Search Results (14)

In this study, we define the $k$-Cullen, $k$-Cullen–Lucas, and Modified $k$-Cullen sequences, and certain terms in these sequences are given. Then, we obtain the Binet formulas, generating functions, summation formulas, etc. In addition, we examine the relations among the terms of the $k$-Cullen, $k$-Cullen–Lucas, Modified $k$-Cullen, Cullen, Cullen–Lucas, Modified Cullen, $k$-Woodall, $k$-Woodall–Lucas, Modified $k$-Woodall, Woodall, Woodall–Lucas, and Modified Woodall sequences. The generating functions were derived and analyzed, especially for cases where Fibonacci numbers were assigned to parameter $k.$ Graphical representations of the generating functions and their logarithmic transformations revealed interesting growth trends and convergence behavior. Further, by multiplying the generating functions with exponential expressions such as $e^k$, we explored the self-similar nature and mirrored dynamics among the sequences. Specifically, it was observed that the Modified Cullen sequence exhibited a symmetric and inverse-like resemblance to the Cullen and Cullen–Lucas sequences, suggesting the presence of deeper structural dualities. Additionally, indefinite integrals of the generating functions were computed and visualized over a range of Fibonacci-indexed k values. These integral-based graphs further reinforced the phenomenon of symmetry and self-similarity, particularly in the Modified Cullen sequence. A key insight of this study is the discovery of a structural duality between the Modified Cullen and standard Cullen-type sequences, supported both algebraically and graphically. This duality suggests new avenues for analyzing generalized recursive sequences through generating function transformations. This observation provides new insight into the structural behavior of generalized Cullen-type sequences. Full article

In most cases, the sizes of the daughter cells of diatoms follow the MacDonald–Pfitzer rule, whereby in many species all diatoms divide once in each generation. In contrast, there are division schemes in which the smaller or larger daughter cell is delayed in its division by one generation and therefore leads to Fibonacci growth. Several properties of diatoms, especially in chain-like colonies, that exhibit such delayed division can be modeled by Lindenmayer systems. These include, above all, the size and orientation of the diatoms. Certain sequences of properties, such as the differences in size indices of neighboring diatoms, are aperiodic and represent self-similar fractal structures. For the division schemes studied, explicit solutions can be found for the number of diatoms of a certain size in each generation. For the experimental differentiation of the division schemes in a diatom chain, in addition to the observation of the division processes over several generations, methods are available that only require the analysis of the structure of a sufficiently large sample. This includes the investigation of the differences in the sizes of neighboring diatoms, the orientations of the diatoms and the frequencies of size indices in a culture. These methods provide a toolbox for investigating diatom properties, applicable to the division models described. Additionally, a mathematical framework is presented that has the potential to be transferable to other properties and other division schemes. Full article

This article is concerned with qualitative and quantitative refinements of the concepts of the log-convexity and log-concavity of positive sequences. A new class of tempered sequences is introduced, its basic properties are established and several interesting examples are provided. The new class extends the class of log-balanced sequences by including the sequences of similar growth rates, but of the opposite log-behavior. Special attention is paid to the sequences defined by two- and three-term linear recurrences with constant coefficients. For the special cases of generalized Fibonacci and Lucas sequences, we graphically illustrate the domains of their log-convexity and log-concavity. For an application, we establish the concyclicity of the points $\left({\displaystyle \frac{a_{2n}}{a_{2n+1}}},{\displaystyle \frac{1}{a_{2n+1}}}\right)$ for some classes of Horadam sequences $(a_n)$ with positive terms. Full article

The Burrows–Wheeler Transform (BWT) is a widely used reversible data compression method, forming the foundation of various compression algorithms and indexing structures. Prior research has analyzed the sensitivity of compression methods and repetitiveness measures to single-character edits, particularly in binary alphabets. However, the impact of such modifications on the compression efficiency of the bijective variant of BWT (BBWT) remains largely unexplored. This study extends previous work by examining the compression sensitivity of both BWT and BBWT when applied to larger alphabets, including alphabet reordering. We establish theoretical bounds on the increase in compression size due to character modifications in structured sequences such as Fibonacci words. Our devised lower bounds put the sensitivity of BBWT on the same scale as of BWT, with compression size changes exhibiting logarithmic multiplicative growth and square-root additive growth patterns depending on the edit type and the input data. These findings contribute to a deeper understanding of repetitiveness measures. Full article

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 $\Delta$-Fibonacci statistical convergence, strong $\Delta$-Fibonacci summability, and $\Delta$-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. Full article

This research addresses the urgent challenges posed by rapid urbanization and climate change through an integrated interdisciplinary approach combining advanced technologies with rigorous scientific exploration. The comprehensive analysis focused on Wuhan, China, spanning decades of meteorological and land-use data to trace extreme urbanization trajectories and reveal intricate temporal and spatial patterns. Employing the innovative 360° radial Fibonacci geometric growth framework, the study facilitated a meticulous dissection of urban morphology at granular scales, establishing a model that combined fixed and mobile observational techniques to uncover climatic shifts and spatial transformations. Geographic information systems and computational fluid dynamics were pivotal tools used to explore the intricate interplay between urban structures and their environments. These analyses elucidated the nuanced impact of diverse morphosectors on local conditions. Furthermore, genetic algorithms were harnessed to distill meaningful relationships from the extensive data collected, optimizing spatial arrangements to enhance urban resilience and sustainability. This pioneering interdisciplinary approach not only illuminates the complex dynamics of urban ecosystems but also offers transformative insights for designing smarter, more adaptable cities. The findings underscore the critical role of green spaces in mitigating urban heat island effects. This highlights the imperative for sustainable urban planning to address the multifaceted challenges of the 21st century, promoting long-term environmental sustainability and urban health, particularly in the context of tomorrow’s climate-adaptive smart cities. Full article

This paper proposes a new meta-heuristic optimization algorithm, the crown growth optimizer (CGO), inspired by the tree crown growth process. CGO innovatively combines global search and local optimization strategies by simulating the growing, sprouting, and pruning mechanisms in tree crown growth. The pruning mechanism balances the exploration and exploitation of the two stages of growing and sprouting, inspired by Ludvig’s law and the Fibonacci series. We performed a comprehensive performance evaluation of CGO on the standard testbed CEC2017 and the real-world problem set CEC2020-RW and compared it to a variety of mainstream algorithms such as SMA, SKA, DBO, GWO, MVO, HHO, WOA, EWOA, and AVOA. The best result of CGO after Friedman testing was 1.6333/10, and the significance level of all comparison results under Wilcoxon testing was lower than 0.05. The experimental results show that the mean and standard deviation of repeated CGO experiments are better than those of the comparison algorithm. In addition, CGO also achieved excellent results in specific applications of robot path planning and photovoltaic parameter extraction, further verifying its effectiveness and broad application potential in practical engineering problems. Full article

Background: Fibonacci patterns and tubular forms both arose early in the phylogeny of multicellular organisms. Tubular forms offer the advantage of a regulated internal milieu, and Fibonacci forms may offer packing efficiencies. The underlying mechanisms behind the cellular genesis of Fibonacci and tubular forms remain unknown. Methods: In a multicellular organism, cells adhere to form a macrostructure and to coordinate further replication. We propose and prove simple theorems connecting cell replication and adhesion to Fibonacci forms and simplicial topology. Results: We identify some cellular and molecular properties whereby the contact inhibition of replication by adhered cells may approximate Fibonacci growth patterns. We further identify how a component $2\to 3$ cellular multiplication step may generate a multicellular structure with some properties of a two-simplex. Tracking the homotopy of a two-simplex to a circle and to a tube, we identify some molecular and cellular growth properties consistent with the morphogenesis of tubes. We further find that circular and tubular cellular aggregates may be combinatorially favored in multicellular adhesion over flat shapes. Conclusions: We propose a correspondence between the cellular and molecular mechanisms that generate Fibonacci cell counts and those that enable tubular forms. This implies molecular and cellular arrangements that are candidates for experimental testing and may provide guidance for the synthetic biology of hollow morphologies. Full article

The aim of this paper is to discuss the emergence of recursive thinking through the famous problem posed by Fibonacci regarding the growth of the rabbit population. This paper qualitatively analyzes and discusses the semiotic aspects raised by the students working with this historical source in the form of a story. From this perspective, the value of the historical problems as socio-cultural references (voices) and of the narrations as mediating factors to enhance students’ learning of new mathematical concepts, such as recursion, is explored in depth. The focus lies on the pivotal role played by the students’ construction of personal senses during in-group mathematical activities, in dialectics with the normative and mathematical meanings. It is highlighted that fostering environments conducive to dialogue among peers, as well as linking various shapes and contexts of knowledge, is necessary. Here, storytelling and history are regarded as fruitful resources aiding students in the gradual construction of a personal sense of mathematical concepts, including recursion. Full article

Amorphous poly(p-vinyl phenol) (PVPh) was added into semicrystalline poly(p-dioxanone) (PPDO) to induce a uniquely novel dendritic/ringed morphology. Polarized-light optical, atomic-force and scanning electron microscopy (POM, AFM, and SEM) techniques were used to observe the crystal arrangement of a uniquely peculiar cactus-like dendritic PPDO spherulite, with periodic ring bands not continuingly circular such as those conventional types reported in the literature, but discrete and detached to self-assemble on each of the branches of the lobs. Correlations and responsible mechanisms for the formation of this peculiar banded-dendritic structure were analyzed. The periodic bands on the top surface and interior of each of the cactus-like lobs were discussed. The banded pattern was composed of feather-like lamellae in random fractals alternately varying their orientations from the radial direction to the tangential one. The tail ends of lamellae at the growth front spawned nucleation cites for new branches; in cycles, the feather-like lamellae self-divided into multiple branches following the Fibonacci sequence to fill the ever-expanding space with the increase of the radius. The branching fractals in the sequence and the periodic ring-banded assembly on each of the segregated lobs of cactus-like dendrites were the key characteristics leading to the formation of this unique dendritic/ringed PPDO spherulite. Full article

Let ${\left(F_n\right)}_{n\ge 0}$ be the sequence of Fibonacci numbers. The order of appearance of an integer $n\ge 1$ is defined as $z\left(n\right):=min\{k\ge 1:n\mid F_k\}$. Let ${\mathcal{Z}}^{\prime }$ be the set of all limit points of $\left\{z\right(n)/n:n\ge 1\}$. By some theoretical results on the growth of the sequence ${(z\left(n\right)/n)}_{n\ge 1}$, we gain a better understanding of the topological structure of the derived set ${\mathcal{Z}}^{\prime }$. For instance, $\{0,1,\frac{3}{2},2\}\subseteq {\mathcal{Z}}^{\prime }\subseteq [0,2]$ and ${\mathcal{Z}}^{\prime }$ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number $t<2$ such that ${\mathcal{Z}}^{\prime }\cap (t,2)$ is the empty set. In this paper, we improve this result by proving that $(\frac{12}{7},2)$ is the largest subinterval of $[0,2]$ which does not intersect ${\mathcal{Z}}^{\prime }$. In addition, we show a connection between the sequence ${\left(x_n\right)}_n$, for which $z\left(x_n\right)/x_n$ tends to $r>0$ (as $n\to \infty$), and the number of preimages of r under the map $m\mapsto z\left(m\right)/m$. Full article

This theoretical article provides a brief description of the model of living systems’ functioning by defining them as self-reproducing information or as self-reproduction of resource flows patterns. It reviews the living systems growth limitation between their development cycles by the Fibonacci sequence. Besides, there are presented systems resource base criteria, necessary for accumulating the resources and their investment. The article also considers the conditions for the formation of various systems strategies. Then we reviewed the principles of elemental analysis of information by a person as a living system according to the considered model. The study also shows the possibility of forming priorities in analyzing information for 16 combinations as maximum. At that, it remains crucial to divide a human’s information analysis between the two hemispheres of the brain. The described combinations of priorities in a person’s information analysis are compared with the existing differential personality models, such as the big five personality traits, the Myers–Briggs type indicator, temperaments model and Honey and Mumford Learning styles. Full article

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth. Full article

We consider a random generalization of the classical Fibonacci substitution. The substitution we consider is defined as the rule mapping, a → baa and b → ab, with probability , and → ba, with probability 1 – p for 0 < p < 1, and where the random rule is applied each time it acts on a . We show that the topological entropy of this object is given by the growth rate of the set of inflated random Fibonacci words, and we exactly calculate its value. Full article