Search Results (16)

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

Tack coat application rates and testing conditions differ among nations and construction conditions because various tack coat materials are available. In this study, newer materials are optimized for addition to porous asphalt pavements exposed to torrential rainfall, which is common in South Korea. Interface shear strength (ISS) tests are used to define the optimum application rates (OARs) of tack coat materials generally used in South Korea, by reference to the Korean Design Standard (KDS), the Korean Construction Standard (KCS), and features of pavement construction and bonding. We performed ISS tests using asphalt mixtures with porosities of 3, 5, and 7% to explore the effect of porosity on shear strength. The ISSs associated with varying tack coat proportions were earlier determined by creating polynomial regression equations. Here, we develop a predictive model using a non-linear function to estimate the OAR of tack coat and compare our approach with the earlier polynomial regression analysis. Based on the ISSs, the golden section search method was applied to define the OARs afforded by the predictive polynomial function. We used the generalized reduced gradient algorithm to construct a nonlinear predictive function using data from the ISS tests. Finally, our comparative analysis showed that the predictive model using the non-linear function was superior to the polynomial model in terms of both error rate and predictive tendency. Full article

The absence of a band gap in graphene makes it of minor interest for field-effect transistors. Layered metal chalcogenides have shown great potential in device applications thanks to their wide bandgap and high carrier mobility. Interestingly, in the ever-growing library of two-dimensional (2D) materials, monolayer InSe appears as one of the new promising candidates, although still in the initial stage of theoretical studies. Here, we present a theoretical study of this material using density functional theory (DFT) to determine the electronic band structure as well as the phonon spectrum and electron-phonon matrix elements. The electron-phonon scattering rates are obtained using Fermi’s Golden Rule and are used in a full-band Monte Carlo computer program to solve the Boltzmann transport equation (BTE) to evaluate the intrinsic low-field mobility and velocity-field characteristic. The electron-phonon matrix elements, accounting for both long- and short-range interactions, are considered to study the contributions of different scattering mechanisms. Since monolayer InSe is a polar piezoelectric material, scattering with optical phonons is dominated by the long-range interaction with longitudinal optical (LO) phonons while scattering with acoustic phonons is dominated by piezoelectric scattering with the longitudinal (LA) branch at room temperature (T = 300 K) due to a lack of a center of inversion symmetry in monolayer InSe. The low-field electron mobility, calculated considering all electron-phonon interactions, is found to be 110 cm²V⁻¹s⁻¹, whereas values of 188 cm²V⁻¹s⁻¹ and 365 cm²V⁻¹s⁻¹ are obtained considering the long-range and short-range interactions separately. Therefore, the calculated electron mobility of monolayer InSe seems to be competitive with other previously studied 2D materials and the piezoelectric properties of monolayer InSe make it a suitable material for a wide range of applications in next generation nanoelectronics. Full article

We study strong optical coupling of metal nanoparticle arrays with dielectric substrates. Based on the Fermi Golden Rule, the particle–substrate coupling is derived in terms of the photon absorption probability assuming a local dipole field. An increase in photocurrent gain is achieved through the optical coupling. In addition, we describe light-induced, mesoscopic electron dynamics via the nonlocal hydrodynamic theory of charges. At small nanoparticle size (<20 nm), the impact of this type of spatial dispersion becomes sizable. Both absorption and scattering cross sections of the nanoparticle are significantly increased through the contribution of additional nonlocal modes. We observe a splitting of local optical modes spanning several tenths of nanometers. This is a signature of semi-classical, strong optical coupling via the dynamic Stark effect, known as Autler–Townes splitting. The photocurrent generated in this description is increased by up to 2%, which agrees better with recent experiments than compared to identical classical setups with up to 6%. Both, the expressions derived for the particle–substrate coupling and the additional hydrodynamic equation for electrons are integrated into COMSOL for our simulations. Full article