Showing papers on "Fibonacci number published in 1996"
TL;DR: In this paper, the formation of primordia is shown to result from successive iterations of two possible simple dynamical systems, and it is demonstrated that this system gives rise to the self organization of the spiral phyllotactic structures.
191 citations
TL;DR: In this paper, it was shown that when each trailer is perpendicular with the previous one the degree of nonholonomy is F n+3 (the (n+3)-th term of the Fibonacci's sequence) and when no two consecutive trailers are perpendicular this degree is n+2.
Abstract: In this paper we study the problem of the car with N trailers. It was proved in previous works ([9], [12]) that when each trailer is perpendicular with the previous one the degree of nonholonomy is
F n+3 (the (n+3)-th term of the Fibonacci's sequence) and that when no two consecutive trailers are perpendicular this degree is n+2. We compute here by induction the degree of non holonomy
in every state and obtain a partition of the singular set by this degree of non-holonomy. We give also for each area a set of vector fields in the Lie Algebra of the control system wich makes a basis of the tangent space.
97 citations
TL;DR: This Letter shows analytically that a subset of the CWFs belonging to general Fibonacci systems are extended from a physical point of view, which widens the notion of extended wave function to include electronic states which are not Bloch functions, and it is a relevant first step to clarify the precise manner in which the quasiperiodic order of fibre systems influences their transport properties.
Abstract: We report on a new class of critical states in the energy spectrum of general Fibonacci systems. By introducing a transfer matrix renormalization technique, we prove that the charge distribution of these states spreads over the whole system, showing transport properties characteristic of electronic extended states. Our analytical method is a first step to find out the link between the spatial structure of critical wave functions and their related transport properties.
92 citations
TL;DR: In this paper, the diffusion properties of initially localized wave packets on the Fibonacci chain were derived and a relation between generalized diffusion exponents and fractal dimensions of the energy spectrum was established.
Abstract: In a perturbative limit, we derive the diffusion properties of initially localized wave packets on the Fibonacci chain. We establish a new relation between generalized diffusion exponents and fractal dimensions of the energy spectrum. We give an argument extending in general to other one dimensional quasiperiodic systems. An illustration is given taking the case of the Harper model.
91 citations
TL;DR: An extension of the Gilbert-Pearson theory of subordinacy is presented, which relates dimensional Hausdorff spectral properties of one-dimensional Schrodinger operators to the behavior of solutions of the correspondingSchrodinger equation.
Abstract: We present an extension of the Gilbert-Pearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of one-dimensional Schrodinger operators to the behavior of solutions of the corresponding Schrodinger equation. We use this theory to analyze these properties for several examples having the singular-continuous spectrum, including sparse barrier potentials, the almost Mathieu operator and the Fibonacci Hamiltonian.
57 citations
TL;DR: The properties of a series of periodic two-dimensional lattices built from pentagons, called Fibonacci pentilings, which resemble aperiodic Penrose tilings, are characterized and computed diffraction patterns from periodic pentilings with moderate size unit cells show decagonal symmetry.
Abstract: To demonstrate that crystallographic methods can be applied to
index and interpret diffraction patterns from well-ordered
quasicrystals that display non-crystallographic 5-fold symmetry, we
have characterized the properties of a series of periodic
two-dimensional lattices built from pentagons, called Fibonacci
pentilings, which resemble aperiodic Penrose tilings. The computed
diffraction patterns from periodic pentilings with moderate size unit
cells show decagonal symmetry and are virtually indistinguishable from
that of the infinite aperiodic pentiling. We identify the vertices and
centers of the pentagons forming the pentiling with the positions of
transition metal atoms projected on the plane perpendicular to the
decagonal axis of quasicrystals whose structure is related to
crystalline η phase alloys. The characteristic length scale of the
pentiling lattices, evident from the Patterson (autocorrelation)
function, is ∼τ 2 times the pentagon edge length,
where τ is the golden ratio. Within this distance there are a finite
number of local atomic motifs whose structure can be
crystallographically refined against the experimentally measured
diffraction data.
57 citations
02 Sep 1996
TL;DR: A characterization of Sturmian morphisms as conjugates of Standard ones is given, to prove that a morphism f generates an infinite word having the same set of factors as a characteristic word generated by a Standard morphism g if and only if f is a conjugate of g.
Abstract: Let A={a, b} be an alphabet. An infinite word on A is Sturmian if it contains exactly n+1 distinct factors of length n for every integer n. A morphism f on A is Sturmian if f(x) is Sturmian whenever x is. A morphism on A is Standard if it is an element of the monoid generated by the two elementary morphisms E, which exchanges a and b, and φ, the Fibonacci morphism defined by φ(a)=ab and φ(b)=a. The set of Standard morphisms is a proper subset of the set of Sturmian morphisms. In the present paper, we give a characterization of Sturmian morphisms as conjugates of Standard ones. Sturmian words generated by Standard morphisms are characteristic words. The previous result allows to prove that a morphism f generates an infinite word having the same set of factors as a characteristic word generated by a Standard morphism g if and only if f is a conjugate of g.
48 citations
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TL;DR: In the thirty years since it was proved that 0, 1 and 144 were the only perfect squares in the Fibonacci sequence, several generalisations have been proved, but many problems remain this paper.
Abstract: In the thirty years since it was proved that 0, 1 and 144 were the only perfect squares in the Fibonacci sequence [1, 9], several generalisations have been proved, but many problems remain. Thus it has been shown that 0, 1 and 8 are the only Fibonacci cubes [6] but there seems to be no method available to prove the conjecture that 0, 1, 8 and 144 are the only perfect powers.
47 citations
01 Jan 1996
TL;DR: In fact, all trigonometric identities can be derived from the basic identity sin2x cos2x = 1 as discussed by the authors, which is the same as the identity used in this paper.
Abstract: Methods for manipulating trigonometric expressions, such as changing sums to products, changing products to sums, expanding functions of multiple angles, etc., are well-known [1], In fact, the process of verifying trigonometric identities is algorithmic (see [2] or [5]). Roughly speaking, all trigonometric identities can be derived from the basic identity sin2x cos2x = 1.
42 citations
TL;DR: It is proved that a complete tree with n nodes can be drawn in n + O(log n√n) area, and a Fibonacci tree withn nodes can been drawn in 1.17n + O (log n √n), which is close to the area requirements for strictly-upward straight-line grid drawings of complete and Fibonca trees.
30 citations
TL;DR: It is demonstrated that periodic coherent-field-induced oscillations (Bloch oscillations) are replaced in Fibonacci superlattices by more complex oscillations displaying quasiperiodic signatures, thus shedding more light onto the very peculiar nature of the electronic states in these systems.
Abstract: We present a detailed study of the dynamics of electronic wave packets in Fibonacci semiconductor superlattices, both in flat band conditions and subject to homogeneous electric fields perpendicular to the layers. Coherent propagation of electrons is described by means of a scalar Hamiltonian using the effective-mass approximation. We have found that an initial Gaussian wave packet is filtered selectively when passing through the superlattice. This means that only those components of the wave packet whose wave numbers belong to allowed subminibands of the fractal-like energy spectrum can propagate over the entire superlattice. The Fourier pattern of the transmitted part of the wave packet presents clear evidences of fractality reproducing those of the underlying energy spectrum. This phenomenon persists even in the presence of unintentional disorder due to growth-induced defects. Finally, we have demonstrated that periodic coherent-field-induced oscillations (Bloch oscillations), which we are able to observe in our simulations of periodic superlattices, are replaced in Fibonacci superlattices by more complex oscillations displaying quasiperiodic signatures, thus shedding more light onto the very peculiar nature of the electronic states in these systems. \textcopyright{} 1996 The American Physical Society.
TL;DR: In this paper, the main properties of the Fibonacci and Lucas numbers are established and congruence properties of L n(k) are studied, leading to a supposedly new characterisation of prime numbers.
Abstract: A particular use of well-known combinatorial expressions for Fibonacci and Lucas numbers gives rise to two interesting classes of integers (namely, the numbersF n(k) andL n(k)) governed by the integral parametersn andk. After establishing the main properties of these numbers and their interrelationship, we study some congruence properties ofL n(k), one of which leads to a supposedly new characterisation of prime numbers. A glimpse of possible generalisations and further avenues of research is also caught.
TL;DR: The results concerning the embedding of trees into recursive circulants are presented and dilation 1 embeddings of Fibonacci trees and full quaternary trees in G are presented.
TL;DR: It is shown that the presence of a single impurity affects all the states of the unperturbed system, reducing the fractal dimension of the spectrum.
Abstract: The effect of substitutional disorder in a Fibonacci chain is studied. In particular it is shown that the presence of a single impurity affects all the states of the unperturbed system, reducing the fractal dimension of the spectrum. Resonant eigenstates are also observed. The consequences of the instability of the spectrum are discussed in the context of the experimental electronic measurements and also the effect of boundary conditions in theoretical calculations.
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TL;DR: In this article, a two-sided analogue of Erdos measure on the 2-torus is defined and the transformation (goldenshift) preserving both Erdos and Lebesgue measures is constructed.
Abstract: We define a two-sided analog of Erdos measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdos measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdos and Lebesgue measures on $\Bbb T^2$ which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdos measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
TL;DR: In this paper, a generalization of the Fibonacci numbers was considered, motivated by the relationship of the Hosoya-Z topological index to the fibonacci number.
Abstract: We consider a generalization of Fibonacci numbers that was motivated by the relationship of the HosoyaZ topological index to the Fibonacci numbers. In the case of the linear chain structures the new higher order Fibonacci numbers
h
F
n
are directly related to the higher order Hosoya-typeZ numbers. We investigate the limitsF
n
/F
n−1
and the corresponding equations, the roots of which allow one to write a general expression forhFn. We also report on the
h
F counting polynomials that give the partition of the
h
F numbers in contributions arising fromk pairs of disjoint paths of lengthh. It is interesting to see that the partitions of
h
F are “hidden” in the Pascal triangle in a similar way to the partitions of the Fibonacci numbers that were discovered some time ago by Hoggatt. We end with illustrations of the recursion formulas for the higher order Hosoya numbers for several families of graphs that are based on the corresponding recursions for the higher Fibonacci numbers.
TL;DR: In this paper, the degenerate perturbation theory was used to study the electronic energy spectra of one-dimensional general two-component Fibonacci lattices, the 3CF lattice, and its generalizations [three-component silver mean, three-component copper mean (3CCM), and the 3CTM lattice in the strong modulation regime.
Abstract: We use degenerate perturbation theory to study the electronic energy spectra of one-dimensional general two-component Fibonacci lattices, the three-component Fibonacci (3CF) lattice, and its generalizations [three-component silver mean (3CSM), three-component copper mean (3CCM)], and the three-component Thue-Morse (3CTM) lattice in the strong modulation regime. To first order in the small parameter t/V, where t is a hopping integral and V is the magnitude of the on-site energy, we obtain the six and five global subband structures of the two-component silver-mean and copper-mean lattices, and the six, seven, seven, and nine global subband structures of the 3CF, 3CSM, 3CCM, and 3CTM lattices, respectively. Further splitting of the subbands appeared when considering the second-order correction. The quantitative predictions for the density of states of these lattices are in good agreement with the numerical results obtained by exact diagonalization. \textcopyright{} 1996 The American Physical Society.
TL;DR: The self-avoiding walk, restricted to a strip, is considered in the context of linguistic combinatorics in this paper, where it is considered as a form of self-defense.
01 Jan 1996
TL;DR: In this article, the authors studied the distributions of integrals of Gaussian processes arising as limiting distributions of test statistics proposed for treating a goodness of fit or symmetry problem and showed that the cumulants of the distributions can be expressed in terms of Fibonacci numbers and Lucas numbers.
Abstract: We study the distributions of integrals of Gaussian processes arising as limiting distributions of test statistics proposed for treating a goodness of fit or symmetry problem. We show that the cumulants of the distributions can be expressed in terms of Fibonacci numbers and Lucas numbers.
01 Jan 1996
TL;DR: In this paper, the question of whether there exists a monochromatic s-term Fibonacci sequence (s ≥ 3) with Rado number RaF(k,s) was posed.
Abstract: In every k-coloring of the first N natural numbers (k ≥ 2), does there exist a monochromatic s-term Fibonacci sequence (s ≥ 3), that is, a sequence f 1 ,f 2 ,…,f 8 where f n + 2 = f n + 1+ f n , n ≥ 1, and f1,f2 are arbitrary natural numbers? In case of existence, the smallest N with this property is called Rado number RaF(k,s) for Fibonacci sequences. For s = 3 the numbers RaF(k, 3) are known as Schur numbers [9]. A similar question for Rado numbers of second order linear recurrences was posed by S. Rabinowitz [12].
TL;DR: In this article, a theoretical calculation of the Raman scattering cross section for plasmonpolaritons in a quasiperiodic superlattice that follows the Fibonacci sequence was carried out.
TL;DR: The enhanced Fibonacci cube (EFC) structure for parallel systems is proposed and it is shown that the parallel prefix sum computation can be efficiently implemented on enhanced fibonacci cubes.
Abstract: We propose the enhanced Fibonacci cube (EFC) structure for parallel systems. It is defined based on the sequence F n = 2F n-2 + 2F n-4 . We show that the enhanced Fibonacci cube contains the Fibonacci cube (FC) as a subgraph and maintains virtually all the desirable properties of the Fibonacci cube. In addition, it is a Hamiltonian graph. We can embed complete binary trees into enhanced Fibonacci cubes with dilation one and with a relatively small expansion. We also propose a series of enhanced Fibonacci cubes EFC (k) , where k is a series number. Each EFC (k) contains an FC of the same order as a subcube. Moreover, each EFC (k) in the series contains any other cube that precedes it as subcubes and the last one in the series is a hypercube of the corresponding order. This series of EFC (k) s provides us with more options for selecting cubes with various sizes. Because EFC is a subgraph of the hypercube, it may find applications in fault-tolerant computing for degraded hypercube computer systems. As an application of EFC, we show that the parallel prefix sum computation can be efficiently implemented on enhanced Fibonacci cubes.
01 Jan 1996
TL;DR: In this paper, all prefixes of Fibonacci words and the subwords of the golden sequence that are of the same length were determined, and the basic structure and properties of these prefixes have been studied.
Abstract: A word w is called an nth order Fibonacci word derived from a pair (a,b) of distinct letters if there exists a finite sequence w1,w2,…,w n of words with w1 = a, w2 = b, wn = w and each wk equals wk−1wk−2 or wk−2wk−1, 3 ≤ k ≤ n. The basic structure and properties of Fibonacci words have been studied in [2–6]. In this paper, we determine all the prefixes of Fibonacci words and the subwords of the golden sequence that are of Fibonacci lengths.
25 Aug 1996
TL;DR: It is shown that this canonical representation of Fibonacci numeration system can be obtained from any representation by a rational function, which is the composition of two subsequential functions that are simply obtained from the system.
Abstract: In the Fibonacci numeration system of order m (m integer ≥2), every integer has a unique canonical representation which has no run of m consecutive l's. We show that this canonical representation can be obtained from any representation by a rational function, which is the composition of two subsequential functions that are simply obtained from the system. The addition of two integers represented in this system can be performed by a subsequential machine. The conversion from a Fibonacci representation to a standard binary representation (or conversely) cannot be realized by a finite-state machine.
01 Jan 1996
TL;DR: In this article, the first few positive values of the integer m were investigated from the point of view of the Zeckendorf decomposition of nFn, and the identity of the Fibonacci-Lucas product was established by proving conjectures based on early cases of A, k, and n.
Abstract: The decomposition of any positive integer N as a sum of positive-subscripted, distinct, nonconsecutive Fibonacci numbers Fk is commonly referred to as the Zeckendorf decomposition ofN (ZD of N, in brief) [10]. This decomposition is always possible and, apart from the equivalent use of Fx instead of F2 (or vice-versa), is unique [8]. In the past years sequences of integers {alb}, where a and b are certain Fibonacci and/or Lucas numbers (Lk), have been investigated from the point of view of the ZD of their terms (e.g., see [3], [4], [5]). The aim of this paper is to extend these studies to sequences {ab}. More precisely, in Section 2 we establish the ZD of mFhFk and ml^^, with h and k arbitrary positive integers (possibly subject to some trivial restrictions), for the first few positive values of the integer m; the ZD of FhLk,FJ?Lk, and FhI?k are also found. In Section 3, after some brief considerations on the ZD of nFn, we analyze certain Fibonacci-Lucas products that emerge from particular choices of n. All the identities presented in this paper have been established by proving conjectures based on behavior that became apparent through the study of early cases of A, k, and n. These conjectures were made with the aid of a multi-precision program including the generation of largesubscripted Fibonacci numbers. On the other hand, once the identities were conjectured, their proofs appeared to be rather easy and similar to one another so that, to save space, we confine ourselves to proving but a few among them; this is done in Section 4. Section 5 provides a glimpse of possible further investigations. It is worth mentioning that formula (1.4) of [4], namely,
01 Sep 1996
TL;DR: In this paper, the idea of unified Fibonacci-type topology was used for construction of wide classes of discrete orthogonal transforms, including Rademacher-Fibonacci, Walsh-Fibrinacci, Haar and Haar-type transforms, etc.
Abstract: The idea of unified Fibonacci-type topology is used for construction of wide classes of discrete orthogonal transforms, including Rademacher-Fibonacci, Walsh-Fibonacci, Haar-Fibonacci-type transforms, etc. Efficient algorithms for proposed transforms directly related with the generalized Fibonacci topology are derived. The generation of discrete wavelets and wavelet packets based on Fibonacci-type recursions is established.
TL;DR: In this article, the transverse field electroabsorption of long-period GaAs/AlGaAs Fibonacci superlattices was investigated for three different realizations of the fibonacci sequence and assessed their bandwidth in photorefractive four-wave mixing experiments.
Abstract: We have investigated the transverse‐field electroabsorption of long‐period GaAs/AlGaAs Fibonacci superlattices for three different realizations of the Fibonacci sequence and assessed their bandwidth in photorefractive four‐wave mixing experiments. The one‐electron density of states exhibits a wide fractal distribution of quasibands, suggesting the ability to tailor bandwidths for optical applications. Many‐electron effects and the inter‐well coupling shift the excitonic oscillator strength to the low‐energy edge of the spectrum in all cases, producing a diffraction bandwidth that is relatively independent of coupling in both weak‐ and strong‐coupling Fibonacci superlattices.
01 Mar 1996
TL;DR: In this article, the lowest conduction band and the highest valence band states of Fibonacci superlattices, formed by different slabs of AlAs and GaAs grown along the (0 0 1) direction, were analyzed.
Abstract: We study the lowest conduction band and the highest valence band states of Fibonacci superlattices, formed by different slabs of AlAs and GaAs grown along the (0 0 1) direction. We employ an empirical tight-binding Hamiltonian including spin-orbit coupling together with the surface Green's function matching method. Second to sixth generation superlattices with different generating layer thicknesses have been analyzed. A selective localization of the spectral strength in the thicker GaAs slab is found for both valence and conduction band states in all the cases studied here.
01 Jan 1996
TL;DR: This paper shows that the additive lagged Fibonacci generators can be parallelized efficiently using the contiguous subsequence technique on any parallel computer and presents an efficient parallelizat ion for hypercubic networks and permutation networks.
Abstract: Srinivaa Aluru School of Computer and Information Science Syracuse University, Syracuse, NY 132444100 email: aluru@top. cis. syr. edu In order to pzxallelize applications that require the use of random numbers, an efficient and good quality parallel random number generator is required. In this paper, we study the parallelization of lagged Fibonacci plus and minus generators, which we collectively call additive lagged Fibonacci generators. Two popular ways of generating a random sequence in parallel are studied the contiguous subsequence technique and the leapfrog technique. We show that the additive lagged Fibonacci generators can be parallelized efficiently using the contiguous subsequence technique on any parallel computer. For the leapfrog technique, we present an efficient parallelizat ion for hypercubic networks and permutation networks.