Showing papers on "Fibonacci number published in 2013"
TL;DR: A survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results, and their median nature that leads to a fast recognition algorithm is discussed.
Abstract: The Fibonacci cube Γ
n
is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results. Their median nature that leads to a fast recognition algorithm is discussed. The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given.
128 citations
TL;DR: For an integer k ≥ 2, the k−generalized Fibonacci sequence (F (k) n )n which starts with 0,..., 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms was considered in this article.
Abstract: For an integer k ≥ 2, we consider the k−generalized Fibonacci sequence (F (k) n )n which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. F. Luca [2] in 2000 and recently D. Marques [3] proved that 55 and 44 are the largest numbers with only one distinct digit (so called repdigits) in the sequences (F (2) n )n and (F (3) n )n, respectively. Further, Marques conjectured that there are no repdigits having at least 2 digits in a k−generalized Fibonacci sequence for any k > 3. In this talk, we report about some arithmetic properties of (F (k) n )n and confirm the conjecture raised by Marques. This is a joint work with Florian Luca.
106 citations
TL;DR: In this paper, a band matrix is introduced and the sequence space where is the k th Fibonacci number for every node is defined, and some inclusion relations concerning this space and its α-, β-, γ-duals are established.
Abstract: In the present paper, we introduce a new band matrix and define the sequence space where is the k th Fibonacci number for every . We also establish some inclusion relations concerning this space and determine its α-, β-, γ-duals. Further, we characterize some matrix classes on the space and examine some geometric properties of this space. MSC:11B39, 46A45, 46B45, 46B20.
86 citations
TL;DR: In this paper, the complex Fibonacci quaternions were investigated and the Binet formula for the generating function and Binet formulas for the matrix representations of the complex quaternion were given.
Abstract: Horadam defined the Fibonacci quaternions and established a few relations for the Fibonacci quaternions. In this paper, we investigate the complex Fibonacci quaternions and give the generating function and Binet formula for these quaternions. Moreover, we also give the matrix representations of them.
82 citations
TL;DR: The focusing properties of diffractive lenses designed using the Fibonacci sequence are studied in this article, where it is demonstrated that these lenses present two equal intensity foci and that the ratio of the two focal distances approaches the golden mean.
Abstract: The focusing properties of diffractive lenses designed using the Fibonacci sequence are studied. It is demonstrated that these lenses present two equal intensity foci and that the ratio of the two focal distances approaches the golden mean. This distinctive optical characteristic is experimentally confirmed. It is suggested that the versatility and potential scalability of these lenses may allow for new applications ranging from X-ray microscopy to THz imaging.
75 citations
TL;DR: In this paper, the authors give an asymptotic estimate of the number of numerical semigroups of a given genus for genus g satisfying f < 3m, where m is the multiplicity and f is the Frobenius number.
Abstract: We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if ng is the number of numerical semigroups of genus g, we prove that
$$\lim_{g \rightarrow \infty} n_g \varphi^{-g} = S $$
where \(\varphi = \frac{1 + \sqrt{5}}{2}\) is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of ng. In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number.
60 citations
TL;DR: In this paper, the split generalized Fibonacci quaternions and split generalized Lucas quaternion were defined and Binet formulas and Cassini identities for them were given, respectively.
Abstract: Starting from ideas given by Horadam in [5] , in this paper, we will define the split Fibonacci quaternion, the split Lucas quaternion and the split generalized Fibonacci quaternion. We used the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations between the split Fibonacci, split Lucas and the split generalized Fibonacci quaternions. Moreover, we give Binet formulas and Cassini identities for these quaternions.
59 citations
TL;DR: This paper presents a strategy for producing high‐quality QMC sampling patterns for spherical integration by resorting to spherical Fibonacci point sets and shows that these patterns, when applied to illumination integrals, are very simple to generate and consistently outperform existing approaches, both in terms of root mean square error (RMSE) and image quality.
Abstract: Quasi-Monte Carlo (QMC) methods exhibit a faster convergence rate than that of classic Monte Carlo methods. This feature has made QMC prevalent in image synthesis, where it is frequently used for approximating the value of spherical integrals (e.g. illumination integral). The common approach for generating QMC sampling patterns for spherical integration is to resort to unit square low-discrepancy sequences and map them to the hemisphere. However such an approach is suboptimal as these sequences do not account for the spherical topology and their discrepancy properties on the unit square are impaired by the spherical projection. In this paper we present a strategy for producing high-quality QMC sampling patterns for spherical integration by resorting to spherical Fibonacci point sets. We show that these patterns, when applied to illumination integrals, are very simple to generate and consistently outperform existing approaches, both in terms of root mean square error (RMSE) and image quality. Furthermore, only a single pattern is required to produce an image, thanks to a scrambling scheme performed directly in the spherical domain.
59 citations
TL;DR: In this article, the authors investigated the Riccati Difference Equation (RCE) solutions of two special types of the RCE and such that their solutions were associated with Fibonacci numbers.
Abstract: In this study, we investigate the solutions of two special types of the Riccati difference equation and such that their solutions are associated with Fibonacci numbers. MSC: 11B39, 39A10, 39A13.
56 citations
TL;DR: This paper investigates an interesting subclass SL of analytic univalent functions in the open unit disc on the complex plane of Fibonacci numbers and presents certain new results for the class SL of functions.
52 citations
TL;DR: In this article, the properties of generalized Fibonacci quaternions and generalized Narayana quaternion are investigated in a generalized quaternian algebra, where they are shown to have properties similar to those of generalized generalized generalized NNs.
Abstract: In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions in a generalized quaternion algebra.
TL;DR: Using the Fibonacci sequence, a new family of DOEs is discovered that inherently behave as bifocal vortex lenses, and where the ratio of the two focal distances approaches the golden mean.
Abstract: Optical vortex beams, generated by Diffractive Optical Elements (DOEs), are capable of creating optical traps and other multi-functional micromanipulators for very specific tasks in the microscopic scale. Using the Fibonacci sequence, we have discovered a new family of DOEs that inherently behave as bifocal vortex lenses, and where the ratio of the two focal distances approaches the golden mean. The disctintive optical properties of these Fibonacci vortex lenses are experimentally demonstrated. We believe that the versatility and potential scalability of these lenses may allow for new applications in micro and nanophotonics.
TL;DR: In this paper, a key-sharing protocol combines entanglement with the mathematical properties of a recursive sequence to allow a realization of the physical conditions necessary for implementation of the no-cloning principle for QKD.
Abstract: Quantum cryptography and quantum key distribution (QKD) have been the most successful applications of quantum information processing, highlighting the unique capability of quantum mechanics, through the no-cloning theorem, to securely share encryption keys between two parties. Here, we present an approach to high-capacity, high-efficiency QKD by exploiting cross-disciplinary ideas from quantum information theory and the theory of light scattering of aperiodic photonic media. We propose a unique type of entangled-photon source, as well as a physical mechanism for efficiently sharing keys. The key-sharing protocol combines entanglement with the mathematical properties of a recursive sequence to allow a realization of the physical conditions necessary for implementation of the no-cloning principle for QKD, while the source produces entangled photons whose orbital angular momenta (OAM) are in a superposition of Fibonacci numbers. The source is used to implement a particular physical realization of the protocol by randomly encoding the Fibonacci sequence onto entangled OAM states, allowing secure generation of long keys from few photons. Unlike in polarization-based protocols, reference frame alignment is unnecessary, while the required experimental setup is simpler than other OAM-based protocols capable of achieving the same capacity and its complexity grows less rapidly with increasing range of OAM used.
TL;DR: This work examines the zero-temperature phase diagram of the two-dimensional Levin-Wen string-net model with Fibonacci anyons in the presence of competing interactions and suggests unusual universality classes.
Abstract: We examine the zero-temperature phase diagram of the two-dimensional Levin-Wen string-net model with Fibonacci anyons in the presence of competing interactions. Combining high-order series expansions around three exactly solvable points and exact diagonalizations, we find that the non-Abelian doubled Fibonacci topological phase is separated from two nontopological phases by different second-order quantum critical points, the positions of which are computed accurately. These trivial phases are separated by a first-order transition occurring at a fourth exactly solvable point where the ground-state manifold is infinitely many degenerate. The evaluation of critical exponents suggests unusual universality classes.
Journal Article•
TL;DR: In this paper, the authors obtained some identities containing generalized Fibonacci and Lucas numbers, some of which are new and some of them are well known, and used them to give some congruences concerning generalized Lucas numbers.
Abstract: In this paper we obtain some identities containing generalized Fibonacciand Lucas numbers. Some of them are new and some are well known.By using some of these identities we give some congruences concerninggeneralized Fibonacci and Lucas numbers such asV 2mn+r ≡ (−(−t)m ) n V r (mod Vm ), U 2mn+r ≡ (− (−t)m ) n U r (mod Vm ), and V 2mn+r ≡ (−t)mn V r (mod Um ), U 2mn+r ≡ (−t)mn U r (mod Um ).
TL;DR: The k-generalized generalized Fibonacci sequence (Fn(k))n as mentioned in this paper is a generalized version of the Fibonaccia sequence, where each term is the sum of the k preceding terms.
TL;DR: A new construction of cross-bifix-free codes is provided which generalizes the construction by Bajic to longer code lengths and to any alphabet size, and the codes are shown to be nearly optimal in size.
Abstract: A cross-bifix-free code is a set of words in which no prefix of any length of any word is the suffix of any word in the set. Cross-bifix-free codes arise in the study of distributed sequences for frame synchronization. We provide a new construction of cross-bifix-free codes which generalizes the construction by Bajic to longer code lengths and to any alphabet size. The codes are shown to be nearly optimal in size. We also establish new results on Fibonacci sequences, which are used in estimating the size of the cross-bifix-free codes.
TL;DR: In this paper, a generalized Delannoy matrix by weighted delannoy numbers is introduced, which is a special case of the generalized Delanoy matrices, while Schroder matrix and Catalan matrix also arise in involving inverses of the general generalized Delanneys matrices.
TL;DR: In this paper, generalized Fibonacci polynomials were presented and generalized Binet's formula and generating function was used to derive the identities. The proofs of the main theorems are based on simple algebra and give several interesting properties involving them.
Abstract: In this study, we present generalized Fibonacci polynomials We have used their Binet’s formula and generating function to derive the identities The proofs of the main theorems are based on special functions, simple algebra and give several interesting properties involving them
TL;DR: A signature theory is developed and characterize all realizable signatures for Fibonacci gates and for bases of arbitrary dimensions it is proved that some slight variations of these counting problems are #P-hard.
TL;DR: In this article, a magnetostatic surface wave (MSSW) propagation in a magnonic quasicrystal (MQC) with Fibonacci type structure was investigated and it was shown that such structure has a greater number of band gaps and narrower pass bands located between them than a periodic structure.
Abstract: This study reports on the experimental investigations of a magnetostatic surface wave (MSSW) propagation in a magnonic quasicrystal (MQC) with Fibonacci type structure. It is shown that such structure has a greater number of band gaps and narrower pass bands located between them than a periodic structure. These features of the MQC and three-wave decay of the MSSW are used in a MQC active ring resonator for the eigenmode selection and dissipative soliton self-generation.
TL;DR: In this paper, a method to solve singularly perturbed differential-difference equations of mixed type, i.e., containing both terms having a negative shift and terms having positive shift in terms of Fibonacci polynomials, is introduced.
Abstract: In this paper, we introduce a method to solve singularly perturbed differential-difference equations of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift in terms of Fibonacci polynomials. Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. First, we present some preliminaries about polynomial interpolation and properties of Fibonacci polynomials then a new approach implementing a collocation method in combination with matrices of Fibonacci polynomials is introduced to approximate the solution of these equations with variable coefficients under the boundary conditions. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving these equations.
TL;DR: In this paper, the authors study probability measures on the unit circle corresponding to orthogonal polynomials whose sequence of Verblunsky coefficients is invariant under the Fibonacci substitution.
TL;DR: In this paper, it was shown that the density of states measure is continuous for almost all pairs of small coupling constants for the square Fibonacci Hamiltonian, and that the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures is also known.
Abstract: We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.
TL;DR: Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers are given, which can be useful also in problems of counting of k-independent sets in graphs.
TL;DR: In this paper, the Fibonacci Life Chart Method (FLCM) was used to model Erikson's eight developmental stages and to identify populations at risk for psychological disorder, which would allow early intervention.
Abstract: The purpose of this study is to describe the use of Fibonacci numbers to model Erikson’s eight developmental stages and to formulate practical clinical implications. Using a new method, called the Fibonacci Life-Chart Method (FLCM), all prospective dates based on the Fibonacci sequence between January 1, 2000 and December 31, 2100 were identified. This study found the FLCM produced a developmental pattern characterized by eight recognizable stages. This finding constitutes a new classification of Erikson’s eight developmental stages. The present research provides support for Erikson’s epigenetic view of predetermined, sequential stages to human development based on the occurrence of Fibonacci numbers in biological cell division and self-organizing systems. This method may help identify populations at risk for psychological disorder, which would allow early intervention. However, a longitudinal study is required to establish its predictive power.
TL;DR: It is proved that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmianspecial words if and only if its longest repeated prefix is a palindrome.
TL;DR: In this paper, a new procedure for the numerical solution of boundary value problems is presented, mainly based on the Fibonacci polynomial expansions, the so-called pseudospectral methods with the collocation method.
Abstract: In this study, we present a new procedure for the numerical solution of boundary value problems. This approach is mainly founded on the Fibonacci polynomial expansions, the so-called pseudospectral methods with the collocation method. The applicability and effectiveness of our proposed approach is shown by some illustrative examples. Then, the results indicate that this method is very effective and highly promising for linear differential equations defined on any subinterval of the real domain.
TL;DR: Some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated and it has been given a new formula for some Lucas numbers.
26 Jan 2013
TL;DR: In this paper, the authors considered a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters.
Abstract: We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters. We show that the spectrum is a Cantor set of zero Lebesgue measure, and discuss its fractal structure and Hausdorff dimension. We also extend some known results on the diagonal and the off-diagonal Fibonacci Hamiltonians.