Showing papers on "Fibonacci number published in 2009"
01 Jan 2009
TL;DR: In this paper, the Diophantine equation x 4 i 6x 2 y 2 + 5y 4 = 16Fni1Fn+1, where (Fn) n¸0 is the Fibonacci sequence.
Abstract: In this paper we study the Diophantine equation x 4 i 6x 2 y 2 + 5y 4 = 16Fni1Fn+1, where (Fn) n¸0 is the Fibonacci sequence and we find a class of such equations having solutions which are determined.
269 citations
TL;DR: In this paper, the numerical errors of such measurements were analyzed, and it was shown that using the Fibonacci lattice would reduce the root mean squared error by at least 40%.
Abstract: The area of a spherical region can be easily measured by considering which sampling points of a lattice are located inside or outside the region. This point-counting technique is frequently used for measuring the Earth coverage of satellite constellations, employing a latitude-longitude lattice. This paper analyzes the numerical errors of such measurements, and shows that they could be greatly reduced if the Fibonacci lattice were used instead. The latter is a mathematical idealization of natural patterns with optimal packing, where the area represented by each point is almost identical. Using the Fibonacci lattice would reduce the root mean squared error by at least 40%. If, as is commonly the case, around a million lattice points are used, the maximum error would be an order of magnitude smaller.
226 citations
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TL;DR: In this article, the definition and discussion of polynomial generalizations of the Fibonacci numbers called δ-Fibonacci Numbers are discussed. And different connections between these numbers and Lucas Numbers are proven.
Abstract: The scope of the paper is the definition and discussion of the polynomial generalizations of the Fibonacci numbers called here δ-Fibonacci numbers. Many special identities and interesting relations for these new numbers are presented. Also, different connections between δ-Fibonacci numbers and Fibonacci and Lucas numbers are proven in this paper.
106 citations
Journal Article•
TL;DR: In this paper, a structure defined on a class of Riemannian manifolds, called by us a Golden Structure, is defined and shown to be a (1, 1)-tensor field P 2 = P + I (which is similar to that satisfied by the Golden Ratio φ ).
Abstract: The Golden Ratio is a fascinating topic that continually generates new ideas. The main purpose of the present paper is to point out and find some applications of the Golden Ratio and of Fibonacci numbers in Differential Geometry. We study a structure defined on a class of Riemannian manifolds, called by us a Golden Structure. A Riemannian manifold endowed with a Golden Structure will be called a Golden Riemannian manifold. Precisely, we say that an (1,1)-tensor field P on a m-dimensional Riemannian manifold ( M, g ) is a Golden Structure if it satisfies the equation P 2 = P + I (which is similar to that satisfied by the Golden Ratio φ )w hereI stands for the (1,1) identity tensor field. First, we establish several properties of the Golden Structure. Then we show that a Golden Structure induces on every invariant submanifold a Golden Structure, too. This fact is illustrated on a product of spheres in an Euclidean space.
96 citations
TL;DR: In this article, a new generalization {qn }, with initial conditions q 0 = 0 and q 1 = 1, which is generated by the recurrence relation qn = aq n-1 + q n-2 (when n is even) or q n = bq n−1+ q n−2 (When n is odd), where a and b are nonzero real numbers.
Abstract: Abstract Consider the Fibonacci sequence having initial conditions F 0 = 0, F 1 = 1 and recurrence relation Fn = F n–1 + F n–2 (n ≥ 2). The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this article, we study a new generalization {qn }, with initial conditions q 0 = 0 and q 1 = 1 which is generated by the recurrence relation qn = aq n–1 + q n–2 (when n is even) or qn = bq n–1 + q n–2 (when n is odd), where a and b are nonzero real numbers. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of {qn } with a = b = 1. Pell's sequence is {qn } with a = b = 2 and the k-Fibonacci sequence is {qn } with a = b = k. We produce an extended Binet's formula for the sequence {qn } and, thereby, identities such as Cassini's, Catalan's, d'Ocagne's, etc.
94 citations
TL;DR: In this article, the authors introduce the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example.
Abstract: We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial ("identity") channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the "golden chain", a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian su(2)_k statistics. The k to infinity limit of the latter yields ordinary SU(2) spin chains.
93 citations
TL;DR: This paper shows how to transform a Fibonacci NLFSR into an equivalentNLFSR in the Galois configuration, in which the feedback can be applied to every bit, thus decreasing the propagation time and increasing the throughput.
Abstract: Conventional nonlinear feedback shift registers (NLFSRs) use the Fibonacci configuration in which the feedback is applied to the last bit only. In this paper, we show how to transform a Fibonacci NLFSR into an equivalent NLFSR in the Galois configuration, in which the feedback can be applied to every bit. Such a transformation can potentially reduce the depth of the circuits implementing feedback functions, thus decreasing the propagation time and increasing the throughput.
93 citations
TL;DR: In this article, the authors introduce a matrix Q h(x ) that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers and present properties of these polynomials.
Abstract: Let h ( x ) be a polynomial with real coefficients. We introduce h ( x ) -Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h ( x ) -Fibonacci polynomials. We also introduce h ( x ) -Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ( x ) that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.
79 citations
TL;DR: In this paper, the authors consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small.
Abstract: We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the coupling constant.
61 citations
TL;DR: The sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say an þ r for fixed integers a and r, are studied to give in a straightforward way several formulas for the sums of such numbers.
54 citations
Book•
01 Oct 2009
TL;DR: The Museum of Harmony and the Golden Section of the Mathematical Museum of Science and Technology as discussed by the authors have been used for many purposes, e.g., for the development of Fibonacci Computers with Irrational Radices and a New Theory of Coding Appendices.
Abstract: The Golden Section Fibonacci and Lucas Numbers Regular Polyhedrons Generalizations of Fibonacci Numbers and the Golden Section Hyperbolic Fibonacci and Lucas Functions Fibonacci and the "Golden" Matrices Algorithmic Measurement Theory Fibonacci Computers Number Systems with Irrational Radices Ternary Mirror-Symmetrical Arithmetic Fibonacci and "Golden" Matrices and a New Theory of Coding Appendices: The Most Important Mathematical Results of the Harmony Mathematics The Museum of Harmony and the Golden Section.
TL;DR: In this paper, Stakhov et al. considered a class of square Fibonacci matrix of order (p+1) whose elements are based on the Fiboni p numbers with determinant equal to +1 or −1, and established generalized relations among the code matrix elements for all values of p.
Abstract: We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or −1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space–time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p = 1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56–66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p = 2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases.
TL;DR: In this article, lower and upper bounds for the number n g of numerical semigroups of genus g were given for the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers.
TL;DR: It is concluded that mirror symmetry in arrays of Fibonacci multilayers is sufficient but not necessary to generate multiple transparent states, opening broader applications of quasiperiodic systems as filters and microcavities of multiple frequencies.
Abstract: In this paper we study the propagation of light through an asymmetric array of dielectric multilayers built by joining two porous silicon substructures in a Fibonacci sequence. Each Fibonacci substructure follows the well-known recursive rule but in the second substructure dielectric layers A and B are exchanged. Even without mirror symmetry, this array gives rise to multiple transparent states, which follow the scaling properties and self-similar spectra of a single Fibonacci multilayer. We apply the transfer matrix formalism to calculate the transmittance. By setting the transfer matrix of the array equal to ± I, the identity matrix, frequencies of perfect light transmission are reproduced in our theoretical calculations. Although the light absorption of porous silicon in the optical range limits our experimental study to low Fibonacci generations, the positions of the transparent states are well predicted by the above-mentioned condition. We conclude that mirror symmetry in arrays of Fibonacci multilayers is sufficient but not necessary to generate multiple transparent states, opening broader applications of quasiperiodic systems as filters and microcavities of multiple frequencies.
TL;DR: In this paper, the authors rigorously analyze Knill's Fibonacci scheme for fault-tolerant quantum computation, which is based on the recursive preparation of Bell states protected by a concatenated error-detecting code.
Abstract: We rigorously analyze Knill's Fibonacci scheme for fault-tolerant quantum computation, which is based on the recursive preparation of Bell states protected by a concatenated error-detecting code. We prove lower bounds on the threshold fault rate of 0.67×10^−3 for adversarial local stochastic noise, and 1.25×10^−3 for independent depolarizing noise. In contrast to other schemes with comparable proved accuracy thresholds, the Fibonacci scheme has a significantly reduced overhead cost because it uses postselection far more sparingly.
TL;DR: In this article, a general and efficient method was proposed to analyze the dipolar modes of aperiodic arrays of metal nanoparticles with ellipsoidal shapes and their electromagnetic coupling with external fields.
Abstract: In this paper, we propose a general and efficient method to analyze the dipolar modes of aperiodic arrays of metal nanoparticles with ellipsoidal shapes and their electromagnetic coupling with external fields We reduce the study of the spectral and localization properties of dipolar modes to the understanding of the spectral properties of an operator L expressing the electric field along the chain in terms of the electric-dipole moments within the electric quasistatic approximation We show that, in general, the spectral properties of the L operator are at the origin of the formation of pseudoband gaps and localized modes in aperiodic chains These modal properties are therefore uniquely determined by the aperiodic geometry of the arrays for a given shape of the nanoparticles The proposed method, which can be easily extended in order to incorporate retardation effects and higher multipolar orders, explains in very clear terms the role of aperiodicity in the particle arrangement, the effect of particle shapes, incoming field polarization, material dispersion, and optical losses Our method is applied to the simple case of linear arrays generated according to the Fibonacci sequence, which is the chief example of deterministic quasiperiodic order The conditions for the resonant excitation of dipolar modes in Fibonacci chains are systematically investigated In particular, we study the scaling of localized dipolar modes, the enhancement of near fields, and the formation of Fibonacci pseudodispersion diagrams for chains with different interparticle separations and particle numbers Far-field scattering cross sections are also discussed in detail All results are compared with the well-known case of periodic linear chains of metal nanoparticles, which can be derived as a special application of our general model Our theory enables the quantitative and predictive understanding of band-gap positions, field enhancement, scattering, and localization properties of aperiodic arrays of resonant nanoparticles in terms of their geometry This is central to the design of metallic resonant arrays that, when excited by an external electromagnetic wave, manifest strongly localized and enhanced near fields
TL;DR: In this paper, two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifold and the Fibonacci manifold, were established.
Abstract: We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci manifolds.
TL;DR: In this article, the m-extension of the Fibonacci and Lucas p-numbers (p⩾ ǫ 0 is integer and m à > 0 is real number) is defined and continuous functions for the m extension of these numbers are obtained using the generalized Binet formulas.
Abstract: In this article, we define the m-extension of the Fibonacci and Lucas p-numbers (p ⩾ 0 is integer and m > 0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers (p = 1, m = 1), Pell and Pell–Lucas numbers (p = 1, m = 2), Fibonacci and Lucas p-numbers (m = 1), Fibonacci m-numbers (p = 1), Pell and Pell–Lucas p-numbers (m = 2) are obtained. Afterwards, we obtain the continuous functions for the m-extension of the Fibonacci and Lucas p-numbers using the generalized Binet formulas. Also we introduce in the article a new class of mathematical constants – the Golden (p, m)-Proportions, which are a wide generalization of the classical golden mean, the golden p-proportions and the golden m-proportions. The article is of fundamental interest for theoretical physics where Fibonacci numbers and the golden mean are used widely.
TL;DR: In this paper, the Pisano period of the k-Fibonacci cyclic sequences is studied and the period length is shown to be π k 2 + 4 for every odd number k.
Abstract: We study here the period-length of the k-Fibonacci sequences taken modulo m. The period of such cyclic sequences is know as Pisano period, and the period-length is denoted by π k ( m ) . It is proved that for every odd number k, π k ( k 2 + 4 ) = 4 ( k 2 + 4 ) .
TL;DR: In this paper, an interface between two non-Abelian quantum Hall states, the Moore-Read state, supporting Ising anyons, and the k=2 non-ABelian spin-singlet state, supported Fibonacci anyons was considered.
Abstract: We consider an interface between two non-Abelian quantum Hall states: the Moore-Read state, supporting Ising anyons, and the k=2 non-Abelian spin-singlet state, supporting Fibonacci anyons. It is shown that the interface supports neutral excitations described by a (1+1)-dimensional conformal field theory with a central charge c=7/10. We discuss effects of the mismatch of the quantum statistical properties of the quasiholes between the two sides, as reflected by the interface theory.
TL;DR: The set partition statistics ls and rb introduced by Wachs and White are investigated and it is shown that the distribution over @P"n(13/2) enumerates certain integer partitions, and the distribution gives q-Fibonacci numbers.
Abstract: We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions that avoid certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, @P"n(13/2), and those avoiding both 13/2 and 123, @P"n(13/2,123). We show that the distribution over @P"n(13/2) enumerates certain integer partitions, and the distribution over @P"n(13/2,123) gives q-Fibonacci numbers. These q-Fibonacci numbers are closely related to q-Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that these q-Fibonacci numbers satisfy q-analogues of many Fibonacci identities. Finally, we indicate how p,q-Fibonacci numbers arising from the bistatistic (ls,rb) give rise to p,q-analogues of identities.
02 Oct 2009
TL;DR: A formula is developed for optimizing the capacitor sizes and improving the performance of the Fibonacci charge pump, with focus on voltage gain and output resistance and including the effects of parasitic capacitances.
Abstract: This paper presents an analysis of two types of integrated charge pumps, Dickson and Fibonacci. The two circuits are compared in slow-switching conditions and at equal area occupation. A formula is developed for optimizing the capacitor sizes and improving the performance of the Fibonacci charge pump. The performance is evaluated with focus on voltage gain and output resistance and including the effects of parasitic capacitances.
TL;DR: In this paper, the generalized Fibonacci and Lucas p-numbers, Fp(n), and their sums, ∑ i = 1 n F p (i ), and the 1-factors of a class of bipartite graphs are derived.
Abstract: In this paper we consider certain generalizations of the well-known Fibonacci and Lucas numbers, the generalized Fibonacci and Lucas p-numbers. We give relationships between the generalized Fibonacci p-numbers, Fp(n), and their sums, ∑ i = 1 n F p ( i ) , and the 1-factors of a class of bipartite graphs. Further we determine certain matrices whose permanents generate the Lucas p-numbers and their sums.
TL;DR: In this article, the binomial, k-binomial, rising, and falling transforms were applied to the k-Fibonacci sequence and many formulas relating the obtained new sequences are presented and proved.
Abstract: In this paper, we apply the binomial, k-binomial, rising, and falling transforms to the k-Fibonacci sequence. Many formulas relating the so obtained new sequences are presented and proved. Finally, we define and find the inverse transforms of the sequences previously obtained.
TL;DR: In this paper, it was shown that every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every 2 terms of different sign differ by at most 3.
Abstract: Abstract We show that every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses. This is an analogue of the Zeckendorf representation.
TL;DR: As the pump pulse energy increases, the excitation-induced dephasing broadens the exciton resonances resulting in a disappearance of sharp features and reduction in peak reflectivity.
Abstract: A detailed experimental and theoretical study of the linear and nonlinear optical properties of different Fibonacci-spaced multiple-quantum-well structures is presented. Systematic numerical studies are performed for different average spacing and geometrical arrangement of the quantum wells. Measurements of the linear and nonlinear (carrier density dependent) reflectivity are shown to be in good agreement with the computational results. As the pump pulse energy increases, the excitation-induced dephasing broadens the exciton resonances resulting in a disappearance of sharp features and reduction in peak reflectivity.
TL;DR: It is shown that, for a scalar random walk system in which the two noise sources have equal variance, the Kalman filter's estimate turns out to be a convex linear combination of the a priori estimate and of the measurements with coefficients suitably related to the Fibonacci numbers.
TL;DR: In this article, the transmission properties of light through the symmetric Fibonacci photonic multilayers were studied, i.e., a binary one-dimensional quasiperiodic structure made up of both positive (SiO 2 ) and negative refractive index materials with a mirror symmetry.
TL;DR: The results broadly generalize and unify recent findings of this type and indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.
Abstract: For $k>1$ and nonnegative integer parameters $a_p, b_p$, $p = 1..k$, we analyze the solutions to the meta-Fibonacci recursion $C(n)=\sum_{p=1}^k C(n-a_p-C(n-b_p))$, where the parameters $a_p, b_p$, $p = 1..k$ satisfy a specific constraint. For $k=2$ we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include the Conolly recursion and all of its generalizations that have been studied to date. We show that the solutions for all the recursions defined by the parameters in these families have a natural combinatorial interpretation: they count the number of labels on the leaves of certain infinite labeled trees, where the number of labels on each node in the tree is determined by the parameters. This combinatorial interpretation enables us to determine various new results concerning these sequences, including a closed form, and to derive asymptotic estimates. Our results broadly generalize and unify recent findings of this type relating to certain of these meta-Fibonacci sequences. At the same time they indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.
TL;DR: In this article, the authors consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant and show that the thickness tends to infinity and the Hausdorff dimension tends to one.
Abstract: We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff di- mension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, t sum of the spec- trum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.