Showing papers on "Fibonacci number published in 1999"
TL;DR: The generating function also enumerates certain lattice paths, plane trees, and directed animals, giving hope that these areas of combinatorics can be applied to enumerating permutations with excluded subsequences.
115 citations
TL;DR: In this paper, the authors considered discrete one-dimensional Schrodinger operators with Sturmian potentials and proved absence of eigenvalues for all elements in the hull for a full-measure set of rotation numbers including the Fibonacci case.
Abstract: We consider discrete one-dimensional Schrodinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case, we prove absence of eigenvalues for all elements in the hull.
108 citations
TL;DR: In this paper, the Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence are discussed and discussed in the context of the FIFO sequence and its relation to the FCA.
Abstract: (1999). The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence. The American Mathematical Monthly: Vol. 106, No. 4, pp. 289-302.
79 citations
TL;DR: In this paper, the effects of the quasiperiodic modulation are intermediate between those of a commensurate potential and a disordered one, and the system exhibits a metal-insulator transition whose position depends both on the strength of the correlations and on the position of the Fermi level.
Abstract: We study analytically one-dimensional interacting spinless fermions in a Fibonacci potential. We show that the effects of the quasiperiodic modulation are intermediate between those of a commensurate potential and a disordered one. The system exhibits a metal-insulator transition whose position depends both on the strength of the correlations and on the position of the Fermi level. Consequently, the conductivity displays a power-law-like size and frequency behavior characterized by a nontrivial exponent.
75 citations
73 citations
TL;DR: In this article, the authors proposed a quasi-phase matching scheme for general quasi-periodic modulation of the nonlinear coefficient, which can be used for various nonlinear devices, such as multiple peak frequency doublers, frequency triplers, and frequency quadruplers.
Abstract: Quasi-phase-matching allows one to arbitrarily phase match a single interaction by periodic modulation of the material nonlinear coefficient. A partial extension is obtained by Fibonacci-based quasi-periodic modulation of the nonlinear coefficient. These Fibonacci-based structures allow for simultaneously phase matching two interactions, provided that their wavevector mismatch ratio obeys selection rules, which are governed by the golden ratio /spl tau/=(1+/spl radic/5)/2. In this paper, we present a novel method for simultaneously phase matching any two nonlinear interactions by general quasi-periodic modulation of the nonlinear coefficient. These quasi-periodic structures, which also include the Fibonacci-based structures as a subgroup, provide greater design flexibility. Our method can be useful for various nonlinear devices, such as multiple-peak frequency doublers, frequency triplers, and frequency quadruplers. We show for two specific devices that similar efficiency, compared to a cascaded device, can be obtained. Furthermore, in contrast to some cascaded devices, these structures can be used in double-pass and standing-wave configurations, since they operate with the same efficiency in both directions of propagation.
65 citations
TL;DR: In this article, the trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and the existence of invariants, and they illustrate the results with generalized Fibonacci chains and show that the well-known Fricke character is not the only invariant that can occur.
Abstract: Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants We illustrate the results with the generalized
Fibonacci chains and show that the well-known Fricke character
I(x,y,z) = x^2 + y^2 + z^2 - 2 x y z - 1 is not the only type of invariant that can occur We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices
65 citations
TL;DR: In this paper, the authors investigated the relationship between the spatial structure and the transport properties of critical normal modes in Fibonacci quasicrystals by using the mass ratio as a tuning parameter.
Abstract: We report on the possibility of modulating the transport properties of critical normal modes in Fibonacci quasicrystals by using the mass ratio as a tuning parameter. The relationship between the spatial structure and the transport properties of these modes is studied analytically in terms of the transmission and Lyapunov coefficients. Power spectrum analysis of the critical modes indicates a complex modulated structure in agreement with previous experimental results, [S0163-1829(99)11337-7].
64 citations
TL;DR: In this paper, the transmission properties of light through the Fibonacci-class quasiperiodic multilayers were studied and the trace map of the propagation matrices, its invariant of motion, and the expression for the transmission coefficient were obtained.
Abstract: We study the transmission properties of light through the Fibonacci-class quasiperiodic multilayers. The trace map of the propagation matrices, its invariant of motion, and the expression for the transmission coefficient are obtained. For the normal incidence of light, the transmission coefficient exhibits a switchlike property in a family of the Fibonacci-class multilayers, while a six-cycle feature in another family is found.
63 citations
TL;DR: In this article, a method that allows one to prove uniform local results for one-dimensional discrete Schrodinger operators with Sturmian potentials was introduced, where the growth rate of eigenfunctions was studied.
Abstract: In this Letter we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrodinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.
52 citations
TL;DR: In this paper, the Lyapunov constant σ(β) = limn→ ∞ | x n|1/n is determined numerically for a wide range of values β, and its dependence on β is observed to be non-smooth.
Abstract: For 0 β*, they grow exponentially. By formulating the problem as a Markov chain involving random matrix products and computing its invariant measure (a fractal) the Lyapunov constant σ(β) = limn→ ∞ | x n|1/n is determined numerically for a wide range of values β, and its dependence on β is observed to be non-smooth. (The limit is defined in the almost sure sense.) This generalizes recent work of Viswanath, who proved σ (1) = 1.131 988 24.... By a simple rescaling, these results also apply to the more general random recurrence x n+1 = α x n± β x n−1 for fixed α and β. These random recurrence relations have links with many fields, including ergodic theory, dynamical systems, heavy–tailed statistics, spectral theory, continued fractions, and condensed matter physics.
TL;DR: In this article, the authors recounted Fibonacci and Lucas Identities and proposed a method for computing the Lucas identity in the context of linear algebra. The College Mathematics Journal: Vol. 30, No. 5, pp. 359-366.
Abstract: (1999). Recounting Fibonacci and Lucas Identities. The College Mathematics Journal: Vol. 30, No. 5, pp. 359-366.
Patent•
21 Jun 1999TL;DR: In this article, an MRI system is employed to acquire image data using a pulse sequence in which k-space is sampled in a Fibonacci spiral trajectory, where a single pulse sequence may be used to sample all of kspace with a single spiral arm, or a plurality of interleaved spiral arms by performing a corresponding series of pulse sequences.
Abstract: An MRI system is employed to acquire image data using a pulse sequence in which k-space is sampled in a Fibonacci spiral trajectory. A single pulse sequence may be used to sample all of k-space with a single spiral arm, or k-space can be sampled with a plurality of interleaved Fibonacci spiral arms by performing a corresponding series of pulse sequences.
TL;DR: The role of quasi-periodic structures with respect to periodic structures, in the form of Cantor Fibonacci sets, and a ''chirped'' set, is discussed in this article.
Abstract: We discuss some of the electromagnetic properties of one-dimensional, metallo-dielectric photonic band gap structures. In our considerations, we include discussions of the transmissive and reflective properties of multilayer stacks, and the density of electromagnetic field modes for the structure. In particular, we highlight and contrast the role of quasi-periodic structures with respect to periodic structures, in the form of Cantor Fibonacci sets, and a `chirped' set.
TL;DR: In this paper, numerically the magnetic properties (magnetization and magnetoresistance) of ultrathin magnetic films (Fe/Cr) grown following the Fibonacci sequence were studied.
Abstract: We study numerically the magnetic properties (magnetization and magnetoresistance) of ultrathin magnetic films (Fe/Cr) grown following the Fibonacci sequence. We use a phenomenological model which includes Zeeman, cubic anisotropy, bilinear, and biquadratic exchange energies. Our physical parameters are based on experimental data recently reported, which contain biquadratic exchange coupling with magnitude comparable to the bilinear exchange coupling. When biquadratic exchange coupling is sufficiently large a striking self-similar pattern emerges.
Posted Content•
[...]
TL;DR: In this paper, the inverse of the n by n Filbert matrix is shown to have integer entries, and it is shown that the corresponding matrix based on Fibonomial coefficients has integer entries.
Abstract: A Filbert matrix is a matrix whose (i,j) entry is 1/F_(i+j-1), where F_n is the nth Fibonacci number. The inverse of the n by n Filbert matrix resembles the inverse of the n by n Hilbert matrix, and we prove that it shares the property of having integer entries. We prove that the matrix formed by replacing the Fibonacci numbers with the Fibonacci polynomials has entries which are integer polynomials. We also prove that certain Hankel matrices of reciprocals of binomial coefficients have integer entries, and we conjecture that the corresponding matrices based on Fibonomial coefficients have integer entries. Our method is to give explicit formulae for the inverses.
TL;DR: The use of Lyapunov exponents for evaluating localization lengths of wave functions in one-dimensional lattices is discussed in this article, where it is shown that it is more practical to calculate this length by using the scaling properties of the trace map of the transfer matrix.
Abstract: The use of Lyapunov exponents for evaluating localization lengths of wave functions in one-dimensional lattices is discussed. As a result, it is shown that it is more practical to calculate this length by using the scaling properties of the trace map of the transfer matrix. This leads to a relationship between localization and the fixed points of the map, which is considered as a dynamical system. The localization length is then defined by a Lyapunov exponent, used in the sense of chaos theory. All these results are discussed for periodic, disordered, and quasiperiodic chains. In particular, the Fibonacci quasiperiodic chain is studied in detail.
TL;DR: In this article, the effects of phason disorder on the dynamical structure factor of Fibonacci chains are studied, and the existence of a coherent phason field in real quasicrystals is addressed.
Abstract: The effects of phason disorder on the dynamical structure factor of Fibonacci chains are studied, and the existence of a coherent phason field in real quasicrystals is addressed. The neutron-scattering response is modeled for coherent and random phasons. The results show that coherent and random phasons can be distinguished for high values of the momentum transfer. However, for both sorts of phasons the response in the acoustic-mode region is quite similar, since the only important quantity is the average length between atoms. In particular, it is shown that a random phason produced in the quasicrystal’s hyperspace leads to a coherent phason field in real space. @S0163-1829~99!00622-0#
TL;DR: In this paper, the authors analyzed the order of convergence of the root-finding iteration functions of Newton's method, Halley's method and their multi-point versions, and derived the asymptotic error constant in terms of special determinants.
Abstract: Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.
TL;DR: In this article, a general theory of quantum waveguide based on transfer matrix method is developed and used to treat periodic stub structures, serial stub structures with a defect stub, and Fibonacci stub structures.
Abstract: The electronic behaviors in quantum wires with serial stubs are studied. A general theory of quantum waveguide based on transfer matrix method is developed and is used to treat periodic stub structures, serial stub structures with a defect stub, and Fibonacci stub structures. A number of interesting physical properties in connection with electronic transmission, energy spectra, and charge density distributions in these structures, are found theoretically. In particular, we find that whether there are periodicity and symmetry in the transmission and energy spectra depends on the commensurability of the length parameters. If one length ratio is incommensurate, then the transmission and energy spectra do not exhibit periodicity and symmetry even for periodic stub structures. In particular, the quasiperiodic behaviors are shown in Fibonacci stub structures proposed by us whenever the length parameters are commensurate. The experimental relevance is also addressed briefly.
TL;DR: In this paper, the optical transmission of one-dimensional binary quasiperiodic dielectric multilayers, which are arranged in Fibonacci sequences along two opposite directions and possess a mirror symmetry, was studied.
Abstract: We study numerically the optical transmission of one-dimensional binary quasiperiodic dielectric multilayers, which are arranged in Fibonacci sequences along two opposite directions and possess a mirror symmetry. We find that the transmission coefficient is unity for all sequences studied at the central wavelength = 0, where 0 = 4nA (B)dA (B), with nA (B) and dA (B) being the index of refraction and thickness of two kinds of layer, respectively. As the number of layers in the sequence increases, more and more perfect transmission peaks appear. We observe a scaling of the transmission spectra with increasing sequence length. These phenomena will find applications in fabrication of multiwavelength narrow-band optical filters.
TL;DR: For the incomplete Fibonacci and incomplete Lucas numbers, which were introduced and studied recently by P. Filliponi [Rend. Circ. Math. as discussed by the authors, 37, 56], the authors derive two classes of generating functions in terms of the familiar fibonacci numbers, respectively.
Abstract: For the incomplete Fibonacci and incomplete Lucas numbers, which were introduced and studied recently by P. Filliponi [Rend. Circ. Math. Palermo (2)45 (1996), 37–56], the authors derive two classes of generating functions in terms of the familiar Fibonacci and Lucas numbers, respectively.
TL;DR: In this article, the spin wave spectra were evaluated in the geometry where the magnetization is perpendicular to the surfaces of the layers of the superlattice by using a transfer-matrix approach.
Abstract: The magnetostatic modes are studied in multilayer structures that exhibit deterministic disorders. Some models that have attracted particular attention are the quasiperiodic magnetic multilayers that obey a substitutional sequence of the Fibonacci type. The spin wave spectra are evaluated in the geometry where the magnetization is perpendicular to the surfaces of the layers of the superlattice by using a transfer-matrix approach. Numerical results are presented for the ferromagnets EuS and Fe and for the antiferromagnet MnF2.
TL;DR: The intimate links between these two factorizations of the Fibonacci infinite word are described and a generalization to characteristic sturmian words is proposed.
TL;DR: In this article, a generalized Fibonacci-type optical superlattice, which can be generated by the concurrent inflation rule A→AmB and B→A (where m=positive integer), has been studied both theoretically and experimentally.
Abstract: Nonlinear optical characterization in a generalized Fibonacci-type optical superlattice, which can be generated by the concurrent inflation rule A→AmB and B→A (where m=positive integer), has been studied both theoretically and experimentally. Quasi-phase-matched second-harmonicgeneration spectrum and direct third-harmonic-generation with high efficiency have been measured in a LiTaO3 superlattice structure with m=2. Two different structures in real space have shown a similarity in reciprocal space.
TL;DR: This is an expository account of a constructive theorem on the shortest linear recurrences of a finite sequence over an arbitrary integral domainR and gives the associated “minimal realization” algorithm, which has a simple control structure and is division-free.
TL;DR: A closed-form solution of structure importance for each component is obtained through the relationship to the Fibonacci sequence with order k, and a complete ordering of the components with respect to their structure importance is obtained.