Golden Ratio Research Articles

Methodology for determining optimal design of funicular arches under point loads and selfweight against in-plane buckling

This paper presents a simple methodology for determining the optimal design of an elastic funicular arch under point loads and selfweight for maximum in-plane buckling capacity. This class of optimization problem has hitherto not been investigated. The cross-section of the arch is assumed to be geometrically similar and constant through its entire length. The arch is assumed to be inextensible. The supporting points of the arch may be either pinned or fixed. The methodology for determining the optimal arch shape involves first finding the funicular shape of the arch under point loads and selfweight. Next, the arch is modelled by the Hencky bar chain model (HBM) for the buckling analysis. The HBM is adopted because it eliminates the requirement for formulating the governing equation for buckling and it is a straightforward model to comprehend and implement in a computer code. Each funicular arch shape is specified by a horizontal force value, and it has to be optimally selected to maximise the buckling capacity of the arch. As it is a single parameter optimization problem, the Golden Section search was used to determine the optimal horizontal force, and hence the optimal funicular arch shape. The methodology is illustrated by solving two example problems: (1) an arch under a point load and selfweight and (2) a symmetric arch under two equal point loads and selfweight. By solving the self-buckling problem of a catenary arch, the maximum spanning capacity of the catenary arch can be obtained as shown herein.

Minimization of Energy Consumption in TDMA-Based Wireless-Powered Multi-Access Edge Computing Networks

Many nodes in Internet of Things (IoT) rely on batteries for power. Additionally, the demand for executing compute-intensive and latency-sensitive tasks is increasing for IoT nodes. In some practical scenarios, the computation tasks of WDs have the non-separable characteristic, that is, binary offloading strategies should be used. In this paper, we focus on the design of an efficient binary offloading algorithm that minimizes system energy consumption (EC) for TDMA-based wireless-powered multi-access edge computing networks, where WDs either compute tasks locally or offload them to hybrid access points (H-APs). We formulate the EC minimization problem which is a non-convex problem and decompose it into a master problem optimizing binary offloading decision and a subproblem optimizing WPT duration and task offloading transmission durations. For the master problem, a DRL based method is applied to obtain the near-optimal offloading decision. For the subproblem, we firstly consider the scenario where the nodes do not have completion time constraints and obtain the optimal analytical solution. Then we consider the scenario with the constraints. By jointly using the Golden Section Method and bisection method, the optimal solution can be obtained due to the convexity of the constraint function. Simulation results show that the proposed offloading algorithm based on DRL can achieve the near-minimal EC.

A nested generalized sidelobe canceller for source counting, localization, and signal separation in reverberant fields

This paper proposes a nested generalized sidelobe canceller (NGSC) for typical array signal processing tasks, including source counting, localization, and signal separation. Multiple blocking matrices are arranged in a nested structure to successively eliminate dominant sources until the remaining signal is predominantly incoherent noise. The microphone signals are dereverberated using a multichannel weighted prediction error algorithm. In source counting, the number of sound sources is determined by tracking the average power of the blocked signal. In source localization, the direction-of-arrival (DOA) is estimated via cosine similarity in conjunction with the golden section search. In signal separation, the estimated DOA enables speech separation in a linearly constrained minimum variance beamformer with postfiltering (LCMV-PF). Monte Carlo simulations are performed to compare the proposed NGSC approach with four baselines, minimum description length, second order statistic of the eigenvalue, multistage Wiener filtering, and multiple signal classification. The results show that NGSC achieves at least 28.80% higher source counting accuracy with a 1.04° lower root mean square degree error than the baselines. The signal-to-distortion ratio achieved by LCMV-PF is 1.52 dB higher than that achieved by the linearly constrained minimum power beamformer and the multichannel Wiener filter.

The Sacramentality of Scripture in the Writings of Johann Gerhard

Abstract: Johann Gerhard (1582–1637) was a Lutheran pastor and university professor in Germany whose writings on scripture stand as some of his most important theological contributions. In these works, he developed the doctrine of scripture in several significant ways. This essay addresses one such development: Gerhard’s understanding of the sacramentality of scripture. Though scripture is not a sacrament properly speaking, the relationship of its external form to its internal divine meaning and purpose presents an interesting sacramental analogy. In speaking of sacramentality, Gerhard is concerned with the status of scripture as a means of grace and the efficacy of the word in itself. This essay will focus on the continuity of Gerhard’s views on the efficacy of scripture, especially in the face of an episode known as the Rahtmannian debate. Two points will be stressed throughout: (1) Gerhard took a consistent approach to the efficacy of scripture throughout his writings and (2) this approach provides a helpful foundation upon which to establish a theology of the sacramentality of scripture.

Towards importance of comprehensive color features analysis using iterative golden ratio proportions for Alphonso mango ripening stage classification by adapting to natural progressive ripening method

The color and texture of the mango fruit are the primary qualitative factors that play a crucial role during fruit sampling for production. For mango fruit industrial processing, we consider the batches of mango fruit to interpret its texture and color properties to determine its ripening stage into unripe, partial ripe, and ripe. To understand the ripening stage of mango fruit, the localized features are computed from the two-level decomposition of fruit using the golden ratio. The features computed from decomposed segments are analyzed using color features to classify the fruit into a specific stage using classifiers, support vector machine, random forest, K nearest neighbor, gradient boost, Adaboost, and decision trees. For experimental analysis, the dataset of 2400 samples of Alphonso mangoes was considered for feature extraction using mean color features from five color spaces and classified with an accuracy of 97.6% with the SVM and Random Forests. The analysis is also conducted with features extracted from the whole fruit and achieved an accuracy of 77.39% with support vector machines using polynomial kernel and random forests. Extended analysis is also done on color features extracted separately from each color space and with specific decomposed slice combinations. From the outcomes achieved, it is evident that the proposed approach achieves promising outcomes and a steep improvement in accuracy over the state-of-the-art models.

Geometric investigation of entrance proportions of houses from the Qajar to the beginning of the early Pahlavi in Dezful City (1789–1979)

Due to the importance of the historical context of Dezful City, investigating historic buildings is necessary. But unfortunately, a large part of it is being destroyed. For this reason, it is essential to discover the guidelines and logic behind the architectural design of these buildings to help their reconstruction or restoration. Regrettably, there is no logical information about the components of this context and the reconstructions are usually done according to taste and based on architecture tested or personal preference. Therefore, it is necessary to find a logic behind these critical components that support the whole plan or parts of it in a particular style. The purpose of this research is to discover and find the logic behind the design of this building in the historical context of Dezful. Finding geometrical proportions can significantly help to reconstruct buildings better and more accurately. For this purpose, 30 historical houses of Dezful were randomly selected, and their input information was drawn in 2D form in AutoCAD. Then, the length, width, and height proportions were measured, and the obtained ratios were compared with the golden proportions and dynamic rectangles. This paper studies entrance based on ratios of 2, 3, 4, 5, 1.25 and φ (the golden section). The results show that golden proportions and proportions derived from dynamic rectangles are present in the wording of all cases. Moreover, the range of ratios in the plan is from 1.34 to 2, and the content of changes in the section is from 1.22 to 2. The results can help architects, and particularly building owners, to reconstruct based on existing logic by discovering the proportions of the entrances of the houses. It is possible to administer a part of the damaged tissue and estimate its dimensions and sizes because the proportions' completeness can help revive half-destroyed historical buildings.

The Threads That Weave Me

The Threads That Weave Me

Golden Ratio Proximal Gradient ADMM for Distributed Composite Convex Optimization

Golden Ratio Proximal Gradient ADMM for Distributed Composite Convex Optimization

The golden ratio in baseball: the influence of historical eras on winning percentages in major league baseball.

The golden section or golden ratio (61.8% or 0.618) is a mathematical phenomenon that appears in art, literature, music and nature with such ubiquity that it is thought to be a fundamental principle of aesthetic organisation. The golden ratio also manifests in sport, particularly as the proportion of wins to losses required to win a Major League Baseball championship. This study extends early work on the golden ratio in baseball by incorporating more than three decades of additional data. This study involved a historically contextualized examination of how winning percentages have changed across the seven historical eras of modern baseball, including analyses of the relative contribution of offensive and defensive statistics to championship winning teams. Data was extracted from Baseball Reference and included statistics for 398 championship winning teams from both the American and National Leagues between 1901 and 2019. Pearson correlation coefficients were computed for winning percentage with indicators of offensive and defensive performance during each era. Main and interaction effects of Era and League on winning percentage were examined using factorial ANOVA, with follow-up analyses examining whether the golden ratio was included in each factor's 95% confidence interval. Our findings suggest that winning percentages for championship teams were closest to the golden ratio during eras where the relative contribution of offense and defense was most closely balanced: the Integration Era (1942-1960) and the Expansion Era (1961-1976). Previous scholarship theorizes that the golden ratio represents an aesthetic ideal or a Gestalt archetype. If this aesthetic theory is applied to sporting competition, these results suggest that baseball may be most aesthetically appealing to fans when offense and defense is balanced in such a way as to ensure that championship teams win 61.8% of their games.

Comparative Evaluation of Golden Proportion and Recurring Esthetic Dental (RED) Proportion in Natural Dental Esthetics

This study explores the differences between the distribution of Golden Proportion and Recurring Esthetic Dental (RED) Proportion in natural maxillary anterior teeth. Photographs of the subject’s anterior teeth were taken from the frontal view. The width ratios of canine to lateral incisor and lateral incisor to central incisor were calculated. The existence of the Golden Proportion was considered when the width ratios between canine to lateral incisor or lateral incisor to central incisor were within the range of 0.55–0.66. To evaluate the existence of RED proportion in each subject, the width ratio of canine to lateral incisor was compared with the width ratio of lateral incisor to central incisor. The distribution of the Golden proportion was observed in 66.7% and the RED proportion was observed in 22.2% of the natural dentition of the study population. The existence of the Golden proportion was significantly higher than the existence of the RED proportion in well-aligned natural dentition in the study population (P < 0.05). Clinicians may choose the Golden proportion over the RED proportion during the maxillary anterior esthetic treatment procedure.

Current sensor-based single MPPT controller using sequential golden section search algorithm for hybrid solar PV generator-TEG in isolated DC microgrid

A current sensor-based single maximum power point tracking (SMPPT) controller is proposed for extracting the maximum power (MP) from a hybrid solar photovoltaic (PV) generator-thermoelectric generator (TEG) supplying an isolated DC microgrid. The proposed SMPPT controller tracks the maximum power points (MPPs) of the hybrid solar PV generator-TEG using the sequential golden section search (SGSS) algorithm. Faster and guaranteed convergence are the major advantages of this SGSS algorithm for tracking the MPPs of hybrid solar PV generator-TEG sequentially. Further, the proposed SMPPT controller requires a single sensing and signal conditioning circuit (SSCC) and a digital controller for successful operation. Thus, it requires less investment cost, reduces control complexity, consumes less power, and improves reliability. Extensive simulations, experiments, and comparative studies have demonstrated the effectiveness of the proposed SMPPT controller using the SGSS algorithm.

Exploring the hidden geometry in the physical of sacred buildings of Iran in the Zandiyeh Era (case study: Vakil Mosque)

ABSTRACT Geometry is one of the most important foundations of Iranian art and architecture, which can be seen in the magnificent buildings left from the past that are important in architecture. One of the most important buildings of Iran from the Zandiyeh Era (250 years ago) is Vakil Mosque, which is located in Shiraz. Vakil Mosque is a sacred and Islamic building in Iran, which is the aim of this article to investigate and find geometry in the architectural body of this building. In the current research, descriptive-analytical methods were used to collect data. Then I studied the geometrical drawings of this mosque, which some of the most important books of Muslim scholars in this field have been studied to obtain the geometry of the plan and elevations of Vakil Mosque, and after many tests and finding acceptable answers, options were proposed as the physical geometry of this sacred building. Finally, I came to the conclusion that the architect or architects of this magnificent building have used their utmost efforts in designing its physical to create golden proportions and regular geometry on it, and this order and geometry is the reason for the glory and strength of this building. The findings of this article can be used along with other similar studies to recognize Iran’s architectural works and restore its original Islamic-Iranian identity.

Study of the length of the anterior base of the skull in children from the Moscow region

Relevance. One of the starting points for calculating the parameters of growth and the ratio of the jaws is considered to be the sagittal size of the anterior base of the skull. To this day, many orthodontists are guided by the average values of the anterior skull base, found in 1972. This article is devoted to the study of the comparison of the average statistical sagittal dimensions of the length of the anterior skull base in children of the Moscow region aged 4 to 16 years with occlusal ratios according to the first Angle class in 2023 with the data proposed in the studies of Dr. Jarabak in 1972.Objective. To compare the average sagittal dimensions of the length of the anterior base of the skull in children of the Moscow region aged from 4 to 16 years with occlusal ratios for the first Engl class in 2023 with the data proposed in the studies of Dr. Jarabak in 1972.Materials and methods. The study included 331 people aged 4 to 16 years. The sagittal dimensions of the length of the anterior base of the skull were calculated in the digital program for the analysis of CBCT in children with occlusive ratios in the first class of male and female Engl on the digital X-ray machine Planmeca ProMax 3D Max X-ray diagnostic centers «Golden Section».Results. Male patients of the Moscow region approach the lower limits of the growth value of the anterior base of the skull only by the age of 12 and reach the confident values of the norm proposed by Dr. Jarabak only by the age of 16. And female patients in the Moscow region, for the most part, do not reach the limits of the norm proposed by Dr. Jarabak, lagging in values by 2–3 mm in a smaller direction.Conclusion. It is necessary to calculate and develop norms for Russian children, taking into account racial characteristics, environmental conditions, nutritional conditions, physical and social development, which differ significantly, but at the same time strongly affect the growth and formation of skeletal structures.

Bregman-Golden Ratio Algorithms for Variational Inequalities

Variational inequalities provide a framework through which many optimisation problems can be solved, in particular, saddle-point problems. In this paper, we study modifications to the so-called Golden RAtio ALgorithm (GRAAL) for variational inequalities—a method which uses a fully explicit adaptive step-size and provides convergence results under local Lipschitz assumptions without requiring backtracking. We present and analyse two Bregman modifications to GRAAL: the first uses a fixed step size and converges under global Lipschitz assumptions, and the second uses an adaptive step-size rule. Numerical performance of the former method is demonstrated on a bimatrix game arising in network communication, and of the latter on two problems, namely, power allocation in Gaussian communication channels and N-person Cournot completion games. In all of these applications, an appropriately chosen Bregman distance simplifies the projection steps computed as part of the algorithm.

“Beating the Drums in God’s Wrestling Arena”: Spirituality Translated into Local Metaphor in Wolof Sufi ʿAjamī Poetry

Abstract How do ʿAjamī scholars translate Sufi spirituality and esoteric concepts into a language accessible to their audience? The body of Senegalese Wolof ʿAjamī poetry often seeks to translate divine concepts into a language that is local, both in time and place. This translation takes place within the local language in all of its literary dimensions. For the Wolof ʿAjamī poet, proverbs, parables, and metaphors are key literary devices that facilitate understanding among listeners. Studying ʿAjamī poems opens a window into a historic locality, teaching the contemporary reader-listener about both Sufi spirituality and the particular contexts in which the poems were produced. Through these metaphors, the poet ʿAjamizes, conferring sacred meaning onto practices and devices that otherwise might be outside the Sufi realm. In the process, metaphors become vessels of divine meaning. This article treats the metaphor as a key literary device whose understanding is paramount for grasping the meaning of the religious message. Treating the poems as social documents and works of literature also offers scholars new possibilities for understanding the everyday life of ordinary users in Senegambia.

Automatic Tracking Method for 3D Human Motion Pose Using Contrastive Learning

Automatic tracking of three-dimensional (3D) human motion pose has the potential to provide corresponding technical support in various fields. However, existing methods for tracking human motion pose suffer from significant errors, long tracking times and suboptimal tracking results. To address these issues, an automatic tracking method for 3D human motion pose using contrastive learning is proposed. By using the feature parameters of 3D human motion poses, threshold variation parameters of 3D human motion poses are computed. The golden section is introduced to transform the threshold variation parameters and extract the features of 3D human motion poses by comparing the feature parameters with the threshold of parameter variation. Under the supervision of contrastive learning, a constraint loss is added to the local–global deep supervision module of contrastive learning to extract local parameters of 3D human motion poses, combined with their local features. After normalizing the 3D human motion pose images, frame differences of the background image are calculated. By constructing an automatic tracking model for 3D human motion poses, automatic tracking of 3D human motion poses is achieved. Experimental results demonstrate that the highest tracking lag is 9%, there is no deviation in node tracking, the pixel contrast is maintained above 90% and only 6 sub-blocks have detail loss. This indicates that the proposed method effectively tracks 3D human motion poses, tracks all the nodes, achieves high accuracy in automatic tracking and produces good tracking results.

Neutrino Mixing Sum Rules and the Littlest Seesaw

In this work, we study the neutrino mixing sum rules arising from discrete symmetries and the class of Littlest Seesaw (LS) neutrino models. These symmetry-based approaches all offer predictions for the cosine of the leptonic CP phase cosδ in terms of the mixing angles, θ13, θ12, θ23, while the LS models also predict the sine of the leptonic CP phase sinδ, as well as making other predictions. In particular, we study the solar neutrino mixing sum rules, arising from charged lepton corrections to tri-bimaximal (TB), bimaximal (BM), golden ratio (GR) and hexagonal (HEX) neutrino mixing, and the atmospheric neutrino mixing sum rules, arising from preserving one of the columns of these types of mixing—for example, the first or second column of the TB mixing matrix (TM1 or TM2)—and we confront them with an up-to-date global fit of the neutrino oscillation data. We show that some mixing sum rules, such as an atmospheric neutrino mixing sum rule arising from a version of neutrino golden ratio mixing (GRa1), are already excluded at 3σ, and we determine the remaining models allowed by the data. We also consider the more predictive LS models (which obey the TM1 sum rules and offer further predictions) based on constrained sequential dominance CSD(n) with n≈3. We compare for the first time the three cases n=2.5, n=3 and n=1+6≈3.45, which are favored by theoretical models, using a new type of analysis to accurately predict the observables θ12, θ23 and δ. We study all the above approaches, solar and atmospheric mixing sum rules and LS models, together so that they may be compared and to give an up-to-date analysis of the predictions of all of these possibilities, when confronted with the most recent global fits.

The golden ratio for cortisol replacement

The golden ratio for cortisol replacement

Discovering golden ratio in the world’s first five-agent network in ancient China

The world’s first five-agent network, also called Wuxing network in ancient China, had been fully established in the second century BC. Surprisingly, the key to cracking the operation of Wuxing network is the golden ratio, the world’s most astonishing number originating from ancient Greece. Wuxing network is composed of five agents located on the vertices of a pentagon such that adjacent agents cooperate with each other, while spaced-apart agents oppose each other. Although it was proposed more than 2000 years ago, it is still an unparalleled network operation protocol. This article reveals the role of the golden ratio in the balance and stability of Wuxing network, and demonstrates how to detect the golden ratio experimentally in Wuxing electronic circuits and in Wuxing formation flight of drones.

Wave-packet effects: a solution for isospin anomalies in vector-meson decay

There is a long-standing anomaly in the ratio of the decay width for ψ(3770)→D0D0¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (3770)\\rightarrow D^0\\overline{D^0}$$\\end{document} to that for ψ(3770)→D+D-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (3770)\\rightarrow D^+D^-$$\\end{document} at the level of 9.5σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$9.5\\,\\sigma $$\\end{document}. A similar anomaly exists for the ratio of ϕ(1020)→KL0KS0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi (1020)\\rightarrow K_\ ext {L}^0K_\ ext {S}^0$$\\end{document} to ϕ(1020)→K+K-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi (1020)\\rightarrow K^+K^-$$\\end{document} at 2.1σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2.1\\,\\sigma $$\\end{document}. In this study, we reassess the anomaly through the lens of a Gaussian wave-packet formalism. Our comprehensive calculations include the localization of the overlap of the wave packets near the mass thresholds and the composite nature of the initial-state vector mesons. The results align within a ∼1σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 1 \\sigma $$\\end{document} confidence level with the Particle Data Group’s central values for a physically reasonable value of the form-factor parameter, indicating a resolution to these anomalies. We also check the deviation of a wave-packet resonance from the Briet–Wigner shape and find that wide ranges of the wave-packet size are consistent with the experimental data.