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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]

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]

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

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]

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]

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.

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]

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]

Motivated by the concept of type-2 fuzzy sets, we introduce a novel framework known as trapezoidal type-2 Pythagorean fuzzy sets (TRT-2-PFSs), an extension of triangular fuzzy sets. Basic operations like addition and scalar multiplication of two TRT-2-Pythagorean fuzzy numbers (TRT-2-PFNs) are defined. We also explore comparative analysis and distance measurements between two TRT-2-PFNs. A methodology for evaluating unknown weight vectors and criteria weights is proposed. Building upon TRT-2-PFSs, an extension of the TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method is developed to address intricate decision-making challenges. Ultimately, the newly introduced TRT-2-PFS-based TODIM technique is employed to tackle multicriteria decision-making (MCDM) problems. [ABSTRACT FROM AUTHOR]

This article presents the circular Pythagorean fuzzy set (C-PFS) model, a generalization of the circular intuitionistic fuzzy set model that improves its performance thanks to the acclaimed extension of intuitionistic fuzzy sets to Pythagorean fuzzy sets. Then, we generalize C-PFSs to produce the novel disc Pythagorean fuzzy sets (D-PFSs). The constituent elements of both C-PFSs and D-PFSs are circular Pythagorean fuzzy values, either with a common or a distinctive radius. We lay out some fundamental algebraic and arithmetic operations on D-PFSs (hence on C-PFSs), namely union, intersection, addition, multiplication, and scalar multiplication, and we explore the main features of these operations. We propose and investigate the properties of the novel circular Pythagorean fuzzy weighted average/geometric aggregation operators. The "COmbinative Distance based ASsesment" approach, which is based on the Hamming and Euclidean distances, is expanded to the D-PFS framework. To justify its implementability, we apply the new methodology to a case study (selection of the best supermarkets to buy fresh fruit for a hotel) and then we compare it to related solutions. [ABSTRACT FROM AUTHOR]

The objective of this paper is to find the necessary and sufficient conditions for a symbolic 3-plithogenic quadruple (to+t1P1 + t2P2 + t3P3, S0 + S1P1+S2P2 + S3P3, ko + k1P1 + k2P2 + k3P3, lo + L1P1 + l2P2 + l3P3) to be a Pythagoras quadruple, i.e. to be a solution for the non-linear Diophantine equation in four variables X² + Y²+ Z² = T². Also, many examples will be illustrated and presented to explain how the theorems work. [ABSTRACT FROM AUTHOR]

The study of fuzzy geometry and its different components has grown in recent years, establishing the formal foundations for its development. This paper is devoted to addressing some metric relations in the fuzzy right triangle; in particular, a version of the Pythagorean theorem and geometric mean theorem are provided in analytical fuzzy geometry. The main results show, under certain conditions of the fuzzy vertices, a subset relation between the fuzzy distances associated with the fuzzy right triangle, which is very similar to the classical statements of the Pythagorean theorem and the geometric mean in Euclidean geometry. [ABSTRACT FROM AUTHOR]

This study uses 'Mathbox' media which is focused and aims to improve students' self-assessment and understanding and is developed using the Codular website. The research method used is Research and Development (RD) with the ADDIE model (Analysis, Design, Development, Implementation, Evaluation). The development of this learning media proved to be valid with an average percentage of 95.45% of the 'Very Good' Media Checker criteria as measured using the Guttmann scale. There was an 11.1% increase in the pre-test to post-test self-assessment scores which were measured using a Likert scale. The difference in the scores of students' understanding tests with the Paired T-test shows that the application of media affects students' understanding test scores. Based on the results obtained, it can be concluded that the designed Kodular web-based Mathbox media is a valid medium and can improve students' self-assessment and understanding of the Pythagorean theorem. Penelitian ini menggunakan media 'Mathbox' yang difokuskan dan bertujuan untuk meningkatkan penilaian dan pemahaman diri siswa, serta dikembangkan menggunakan website Codular. Metode penelitian yang digunakan adalah Research and Development (RD) dengan model ADDIE (Analysis, Design, Development, Implementation, Evaluation). Pengembangan media pembelajaran ini terbukti valid dengan persentase rata-rata 95,45% kriteria validasi media 'Sangat Baik' yang diukur menggunakan skala Guttmann. Ada peningkatan 11,1% pada skor self-assessment pre-test hingga post-test yang diukur dengan menggunakan skala Likert. Perbedaan nilai tes pemahaman siswa dengan Paired T-test menunjukkan bahwa penerapan media berpengaruh terhadap nilai tes pemahaman siswa. Berdasarkan hasil yang diperoleh, dapat disimpulkan bahwa media Mathbox berbasis web Kodular yang dirancang merupakan media yang valid dan dapat meningkatkan penilaian diri dan pemahaman siswa terhadap teorema Pythagoras.

A Fermatean fuzzy set (FRFS) is the extension of a fuzzy set, an intuitionistic fuzzy set, and a Pythagorean fuzzy set, and is used in different fields. Unlike other fuzzy structures, the sum of cubes of membership grades in FRFSs approximates a unit interval, increasing uncertainty. In this study, we intend to provide unique operational rules and aggregation operators (AOs) inside a Fermatean fuzzy environment. To develop a fair remedy for the membership degree and non-membership degree features of "Fermatean fuzzy numbers (FRFNs)", our solution introduces new neutral or fair operating principles, which include the concept of proportional distribution. Based on the suggested operating principles, we provide the "Fermatean fuzzy fairly weighted average operator and the Fermatean fuzzy fairly ordered weighted averaging operator". Our suggested AOs provide more generalized, reliable, and exact data than previous techniques. Combining the recommended AOs with multiple decision-makers and partial weight information under FRFSs, we also devised a technique for "multi-criteria decision-making". To illustrate the application of our novel method, we provide an example of the algorithm's effectiveness in addressing decision-making challenges. [ABSTRACT FROM AUTHOR]

With the increase of complex information in applications of decision making problems, the use of probabilistic hesitant fuzzy set structure has expanded. Therefore, this paper aims to present two new operators namely q-rung orthopair probabilistic hesitant fuzzy hybrid weighted arithmetic and geometric (q-ROPHHWAG) operator and q-rung orthopair probabilistic hesitant fuzzy hybrid ordered weighted arithmetic and geometric (q-ROPHHOWAG) operator for q>0. The presented operators are better than existing operators in many respects as adding a new parameter, having more flexible structure and presenting comparative analysis in its own. Moreover, we mention from some properties of the proposed operators. In addition to, we give an algorithm and example to indicate effective, reality and flexible of presented method and operators. Then, we solve an example over Pythagorean probabilistic hesitant fuzzy sets with our operators and the results are agreement and the offered operators have superior effect than other operators. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new multi-attribute decision-making (MADM) method based on probabilistic dual hesitant Pythagorean fuzzy (PDHPF) sets. The existing PDHPF power weighted Hamy mean operator has the drawback that if one PDHPF element among the PDHPF elements whose non-membership grade equals 0, then the non-membership grade of the aggregated PDHPF element is equal to 0. Thus, the existing MADM method based on the PDHPF power weighted Hamy mean operator has the drawback that it cannot distinguish the ranking orders of alternatives in some situations. To overcome these drawbacks of the existing MADM method, this paper proposes the PDHPF improved power weighted averaging (PDHPFIPWA) operator and proposes a MADM method based on the proposed PDHPFIPWA operator. The proposed MADM method can overcome the drawbacks of the existing MADM method. It offers us a very useful approach for MADM in PDHPF environments. [ABSTRACT FROM AUTHOR]

Copyright of Gaceta de la Real Sociedad Matematica Espanola is the property of Real Sociedad Matematica Espanola and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper presents an improved current control strategy for a three-phase voltage source converter connected to distorted grid. The proposed controller can successfully compensate the induced harmonic currents, while suppressing possible active power ripple. To manage various objectives and deal with tough restrictions, model predictive control can offer a promising solution. However, a modified extended complex Kalman filter is introduced to estimate the positive and negative fundamental components of the grid voltages which are used with the real and reactive power references to generate pure sinusoidal reference currents without the need for a phase-locked loop (PLL), avoiding its challenges under abnormal grid conditions. An improved space vector-based model predictive control (SVM-MPC) is proposed to regulate the real and reactive power components. The main target of this controller is to maintain harmonic free grid currents, even in unbalanced and distorted grid conditions, with enhanced performance in terms of fast dynamic response and good steady-state behavior. The proposed SVM MPC uses Pythagoras theorem to determine duty factors, avoiding trigonometric functions, which improves the computational burden and implementation complexity. In addition, fixed switching frequency can be ensured. Simulation and experimental results are carried out to validate the effectiveness and the practical feasibility of the proposed controller under distorted grid conditions. [ABSTRACT FROM AUTHOR]

There are many genetic and environmental factors that affect cognitive development. Music education can also be considered as one of the environmental factors. Some researchers emphasize that Music is an action that requires meta-cognitive functions such as mathematics and chess, and supports spatial intelligence. The effect of music on cognitive development in early childhood was examined with the Pythagorean Fuzzy Sets(PFS) method defined by Yager. In this study, PFS was created from experts' opinions and an algorithm was given according to PFS. The results of the algorithm supported the data of the experts on the development of spatial-temporal skills of music education given in early childhood. In the algorithm, the ranking has been done with the Expectation Score Function. The rankings obtained from the algorithm overlap with the experts' rankings. [ABSTRACT FROM AUTHOR]

Finding the best transportation project and logistic service provider is one for the most important aspects of the development of a country. This task becomes more complicated from time to time as different criteria are involved. Hence, this paper proposes an approach to the linguistic three-way decision-making (TWDs) problem for selecting sustainable transportation investments and logistic service providers with unknown criteria and expert weight information. To this end, we first propose a new tool, the Pythagorean double hierarchy linguistic term sets (PyDHLTSs), which is a combination of first hierarchy linguistic term sets and second hierarchy linguistic term sets which can describe uncertainty and fuzziness more flexibly in decision-making (DM) problems. In addition, we propose some aggregation operators and basic operational laws for PyDHLTSs. A new decision-making technique for PyDHLTSs based on decision-theoretic rough sets (DTRSs) is proposed in the three-way decisions. Next, the conditional probability is computed using grey relational analysis in a PyDHLTSs environment, which improves decision-making. The loss function is computed by using the proposed aggregation operator, and the decision's results are determined by the minimum-loss principle. Finally, a real-world case study of a transportation project and logistic service provider is considered to demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

This article develops a colorimetric equation and a colorimetric model to create a smartphone application that identifies the ripening stage of the lady finger banana (LFB) (Musa AA group 'Kluai Khai', กล้วยไข่ "gluay kai" in Thai). The mobile application photographs an LFB, automatically analyzes the color of the banana, and tells the user the number of days until the banana ripens and the number of days the banana will remain edible. The application is called the Automatic Banana Ripeness Indicator (ABRI, pronounced like "Aubrey"), and the rapid analysis that it provides is useful to anyone involved in the storage and distribution of bananas. The colorimetric equation interprets the skin color with the CIE L*a*b* color model in conjunction with the Pythagorean theorem. The colorimetric model has three parts. First, COCO-SSD object detection locates and identifies the banana in the image. Second, the Automatic Power-Law Transformation, developed here, adjusts the illumination to a standard derived from the average of a set of laboratory images. After removing the image background and converting the image to L*a*b*, the data are sent to the colorimetric equation to calculate the ripening stage. Results show that ABRI correctly detects a banana with 91.45% accuracy and the Automatic Power-Law Transformation correctly adjusts the image illumination with 95.72% accuracy. The colorimetric equation correctly identifies the ripening stage of all incoming images. ABRI is thus an accurate and robust tool that quickly, conveniently, and reliably provides the user with any LFB's ripening stage and the remaining days for consumption. [ABSTRACT FROM AUTHOR]

The MULTIMOORA is a relatively new multi-attribute decision-making (MADM) technique containing three major parts. It has partial and complete aggregation information. The MULTIMOORA procedure contains three major parts: the full multiplicative shape, ratio system, and reference point. Furthermore, the Muirhead mean (MM) operator can easily capture the interaction between any number of preferences. The main feature of this model is to evaluate the novel theory of complex Pythagorean fuzzy uncertain linguistic (CPFUL) settings and utilize their useful properties ("algebraic laws, score value, and accuracy value"). Additionally, we combined the main theory of MM operators with information taken from the theory of CPFUL to diagnose the major idea of the CPFUL Muirhead mean (CPFULMM), CPFUL weighted Muirhead mean (CPFULWMM), CPFUL dual Muirhead mean (CPFULDMM), and CPFUL dual weighted Muirhead mean (CPFULDWMM) operators. We then evaluated their valuable properties ("idempotency, Boundedness, and monotonicity"). Moreover, to enhance the worth and feasibility of the diagnosed approach, we elaborated the extension of the MULTIMOORA method under CPFUL information. We then diagnosed a procedure of the MADM scenario whilst considering the evaluated operators using the CPFUL set theory. Lastly, to compare the diagnosed information with some old operators, we used numerical examples to illustrate the flexibility and feasibility of the presented information. [ABSTRACT FROM AUTHOR]

Background & Objective: Invasive breast carcinoma (IBC) is the most commonly diagnosed cancer among women in India. The conventional visual method of evaluation of Tumor-Stroma Ratio (TSR) and Stromal Tumor-Infiltrating Lymphocytes (sTIL) appears to be subjective. The present study aims to evaluate the utility of the indigenously designed square grid method for the evaluation of tumor-stroma ratio and stromal tumor-infiltrating lymphocytes in invasive breast carcinoma by assessing the inter-observer variability. Methods: This was a retrospective study conducted at a rural tertiary care referral institute from July 2018 to June 2020. In each case, microphotographs were taken from 10 representative fields in H&E-stained sections for evaluating TSR in low-power and sTIL in high-power. Both the parameters were evaluated employing an indigenously designed square grid applied onto microphotographs in the power-point slides by making use of principles of the Pythagorean theorem. Both parameters were separately evaluated by two pathologists. Cohen kappa statistics was the statistical tool used to analyze inter-observer variability. Results: Thirty cases were analyzed. Invasive breast carcinoma of no special type (IBC-NST) was the most common histopathological type (26 cases (86.67%)). For TRS evaluation, a Kappa value of 0.78 suggested substantial agreement with an agreement of 91.67%. For sTIL evaluation, a Kappa value of 0.51 suggested moderate agreement with an agreement of 88.33%. The P-values were statistically highly significant (P<0.001). Conclusion: Square grid method is a novel technique for evaluating TSR and sTIL in invasive breast carcinoma. It can be considered an example of the application of Pythagoras' theorem in Pathology. [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. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Inquiry Based Activities (JIBA) / Araştırma Temelli Etkinlik Dergisi (ATED) is the property of Journal of Inquiry Based Activities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This study aims to determine the mathematical representation ability of class IX students with visual, auditory, and kinesthetic learning styles on the Pythagoras Theorem material at SMPN 1 (public junior high school) Gunungsari Indonesia in the 2021/2022 academic year. This type of research is descriptive with a quantitative approach. The populations in this study were all students of class IX SMPN 1 Gunungsari. The research sampling technique used purposive sampling, and the sample was class IX-H. The instruments used are learning style questionnaires, mathematical representation ability tests, and interview guidelines. The results of this study indicate that the mathematical representation ability of students with visual learning styles is 35.56% in the low category, or students have yet to be able to meet the overall visual, symbolic, and verbal representation indicators. Furthermore, the mathematical representation ability of students with auditory and kinesthetic learning styles is 49.86% and 41.87% in the medium category. Respectively, students have been able to meet the indicators of visual and symbolic representation quite well but have yet to be able to fulfill the visual and symbolic representation and verbal representation indicators.

This paper presents a variational model for the denoising of wrapped phase images. By enforcing the required Pythagorean trigonometric identity between the real and imaginary components of the signal, this model improves the signal-to-noise ratio of the restored signal. To preserve phase map discontinuities, the model is based on total variation. The existence and uniqueness of the model's solution are demonstrated using standard techniques. In addition, the convergence of a rapid fixed-point method to determine the numerical solution is demonstrated. Experiments on both synthetic and actual patterns validate the model's performance. [ABSTRACT FROM AUTHOR]

This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG) in the context of T -norm and S -conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the "min" and "max" operations, respectively. However, in this work, we develop a theory that utilizes the T -norm for "min" and the S -conorm for "max", providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the T -norm and S -conorm, and discuss important properties associated with them. [ABSTRACT FROM AUTHOR]

Objectives: Recent research has suggested the contour of the prosthesis and the vertical height of the peri‐implant mucosa as important parameters that can influence the long term health and stability of the peri‐implant tissue. In particular, overcontouring of the prosthesis has been correlated with an increased risk for peri‐implantitis, while reduced soft tissue height has been associated with marginal bone loss, recession, and other soft tissue complications. Although these two parameters have been investigated as independent in the current literature, clinical experience points toward a close interrelation between transmucosal tissue height and prosthesis contour angle. It is often found that a reduced vertical height of the implant supracrestal complex is the main reason for overcontouring of the prosthesis. At the same time, achieving a favorable contour of 30o or less is not possible unless the clinician has ensured an adequate vertical height of the soft tissue. The purpose of this short communication is to establish the relation between tissue vertical height and prosthesis contour by utilizing a theoretical geometry equation based on the Pythagorean theorem. In doing so, one can use the dimensions of the implant as well as those of the prosthesis at the mucosal margin to calculate the essential vertical height for achieving a favorable prosthesis contour. Conclusions: As the treatment plan of the implant supracrestal complex is "top‐down," in case of deficient vertical height, subcrestal placement of the implant should be considered to achieve a proper prosthesis contour. [ABSTRACT FROM AUTHOR]

Copyright of International Journal of Chinese Language Teaching is the property of Clifford Publishing and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Introduction: Gaining access to the kidney is crucial step in percutaneous nephrolithotomy (PCNL); it has a steep learning curve. Objective: Describe the mathematical method to predict renal puncture angle and distance based on preoperative computed tomography (CT) measurements. Then evaluating how it correlates with measured values. Patients and Methods: The study was prospectively designed. After ethical committee approval, the study uses data from preoperative CT to construct a triangle so we can estimate puncture depth and angle. A triangle of three points, the first is point of entry to the pelvicalyceal system (PCS), the second is point on the skin perpendicular to it, and the third where the needle punctures the skin. The needle travel is estimated using the Pythagorean theorem and puncture angle using the inverse sine function. We evaluated 40 punctures in 36 PCNL procedures. After PCS puncture using fluoroscopy-guided triangulation, we measured the needle travel distance and angle to the horizontal plane. Then compared the results with mathematically estimated values. Results: We targeted posterior lower calyx in 21 (70%) case. The correlation between measured and estimated needle travel distance with Rho coefficient of 0.76 with P < 0.001. The mean difference between the estimated and the measured needle travel was - 0.37 ± 1.2 cm (-2.6-1.6). Measured and estimated angle correlate with Rho coefficient of 0.77 and P < 0.001. The mean difference between the estimated and the measured angle was 2° ± 8° (-21°-16°). Conclusion: Mathematical estimation of needle depth and angle for gaining access to the kidney correlates well with measured values. [ABSTRACT FROM AUTHOR]

Multiple Criteria Decision Analysis (MCDA) has been widely investigated and successfully applied to many fields,owing to its great capability of modeling the process of actual decision-making problems and establishing proper evaluation and assessment mechanisms. With the development of management and economics, real-world decision-making problem are becoming diversified and complicated to an increasing extent, especially within a changeable and unpredictable enviroment. Multi-criteria is a decision-making technique that explicitly evaluates numerous contradictory criteria. TOPSIS is a well-known multi-criteria decision-making process. The goal of this research is to use TOPSIS to solve MCDM problems in a Pythagorean fuzzy environment. The distance between two Pythagorean fuzzy numbers is utilized to create the model using the spherical distance measure. To construct a ranking order of alternatives and determine the best one,the revised index approach is utilized. Finally, we look at a set of MCDM problems to show how the proposed method and approach work in practice. In addition, it shows comparative data from the relative closeness and updated index methods. [ABSTRACT FROM AUTHOR]

The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous algebraic properties. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. We also introduce the concept of Pythagorean fuzzy semi-prime ideals and give a detailed exposition of its different algebraic characteristics. In the end, we characterized regular rings by virtue of Pythagorean fuzzy ideals. [ABSTRACT FROM AUTHOR]

In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into two classes according to the degrees of their derivatives' factors, we introduce auxiliary control points to reconstruct the internal algebraic structure for both classes. We prove that a sextic curve is completely characterized by the lengths of legs and angles formed by the legs of their Bézier control polygons. As such conditions are invariant under rotations and translations, we call them the geometric characteristics of sextic PH curves. We demonstrate that the geometric characteristics form the basis for an easy and intuitive method for identifying sextic PH curves. Benefiting from our results, the computations of the parameters of cusps and/or inflection points can also be simplified. [ABSTRACT FROM AUTHOR]

Copyright of Ricercare is the property of Universidad EAFIT and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we study the notion of Pythagorean fuzzy Γ-hyperideals in Γ-hypersemigroups. We also study Pythagorean fuzzy Γ-subsemihypergroup and Pythagorean fuzzy bi Γ-hyperideals in Γ-hypersemigroups and discuss some properties. Relations between these Γ-hyperideals are also studied. Inverse image of Pythagorean fuzzy Γ-subsemihypergroup and Pythagorean fuzzy bi Γ-hyperideals are discussed. [ABSTRACT FROM AUTHOR]

This research aims to describe the mathematical communication skills in writing of junior high school students in grade VIII in solving the problems of Theorem Pythagoras. The method used in this study is a qualitative descriptive method. The subjects of this study numbered 31 students and were categorized in groups of very high, high, medium, low and very low female students. The results showed that students who had mathematical communication skills in a very high group were students who were very able to meet the four indicators tested with an average percentage of 83%. The mathematical communication skills of students in the high group are students who are able to meet the four indicators tested with an average total percentage of 70%. The mathematical communication skills of students in the moderate group are students who are sufficiently able to meet the four indicators tested with an average total percentage of 51%. The mathematical communication skills of students in the lower group were students who were less able to meet the four indicators tested with an average total percentage of 32%. The mathematical communication skills of students in the very low group were students who had not been able to meet the four indicators tested with an average total percentage of 17%.

The paper investigates a number of incomplete exact roots of a series of natural numbers, in relation to the Pythagorean theorem that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the legs. It uses the fact that the equation x^2+y^2=z^n,n=2,3,4,.. always has an solution (x,y,z) in integer numbers x,y,z∈Z={0,±1,±2,…}.

In this study, a novel Pythagorean fuzzy aggregation operator is presented by combining the concepts of Aczel–Alsina (A A) T-norm and T-conorm operations for multiple attribute group decision-making (MAGDM) challenge for the superiority and inferiority ranking (SIR) approach. This approach has many advantages in solving real-life problems. In this study, the superiority and inferiority ranking method is illustrated and showed the effectiveness for decision makers by using multicriteria. The Aczel–Alsina aggregation operators on interval-valued IFSs, hesitant fuzzy sets (HFSs), Pythagorean fuzzy sets (PFSs), and T-spherical fuzzy sets (TSFSs) for multiple attribute decision-making (MADM) issues have been proposed in the literature. In addition, we propose a Pythagorean fuzzy Aczel–Alsina weighted average closeness coefficient (PF − A A − WA − C C) aggregation operator on the basis of the closeness coefficient for MAGDM challenges. To highlight the relevancy and authenticity of this approach and measure its validity, we conducted a comparative analysis with the method already in vogue. [ABSTRACT FROM AUTHOR]