Search Results (19)

Stochastically forced nonlinear wave systems are commonly associated with complex dynamical behavior, although little is known about the general interaction of nonlinear dispersion, irrational forcing frequencies, and multiplicative noise. To fill this gap, we consider a generalized stochastic SIdV equation and examine the effects of deterministic and stochastic influences on the long-term behavior of the equation. The PDE was modeled using a stochastic traveling-wave transformation that simplifies it into a planar system, which was studied using Darboux-seeded constructions, Poincaré maps, bifurcation patterns, Lyapunov exponents, recurrence plots, and sensitivity diagnostics. We discovered that natural, implicit, and unique seeds produce highly diverse transformed wave fields exhibiting both irrational and golden-ratio forcing, controlling the transition from quasi-periodicity to chaos. Stochastic perturbation is demonstrated to suppress as well as to amplify chaotic states, based on noise levels, altering attractor geometry, predictability, and multistability. Meanwhile, OGY control is demonstrated to be able to stabilize chosen unstable periodic orbits of the double-well regime. A stochastic bifurcation analysis was performed with respect to noise strength $\sigma$, revealing that the attractor structure of the system remains robust under stochastic excitation, with noise inducing only bounded fluctuations rather than qualitative dynamical transitions within the investigated parameter regime. These findings demonstrate that the emergence, deformation, and controllability of complex oscillatory patterns of stochastic nonlinear wave models are jointly controlled by nonlinear structure, external forcing, and noise. Full article

Loquat fruit is valued for its pleasant taste and favorable ripening period. However, its delicate texture and high perishability make it highly vulnerable to damage during packaging, so the fruit is usually packed by hand. Developing a fruit-sizing machine could increase commercial market opportunities. Automated mass detection reduces manual sorting errors and labor requirements. Overall, it enhances grading accuracy, speed, and uniformity in loquat processing. It also helps distinguish between ripe, underripe, and overripe fruits through subtle mass differences. Mass modeling has proven to be an effective baseline approach for the development and optimization of grading machines, and its efficiency has been demonstrated across different fruit types. Here, we present a comparative analysis of various models for mass modeling of six international and Italian loquat varieties (“Algerie,” “Peluche,” “Golden Nugget,” “Virticchiara,” “Nespolone di Trabia,” and “Claudia”) cultivated in southern Italy. On fifty fruits per variety, singular mass and spatial diameters [longitudinal (DL), maximum transverse (DT1), and minimum transverse (DT2) were measured. Linear and non-linear regression analyses, including quadratic, polynomial, and cubic models, were applied to both the complete dataset and individual varieties. A set of predictors was used, including DL (length), DT1 (width), and DT2 (thickness), ellipsoid and oblate spheroid volume. Model performance was evaluated based on higher R² values, and lower RMSE and MBE values. The best general model was obtained using an ellipsoidal volume (R² = 0.97, RMSE = 2.76). Both linear and cubic models demonstrated high suitability across all varieties, with ellipsoidal volume emerging as the most effective predictor. Conversely, (DL) based models were the least suitable, yielding the lowest (R² = 0.41) values in “Virticchiara.” The developed general and specific-variety models and equations provide a solid foundation for establishing high-performance systems for mass and size estimation, which can be effectively integrated into a fruit sizer machine. Full article

In the general linear Lie algebra of continuous linear transformations in n dimensions, we show that unequal Abelian scaling transformations on the components of a vector can stabilize the system information in the presence of Markov component transformations on the vector, which, alone, would lead to increasing entropy. The more interesting results follow from seeking Diophantine (integer) solutions, with the result that the system can be stabilized with constant information for each of a set of entropy rates ($k=\mathrm{1,2},3,\dots$ ). The first of these—the simplest—where $k=1$, results in the Fibonacci sequence, with information determined by the olden mean, and Fibonacci interpolating functions. Other interesting results include the fact that a new set of higher order generalized Fibonacci sequences, functions, golden means, and geometric patterns emerges for $k=2, 3, \dots$ Specifically, we define the $k$^th order golden mean as ${\Phi }_k=k/2+\sqrt{{(k/2)}^2+1}$ for $k =1, 2, 3, \dots .$. One can easily observe that one can form a right triangle with sides of 1 and $k/2$ and that this will give a hypotenuse of $\sqrt{{(k/2)}^2+1}$. Thus, the sum of the $k/2$ side plus the hypotenuse of these triangles so proportioned will give geometrically the exact value of the golden means for any value of k relative to the third side with a value of unity. The sequential powers of the matrix $(k^2+1,k,k,1)$ for any integer value of $k$ provide a generalized Fibonacci sequence. Also, using the general equation expressed as ${\Phi }_k={\displaystyle \frac{k}{2}}+\sqrt{{(k/2)}^2+1}$ for $k =\mathrm{1,2},3, \dots ,$ one can easily prove that ${\Phi }_k=k+1/{\Phi }_k$ which is a generalization of the familiar equation expressed as $\Phi =1+1/\Phi$. We suggest that one could look for these new ratios and patterns in nature, with the possibility that all of these systems are connected with the retention of information in the presence of increasing entropy. Thus, we show that two components of the general linear Lie algebra ($GL(n,\mathbb{R})$), acting simultaneously with certain parameters, can stabilize the information content of a vector over time. Full article

We develop a symmetry-based variational theory that shows the coarse-grained balance of work inflow to heat outflow in a driven, dissipative system relaxed to the golden ratio. Two order-2 Möbius transformations—a self-dual flip and a self-similar shift—generate a discrete non-abelian subgroup of $\mathrm{PGL}(2,\mathbb{Q}\left(\sqrt{5}\right))$. Requiring any smooth, strictly convex Lyapunov functional to be invariant under both maps enforces a single non-equilibrium fixed point: the golden mean. We confirm this result by (i) a gradient-flow partial-differential equation, (ii) a birth–death Markov chain whose continuum limit is Fokker–Planck, (iii) a Martin–Siggia–Rose field theory, and (iv) exact Ward identities that protect the fixed point against noise. Microscopic kinetics merely set the approach rate; three parameter-free invariants emerge: a $62\%:38\%$ split between entropy production and useful power, an RG-invariant diffusion coefficient linking relaxation time and correlation length ${\mathcal{D}}_{\alpha }={\xi }^z/\tau$, and a $\vartheta ={45}^{\circ }$ eigen-angle that maps to the golden logarithmic spiral. The same dual symmetry underlies scaling laws in rotating turbulence, plant phyllotaxis, cortical avalanches, quantum critical metals, and even de-Sitter cosmology, providing a falsifiable, unifying principle for pattern formation far from equilibrium. Full article

In this work, we generalize and describe the golden ratio in multi-dimensional vector spaces. We also introduce the concept of the law of similarity for multidimensional vectors. Initially, the law of similarity was derived for one-dimensional vectors. Although it operated with the values of the ratio of the parts of the whole, it created linear dimensions (a line is one-dimensional). The presented concept of the general golden ratio (GGR) for the vectors in a multidimensional space is described in detail with equations. It is shown that the GGR is a function of one or more angles, which is the solution to the golden equation described in this work. The main properties of the GGR are described, with illustrative examples. We introduce and discuss the concept of the golden pair of vectors, as well as the concept of a set of similarities for a given vector. We present our vision on the theory of the golden ratio for triangles and describe similarity triangles in detail and with illustrative examples. Full article

In this research, we proposed a new concept called as the $\mathcal{H}$-Nacci sequence. The $\mathcal{H}$-Nacci sequence (Fibonacci sequences of length h) is a collection of numbers developed from the coefficients of the generalized m-th Fibonacci equation. After that, we determined the golden ratio for each type of $\mathcal{H}$-Nacci sequence, which also coincided with an existing Fibonacci sequence. As each Fibonacci sequence has a unique advantage, first of all, we have applied the $\mathcal{H}$-Nacci sequence to the virus mutation process to show the key benefits of the $\mathcal{H}$-Nacci sequence, and then we found the Fibonacci risk model to analyze the risk factor of each mutant virus using the $\mathcal{H}$-Nacci sequence. Full article

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained. Full article

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum mechanical picture. On the other hand, it seems very unlikely that the similarities should be just coincidental. The interpretation of the author is that this can be a golden opportunity to investigate the interplay between these two theories. Full article

This paper proposes a dynamical approach to determine the optimal values of the parameters used in each iteration of the symmetric successive over-relaxation (SSOR), accelerated over-relaxation (AOR), and symmetric accelerated over-relaxation (SAOR) methods for solving linear equation systems. When the optimal values of the parameters in the SSOR, AOR, and SAOR are hard to determine as some fixed values, they are obtained by minimizing the merit functions, which are based on the maximal projection technique between the left- and right-hand-side vectors, which involves the input vector, the previous step values of the variables, and the parameters. The novelty is a new concept of the dynamical optimal values of the parameters, instead of the fixed values and the maximal projection technique. In a preferred range, the optimal values of the parameters can be quickly determined by using the golden section search algorithm with a loose convergence criterion. Without knowing and having the theoretical optimal values in general, the new methods might provide an alternative and proper choice of the values of the parameters for accelerating the convergence speed. Numerical testings of the linear Poisson equation discretized to a matrix–vector form and a Lyapunov equation form were used to assess the performance of the DOSSOR, DOAOR, and DOSAOR dynamical optimal methods. Full article

Intestinal failure is defined as the inability to absorb the minimum of macro and micronutrients, minerals and vitamins due to a reduction in gut function. In a subpopulation of patients with a dysfunctional gastrointestinal system, treatment with total or supplemental parenteral nutrition is required. The golden standard for the determination of energy expenditure is indirect calorimetry. This method enables an individualized nutritional treatment based on measurements instead of equations or body weight calculations. The possible use and advantages of this technology in a home PN setting need critical evaluation. For this narrative review, a bibliographic search is performed in PubMed and Web of Science using the following terms: ‘indirect calorimetry’, ‘home parenteral nutrition’, ‘intestinal failure’, ‘parenteral nutrition’, ‘resting energy expenditure’, ‘energy expenditure’ and ‘science implementation’. The use of IC is widely embedded in the hospital setting but more research is necessary to investigate the role of IC in a home setting and especially in IF patients. It is important that scientific output is generated in order to improve patients’ outcome and develop nutritional care paths. Full article

Pumping systems, especially those used in the water supply sector and in industrial and hydroelectric facilities, are commonly infested by the golden mussel. This causes an increase in maintenance operations (e.g., system shutdowns for cleaning) that can generate an increased energy cost. The geographical expansion of the golden mussel in Latin America presents an economic risk, not only to the ecosystem in general, but also to the energy sector. The imminence of its spread in the Amazon region, one of the main river basins in South America, is cause for concern with regard to the problems that bioinvasion of this species can cause. Given the absence of studies on the loss of energy efficiency in pumping systems impacted by the golden mussel, this study proposes a methodology to estimate the increase in energy consumption and costs of pumping under such bioinfestation. For the standardization of the methodology and development of mathematical calculations (both novel and improved equations), data from the literature (the growth of the golden mussel as a function of infestation time) and an analysis of the dimensions (length and height) of a sample of mussels available in the laboratory were considered. These data were used to calculate the roughness generated by the mussel infestation in the pumping suction and discharger pipe, which was necessary to determine the loss of energy efficiency (load loss, power consumption, and cost of pumping) resulting from the increase in energy consumption for pumping. This methodology was applied to a pumping station representative of the Brazilian Amazon as a case study. The results show an average increase in economic indicators (consumption and cost of pumping) after the system undergoes bioinfestation. This total increase corresponded to 19% and 44% in the first and second years, respectively, achieving a stabilization of the increase in the cost of pumping at 46%, in the 30 months of operation. Our results demonstrate the pioneering nature of the proposal, since these are the first quantitative data on the energy efficiency of pumping systems associated with bioinfestation by the golden mussel. These results can also be used to estimate the increase in costs caused by golden mussel bioinfestation in the raw water pumping systems of other facilities. Full article

The paper is written to demonstrate the applicability of the notion of triangulation typically used in social sciences research to computationally enhance the mathematics education of future K-12 teachers. The paper starts with the so-called Brain Teaser used as background for (what is called in the paper) computational triangulation in the context of four digital tools. Computational problem solving and problem formulating are presented as two sides of the same coin. By revealing the hidden mathematics of Fibonacci numbers included in the Brain Teaser, the paper discusses the role of computational thinking in the use of the well-ordering principle, the generating function method, digital fabrication, difference equations, and continued fractions in the development of computational algorithms. These algorithms eventually lead to a generalized Golden Ratio in the form of a string of numbers independently generated by digital tools used in the paper. Full article

Formulation is an ancient concept, although the word has been used only recently. The first formulations made our civilization advance by inventing bronze, steel, and gunpowder; then, it was used in medieval alchemy. When chemistry became a science and with the golden age of organic synthesis, the second formulation period began. This made it possible to create new chemical species and new combinations “à la carte.” However, the research and developments were still carried out by trial and error. Finally, the third period of formulation history began after World War II, when the properties of a system were associated with its ingredients and the way they were assembled or combined. Therefore, the formulation and the systems’ phenomenology were related to the generation of some synergy to obtain a commercial product. Winsor’s formulation studies in the 1950s were enlightening for academy and industries that were studying empirically surfactant-oil-water (SOW) systems. One of its key characteristics was how the interfacial interaction of the adsorbed surfactant with oil and water phases could be equal by varying the physicochemical formulation of the system. Then, Hansen’s solubility parameter in the 1960s helped to reach a further understanding of the affinity of some substances to make them suitable to oil and water phases. In the 1970s, researchers such as Shinoda and Kunieda, and different groups working in Enhanced Oil Recovery (EOR), among them Schechter and Wade’s group at the University of Texas, made formulation become a science by using semiempirical correlations to attain specific characteristics in a system (e.g., low oil-water interfacial tension, formulation of a stable O/W or W/O emulsion, or high-performance solubilization in a bicontinuous microemulsion system at the so-called optimum formulation). Nowadays, over 40 years of studies with the hydrophilic-lipophilic deviation equation (HLD) have made it feasible for formulators to improve products in many different applications using surfactants to attain a target system using HLD in its original or its normalized form, i.e., HLD_N. Thus, it can be said that there is still current progress being made towards an interdisciplinary applied science with numerical guidelines. In the present work, the state-of-the-art of formulation in multiphase systems containing two immiscible phases like oil and water, and therefore systems with heterogeneous or micro-heterogeneous interfaces, is discussed. Surfactants, from simple to complex or polymeric, are generally present in such systems to solve a wide variety of problems in many areas. Some significant cases are presented here as examples dealing with petroleum, foods, pharmaceutics, cosmetics, detergency, and other products occurring as dispersions, emulsions, or foams that we find in our everyday lives. Full article

The paper promotes the notion of computational experiment supported by a multi-tool digital environment as a means of the development of new mathematical knowledge in the context of education. The main study of the paper deals with the issues of teaching this knowledge to secondary teacher candidates within a graduate capstone mathematics education course. The interplay of mathematics and education is considered through the lens of using technology to enhance one’s mathematical background by advancing ideas from mostly known to genuinely unknown. In this paper, the knowns consist of Fibonacci numbers, Pascal’s triangle, and continued fractions; among the unknowns are Fibonacci-like polynomials and generalized golden ratios in the form of cycles of various lengths. The paper discusses the interplay of pragmatic and epistemic uses of digital tools by the learners of mathematics. The data for the study were collected over the years through solicited comments by teacher candidates enrolled in the capstone course. The main results indicate the candidates’ appreciation of the need for deep mathematical knowledge as an instrument of the modern-day pedagogy aimed at making high schoolers interested in the subject matter. Full article

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given. Full article