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This document provides a summary of a special issue of the journal Mathematics, which features research papers in the field of algebraic, analytic, and computational number theory. The papers cover a wide range of topics, including generalized Lucas pseudoprimality, Fibonacci numbers, piecewise linear difference equations, properties of Euler's function, regularities in ordered semihypergroups, zero-density results for automorphic functions, properties of Fibonacci finite operator quaternions, and more. The authors of the special issue believe that the techniques used in these papers have the potential to yield new results in these areas of mathematics. [Extracted from the article]

This article discusses the benefits of integrating storytelling and narrative techniques into teaching practices across various disciplines. The papers in this Special Issue highlight the transformative power of narrative in enhancing educational experiences, particularly in complex subjects like protein biosynthesis, geometry, recursive thinking, optics, and scientific phenomena. The studies demonstrate how storytelling can demystify complex concepts, engage students, aid in retention and application of knowledge, and foster interdisciplinary learning. Overall, the integration of storytelling in education can enhance students' understanding, retention, and enthusiasm, leading to more effective learning experiences across various disciplines. [Extracted from the article]

We define and study the notion of the golden ratio of order k = 0, denoted ϕk, as a generalized form of the golden ratio ϕ for any real number k = 0. We show that similar to the special case of ϕ and its conjugate ϕ, ϕk and ϕk are the two distinct roots of a quadratic polynomial for any fixed real k = 0. We express some numerical and algebraic properties of ϕk and ϕk and write their relations to ϕ and ϕ, respectively, with some examples for some special values of k. In particular, it is shown that ϕk = ϕ and ϕk = ϕ if and only if k = 0. We show that Z[(k + 1)ϕk] is a subring of the ring Z[ϕ] for any nonnegative integer k. We will define the golden rectangle of order k (or k-golden rectangle for short) with a class of examples for all k = 0. We also discuss some cases of two Fibonacci numbers in connection to the golden ratio. We will show that the ratio of height to width of the pages of the Gutenberg Bible is the golden ratio of order k ϕ= 0. Actually, some erroneous ideas and examples of disputed observations related to the golden ratio are good reasons to apply ϕk to improve the measurements regarding ?ϕ for some k ϕ= 0. Finally, we end the paper by posing a question related to the Penrose tiling and quasicrystals in connection to the golden ratio of order k > 0. [ABSTRACT FROM AUTHOR]

Epochs-making events of history escalated the paradigms of pedagogical writings of the organizational realm and served as a heuristic tool for the development of theories. The centre of this paper is to erect the bridge linking all junctures of ancient times and transfigured anatomy of organizational slants. The first segment of the paper seeds the soil of contents to our arguments principally for the readers who are entering the realm of chronological demarcations for the first time. Pithy accounts of organizational theories are delineated regarding how paradoxical pillars of faiths towards all institutions disoriented the entire organizational domains and perspectives and bestow the discourse with three Centrum themes. Scholars spared no effort to expound on the introduction of the Fibonacci family and its inexplicable effects in the patterns of nature, music, architecture, and painting even in the human body. This pedagogical attempt proffers matchless currency by enunciating the expression of golden Fibonacci numbers; golden Fibonacci recursive formula as a metaphorical delineation to modernism, symbolic and post-modernist organizational theories. The golden rectangle comparison to perceptions of four research paradigms; the golden Fibonacci ratio concerning fathoming the ontological and epistemological paradigms; the golden angle exemplification to decipher the reflective organizations; and the golden spiral allegorical attempt at reflective organizations is the eminent pronouncement of this paper. Prestigious contemporary constituent; corporate aesthetic reviews engrossed the contemplation of organizational scientists during the last decade. In this research, we ventured to comprehend corporate aesthetic capital over instrumental capital with the supremacy of organizational "crisis of representation". [ABSTRACT FROM AUTHOR]

The paper aims to extend the classical two-dimensional (2D) Fibonacci spiral into three-dimensional (3D) space by using geometric constructions starting from cubic Fibonacci identities and relying on affine maps and parametrizations of the curves. We have already performed a comprehensive survey of cubic Fibonacci identities, which, to our surprise, uncovered only a handful of homogenous cubic identities. Obviously, the goal here is to show how one could use a particular homogenous cubic Fibonacci identity for generating 3D geometric designs similar in spirit to the way the classical Fibonacci spiral is built in 2D starting from a quadratic Fibonacci identity. This made us realize that for any cubic identity there are many different ways of packing cuboids, while only an insignificant fraction of those possible tilings might allow a smooth spiral-like curve to be drawn through them. After reviewing the state of the art, we present accurate details on ways to construct such 3D spirals using affine maps. We go on to prove the continuity and smoothness of such 3D spirals by giving a parametrization of the intersection of the surfaces that define the curves. Throughout the paper, we visualize the resulting 3D spirals by generating geometrically correct stereoscopic views. Finally, it is to be mentioned that the recursive 3D packing of cuboids tends to lead to fractal structures, which will need further investigations. [ABSTRACT FROM AUTHOR]

In this paper, dual bicomplex Balancing and Lucas-Balancing numbers are defined, and some identities analogous to the classic properties of the Fibonacci and Lucas sequences are produced. We give the relationship between these numbers and Pell and Pell-Lucas numbers. From these, the basic bicomplex properties for the norm and its conjugate of these numbers are also developed. These in turn lead to the Binet formula, the generating functions and exponential generating functions, which are important concepts for number sequences. The Cassini identity, which is important for number sequences, actuallyemerged to solve the famous Curry paradox. We calculated the Cassini, Catalan, Vajda and d'Ocagne identities for these numbers. [ABSTRACT FROM AUTHOR]

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. [ABSTRACT FROM AUTHOR]

Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10dFm + Lk and Fn = 10dLm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results. [ABSTRACT FROM AUTHOR]

In the present paper, K denotes a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of sequences, infinite series and infinite matrices are in K. Following Kangro [2, 3, 4], we introduce the concept of boundedness with speed A or A-boundedness. We then obtain a characterization of the matrix class (mx.nF), where mx denotes the set of all A-bounded sequences in K. We conclude the paper with a remark about the matrix class (cx, rn), where cx denotes the set of all A-convergent sequences in K. [ABSTRACT FROM AUTHOR]

In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz–Sobolev spaces. What matters here is the behavior of the integral, not the space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identity of these numbers. [ABSTRACT FROM AUTHOR]

This paper aims to exemplify the use of the golden ratio principle for the creation of a continuous print. The ratio of 1:1.618 signifies the bases for the creation of the componing elements of the print and corresponds to the rapport. The print is achieved using the graphics program Adobe Illustrator®, starting with the drawing of the rectangle in which the circles, which are the components of the print, will be framed. Adobe Illustrator® is a program that specializes in creating and editing vector graphics. This program has become one of the most popular software in the industry for creating vector graphics that can be scaled and edited without losing resolution or clarity. The program allows you to draw both simple and complex shapes using various tools and palettes, logos, sketches, illustrations. Objects created in Adobe Illustrator can be printed or published exactly as they were made. The subsequent use of the print as well as of the single base element is multiple, this paper describes needed steps for creating a continuous print, but the single initial element can be used for a placed print in a smaller size, such as logo which is the most representative form of product identification. A closer look reveals that the Fibonacci sequence has found its usefulness in a variety of activities, including art, sculpture, architecture, music, and graphics. [ABSTRACT FROM AUTHOR]

The paper presents an algorithm for harvesting optimal power points (OPP) utilizing the direct power control (DPC) technique of the dual-fed induction generator (DFIG) wind turbine. The conventional Fibonacci search algorithm (FSA) technique cannot find the exact global peak (GP) strategy and may fail. It proposes a new modified Fibonacci search algorithm (MFSA) to extract (OPP) with variable-speed wind energy turbines (VSWET), which is very important for many people and companies. Among the other different algorithms, the MFSA is more efficient and has a simple strategy to track the optimal power point that depends on calculating the effective speed of the Ωm in an interval [a, b]. The research paper suggests modifying the Fibonacci-based search on the rotor side conversion (RSC) of a 7.5 KW DFIG in the WECS wind energy conversion strategy. The simulation uses the built-in MATLAB/SIMULINK (R2023a) software. It demonstrates the ease of implementation, lower total harmonic distortion THD and efficiency since it uses a step size that is not fixed based on the Fibonacci sequence. [ABSTRACT FROM AUTHOR]

A specially multiplicative arithmetic function is the Dirichlet convolution of two completely multiplicative arithmetic functions. The aim of this paper is to prove explicitly that two mathematical objects, namely (a, b)-Fibonacci sequences and specially multiplicative prime-independent arithmetic functions, are equivalent in the sense that each can be reconstructed from the other. Replacing one with another, the exploration space of both mathematical objects expands significantly [ABSTRACT FROM AUTHOR]

Frank matrix is one of the popular test matrices for eigenvalue routines because it has well-conditioned and poorly conditioned eigenvalues. In this paper, we investigate the bounds for the maximum eigenvalues of the special cases of the generalized Frank matrices which are called Fibonacci-Frank and Lucas-Frank matrices. Then, we obtain the Euclidean norms and the upper bounds for the spectral norms of these matrices. [ABSTRACT FROM AUTHOR]

In this paper, we derive the well-defined solutions to a θ -dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results. [ABSTRACT FROM AUTHOR]

The Capsule Reviews are intended to provide a short succinct review of each paper in the issue in order to bring to a wider readership. The Capsule Reviews were compiled by Fairouz Kamareddine. Professor Kamareddine is an Associate Editor of The Computer Journal and is based in the Department of Mathematical and Computer Sciences at Heriot-Watt University, Edinburgh, UK. [ABSTRACT FROM PUBLISHER]

In this paper, we define a function Q which gives the sequence of partial sums of the Fibonacci quaternion sequence (Q n). We provide an identity about Q k which is the sequence obtained by applying Q to (Q n) k -times. [ABSTRACT FROM AUTHOR]

The fabrication of artificial heterostructures is mainly based on substitution systems. We present simple ways to construct double-sided versions of the Fibonacci, Prouhet-Thue-Morse, paperfolding, period doubling and Golay-Rudin-Shapiro sequences. We also construct a generic instance of the two-dimensional Prouhet-Thue-Morse structure and explore its symbolic complexity. The complexity turns out to be polynomial and hence, the entropy goes to zero. [ABSTRACT FROM AUTHOR]

P. Erdős et al. proved in 1990 that every nontrivial number has a continuum of expansions on two-letter alphabets in every base smaller than the Golden ratio, and that this property fails for the Golden ratio base. It was shown in a recent paper of Baiocchi et al. that if we replace the powers of the Golden ratio by the closely related Fibonacci sequence, then the resulting integer base expansions still have the continuum expansion property. The proof heavily relied on the special properties of the Golden ratio. The difficulty came from the fact that the new expansions do not have any more the ergodic structure of non-integer base expansions. In this paper we introduce a new "quasi-ergodic" approach that allows us to handle many more general cases. We apply this approach to Baker's generalized Golden ratios. • In small bases on two-letter alphabets each number has a continuum of expansions. • This property fails in the limiting case of the Golden ration base. • But the property was restored for the closely related Fibonacci expansions. • By a new approach we extend the last result for larger alphabets. [ABSTRACT FROM AUTHOR]

In this paper, we study the zeros of lacunary-type polynomials with complex coefficients. Here we present some results to locate the zeros of lacunary-type polynomials and discuss their importance with respect to existing results comparatively. [ABSTRACT FROM AUTHOR]

In this paper, it is given a generalization of the relation of the repeating decimals which displays the successive terms of the Fibonacci sequence and Pascal's rows. Additionally, two types of fractions that contain successive terms of the Padovan sequence in their decimal representation are given, by reading from left to right and giving numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized anti-symmetry laws in groupoids, and we apply this concept to several algebraic structures. Moreover, we show that every Fibonacci sequence on (C, *) is periodic. [ABSTRACT FROM AUTHOR]

In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: ...xn+1 = Lm+2+Lm+1yn􀀀1 Lm+3+Lm+2yn􀀀1; yn+1 = Lm+2+Lm+1zn􀀀1 Lm+3+Lm+2zn􀀀1; zn+1 = Lm+2+Lm+1wn􀀀1 Lm+3+Lm+2wn􀀀1; wn+1 = Lm+2 +Lm+1xn􀀀1 Lm+3 +Lm+2xn􀀀1: where n 2 N0, fLmg+¥ m=􀀀¥ is Lucas sequence and the initial conditions x􀀀1, x0, y􀀀1, y0, z􀀀1, z0, w􀀀1, w0 are arbitrary real numbers such that v􀀀i 6= 􀀀 Lm+3 Lm+2, where v􀀀i = x􀀀i;y􀀀i; z􀀀i;w􀀀i, i = 0;1 and m 2 Z. [ABSTRACT FROM AUTHOR]

In this paper, by using generating functions for the Chebyshev polynomials, we have obtained the convolution formulas involving the bi-periodic Fibonacci and Lucas polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we find all k-generalized Fibonacci numbers which are powers of 10 as a special case of almost repdigits. In the second part of the paper, by using the roots of the characteristic polynomial of the k-generalized Fibonacci sequence, we introduce k-generalized tiny golden angles and show the feasibility of this new type of angles in application to magnetic resonance imaging. [ABSTRACT FROM AUTHOR]

In this paper, we look at Diophantine triples with values in three different binary recurrence sequences. These are the Fibonacci and Pell sequences and the sequence of one more of powers of a given prime p. The novelty of the article is the appearance of three different sequences, as up to now the analogous problem had been investigated only for one sequence. [ABSTRACT FROM AUTHOR]

In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group (Zn, +), where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence is a well-known sequence of numbers with numerous applications in mathematics, computer science, and other fields. In recent years, there has been growing interest in studying Fibonacci-like sequences on elliptic curves. These sequences have a number of exciting properties and can be used to build new encryption systems. This paper presents a further generalization of the Fibonacci sequence defined on elliptic curves. We also describe an encryption system using this sequence which is based on the discrete logarithm problem on elliptic curves. [ABSTRACT FROM AUTHOR]

In this paper, we use the method of error estimation to consider the reciprocal sums of products of any m th-order linear recurrence sequences { u n } . Specifically, we find that a series of sequences are "asymptotically equivalent" to the reciprocal sums of products of any m th-order linear recurrence sequences { u n } . [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper, using some properties of Fibonacci numbers, reciprocal polynomials for Fibonacci polynomials, and Legendre symbol, we establish some congruences involving central binomial coefficient modulo p and p². We also give some new identities for hyperbolic functions. [ABSTRACT FROM AUTHOR]

Let n be the set of un-oriented and rational links with crossing number n , a precise formula for | n | was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let Λ n be the set of oriented rational links with crossing number n and let Λ n (d) be the set of oriented rational links with crossing number n (n ≥ 2) and deficiency d. In this paper, we derive precise formulas for | Λ n | and | Λ n (d) | for any given n and d and show that Λ n (d) = F n − d − 1 (d) + 1 + (− 1) n d 2 F ⌊ n 2 ⌋ − ⌊ d + 1 2 ⌋ (⌊ d 2 ⌋) , where F n (d) is the d th convolution of the convolved Fibonacci sequences. [ABSTRACT FROM AUTHOR]

In this paper, a bipolar chaotic Toeplitz measurement matrix optimization algorithm for alternating optimization is presented. The construction of measurement matrices is one of the key techniques for compressive sensing from theory to engineering applications. Recent studies have shown that bipolar chaotic Toeplitz matrices, constructed by combining the intrinsic determinism of bipolar chaotic sequences with the advantages of Toeplitz matrices, have significant advantages over other measurement matrices in terms of memory overhead, computational complexity, and hard implementation. However, problems such as strong correlation and large interdependence coefficients between measurement matrices and sparse dictionaries may still exist in practical applications. To address this problem, we propose a new bipolar chaotic Toeplitz measurement matrix alternating optimization algorithm. Firstly, by introducing the structure matrix, the optimization problem of the measurement matrix is transformed into the optimization problem of the generating sequence, thus ensuring that the optimization process does not destroy the structural properties of the matrix; then, constraints are added to the values of the generating sequence during the optimization process, so that the optimized measurement matrix still maintains the bipolar properties. Finally, the effectiveness of the optimization algorithm in this paper is verified by simulation experiments. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix can effectively reduce the reconstruction error and improve the reconstruction probability. [ABSTRACT FROM AUTHOR]

In this paper, inspiring Hosoya's triangle, we define a new Narayana triangle. Then, we represent this Narayana triangle geometrically on the plane. In addition, we give some identities and properties of the new Narayana triangle. [ABSTRACT FROM AUTHOR]

In this paper, we gave some new identities and super-congruences involving binomial coefficient, and some interesting congruences related to Fibonacci and Lucas sequence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduced m-extension dual complex Fibonacci p-numbers. We established the properties of mextension dual complex Fibonacci p-numbers. They are connected to complex Fibonacci numbers, complex Fibonacci p-numbers and dual complex Fibonacci p-numbers. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm is proposed to compute the inverse of an invertible matrix. The new algorithm is a generalization of the algorithms based on the well-known Schultz-type iterative methods. We show that the convergence order of the new method is a linear combination of the Fibonacci sequence and also is powerful and efficient in finding and keeping sparsity of the obtained approximate inverse of sparse matrices. The convergence of the algorithm is analysed and some applications are studied. It is shown that the proposed algorithm can be used for computing an approximation of the Moore–Penrose inverse of matrices. Numerical examples are provided to verify the feasibility and effectiveness of the new method. [ABSTRACT FROM AUTHOR]

In this paper, we aim to provide new and simple proof for the determinant characterization of moment sequences with discrete measures by investing in generalized Fibonacci sequences. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized Fibonacci like sequences {tk,n}k∈{2,3},n∈N with arbitrary initial seed and give some new and well-known identities like Binet’s formula, trace sequence, Catalan’s identity, generating function, etc. Further, we study various properties of these generalized sequences, establish a recursive matrix and relationships with Fibonacci and Lucas numbers and sequence of Fibonacci traces. In this study, we examine the nature of identities and recursive matrices for arbitrary initial values. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized continued fractions for the expression of bi-periodic Fibonacci ratios. [ABSTRACT FROM AUTHOR]

Recently, the most general form of the quaternion algebra on 3 parameters (3PGQ) has been introduced, which prompted us to look for some properties associated with this algebra, which will be called kλ1,λ2,λ3 in this article. This paper consists of two main parts: the first part focuses on derivations, while the second part deals with the Fibonacci sequence, and the properties of both parts are related to the algebra kλ1,λ2,λ3. First we determine the derivations of the algebra kλ1,λ2,λ3 and get the algebra Der(kλ1,λ2,λ3) of derivations of kλ1,λ2,λ3. Then we were able to obtain generalized derivations that have been studied by mathematicians in the context of algebras of certain normed spaces, and semi-prime and prime rings. In the second part, we introduce some properties of the Fibonacci quaternions in the generalized quaternion algebra kλ1,λ2,λ3, which allow us to assume that the algebra kλ1,λ2,λ3 is not always a division algebra. [ABSTRACT FROM AUTHOR]