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Showing papers on "Fibonacci number published in 1995"


Journal ArticleDOI
TL;DR: In this article, the authors give new examples of discrete Schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum, where the hull X of the potential is strictly ergodic, which implies that there is a generic set in X for which the operator has no eigenvalues.
Abstract: We give new examples of discrete Schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potentialx in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is a z ∈ X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in X. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all x ∈ X if X derives from a primitive substitution. For potentials defined by circle maps, x_n = 1_J (θ_0 + nα), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every half-open interval J.

186 citations


Book ChapterDOI
TL;DR: In this article, the authors review the recent developments in the theory of the tight-binding Schrodinger equation for a class of deterministic ergodic potentials, which are generated by substitutional sequences, like the Fibonacci or the Thue-Morse sequence.
Abstract: We review the recent developments in the theory of the one-dimensional tight-binding Schrodinger equation for a class of deterministic ergodic potentials. In the typical examples the potentials are generated by substitutional sequences, like the Fibonacci or the Thue-Morse sequence. We concentrate on rigorous results which will be explained rather than proved. The necessary mathematical background is provided in the text.

69 citations


Book ChapterDOI
01 Jul 1995
TL;DR: A brief introduction to the Fibonacci language is provided to present its features, which are particularly suited to modeling complex databases.
Abstract: Fibonacci is an object-oriented database programming language characterized by static and strong typing, and by new mechanisms for modeling databases in terms of objects with roles, classes, and associations. A brief introduction to the language is provided to present those features, which are particularly suited to modeling complex databases. Examples of the use of Fibonacci are given with reference to the prototype implementation of the language.

67 citations


Book ChapterDOI
05 Jul 1995
TL;DR: In this article, it was shown that for a pattern P and text T of descriptive sizes m, n, an occurrence of P in T can be found (if there is any) in time polynomial with respect to n.
Abstract: We consider strings which are succinctly described The description is in terms of straight-line programs in which the constants are symbols and the only operation is the concatenation Such descriptions correspond to the systems of recurrences or to context-free grammars generating single words The descriptive size of a string is the length n of a straight-line program (or size of a grammar) which defines this string Usually the strings of descriptive size n are of exponential length Fibonacci and Thue-Morse words are examples of such strings We show that for a pattern P and text T of descriptive sizes m, n, an occurrence of P in T can be found (if there is any) in time polynomial with respect to n This is nontrivial, since the actual lengths of P and T could be exponential, and none of the known string-matching algorithms is directly applicable Our first tool is the periodicity lemma, which allows to represent some sets of exponentially many positions in terms of feasibly many arithmetic progressions The second tool is arithmetics: a simple application of Euclid algorithm Hence a textual problem for exponentially long strings is reduced here to simple arithmetics on integers with (only) linearly many bits We present also an NP-complete version of the pattern-matching for shortly described strings

51 citations


Journal Article
TL;DR: A textual problem for exponentially long strings is reduced here to simple arithmetics on integers with (only) linearly many bits, which allows to represent some sets of exponentially many positions in terms of feasibly many arithmetic progressions.
Abstract: We consider strings which are succinctly described. The description is in terms of straight-line programs in which the constants are symbols and the only operation is the concatenation. Such descriptions correspond to the systems of recurrences or to context-free grammars generating single words. The descriptive size of a string is the length n of a straight-line program (or size of a grammar) which defines this string. Usually the strings of descriptive size n are of exponential length. Fibonacci and Thue-Morse words are examples of such strings. We show that for a pattern P and text T of descriptive sizes m, n, an occurrence of P in T can be found (if there is any) in time polynomial with respect to n. This is nontrivial, since the actual lengths of P and T could be exponential, and none of the known string-matching algorithms is directly applicable. Our first tool is the periodicity lemma, which allows to represent some sets of exponentially many positions in terms of feasibly many arithmetic progressions. The second tool is arithmetics: a simple application of Euclid algorithm. Hence a textual problem for exponentially long strings is reduced here to simple arithmetics on integers with (only) linearly many bits. We present also an NP-complete version of the pattern-matching for shortly described strings.

49 citations


Journal ArticleDOI
TL;DR: New decompositional methods based on Fibonacci p-codes for computing the output for different stack filters are presented and the computational complexities of these methods are studied.

49 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the electronic motion in quasiperiodic systems (the Harper model, the Fibonacci chain, two-and three-dimensional fibonacci quasilattices) in the framework of a tight-binding Hamiltonian.
Abstract: The electronic motion in quasiperiodic systems (the Harper model, the Fibonacci chain, two- and three-dimensional Fibonacci quasilattices) is studied, in the framework of a tight-binding Hamiltonian. The spreading with time of the wavepacket is described in terms of the behaviour of the autocorrelation function C(t). It is found that, in all cases, C(t) approximately t- delta . For the Harper model with lambda <2, the motion of the electron is ballistic ( delta =1), which goes against a previous estimate of delta =0.84. We show that this discrepancy is due to the neglect of a logarithmic contribution in the scaling analysis. For the Harper model with lambda =2 and the Fibonacci chain, the motion is non-ballistic with 0< delta <1. For the higher-dimensional Fibonacci quasilattices, C(t) exhibits a transition from a ballistic to a non-ballistic behaviour, upon varying the modulation strength of the quasiperiodicity. The relation between C(t) and the fractal dimensions of the spectral measure is also studied.

47 citations


Journal ArticleDOI
TL;DR: In this paper, the spectral properties of a tight-binding hamiltonian on the Fibonacci chain were characterized analytically as completely as possible, such as the spectral measure, the bandwidth distribution, the Lebesgue measure exponent, the Hausdorff dimension, the multifractal scaling, the gaps distribution as well as the long time return probability.
Abstract: We solve the approximate renormalisation group found by Qiu Niu and Franco Nori(Phys. Rev. Lett. 57 2057(1986)) for a tight-binding hamiltonian on the Fibonacci chain. This enables us to characterize analytically as completely as possible the spectral properties of this model, such as the spectral measure, the bandwidth distribution, the Lebesgue measure exponent, the Hausdorff dimension, the multifractal scaling, the gaps distribution as well as the long time return probability. Our results, qualitatively and quantitatively complete and unify previous works on similar models.

47 citations


Journal ArticleDOI
TL;DR: A parallel decompositional algorithm and VLSI architecture is proposed for computation of the output of a stack filter over a single window of input samples using Fibonacci p-codes and for a subclass of positive Boolean functions.
Abstract: A parallel decompositional algorithm and VLSI architecture is proposed for computation of the output of a stack filter over a single window of input samples using Fibonacci p-codes. For a subclass of positive Boolean functions, a more efficient parallel algorithm and VLSI architecture for running stack filtering is also presented. The area-time complexities of the proposed designs are estimated. >

45 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order l have a finite invariant measure if l ∈ (1 l 1) where l1 is some number strictly greater than 2.
Abstract: We prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order l have a finite absolutely continuous invariant measure if l ∈ (1 l1) where l1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case l = 2.

42 citations


Journal ArticleDOI
TL;DR: This method applies a recently developed Hadamard matrix-based technique to describe elements of I(T) in terms of edge-disjoint packings of subtrees in T, and thereby complements earlier more algebraic treatments.
Abstract: Linear invariants are useful tools for testing phylogenetic hypotheses from aligned DNA/ RNA sequences, particularly when the sites evolve at different rates. Here we give a simple, graph theoretic classification for each phylogenetic tree T, of its associated vector space I(T) of linear invariants under the Jukes–Cantor one-parameter model of nucleotide substitution. We also provide an easily described basis for I(T), and show that if T is a binary (fully resolved) phylogenetic tree with n sequences at its leaves then: dim[I(T)] = 4n − F2n−2 where Fn is the nth Fibonacci number. Our method applies a recently developed Hadamard matrix-based technique to describe elements of I(T) in terms of edge-disjoint packings of subtrees in T, and thereby complements earlier more algebraic treatments. Key words: Phylogenetic invariants; trees; forests; Hadamard matrix; Jukes–Cantor model

Journal ArticleDOI
TL;DR: In this paper, Yu et al. studied three-dimensional compact orientable hyperbolic manifolds connected with the Fibonacci groups and proved that the group F(2, m) is finite if and only if m = 1,2,3, 4,5, 7.817.
Abstract: A. Yu. Vesnin and A. D. Mednykh UDC 515.16 + 512.817.7 This article is devoted to the study of three-dimensional compact orientable hyperbolic manifolds connected with the Fibonacci groups. The Fibonacci groups F(2, m) = (Zl, z2,..., z,n : ziZi+l = zi+2, { mod m) were introduced by J. Conway [1]. The first natural question connected with these groups was whether they are finite or not [1]. It is known from [2-6] that the group F(2, m) is finite if and only if m = 1,2,3, 4,5, 7. Some algebraic generalizations of the groups

Journal ArticleDOI
TL;DR: In this paper, the magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures are considered, including the Fibonacci chain and the tenfold symmetric triangular tiling.
Abstract: The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.

Journal ArticleDOI
TL;DR: In this article, four and five dimensional Cantor sets are analyzed in relation to two different Fibonacci series, and connections to classical and quantum mechanical statistics are outlined, with a focus on the connection between quantum and quantum mechanics.
Abstract: Four and five dimensional Cantor sets are analysed in relation to two different Fibonacci series. Connections to classical and quantum mechanical statistics are outlined.


Journal ArticleDOI
TL;DR: In this paper, the authors studied the energy spectra of one-dimensional quasiperiodic Fibonacci and Thue-Morse systems and analyzed the fractal character of the energy spectrum of these systems through their integrated density of states and fractal dimensionality.
Abstract: We study the density of states, the distribution of energy spacings, and the transmission coefficient of one-dimensional quasiperiodic Fibonacci and Thue-Morse systems. We consider arrays of \ensuremath{\delta} potentials with constant separation and two potential strengths, and tight-binding systems with constant nearest-neighbor couplings and two different on-site energies. The quasiperiodicity lies in the arrangement of the two possible values of either the potential strengths or the on-site energies. We analyze the fractal character of the energy spectra of these systems through their integrated density of states and fractal dimensionality. We study the average with respect to energy of the transmission coefficient, which turns out to be a good way to measure the regularity of the system.

Journal ArticleDOI
TL;DR: In this article, a proof by computer that the Fibonacci group F(2, 9) is automatic is given. But this proof is based on the assumption that the group generators have infinite order.
Abstract: This note contains a report of a proof by computer that the Fibonacci group F(2, 9) is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that the group generators have infinite order, which of course implies that the group itself is infinite.

Journal ArticleDOI
TL;DR: An orthorhombic phase with 1.23 nm, b = 1.24 nm and c = 3.07 nm has been found in the Al12Fe2Cr alloy as mentioned in this paper.
Abstract: An orthorhombic phase with a = 1.23 nm, b = 1.24 nm and c = 3.07nm has been found in the Al12Fe2Cr alloy. The distribution of strong diffraction spots in the [010], [100] and [001] electron diffraction patterns (EDPs) corresponds to the tenfold, the 2D twofold, and the 2P twofold EDPs respectively of a decagonal quasicrystal. Therefore this orthorhombic phase is called a decagonal approximant. However, unlike other orthorhombic approximants, its parameter c = 3.07 nm does not agree with any of the values obtained by substituting a rational ratio of two consecutive Fibonacci numbers, such as 1/1, 2/1, 3/2, 5/3, etc., for the irrational τ in one of the quasiperiodic directions in a decagonal quasicrystal. Its [010] high-resolution electron microscopy image consists of a network of 72° and 36° rhombi whose vertices are surrounded by a decagon of image points. The ratio of the thick to thin rhombi is 4 to 1 rather than a Fibonacci ratio. This is a new type of decagonal approximant. On ordering, a C-c...

Journal ArticleDOI
TL;DR: In this paper, the authors used real-space renormalization-group techniques to derive the local density of phonon states, the lattice specific heat, and the transmission coefficient of a particle through a layered quasicrystal whose ends are fixed.
Abstract: Calculations are presented for the local density of phonon states scrD(${\mathrm{\ensuremath{\omega}}}^{2}$), the lattice specific heat ${\mathit{C}}_{\mathit{v}}$, and the transmission coefficient of a particle through a layered quasicrystal whose ends are fixed. In this paper, scrD(${\mathrm{\ensuremath{\omega}}}^{2}$) is obtained using a systematic decimation of the equations of motion for the atoms on a chain (where the two-dimensional plane is simulated by an atom) with the use of real-space renormalization-group techniques. The renormalized spring coupling constants for the atomic arrangement with the silver and golden means become invariant after the first decimation. However, the copper mean arrangement only becomes invariant after two decimations. We analyze the effect of this behavior in calculating the limiting case of a periodic chain from the Fibonacci series. We compare the generalized Fibonacci lattice with a lattice whose coupling constants are arranged in the Thue-Morse sequence. For the Fibonacci lattices, there is a significant difference between the copper mean results for the low-frequency density of states and those for the gold and silver lattices. This difference leads to a significant change in the specific heat for the copper relative to the periodic lattice. The density of states for the Thue-Morse chain has a unique low-frequency behavior and this also leads to a significant change in its specific heat at low temperature compared with a periodic lattice of the same length.

Posted Content
TL;DR: This note contains a report of a proof by computer that the Fibonacci group F(2, 9) is automatic, and can be used to solve the word problem in the group.
Abstract: This note contains a report of a proof by computer that the Fibonacci group F(2,9) is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that the group generators have infinite order, which of course implies that the group itself is infinite.

Proceedings ArticleDOI
25 Oct 1995
TL;DR: This paper proposes a new network topology called extended Fibonacci cube (EFC/sub 1/) based on the same sequence F(i)=F( i-1)+F(i-2) as the regular fibonacci sequence but with different initial values.
Abstract: The Fibonacci cube (FC) is an interconnection network that possesses many desirable properties that are important in network design and application. However, most Fibonacci cubes (more than two third of all) are not Hamiltonian. In this paper, we propose a new network topology called extended Fibonacci cube (EFC/sub 1/) based on the same sequence F(i)=F(i-1)+F(i-2) as the regular Fibonacci sequence but with different initial values. We also propose a series of extended Fibonacci cubes with even number of nodes. Any extended Fibonacci cube (EFC/sub k/, with k/spl ges/1) in the series contains the node set of any other cube that precedes EFC/sub k/ in the series. We show that any extended Fibonacci cube maintains virtually all the desirable properties of the Fibonacci cube. EFC/sub k/'s can be considered as flexible versions of incomplete hypercubes which eliminate the restriction on the number of nodes and thus makes it possible to construct parallel machines with arbitrary sizes.


01 Jan 1995
TL;DR: In this article, it was proved that the set n 2Fni1; 2Fn+1, 2F 3 n Fn+1Fni+2, 2 Fn+Ln+2Fni 1; 2 Fni 2 n+1 i F 2 n ) o
Abstract: Let n be an integer. A set of positive integers is said to have the property D(n) if the product of its any two distinct elements increased by n is a perfect square. In this paper, the sets of four numbers represented in terms of Fibonacci numbers with the property D(F 2 n ) and D(L 2 ), where (Fn) is the Fibonacci sequence and (Ln) is the Lucas sequence, are constructed. Among other things, it is proved that the set n 2Fni1; 2Fn+1; 2F 3 n Fn+1Fn+2; 2Fn+1Fn+2Fn+3(2F 2 n+1 i F 2 n ) o

Journal ArticleDOI
TL;DR: In this article, the theoretical electronic structure of Fibonacci superlattices of narrow-gap III-V semiconductors is described within the envelope-function approximation in a two-band model.
Abstract: We report the theoretical electronic structure of Fibonacci superlattices of narrow-gap III-V semiconductors. The electron dynamics is accurately described within the envelope-function approximation in a two-band model. Quasiperiodicity is introduced by considering two different III-V semiconductor layers and arranging them according to the Fibonacci series along the growth direction. The resulting energy spectrum is then found by solving exactly the corresponding effective-mass (Dirac-like) wave equation using transfer-matrix techniques. We find that a self-similar electronic spectrum can be seen in the band structure. Electronic transport properties of samples are also studied and related to the degree of spatial localization of electronic envelope functions via the Landauer resistance and Lyapunov coefficient. As a working example, we consider type II InAs/GaSb superlattices and discuss in detail our results in this system.

Journal ArticleDOI
J.W. Feng1, Guojun Jin1, A. Hu1, S.S. Kang1, S. S. Jiang1, D. Feng1 
TL;DR: The magnetostatic modes in a Fibonacci multilayer consisting of alternating magnetic and nonmagnetic layers are studied and the constant of motion, depending on both the in-plane wave vector and the frequency of the mode, is explicitly obtained and used to describe general features of the frequency spectra.
Abstract: We study the magnetostatic modes in a Fibonacci multilayer consisting of alternating magnetic and nonmagnetic layers. The constant of motion, depending on both the in-plane wave vector and the frequency of the mode, is explicitly obtained and used to describe general features of the frequency spectra. Furthermore, spin wave spectra and precession amplitudes of magnetization for a finite Fibonacci multilayer are numerically calculated by the transfer matrix method. For a given in-plane wave vector, the distribution of frequency exhibits a triadic Cantor structure with large gaps in the low-frequency region, small gaps in the high-frquency region, and many ``isolated'' modes in the gaps. The gaps strongly depend on the in-plane wave vector and the thicknesses of the magnetic and nonmagnetic layers. We find three types of states in the quasiperiodic direction: extended states in the high-frequency region near the upper band edge, critical states in the triadic subbands, and surface states in gaps. Besides the conventional Damon-Eshbach surface mode localized at two opposite surfaces of the multilayer for positive and negative wave vectors, respectively, another kind of surface modes are discovered. When the wave vector is reversed, these modes are still localized at the same sides of the multilayer.

Dissertation
01 Jan 1995
TL;DR: In this article, the authors develop the theory of semi-primary lattices, a class of modular lattices including abelian subgroup lattices and invariant subspace lattices in which an integer partition is assigned to every element and every interval.
Abstract: We develop the theory of semi-primary lattices, a class of modular lattices, including abelian subgroup lattices and invariant subspace lattices, in which an integer partition is assigned to every element and every interval. Flags in these lattices give rise to chains of partitions, which may be encoded as tableaux. In certain of these lattices, Steinberg and van Leeuwen respectively have shown that relative positions and cotypes, which describe configurations of elements in flags, are generically computed by the well known Robinson-Schensted and evacuation algorithms on standard tableaux. We explore extensions of this to semi-primary lattices: we consider the nongeneric configurations, leading to nondeterministic variations of the Robinson-Schensted and evacuation tableau games, and consider exact and asymptotic enumeration of the number of ways to achieve certain configurations. We also introduce other configuration questions leading to new tableau games, and develop a number of deterministic and nondeterministic tableau operators that can be combined to describe the generic and degenerate configurations of flags undergoing various transformations. We also look at similar problems in the class of modular lattices whose complemented intervals have height at most 2, such as Stanley's Fibonacci lattice Z(r). Here the generic relative position is related to Fomin's analogue of the Robinson-Schensted correspondence in Z(1). (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

Journal ArticleDOI
TL;DR: In this paper, the theoretical electronic structure of Fibonacci superlattices of narrow-gap III-V semiconductors is described within the envelope-function approximation in a two-band model.
Abstract: We report theoretical electronic structure of Fibonacci superlattices of narrow-gap III-V semiconductors. Electron dynamics is accurately described within the envelope-function approximation in a two-band model. Quasiperiodicity is introduced by considering two different III-V semiconductor layers and arranging them according to the Fibonacci series along the growth direction. The resulting energy spectrum is then found by solving exactly the corresponding effective-mass (Dirac-like) wave equation using tranfer-matrix techniques. We find that a self-similar electronic spectrum can be seen in the band structure. Electronic transport properties of samples are also studied and related to the degree of spatial localization of electronic envelope-functions via Landauer resistance and Lyapunov coefficient. As a working example, we consider type II InAs/GaSb superlattices and discuss in detail our results in this system.

Journal ArticleDOI
TL;DR: In this article, it was shown that in a lattice with absorbing elements, the diffraction from a phase grating cannot always be indexed with a successive pair of Fibonacci numbers.

Posted Content
TL;DR: In this article, the Bordelaise philosophy was used to enumerate self-avoiding walks in a $[0, 1] \times (- \infty, √ √ n) time.
Abstract: The Bordelaise philosophy, or rather a juvenile version of it, is used to enumerate self avoiding walks in a $[0,1] \times (- \infty, \infty)$.

Journal Article
TL;DR: In this paper, the authors established the relevant uniqueness statements for ratios of Fibonacci numbers and Lucas numbers for any positive integer satisfying the recurrence relation un+2 = un+l + un, n>\\.
Abstract: mirroring a well-known feature of Fibonacci numbers (see Theorem 2.5). It was pointed out in [1] that (0.2) could itself be used to disprove the corresponding assertion for the 1cm; precisely, if lcm(a, b) = £, then lcm(Ma, Mb) Mt only in the trivial cases a\\b or b\\a. The argument rested on a uniqueness theorem for the expression of rational numbers as a ratio of two members of the {Mn} sequence. However, the authors did not establish the corresponding negative results for lcm(i^, Fb), lcm (Z,a, Lb). In this paper the gap is filled, precisely by establishing the relevant uniqueness statements for ratios of Fibonacci numbers and Lucas numbers. It turns out that much of the work can be done for arbitrary sequences {un} of positive integers satisfying the recurrence relation un+2 = un+l + un, n>\\. Such sequences are, in a sense, classified by their initial values uh u2. However, to discuss the classification, it is better to take the sequences backward with respect to n, that is, to allow n to take any integer value, although the principal results are all to be concerned with positive values of n. Then the Fibonacci sequence {Fn} belongs to the special class given by u0 = 0. Another interesting class, from our point of view, is given by 0 < u0 < ux. The Lucas sequence {Ln} seems, to us, to belong to a singleton class.