Fibonacci Numbers Research Articles

Pure Point Diffraction and Almost Periodicity

AbstractThis article deals with pure point diffraction and its connection to various notions of almost periodicity. We explain why the Fibonacci chain does not fit into the classical concept of Bohr almost periodicity and how it fits into the classes of mean, Besicovitch and Weyl almost periodic point sets. We report on recent results which characterize pure point diffraction as mean almost periodicity of the underlying structure, and discuss how the complex amplitudes fit into this picture.

Improving the functionality of biosensors through the use of periodic and quasi-periodic one-dimensional phononic crystals

Abstract Resonant acoustic band gap materials have steered a new sensing technology era. This study is presented to investigate of the one-dimensional (1D) phononic crystals (PnCs), involving periodic, as well as quasi-periodic 1D layered PnCs represented as a highly sensitive biosensor to detect and monitor the quality of milk. In this regard, the numerical findings show that the examined periodic PnCs structure outperformed the quasi-periodic structure. In particular, it provides a wider phononic band gap and greater sensitivity as well. In addition, the quasi-periodic design (especially Fibonacci sequence S4) introduces multiple resonance peaks via transmission spectra, which may lead to some conflicts during the detection process. The findings reveal that the frequency of the resonant peak can effectively change with varied milk solution concentrations and temperatures. The optimized sensor is capable of differentiating between concentrations ranging between 0 and 50 % with a 10 % step, and it can also differentiate between temperatures, which range between 5 °C and 50 °C. This makes it ideal for precise detection of other liquids and solutions. The sensor performs efficiently for all milk solution concentrations. Here, the findings demonstrated that the examined defective PnC structure exhibited the most favorable sensitivity of the value of 94.34 MHz, so it showed the highest sensitivity when varying milk concentrations. In addition, the configurated sensor provided high QF and FOM values of 3,853.645161 and 157.42, respectively. On the other hand, the sensor performs very well for all temperatures of the milk solution. As such, the S 4 quasi-periodic structure is characterized as the optimal sensor structure when varying temperatures, introducing a sensitivity of 4.78 MHz/°C, QF of 4,278.521, and FOM of 7.48 °C−1.

On (k,p)-Fibonacci numbers and matrices

In this paper, some relations between the powers of any matrices $X$ satisfying the equation $X^k-pX^{k-1}-(p-1)X-I=\bf{0}$ and $(k,p)$-Fibonacci numbers are established with $k\geq2$. First, a result is obtained to find the powers of the matrices satisfying the condition above via $(k,p)$-Fibonacci numbers. Then, new properties related to $(k,p)$-Fibonacci numbers are given. Moreover, some relations between the sequence $\{F_{3,s}(n)\}$ and the generalized Fibonacci sequence $\{U_n(p,q)\}$ are also examined.

Revisiting some r-Fibonacci sequences and Hessenberg matrices

The relationship between different generalizations of Fibonacci numbers and matrices is common in the literature. However, the basic relation of such sequences with Hessenberg matrices is often not properly explored. In this work we revisit some classic results and present some applications in recent contexts.

Inverse tangent series via telescoping sums

When first learning about infinite series, students typically are shown some examples for which the partial sums can be simplified by taking advantage of telescoping sums. In this paper, we present many examples of such series, all involving the inverse tangent function and most of which involve the Fibonacci and Lucas numbers. Most of the series presented here have appeared in various papers (see the references), but the authors are usually working in an abstract setting which makes it difficult for students to follow the basic ideas. We seek to make these results accessible to a wider audience.

More on the Fascinating Characterizations of Mulatu’s Numbers

Background The discoveries of Mulatu’s numbers, better known as Mulatu’s sequence, represent revolutionary contributions to the mathematical world. His best-known work is Mulatu’s sequence, in which each new number is the sum of the two preceding numbers. When various operations and manipulations are performed on the numbers in this sequence, beautiful and incredible patterns begin to emerge. This study aimed to identify novel characterizations of Mulatu’s numbers. Methods This study employed a multi-faceted approach to investigate characterizations of Mulatu’s numbers. Mathematical proof techniques such as principle of mathematical induction, proof by contradiction and direct proof were utilized to substantiate findings. GNU Octave (version 9, (1)) software was applied to verify the existence, classify, and conduct computational investigations of findings prior to formal proofs. Results In this study, we provided several characterizations of Mulatu’s numbers. We also investigated the properties and patterns of these fascinating numbers. Moreover, we have also shown that, similar to Fibonacci’s numbers, Mulatu’s numbers also give the so-called golden ratio, which is most applicable in numerical optimization. Furthermore, we formulated a relation among Mulatu’s numbers, Fibonacci numbers, and Lucas numbers. Finally, we provided a generating function for the Mulatu numbers. Conclusions In this study, we uncovered novel characterizations of Mulatu’s numbers and introduced a generating function for them. We investigated relationship between Mulatu’s numbers and the golden ratio. The results discussed offer valuable insights and enhance our understanding of their properties. Furthermore, these findings play a vital role in the boarder context of mathematical sequences, contributing significantly to the field.

The Pell Tower and Ostronometry

Conway and Ryba considered a table of bi-infinite Fibonacci sequences and discovered new interesting patterns. We extend their considerations to tables that are defined by the recurrence $X_{n+1}=dX_n+X_{n-1}$ for natural numbers $d$. In our search for new patterns we run into a Red Wall and exotic numeration systems.

SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

Abstract Using the special value at $u=1$ of Artin–Ihara L-functions, we associate to every $\mathbb {Z}$ -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$ -cover, our result gives us back Lengyel’s calculation of the p-adic valuations of Fibonacci numbers.

A Mathematical Problem Formulation Based on Number Theory

A branch of mathematics known as number theory is primarily concerned with the study of whole numbers. In that sense, number theory is used less in design than it is in analytics, computation, and other fields. Its inability to be used directly in any application was the issue. However, when combined with the computing power of modern PCs, number theory provides intriguing solutions to real-world issues. It serves a variety of functions in several industries, including computing, numerical analysis, and cryptography. The study of whole numbers is the primary focus of number theory. The fundamental structure of number theory has been gradually improved as a result of the dedication made by mathematicians throughout history to advancing the study of numbers, and as a result, a comprehensive and integrated field has been established. Number theory is a foundational subject that influences many other subjects and has a big impact on teachers. Number theory has been applied in statistics in a few fascinating ways. This overview paper's goal is to highlight particular noteworthy uses of this type. In number theory, indivisible numbers make up an interesting and challenging field of study. The main component of number theory is structured by diophantine equations. A Diophantine equation is an equation that needs necessary arrangements. This paper's first section looks at a few issues related to indivisible numbers and the function of Diophantine equations in Plan Theory. It is explained why Fibonacci and Lucas numbers commit to a semi-remaining Metis structure.

Frequency Division Multiple Access with High Performance Based on Several Defect Resonators According to the Fibonacci Sequence in 1D Photonic Star Waveguide Structure

Frequency Division Multiple Access with High Performance Based on Several Defect Resonators According to the Fibonacci Sequence in 1D Photonic Star Waveguide Structure

A partial recurrence Fibonacci link

The purpose of this note is to develop a conjecture for a Fibonacci number generating function in terms of the elements of a second-order two parameter partial recurrence relation which arose in an operations research problem on Poisson distributed lead time in inventory control.

New Properties and Matrix Representations on Higher-Order Generalized Fibonacci Quaternions with q-Integer Components

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices.

Tiling of dominoes with ranked colors

Several articles deal with tilings with various colors and shapes. In this paper, we present a new type of tiling problem of a (1×n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\ imes n)$$\\end{document}—board where the colors have a prescribed order of preference and the size of colored dominoes is bounded by (1×s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\ imes s)$$\\end{document}. We show that the total number of tilings can be given as a linearly recurrent sequence of order ks, and at the same time by a higher order self-convolution of s-generalized Fibonacci sequences.

Free vibration analysis of laminated composite shallow curved beam with various boundary conditions

This study introduces a novel Ritz approximation function to analyze the free vibrations of laminated compositeshallow curved beams. The methodology utilizes polynomial functions-based Fibonacci series to develop theRitz’s approximation function. The displacement field aligns with the high-order shear deformation theorydesigned explicitly for shallow curved beams. The problem’s governing equation is established by applying theLagrange equation, providing a comprehensive framework for the subsequent analysis. The study meticulouslyexamines the convergence rate and accuracy of the proposed approximation function. Numerical investigationsare conducted to determine the natural frequencies of curved beams. Factors such as curvature-to-length ratio,length-to-thickness ratio, boundary conditions, and material anisotropy are carefully considered. Furthermore,extensive survey results concerning the natural frequencies of curved beams are presented. These results serveas valuable reference data for future investigations in this area, thereby contributing to the academic discourseon this subject matter.

Josephson effect in a Fibonacci quasicrystal

Quasiperiodicity has recently been proposed to enhance superconductivity and its proximity effect. Simultaneously, there has been significant experimental progress in the fabrication of quasiperiodic structures, including in reduced dimensions. Motivated by these developments, we use microscopic tight-binding theory to investigate the DC Josephson effect through a ballistic Fibonacci chain attached to two superconducting leads. The Fibonacci chain is one of the most-studied examples of quasicrystals, hosting a rich multifractal spectrum, containing topological gaps with different winding numbers. We study how the Andreev-bound states (ABS), current-phase relation, and the critical current depend on the quasiperiodic degrees of freedom, from short to long junctions. While the current-phase relation shows a traditional 2π sinusoidal or sawtooth profile, we find that the ABS develop quasiperiodic oscillations and that the Andreev reflection is qualitatively altered, leading to quasiperiodic oscillations in the critical current as a function of junction length. Surprisingly, despite earlier proposals of quasiperiodicity enhancing superconductivity compared to crystalline junctions, we do not, in general, find that it enhances the critical current. However, we find significant current enhancement for reduced interface transparency because of the modified Andreev reflection. Furthermore, by varying the chemical potential, e.g., by an applied gate voltage, we find a fractal oscillation between superconductor-normal metal-superconductor (SNS) and superconductor-insulator-superconductor (SIS) behavior. Finally, we show that the winding of the subgap states leads to an equivalent winding in the critical current, such that the winding numbers, and thus the topological invariant, can be determined. Published by the American Physical Society 2024

Decompositions of Hyperbolic Kac–Moody Algebras with Respect to Imaginary Root Groups

We propose a novel way to define imaginary root subgroups associated with (timelike) imaginary roots of hyperbolic Kac–Moody algebras. Using in an essential way the theory of unitary irreducible representation of covers of the group SO(2, 1), these imaginary root subgroups act on the complex Kac–Moody algebra viewed as a Hilbert space. We illustrate our new view on Kac–Moody groups by considering the example of a rank-two hyperbolic algebra that is related to the Fibonacci numbers. We also point out some open issues and new avenues for further research, and briefly discuss the potential relevance of the present results for physics and current attempts at unification.

Dynamic continualization of mechanical metamaterials with quasi-periodic microstructure.

This article focuses on characterizing a class of quasi-periodic metamaterials created through the repeated arrangement of an elementary cell in a fixed direction. The elementary cell consists of two building blocks made of elastic materials and arranged according to the generalized Fibonacci sequence, giving rise to a quasi-periodic finite microstructure, also called Fibonacci generation. By exploiting the transfer matrix method, the frequency band structure of selected periodic approximants associated with the Fibonacci superlattice, i.e. the layered quasi-periodic metamaterial, is determined. The self-similarity of the frequency band structure is analysed by means of the invariants of the symplectic transfer matrix as well as the transmission coefficients of the finite clusters of Fibonacci generations. A high-frequency continualization scheme is then proposed to identify integral-type or gradient-type non-local continua. The frequency band structures obtained from the continualization scheme are compared with those derived from the Floquet-Bloch theory to validate the proposed scheme. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1).'

Optical properties of the 1D quasiperiodic photonic crystals comprising gyroidal superconductor for THz applications

The present research proposes a quasiperiodic 1D photonic crystal structure, which is designed with a successive arrangement of gyroidal geometry-based Bismuth Strontium Calcium Copper Oxide (BSCCO) superconducting material and SiO2 following a 10th order of Fibonacci sequences. The numerical simulation is carried out using the transfer matrix method to study the transmittance characteristics over a broad band of THz frequency range. The effects of various structural parameters such as filling fraction of BSCCO in the gyroidal layer, hosting materials of the gyroidal layer, and the Fibonacci sequence order on the transmission spectrum are extensively analyzed. It is observed that by suitably dealing with these parameters, we can easily control the number of stop bands, and the cutoff frequency as well. Therefore, the designed structure can be considered as an apt candidate for multichannel filtering applications in the THz frequency regime.

A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

An image encryption based on Fibonacci sequence and fusion of advanced encryption standard-least significant bit method

An image encryption based on Fibonacci sequence and fusion of advanced encryption standard-least significant bit method