Showing papers on "Fibonacci number published in 2011"
TL;DR: In this paper, the authors consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant and show that the thickness tends to infinity and the Hausdorff dimension tends to one.
Abstract: We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents.
93 citations
TL;DR: It is proved that A n is invertible for n > 2, and B n isInverted for any positive integer n .
80 citations
TL;DR: Several new properties of Fibonacci sequences are studied and a sequence closely related to these sequences which can be regarded as a generalization of Lucas sequence of the first kind is investigated.
79 citations
TL;DR: In this article, the authors investigated the electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice and found that the defect mode appeared inside the zero-k¯ gap, which can be applicable to control the electron transport.
Abstract: We investigate electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice. It is found that such structure can possess a zero-k¯ gap which exists in all Fibonacci sequences. Different from Bragg gap, zero-k¯ gap associated with Dirac point is less sensitive to the incidence angle and lattice constants. The defect mode appeared inside the zero-k¯ gap has a great effect on transmission, conductance, and shot noise, which can be applicable to control the electron transport.
73 citations
TL;DR: In this article, the electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice was investigated, and it was found that such structure can possess a zero-$\bar{k}$ gap which exists in all Fibinacci sequences Different from Bragg gap associated with Dirac point is less sensitive to the incidence angle and lattice constants, which can be applicable to control the electron transport.
Abstract: We investigate electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice It is found that such structure can possess a zero-$\bar{k}$ gap which exists in all Fibonacci sequences Different from Bragg gap, zero-$\bar{k}$ gap associated with Dirac point is less sensitive to the incidence angle and lattice constants The defect mode appeared inside the zero-$\bar{k}$ gap has a great effect on transmission, conductance and shot noise, which can be applicable to control the electron transport
68 citations
TL;DR: This paper investigates some basic geometric properties for the class KSL of functions f analytic in the open unit disc @D={z:|z|<1} (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that f(0)=0,f^'(0)=(1-5)/2 is such that |@t| fulfils the golden section of the segment [0,1].
Abstract: This paper investigates some basic geometric properties for the class KSL of functions f analytic in the open unit disc @D={z:|z|<1} (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that f(0)=0,f^'(0)=1andzf^''(z)f^'(z)@[email protected][email protected]^2z^[email protected]@t^2z^2([email protected][email protected]), where, the number @t=(1-5)/2 is such that |@t| fulfils the golden section of the segment [0,1]. Some relevant remarks and useful connections of the main results are also pointed out.
59 citations
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TL;DR: In this article, the authors give an asymptotic estimate of the number of numerical semigroups of a given genus, and show that the proportion of numerical semiigroups satisfying the Frobenius number approaches 1 as ρ ≥ 3m, where ρ is the golden ratio.
Abstract: We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if $n_g$ is the number of numerical semigroups of genus $g$, we prove that $n_g$ tends to $S \phi^g$, where $\phi$ is the golden ratio, and $S$ is a constant, resolving several related conjectures concerning the growth of $n_g$. In addition, we show that the proportion of numerical semigroups of genus $g$ satisfying $f < 3m$ approaches 1 as $g \rightarrow \infty$, where $m$ is the multiplicity and $f$ is the Frobenius number.
54 citations
TL;DR: The bivariate Fibonacci and Lucas p –polynomials are studied and some properties of these polynomials are obtained.
51 citations
TL;DR: It is found that a large number of photonic stopbands can occur at the dielectric Fibonacci multilayers, and the number of absorption peaks can be tuned effectively and enlarged significantly.
Abstract: Absorption properties in one-dimensional quasiperiodic photonic crystal composed of a thin metallic layer and dielectric Fibonacci multilayers are investigated. It is found that a large number of photonic stopbands can occur at the dielectric Fibonacci multilayers. Tamm plasmon polaritons (TPPs) with the frequencies locating at each photonic stopband are excited at the interface between the metallic layer and the dielectric layer, leading to almost perfect absorption for the energy of incident wave. By adjusting the length of dielectric layer with higher refractive-index or the Fibonacci order, the number of absorption peaks can be tuned effectively and enlarged significantly.
50 citations
TL;DR: The class SLM α being closely related to the classes of starlike and convex functions, some basic techniques are applied to investigate certain interesting properties for this class of functions.
48 citations
TL;DR: It is proved that the number of vertices of degree k in @C"n and @L"n is @?"i"="0^k(n-2ik-i)( i+1n-k-i+1) and @?" i"=" 0^k[2(i2i+k-n)(n- 2i-1k- i)+(i-12i+ k-n)], respectively".
TL;DR: The lower and upper bounds for the spectral norms of the matrices A = [ F ( mod ( j - i , n ) ] i, j = 1 n and B = [ L ( mod( j- i, n) ] i , j =1 n are established.
01 Apr 2011
TL;DR: In this article, it was shown that there is no integer ε ≥ 3 such that the sum of two consecutive Fibonacci numbers is a Fibonach number. But this is not the case for any integer ϵ > 0.
Abstract: Here, we show that there is no integer $s\ge 3$ such that the sum of $s$th powers
of two consecutive Fibonacci numbers is a Fibonacci number.
TL;DR: The results by Ohtsuka and Nakamura, who treated the partial infinite sum for all positive integers n as a function of Fibonacci Zeta, are generalized.
Abstract: Abstract The Fibonacci Zeta functions are defined by . Several aspects of the function have been studied. In this article we generalize the results by Ohtsuka and Nakamura, who treated the partial infinite sum for all positive integers n.
19 Dec 2011
TL;DR: In this paper, a simple algebraic approach to synthesis Fibonacci Switched Capacitor Converters (SCC) was developed, which reduces the power losses by increasing the number of target voltages.
Abstract: A simple algebraic approach to synthesis Fibonacci Switched Capacitor Converters (SCC) was developed. The proposed approach reduces the power losses by increasing the number of target voltages. The synthesized Fibonacci SCC is compatible with the binary SCC and uses the same switch network. This feature is unique, since it provides the option to switch between the binary and Fibonacci target voltages, increasing thereby the resolution of attainable conversion ratios. The theoretical results were verified by experiments.
TL;DR: In this article, a brief history of Fibonacci's life and career is presented, along with a wide variety of material accessible to college and even high school mathematics students and teachers at all levels.
Abstract: This article deals with a brief history of Fibonacci's life and career. It includes Fibonacci's major mathematical discoveries to establish that he was undoubtedly one of the most brilliant mathematicians of the Medieval Period. Special attention is given to the Fibonacci numbers, the golden number and the Lucas numbers and their fundamental properties with enlightening examples. A large number of exciting applications of these numbers to mathematical, physical, biological and engineering sciences is included. It also contains a wide variety of material accessible to college and even high school mathematics students and teachers at all levels. Included is also mathematical information that puts the professionals and prospective mathematical scientists at the forefront of current research.
TL;DR: Theoretical analysis shows that the diffraction pattern of CFGs is composed of fractal distributions of impulse rings, which should be of great theoretical interest and shows potential to be further developed into practical applications, such as in laser measurement with wideband illumination.
Abstract: We introduce circular Fibonacci gratings (CFGs) that combine the concept of circular gratings and Fibonacci structures. Theoretical analysis shows that the diffraction pattern of CFGs is composed of fractal distributions of impulse rings. Numerical simulations are performed with two-dimensional fast Fourier transform to reveal the fractal behavior of the diffraction rings. Experimental results are also presented and agree well with the numerical results. The fractal nature of the diffraction field should be of great theoretical interest, and shows potential to be further developed into practical applications, such as in laser measurement with wideband illumination.
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TL;DR: The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients as mentioned in this paper, which can be used to verify and discover identities involving harmonic numbers and derangement numbers.
Abstract: We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.
TL;DR: In this article, for complex linear homogeneous recursive sequences with constant coefficients, a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms of the classical Fibonacci sequence is given.
Abstract: For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincare’s theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.
TL;DR: The sequence follows 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. and represents a trend observable in many natural settings, including inheritance patterns, the design of flowers and the branching of leaves.
Abstract: T thirteenth century mathematician, Leonardo of Pisa, nicknamed “Fibonacci” described a sophisticated number series that was subsequently named after him. The sequence follows 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. and represents a trend observable in many natural settings. These include inheritance patterns, the design of flowers and the branching of leaves (phyllotaxis) [1]. Its widespread existence is further signified by its ability to derive the ‘Golden Ratio’ also known as ‘nature’s m i o
TL;DR: This paper considers the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers α that are generalizations of most of the Fibonacci-like sequences, and shows that these sequences satisfy a linear recurrence relationship when considered modulo k, even though the sequences themselves do not.
Abstract: It is well-known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational α. In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers α. These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r-Fibonacci sequences, and the Pell sequence, to name a few. We show that these sequences satisfy a linear recurrence relation when considered modulo k, even though the sequences themselves do not. We then employ this recurrence relation to determine the generating functions and Binet-like formulas. We end by discussing the convergence of the ratios of the terms of the sequences.
01 Jan 2011
TL;DR: In this paper, a family of conditional sequences is defined by the recurrence relation un = aun 1 + bun 2 if n is even, un = cun 1+dun 2 ifn is odd, with initial conditions u0 = 0 and u1 = 1, where a,b,c and d are non-zero numbers.
Abstract: In this paper, we deal with two families of conditional sequences. The first family consists of generalizations of the Fibonacci sequence. We show that the Gelin-Cesaro identity is satisfied. Also, we define a family of conditional sequences {un} by the recurrence relation un = aun 1 + bun 2 if n is even, un = cun 1+dun 2 if n is odd, with initial conditions u0 = 0 and u1 = 1, where a,b,c and d are non-zero numbers. Many sequences in the literature are special cases of this sequence. We find the generating function of the sequence and Binet's formula for odd and even subscripted sequences. Then we show that the Catalan and Gelin-Cesaro identities are satisfied by the indices of this generalized sequence.
Journal Article•
TL;DR: In this article, a short survey on the problem of counting the number of idempotents of monoids of transformations on a finite chain is presented, in particular transformations that preserve or reverse either the order or the orientation.
Abstract: We consider various classes of monoids of transformations on a finite chain, in particular transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested in computing both their cardinals and the number of their idempotents. In this note we present a short survey on these questions which have been approached by various authors and close the problem by computing the number of idempotents of those monoids not considered before. Fibonacci and Lucas numbers play an essential role in the last computations. 2000 Mathematics Subject Classification: Primary 20M17; Secondary 20M20, 05A10.
TL;DR: In this paper, the authors prove dynamical upper bounds for discrete one-dimensional Schrodinger operators in terms of various spacing properties of the eigenvalues of finite-volume approximations.
Abstract: We prove dynamical upper bounds for discrete one-dimensional Schrodinger operators in terms of various spacing properties of the eigenvalues of finite-volume approximations. We demonstrate the applicability of our approach by a study of the Fibonacci Hamiltonian.
TL;DR: In this paper, the Binet-Fibonacci formula for Fibonacci numbers is treated as a q-number (and q-operator) with Golden ratio bases $q=\phi$ and $Q=-1/\phi$.
Abstract: The Binet-Fibonacci formula for Fibonacci numbers is treated as a q-number (and q-operator) with Golden ratio bases $q=\phi$ and $Q=-1/\phi$. Quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given just by Fibonacci numbers. Ratio of successive energy levels is found as the Golden sequence and for asymptotic states it appears as the Golden ratio. This why we called this oscillator as the Golden oscillator. By double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived.
TL;DR: In this paper, it was shown that for any distinct positive integers s1, s2, s3 corresponding to algebraic α, β with |β| < 1 and αβ = −1, the numbers Φ2s1, Φ 2s2 and Φϵ2s3 are algebraically independent over Q if and only if at least one of s 1, s 2, s 3 is even.
Abstract: Recently, the authors have shown that in the case of algebraic α, β the reciprocal sums Φ2, Φ4, Φ6 are algebraically independent over Q and that for any s > 3 every Φ2s can be expressed explicitly as an algebraic function of Φ2, Φ4, and Φ6. In this paper we prove that for any distinct positive integers s1, s2, s3 corresponding to algebraic α, β with |β| < 1 and αβ = −1 the numbers Φ2s1 , Φ2s2 and Φ2s3 are algebraically independent over Q if and only if at least one of s1, s2, s3 is even. In particular, for β = (1 − √ 5)/2 and β = 1 − √ 2, our results on Φ2s are applicable to reciprocal sums on Fibonacci numbers Un = Fn and on Pell numbers Un = Pn, respectively.
Journal Article•
TL;DR: In this paper, a bijective proof is given for the following theorem: the number of compositions of n into odd parts is the same as that of n + 1 into parts greater than one.
Abstract: A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n + 1 into parts greater than one. Some commentary about the history of partitions and compositions is provided.
TL;DR: Using the method of Laplace expansion to evaluate the determinant tridiagonal matrices, a kind of determinants are constructed to give new proof of the Fibonacci identities.
TL;DR: In this article, 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.