Showing papers on "Fibonacci number published in 2012"
TL;DR: In this paper, the authors investigated the Fibonacci and Lucas quaternions and derived the generating functions and Binet formulas for these two types of quadratic functions. And they also derived some sums formulas for them.
Abstract: In this paper, we investigate the Fibonacci and Lucas quaternions. We give the generating functions and Binet formulas for these quaternions. Moreover, we derive some sums formulas for them.
163 citations
TL;DR: In this article, the physics of interacting Fibonacci anyons can be studied with strongly interacting Rydberg atoms in a lattice, when due to the dipole blockade the simultaneous laser excitation of adjacent atoms is forbidden.
Abstract: The physics of interacting Fibonacci anyons can be studied with strongly interacting Rydberg atoms in a lattice, when due to the dipole blockade the simultaneous laser excitation of adjacent atoms is forbidden. The Hilbert space maps then directly on the fusion space of Fibonacci anyons. Interactions between anyons are generated and controlled by the intensity and frequency of the excitation laser. Fusion outcomes of neighboring anyons can be determined experimentally via the measurement of three-point correlations among three consecutive atoms. Our work shows that a Rydberg lattice gas constitutes a natural physical platform for the experimental exploration of topological quantum liquids of non-Abelian anyons.
123 citations
TL;DR: In this article, it was shown that for a less than the fourth power 4 of the golden ratio, c(a) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal.
Abstract: This paper calculates the function c(a) whose value at a is the inmum of the size of a ball that contains a symplectic image of the ellipsoidE(1;a). (Here a 1 is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of c(a) is surprisingly rich. The volume constraint implies that c(a) is always greater than or equal to the square root of a, and it is not hard to see that this is equality for large a. However, for a less than the fourth power 4 of the golden ratio, c(a) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval [ 4 ; 7] we nd c(a) = (a + 1)=3. For a 7, the function c(a) coincides with the square root except on a nite number of intervals where it is again piecewise linear. The embedding constraints coming from embedded contact homology give rise to another capacity function cECH which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that cECH(a) c(a) for all a. We show here that cECH(a) c(a) for all a.
97 citations
01 Jan 2012
TL;DR: In this paper, the Diophantine equation F (k) n = 2 m in positive integers n;k;m is studied and the powers of two that appear in k generalized generalized Fibonacci sequences are investigated.
Abstract: The k generalized Fibonacci sequence F (k) n n resembles the Fi- bonacci sequence in that it starts with 0;:::; 0; 1 (k terms) and each term af- terwards is the sum of the k preceding terms. In this paper, we are interested in nding powers of two that appear in k generalized Fibonacci sequences; i.e., we study the Diophantine equation F (k) n = 2 m in positive integers n;k;m
66 citations
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TL;DR: In this paper, the properties of generalized Fibonacci quaternions and FPNs were investigated, and some properties of the generalized FPNN quaternion were investigated.
Abstract: In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.
59 citations
TL;DR: The question whether Q d ( f ) is an isometric subgraph of Q d is studied and the question is completely solved for strings f of length at most five and for strings consisting of at most three blocks.
59 citations
TL;DR: Gaudet et al. as discussed by the authors generalized this result to a combinatorial setting and showed that the distribution of the number of positive and negative summands converges to a bivariate normal with computable, negative correlation.
56 citations
19 May 2012
TL;DR: This work presents the first pointer-based heap implementation with time bounds matching those of Fibonacci heaps in the worst case, and uses the pigeonhole principle in place of the redundant counter mechanism.
Abstract: We present the first pointer-based heap implementation with time bounds matching those of Fibonacci heaps in the worst case. We support make-heap, insert, find-min, meld and decrease-key in worst-case O(1) time, and delete and delete-min in worst-case O(lg n) time, where n is the size of the heap. The data structure uses linear space. A previous, very complicated, solution achieving the same time bounds in the RAM model made essential use of arrays and extensive use of redundant counter schemes to maintain balance. Our solution uses neither. Our key simplification is to discard the structure of the smaller heap when doing a meld. We use the pigeonhole principle in place of the redundant counter mechanism.
51 citations
28 May 2012
TL;DR: It is found experimentally that orderings that result in smaller exact B DDs have a strong tendency to produce tighter bounds in relaxation BDDs, and it is shown that the width of an exact BDD can be given a theoretical upper bound for certain classes of graphs.
Abstract: The ordering of variables can have a significant effect on the size of the reduced binary decision diagram (BDD) that represents the set of solutions to a combinatorial optimization problem. It also influences the quality of the objective function bound provided by a limited-width relaxation of the BDD. We investigate these effects for the maximum independent set problem. By identifying variable orderings for the BDD, we show that the width of an exact BDD can be given a theoretical upper bound for certain classes of graphs. In addition, we draw an interesting connection between the Fibonacci numbers and the width of exact BDDs for general graphs. We propose variable ordering heuristics inspired by these results, as well as a k-layer look-ahead heuristic applicable to any problem domain. We find experimentally that orderings that result in smaller exact BDDs have a strong tendency to produce tighter bounds in relaxation BDDs.
49 citations
TL;DR: In this paper, two omnidirectional photonic band gaps realized by one-dimensional (1D) Fibonacci quasi-periodic structure which is composed of superconductor and two kinds of isotropic dielectric are theoretically studied by the transfer matrix method.
46 citations
TL;DR: In this paper, it was shown that for small values of the coupling constant V, this measure is exact-dimensional and the almost everywhere value dV of the local scaling exponent is a smooth function of V, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero.
Abstract: We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V, this measure is exact-dimensional and the almost everywhere value dV of the local scaling exponent is a smooth function of V, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.
TL;DR: In this article, the moments of the weight function of the Askey-Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors.
Abstract: We explain how the moments of the (weight function of the) Askey-Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant than the formula given in their earlier paper. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the number of permutations, and the number of matchings.
01 Jan 2012
TL;DR: In this paper, the main purpose of the paper is to study some sums of powers of Fibonacci polynomials and Lucas polynomial, and give several interesting identities.
Abstract: The main purpose of this paper is to study some sums of powers of Fibonacci polynomials and Lucas polynomials, and give several interesting identities. Finally, using these identities we shall prove a conjecture proposed by R. S. Melham in [4].
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TL;DR: In this paper, the average number of summands needed for integers in [G n, G n+1] is shown to have Gaussian fluctuations about the mean of the decay ratio.
Abstract: An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is now known as Zeckendorf's theorem [21], and similar decompositions exist for many other sequences ${G_{n+1} = c_1 G_{n} + ... + c_L G_{n+1-L}}$ arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker [15] proved the average number of summands needed for integers in $[G_n, G_{n+1})$ is on the order of $C_{{\rm Lek}} n$ for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean.
Kolo$\breve{{\rm g}}$lu, Kopp, Miller and Wang [17, 18] recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all $m \in [G_n, G_{n+1})$ for special choices of the $c_i$'s. Specifically, we study the case where each $c_i \in {0,1}$ and there is a $g$ such that there are always exactly $g-1$ zeros between two non-zero $c_i$'s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than $g$, and the probability of a gap of length $j > g$ decays geometrically, with the decay ratio equal to the largest root of the recurrence relation. These methods are combinatorial and apply to related problems; we end with a discussion of similar results for far-difference (i.e., signed) decompositions.
TL;DR: In this article, the cube polynomial of a graph is defined for the number of induced k-dimensional hypercubes (k?0) in a graph, and several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers.
Abstract: The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k?0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.
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TL;DR: In this paper, a new construction of cross-bifix-free codes is presented, which generalizes the construction in Bajic (2007) to longer code lengths and to any alphabet 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 in Bajic (2007) 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, that are used in estimating the size of the cross-bifix-free codes.
01 Jan 2012
TL;DR: In this article, a class of determinants, called n-determinants, is defined and a generalization of theorem of the product of two determinants is obtained, and the upper Hessenberg determinants are shown to belong to this class.
Abstract: We examine relationships between two minors of order n of some matrices of n rows and n + r columns. This is done through a class of determinants, here called n-determinants, the investigation of which is our objective. As a consequence of our main result we obtain a generalization of theorem of the product of two determinants. We show the upper Hessenberg determinants, with 1 on the subdiagonal, belong to our class. Using such determinants allow us to represent terms of various recurrence sequences in the form of determinants. We illustrate this with several examples. In particular, we state a few determinants, each of which equals a Fibonacci number. Also, several relationships among terms of sequences defined by the same recurrence equation are derived.
TL;DR: In this article, a general numerical procedure for the accurate evaluation of the geometric view factors between equally sized spheres in a randomly packed assembly is proposed, which is a combination of the Tanaka integral expression for the view factor between two equal spheres, the product form of the Fibonacci integration scheme for integrals on spheres, and the non-uniform variable scaling.
Journal Article•
TL;DR: In this article, the authors find all base 10 repdigits which are sums of three Fibonacci numbers in AMS subject classifications: Primary 11D61; Secondary 11A67, 11B39.
Abstract: In this paper, we find all base 10 repdigits which are sums of three Fibonacci numbers. AMS subject classifications: Primary 11D61; Secondary 11A67, 11B39
TL;DR: In this article, the transmission properties of Fibonacci quasi-periodic one-dimensional photonic crystals (1DPCs) containing superconducting material are theoretically investigated based on the transfer matrix method.
TL;DR: It is shown that the topological entropy of this object is given by the growth rate of the set of inflated generalised random Fibonacci words.
Abstract: The generalised random Fibonacci chain is a stochastic extension of the classical Fibonacci substitution and is defined as the rule that is mapping \({0\mapsto 1}\) and \({1 \mapsto 1^i01^{m-i}}\) with probability pi, where pi ≥ 0 with \({\sum_{i=0}^m p_i =1}\) , and where the random rule is applied each time it acts on a 1. We show that the topological entropy of this object is given by the growth rate of the set of inflated generalised random Fibonacci words.
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TL;DR: In this paper, the left and right real matrix representations for the complex quaternions were investigated and some examples in the special case of the complex Fibonacci quaternion were given.
Abstract: Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and we give some examples in the special case of the complex Fibonacci quaternions.
14 May 2012
TL;DR: The use of a Fibonacci scale, and possibly other non-linear scales, is likely to affect the effort estimates towards lower values compared to linear scales.
Abstract: Background: The estimation technique Planning Poker is common in agile software development. The cards used to propose an estimate in Planning Poker do not include all numbers, but for example only the numbers 0, ½, 1, 2, 3, 5, 8, 13, 20, 40 and 100. We denote this, somewhat inaccurately, a Fibonacci scale in this paper. In spite of the widespread use of the Fibonacci scale in agile estimation, we do not know much about how this scale influences the estimation process. Aim: Better understanding of the effect of going from a linear scale to a Fibonacci scale in effort estimation. Method: We conducted two empirical studies. In the first study, we gave computer science students the same estimation task. Half of the students estimated the task using the Fibonacci scale and the other half a linear scale. The second study included four estimation teams, each composed of four software professionals, estimating the effort to complete the same ten tasks. Two of the teams estimated the first five tasks using the Fibonacci scale and the last five using the linear scale. The two other teams used the scales in the opposite sequence. Results: We found a median decrease in the effort estimates of 60% (first study) and 26% (second study) when using a Fibonacci scale instead of the traditional linear scale. The scale difference in the effort estimates decreased as the developers' skill increased. Conclusion: The use of a Fibonacci scale, and possibly other non-linear scales, is likely to affect the effort estimates towards lower values compared to linear scales. A possible explanation for this scale-induced effect is that people tend to be biased towards toward the middle of the provided scale, especially when the uncertainty is substantial. The middle value is likely to be perceived as lower for the Fibonacci than for the line
01 Jan 2012
TL;DR: In this paper, it was proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonca numbers, which in turn yields a closed formula for the index.
Abstract: In the language of mathematical chemistry, Fibonacci cubes can be defined as the resonance graphs of fibonacenes. Lucas cubes form a symmetrization of Fibonacci cubes and appear as resonance graphs of cyclic polyphenantrenes. In this paper it is proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonacci numbers which in turn yields a closed formula for the Wiener index of Fibonacci cubes. Asymptotic behavior of the average distance of Fibonacci cubes is obtained. The generating function of the sequence of ordered Hosoya polynomials of Fibonacci cubes is also deduced. Along the way, parallel results for Lucas cubes are given.
Journal Article•
TL;DR: This work introduces time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation.
Abstract: We report on a numerical experiment in which we use time-dependent potentials to braid non-Abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where , the ratio of particles to flux quanta, is near 1=2, 1, or 3=2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for near 1 and 3=2, with Berry matrices, respectively, consistent with Ising and Fibonacci anyons. Near 1⁄4 1=2, the braids commute.
TL;DR: It is proved that the Fibonacci setFn has minimal in the sense of order quartered Lp discrepancy for all p 2 (1;1) and this in turn implies that certain twofold symmetrizations of the Fib onacci setfn are optimal with respect to the standard L p discrepancy.
18 Feb 2012
TL;DR: The main purpose of as mentioned in this paper is to define a new regular matrix by us- ing Fibonacci numbers and investigate its matrix domain in the classical sequence spaces 'p,1,c and c0, where 1 p < 1.
Abstract: The main purpose of this paper is to define a new regular matrix by us- ing Fibonacci numbers and to investigate its matrix domain in the classical sequence spaces 'p,'1,c and c0, where 1 p < 1.
01 Jan 2012
TL;DR: In this article, it was shown that if Fn · · ·Fn+(k−1) is a repdigit, with at least two digits, then (k, n) = (1, 10).
Abstract: Let (Fn)n≥0 be the Fibonacci sequence. In 2000, F. Luca proved that F10 = 55 is the largest repdigit (i.e. a number with only one distinct digit in its decimal expansion) in the Fibonacci sequence. In this note, we show that if Fn · · ·Fn+(k−1) is a repdigit, with at least two digits, then (k, n) = (1, 10).
TL;DR: In this article, the authors construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the Levin-Wen model for doubled Fibonacci anyons.
Abstract: We construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the Levin-Wen model for doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum error-correcting code defined by the ground states of this model (the Fibonacci code). We quantify the complexity of these circuits with gate counts using different universal gate sets and find these measurements become significantly easier to perform if $n$-qubit Toffoli gates with $n=3,\phantom{\rule{0.28em}{0ex}}4$, and 5 can be carried out directly. In addition to measurement circuits, we construct simplified quantum circuits requiring only a few qubits that can be used to verify that certain self-consistency conditions, including the pentagon equation, are satisfied by the Fibonacci code.