Fibonacci Numbers Research Articles

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.

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

On a Matrix Formulation of the Sequence of Bi-Periodic Fibonacci Numbers

On a Matrix Formulation of the Sequence of Bi-Periodic Fibonacci Numbers

A New Coding Algorithm Using Fibonacci Numbers, Linear Mappings, and Change of Basis

A New Coding Algorithm Using Fibonacci Numbers, Linear Mappings, and Change of Basis

Billiard partitions, Fibonacci sequences, SIP classes, and quivers

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d d -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths.

Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs

Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between the large value to the small value. A Fibonacci sequence is a sequence of numbers that has a unique shape. The first term of this sequence of numbers is 1, then the second term is also 1, then for the third term it is determined by adding the two previous terms so that a sequence of numbers with a certain pattern is obtained. Pascal's triangle is a geometric rule of binomial coefficients in a triangle. Shapes that resemble fractal geometry, golden ratios and Fibonacci numbers are found in many places in this realm, for example the shapes of various plants, animals, as well as in humans themselves, even the universe. This fractal geometry can also be used to design batik motif creations, architectural design or musical art. This paper describes how a fractal object can be made with geometric transformations that can be used to develop batik motif creations. And also look at the relationship between fractal geometry, the golden ratio, Fibonacci numbers, and Pascal's triangles and the shapes of these shapes that exist in this nature.

Robust Left-Right Hashing Scheme for Ubiquitous Computing

Ubiquitous computing systems possess the capability to collect and process data, which is subsequently shared with other devices. These systems encounter resource challenges such as memory constraints, processor speed limitations, power consumption considerations, and the availability of data storage. Therefore, maintaining data access and query processing speed in ubiquitous computing is challenging. Hashing is crucial to search operations and has caught the interest of many researchers. Several hashing techniques have been proposed and Cuckoo Hashing is found efficient to use in several applications. There are two variants of Cuckoo Hashing: Parallel Cuckoo Hashing and Sequential Cuckoo Hashing. Cuckoo Hashing suffers from challenges like high insertion latency, inefficient memory usage, and data migration. This paper proposes two hashing schemes: Left-Right Hashing and Robust Left-Right Hashing that successfully address and solve the major challenges of Sequential Cuckoo Hashing. The proposed schemes adopt the Combinatorial Hashing technique after modification and use this with a new collision resolution technique called Left-Right Random Probing. Left-Right Random Probing is a variant of random probing and uses prime numbers and Fibonacci numbers. In addition, this paper proposes a new performance indicator, degree of dexterity to estimate the performance of hashing techniques. Sequential Cuckoo Hashing suffers from hidden switching costs which are identified and its estimation is given by a new performance indicator called, T.R.C./Key. Performance of Sequential Cuckoo Hashing is order dependent.

Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty.

Shared market opinions and beliefs by market participants generate a set of constraints that mediate information through a not-so-unstable system of expected target prices. Price trajectories, within these sets of constraints, confirm or disprove the likelihood of participant expectations and cannot, de facto, be considered permutable, as literature has shown, since their inner structure is dynamically affected by their own progress, suggesting per se the presence of both heat and cycles. This study described and discussed how trajectories are built using different alphabets and suggests that prices follow an ergodic course within structurally similar tessellation classes. It is reported that the courses of price moves are self-similar due to their a priori structure, and they do not need to be complete in order to create the conditions, in resembling conditions, for the appearance of the well-known and commonly used Fibonacci ratios between price trajectories. To date, financial models and engineering are mostly based on the mathematics of randomness. If these theoretical findings need empirical validation, such a potential infrastructure of ratios would suggest the possibility for a superstructure to exist, in other words, the emergence of exploitable patterns.

Golden lichtenberg algorithm: a fibonacci sequence approach applied to feature selection

Golden lichtenberg algorithm: a fibonacci sequence approach applied to feature selection

A new notion of convergence defined by weak Fibonacci lacunary statistical convergence in normed spaces

Abstract The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces.