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Showing papers on "Fibonacci number published in 2020"


Journal ArticleDOI
TL;DR: In this paper, the authors explore the localization properties of waves in a novel family of quasiperiodic chains obtained when continuously interpolating between two paradigmatic limits: the Aubry-Andre model and the Fibonacci chain.
Abstract: Conduction through materials crucially depends on how ordered the materials are. Periodically ordered systems exhibit extended Bloch waves that generate metallic bands, whereas disorder is known to limit conduction and localize the motion of particles in a medium1,2. In this context, quasiperiodic systems, which are neither periodic nor disordered, demonstrate exotic conduction properties, self-similar wavefunctions and critical phenomena3. Here, we explore the localization properties of waves in a novel family of quasiperiodic chains obtained when continuously interpolating between two paradigmatic limits4: the Aubry–Andre model5,6, famous for its metal-to-insulator transition, and the Fibonacci chain7,8, known for its critical nature. We discover that the Aubry–Andre model evolves into criticality through a cascade of band-selective localization/delocalization transitions that iteratively shape the self-similar critical wavefunctions of the Fibonacci chain. Using experiments on cavity-polariton devices, we observe the first transition and reveal the microscopic origin of the cascade. Our findings offer (1) a unique new insight into understanding the criticality of quasiperiodic chains, (2) a controllable knob by which to engineer band-selective pass filters and (3) a versatile experimental platform with which to further study the interplay of many-body interactions and dissipation in a wide range of quasiperiodic models. The localization properties of waves in the quasiperiodic chains described by the Aubry–Andre model and Fibonacci model are investigated. Passing from one model to the other, the system develops a cascade of delocalization transitions.

77 citations


Journal ArticleDOI
01 Jan 2020
TL;DR: The proposed algorithm is inspired by the golden ratio of plant and animal growth which is formulated by the well-known mathematician Fibonacci and is compared with those of 11 well-regarded state-of-the-art optimization algorithms.
Abstract: A novel parameter-free meta-heuristic optimization algorithm known as the golden ratio optimization method (GROM) is proposed. The proposed algorithm is inspired by the golden ratio of plant and animal growth which is formulated by the well-known mathematician Fibonacci. He introduced a series of numbers in which a number (except the first two numbers) is equal to the sum of the two previous numbers. In this series, the ratio of two consecutive numbers is almost the same for all the numbers and is known as golden ratio. This ratio can be extensively found in nature such as snail lacquer part and foliage growth of trees. The proposed approach employed this golden ratio to update the solutions in an optimization algorithm. In the proposed method, the solutions are updated in two different phases to achieve the global best answer. There is no need for any parameter tuning, and the implementation of the proposed method is very simple. In order to evaluate the proposed method, 29 well-known benchmark test functions and also 5 classical engineering optimization problems including 4 mechanical engineering problems and 1 electrical engineering problem are employed. Using several test functions, the performance of the proposed method in solving different problems including discrete, continuous, high dimension, and high constraints problems is testified. The results of the proposed method are compared with those of 11 well-regarded state-of-the-art optimization algorithms. The comparisons are made from different aspects such as the final obtained answer, the speed and behavior of convergence, and CPU time consumption. Superiority of the purposed method from different points of views can be concluded by means of comparisons.

67 citations



Journal ArticleDOI
TL;DR: In this article, the q − Lucas hybrid numbers and the q- Fibonacci hybrid numbers were defined and some algebraic properties of the q−Lucas hybrid number were given.
Abstract: In the paper, we define the q − Fibonacci hybrid numbers and the q − Lucas hybrid numbers, respectively. Then, we give some algebraic properties of q − Fibonacci hybrid numbers and the q − Lucas hybrid numbers.

33 citations


Journal ArticleDOI
01 Sep 2020-Optik
TL;DR: In this article, the transmittance spectrum in a quasi-periodic one-dimensional photonic crystal (1D-PC) consisting of superconducting materials (HgBa 2 Ca 2 Cu 3 O 8 + δ ) and semiconductor (GaAs) was calculated.

27 citations


Journal ArticleDOI
22 Jun 2020-Symmetry
TL;DR: A new class of q-starlike functions associated with k-Fibonacci numbers is defined, a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) under rotations.
Abstract: In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant.

26 citations


Journal ArticleDOI
TL;DR: It is shown how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata, and it is argued that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.
Abstract: We show how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata. We start by recalling the relationship between first-order logic and finite automata, and use this relationship to solve several problems involving sums of numbers defined by their base-2 and Fibonacci representations. Next, we turn to harder results. Recently, Cilleruelo, Luca, & Baxter proved, for all bases b ≥ 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome (Cilleruelo et al., Math. Comput. 87, 3023–3055, 2018). However, the cases b = 2, 3, 4 were left unresolved. We prove that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem of palindromes as an additive basis. We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.

22 citations


Journal ArticleDOI
TL;DR: In this paper, the transmittance characteristics of one dimensional quasi-periodic photonic crystals are investigated and the effect of volume density of the plasma and the volume fraction of the nanocomposite material on the properties of the photonic band gaps are studied in detail.
Abstract: In this communication, the transmittance characteristics of one dimensional quasi-periodic photonic crystals is investigated. The proposed structure consists basically of plasma and nanocomposite layers arranged in accordance to the Fibonacci sequence. The effect of volume density of the plasma and the volume fraction of the nanocomposite material on the properties of the photonic band gaps are studied in detail. The effect of layers thicknesses and the sequence number are also taken into account. We believe that such structures can be potentially beneficial in many novel future applications such as multichannel and pass band filters and optical switches. Additionally, such photonic crystals can be used to design tunable optical devises through controlling the plasma and nanoparticle densities.

22 citations


Journal ArticleDOI
TL;DR: Fibonacci or bifocal terahertz (THz) imaging is demonstrated experimentally employing silicon diffractive zone plate (SDZP) in a continuous wave mode as mentioned in this paper.
Abstract: Fibonacci or bifocal terahertz (THz) imaging is demonstrated experimentally employing silicon diffractive zone plate (SDZP) in a continuous wave mode. Images simultaneously recorded in two different planes are exhibited at 0.6 THz frequency with the spatial resolution of wavelength. Multi-focus imaging operation of the Fibonacci lens is compared with a performance of the conventional silicon phase zone plate. Spatial profiles and focal depth features are discussed varying the frequency from 0.3 THz to 0.6 THz. Good agreement between experimental results and simulation data is revealed.

22 citations


Journal ArticleDOI
TL;DR: Fingerprints of the aperiodicity in the hoppings are found when distributional measures such as the Shannon and von Neumann entropies, the Inverse Participation Ratio, the Jensen-Shannon dissimilarity, and the kurtosis are computed, which allow assessing informational and delocalization features arising from these protocols and understanding the impact of linear and nonlinear correlations of the jump sequence in a quantum walk as well.
Abstract: We analyze a set of discrete-time quantum walks for which the displacements on a chain follow binary aperiodic jumps according to three paradigmatic sequences: Fibonacci, Thue-Morse, and Rudin-Shapiro. We use a generalized Hadamard coin, C[over ]_{H}, as well as a generalized Fourier coin, C[over ]_{K}. We verify the QW experiences a slowdown of the wave packet spreading, σ^{2}(t)∼t^{α}, by the aperiodic jumps whose exponent, α, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely, (1) while the superdiffusive regime (1<α<2) is predominant, α displays an unusual sensibility with the type of coin operator where the more pronounced differences emerge for the Rudin-Shapiro and random protocols and (2) even though the angle θ of the coin operator is homogeneous in space and time, there is a nonmonotonic dependence of α with θ. Fingerprints of the aperiodicity in the hoppings are also found when distributional measures such as the Shannon and von Neumann entropies, the Inverse Participation Ratio, the Jensen-Shannon dissimilarity, and the kurtosis are computed, which allow assessing informational and delocalization features arising from these protocols and understanding the impact of linear and nonlinear correlations of the jump sequence in a quantum walk as well. Finally, we argue the spin-lattice entanglement is enhanced by aperiodic jumps.

19 citations


Journal ArticleDOI
TL;DR: An efficient method for finding the approximate solution of fractional delay differential equations based on hybrid functions of block-pulse and fractional-order Fibonacci polynomials is proposed, which includes the Hutchinson model which describes the rate of population growth.
Abstract: The aim of the current paper is to propose an efficient method for finding the approximate solution of fractional delay differential equations. This technique is based on hybrid functions of block-pulse and fractional-order Fibonacci polynomials. First, we define fractional-order Fibonacci polynomials. Next, using Fibonacci polynomials of fractional-order, we introduce a new set of basis functions. These new functions are called fractional-order Fibonacci-hybrid functions (FFHFs) which are appropriate for the approximation of smooth and piecewise smooth functions. The Riemann–Liouville integral operational matrix and delay operational matrix of the FFHFs are obtained. Then, using these matrices and collocation method, the problem is reduced to a system of algebraic equations. Using Newton’s iterative method, we solve this system. Some examples are proposed to test the efficiency and effectiveness of the present method. Given the application of these kinds of fractional equations in the modeling of many phenomena, a numerical example of this work includes the Hutchinson model which describes the rate of population growth.

Journal ArticleDOI
TL;DR: In this article, the Fibonacci and Lucas hybrinomials, i.e., polynomials which are generalization of complex, hyperbolic and dual numbers, are introduced and studied.
Abstract: The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we introduce and study the Fibonacci and Lucas hybrinomials, i.e. polynomials, which are a generalizati...

Journal ArticleDOI
21 Apr 2020
TL;DR: In this paper, it was shown that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c c c … c. The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.
Abstract: In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

Posted Content
TL;DR: In this paper, the ideal non-abelian many-body spectral theory of ideal anyons has been developed, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on $N$ strands.
Abstract: We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on $N$ strands. Allowing for arbitrary rank (dependent on $N$) and non-abelian representations, and letting $N \to \infty$, this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincare and Hardy inequalities) and a local exclusion principle (also implying a Lieb-Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.

Journal ArticleDOI
TL;DR: In this paper, the authors studied non-equilibrium dynamics in driven 1+1D conformal field theories with periodic, quasi-periodic, and random driving, and showed that the non-heating phase of a CFT with a Fibonacci sequence exhibits self-similarity and can be arbitrarily small.
Abstract: In this paper and its sequel, we study non-equilibrium dynamics in driven 1+1D conformal field theories (CFTs) with periodic, quasi-periodic, and random driving We study a soluble family of drives in which the Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength The resulting time evolution is then captured by a Mobius coordinate transformation In this Part I, we establish the general framework and focus on the first two classes In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement/energy evolution in different phases, ie the heating, non-heating phases and the phase transition between them In quasi-periodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence We find that (i) the non-heating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the non-heating fixed point, where the entanglement entropy/energy oscillate at the Fibonacci numbers, but grow logarithmically/polynomially at the non-Fibonacci numbers; (v) for certain choices of driving Hamiltonians, the non-heating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasi-crystal In addition, another quasi-periodically driven CFT with an Aubry-Andre like sequence is also studied We compare the CFT results to lattice calculations and find remarkable agreement

Posted Content
TL;DR: This work considers a two-dimensional quantum memory of qubits on a torus which encode the extended Fibonacci string-net code, and devise strategies for error correction when those qubits are subjected to depolarizing noise.
Abstract: We consider a two-dimensional quantum memory of qubits on a torus which encode the extended Fibonacci string-net code, and devise strategies for error correction when those qubits are subjected to depolarizing noise. Building on the concept of tube algebras, we construct a set of measurements and of quantum gates which map arbitrary qubit errors to the string-net subspace and allow for the characterization of the resulting error syndrome in terms of doubled Fibonacci anyons. Tensor network techniques then allow to quantitatively study the action of Pauli noise on the string-net subspace. We perform Monte Carlo simulations of error correction in this Fibonacci code, and compare the performance of several decoders. For the case of a fixed-rate sampling depolarizing noise model, we find an error correction threshold of 4.75% using a clustering decoder. To the best of our knowledge, this is the first time that a threshold has been estimated for a two-dimensional error correcting code for which universal quantum computation can be performed within its code space.

Journal ArticleDOI
24 Mar 2020
TL;DR: In this paper, closed forms of the sum formulas for the squares of generalized Fibonacci numbers are presented, and the proofs to indicate how these formulas, in general, were discovered.
Abstract: In this paper, closed forms of the sum formulas for the squares of generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

Journal ArticleDOI
TL;DR: In this article, the authors proposed the use of MoS2 monolayers as part of the defect in Fibonacci and Thue-Morse defective quasiphotonic crystals (DQPCs) to create an adjustable defect mode with relatively high absorption.
Abstract: In recent years, two-dimensional materials such as MoS2 monolayers have attracted a lot of attention due to their high absorption. This study proposes the use of MoS2 monolayers as part of the defect in Fibonacci and Thue-Morse defective quasiphotonic crystals (DQPCs) to create an adjustable defect mode with relatively high absorption. The wavelength adjustability of the defect mode is investigated by parameters such as generation number, periodicity, and type of sequence used in DQPCs. The results revealed that using DQPCs can help enhance absorption and enables the adjustment of the defect mode as its generation number changes. Moreover, a Fibonacci DQPC generally requires much fewer layers in comparison to a Thue-Morse DQPC. To conclude, it is possible to achieve a wavelength-adjustable absorption of more than 90% with a Fibonacci DQPC.

Journal ArticleDOI
TL;DR: In this article, the authors studied the behavior of the difference equation for arbitrary positive real numbers and gave the solution of some special cases of this equation, where the initial conditions $x-2,\ x-1, \x-0,\x-1},\ x 0} are arbitrary real numbers, and $a,b,c,d$ are positive constants.
Abstract: In this paper, we study the behavior of the difference equation $x_{n+1}=ax_{n}+\dfrac{bx_{n}x_{n-1}}{cx_{n-1}+dx_{n-2}},~n=0,1,\ldots,$ where the initial conditions $x_{-2},\ x_{-1},\ x_{0}$ are arbitrary positive real numbers and $a,b,c,d$ are positive constants. Also, we give the solution of some special cases of this equation.


Posted Content
TL;DR: In this article, the authors construct volume-filling embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost-toric varieties.
Abstract: This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost-toric varieties. The construction uniformly recovers the sharp sequences for the Fibonacci Staircase of McDuff-Schlenk, the Pell Staircase of Frenkel-Muller and the Cristofaro-Gardiner-Kleinman's Staircase, and adds new infinite sequences of sharp ellipsoid embeddings. In addition, we initiate the study of symplectic tropical curves for almost-toric fibrations and emphasize the connection to quiver combinatorics.

Journal ArticleDOI
TL;DR: In this paper, a generic quantum-net wave function with two tuning parameters dual with each other is proposed, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled Potts models.
Abstract: The non-Abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wave function with two tuning parameters dual with each other, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled ${\ensuremath{\phi}}^{2}$-state Potts models, where $\ensuremath{\phi}=(\sqrt{5}+1)/2$ is the golden ratio. By developing the tensor network representation of this wave function on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-Abelian Fibonacci topological phase is enclosed by three distinct nontopological phases and their dual phases of a single ${\ensuremath{\phi}}^{2}$-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those nontopological phases.

Journal ArticleDOI
TL;DR: This work considers k-Fibonacci cubes, which are obtained as subgraphs of Fibonaccia as well as hypercubes and Fibonacci cube as classical models for interconnection networks with interesting graph theoretic properties.
Abstract: Hypercubes and Fibonacci cubes are classical models for interconnection networks with interesting graph theoretic properties. We consider k-Fibonacci cubes, which we obtain as subgraphs of Fibonacc...

Journal ArticleDOI
TL;DR: In this article, results from different disciplines of science were compared to show their intimate interweaving with each other having in common the golden ratio φ and its fifth power φ5.
Abstract: In this contribution results from different disciplines of science were compared to show their intimate interweaving with each other having in common the golden ratio φ respectively its fifth power φ5. The research fields cover model calculations of statistical physics associated with phase transitions, the quantum probability of two particles, new physics of everything suggested by the information relativity theory (IRT) including explanations of cosmological relevance, the e-infinity theory, superconductivity, and the Tammes problem of the largest diameter of N non-overlapping circles on the surface of a sphere with its connection to viral morphology and crystallography. Finally, Fibonacci anyons proposed for topological quantum computation (TQC) were briefly described in comparison to the recently formulated reverse Fibonacci approach using the Janicko number sequence. An architecture applicable for a quantum computer is proposed consisting of 13-step twisted microtubules similar to tubulin microtubules of living matter. Most topics point to the omnipresence of the golden mean as the numerical dominator of our world.

Journal ArticleDOI
TL;DR: A new method of characteristic sequences based on linear algebra technique is introduced that provides an efficient tool for computing the length function in non-associative case and an upper bound for the length of an arbitrary locally complex algebra is obtained.

Journal ArticleDOI
TL;DR: For an integer d ≥ 2 which is square-free, this paper showed that there is at most one value of the positive integer x participating in the Pell equation x 2 − d y 2 = ± 1, which is a k-generalized Fibonacci number, with a couple of parametric exceptions which they completely characterize.

Journal ArticleDOI
TL;DR: In this article, the authors introduce a class of Parity-time symmetric elastodynamic metamaterials (Ed-MetaMater) whose Hermitian counterpart exhibits a frequency spectrum with unfolding (fractal) symmetries.
Abstract: We introduce a class of Parity-Time symmetric elastodynamic metamaterials (Ed-MetaMater) whose Hermitian counterpart exhibits a frequency spectrum with unfolding (fractal) symmetries. Our study reveals a scale-free formation of exceptional points (EP) whose density is dictated by the fractal dimension of their Hermitian spectra. Demonstrated in a quasi-periodic Aubry-Harper, a geometric H-tree-fractal, and an aperiodic Fibonacci Ed-MetaMater, the universal route for EP-formation is established via a coupled mode theory model with controllable fractal spectrum. This universality will enable the rational design of novel Ed-MetaMater for hypersensitive sensing and elastic wave control.

Posted Content
TL;DR: In this article, some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials have been derived, which involve hypergeometric functions of the type $_2F_{1}(z)$ for certain $z.
Abstract: This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain $z$. Several new connection formulae between some famous polynomials such as Fibonacci, Lucas, Pell, Fermat, Pell-Lucas, and Fermat-Lucas polynomials are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.

Journal ArticleDOI
TL;DR: In this article, all possible determinants of Bohemian Hessenberg matrices with subdiagonal fixed-to-one determinants are enumerated, and several conjectures recently stated by Corless and Thornton follow from their results.
Abstract: A matrix is Bohemian if its elements are taken from a finite set of integers. We enumerate all possible determinants of Bohemian upper Hessenberg matrices with subdiagonal fixed to one, and consider the special case of families of matrices with only zeros on the main diagonal, whose determinants proved to be related to a generalization of Fibonacci numbers. Several conjectures recently stated by Corless and Thornton follow from our results.

Journal ArticleDOI
TL;DR: P-rational real quadratic fields are characterized in terms of generalized Fibonacci numbers to give numerical evidence to a conjecture of Greenberg asserting the existence of p-rational multi-quadratic fields of arbitrary degree.
Abstract: We characterize p-rational real quadratic fields in terms of generalized Fibonacci numbers. We then use this characterization to give numerical evidence to a conjecture of Greenberg asserting the existence of p-rational multi-quadratic fields of arbitrary degree $$2^{t}$$ , $$t\ge 1$$ .