Showing papers on "Fibonacci number published in 2014"
TL;DR: In this article, it was shown that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support Ising non-Abelian anyons.
Abstract: Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green’s observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here, we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that—unlike Ising anyons—allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane’s construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a twofold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types, yet depending on nonuniversal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.
251 citations
TL;DR: Imaging the polariton modes both in real and reciprocal space, features characteristic of their fractal energy spectrum such as the opening of minigaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states are observed.
Abstract: We report on the study of a polariton gas confined in a quasiperiodic one-dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of minigaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrodinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.
136 citations
01 Jan 2014
TL;DR: In this article, a Binet-style formula that can be used to produce the k-generalized Fibonacci numbers (that is, the Tribonaccis, Tetranaccis etc.).
Abstract: In this paper, we present a Binet-style formula that can be used to produce the k-generalized Fibonacci numbers (that is, the Tribonaccis, Tetranaccis, etc.). Furthermore, we show that in fact one needs only take the integer closest to the first term of this Binet-style formula in order to generate the desired sequence.
94 citations
TL;DR: In this article, generalized Fibonacci sequences and the initial value of the bilinear difference equation are represented as well as some applications concerning a two-dimensional system of BLE equations.
Abstract: Well-defined solutions of the bilinear difference equation are represented in terms of generalized Fibonacci sequences and the initial value. Our results extend and give natural explanations of some recent results in the literature. Some applications concerning a two-dimensional system of bilinear difference equations are also given.
83 citations
25 Nov 2014
TL;DR: In this article, the problem of expressing a term of a given non-degenerate binary recurrence sequence as a linear combination of a factorial and an S unit whose coefficients are bounded was studied.
Abstract: In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an \(S\)-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an \(S\)-unit associated to the set of primes \(\{2,3,5,7\}\).
81 citations
TL;DR: The problem of compiling quantum operations into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model is addressed and a probabilistically polynomial algorithm is developed that approximates any given single-qubit unitary to a desired precision by an asymptotically depth-optimal braid pattern.
Abstract: We address the problem of compiling quantum operations into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model. We classify the single-qubit unitaries that can be represented exactly by Fibonacci anyon braids and use the classification to develop a probabilistically polynomial algorithm that approximates any given single-qubit unitary to a desired precision by an asymptotically depth-optimal braid pattern. We extend our algorithm in two directions: to produce braids that allow only single-strand movement, called weaves, and to produce depth-optimal approximations of two-qubit gates. Our compiled braid patterns have depths that are 20 to 1000 times shorter than those output by prior state-of-the-art methods, for precisions ranging between 10(-10) and 10(-30).
77 citations
TL;DR: A possible simplified version of the quantum Arnold image scrambling is presented based on 3 theorems and a corollary which represent the particularities of binary arithmetic.
Abstract: We investigate the quantum Arnold image scrambling proposed by Jiang et al. (Quantum Inf Process 13(5):1223---1236, 2014). It is aimed to realize Arnold and Fibonacci image scrambling in quantum computer. However, the algorithm does not perceive the particularities of "mod $$2^{n}$$2n," multiply by 2, and subtraction in binary arithmetic. In this paper, a possible simplified version is presented based on 3 theorems and a corollary which represent the particularities of binary arithmetic. The theoretical analysis indicates that the network complexity is dropped from 140n$$\sim $$~168n to 28n$$\sim $$~56n and the unitarity of circuits is not destroyed.
64 citations
TL;DR: This work defines a generalization of Lucas sequence by the recurrence relation l m = bl m - 1 + l m - 2 (if m is even) and gives some relations between this sequence and the generalized Fibonacci sequence m = 0 ∞ which is defined in Edson and Yayenie (2009).
58 citations
TL;DR: The solutions are presented in terms of Fibonacci numbers for twelve systems of these fourteen systems of difference equations, with nonzero real initial values x 0 and y 0.
55 citations
TL;DR: In this paper, the Fibonacci generalized quaternions are introduced and well-known identities related to the fibonacci and Lucas numbers are used to obtain the relations regarding these quaternion.
Abstract: In this paper, the Fibonacci generalized quaternions are introduced. We use the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the Fibonacci generalized quaternions are classified by considering the special cases of quaternionic units.
53 citations
TL;DR: In this article, the Hyers-Ulam stability of the linear functional equation of third order was proved for the Fibonacci numbers. But this was in a vector space.
Abstract: Given a vector space , we investigate the solutions of the linear functional equation of third order , which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.
TL;DR: In this article, it was shown that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers, where F 1 = 1, F 2 = 2 and F n+1 = F n + F n − 1.
TL;DR: It is shown that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous "Fibonacci" anyon, whose non- Abelian statistics is powerful enough for universal TQC.
Abstract: The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this Letter, we consider a simple bilayer fractional quantum Hall system with the 1/3 Laughlin state in each layer. We show that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous "Fibonacci" anyon, whose non-Abelian statistics is powerful enough for universal TQC. Our analysis rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions. We provide evidence that the transition can be continuous, at which point the charge gap remains open while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at ν=2/3 in bilayers, as past experiments may not have definitively ruled them out.
TL;DR: In this article, the Hyers-Ulam stability of the generalized Fibonacci functional equation was shown to hold for any ε > 0, where and h are given functions.
Abstract: We prove the Hyers-Ulam stability of the generalized Fibonacci functional equation , where and h are given functions.
TL;DR: In this article, infinite sums derived from the reciprocals of the generalized Fibonacci numbers were derived for the first time, and some new and interesting identities for the generalized FPNs were obtained.
Abstract: We consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.
TL;DR: In this paper, the identities for k-Fibonacci numbers were obtained by using Binet's formula, and another expression for the general term of the sequence, using the ordinary generating function, was provided.
Abstract: We obtain some identities for k-Fibonacci numbers by using its Binet’s formula. Also, another expression for the general term of the sequence, using the ordinary generating function, is provided.
TL;DR: In this article, the authors found that FexTaS2 crystals with x = 1/4 and 1/3 exhibit complicated antiphase and chiral domain structures related to ordering of intercalated Fe ions with 2a × 2a and √ 3a × √3a superstructures, respectively.
Abstract: Common mathematical theories can have profound applications in understanding real materials. The intrinsic connection between aperiodic orders observed in the Fibonacci sequence, Penrose tiling, and quasicrystals is a well-known example. Another example is the self-similarity in fractals and dendrites. From transmission electron microscopy experiments, we found that FexTaS2 crystals with x = 1/4 and 1/3 exhibit complicated antiphase and chiral domain structures related to ordering of intercalated Fe ions with 2a × 2a and √3a × √3a superstructures, respectively. These complex domain patterns are found to be deeply related with the four color theorem, stating that four colors are sufficient to identify the countries on a planar map with proper coloring and its variations for two-step proper coloring. Furthermore, the domain topology is closely relevant to their magnetic properties. Our discovery unveils the importance of understanding the global topology of domain configurations in functional materials.
TL;DR: In this paper, the authors considered the Fibonacci Hamiltonian model of one-dimensional quasicrystals, and provided a detailed description of its spectrum and spectral properties (namely, the optimal Holder exponent of the integrated density of states, the dimension of the density of state measure, the spectrum, and the upper transport exponent) for all values of the coupling constant.
Abstract: We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and provide a detailed description of its spectrum and spectral characteristics (namely, the optimal Holder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) for all values of the coupling constant (in contrast to all previous quantitative results, which could be established only in the regime of small or large coupling).
In particular, we show that the spectrum of this operator is a dynamically defined Cantor set and that the density of states measure is exact-dimensional; this implies that all standard fractal dimensions coincide in each case. We show that all the gaps of the spectrum allowed by the gap labeling theorem are open for all values of the coupling constant. Also, we establish strict inequalities between the four spectral characteristics in question, and provide the exact large coupling asymptotics of the dimension of the density of states measure (for the other three quantities, the large coupling asymptotics were known before).
A crucial ingredient of the paper is the relation between spectral properties of the Fibonacci Hamiltonian and dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We establish exact identities relating the spectral and dynamical quantities, and show the connection between the spectral quantities and the thermodynamic pressure function.
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TL;DR: There exists an aperiodic infinite binary word avoiding the pattern x x x^R, which is the first avoidability result concerning a nonuniform morphism proven purely mechanically.
Abstract: We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010..., the fixed point of the morphism 0 -> 01 and 1 -> 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth. As an application of our method we prove a new result: there exists an aperiodic infinite binary word avoiding the pattern x x x^R. This is the first avoidability result concerning a nonuniform morphism proven purely mechanically.
TL;DR: In this paper, it was shown that the distribution of the number of summands converges to a Gaussian as n → ∞, and generalizations to related decompositions.
Abstract: A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers \(\{F_{n}\}_{n=1}^{\infty }\); Lekkerkerker proved that the average number of summands for integers in [F n , F n+1) is \(n/(\varphi ^{2} + 1)\), with φ the golden mean. Interestingly, the higher moments seem to have been ignored. We discuss the proof that the distribution of the number of summands converges to a Gaussian as n → ∞, and comment on generalizations to related decompositions. For example, every integer can be written uniquely as a sum of the ± F n ’s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely \(-(21 - 2\varphi )/(29 + 2\varphi ) \approx -0.551058\).
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TL;DR: In this article, the Baker-Davenport reduction method in diophantine approximation was used to search for powers of 2 which are sums of two $k-$Fibonacci numbers.
Abstract: For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n}$ be the $k-$Fibonacci sequence which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for powers of 2 which are sums of two $k-$Fibonacci numbers. The main tools used in this work are lower bounds for linear forms in logarithms and a version of the Baker--Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of \cite{BL2} and \cite{BL13}.
TL;DR: In this paper, a fiber-optic quasi-periodic layer and a resonant metal back reflector are used to achieve high absorption in the mid-infrared range.
Abstract: A heterostructure containing a Fibonacci quasi-periodic layer and a resonant metal back reflector is proposed, which can realize wideband absorption. The Fibonacci layer is composed of graphene-based hyperbolic metamaterials and isotropic media to obtain wideband absorption. To enhance absorption, an impedance-matching layer is put on top of the Fibonacci layer. It is shown to absorb roughly 90% of all available electromagnetic waves in an 11 terahertz absorption bandwidth for a transverse magnetic mode at normal angle incidence. The absorption bandwidth is affected by the reflection band gap. Compared with some previous designs, our proposed structure has a larger absorption bandwidth and higher absorption in the mid-infrared range. The results should be valuable in the design of infrared stealth and broadband optoelectronic devices.
TL;DR: In this paper, a new kind of bifocal kino-form lenses based on the Fibonacci sequence was proposed, and the focusing properties of these DOEs were analytically studied and compared with binary-phase FLs.
Abstract: In this paper, we present a new kind of bifocal kinoform lenses in which the phase distribution is based on the Fibonacci sequence. The focusing properties of these DOEs coined Kinoform Fibonacci lenses (KFLs) are analytically studied and compared with binary-phase Fibonacci lenses (FLs). It is shown that, under monochromatic illumination, a KFL drives most of the incoming light into two single foci, improving in this way the efficiency of the FLs. We have also implemented these lenses with a spatial light modulator. The first images obtained with this type of lenses are presented and evaluated.
TL;DR: In this paper, the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and a generic construction for classes of such identities is provided.
Abstract: We show that the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities The utilization of the Fibonacci number formula gives us an upper bound on the length of these 2-variable semigroup identities
TL;DR: A new version of the Kudryashov's method for solving non-integrable p roblems in mathematical physics is presented in this article, where exact solutions of the heat conduction equation and K(m, n) equation with generalized evolution are obtained.
Abstract: A new version of the Kudryashov's method for solving non-integrable p roblems in mathematical physics is presented in this paper. New exact solutions of the heat conduction equation and K(m, n) equation with generalized evolution are obtained by using this method. The solutions gained from the proposed method have been verifi ed with obtained by the (G 0 /G)-expansion method.
TL;DR: In this paper, generalized symmetric functions are used to find explicit formulas of the Fibonacci numbers, and of the Tchebychev polynomials of first and second kinds.
Abstract: In this paper, we calculate the generating functions by using the con- cepts of symmetric functions. Although the methods cited in previous works are in principle constructive, we are concerned here only with the question of manipulating combinatorial objects, known as symmetric op- erators. The proposed generalized symmetric functions can be used to find explicit formulas of the Fibonacci numbers, and of the Tchebychev polynomials of first and second kinds.
TL;DR: In this paper, explicit formulae of spectral norms for g -circulant matrices are investigated and some numerical tests are presented to verify the results.
TL;DR: In this article, the invertibility of circulant-type matrices with the -Fibonacci and Lucas numbers is discussed. But the authors focus on the transformation matrices.
Abstract: Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with the -Fibonacci and -Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).
28 Jul 2014
TL;DR: In this article, the authors proved Bruckman and Anderson's conjecture by studying the algebraic group G : x 2 − 5y 2 = 1 and relating Z(p) to the order of � = (3/2, 1/2) ∈ G(Fp).
Abstract: For a prime p, let Z(p) be the smallest positive integer n so that p divides Fn, the nth term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for �(m), the density of primes p for which m|Z(p) on the basis of numerical evidence. We prove Bruckman and Anderson's conjecture by studying the algebraic group G : x 2 − 5y 2 = 1 and relating Z(p) to the order of � = (3/2,1/2) ∈ G(Fp). We are then able to use Galois theory and the Chebotarev density theorem to compute �(m).
TL;DR: A general construction method for cross-bifix-free sequences based on kernels, applicable to a limited number of so-called “regular kernel sets”, is proposed and properties of such sequences with an outline for further research are discussed.
Abstract: Cross-bifix-free sets are sets of bifix-free sequences with the property that no prefix of any sequence is a suffix of any other sequence. This paper presents a general construction method for cross-bifix-free sequences based on kernels. The cardinality of cross-bifix-free sets follows the Fibonacci progression. A simplified method, applicable to a limited number of so-called "regular kernel sets", is proposed as well. Properties of such sequences with an outline for further research are discussed.