Showing papers on "Fibonacci number published in 2021"
TL;DR: This work presents a new algorithm for image encryption using a hyperchaotic system and Fibonacci Q-matrix, which achieved an excellent security level and outperformed the existing image encryption algorithms.
Abstract: In the age of Information Technology, the day-life required transmitting millions of images between users. Securing these images is essential. Digital image encryption is a well-known technique used in securing image content. In image encryption techniques, digital images are converted into noise images using secret keys, where restoring them to their originals required the same keys. Most image encryption techniques depend on two steps: confusion and diffusion. In this work, a new algorithm presented for image encryption using a hyperchaotic system and Fibonacci Q-matrix. The original image is confused in this algorithm, utilizing randomly generated numbers by the six-dimension hyperchaotic system. Then, the permutated image diffused using the Fibonacci Q-matrix. The proposed image encryption algorithm tested using noise and data cut attacks, histograms, keyspace, and sensitivity. Moreover, the proposed algorithm’s performance compared with several existing algorithms using entropy, correlation coefficients, and robustness against attack. The proposed algorithm achieved an excellent security level and outperformed the existing image encryption algorithms.
52 citations
TL;DR: In this paper, a new trapezoidal longitudinal outer fin distributed by Fibonacci sequence is proposed due to the low thermal conductivity of phase change material, which can ensure the excellent solidification performance.
34 citations
TL;DR: In this paper, the authors introduce the multiple manifestations of multifractality in the Fibonacci chain, a 1D paradigm for quasicrystals, and discuss some important generalizations of the basic tight-binding models, and closely related models for other quasiperiodic systems.
Abstract: The distinctive electronic properties of quasicrystals stem from their long range structural order, with invariance under rotations and under discrete scale change, but without translational invariance. This review introduces the multiple manifestations of multifractality in the Fibonacci chain, a 1D paradigm for quasicrystals. Quasiperiodic systems have been recently of interest furthermore for their topological properties, described in this review for the simplest 1D case. Some important generalizations of the basic tight-binding models, and closely related models for other quasiperiodic systems are also discussed.
32 citations
TL;DR: In this paper, a computational method for the solution of the fractional optimal control problems subject to fractional systems with equality and inequality constraints is presented, and the proposed procedure is shown to be efficient.
Abstract: This study presents a computational method for the solution of the fractional optimal control problems subject to fractional systems with equality and inequality constraints. The proposed procedure...
31 citations
TL;DR: A numerical method to find out the approximate solution for fractional integro-differential equations of variable order (FIDE-VO) is formulated and the applicability and accuracy of the recommended method will be demonstrated through some numerical applications.
30 citations
TL;DR: The only Fibonacci numbers that are concatenations of two repdigits are 13, 21, 34, 55, 89, 144, 233, 377 as discussed by the authors, and 14.
Abstract: We show that the only Fibonacci numbers that are concatenations of two repdigits are 13, 21, 34, 55, 89, 144, 233, 377.
24 citations
01 Jun 2021
TL;DR: In this paper, the authors introduced some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function.
Abstract: The aim of the present paper is to introduce some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function $$\phi (s)= \frac{2}{1+e^{-s} },\,s\ge 0$$
in the open unit disc $${\mathfrak {D}}$$
. We investigate the upper bounds on initial coefficients for functions of the form $$g_{\phi }(z)=z+\sum
olimits _{j=2}^{\infty }\phi (s)d_jz^j$$
, in these newly introduced special families and also discuss the Fekete–Szego problem. Some interesting consequences of the results established here are also indicated.
23 citations
Posted Content•
TL;DR: In this paper, the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines are constructed.
Abstract: We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup $\mathcal{H}_3$ fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.
22 citations
TL;DR: In this article, a new and efficient operational matrix method based on the amalgamation of Fibonacci wavelets and block pulse functions is proposed for the solutions of time-fractional telegraph equations with Dirichlet boundary conditions.
Abstract: In this article, a new and efficient operational matrix method based on the amalgamation of Fibonacci wavelets and block pulse functions is proposed for the solutions of time-fractional telegraph equations with Dirichlet boundary conditions. The Fibonacci polynomials and the corresponding wavelets along with their fundamental properties are briefly studied at first. These functions along with their nice characteristics are then utilized to formulate the Fibonacci wavelet operational matrices of fractional integrals. The proposed method reduces the fractional model into a system of algebraic equations, which can be solved using the classical Newton iteration method. Approximate solutions of the time-fractional telegraph equation are compared with the recently appeared Legendre and Sinc-Legendre wavelet collocation methods. The numerical outcomes show that the Fibonacci technique yields precise outcomes and is computationally more effective than the current ones.
19 citations
TL;DR: In this article, the typical collocation method and the tau method are employed for obtaining the desired n. The spectral solutions for the nonlinear fractional Klein-Gordon equation were developed.
Abstract: This paper is devoted to developing spectral solutions for the nonlinear fractional Klein–Gordon equation. The typical collocation method and the tau method are employed for obtaining the desired n...
19 citations
TL;DR: In this paper, the interplay of the non-Herimiticity and the cascade of delocalization transition in a quasiperiodic chain is studied, and the results show that the nonhermiticity provides an additional knob to control the cascade.
Abstract: In this paper, the interplay of the non-Herimiticity and the cascade of delocalization transition in a quasiperiodic chain are studied. The study is applied in the non-Hermitian interpolating Aubry-Andr\'e-Fibonacci (IAAF) model, which combines the non-Hermitian Aubry-Andr\'e (AA) model and the non-Hermitian Fibonacci model through a varying parameter, and the non-Hermiticity in this model is introduced by nonreciprocal hopping. In the non-Hermitian AA limit, the system undergoes a delocalization transition by tuning the potential strength. At the critical point, the spatial distribution of the critical state shows a self-similar structure with the relative distance between the peaks being the Fibonacci sequence, and the finite-size scaling of the inverse participation ratios $(\mathrm{IPRs})$ of the critical ground state with lattice size $L$ shows that ${\mathrm{IPR}}_{g}\ensuremath{\propto}{L}^{\ensuremath{-}0.1189}$. In the non-Hermitian Fibonacci limit, we find that the system is always in the extended phase. Along the continuous deformation from the non-Hermitian AA model into the non-Hermitian Fibonacci model in the IAAF model, the cascade of the delocalization transition is found, but only a few plateaux appear. Moreover, the self-similar structure of spatial distribution for the critical modes along the cascade transition is also found. In addition, we find that the delocalization transition and the real-complex transition for the excited states happen at almost the same parameter. Our results show that the non-Hermiticity provides an additional knob to control the cascade of the delocalization transition besides the on-site potential.
01 Mar 2021
TL;DR: In this article, an algorithmic approach to assess the potential of hybrid topological quantum computers is developed, allowing one to remove excess non-topological gates and reduce the impact of noise on the compiled braid circuits.
Abstract: An algorithmic approach to assess the potential of hybrid topological quantum computers is developed, allowing one to remove excess non-topological gates and reduce the impact of noise on the compiled braid circuits.
TL;DR: In this paper, a numerical scheme based on polynomials and finite difference method is developed for numerical solutions of two-dimensional linear and nonlinear Sobolev equations, which is applied for discretization of time derivative whereas space derivatives are approximated by two dimensional Lucas polynomial.
Abstract: A numerical scheme based on polynomials and finite difference method is developed for numerical solutions of two-dimensional linear and nonlinear Sobolev equations. In this approach, finite difference method is applied for the discretization of time derivative whereas space derivatives are approximated by two-dimensional Lucas polynomials. Applying the procedure and utilizing finite Fibonacci sequence, differentiation matrices are derived. With the help of this technique, the differential equations have been transformed to system of algebraic equations, the solution of which compute unknown coefficients in Lucas polynomials. Substituting the unknowns constants in Lucas series, required solution of targeted equation has been obtained. Performance of the method is verified by studying some test problems and computing E2, E
$$_{\infty }$$
and Erms (root mean square) error norms. The obtained accuracy confirms feasibility of the proposed technique.
01 Jul 2021
TL;DR: A generalization of the Pell sequence is investigated called the k-generalized Pell sequence which is generated by a recurrence relation of a higher order.
Abstract: The Pell sequence $(P_n)_{n=0}^{\infty }$ is the second order linear recurrence defined by $P_n=2P_{n-1}+P_{n-2}$ with initial conditions $P_0=0$ and $P_1=1$ In this paper, we investigate a generalization of the Pell sequence called the $k$-generalized Pell sequence which is generated by a recurrence relation of a higher order We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced
TL;DR: Fractional order reaction advection sub-diffusion equation containing Caputo and Riemann–Liouville fractional derivatives has been solved and the effects due to change in various parameters presented in the considered model with the graphical representation have been discussed.
TL;DR: In this article, it was shown that the k-generalized Fibonacci numbers have at most four digits for concatenations of two repdigits, and that the concatenation has at most two digits.
Abstract: We show that the k-generalized Fibonacci numbers that are concatenations of two repdigits have at most four digits.
TL;DR: In this article, the complex type k-Fibonacci numbers were defined and the relationship between the k-step Fibonacci number and the complex-type k-fibonach number was investigated.
Abstract: In this article, we define the complex-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the complex-type k-Fibonacci numbers. Also, we obtain miscel...
TL;DR: In this paper, the authors investigate whether anomalous diffusion present in the noninteracting system survives in the presence of interactions and establish connections to a possible transition towards a localized phase.
Abstract: Quantum dynamics on quasiperiodic geometries has recently gathered significant attention in ultracold atom experiments where nontrivial localized phases have been observed. One such quasiperiodic model is the so-called Fibonacci model. In this tight-binding model, noninteracting particles are subject to on-site energies generated by a Fibonacci sequence. This is known to induce critical states, with a continuously varying dynamical exponent, leading to anomalous transport. In this work, we investigate whether anomalous diffusion present in the noninteracting system survives in the presence of interactions and establish connections to a possible transition towards a localized phase. We investigate the dynamics of the interacting Fibonacci model by studying real-time spread of density-density correlations at infinite temperature using the dynamical typicality approach. We also corroborate our findings by calculating the participation entropy in configuration space and investigating the expectation value of local observables in the diagonal ensemble.
TL;DR: In this article, the performance of bio-inspired composite laminates based on the Fibonacci sequence for impact resistant applications is analyzed by considering ballistic impact loading with six different impact velocities, out of which three are below the ballistic limit.
TL;DR: In this article, all the solutions of the Diophantine equation P n ( k ) = F m ( l ) in positive integers n, k, m, l with k, l ≥ 2 were found.
17 Sep 2021
TL;DR: In this paper, the authors investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field.
Abstract: We investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-Andr\'e and the Fibonacci models. We find that stronger many-body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation of the quasiperiodic modulation. As a result, the position of the many-body localization transition depends on the interaction strength for an intermediate degree of deformation. Moreover, in the region of parameter space where the single-particle spectrum contains both localized and extended states, many-body interactions induce an anomalous effect: weak interactions localize the system, whereas stronger interactions enhance ergodicity. We map the model's localization phase diagram using the decay of the quenched spin imbalance in relatively long chains. This is accomplished employing a time-dependent variational approach applied to a matrix product state decomposition of the many-body state. Our model serves as a rich playground for testing many-body localization under tunable potentials.
TL;DR: In this paper, a Fibonacci composite inspired by Liquidambar formosana is proposed to show that the structural geometry of stiff skeleton and soft inclusion governs the toughening performance of cellular composites.
TL;DR: In this article, a family of regular and biunivalent functions of the type in the open unit disc is presented and the upper bounds on initial coefficients and also the functional of Fekete-Szego for functions in this family are estimated.
Abstract: By using - Fibonacci numbers, we present a comprehensive family of regular and biunivalent functions of the type in the open unit disc . We estimate the upper bounds on initial coefficients and also the functional of Fekete-Szego for functions in this family. We also discuss few interesting observations and provide relevant connections of the result investigated.
TL;DR: In this paper, a wavelet collocation method based on Fibonacci wavelets is proposed to solve the dual-phase-lag (DPL) bioheat transfer model in multilayer skin tissues during hyperthermia treatment.
Abstract: In this article, a novel wavelet collocation method based on Fibonacci wavelets is proposed to solve the dual-phase-lag (DPL) bioheat transfer model in multilayer skin tissues during hyperthermia treatment. Firstly, the Fibonacci polynomials and the corresponding wavelets along with their fundamental properties are briefly studied. Secondly, the operational matrices of integration for the Fibonacci wavelets are built by following the celebrated approach of Chen and Haiso. Thirdly, the proposed method is utilized to reduce the underlying DPL model into a system of algebraic equations, which has been solved using the Newton iteration method. Towards the culmination, the effect of different parameters including the tissue-wall temperature, time-lag due to heat flux, time-lag due to temperature gradient, blood perfusion, metabolic heat generation, heat loss due to diffusion of water, and boundary conditions of various kinds on multilayer skin tissues during hyperthermia treatment are briefly presented and all the outcomes are portrayed graphically.
TL;DR: In this article, the Fibonacci collocation method is used for approximately solving a class of nonlinear Pantograph differential equations with initial conditions, where the problem is firstly reduced into a nonlinear algebraic system via collocation points, and then the unknown coefficients of the approximate solution function are calculated.
Abstract: In this paper, Fibonacci collocation method is firstly used for approximately solving a class of systems of nonlinear Pantograph differential equations with initial conditions. The problem is firstly reduced into a nonlinear algebraic system via collocation points, later the unknown coefficients of the approximate solution function are calculated. Also, some problems are presented to test the performance of the proposed method by using the absolute error functions. Additionally, the obtained numerical results are compared with exact solutions of the test problems and approximate ones obtained with other methods in the literature.
TL;DR: The irregularity polynomials specialize to the Albertson index and also provide additional information about the higher moments of $$|deg(u)-deg(v)|$$ in these families of graphs as mentioned in this paper.
Abstract: Irregularity of a graph is an invariant measuring how much the graph differs from a regular graph. Albertson index is one measure of irregularity, defined as the sum of $$|deg(u)-deg(v)|$$
over all edges uv of the graph. Motivated by a recent result on the irregularity of Fibonacci cubes, we consider irregularity polynomials and determine them for Fibonacci and Lucas cubes. These are graph families that have been studied as alternatives for the classical hypercube topology for interconnection networks. The irregularity polynomials specialize to the Albertson index and also provide additional information about the higher moments of $$|deg(u)-deg(v)|$$
in these families of graphs.
06 Feb 2021
TL;DR: In this paper, higher order Fibonacci quaternions were introduced with the help of higher order numbers, and they generalize the Fiboni-quaternions studied by Horadam and Halici and give recurrence relation, Binet formula, generating and exponential generating functions, and some other algebraic properties.
Abstract: In this paper, with the help of higher order Fibonacci numbers, we introduce higher order Fibonacci quaternions that generalize the Fibonacci quaternions studied by Horadam and Halici We give recurrence relation, Binet formula, generating and exponential generating functions and some other algebraic properties of higher order Fibonacci quaternions
TL;DR: The Fibonacci-run graph as discussed by the authors is a special case of the hypercube graph with a restriction on the runlength of the nodes. But it does not have the same number of vertices as hypercubes, but fewer edges and different graph theoretical properties.
TL;DR: It is shown that the Ehrhart polynomial of these polytopes, and therefore the Hilbert series of the ideals, depends only on the number of leaves of the underlying binary tree, and not on the topology of the tree itself.