pj(n)

Abstract: In this paper we generalise the results of Asveld [1] for Gn. His method and notation will be followed where practicable. Our main objective is to extend his polynomials pj(n) for generalised Fibonacci numbers.