Showing papers on "Fibonacci number published in 2010"

Measurement of Areas on a Sphere Using Fibonacci and Latitude–Longitude Lattices

Abstract: The area of a spherical region can be easily measured by considering which sampling points of a lattice are located inside or outside the region. This point-counting technique is frequently used for measuring the Earth coverage of satellite constellations, employing a latitude–longitude lattice. This paper analyzes the numerical errors of such measurements, and shows that they could be greatly reduced if the Fibonacci lattice were used instead. The latter is a mathematical idealization of natural patterns with optimal packing, where the area represented by each point is almost identical. Using the Fibonacci lattice would reduce the root mean squared error by at least 40%. If, as is commonly the case, around a million lattice points are used, the maximum error would be an order of magnitude smaller.

Binet's formula for the tribonacci sequence

Abstract: The terms of a recursive sequence are usually defined by a recurrence procedure; that is, any term is the sum of preceding terms. Such a definition might not be entirely satisfactory, because the computation of any term could require the computation of all of its predecessors. An alternative definition gives any term of a recursive sequence as a function of the index of the term. For the simplest nontrivial recursive sequence, the Fibonacci sequence, Binet's formula [1] _ un = (l//5)(a" B")

Resistance distance in wheels and fans

Abstract: The wheel graph is the join of a single vertex and a cycle, while the fan graph is the join of a single vertex and a path. The resistance distance between any two vertices of a wheel and a fan is obtained. The resistances are related to Fibonacci numbers and generalized Fibonacci numbers. The derivation is based on evaluating determinants of submatrices of the Laplacian matrix. A combinatorial argument is also illustrated. A connection with the problem of squaring a rectangle is described.

On the Properties of k-Fibonacci Numbers

Abstract: In this paper, we obtain some new identities for k-Fibonacci numbers. Moreover the identities including generating functions for kFibonacci numbers have been obtained by Binet’s Formula, also divisibility properties of these numbers have been investigated. Mathematics Subject Classification: 11B39, 11B83

Distributed algorithms for ultrasparse spanners and linear size skeletons

Abstract: We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an $${O(2^{{\rm log}^{*} n} {\rm log} n)}$$ -spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size, namely $${O(n(\log \log n)^{\phi})}$$ , where $${\phi = (1 + \sqrt{5})/2}$$ is the golden ratio. As the distance increases the multiplicative distortion of a Fibonacci spanner passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.

On the Usefulness of Fibonacci Compression Codes

Abstract: Recent publications advocate the use of various variable length codes for which each codeword consists of an integral number of bytes in compression applications using large alphabets. This paper shows that another tradeoff with similar properties can be obtained by Fibonacci codes. These are fixed codeword sets, using binary representations of integers based on Fibonacci numbers of order m ≥ 2. Fibonacci codes have been used before, and this paper extends previous work presenting several novel features. In particular, the compression efficiency is analyzed and compared to that of dense codes, and various table-driven decoding routines are suggested.

On the number of summands in Zeckendorf decompositions

Abstract: Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many summands are needed. Using a continued fraction approach, Lekkerkerker proved that the average number of such summands needed for integers in $[F_n, F_{n+1})$ is $n / (\varphi^2 + 1) + O(1)$, where $\varphi = \frac{1+\sqrt{5}}2$ is the golden mean. Surprisingly, no one appears to have investigated the distribution of the number of summands; our main result is that this converges to a Gaussian as $n\to\infty$. Moreover, such a result holds not just for the Fibonacci numbers but many other problems, such as linear recurrence relation with non-negative integer coefficients (which is a generalization of base $B$ expansions of numbers) and far-difference representations. In general the proofs involve adopting a combinatorial viewpoint and analyzing the resulting generating functions through partial fraction expansions and differentiating identities. The resulting arguments become quite technical; the purpose of this paper is to concentrate on the special and most interesting case of the Fibonacci numbers, where the obstructions vanish and the proofs follow from some combinatorics and Stirling's formula; see [MW] for proofs in the general case.

Thirty-Three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra

Abstract: This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lovasz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for seminar presentations. Table of Contents: Fibonacci numbers, quickly; Fibonacci numbers, the formula; The clubs of Oddtown; Same-size intersections; Error-correcting codes; Odd distances; Are these distances Euclidean?; Packing complete bipartite graphs; Equiangular lines; Where is the triangle?; Checking matrix multiplication; Tiling a rectangle by squares; Three Petersens are not enough; Petersen, Hoffman-Singleton, and maybe 57; Only two distances; Covering a cube minus one vertex; Medium-size intersection is hard to avoid; On the difficulty of reducing the diameter; The end of the small coins; Walking in the yard; Counting spanning trees; In how many ways can a man tile a board?; More bricks--more walls?; Perfect matchings and determinants; Turning a ladder over a finite field; Counting compositions; Is it associative?; The secret agent and umbrella; Shannon capacity of the union: a tale of two fields; Equilateral sets; Cutting cheaply using eigenvectors; Rotating the cube; Set pairs and exterior products; Index. (STML/53)

Photonic transmission spectra in one-dimensional fibonacci multilayer structures containing single-negative metamaterials

Abstract: We investigate the transmission properties of the Fibonacci quasiperiodic layered structures consisting of a pair of double positive (DPS), epsilon-negative (ENG) or/and mu-negative (MNG) materials. It is found that there exist the polarization-dependent transmission gaps which are invariant with a change of scaling and insensitive to incident angles. Analytical methods based on transfer matrices and efiective medium theory have been used to explain the properties of transmission gaps of DPS-MNG, DPS-ENG and ENG- MNG Fibonacci multilayer structures.

Omnidirectional bandgaps in Fibonacci quasicrystals containing single-negative materials

Abstract: The band structure and bandgaps of one-dimensional Fibonacci quasicrystals composed of epsilon-negative materials and mu-negative materials are studied. We show that an omnidirectional bandgap (OBG) exists in the Fibonacci structure. In contrast to the Bragg gaps, such an OBG is insensitive to the incident angle and the polarization of light, and the width and location of the OBG cease to change with increasing Fibonacci order, but vary with the thickness ratio of both components, and the OBG closes when the thickness ratio is equal to the golden ratio. Moreover, the general formulations of the higher and lower band edges of the OBG are obtained by the effective medium theory. These results could lead to further applications of Fibonacci structures.

An LSB Data Hiding Technique Using Prime Numbers

Abstract: In this paper, a novel data hiding technique is proposed, as an improvement over the Fibonacci LSB data-hiding technique proposed by Battisti et al. First we mathematically model and generalize our approach. Then we propose our novel technique, based on decomposition of a number (pixel-value) in sum of prime numbers. The particular representation generates a different set of (virtual) bit-planes altogether, suitable for embedding purposes. They not only allow one to embed secret message in higher bit-planes but also do it without much distortion, with a much better stego-image quality, and in a reliable and secured manner, guaranteeing efficient retrieval of secret message. A comparative performance study between the classical Least Significant Bit (LSB)method, the Fibonacci LSB data-hiding technique and our proposed schemes has been done. Analysis indicates that image quality of the stego-image hidden by the technique using Fibonacci decomposition improves against that using simple LSB substitution method, while the same using the prime decomposition method improves drastically against that using Fibonacci decomposition technique. Experimental results show that, the stego-image is visually indistinguishable from the original cover-image.

Finding Matching Initial States for Equivalent NLFSRs in the Fibonacci and the Galois Configurations

Abstract: The Fibonacci and the Galois configurations of nonlinear feedback shift registers (NLFSRs) are considered. In the former, the feedback is applied to the input bit of the shift register only. In the latter, the feedback can potentially be applied to every bit. The sufficient conditions for equivalence of NLFSRs in the Fibonacci and the Galois configurations have been formulated previously. The equivalent NLFSRs in different configurations normally have to be initialized to different states to generate the same output sequences. The mapping between the initial states of two equivalent NLFSRs in the Fibonacci and the Galois configurations is derived in this paper.

Enhancement of the 1.54 μm Er3+ emission from quasiperiodic plasmonic arrays

Abstract: Periodic and Fibonacci Au nanoparticle arrays of varying interparticle separations were fabricated on light emitting Er:SiNx films using electron beam lithography. A 3.6 times enhancement of the photoluminescence (PL) intensity accompanied by a reduction in the Er3+ emission lifetime at 1.54 μm has been observed in Fibonacci quasiperiodic arrays and explained with radiating plasmon theory. Our results are further supported by transmission measurements through the Fibonacci and periodic nanoparticle arrays with interparticle separation in the 25–500 nm range. This work demonstrates the potential of quasiperiodic nanoparticle arrays for the engineering of light emitting devices based on the silicon technology.

On the periods of generalized Fibonacci recurrences

Abstract: We give a simple condition for a linear recurrence (mod 2^w) of degree r to have the maximal possible period 2^(w-1).(2^r-1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for 3-term linear recurrences defined by trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.

Resonant tunnelling in a Fibonacci bilayer graphene superlattice

Abstract: The transmission coefficients (TCs) and angularly averaged conductance for quasi-particle transport are studied for a bilayer graphene superlattice arranged according to the Fibonacci sequence. The transmission is found to be symmetric around the superlattice growth direction and highly sensitive to the direction of the quasi-particle incidence. The transmission spectra are fragmented and appear in groups due to the quasi-periodicity of the system. The average conductance shows interesting structures sharply dependent on the height of the potential barriers between two graphene strips. The low-energy conductance due to Klein transmission is substantially modified by the inclusion of quasi-periodicity in the system.

A digital IC Random Number Generator with logic gates only

Abstract: Random Number Generators with logic gates only are popular among digital IC designers in terms of their speed compatibility and uncomplicated integration to digital platforms. To the best of our knowledge, this paper presents the first ASIC implementation of a Random Number Generator based on Fibonacci and Galois ring oscillators. Prototypes have been designed and fabricated by using HHNEC's 0.25µm eFlash process with a supply voltage of 2.5V. Fibonacci and Galois ring oscillators are implemented in combined configuration. A combined configuration, which consists of a Fibonacci ring oscillator with 16 inverters and a Galois ring oscillator with 32 inverters, occupies 0.0048mm2 and dissipates 2.5mW of power which is quite small compared to other well-known random number generators based on digital circuitry. IC design level experiences, measurements, analysis of measurements and statistical test results are also demonstrated. Furthermore, we propose to use several of these oscillators in an xored configuration, in order to speed up and improve the quality of the generated bit stream. We could achieve fulfilled test results from NIST 800-22 test suit after Von Neumann corrector for 7 xored Fibonacci and Galois ring oscillators with a sampling frequency of 125MHz and 31.25Mbps throughput. In addition, increasing the number of xored Fibonacci and Galois ring oscillators from 7 to 8 also fulfills the tests of NIST 800-22 at the same sampling frequency however, without any further post processing. Thus, 125Mbps of throughput, which is the highest data rate to date with fulfilled test results, could be obtained.

Fibonacci, Littler, and the Hand: A Brief Review

Abstract: In a landmark paper published in 1973, the eminent hand surgeon J. William Littler, MD, proposed two mathematical relationships between the anatomic and functional geometry of the hand. His proposal that the motion of the tips of the fingers follow an equiangular spiral has been experimentally supported. Studies have not supported his other idea that the lengths of the phalanges follow a Fibonacci series. This review, after providing the necessary mathematical background, reexamines Littler's claims, presents the associated studies, and re-evaluates their conclusions. Our analysis shows that the functional lengths of the phalanges of the little finger actually do follow a Fibonacci series and that the functional lengths of the index, long, and ring fingers follow a mathematical relative of the Fibonacci series.

Some identities involving the fibonacci polynomials

Abstract: for » = (>, 1,2,.... If x = 1, then the sequence F(l) is called the Fibonacci sequence, and we shall denote it by F = {F„). The various properties of {Fn) were investigated by many authors. For example, Duncan [1] and Kuipers [3] proved that QogFJ is uniformly distributed mod 1. Robbins [4] studied the Fibonacci numbers of the forms px ±1 and px ± 1, where p is a prime. The second author [5] obtained some identities involving the Fibonacci numbers. The main purpose of this paper is to study how to calculate the summation involving the Fibonacci polynomials:

Fractal spectrum of charge carriers in quasiperiodic graphene structures.

Abstract: In this work we investigate the interaction of charge carriers in graphene with a series of p-n-p junctions arranged according to a deterministic quasiperiodic substitutional Fibonacci sequence. The junctions create a potential landscape with quantum wells and barriers of different widths, allowing the existence of quasi-confined states. Spectra of quasi-confined states are calculated for several generations of the Fibonacci sequence as a function of the wavevector component parallel to the barrier interfaces. The results show that, as the Fibonacci generation is increased, the dispersion branches form energy bands distributed as a Cantor-like set. Besides, for a quasiperiodic set of potential barriers, we obtain the electronic tunneling probability as a function of energy, which shows a striking self-similar behavior for different generation numbers.

Fibonacci-Like Sequence and its Properties

Abstract: The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. In this paper, we study Fibonacci-Like sequence that is defined by the recurrence 12 0 1 ,f or all 2, 2, 2 nn n SS S n S S −− =+ ≥ = =. The associated initial conditions are the sum of initial conditions of Fibonacci and Lucas sequences respectively. We shall define Binet’s formula and generating function of Fibonacci-Like sequence. Mainly, Inducion method and Binet’s formula will be used to establish properties of Fibonacci-Like sequence.

On The Generalized Fibonacci And Pell Sequences By Hessenberg Matrices.

Abstract: In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de…ned by the recurrence relation, for n 1 Pn+1 = 2Pn + Pn 1 (1.2) where P0 = 0; P1 = 1: The well-known Fibonacci and Pell numbers can be generalized as follow: Let A be nonzero, relatively prime integers such that D = A +4 6= 0: De…ne sequence fung by, for all n 2 (see [17]), un = Aun 1 + un 2 (1.3) where u0 = 0; u1 = 1: If A = 1; then un = Fn (the nth Fibonacci number). If A = 2; then un = Pn ( the nth Pell number). An alternative is to let the roots of the equation t At 1 = 0 be, for n 0 un = n n : The sequence fung have studied by several authors (see [6], [1]). The following identities can be found in [6], [1]:

On the sum of powers of two consecutive Fibonacci numbers

Abstract: Let $(F_{n})_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_{n}$, for $n\geq 0$, where $F_{0}=0$ and $F_{1}=1$. In this note, we prove that if $s$ is an integer number such that $F_{n}^{s}+F_{n+1}^{s}$ is a Fibonacci number for all sufficiently large integer $n$, then $s=1$ or 2.

An LSB Data Hiding Technique Using Natural Numbers

Abstract: In this paper, a novel data hiding technique is proposed, as an improvement over the Fibonacci LSB data-hiding technique proposed by Battisti et al,based on decomposition of a number (pixel-value) in sum of natural numbers. This particular representation again generates a different set of (virtual) bit-planes altogether, suitable for embedding purposes. We get more bit-planes than that we get using Prime technique.These bit-planes not only allow one to embed secret message in higher bit-planes but also do it without much distortion, with a much better stego-image quality, and in a reliable and secured manner, guaranteeing efficient retrieval of secret message. A comparative performance study between the classical Least Significant Bit(LSB) method, the Fibonacci LSB data-hiding technique and the proposed schemes indicate that image quality of the stego-image hidden by the technique using the natural decomposition method improves drastically against that using Prime and Fibonacci decomposition technique. Experimental results also illustrate that, the stego-image is visually indistinguishable from the original cover-image. Also we show the optimality of our technique.