Showing papers on "Fibonacci number published in 2018"
TL;DR: The definition of absolute equivalence is given to investigate the transformation between Galois nonlinear feedback shift registers (NLFSRs) and Fibonacci NLFSRs and some interesting results are presented.
68 citations
TL;DR: The main purpose of as discussed by the authors is to construct new families of special inequalities with their generating functions, which are related to many well-known numbers, such as Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers.
Abstract: The main purpose of this paper is to construct new families of special
numbers with their generating functions. These numbers are related to many
well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas
numbers, Stirling numbers of the second kind and central factorial numbers.
Our other inspiration of this paper is related to the Golombek's problem
[15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are
not only related to the Golombek's problem, but also computation of the
negative order Euler numbers. We compute a few values of the numbers which
are given by tables. We give some applications in probability and
statistics. That is, special values of mathematical expectation of the
binomial distribution and the Bernstein polynomials give us the value of our
numbers. Taking derivative of our generating functions, we give partial
differential equations and also functional equations. By using these
equations, we derive recurrence relations and some formulas of our numbers.
Moreover, we come up with a conjecture with two open questions related to
our new numbers. We give two algorithms for computation of our numbers. We
also give some combinatorial applications, further remarks on our new
numbers and their generating functions.
47 citations
TL;DR: In this article, the authors introduced and investigated new subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers, and determined Fekete-Szegö inequalities for these function classes.
Abstract: Abstract In this paper, we introduce and investigate new subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers. Furthermore, we find estimates of first two coefficients of functions in these classes. Also, we determine Fekete-Szegö inequalities for these function classes.
47 citations
10 Jul 2018
TL;DR: In this paper, the authors present a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations.
Abstract: Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.
37 citations
TL;DR: In this article, a model of interacting Majorana fermions was introduced to describe a superconducting phase with a topological order characterized by the Fibonacci topological field theory.
Abstract: We introduce a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory, which is based on a SO(7)_{1}/(G_{2})_{1} coset factorization, leads to a solvable one-dimensional model that is extended to two dimensions using a network construction. In addition to providing a description of the Fibonacci phase without parafermions, our theory predicts a closely related "anti-Fibonacci" phase, whose topological order is characterized by the tricritical Ising model. We show that Majorana fermions can split into a pair of Fibonacci anyons, and propose an interferometer that generalizes the Z_{2} Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.
35 citations
TL;DR: This work aims to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations.
Abstract: Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.
28 citations
TL;DR: In this article, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions.
27 citations
TL;DR: Application of the proposed Fibonacci Indicator Algorithm in a wide set of benchmark functions has asserted its capability to deal with difficult optimization problems.
27 citations
TL;DR: The main objective of this paper is to demonstrate that Lucas polynomial based method is convenient for 1D and 2D nonlinear problems.
27 citations
TL;DR: In this article, the spectral type of the corresponding diffraction measure is analyzed for its geometric realisation with prototiles (intervals) of natural length, and all of them have singular diffraction, either of pure point type or essentially singular continuous.
Abstract: The family of primitive binary substitutions defined by $$1 \mapsto 0 \mapsto 0 1^m$$
with $$m\in \mathbb {N}$$
is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation (
$$m=1$$
), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous.
26 citations
TL;DR: In this article, the authors studied the problem of finding the n-th and m-th Fibonacci numbers under the assumption that n ≡ m ( mod 2 ) and p ≥ 2.
TL;DR: In this paper, it was shown that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-dy^2=\pm 1$ which is a Fibonacci number.
Abstract: For an integer $d>2$ which is not a square, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-dy^2=\pm 1$ which is a Fibonacci number.
TL;DR: McDuff and Schlenk as discussed by the authors showed that when the ellipsoid is close to round, the answer is given by an infinite staircase determined by the odd-index Fibonacci numbers.
Abstract: McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers. We show that this result still holds in higher dimensions when we "stabilize" the embedding problem.
TL;DR: Fibonacci or bifocal terahertz (THz) imaging is demonstrated experimentally employing a silicon diffractive zone plate in continuous wave mode with good agreement between experimental results and simulation data.
Abstract: Fibonacci or bifocal terahertz (THz) imaging is demonstrated experimentally employing a silicon diffractive zone plate in continuous wave mode. Images simultaneously recorded in two different planes are exhibited at 0.6 THz frequency with the spatial resolution of wavelength. Multifocus 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 to 0.6 THz. Good agreement between experimental results and simulation data is revealed.
TL;DR: In this article, the authors consider a prototypical integrable spin system, the spin-$1/2$ transverse field Ising model in one dimension, in a pulsed magnetic field.
Abstract: Driven many-body quantum systems where some parameter in the Hamiltonian is varied quasiperiodically in time may exhibit nonequilibrium steady states that are qualitatively different from their periodically driven counterparts. Here we consider a prototypical integrable spin system, the spin-$1/2$ transverse field Ising model in one dimension, in a pulsed magnetic field. The time dependence of the field is taken to be quasiperiodic by choosing the pulses to be of two types that alternate according to a Fibonacci sequence. We show that a steady state emerges after an exponentially long time when local properties (or equivalently, reduced density matrices of subsystems with size much smaller than the full system) are considered. We use the temporal evolution of certain coarse-grained quantities in momentum space to understand this nonequilibrium steady state in more detail and show that unlike the previously known cases, this steady state is neither described by a periodic generalized Gibbs ensemble nor by an infinite temperature ensemble. Finally, we study a toy problem with a single two-level system driven by a Fibonacci sequence; this problem shows how sensitive the nature of the final steady state is to the different parameters.
TL;DR: In this article, the algebraic properties of bicomplex Fibonacci quaternions which are connected with BICOMplex numbers are investigated and Binet's formula, Cassini's identity, Catalan's identity and real representation of these quaternion are given.
Abstract: In this paper, bicomplex Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex Fibonacci quaternions which are connected with bicomplex numbers and Fibonacci numbers are investigated. Furthermore, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and real representation of these quaternions are given.
TL;DR: In this article, an invariant function of the circular frequency, the Kohmoto's invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index.
Abstract: Harmonic axial waves in quasiperiodic-generated structured rods are investigated. The focus is on infinite bars composed of repeated elementary cells designed by adopting generalised Fibonacci substitution rules, some of which represent examples of one-dimensional quasicrystals. Their dispersive features and stop/pass band spectra are computed and analysed by imposing Floquet–Bloch conditions and exploiting the invariance properties of the trace of the relevant transfer matrices. We show that for a family of generalised Fibonacci substitution rules, corresponding to the so-called precious means, an invariant function of the circular frequency, the Kohmoto’s invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index. Other parts of the spectrum are instead occupied by almost constant ultrawide band gaps. The Kohmoto’s invariant also explains the existence of particular frequencies, named canonical frequencies, associated with closed orbits on the geometrical three-dimensional representation of the invariant. The developed theory represents an important advancement towards the realisation of elastic quasicrystalline metamaterials.
TL;DR: For an integer c having at least two representations as a difference between a k-generalized Fibonacci number and a power of 2 for any fixed ǫ ≥ 4, the work of as mentioned in this paper extends previous work from Ddamulira et al.
Abstract: For an integer $$ k \ge 2 $$
, let $$ \{F^{(k)}_{n} \}_{n\ge 0}$$
be the k–generalized 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 find all integers c having at least two representations as a difference between a k–generalized Fibonacci number and a power of 2 for any fixed $$k \ge 4$$
. This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411–421, 2017. https://doi.org/10.1007/s12044-017-0338-3
) for the case $$k=2$$
and Bravo et al. (Bull Korean Math Soc 54(3): 069–1080, 2017. https://doi.org/10.4134/BKMS.b160486
) for the case $$k=3$$
.
TL;DR: In this article, the optical refractive index of the MoS2 monolayer was determined based on the Lorentz-Drude-Gauss model and the effects of Fibonacci order, periodicity, incident light angle and polarization were included in the calculations.
Abstract: 2D molybdenum disulfide MoS2, has represented potential applications in optoelectronic devices based on their promising optical absorption responses. However, for practical applications, absorption should increase furthermore in a wide wavelength window. In this paper, we design Fibonacci photonic crystals (PCs) based on Si, SiO2 and MoS2 monolayer and we calculate their absorption responses based on the transfer matrix method. The optical refractive index of the MoS2 monolayer was determined based on the Lorentz–Drude–Gauss model. Effects of Fibonacci order, periodicity, incident light angle and polarization are included in our calculations. Finally, an absorption as large as 90% in a wide optical wavelength range is achieved for both polarizations and incident angle down to 60°. Our results are useful for designing photonic devices with high absorption efficiency.
TL;DR: A new method is shown to prove real-rootedness of the independence polynomials of certain families of trees and of centipedes, caterpillars, and Zhu’s theorem.
TL;DR: This paper proposes a high-rate and high-capacity measurement-device-independent quantum key distribution (MDI-QKD) protocol with Fibonacci-valued and Lucas-valued orbital angular momentum (OAM) entangled states in free space that can distill multiple fibonacci key matrices from the same block of data, thus reducing the statistical fluctuations in the sample and increasing the final QKD rate.
Abstract: This paper proposes a high-rate and high-capacitymeasurement-device-independent quantum key distribution (MDI-QKD)protocol with Fibonacci-valued and Lucas-valued orbital angularmomentum (OAM) entangled states in free space. In the existingMDI-OAM-QKD protocols, the main encoding algorithm handles encodednumbers in a bit-by-bit manner. To design a fast encoding algorithm,we introduce a Fibonacci matrix coding algorithm, by which, encodednumbers are separated into segments longer than one bit. By doingso, when compared to the existing MDI-OAM-QKD protocols, the newprotocol can effectively increase the key rate and the codingcapacity. This is because Fibonacci sequences are used in preparingOAM entangled states, reducing the misattribution errors (which slowdown the execution cycle of the entire QKD) in QKD protocols.Moreover, our protocol keeps the data blocks as small as possible,so as to have more blocks in a given time interval. Mostimportantly, our proposed protocol can distill multiple Fibonaccikey matrices from the same block of data, thus reducing thestatistical fluctuations in the sample and increasing the final QKDrate. Last but not the least, the sender and the receiver can omitclassical information exchange and bit flipping in the secure keydistillation stage.
18 Dec 2018
TL;DR: In this article, the structural properties of the Fibonacci polynomials and numbers and their identities were investigated. And the main results were obtained by using a new second-order nonlinear recursive sequence, the properties of power series, and combinatorial methods.
Abstract: The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods.
TL;DR: An experimental realisation of the Fibonacci square grid structure in a molecular overlayer is shown, by selective adsorption at specific sites, and the use of molecules as a surface chemical probe to identify atomic species on similar metallic alloy surfaces is demonstrated.
Abstract: Quasicrystals differ from conventional crystals and amorphous materials in that they possess long-range order without periodicity. They exhibit orders of rotational symmetry which are forbidden in periodic crystals, such as five-, ten-, and twelve-fold, and their structures can be described with complex aperiodic tilings such as Penrose tilings and Stampfli–Gaehler tilings. Previous theoretical work explored the structure and properties of a hypothetical four-fold symmetric quasicrystal—the so-called Fibonacci square grid. Here, we show an experimental realisation of the Fibonacci square grid structure in a molecular overlayer. Scanning tunnelling microscopy reveals that fullerenes (C60) deposited on the two-fold surface of an icosahedral Al–Pd–Mn quasicrystal selectively adsorb atop Mn atoms, forming a Fibonacci square grid. The site-specific adsorption behaviour offers the potential to generate relatively simple quasicrystalline overlayer structures with tunable physical properties and demonstrates the use of molecules as a surface chemical probe to identify atomic species on similar metallic alloy surfaces. Quasicrystals possess long range order but no translational symmetry, and rotational symmetries that are forbidden in periodic crystals. Here, a fullerene overlayer deposited on a surface of an icosahedral intermetallic quasicrystal achieves a Fibonacci square grid structure, by selective adsorption at specific sites.
01 Jan 2018
TL;DR: In this article, the authors examined various biological structures based on Fibonacci numbers and found that the energy sources should be connected at both ends of DNA structure; single source is not suitable for energy transmission and the array form of eye's receptor cell is enabled to capture the clocking conduction, localization and delocalization nature of field.
Abstract: Although the notion of an integrating equation of life has yet to be discovered, the Fibonacci order may institute a basis for such a growth. We examined various biological structures based on Fibonacci numbers. We have observed that (i) for wasp Fibonacci’s sequence increases the information amount. (ii) The energy sources should be connected at both ends of DNA structure; single source is not suitable for energy transmission. (iii) Array form of eye’s receptor cell is enabled to capture the clocking conduction, localization and delocalization nature of field. We also identified the entire resonance peaks for every reported structure. Fibonacci-based structures may be used in biomedical applications like as to understand the signal propagation along the structures.
01 Apr 2018
TL;DR: By applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, some identities and relations are derived including the combinatorsial numbers, the apostol-Bernoulli polynomes and numbers, and the Euler numbers and Apostol-Euler numbers.
Abstract: In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y
1
n
,
k
;
λ
. Finally, we make some remarks and observations regarding these identities and relations.
01 Jun 2018
TL;DR: In this paper, the authors derived expressions for sums of first, second, third and fourth powers of Fibonacci and Lucas numbers and their alternating versions and presented new results for the first and fourth order power sums.
Abstract: Abstract: We derive expressions for sums of first, second, third and fourth powers of Fibonacci and Lucas numbers and their alternating versions. On our way of exploration we rediscover some known results and present new. Focusing on third and fourth order power sums, our findings complete those of Clary and Hemenway, Melham and Adegoke.
TL;DR: Through an appreciation of some mathematical definitions and recent theorems, the current evolutionary content of mathematical formulations discussed over this writing can be understood and the verification of numerical and algebraic behavior correspondingly to Generalized Fibonacci model is indicated.
Abstract: This paper discusses a proposal for exploration and verification of numerical and algebraic behavior correspondingly to Generalized Fibonacci model. Thus, it develops a special attention to the class of Fibonacci quaternions and Fibonacci octonions and with this assumption, the work indicates an investigative and epistemological route, with assistance of software CAS Maple. The advantage of its use can be seen from the algebraic calculation of some Fibonacci’s identities that showed unworkable without the technological resource. Moreover, through an appreciation of some mathematical definitions and recent theorems, we can understand the current evolutionary content of mathematical formulations discussed over this writing. On the other hand, the work does not ignore some historical elements which contributed to the discovery of quaternions by the mathematician William Rowan Hamilton (1805 – 1865). Finally, with the exploration of some simple software’s commands allows the verification and, above all, the comparison of the numerical datas with the theorems formally addressed in some academic articles.
09 Oct 2018
TL;DR: In this article, a new 2-Fibonacci polynomials were defined by using terms of a new family of Fibonacci numbers given in [4].
Abstract: In the present study, we define new 2-Fibonacci polynomials by using terms of a new family of Fibonacci numbers given in [4]. We show that there is a relationship between the coefficient of the 2-Fibonacci polynomials and Pascal’s triangle. We give some identities of the 2-Fibonacci polynomials. Afterwards, we compare the polynomials with known Fibonacci polynomials. We also express 2-Fibonacci polynomials by the Fibonacci polynomials. Furthermore, we prove some theorems related to the polynomials. Also, we introduce the derivative of the 2-Fibonacci polynomials.
TL;DR: The remarkable unitary braid group representations associated with Majorana fermions are studied to study the relationships between topology, quantum computing, and the properties of Fermions.
Abstract: This paper is an introduction to relationships between topology, quantum computing, and the properties of Fermions. In particular, we study the remarkable unitary braid group representations associated with Majorana fermions.
TL;DR: The notion of (a, b ) -cores is closely related to rational (a, b )-Dyck paths via the bijection due to J. Anderson as mentioned in this paper.