Search

Summary: We give an infinite number of proofs of the Pythagorean theorem. Some can be classified as 'self-similar proofs'. [ABSTRACT FROM AUTHOR]

The purpose of this work is to propose a new efficient initial guess algorithm rather than the conventional Newton method to meet the adjoint terminal conditions, rapidly. Thus, the focus of this paper is to use better algorithms for obtaining refined initial guesses with shooting method to solve the coupled boundary value problems. The boundary value problem is formulated as a system of equations i.e. initial value problems with one unknown initial condition. The efficiency and accuracy of the Shooting method is enhanced by improving the initial guess and then solving the problem iteratively. [ABSTRACT FROM AUTHOR]

The fractional factor implicates the characteristics of fractional flow in network data transmission, and it is a crucial tool for analyzing network information transfer. When there is uncertain information in the network, the corresponding network graph should be characterized by fuzzy graphs, in which the vertex membership function (MF) describes the uncertainty of sites, and the edge membership reveals the uncertainty of channels. The previous work introduced the concept of fuzzy fractional factor (FFF) on fuzzy graphs, but the correlated concepts are still open on other fuzzy graph classes. In order to overcome this defect, in this contribution, the concept of fuzzy fractional factor is extended to intuitionistic fuzzy graph, Pythagorean fuzzy graph, and picture fuzzy graph. Sign-alternating walk and increasing walk are extended to the corresponding settings, and the transformation operations are redefined in light of various situations. By means of constructive approaches, the corresponding theoretical results are further generalized in these settings, which characterize the existence of (resp. maximum) fuzzy fractional factors in different kinds of fuzzy graphs. [ABSTRACT FROM AUTHOR]

Summary: This paper provides a review of proofs of Heron's formula for the area of a plane triangle. We groups proofs together into analogy/experimental evidence, geometry (incircle radius multiplied by semi-perimeter), algebra (equivalence with the Pythagorean Theorem), trigonometric proofs (often via the cosine rule), and specialization from a more general theorem. Lastly, we discuss a neglected paper by Baker of 1885 that lists 110 distinct formulae for the area of a plane triangle and relate some of these formulae to those used in proving Heron's formula. [ABSTRACT FROM AUTHOR]

The current work presents unconventional but instructionally useful approaches to data analysis and modelling that allow for measurements of the acceleration due to gravity, g, using free-falling motion and the popular Tracker software. The two techniques detailed in this article illustrate the employability of the Pythagorean Theorem and the inverse tangent function in modelling free-falling motion that likewise, expectedly, permit for the determination of g. These two mathematical concepts are brought to relevance by strategically situating the origin of the coordinate system at an appropriate point. The models returned values of the gravitational acceleration that are comparable to that by the traditional technique, the one directly based on the vertical position-time data. Additionally, the unconventional free-fall data analyses presented in this paper suggest extra and insightful activities for the students, exercises on a notably less-automated nonlinear curve fitting on a spreadsheet (MS Excel) at most. Ultimately, by applying familiar mathematical concepts associated with a right triangle to nontraditional contexts like free-fall, this laboratory or lecture activity is perceived to spark greater interest of the students and enhance learning by unifying theory and experimental data. [ABSTRACT FROM AUTHOR]

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2 + 1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras. [ABSTRACT FROM AUTHOR]

Eighth-grade students are expected to master the Pythagorean theorem. This understanding provides a strong foundation for various mathematical concepts. Besides, the Pythagorean theorem serves as one of the prerequisite subjects for further courses. However, these students have been reported to have low cognitive levels and encounter difficulty in learning the Pythagorean theorem. Therefore, this study explores the achievement of students' cognitive levels in solving Pythagorean theorem problems based on revised Bloom's taxonomy and the underlying factors. This research employed qualitative methods with multiple case study types. Prospective participants were 35 eighth-grade junior high school students. For the instrument, this study used semi-structured interview guidelines, written tests of problem-solving, indicator rubrics, audio-visual recording devices, and researcher notes. The data were garnered from the participants’ responses to written tests on the Pythagorean theorem, the researchers' notes, as well as the audio-visual recordings of the interviews between the researchers and the subjects. The garnered data were analyzed using techniques proposed by Miles and Huberman. The subsequent stages of the research process include data reduction, data presentation, and conclusion drawing. The findings indicate that learning experience presents a pivotal role in the achievement of higher cognitive levels. A supportive learning environment, including resources that assist in learning, can influence students' problem-solving ability to reach higher cognitive levels. Furthermore, providing various forms of problems can help hone the subject's cognitive level. Such diverse learning experiences can broaden the subject's horizons and help achieve higher cognitive levels. BAHASA INDONESIA ABSTRACT: Siswa kelas VIII diharapkan mampu untuk menguasai teorema pythagoras. Pemahaman tersebut menciptakan landasan yang kuat untuk berbagai konsep matematika dan menjadi salah satu materi pelajaran prasyarat untuk materi selanjutnya. Namun, siswa kelas VIII memiliki level kognitif rendah dan mengalami kesulitan dalam mempelajari teorema pythagoras. Tujuan penelitian ini adalah untuk menganalisis ketercapaian level kognitif siswa dalam memecahkan masalah teorema pythagoras berdasarkan taksonomi bloom revisi serta faktor penyebabnya. Penelitian ini menggunakan metode kualitatif dengan jenis studi kasus jamak. Calon subjek adalah siswa SMP kelas VIII sebanyak 35 dan terpilih 16 subjek. Instrumen yang digunakan adalah pedoman wawancara semi terstruktur, tes tertulis pemecahan masalah, rubrik indikator, alat rekam audio visual, dan catatan peneliti. Data yang digunakan adalah hasil tes tertulis subjek dalam memecahkan permasalahan teorema pythagoras, catatan peneliti, dan hasil rekaman wawancara antara peneliti dengan subjek. Penelitian ini menggunakan teknik analisis data interaktif miles dan huberman. Langkah berikutnya setelah pengumpulan data adalah reduksi data, penyajian data, dan penarikan kesimpulan. Hasil penelitian adalah pengalaman belajar menjadi faktor krusial dalam ketercapaian level kognitif yang lebih tinggi. Subjek yang memiliki pengalaman belajar lebih banyak dapat mencapai level kognitif yang lebih tinggi. Lingkungan belajar yang mendukung, termasuk sumber daya yang membantu dalam belajar, dapat mempengaruhi kemampuan pemecahan masalah siswa untuk mencapai level kognitif yang lebih tinggi. Selain itu, pemberian berbagai bentuk masalah dapat membantu mengasah level kognitif subjek. Pengalaman belajar yang beragam tersebut dapat memperluas wawasan subjek dan membantu mencapai level kognitif yang lebih tinggi.

It is a well-known fact that groups are the only algebraic structures having a single binary operation that is mathematically so perfect that it is impossible to introduce a richer structure within it. The main purpose of this study is to introduce the notion of the Pythagorean neutrosophic triplet (PNT) which is the generalization of neutrosophic triplet (NT). The PNT is an algebraic structure of three ordered pairs that satisfy several properties under the binary operation (B-Operation) "*". Furthermore, we used the PNTs to introduce the novel concept of a Pythagorean neutrosophic triplet group (PNTG). The algebraic structure (AS) of PNTG is different from the neutrosophic triplet group (NTG). We discussed some properties, related results, and particular examples of these novel concepts. We further studied Pythagorean neutro-homomorphism, Pythagorean neutro-isomorphism, etc., for PNTGs. Moreover, we discussed the main distinctions between the neutrosophic triplet group (NTG) and the PNTG. [ABSTRACT FROM AUTHOR]

The transportation problem is one of the most significant mathematical programming applications that appears in various real-world decision-making problems. In an actual scenario, the supply, demand, and cost parameters of a transportation problem cannot be exactly quantified due to market instability. To deal with such types of impreciseness, the researchers have widely used fuzzy numbers and their extensions. Pythagorean fuzzy set theory is a prominent tool for handling uncertain and vague information in complex decision-making situations. This paper aims to develop a solution approach to solve the transportation problem with uncertainty in input parameters by incorporating Pythagorean fuzzy numbers. To do so, first, a new score function is proposed to rank Pythagorean fuzzy numbers more efficiently. A comparative study highlights some flaws in existing score functions, which depicts the advantages of the proposed score function over existing ones. Afterward, we solve the Pythagorean fuzzy transportation problem using the proposed score function. The solution technique is demonstrated with the help of some numerical examples. In addition, a comparative study is also included to show the efficacy of the proposed approach over existing ones. [ABSTRACT FROM AUTHOR]

The origination of graphs with neutrosophic type where membership of indeterminacy expels the vague results, by increasing the accuracy is used to extend application through the graphical environment. Since it is an extension of the intuitionistic type, there comes an immediate need to extend its findings and application to the neutrosophic type. Reversing the conditions of neutrosophic graphs by introducing the anti-behavior properties will produce an adequate number of new results and data, breaking the backlog in approaching decision-making problems and other real-world applications. This research aims to recognize the complementation concept in the Pythagorean co-neutrosophic graph, which has not been dealt with yet. The co-neutrosophic graph is the reversal concept of neutrosophic graphs, where the vertex and edge membership conditions are reversed, but the total sum of these memberships remains the same. Here, the discussion about complementation, cocomplementation, and its properties are carried out on a Pythagorean co-neutrosophic Graph. As a result, an application with improved accuracy result will be obtained as an outcome. [ABSTRACT FROM AUTHOR]

The famous Pythagorean theorem relating a team's winning percentage over a season to it's total runs scored and surrendered has been the subject of much attention, in both popular culture and academia. In academia, a main focus has been to find the optimal value for the exponent in the formula, originally taken to be 2 in Bill James' work. The optimal value has been obtained in many published works via regression or other statistical estimation techniques, and calibrated to various sports to reflect the type of play in each sport. In this short article, we borrow from ideas in quantitative finance and risk management to estimate the distribution of these Pythagorean exponents, viewing them as random variables that can evolve over time. We carry out our analysis in a manner similar to that of implied volatility from the Black‐Scholes formula. We present the result of our analysis of 150 years of team data via the Lahman package in R, and also provide dynamic models for an evolving measure of the Pythagorean exponent. This includes suggestions for modeling credit default as well as contract valuation in sports via macro variables in connection with historically implied parameters. [ABSTRACT FROM AUTHOR]

Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple. [ABSTRACT FROM AUTHOR]

Purpose: This study aimed to evaluate the association between the progression of medial joint space (MJS) narrowing, medial meniscus extrusion (MME) and clinical scores and the tibial tunnel position in pullout repairs for medial meniscus posterior root tears (MMPRTs). Methods: This retrospective study examined 54 patients. Changes in MJS (ΔMJS), MME (ΔMME) and clinical scores and their relationship with the tibial tunnel position were evaluated using correlation coefficients. The distance from the anatomical to technical attachment position in the tibial tunnel position was measured in the anterior and medial directions, and the direct distance was measured using the Pythagorean theorem. Results: The mean ΔMJS and ΔMME were 0.6 ± 0.8 and 1.3 ± 1.3 mm, respectively, and the mean anterior, medial and direct distances were 1.4 ± 2.3, 2.2 ± 1.7 and 3.4 ± 1.7 mm, respectively. ΔMJS had a significant positive correlation with the medial (r = 0.580, p < 0.001) and direct (r = 0.559, p < 0.001) distances, while ΔMME had a significant positive correlation with direct distance (r = 0.295, p = 0.030). Several clinical scores were significantly negatively correlated with these distances. Conclusion: In transtibial pullout repair for MMPRTs, accurate tibial tunnel position delayed the progression of MJS narrowing and MME, leading to improved clinical outcomes. The progression of MJS narrowing was associated with the mediolateral direction of the tibial tunnel position, while the clinical scores were associated with the anteroposterior direction of the tibial tunnel position. These findings indicate the need to orient the tip of the guide in a more posterolateral direction when creating the tibial tunnel. Level of Evidence: Level IV. [ABSTRACT FROM AUTHOR]

We introduce the new type neutrosophic interval-valued set (NIVS) problems relevant to multiple attribute decision making (MADM). Pythagorean interval-valued fuzzy set (PIVFS) and neutrosophic set (NS) can be extended into new type neutrosophic interval-valued set. We discusses new type neutrosophic interval-valued weighted averaging (new type NIVWA), new type neutrosophic interval-valued weighted geometric (new type NIVWG), generalized new type neutrosophic interval-valued weighted averaging (new type GNIVWA) and generalized new type neutrosophic interval-valued weighted geometric (new type GNIVWG). A number of algebraic properties of new type NIVSs have been established such as associativity, distributivity and idempotency. Using expert judgments and criteria, we will be able to decide which options are the most appropriate. Several of the proposed and current models are also compared in order to demonstrate the reliability and usefulness of the models under study. Additionally, the findings of the study are fascinating and intriguing. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. This article explores generalizations of the Pythagorean theorem using classical Euclidean plane geometry tools. It introduces innovative extensions that shed new light on the theorem's geometric relationships. [ABSTRACT FROM AUTHOR]

Green supply chain management concerns the incorporation of globally sustainable methods into supply chain management techniques. The major goal is to decrease the environmental impact of the entire supply chain to increase the social, ecological, and economic relationships with other countries for business. In this manuscript, we aim to compute the Dombi operational laws based on the complex Pythagorean fuzzy (CPF) information. Furthermore, we construct the Dombi aggregation operators under the consideration of the CPF information, such as the CPF power Dombi averaging (CPFPDA) operator, CPF power weighted Dombi averaging (CPFPWDA) operator, CPF power Dombi geometric (CPFPDG) operator, and CPF power weighted Dombi geometric (CPFPWDG) operator. Some flexible properties for the invented operators are also discussed. Additionally, we develop the multi-attribute decision-making (MADM) method to evaluate the problem of green supply chain management based on the invented operators for CPF information. Finally, we use some numerical examples to show the supremacy and validity of the developed techniques by comparing their ranking results with the obtaining ranking results of the existing techniques. [ABSTRACT FROM AUTHOR]

This research delves into the exploration of Pythagorean fuzzy sublattices and Pythagorean fuzzy ideals within the context of lattice theory. Through a rigorous analysis of structural theorems concerning these concepts derived from Pythagorean fuzzy sets, we uncover significant parallels with classical theory. Additionally, we investigate the behavior of Pythagorean fuzzy ideals under lattice homomorphisms. Our findings shed light on the applicability and utility of Pythagorean fuzzy theory in lattice-based structures, offering insights into their properties and relationships. [ABSTRACT FROM AUTHOR]

The low completeness of mathematics learning on the material of the Pythagorean theorem is wavered by the disinteractive interest of students in the process of learning activities. The purpose of this study was to obtain a description of the implementation of the Problem Based Learning model to improve the learning outcomes of grade VIII students of Daarul Quran Colomadu Junior High School on the Pythagorean Theorem material. This type of research is classroom action research that refers to Kemmis and Mc. Taggart's design, namely planning, implementing actions, observation and reflection. The study was conducted in two cycles. The subjects of this study were grade VIII students of Daarul Quran Colomadu totaling 40 students. Data collection techniques are obtained from data from teacher and student observations. This research was conducted in two cycles and each cycle was carried out in two meetings. The results showed that learning using the Problem Based Learning type cooperative learning model can improve student learning outcomes as shown by an increase in the number of students who completed in the first cycle by 31 students (70%) and the number of students incomplete by 9 people (30%). In the second cycle, the number of students who were declared complete increased by 37 students with a percentage of 90% and 3 students declared incomplete with a percentage of 10%.

To investigate a novel approach for establishing the transverse pedicle angle (TPA) of the lower lumbar spine using preoperative digital radiography (DR). Computed Tomography (CT) datasets of the lower lumbar were reconstructed using MIMICS 17.0 software and then imported into 3-matic software for surgical simulation and anatomical parameter measurement. A mathematical algorithm of TPA based on the Pythagorean theorem was established, and all obtained data were analyzed by SPSS software. The CT dataset from 66 samples was reconstructed as a digital model of the lower lumbar vertebrae (L3-L5), and the AP length/estimated lateral length for L3 between the right and left sides was statistically significant (P = 0.015, P = 0.005). The AP length of the right for L4 was smaller than that of the left after a paired t test was executed (P = 0.006). Both the width of the pedicle and the length of the pedicle (P2C1) were consistent with TPA (L3

In this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality ⊥ B ϵ . First, we show the relation between linear functionals and hyperplanes by means of approximate Birkhoff orthogonality. Second, we establish the sufficient conditions for Birkhoff orthogonality and approximate Birkhoff orthogonality. Then some sufficient and necessary conditions for isosceles orthogonality and Pythagorean orthogonality of bounded operators are shown. In addition, we characterize the inner product space in terms of functional orthogonality. [ABSTRACT FROM AUTHOR]

Purpose: To determine if the changes in stereoacuity and aniseikonia, following bilateral implantation of presbyopia correcting intraocular lenses could be predicted from preoperative measurements of higher order aberrations (HOAs), axial lengths (AL), refractive errors (RE) and corrected visual acuities (CVAs). Patients and Methods: Stereoacuity (Randot tests, @6m & 40cm, in steps of 20 arcsecs") vertical and horizontal aniseikonia (Awaya test @6m, in steps of 1%) with best correction and HOAs (Shack-Hartmann aberrometer) were measured before, 3 and 6 months after uncomplicated bilateral phacoemulsification. Twenty patients (I) underwent a mix-and-match procedure (Tecnis MF, ZKB00 in one eye and ZLB00 in the other), 17 (II) were implanted with a trifocal (AT LISA 839 triMP) and 18 (III) with a one-piece diffractive (Synergy OU) intraocular lens. The resultant aniseikonia (AR) of vertical and horizontal pairs of aniseikonia measurements was calculated using the Pythagorean theorem. Twenty untreated age/gender matched cases were recruited as controls (IV). Results: The key results (p < 0.001) were a) stereoacuity at distance (SAD) and near (SAN) improved, AR reduced in groups I, II & III remaining unchanged in group IV; b) some significant intergroup differences in SAD, SAN & AR were detected at postop; c) at 6 months postop, changes (Δ=pre- minus postoperative value) correlated with preoperative values (x). Linear regression revealed, I ΔSAD=0.66x-57.47 [0.832, ± 66.4], ΔSAN=0.96x-34.59 [0.821, ± 16.9], ΔAR=0.93AR-2.12 [0.795, ± 1.4] II ΔSAD=0.79x-62.91 [0.916, ± 38.1], ΔSAN=0.96x-31.49 [0.892, ± 8.0], ΔAR=0.91AR-0.91 [0.839, ± 1.3] III ΔSAD=0.67x-35.50 [0.991, ± 23.7], ΔSAN=0.88x-38.51[0.988, ± 10.6], ΔAR=0.86AR-0.96 [0.900, ± 1.3]. Figures in parentheses are the corresponding rs and ±limits of agreement between actual and estimated values. Definitive overarching associations connecting interocular differences in HOAs, AL, RE, and CVAs with SAD, SAN and AR were not found. Conclusion: Changes in stereoacuity and aniseikonia can be predicted using preoperative values. ΔSAN can be predicted within ± 1, and ΔAR within ± 2, scale divisions. In group III ΔSAD can be predicted within ± 1, and in group I ± 3, scale divisions. [ABSTRACT FROM AUTHOR]

Green Human Resource Management (GHRM) state a determination of the association using crossing points of employees to stimulate environment performance activity, increase the employee awareness and sustainable activities, consequently, increasing the employee awareness towards environmental challenges. The hotel industry is developing quickly in emerging nations owing to an upsurge in the tourism business; but, conversely, the hotel industry is mainly growing the problem of the environment. As a result, owing to the enormous amount of conservation problems that hotel business has faced, there is a growing potency to pay an accurate response to environmental problems and performing sustainable industry performance like the adoption of GHRM practice provides a win-win situation for its stakeholders and the organization. Accordingly, it indicates the requirement to scrutinize how GHRM performs will augment the environment in the hotel business. This manuscript models the design of GHRM using Pythagorean Neutrosophic Bonferroni Mean (GHRM-PNBM) approach. The presented GHRM-PNBM method objectives are to evaluate the limitation of hotel GRHM. Moreover, the presented technique constructs an expert system analysis technique for assessing the performance of hotel GHRM. Adaptive optimization of hotel GHRM assessment can be done using the PNBM technique, and the parameter selection method can be done using Quasi-Oppositional-Teaching-Learning-Based Optimization (QTLBO) method. The empirical analysis reports that the performance calculation of hotel GHRM has good confidence level and high accuracy. [ABSTRACT FROM AUTHOR]

Road traffic accidents (RTAs) pose a significant hazard to the security of the general public, especially in developing nations. A daily average of more than three thousand fatalities is recorded worldwide, rating it as the second most prevalent cause of death among people aged 5–29. Precise and reliable decisionmaking techniques are essential for identifying the most effective approach to mitigate road traffic incidents. This research endeavors to investigate this specific concern. The Fermatean fuzzy set (FFS) is a strong and efficient method for addressing ambiguity, particularly when the concept of Pythagorean fuzzy set fails to provide a solution. This research presents two innovative aggregation operators: the Fermatean fuzzy ordered weighted averaging (FFOWA) operator and the Fermatean fuzzy dynamic ordered weighted geometric (FFOWG) operator. The salient characteristics of these operators are discussed and important exceptional scenarios are thoroughly delineated. Furthermore, by implementing the suggested operators, we develop a systematic approach to handle multiple attribute decisionmaking (MADM) scenarios that involve Fermatean fuzzy (FF) data. In order to show the viability of the developed method, we provide a numerical illustration encompassing the determination of the most effective approach to alleviate road traffic accidents. Lastly, we conduct a comparative evaluation of the proposed approach in relation to a number of established methodologies. [ABSTRACT FROM AUTHOR]

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x^2 + y^2 = z^2. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3, 4, 5) and (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic \neq 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring. [ABSTRACT FROM AUTHOR]

Green supplier selection is an important debate in green supply chain management (GSCM), attracting global attention from scholars, especially companies and policymakers. Companies frequently search for new ideas and strategies to assist them in realizing sustainable development. Because of the speculative character of human opinions, supplier selection frequently includes unreliable data, and the interval-valued Pythagorean fuzzy soft set (IVPFSS) provides an exceptional capacity to cope with excessive fuzziness, inconsistency, and inexactness through the decision-making procedure. The main goal of this study is to come up with new operational laws for interval-valued Pythagorean fuzzy soft numbers (IVPFSNs) and create two interaction operators-the intervalvalued Pythagorean fuzzy soft interaction weighted average (IVPFSIWA) and the interval-valued Pythagorean fuzzy soft interaction weighted geometric (IVPFSIWG) operators, and analyze their properties. These operators are highly advantageous in addressing uncertain problems by considering membership and non-membership values within intervals, providing a superior solution to othermethods. Moreover, specialist judgments were calculated by the MCGDM technique, supporting the use of interaction AOs to regulate the interdependence and fundamental partiality of green supplier assessment aspects. Lastly, a statistical clarification of the planned method for green supplier selection is presented. [ABSTRACT FROM AUTHOR]

This article explores the refractive index of hydrated superabsorbent polymer (SAP) spheres, which are polymers that can absorb large amounts of water and become nearly invisible when submerged. The authors developed a method to estimate the refractive index of fully hydrated SAP spheres using a computer model and a simple experiment. They discovered that the refractive index of the spheres closely resembled that of water. This information could be valuable for educational purposes, particularly in classroom laboratory activities. The study utilized affordable tools such as a smartphone camera and spreadsheet software, making it accessible for educational use. The researchers also propose the possibility of conducting further experiments with different liquids. The spreadsheet model employed in the study is available upon request. [Extracted from the article]

After analyzing the existing ranking methods of Pythagorean fuzzy numbers (PFNs) by several examples, some shortcomings of the existing ranking methods are pointed out. To conquer such shortcomings, we propose a novel approach to rank PFNs by using power index measure and score function of PFN. Simultaneously, the rationality of the proposed ranking method is analyzed theoretically. Besides, the techniques of aggregating Pythagorean fuzzy information are investigated. To expand the practical application scope of the DOWA operator to Pythagorean fuzzy environments, we present two kinds of Pythagorean fuzzy monotonic DOWA (PFMDOWA) operators by utilizing the power index measure of PFN. If the weights of attributes are considered, we further develop two kinds of Pythagorean fuzzy hybrid monotonic DOWA (PFHMDOWA) operators. The main advantage of the PFMDOWA operator and PFHMDOWA operator is that the weights associated to the operators can be generated and adjusted dynamically. Lastly, a decision making example is given to verify the flexibility and rationality of the presented techniques. [ABSTRACT FROM AUTHOR]

This study employs a Pythagorean Neutrosophic Bonferroni Mean (PNBM) - Decision Making Trial and Evaluation Laboratory (DEMATEL) approach to analyze barriers faced by young entrepreneurs in Digital Social Innovation (DSI). Pythagorean Neutrosophic Set (PNS) enriches the analysis, accommodating uncertainties in the complex socio-economic context. Limited Access to Funding emerges as the most influential barrier, showcasing its pivotal role in impacting other DSI challenges. Regulatory and Compliance Challenges are identified as interdependent, emphasizing their interconnected nature with broader barriers. Neutrosophic elements elucidate the uncertainties surrounding financial constraints, regulatory frameworks, and mentorship dynamics. The study contributes to a nuanced understanding of DSI challenges and pioneers the application of neutrosophic logic in socio-economic research. It advocates for a more inclusive decision-making methodology, fostering adaptability in addressing the indeterminate nature of barriers faced by young entrepreneurs navigating the digital social innovation landscape. The findings aim to enhance support systems, fostering a conducive environment for DSI initiatives and encouraging future research in neutrosophic decision-making methodologies. [ABSTRACT FROM AUTHOR]

The development of technology today has a great influence on human life. This influence also includes the use of technology in education, such as the utilization of learning media connected to information technology. This research was conducted to validate the use of learning media in the form of Android applications with the help of iSpring Suite in learning Pythagorean Theorem material. The research method used is RnD by applying the ADDIE model. The research subjects consisted of 3 experts, one practitioner, and 5 grade VIII students. Data collection involved qualitative and quantitative data. The results of the analysis showed that the experts gave an average assessment of 92.09% with the category "Very Valid." Trials conducted on practitioners and small groups also showed a "Very Valid" assessment with an average of 90%. Overall, it can be concluded that the use of Android-based learning media with iSpring Suite is appropriate and effective in achieving learning objectives. With this learning media that has been proven valid and effective, it can be an interesting alternative to improve the quality of mathematics learning in the classroom. Integrating technology in learning can also provide a more interactive and interesting learning experience for students. Thus, the results of this study make a positive contribution to answering the challenges of education in the digital era, enriching learning methods, and increasing the attractiveness of learning materials.

This article discusses a generalization of de Gua's theorem, which is a three-dimensional analogue of the Pythagorean theorem. The theorem states that if a tetrahedron has a right-angled corner, then the square of the area of the face opposite that corner equals the sum of the squares of the areas of the other three faces. The article presents an extension of this theorem to an arbitrary tetrahedron and provides a vector proof. The article also introduces the concept of the areal vector of a triangle and presents a lemma related to the theorem. The given text provides a proof of Theorem 1, which is a generalization of the Pythagorean theorem in space. The proof involves applying Lemma 1 and using vector calculations. The text also includes a diagram (Figure 2) to illustrate the theorem. [Extracted from the article]

This article explores the differences between the equal-tempered scale, the Pythagorean scale, and the just scale in music. It explains how these scales are constructed using geometric methods and how they differ in terms of the ratios between notes. The article also discusses the concept of cents as a measure of the distance between notes and provides a table comparing the ratios and cents for the notes of the C major scale in each scale. Overall, the article highlights the historical and mathematical aspects of musical scales and their impact on our perception of music. [Extracted from the article]

With the growing popularity of movies as a source of entertainment and relaxation in modern society, movie recommendation systems have become increasingly important for helping viewers navigate a vast selection of movie products. However, existing methods may not accurately capture the preferences and opinions of viewers. To address this gap, we propose a novel approach that utilizes the Pythagorean hesitant fuzzy distance measure in combination with the VIKOR method to deal with the extracted review information and evaluate the similarity between different movies in a given genre. We then transform this information into Pythagorean hesitant fuzzy attribute evaluation terms using the Probabilistic Linguistic Term Set (PLTS) and apply conflict degree theory to calculate similarity scores. The algorithm proposed in this paper employs the VIKOR method to select recommended movie products that match the preferences of users and satisfy their needs. Comparative analysis demonstrates its reliability and stability. In addition, according to the comparative data, its performance is better than the comparable method in the selection of alternative schemes, which highlights its contribution in the field of movie recommendation systems. [ABSTRACT FROM AUTHOR]

This article examines Pythagorean neurosophic vague set (PyNVS) problems relevant to multiple attribute decision-making (MADM). Pythagorean vague set (PyVS) and neutrosophic set (NS) can be generalized into Pythagorean neutrosophic vague set (PyNVS). We discuss log Pythagorean neutrosophic vague weighted averaging (log PyNVWA), logarithmic Pythagorean neutrosophic vague weighted geometric (log PyNVWG), log generalized Pythagorean neurosophic vague weighted averaging (log GPyNVWA) and log generalized Pythagorean neutrosophic vague weighted geometric (log GPyNVWG). In this article, we define the Euclidean distance (ED), Hamming distance (HD), operator laws, and flowchart using an algorithm. By analyzing log PyNVS through algebraic operations, we discuss its properties. They can identify the best option more quickly and understand the practicalities better. An illustrative example of this is the fusion of computer science and machine tool technology in agriculture. Furthermore, there are autonomous robot tractors and soil sterilization robots that can harvest crops, weed, and take photos of seed planting with seedlings. A random selection of five farmers (alternatives) has been made. Climate, water, soil, disease, and flooding are all criteria to consider when choosing a farmer. Our goal is to narrow down the options by comparing expert judgments with the criteria. [ABSTRACT FROM AUTHOR]

A composition law, inspired by the Farey addition, is introduced on the set of Pythagorean triples. We study some of its properties as well as two symmetric matrices naturally associated to a given Pythagorean triple. Several examples are discussed, some of them involving the degenerated Pythagorean triple (1, 0, 1). The case of Eisenstein triples is also presented. [ABSTRACT FROM AUTHOR]

The 2 × 2 identity matrix, $${I_2} = \left(\begin{gathered}{\rm{1 \,\,\,0}} \hfill \\ {\rm{0 \,\,\,1}} \hfill \\ \end{gathered} \right)$$ , has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples. [ABSTRACT FROM AUTHOR]

Carlo Cellucci's book, "The Making of Mathematics - Heuristic/Philosophy of Mathematics," challenges the traditional view of mathematics as theorem proving through the axiomatic method. Cellucci argues that mathematics is best understood as problem solving through the analytic method. He emphasizes the importance of reconnecting the context of discovery with that of justification, allowing for a more complete understanding of mathematical arguments. Cellucci also critiques textbooks that omit the real mathematical process, advocating for the inclusion of heuristics in problem solving. Overall, his work has the potential to reshape the way mathematics is presented and understood in the future. [Extracted from the article]

There are many ways to prove Heron's formula, which allows us to find the area of a triangle given the lengths of its three sides. In this paper, we prove Heron's formula by constructing a square with the length of the shortest side of the triangle as its diagonal and using the Pythagorean theorem. [ABSTRACT FROM AUTHOR]

This research proposed a methodology with the objective of promoting a meaningful learning of the Pythagorean Theorem through Didactic Sequence, encompassing digital resources, in view of the need for teaching strategies for this theorem that can provide the student with the association of the concepts seen in class. with your everyday life. The study, with a qualitative bias, consisted of exploratory research, through a conversation with Mathematics teachers. Next, a documentary search was carried out to list the skills in the National Common Curricular Base and to identify the digital tool most used by students according to the Continuous National Household Sample Survey. From then on, a Didactic Sequence was elaborated and evaluated by teachers, who found that the proposed methodology, permeated by digital technologies, by approaching in a playful way situations of applicability of the Pythagorean Theorem, related to accessibility, could make learning more attractive and significant. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new generalized multifarious radical reciprocal functional equation by generalizing the equation employed by Narasimman et al. in [5] and combining three classical Pythagorean means arithmetic, geometric and harmonic. Also, we illustrate the geometrical interpretation. Mainly, we find its general solution and stabilities related to Ulam problem in modular spaces by using fixed point approach. [ABSTRACT FROM AUTHOR]

In this paper, the concept of bipolar Pythagorean fuzzy neutrosophic sets (BPNSs) will be introduced which represents an innovative synthesis, incorporating bipolarity, Pythagorean fuzzy sets, and neutrosophic sets to provide a comprehensive solution to the multifaceted issues encountered in decision-making challenges. BPNS is proposed to enhance the expressiveness and flexibility of the model by incorporating both positive and negative preferences that can lead to more accurate and realistic modeling of complex decision scenarios. The essential idea of BPNSs is defined and some definitions for instance complement, intersection, and union are described. The basic operation, the derivation of its properties, and the numerical example of the method are integrated to execute the proposed model. [ABSTRACT FROM AUTHOR]