Narrow Results

The stability in stock markets is the theme of this work. It has been demonstrated that the random walk theory alone is insufficient to explain the dynamic behavior of asset or stock prices. Deviations from the random walk theory reveal collective behaviors that produce waves and patterns. Research has shown that the Fibonacci sequence and the golden ratio emerge in such dynamic systems, representing states of minimal stability. Their sustained stability is ensured by the presence of Landau damping within the system. The complex dynamics of the stock market can be analyzed by considering the exchanged shares as fluctuating and the unexchanged shares as nonfluctuating entities. The traders exhibit a kind of group oscillations that resemble the waves in physical plasma. At steady state, the waves can be expressed by a cosine term, and at the least stable state, the dynamics involves the golden ratio, such that the cosine of $36^{\circ }$ is equal to half of the golden ratio. Using the trigonometric cosine formula, it is possible to obtain other angles which are the multiples of $9^{\circ }$. They can be expressed in terms of the golden ratio, and they stand as Fibonacci angles. The stabilization is achieved by a mechanism so-called Landau damping, and the waves thus created are called Elliott waves, and they keep the system near the instability border. It was found that these angles appear quite often in the motive and corrective Elliott waves in the weekly price change of crude oil between January 2001 and June 2023. The percent occurrence of these angles increases through oscillations in motive waves with peaks at 18°, $45^{\circ }$ and 72°. The corrective wave has a maximum peak at 36°, and the percent occurrence decreases through oscillations having smaller peak values at 54° and 72°. The highest values are such that the motive waves appear at 50% in the bull market, and the corrective waves at 27.8% in the bear market.

A new recursive top-down algorithm for the construction of a unique Huffman tree is introduced. We show that the prefix codes generated from the Huffman tree are unique and the weighted path length is optimal. Initially we have not imposed any restriction on the maximum length (the number of bits) a prefix code can take. But if buffering of the source is required, we have to put a restriction on the length of the prefix code. In this context we extend the top-down recursive algorithm for generating length-limited prefix codes.

The relationship between period-doubling bifurcations and Feigenbaum's constants has been studied for nearly 40 years and this relationship has helped uncover many fundamental aspects of universal scaling across multiple nonlinear dynamical systems. This paper will combine information entropy with symbolic dynamics to demonstrate how period doubling can be defined using these tools alone. In addition, the technique allows us to uncover some unexpected, simple estimates for Feigenbaum's constants which relate them to log 2 and the golden ratio, φ, as well as to each other.

Maximum possible lengths of short words in unavoidable sets of order no more than n have the form log n + O(log log n). The respective log bases of the upper and lower bounds of the shortest and second shortest words are (for a two-letter alphabet) 2 and τ, the Golden Ratio. The latter result comes through identifying certain bases of free monoids.

In his fascinating book Wonders of Numbers, Clifford Pickover introduces the Ana sequence and fractal, two self-referential constructions arising from the use of language. This paper answers Pickover's questions on the relative composition of sequence terms and the dimension of the fractal. In the process, it introduces a novel way of obtaining fractals from iterative set operations. Also, it presents a beautiful variant of the Ana constructions involving the golden ratio. In conclusion, it suggests ways of constructing similar fractals for the Morse-Thue and "Look and Say" sequences.

An autocorrelation function is obtained on the base of the recurrence relation formalism, whose continued fraction form corresponds to that of the golden ratio. It turns out that this Golden Ratio (GR) autocorrelation is known in science and obeys all necessary conditions, in contrast to the exponential autocorrelation function. Using the Kubo approach it is shown how exponential correlations appear in the linear response theory as a result of non-Hermitian relaxation of the system.

The main objective for the next generation wireless network is the offer of a high data rate when the user is on the move. The key element that offers continuous connectivity is the handoff. In this paper, we propose a handoff prediction model, which can predict handoff behavior of the user well in advance and reduce the latency in the handoff operation. The prediction model is validated with real life scenario both for the pedestrian user and the vehicle user, traveling at a speed of 80km/h. The experimental result verifies the capability of the proposed algorithm to predict the future sample with accuracy and minimum latency. Simulation results demonstrate the proposed system outperforming the existing system compared to the probability of the handoff detection and minimizing the false alarm probability. There is also the fact of the proposed algorithm not requiring any additional hardware for predicting the mobility of the user.

In this study, a novel hybrid fuzzy decision-making model is constructed for the effective omnichannel strategy selection of financial services. The first phase of this model is related to inputting missing expert decisions for the quality function deployment (QFD) stages and omnichannel service strategies. The QFD stages of financial services are then weighted by bipolar $q$-rung orthopair fuzzy ($q$-ROF) multi-stepwise weight assessment ratio analysis ($M$-SWARA) based on the golden ratio. The QFD-based omnichannel strategy alternatives for financial services are then ranked using bipolar $q$-ROF ELECTRE. These calculations are also performed by considering PFSs and IFSs. Finally, the TOPSIS methodology is used to rank the alternatives so that comparative results can be obtained. The main contribution of this study is the creation of effective omnichannel strategies to improve financial services using a novel hybrid fuzzy decision-making methodology based on bipolar $q$-rung orthopair fuzzy sets ($q$-ROFSs), $M$-SWARA, ELECTRE, and the imputation of expert evaluations with collaborative filtering. The analysis results obtained will facilitate determination of the most appropriate omnichannel strategies for businesses to provide effective financial services. In this manner, companies will be able to determine appropriate marketing strategies without incurring excessive costs. Because the analysis results are the same for all fuzzy sets, the proposed model is coherent and reliable. The findings demonstrate that online financial services are the most critical strategy for improving omnichannel. Thus, companies should prioritize online channels to provide financial services that are more effective.

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd $k$ describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number $F_k$. In the limit $k\to 0$, Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, $R$-matrices, geometry of hydrodynamic images and quantum computations are discussed.

The set of maximally fermion–boson entangled Bell super-coherent states is introduced. A superposition of these states with separable bosonic coherent states, represented by points on the super-Bloch sphere, we call the Bell-based super-coherent states. Entanglement of bosonic and fermionic degrees of freedom in these states is studied by using displacement bosonic operator. It acts on the super-qubit reference state, representing superposition of the zero and the one super-number states, forming computational basis super-states. We show that the states are completely characterized by displaced Fock states, as a superposition with non-classical, the photon added coherent states, and the entanglement is independent of coherent state parameter $\alpha$ and of the time evolution. In contrast to never orthogonal Glauber coherent states, our entangled super-coherent states can be orthogonal. The uncertainty relation in the states is monotonically growing function of the concurrence and for entangled states we get non-classical quadrature squeezing and representation of uncertainty by ratio of two Fibonacci numbers. The sequence of concurrences, and corresponding uncertainties $\hslash F_n/F_{n+1}$, in the limit $n\to \infty$, convergent to the Golden ratio uncertainty $\hslash /\phi$, where $\phi =\frac{1+\sqrt{5}}{2}$ is found.

Given a field $K$ and $n>1$, we say that a polynomial $f\in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of characterizing, for given $d,n>1$, fields $K$ such that there exists $f\in K[x]$ of degree $d$ with newly reducible $n$th iterate, and the similar problem for fields admitting infinitely many such $f$. We give results in the cases $(d,n)\in \{(2,3),(3,2),(4,2)\}$ as well as for $(d,2)$ when $d\equiv 2mod4$. In particular, we show that for all these $(d,n)$ pairs, there are infinitely many monic $f\in \mathbb{Z}[x]$ of degree $d$ with newly reducible $n$th iterate over $\mathbb{Q}$. Curiously, the minimal polynomial $x^2-x-1$ of the golden ratio is one example of $f\in \mathbb{Z}[x]$ with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.

A model of the growth curve of microorganisms was proposed, which reveals a relationship with the number of a ‘golden section’, 1.618…, for main parameters of the growth curves. The treatment mainly concerns the ratio of the maximum asymptotic value of biomass in the phase of slow growth to the real value of biomass accumulation at the end of exponential growth, which is equal to the square of the ‘golden section’, i.e., 2.618. There are a few relevant theorems to explain these facts. New, yet simpler, methods were considered for determining the model parameters based on hyperbolic functions. A comparison was made with one of the alternative models to demonstrate the advantage of the proposed model. The proposed model should be useful to apply at various stages of fermentation in scientific and industrial units. Further, the model could give a new impetus to the development of new mathematical knowledge regarding the algebra of the ‘golden section’ as a whole, as well as in connection with the introduction of a new equation at decomposing of any roots with any degrees for differences between constants and/or variables.

The study of new error correcting codes has raised attention in the last years, especially because of their use in cryptosystems that are resistant to attacks running on quantum computers. In 2006, while leaving a more in-depth analysis for future research, Stakhov gave some interesting ideas on how to exploit Fibonacci numbers to derive an original error correcting code with a compact representation. In this paper, we provide an explicit formula to compute the redundancy of Stakhov codes, we identify some flows in the initial decoding procedure described by Stakhov, whose crucial point is to solve some non-trivial Diophantine equations, and provide a detailed discussion on how to avoid solving such equations in some cases and on how to detect and correct errors more efficiently.

The given paper discusses an intuitionistic fuzzy discrete time-space optimization model while taking into account the COVID-19 pandemic. The novelty of this study is characterized by the proposal of a distinctive multi-objective intuitionistic fuzzy transportation model in which the demand function varies with time as the number of infected, exposed and susceptible persons increase. Forecasting of demand is executed using the epidemic diffusion model. Also, a new approach to defuzzify triangular intuitionistic fuzzy numbers, depending upon the concept of the golden ratio, is presented. Intuitionistic fuzzy programming approach is used to obtain the pareto-optimal solution. A real-life numerical illustration is explained using the proposed methodology to examine its practical suitability. Outperforming the existing traditional method of solving a transportation problem verifies its proficiency.

The golden ratio (GR) is an irrational number (close to 1.618) that repeatedly occurs in nature as well as in masterpieces of art. The GR has been considered a proportion perfectly representing beauty since ancient times, and it was investigated in several scientific fields but with conflicting results. This study aims at investigating if this proportion is associated with a judgment of beauty independently of the type of the stimulus, and the factors that may affect this aesthetic preference. In Experiment 1, an online psychophysical questionnaire was administered to 256 volunteers asked to choose among three possible proportions between the parts of the same stimulus (GR, 1.5, and 1.8). In Experiment 2, we recorded eye movements of 15 participants who had to express an aesthetic judgment on the same stimuli as Experiment 1. The results revealed a slight overall preference for GR (53%, p < 0.001), with higher preferences for stimuli representing humans, anthropomorphic sculptures, and paintings, regardless of the cultural level. In Experiment 2, a shorter dwell time was significantly associated with a better aesthetic judgment (p = 0.005), suggesting the possibility that GR could be associated with easier visual processing, and it could be hence considered as a visual affordance.

Chapter 9 addresses the foundational issues as they relate to the topological distinctions found in the Mereon Trefoil in particular and the Mereon geometry in general. Kauffman presents how the non-standard version of the trefoil knot seen tied by tracing the arcing angles of the Matrix is a higher energy version of the alternating weave of the 3,2 trefoil knot. Geospherical in nature, links are made to the pentagon, Petersen graph, the Five Elements. So too, connections are made to Fourier knots, and how the Mereon Trefoil form by connecting double helices. Connections are established between the pattern of DNA recombination, Spencer-Brown’s ‘Laws of Form’, and Shea Zellweger’s Logical Garnet. The assertion that the shape of the universe is dodecahedral is linked with how its construction might arise from the trefoil knot as the unfolding of a five-fold symmetry. It is presented how this would make the Mereon knot a complex but simple ‘cosmic loop’ around the whole universe, thus making knot theory universally applicable.