Golden Ratio Research Articles

Chaotic opposition Golden Sinus Algorithm for global optimization problems

Optimization techniques are required to find the best solutions to challenging problems in many engineering disciplines. Metaheuristic algorithms that effectively increase search performance using various evolutionary strategies have become increasingly popular in recent years. The Golden Sinus Algorithm (GoldSa) is a population-based optimization algorithm that uses the sine function and the golden ratio. In this study, a chaotically enhanced opposition-based Golden Sinus Algorithm (Co-GoldSa) has been proposed to improve the efficiency of the exploitation and exploration ability of the GoldSa method. While designing this approach, it is first necessary to analyze the chaotic behavior of the GoldSa parameters. For this purpose, three chaotic GoldSa methods have been developed using eight chaotic maps to determine the effect of the chaotic maps on the parameters with different behaviors. Secondly, the opposition-based learning strategy is adjusted to the cGoldSa to enhance the searching ability. To investigate the proficiency of the proposed Co-GoldSa method, it has been examined with well-known and newly introduced metaheuristic approaches on benchmark functions and classical engineering design problems. Besides, an efficient framework of the multilevel thresholding image segmentation has been presented based on the Co-GoldSa method since the efficient processing of pathological images is quite important in medical diagnostics. The experimental outcomes reveal the superiority of the proposed method in solving global optimization problems, image segmentation, and engineering problems. Thus, the outcomes of the benchmark functions, image segmentation, and classical engineering problems support that the proposed Co-GoldSa approach can be considered a promising method for resolving challenging optimization problems.

Golden ratio method for solving monotone variational inequality problems in Hadamard spaces

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

Blind frequency offset estimation method based on a minimum phase correction error for a full spectrum modulated NFDM system.

To improve the spectral efficiency of a full spectrum modulated nonlinear frequency division multiplexing (FS-NFDM) system, a blind frequency offset estimation (FOE) method has been proposed. The approach based on the minimum phase correction error can achieve high estimation accuracy of sub-MHz without need of any training symbols. Furthermore, in order to reduce the computational complexity, an eigenvalue-shift method is used to get a coarse search interval of FO, and then the one-dimensional optimization algorithm based on golden section search and parabolic interpolation is used to get the optimal FOE for the coarse search interval. The feasibility and reliability of the proposed blind FOE approach have been demonstrated in both BTB and fiber transmission scenarios. Compared with the grid search method, the proposed solving scheme can save hundreds of times of the searches. The experimental results reveal that the proposed method is robust to the amplified spontaneous emission noise and phase noise and has the capabilities of a wide FOE range and a high FOE accuracy.

Preconditioned golden ratio primal-dual algorithm with linesearch

Preconditioned golden ratio primal-dual algorithm with linesearch

Using the Golden Section to Approximate π

Summary We derive one specific example of a Machin-like formula for approximating π using the unit circle and the reciproval of the golden section. The efficiency of our derived formula is compared to other, similar formulas using Lehmer’s measure.

Nano Sensor Empowered Livestock Monitoring System Using Golden Section Optimized Recurrent Neural Network IoT Platform

Nano Sensor Empowered Livestock Monitoring System Using Golden Section Optimized Recurrent Neural Network IoT Platform

Optimizing bimetallic plasmonic structures for enhanced surface plasmon resonance sensing through golden ratio design

Optimizing bimetallic plasmonic structures for enhanced surface plasmon resonance sensing through golden ratio design

To Approach the Factors of Influencing Consumer Preferences for Corrective Eyewear Design

Correctional eyeglasses used for myopia, hyperopia, astigmatism, and presbyopia require adaptation and prolonged wearing, similar to assistive devices. Unlike decorative glasses, which can be freely changed or not worn as desired and coordinated with overall attire, wearing eyeglasses alters facial appearance. The aesthetics, comfort, and functional quality of eyeglasses directly affect the wearer's willingness, thus impacting visual health and quality of life. Therefore, corrective eyeglass frames need to balance between aesthetic appearance and visual health. This study employs methods such as the Kano Model, Important Performance Analysis (IPA), and Choquet integral to explore the key preference factors for the design of corrective eyeglass frames. Differing from traditional 5-point equidistant semantic questionnaires, this study employs a non-equal interval semantic questionnaire with a golden ratio scale to differentiate adjacent semantic importance ratios, highlighting weighted emphasis on aesthetics. The results provide reference for eyeglass designers and related industries. It is expected to design eyewear that better meets the needs of the Taiwanese population, encouraging people to wear corrective eyewear willingly, thereby further maintaining visual health, improving quality of life, enhancing visual health, and quality of life.

Quest for the golden ratio universality class.

Using mode-coupling theory, the conditions for all allowed dynamical universality classes for the conserved modes in one-dimensional driven systems are presented in closed form as a function of the stationary currents and their derivatives. With an eye on the search for the golden ratio universality class, the existence of some families of microscopic models is ruled out a priori by using an Onsager-type macroscopic current symmetry. In particular, if the currents are symmetric or antisymmetric under the interchange of the conserved densities, then at equal mean densities the golden modes can only appear in the antisymmetric case and if the conserved quantities are correlated, but not in the symmetric case where at equal densities one mode is always diffusive and the second may be either Kardar-Parisi-Zhang (KPZ), modified KPZ, 3/2-Lévy, or also diffusive. We also show that the predictions of mode-coupling theory for a noisy chain of harmonic oscillators are exact.

Fault distance measurement method for wind power transmission lines based on improved NSGA II

A fault distance measurement method that is suitable for long-distance wind power transmission lines is proposed by this paper. By combining the voltage and current at both ends of the line during normal operation, a dynamic line parameter correction method based on improved NSGA II is proposed to ensure measurement errors caused by parameter changes are avoided. Based on the construction of the fault distance measurement function, a global optimization method that is based on adaptive golden section search is proposed, where only the power frequency measurement data at both ends of the bus is required and synchronization is not required for the measurement of fault distance. A large number of experimental results demonstrate that the method this paper proposes is highly adaptable, with a relative distance measurement error of less than 0.2 km, and high accuracy is maintained in distance measurement even with certain degrees of noise interference.

Using the golden ratio in detecting the aesthetics of Mosque facades: The case of contemporary Mosques in Erbil city

From ancient Greece till the present, the golden ratio has been considered one of the most significant criteria of beauty in art and architecture. Architectural forms are part of the visual language that is bound by the rules of geometry and proportion. This study aims to identify the extent to which contemporary mosques in the city of Erbil utilize the golden ratio in the aesthetics of their facades. The research attempts to answer the question: was the golden ratio implemented in designing the facade of contemporary mosques in Erbil? The study adopted a mixed quantitative and qualitative analysis method. To achieve the study's aim and answer its questions, six mosques built in the last decade in Erbil were selected. Qualitatively, survey and documentation followed. Quantitatively, Phi matrix software, AutoCAD, and graphic software were applied as a mathematical approach. The study concluded that only one case used the golden ratio in the design of its façade by 100%, while it did not exceed 75% in most other mosques. The study draws the attention of designers and architects in the field of mosque architecture to the adoption of the golden ratio in the designs of mosque facades because of its role in achieving the aesthetic aspect.

The General Parametric Equation of Pythagoras Theorem and The General Connectedness Theorem

In this article, the Quarter Squares Rule is used to prove that it also satisfies the Pythagoras Theorem. Using this proof, it will be shown that there are parametric equations among sides and the radius of inner circle of a right triangle. Thus, it is proved that a quadratic equation which has some definite properties is connected with a right triangle. After that, firstly, utilizing this connection, it will be reached the general parametric equation of The Pythagoras Theorem. Secondly, it is going to be identified the general connectedness theorem. Further, it is going to be given relation between golden ratio and Earth’s axial tilt angle as an interesting example.

Smile aesthetics in Pakistani population: dentist preferences and perceptions of anterior teeth proportion and harmony

BackgroundThis study aimed to evaluate dentist perceptions of attractive smiles in the Pakistani population, considering different dental proportions.MethodsMaxillary casts and digital images were used to create symmetrical representations of anterior teeth. dentists’ preferences for good and bad teeth proportions, width/height ratios, and various dental proportions (golden, recurring esthetic dental (RED), golden percentage, Preston, and local/observed) were assessed using one sample and paired t-test. The Chi-square test was used to determine the gender disparities and factors affecting smile attractiveness. A p-value of ≤ 0.05 was taken as significant.ResultsThe RED proportion emerged as the preferred choice for normal-sized teeth, with specialists and general dentists favoring it over the golden proportion. For tall teeth, the golden proportion was predominantly preferred. The golden percentage received limited preference for aesthetic smile construction.ConclusionsThe smiles created using the principles of RED proportion were opted as the most attractive by local dentists. Factors such as tooth arrangement, color, and midline were highlighted as essential considerations in aesthetic smile construction.

Computational modelling of the key stages of placing bioresorbable polymeric coronary stent

Cardiovascular diseases are one of the main problems in the world in medicine. It ranks first in the list of causes of fatal accidents, in addition, more than a half of cases are a result of ischemic heart diseases. One of the most effective method of treatment is coronary stenting. The main purpose of the study is to outline bioresorbable polymeric coronary stent’s behaviour using computational modelling and to demonstrate benefits of application of polymer materials instead of standard metallic alloys. During the research the main stages of coronary stenting that are necessary for general understanding of process were analyzed: the strain after crimping, the shape of angioplasty balloon after deployment and the stress-strain state of vessel and atherosclerotic plaque after coronary stent deployment. PLLA (poly-L-lactic acid) was taken as a material for the research. It allows to create stent construction that will provide the required lumen of the vessel during rehabilitation. Moreover, with the progress of poly-L-lactic acid vasomotor function will restore thereby forming «golden section». The results showed the possibility of use of approaches that were given in this paper for examination of polymer material’s behaviour with coronary stenting. It can be concluded that polymer bioresorbable coronary stents in medicine are promising for further application for improvement of quality of invasive coronary operation.

Generalized Freudenthal duality for rotating extremal black holes

Freudenthal duality (FD) is a non-linear symmetry of the Bekenstein-Hawking entropy of extremal dyonic black holes (BHs) in Maxwell-Einstein-scalar theories in four space-time dimensions realized as an anti-involutive map in the symplectic space of electric-magnetic BH charges. In this paper, we generalize FD to the class of rotating (stationary) extremal BHs, both in the under- and over-rotating regime, defining a (generalized) rotating FD (generally, non-anti-involutive) map (RFD), which also acts on the BH angular momentum. We prove that the RFD map is unique, and we compute the explicit expression of its non-linear action on the angular momentum itself. Interestingly, in the non-rotating limit, RFD bifurcates into the usual, non-rotating FD branch and into a spurious branch, named “golden” branch, mapping a non-rotating (static) extremal BH to an under-rotating (stationary) extremal BH, in which the ratio between the angular momentum and the non-rotating entropy is the square root of the golden ratio. Finally, we investigate the possibility of inducing transitions between the under- and over- rotating regimes by means of RFD, obtaining a no-go result.

Ассимиляционный потенциал природы и территорий: нетривиальные следствия управления устойчивым эколого-экономическим развитием

Ecological-economic systems, which are in a state of unstable dynamic equilibrium under the influence of natural and anthropogenic factors, strive for sustainability if human economic activity is able to maintain the assimilation potential at the level of reproduction of natural resources and environmental quality by devoting part of the income to the environmental agenda. Within the framework of the law of balanced environmental management T=1.618·t·a, sustainable environmental and economic development is achieved by allocating funds at the level (t) of the GDP growth rate (T) to the environmental agenda for the regions. The value of the ecological situation coefficient (ESC) in the law of balanced environmental management for the Russian Federation varies around the golden numbers 0.618, 1.618, 2.618, depending on the level of economic development in the biosphere, artificial maintenance of the quality of the environment in it, as well as in the future possibility of terraforming the solar planets systems. An indicator of sustainable, harmonious environmental and economic development is a linear relationship between the assimilation potential (a) and the EES at the level of golden numbers 0.618, 1.618, 2.618. In this case, sustainable environmental and economic development can be achieved in any environmental condition as long as balance is maintained between the rate of economic growth and allocations of funds for the restoration of the assimilation function of nature. But they must be directed to the environmental agenda in strict accordance with the golden proportion relative to the main directions of environmental activities, even in conditions of transformed (converted) nature, and in the near future, in conditions of terraforming. However, then all development costs will go exclusively to maintaining the habitat in multiples of the golden numbers of 1.618 and 2.618 in the coefficient of the law of balanced environmental management.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Frontal facial analysis of female celebrity attractiveness standards through artificial intelligence

The contemporary significance of celebrities’ facial aesthetics underscores their heightened importance in shaping attractiveness standards. This retrospective study aimed to investigate the impact of patterns on aesthetic canons in the profile views of female celebrities, using artificial intelligence. The study sought to compare different races and propose standards for attractive faces.In this retrospective cohort study, a Python-based algorithm was used to analyze frontal patterns and evaluate their influence on aesthetic norms in publicly accessible images of female global celebrities. Ten ideal angular or proportional measures were gathered from the literature, and were trained to serve as a benchmark for the analysis of facial attractiveness. Demographic characteristics were described statistically. A one-way ANOVA test was employed to assess data distribution. Differences in means between groups were evaluated using nonparametric independent-sample tests, with statistical significance set at < 0.05.The study involved facial analyses for 115 female celebrities. It revealed variations in facial features among races. The mean golden ratio differed, with African and Asian individuals showing lower ratios. Symmetry varied, with Latin and Caucasian faces considered the most symmetrical. The zygomatic-to-mandibular width ratio was similar across races, with a ratio close to 80% being associated with more attractive faces. Differences in nose-to-mouth ratio, lips, alar base width, and chin angle were noted among race groups.The study concluded that, regardless of race, an attractive female face is characterized by specific ratios and angles. Facial symmetry, though desirable, is not strictly necessary. Irrespective of race background, an appealing female face is characterized by a zygomatic-to-mandibular width ratio nearing 80%, a mid-facial third that is slightly larger than the lower third, and a distinctive chin angle of approximately 138°, contributing to a trapezoidal facial shape. The findings contribute valuable insights into attractiveness standards and the impact of frontal patterns on aesthetic canons in female celebrities.

Evaluating the validity of recurring esthetic dental proportion in young adults

Objectives: An important aspect of cosmetic dentistry is developing a geometric or numerical proportion to associate the consecutive width of maxillary anterior teeth. New hypotheses in this area include the Golden proportion, golden percentage, and the recurrent esthetic dental (RED) proportion. This study used digital photos and computer analysis to examine the existing RED proportion associated with the widths of maxillary anterior teeth in people with permanent dentition. Material and Methods: Sixty young adults (28 males and 32 females) were photographed from the front. Digital measurements were taken of all maxillary anterior teeth. Results: To determine whether the RED proportion exists in the permanent dentition, the mean width proportion of the maxillary lateral incisor to central incisor and maxillary canine to lateral incisor was established. The average lateral incisor to central incisor width proportion of the right and left side was determined to be 70% and 71%, respectively. The average canine to lateral incisor width proportion was determined to be 80% and 78%, respectively. Conclusion: The maxillary anterior teeth’s consecutive widths did not exhibit RED proportions in the data that were examined. People’s anterior tooth proportions differ based on their race and ethnicity. Thus, the RED ratio is not the only standard for rebuilding aesthetically acceptable smiles globally.

Secrecy Enhancement of relay cooperative NOMA network based on user behavior

From the perspective of information security, relays exhibit varying levels of trustworthiness when forwarding information. For screening and eliminating the malicious users of cooperative NOMA relay networks, a reputation-based algorithm is proposed, in which the user relays (URs) are categorized into three types based on their behaviors—malicious users, speculative users, and honest users. Meanwhile, in order to minimize information leakage at the relay, cooperative interference signals are transmitted by far user (FU) and near user (NU) via the uplink while the base station (BS) is also transmitting signals to the relay. This approach increases inter-user interference (IUI) to confuse the eavesdropping capability of the relay. To verify the superiority of the proposed scheme, an asymptotic lower bound expression for the ergodic secrecy rate (ESR) of the downlink is derived to characterize the secrecy performance of the proposed scheme. Furthermore, in the case of limited system power, a joint power allocation algorithm based on the golden section method is also proposed to further enhance the security performances of the system. The simulation results validate the accuracy of the derived analysis results and demonstrate that the proposed scheme has a significant improvement in the ESR compared with the traditional scheme.