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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 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 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.

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.