Golden Ratio Research Articles

Relationship between the Ideal Female Buttock and the Golden Ratio.

Relationship between the Ideal Female Buttock and the Golden Ratio.

Comparison of craniofacial anthropometric measurements with the golden ratio in young adults

Comparison of craniofacial anthropometric measurements with the golden ratio in young adults

An integrated expert recommender system approach to environmental service priorities in renewable energy

The purpose of this study is to analyze the investment success of renewable energy generation projects design. A novel model has been constructed for this purpose. At the first stage, collaborative filtering methodology is taken into consideration to complete missing evaluations. After that, M-SWARA based on QUSFSs with golden cut is used to compute the weights of these factors. Finally, the components of the service design are ranked by TOPSIS approach. The main contribution of the paper is that a new methodology (M-SWARA) has been created in this study by making improvements to SWARA. With the help of this new model, causal directions between the indicators can also be examined. Similarly, collaborative filtering methodology is taken into consideration to complete missing evaluations. In this process, the decision makers are allowed to leave the questions they wanted blank. This situation is considered as the superiority of the proposed model compared to many previous models in the literature. The findings indicate that cost is the most significant factor for the success of renewable energy investments because it gets the highest weight (.261). The ranking results also demonstrate that product is the most essential component of the service design of renewable energy investments. Therefore, solving the high-cost problem is of vital importance to increase these investments. First, renewable energy companies can reduce costs with more effective financial management. To carry out this process effectively, a finance department consisting of qualified personnel is needed. Thanks to this team, current situations in the financial markets will be better followed and this will play an important role in reducing costs.

Golden Ratio Realistically and Theoretically

Golden Ratio Realistically and Theoretically

A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

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 Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-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.

On Quark–Lepton Mixing and the Leptonic CP Violation

In the absence of a Grand Unified Theory framework, connecting the values of the mixing parameters in the quark-and-lepton sector is a difficult task, unless one introduces ad hoc relations among the matrices that diagonalize such different kinds of fermions. In this paper, we discuss in detail the possibility that the PMNS matrix is given by the product UPMNS=VCKM★T★, where T comes from the diagonalization of a see-saw like mass matrix that can be of a Bimaximal (BM), Tri-Bimaximal (TBM) and Golden Ratio (GR) form, and identify the leading corrections to such patterns that allow for a good fit to the leptonic mixing matrix as well as to the CP phase. We also show that the modified versions of BM, TBM and GR can easily accommodate the solar and atmospheric mass differences.

Method for determining phase shift using two-phase interpolation transformation

This study focuses on a method for accurately determining the phase shift between two harmonic signals. The proposed technique compares the waveform of the combined signal, which is obtained by applying a two-phase interpolation transformation, with a set of predefined reference functions. This study aims to develop a method for accurately measuring the phase shift between two harmonic signals to reduce the measurement errors caused by phase asymmetry in signal transmission channels and minimize the costs associated with parameter control. The sources of error in the proposed measurement method are identified and addressed. The task involves setting up the measurement objective to determine the phase shift between two harmonic signals. This section presents the analyses of known methods for measuring phase shifts using of analog-to-digital conversion. The next step involves selecting indicators and criteria to determine how closely the investigated signal matches the reference function. Then, we propose the synthesis of an algorithm to find the extremum of the sum of squares deviations for the set of reference functions and the investigated signal, relying on the golden ratio method. Finally, the paper will analyze possible sources of error that could affect the measurement outcome. The methods used are the following: methodology for conducting digital signal processing and measuring error estimation, numerical methods for extremum search, and methodology for single-factor experiments. The following results were obtained. A proposed compensation method for measuring phase shift is based on comparing the shape of the normalized signal, which is obtained as a result of the summation of harmonic signals after their semiperiodical transformation, with a set of normalized reference functions synthesized by computational means. A list of measurement and auxiliary operations, which should be ideally conducted to implement this measurement method, has been determined. An analysis of the components of measurement errors was conducted. Conclusions. The scientific novelty of the obtained results lies in the following: a method for measuring the phase shift of a signal has been developed, which, in our opinion, should be classified as a compensatory measurement method. This will significantly reduce the error component caused by the phase asymmetry of signal transmission channels and reduce the costs of controlling parameters (up to 10%) without decreasing the quality of control; a list of the measuring and auxiliary operations necessary for the implementation of the proposed measurement method has been identified; sources of errors have been determined.

Analysis of green energy-oriented sustainable development goals for emerging economies

Countries should focus on the improvements to achieve sustainable development goals. However, taking these actions create some operational costs. Thus, it is appropriate for rising nations to prioritize sustainable development objectives. Due to this condition, more critical sustainable development goals should be identified. Accordingly, this study evaluates green energy-oriented sustainable development goals for emerging economies. Within this context, a model is generated by considering the golden cut and bipolar q-rung orthopair fuzzy sets (q-ROFSs). The novelty of this study is to identify key green energy-oriented sustainable development goals for the improvement of emerging economies by an original model based on the golden cut and bipolar q-ROF decision making approach. All results are similar, indicating the proposed model's validity and coherency. It is concluded that increasing renewable energy is the most significant target with a weight of 0.235 regarding green energy-oriented sustainable development. Cooperation and efficiency are other significant issues for this condition, with weights of 0.198 and 0.196. It is also identified that Turkey and Mexico are the most successful countries in reaching green energy-oriented sustainable development goals. Increasing the share of renewable energy in total energy production and consumption is important for emerging countries. For this situation, governments should provide subsidies for renewable energy investors to provide a cost advantage. In this context, issues such as tax cuts and loans with low-interest rates can increase the motivation of clean energy investors.

Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs

Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between the large value to the small value. A Fibonacci sequence is a sequence of numbers that has a unique shape. The first term of this sequence of numbers is 1, then the second term is also 1, then for the third term it is determined by adding the two previous terms so that a sequence of numbers with a certain pattern is obtained. Pascal's triangle is a geometric rule of binomial coefficients in a triangle. Shapes that resemble fractal geometry, golden ratios and Fibonacci numbers are found in many places in this realm, for example the shapes of various plants, animals, as well as in humans themselves, even the universe. This fractal geometry can also be used to design batik motif creations, architectural design or musical art. This paper describes how a fractal object can be made with geometric transformations that can be used to develop batik motif creations. And also look at the relationship between fractal geometry, the golden ratio, Fibonacci numbers, and Pascal's triangles and the shapes of these shapes that exist in this nature.

About the Terms „Golden Section" and "Gilt Edges" in Lithuanian

About the Terms „Golden Section" and "Gilt Edges" in Lithuanian

К оценке геодинамической устойчивости геосфер Земли

The Earth, as a system, is characterized by certain average state parameters (density, temperature). These parameters cannot but depend on similar states of internal geospheres (subsystems), which follow the unified law of increasing entropy of the cooling Earth from the center in the direction of the Earth's radius. As a model for assessing the dynamic stability of the Earth's internal geospheres relative to each other, the principle of the golden proportion (golden section) was used with the given parameters of density, temperature, power of the geospheres to the extreme ratios of their states at points (0.618; 1.618; 2.618), which determine the conditions for the stable dynamic equilibrium of the compared parameters of geospheres. And if these relations satisfied the known theoretical, model, geophysical data, then the conclusion was made that the geospheres in the compared parameter relations were in a state close to a stable dynamic equilibrium of the exchange of matter and energy between them. Otherwise, it was recognized that the compared geospheres of the Earth were in a state of either unstable dynamic equilibrium with an assessment of the level of this deviation from the golden numbers, or the known values of the density and temperature of the geospheres, as well as their structure, should be subject to clarification. Average model estimates of the parameters of the Earth's internal geospheres reach a stable dynamic equilibrium at a level of deviation from the golden numbers of up to 2.72 %. This allows estimating the model duration of the modern geological activity of the Earth for another 124 million years until the formation of a new state of the existing continents. The instability of the dynamic equilibrium of density and temperature of matter is established at the boundaries: continental and oceanic crust; oceanic crust and Golitsin layer; at the boundary between the outer part of the Earth's core and the lower mantle; inside the Earth's core at the level of identified transition zones in the four-layer core model. The presence of spheres with a high core density from 16.05 to 26.45 g/cm3 suggests the stratification of its central part into essentially gold and platinoid. The presence of instability boundaries in the dynamic state of geospheres is in good agreement with the concept of plate tectonics and with the depths of plume origin. It can be assumed that the relative equilibrium of the internal structure of the Earth will continue for about 124 million years until the beginning of the formation of a new Pangea Proxima, which, according to Cristofor Scotese, will arise in 200 million years.

Preparation and Properties of Brake Friction Materials Reinforced with Coconut Fiber and Dypsis Lutescens Fiber.

Brake friction material reinforced with coconut fiber and dypsis lutescens fiber was designed and prepared in this study. Specimens incorporating 0-8 wt.% of coconut fibers or dypsis lutescens fibers were fabricated. The effect of the content of these reinforcing fibers on the overall properties of brake friction materials was systematically investigated. The results indicate that the inclusion of reinforcing fibers in the formulation of brake friction materials can improve the physical properties and friction and wear properties of brake friction materials. The specimen incorporating 6 wt.% plant fiber obtained the optimal comprehensive performance with excellent fade resistance and recovery properties, and better wear resistance. In order to further investigate their performance, nine hybrid fiber brake friction materials were designed using the golden section method and orthogonal test method. The study indicated that the F-6 hybrid fiber-reinforced brake friction materials have better physical properties, thermal degradation resistance, recovery properties, and abrasion resistance than the single-fiber-reinforced brake friction materials. This study provides new concepts for the preparation of fiber-reinforced brake friction materials as well as formulation optimization.

Employing SHlNO_Glaze technology using local materials and applying it to high-temperature clays

This research is concerned with studying (SHlNO_Glaze technology using local materials), which consists of five chapters: The first chapter included a presentation of the research problem to find out what variables the technique causes when added to the surface of pottery? While the importance of the research came to increase scientific knowledge, in addition to the scarcity of its use, which makes it surrounded by a kind of mystery about the extent of its color and texture transformations, while the research aims to reveal the technical effects resulting in ceramic glass. The limits of the research include the addition of red earth and potassium feldspar. Calcium feldspar and soda ash, in addition to colored oxides, are high-temperature clays (1250 degrees Celsius). While the second chapter contained the theoretical framework and previous studies, this chapter reviewed the clays and their uses, as well as the types of clays found in their locations. It reviewed the structure of the glass and its compatibility with the pottery body and the classification of the glass in terms of chemical composition and maturation temperatures. This chapter also reviewed the coloring oxides through which it was identified. The researcher studied the oxides of copper, cobalt, iron, and tin, which determine the color and texture. This chapter also reviewed the techniques of special effects in porcelain glass, the firing atmospheres, and the types of reduction. The third chapter included the selection of materials added to the glass mixture based on colored oxides, which are considered the most common coloring oxides for potters, which are copper oxide (CuO), cobalt oxide (CoO), tin oxide (zn), and iron oxide (feo). In the golden ratios to the glass mixture, while fixing the ratios of the glass mixture and limiting the variable only to the chino, as it was applied to kaolin clay, the number of glazed models reached (12) models. As for the fourth chapter, it included the research sample, and the number of samples reached (12) samples representing the majority of the research population. They were reviewed with percentages and temperatures and were discussed according to each sample and its variable. The fifth chapter included conclusions, including :Adding recommendations, suggestions, sources, appendices, and a summary in English.

There Is no Universal Standard of Beauty.

This article discusses the various factors that contribute to facial beauty and the different contexts in which they are significant. The authors examine the roles of the golden ratio, ideal proportions, symmetry, and specific facial features in what constitutes an attractive face. They also investigate the influence of cultural and evolutionary perspectives on facial beauty and the impact of social media on our perception of beauty. The article provides a comprehensive overview of the topic and emphasises the significance of facial beauty and its impact on our lives. The authors acknowledge that the concept of beauty is difficult to define with absolute precision, and that the objective-subjective argument regarding facial beauty remains unsolved. However, they argue that physical attributes undoubtedly influence perceptions of attractiveness, but beauty extends beyond surface-level features. They also suggest that embracing diversity in beauty standards fosters a more inclusive society that celebrates the unique qualities of every person.Level of Evidence V This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

Evaluation of the Golden Proportion, Golden Percentage, and Recurring Esthetic Dental Proportion in Kenyans of African Descent.

To evaluate the validity of the Golden Proportion, Golden Percentage, and Recurring Esthetic Dental (RED) Proportion among Kenyans of African descent with naturally well-aligned teeth. Standardized frontal photographic images of the smiles of 175 participants aged 18-35 years were obtained, and Adobe Photoshop was used to analyze and measure the frontal widths of the maxillary central and lateral incisorsand canines in triplicate. The average teeth widths were calculated to determine the existence of the Golden Proportion, Golden Percentage, and RED Proportion, and their validity using independent sample t-tests to compare the differences in the mean teeth widths at α < 0.05. The number of male and female participants was 107 (61.1%) and 68 (38.9%), respectively. The Golden Proportion between the maxillary central and lateral incisors was found in 4.0% on the right and 2.8% on the left of all the participants, but between the maxillary lateral incisors and canines was found in only 0.6% on the right of male participants (p < 0.0001). The RED Proportion between the maxillary lateral and central incisors was in the range of 67%-70%, and between the canines and lateral incisors was 82%-84% (p < 0.0001). The proportion of RED was not constant, and gradually increased distally. The Golden Percentage of 15% was observed in the lateral incisors bilaterally; however, in the central incisors and the canines, the Golden Percentage was 22% and 12%, respectively. The Golden and RED Proportions were invalid determinants of anterior teeth proportions. The Golden Percentage existed only in the lateral incisors. The Golden Proportion, RED Proportion, and Golden Percentage theories may not be applicable to all populations when designing smiles. Racial and ethnic backgrounds are important considerations to establish objective quantifiable values of anterior tooth proportions that are beneficial for esthetic restorations.

Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model

Fluid flow throttling is common in industrial and building services engineering. Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane, as the tunnel has a lower lane number than in the roadway normal segment. To predict the effects of tunnel throttling of annular freeway vehicular flow, a three-lane continuum model is developed. Lane III of the tunnel is completely blocked due to the need of tunnel rehabilitation, etc. There exists mandatory net lane-changing rate from lane III to lane II just upstream of the tunnel entrance, which is described by a model of random number generated through a golden section analysis. The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form. This paper assumes that the annular freeway has a total length of 100 km, a two-lane tunnel of length 2 km with a speed limit of 80 km/h. The free flow speeds on lanes I, II and III are assumed to be 110, 100 and 90 km/h respectively. Based on the three-lane continuum model, numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy. Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling. Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance. The vehicle fuel consumption is lane number dependent, distributes with initial density concavely, ranging from 5.56 to 8.00 L. Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.

On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains

On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains

AI-enhanced "Two-thirds Guidelines" for Lipolifting: Addressing Multiple Hallmarks of Facial Aging.

Facial aging involves complex changes such as volume loss, ligament weakening, and skin quality alterations. The "two-thirds guidelines" emerge as a novel strategy to combat these aging signs, drawing from an extensive analysis of 2800 facial fat grafting procedures conducted over two decades. Guided by facial lipolifting data, including patient age, fat type (microfat and nanofat), and injection depth, this study devises a systematic framework for multilayer fat rejuvenation and ligament restoration. The two-thirds guidelines advocate injecting two-thirds of the patient's age for microfat and one-third for nanofat, with specific injection codes for lower, middle, and upper facial regions. A prospective study involving 400 patients confirms the efficacy of the two-thirds guidelines. However, applicability may vary for patients outside SD ranges, particularly concerning facial proportions and body mass index. Patients within the golden ratio range (1.4-1.9) report high satisfaction rates and a 50% fat graft uptake, with minimal complications. For patients outside this range, an artificial intelligence (AI) program was implemented. The two-thirds guidelines offer a comprehensive approach to facial rejuvenation, addressing volume loss, ligament weakening, and skin quality. They are applicable in early aging stages, promising enduring and natural outcomes while mitigating effects of weight fluctuations. These guidelines provide a safe, replicable, and adaptable approach to facial fat grafting, either standalone or in combination with facelift techniques, with minimized overfilling risks. A dataset obtained from 2800 patients serves as the foundation for developing an AI program tailored to aid doctors in diagnosing and treating similar cases.

ϕ−Multiplicative Calculus

In this paper, we present a novel mathematical framework termed “ϕ-multiplicative calculus”, which serves as a Golden Fibonacci calculus to fundamental concepts in multiplicative calculus. This innovative calculus introduces a parameter ϕ (Golden ratio), offering a nuanced extension of traditional calculus. Our work encompasses the establishment of ϕ-multiplicative calculus and the demonstration of essential theorems concerning derivatives, integrals, and their operation properties within this mathematical framework. The paper contributes to the academic discourse by providing a comprehensive exploration of the proposed ϕ-multiplicative calculus, presenting a robust foundation for further investigations in this specialized mathematical domain.

Rapid computation of Hopf bifurcation points of continuous and discrete systems through minimization

For an autonomous nonlinear system, the Hopf bifurcation point along the equilibrium path is a critical feature that indicates whether the values of the parameters change from exhibiting fixed-point behavior to having a periodic orbit. To solve these problems, we developed a method of transforming an eigenvalue problem based on the Jacobian matrix at equilibrium into a minimization problem, enabling the rapid identification of a solution. Specifically, this generalized eigenvalue problem is solved by identifying the vector variable after reducing the number of eigenequations by one in the nonhomogeneous linear system. This can be achieved by normalizing the value of a selected nonzero component of the eigenvector and then moving the column containing this component to the other side of the equation. An appropriate merit function was established in terms of the Euclidean norm of the eigenequation, and this merit function was minimized using the golden section search algorithm to determine the eigenparameters of the bifurcation point. The accuracy of the method for identifying the parameter values and the corresponding imaginary eigenvalues at the Hopf bifurcation points was evaluated for numerous examples for both the continuous and discrete systems. The method was both fast and accurate. Moreover, its stability in the presence of noise was investigated, and the method was robust.