Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and mean-ingful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbersand Lucas numbers are obtained.

The Some Sum Formula for Generalized Fibonacci Numbers

This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.

Summation of Reciprocals Related to Square of Products of Fibonacci Numbers

By applying the method of on summation by parts,the purpose of this paper is to give several reciprocal summations related to squares of products of the Fibonacci numbers.

The Infinite Sum of Reciprocal of the Fibonacci Numbers

In this paper,we consider infinite sums of the reciprocals of the Fibonacci numbers.Then applying the floor function to the reciprocals of this sums,we obtain a new identity involving the Fibonacci numbers.

On the Norms of r-Toeplitz Matrices Involving Fibonacci and Lucas Numbers

Let us define to be a r-Toeplitz matrix. The entries in the first row of are or;where Fn and Ln denote the usual Fibonacci and Lucas numbers, respectively. We obtained some bounds for the spectral norm of these matrices.

Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers

In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].

On the Norms of r-Hankel Matrices Involving Fibonacci and Lucas Numbers

Let us define A=Hr=(aij)?to be n×n?r-Hankel matrix. The entries of matrix A are Fn=Fi+j-2?or Ln=Fi+j-2?where Fn?and Ln?denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact value of matrix A’s spectral norm. These kinds of matrices have connections with signal and image processing, time series analysis and many other problems.

k-Order Fibonacci Polynomials on AES-Like Cryptology

The Advanced Encryption Standard(AES)is the most widely used symmetric cipher today.AES has an important place in cryptology.Finite field,also known as Galois Fields,are cornerstones for understanding any cryptography.This encryption method on AES is a method that uses polynomials on Galois fields.In this paper,we generalize the AES-like cryptology on 2×2 matrices.We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm.So,this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.

Do Planetary Transits Predict Synchronicity Experience?

Synchronicity involves the experience of personal meaning entangled with ambiguous coincidences in time. Ambiguity results from incomplete information about the chances of various events occurring. The problem that this study addresses is the lack of empirical research on synchronicity. This study sought to address this problem by exploring the astrological hypothesis that planetary transits predict synchronicity events. Synchronicities were compared with the probability distributions of planetary transits. In comparison with the base rate prediction, planetary transits were not a significant predictor of synchronicity events. The findings of this study provide new insight into the complex, multifaceted, and ambiguous phenomenon of synchronicity. The concept of ambiguity tolerance plays a significant role in synchronicity research since ambiguity cannot be completely eliminated.

A Different Approach to High-Tc Superconductivity: Indication of Filamentary-Chaotic Conductance and Possible Routes to Room Temperature Superconductivity

The empirical relation of between the transition temperature of optimum doped superconductors Tco and the mean cationic charge , a physical paradox, can be recast to strongly support fractal theories of high-Tc superconductors, thereby applying the finding that the optimum hole concentration of σo = 0.229 can be linked with the universal fractal constant δ1 = 8.72109… of the renormalized quadratic Hénon map. The transition temperature obviously increases steeply with a domain structure of ever narrower size, characterized by Fibonacci numbers. However, also conventional BCS superconductors can be scaled with δ1, exemplified through the energy gap relation kBTc ≈ 5Δ0/δ1, suggesting a revision of the entire theory of superconductivity. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possibly ferroelastic nature delivering nano-channels for very fast charge transport, in common for both high-Tc superconductor and organic-inorganic halide perovskite solar materials. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of “bulk” CuO1﹣x samples. In addition, cesium substitution in the Tl-1223 compound is an option.

On optimal binary signed digit representations of integers

Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.

DIFFERENTIAL SUBORDINATIONS AND α-CONVEX FUNCTIONS

This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = {z ： ｜z｜〈 1} satisfying the conditions that f（0） =0, f＇（0）= 1, and α（1＋zf＇＇（z）/f＇（z）＋（1-α）zf＇（z）/f（z）∈p（U）for all z ∈ U, where αis a real number and p（z）=1＋r^2z^2/1-Tz-T^2z^2（z∈ U）. The number T = （1 -√5）/2 is such that T^2 = 1 ＋ T. The class SFLMa introduced by J. Dziok, R.K. Raina, and J. Sokot [3, Appl. Math. Comput. 218 （2011）, 996-1002] is closely related to the classes of starlike and convex functions. The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.

A Note on the Euclidean Algorithm

The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper bounds have been found for this problem. Here, we provide a sharp upper bound for a function which has a direct relation to the numbers whom the greatest common divisor we are trying to calculate. We mainly use some features of Fibonacci numbers as our tools.

Squares from D(-4)and D(20)Triples

We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.

Complete Solution of Diophantine Pairs Induced by Some Fibonacci Formula

A set[ai,a2,...,am)of positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all 1≤i<j≤m.Let(a,b,c)be the Diophantine triple with c>max(a,b].In this paper,we find the condition for the reduction of third element c,and using this result,we prove the extendibility of Diophantine pair[F_(k)-1F_(k+1),F_(k-2)F_(k+2)],where Fn is the n-th Fibonacci number.

Golden Quartic Polynomial and Moebius-Ball Electron

A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (ge = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.

Fractal Analyses Reveal the Origin of Aesthetics in Chinese Calligraphy

Chinese calligraphy is a thousand-year-old writing art. The question of how Chinese calligraphy artworks convey emotion has cast its spell over people for millennia. Calligraphers' joys and sorrows were expressed in the complexity of the character strokes, style variations and general layouts. Determining how Chinese calligraphy aesthetic patterns emerged from the general layout of artworks is a challenging objective for researchers. Here we investigate the statistical fluctuation structure of Chinese calligraphy characters sizes using characters obtained from the calligraphy artwork "Preface to the Poems Collected from the Orchid Pavilion" which was praised as the best running script under heaven. We found that the character size distribution is a stretched exponential distribution. Moreover, the variations in the local correlation features in character size fluctuations can accurately reflect expressions of the calligrapher's complex feelings. The fractal dimensions of character size fluctuations are close to the Fibonacci sequence. The Fibonacci number is first discovered in the Chinese calligraphy artworks, which inspires the aesthetics of Chinese calligraphy artworks and maybe also provides an approach to creating Chinese calligraphy artworks in multiple genres.

Topological Properties of Fibonacci Networks

The Fibonacci numbers are the numbers defined by the linear recurrence equation, in which each subsequent number is the sum of the previous two. In this paper, we propose Fibonacci networks using Fibonacci numbers. The analyticai expressions involving degree distribution, average path lengh and mean first passage time are obtained. This kind of networks exhibits the smail-world characteristic and follows the exponential distribution. Our proposed models would provide the vaiuable insights into the deterministicaily delayed growing networks.

Fibonacci and Lucas Congruences and Their Applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L2mn＋k ≡（-1）（m＋1）nLk（modLm）, F2mn＋k ≡（-1）（m＋1）nFk （modLm）, L2mn＋k ≡ （-1）mn Lk（mod Fm） and F2mn＋k≡ （-1）mn Fk （mod Fm）. By the achieved identities, divisibility properties of Fibonacci and Lueas numbers are given. Then it is proved that there is no Lucas number Ln such that Ln = L2ktLmx2 for m 〉 1 and k≥1. Moreover it is proved that Ln = LmLr is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.

Self-assembly of Fibonacci number spirals in amyloid-like nanofibril films

Fibonacci number spiral patterns can be found in nature,particularly in plants,such as the sunflowers and phyllotaxis.Here,we demonstrated this pattern can be reproduced spontaneously within self-assembling peptide nanofibril films.By high-temperature water vapor annealing of an amorphous film containing both peptide and cationic diamines,well-defined amyloid-like nanofibrils can be assembled spontaneously,during which the nanofibrils will hierarchically stack with each other following the Fibonacci number patterns.The formation of the patterns is a selftemplated process,which involves stepwise chiral amplification from the molecular scale to the nano-and micro-scales.Moreover,by controlling the diameter,length,and handedness of the nanofibrils,various complex hierarchical structures could be formed,including vertically aligned nanoarray,mesoscale helical bundles,Fibonacci number spirals,and then helical toroids.The results provide new insights into the chiral self-assembly of simple biological molecules,which can advance their applications in optics and templated synthesis.