Showing papers on "Fibonacci number published in 1991"
TL;DR: It is shown that dynamically accessible lattices are characterized by pairs of consecutive Fibonacci numbers.
Abstract: The geometry of a flux lattice pinned by superconducting layers is studied. Under variation of magnetic field the lattice undergoes an infinite sequence of continuous transitions corresponding to different ways of selection of shortest distances. All possible lattices form a hierarchical structure identified as the hierarchy of Farey numbers. It is shown that dynamically accessible lattices are characterized by pairs of consecutive Fibonacci numbers.
86 citations
TL;DR: In this paper, it was shown that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the tight-binding Hamiltonian.
Abstract: Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianHψ(n)=ψ(n+1)+ψ(n−1)+λv(n)ψ(n),neℤ,ψel2(ℤ),λeℝ, wherev(n)=[(n+1)α]−[nα],[x] denoting the integer part ofx and α the golden mean\((\sqrt 5 --1)/2\), give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.
73 citations
Journal Article•
66 citations
TL;DR: The main result being the characterization of a class of morphisms whose stationary words are Sturmian, which is based on Marton Kosa's (1987) conjectures.
58 citations
TL;DR: In this paper, a hierarchical system of quasi-bifurcations of soft lattices subjected to strong compression is studied, and the deterministic "principle of maximal denominator for quasi bifurcation is proven analytically, thus explaining the universality of the appearance of Fibonacci numbers for dynamically accessible lattices.
Abstract: Soft lattices subjected to strong compression are studied. Their equilibrium states form a hierarchical system of quasi-bifurcations. Underlying SL(2, Z) ⊗ Z2 symmetry of the problem is revealed. The deterministic "principle" of maximal denominator for quasi-bifurcations is proven analytically, thus explaining the universality of the appearance of Fibonacci numbers for dynamically accessible lattices. The symmetry provides the relation between the hierarchical structure and the Cayley tree with branching number 3 found by Koch and Rothen.
55 citations
TL;DR: In this paper, the transmission spectra were experimentally studied in third sound propagation through a Fibonacci lattice, and thin aluminum films were used to configure the lattice on a glass substrate.
Abstract: Transmission spectra were experimentally studied in third sound that was propagated through a Fibonacci lattice. Thin aluminum films were used to configure the Fibonacci lattice on a glass substrat...
34 citations
31 citations
Proceedings Article•
01 Jan 1991
30 citations
01 Jan 1991
TL;DR: It is well known that for x2 − x-1 = 0, the two roots are (1 \pm \sqrt 5 )/2\), and that ==================>>\s€n} = \frac{{{L_n} \pm 1/5 {F_n})^n}€n, where Ln are the Lucas numbers and Fn the Fibonacci numbers.
Abstract: It is well known that, for x2 – x – 1 = 0, the two roots are \((1 \pm \sqrt 5 )/2\), and that
$${(\frac{{1 \pm \sqrt 5 }}{2})^n} = \frac{{{L_n} \pm \sqrt 5 {F_n}}}{2}$$
(1)
where Ln are the Lucas numbers and Fn the Fibonacci numbers. Identities (1) are called “de Moivre-type identities” [1].
28 citations
TL;DR: In this paper, the authors study tunneling of Dirac particles through Kronig-Penney models on general lattices, computing their transmission coefficient, and compare their results to the non-relativistic ones, and find a shrinkage of the spectrum similar to that of periodic systems.
TL;DR: It is shown that the normalization can be realized by means of infinite automata, and it is proved that it is possible to add to integers written in the Fibonacci numeration system of order m by Means of a finite-state automaton.
Abstract: Finite-state automata are used as a simple model of computation since only a finite memory is needed. The problem of passing from any representation to the normal representation of an integer within the Fibonacci numeration system, which is called the process of normalization, is addressed. It is shown that the normalization can be realized by means of infinite automata. More precisely, this function can be obtained by the composition of two subsequential transducers that are simply obtained from the linear recurrence definition of the basis of the Fibonacci system, one processing words from left to right and the other from right to left. The normalization, although not a sequential process, can be obtained in two sequential passes. It is proved that it is possible to add to integers written in the Fibonacci numeration system of order m by means of a finite-state automaton. The conversion from a Fibonacci representation to the standard binary representation cannot be realized by a finite-state automaton. >
01 Jan 1991
TL;DR: In this paper, the generalized Fibonacci lattices were studied using the dynamical system technique, for which the off-diagonal tight-binding model was employed, and the matrix and trace maps were obtained and investigated in a unified way.
Abstract: Electronic properties of the generalized Fibonacci lattices are studied using the dynamical system technique, for which the off-diagonal tight-binding model is employed. The matrix and trace maps are obtained and investigated in a unified way. It is found that the energy spectra are Cantor-like and the wavefunctions are critical at many energies. For some systems, it is also shown that there are extended and localized wavefunctions. In addition, according to the degree of spatial extension or localization, two other types of wavefunctions are further distinguished, of which one has the tendency to be extended and the other has the tendency to be localized.
TL;DR: Presentation d'une methode du groupe de renormalisation dans l'espace reel pour calculer les fonctions de Green locales and moyennes exactes pour les reseaux de Fibonacci generalises.
Abstract: We present a real-space renormalization-group method for calculating the exact local and average Green's functions for generalized Fibonacci lattices. The fundamental requirement of the present method is that one should be able to split the original lattice into a finite number of sublattices, all of which need not be simple scaled versions of the parrent lattice. In this method we determine exactly the renormalized local environment, up to infinite order, of any arbitrarily chosen site in the original lattice. We have also been able to classify the generalized Fibonacci chains into two groups, depending on their spectral properties. For one of the groups (silver-mean class) eigenstates are shown to be critical, whereas, the other group (copper-mean class) has some states extended. This method is readily applicable to any self-similar lattice in one and higher dimensions.
TL;DR: In this article, the generalized Fibonacci lattices were studied using the dynamical system technique, for which the off-diagonal tight-binding model was employed, and the matrix and trace maps were obtained and investigated in a unified way.
Abstract: Electronic properties of the generalized Fibonacci lattices are studied using the dynamical system technique, for which the off-diagonal tight-binding model is employed. The matrix and trace maps are obtained and investigated in a unified way. It is found that the energy spectra are Cantor-like and the wavefunctions are critical at many energies. For some systems, it is also shown that there are extended and localized wavefunctions. In addition, according to the degree of spatial extension or localization, two other types of wavefunctions are further distinguished, of which one has the tendency to be extended and the other has the tendency to be localized.
TL;DR: It is found that three kinds of branching patterns alternately appear in the spectrum, which displays a kind of self-similarity very different from the trifurcating self-Similarity of one-dimensional Fibonacci quasilattices.
Abstract: Using a decomposition-decimation method based on the renormalization-group technique, we have studied the spectral structure of a class of one-dimensional three-tile quasiperiodic lattice models, for which the (concurrent) substitution rules are S --> M, M --> L, and L --> LS, where S, M, and L represent, respectively, the short, medium, and long tiles. Branching rules for the electronic energy spectrum have been analytically obtained and confirmed by numerical simulations. It is found that three kinds of branching patterns alternately appear in the spectrum, which displays a kind of self-similarity very different from the trifurcating self-similarity of one-dimensional Fibonacci quasilattices.
TL;DR: The generic trace map is established for binary quasiperiodic lattices represented by any irrational number through a generalization of the renormalization-group method of Kohmoto, Kadanoff, and Tang for the Fibonacci lattice.
Abstract: The generic trace map is established for binary quasiperiodic lattices represented by any irrational number. This is a generalization of the renormalization-group method of Kohmoto, Kadanoff, and Tang (KKT) for the Fibonacci lattice. All the trace maps for any quasiperiodic lattice preserve the same invariant surface first discovered by KKT. There is a special set of points on the invariant surface, which we call the invariant six-cycle. These are fixed points of the trace maps. We are able to obtain representations of the trace maps or scaling transformations on the invariant six-cycle. This enables us to determine the periods of the trace maps, which are very important in order to know the scaling property of the electronic wave function and energy spectrum at the band center.
TL;DR: In this paper, a unified trace map and the Landauer resistance were obtained for a class of one-dimensional quasiperiodic lattices and the energy spectra of such lattices were analyzed.
TL;DR: It is shown that the energy spectra do not have uniform scalings and that for some systems the wave functions are extended or localized in certain energy regions.
Abstract: We study the quantum Heisenberg-Ising models on generalized Fibonacci lattices by the dynamical-maps technique, in which the nearest-neighbor Ising interactions take two values that follow successively the generalized Fibonacci sequences. The energy spectra are Cantor-like, and the wave functions are generally critical. It is further shown that the energy spectra do not have uniform scalings and that for some systems the wave functions are extended or localized in certain energy regions. In addition, we also obtain the critical lines of the quasiperiodic quantum Heisenberg-Ising models.
TL;DR: Etude de la transmission et de the reflexion d'une onde plane a travers un arrangement unidimensionnel de N potentiels de fonction δ de longueur egales situees sur une suite de Fibonacci x n =n+u [n/τ], n=1, 2,..., N, dans the limite N→∞
Abstract: Etude de la transmission et de la reflexion d'une onde plane a travers un arrangement unidimensionnel de N potentiels de fonction δ de longueur egales situees sur une suite de Fibonacci x n =n+u [n/τ], n=1, 2,..., N, dans la limite N→∞
TL;DR: In this article, the local electronic properties of a family of generalized Fibonacci lattices associated with the sequences which are given by the inflation rule (A, B) --> (AB(n),A), where n is an arbitrary positive integer greater than one.
Abstract: We study the local electronic properties of a family of generalized Fibonacci lattices associated with the sequences which are given by the inflation rule (A, B) --> (AB(n),A), where n is an arbitrary positive integer greater than one. A unified real-space renormalization-group approach is presented to calculated the local Green function and the local density of states at any given site.
TL;DR: In this paper, a new exact real-space renormalization group approach is presented for the local phonon properties of any given site in an infinite Fibonacci chain, and it is found that the local densities of states for different sites are different from each other, although they have the same structure.
Abstract: A new exact real-space renormalization group approach is presented for the local phonon properties of any given site in an infinite Fibonacci chain. It is found that the local phonon densities of states for different sites are different from each other, although they have the same structure. Results also suggest that the local phonon spectrum is a Cantor set.
TL;DR: It is proved that two infinite families of polynomials naturally associated to Fibonacci Lattices have only real zeros and this implies the log-concavity of several combinatorial sequences arising from fibonacci lattices.
Abstract: We prove that two infinite families of polynomials naturally associated to Fibonacci Lattices have only real zeros and give combinatorial interpretations to these polynomials. This, in particular, implies the log-concavity of several combinatorial sequences arising from Fibonacci Lattices and generalizes a result obtained by R. Stanley.
Journal Article•
TL;DR: Several generalizations of the standard test for an odd composite integer n to be a pseudoprime of the mth kind are considered, including the generalized Lucas numbers and the Dickson polynomials.
08 Apr 1991
TL;DR: The paper gives a general definition for the concept of strong Dickson pseudoprimes which contains as special cases the Carmichael numbers and the strong Fibonacci pseudopRimes and deduce some properties for their elements.
Abstract: The paper gives a general definition for the concept of strong Dickson pseudoprimes which contains as special cases the Carmichael numbers and the strong Fibonacci pseudoprimes. Furthermore, we give necessary and sufficient conditions for two important classes of strong Dickson pseudoprimes and deduce some properties for their elements. A suggestion of how to improve a primality test by Baillie&Wagstaff concludes the paper.