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The article explores the relationship between Euclidean geometry and calculus, speculating on what would have happened if calculus had been developed before geometry. It discusses the application of Apollonius's theorem and Stewart's theorem in geometry, as well as the work of Heron of Alexandria. The text also discusses the law of cosines and Newton's assumption about the length of the angle bisector. It explores the generalization of the Pythagorean theorem in three dimensions and discusses a method called de Gua's theorem for deriving classical theorems in Euclidean geometry. The text acknowledges the contributions of others in developing this method. [Extracted from the article]

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]

Prioritization is usually required in problems involving multi-attribute group decision-making (MAGDM). Several strategies and procedures have been introduced in fuzzy systems to apply prioritization. This study examines the MAGDM problem in a Pythagorean fuzzy (PF) setting with varying amounts of demand for specialists and attributes. We regard the novel Aczel Alsina aggregation operators (AOs) as the most addition to fuzzy mathematics that can deal with uncertainties significantly. We suggest a few PF AOs based on Aczel Alsina t-norm and t-conorm, including the PF-prioritized Aczel Alsina averaging (PFPAAA) and PF-prioritized Aczel Alsina geometric (PFPAAG) operators. It is proven that these AOs fulfil the aggregation criteria by investigating the properties of monotonicity, boundedness, and idempotency. The weights for prioritization are derived from the knowledge of experts, and the proposed operators can capture the phenomenon of prioritization among the aggregated arguments. The proposed AOs are then applied to assess fire extinguishers using a MAGDM technique. The importance of PFPAAA and PFPAAG operators is verified by comparing the proposed AOs with other well-known AOs. [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]

Ultrastructural analysis of muscular biopsy is based on images of longitudinal sections of the fibers. Sometimes, due to experimental limitations, the resulting sections are instead oblique, and no accurate morphological information can be extracted with standard analysis methods. Thus, the biopsy is performed again, but this is too invasive and time‐consuming. In this study, we focused our attention on the sarcomere's shape and we investigated which is the structural information that can be obtained from oblique sections. A routine was written in MATLAB to allow the visualization of how a sarcomere's section appears in ultrastructural images obtained by Transmission Electron Microscopy (TEM) at different secant angles. The routine was used also to analyze the intersection between a cylinder and a plane to show how the Z‐bands and M‐line lengths vary at different secant angles. Moreover, we explored how to calculate sarcomere's radius and length as well as the secant angle from ultrastructural images, based only on geometrical considerations (Pythagorean theorem and trigonometric functions). The equations to calculate these parameters starting from ultrastructural image measurements were found. Noteworthy, to obtain the real sarcomere length in quasi‐longitudinal sections, a small correction in the standard procedure is needed and highlighted in the text. In conclusion, even non‐longitudinal sections of skeletal muscles can be used to extrapolate morphological information of sarcomeres, which are important parameters for diagnostic purposes. [ABSTRACT FROM AUTHOR]

In this paper, we find necessary and sufficient conditions for a weak fuzzy complex integer triple (X, Y, Z) to be a pythagoras triple, and for an anti-weak fuzzy complex integer triple (X, Y, Z) to be a Pythagoras triple, where we prove that the non-linear Fermat's Diophantine equation has three different types of solutions according to the value of 𝜀². All types will be solved and discussed in terms of theorems and examples that explains how the algorithms work. [ABSTRACT FROM AUTHOR]

The objective of this study is to introduce a novel operation called pro-addition for Pythagorean triplets, which extends to various groups of Pythagorean triplets through a new generation process. This operation works in tandem with conventional addition and serves to establish either the closure property within a set or establish connections between sets belonging to different groups. It is our sincere aspiration that this article proves valuable to scholars, students, and math educators in their pursuit of exploring new avenues in this field. [ABSTRACT FROM AUTHOR]

This is a seminar paper on the article picture perfect numbers by Joseph L. Pe that was published in the journal mathematical spectrum in 2008. This paper begins with a discussion of the definition of the more familiar concept of perfect numbers, then proceeds to a discussion of the picture-perfect numbers as defined by Joseph L. Pe, and winds up with a discussion on how to obtain a picture-perfect number using the Andersen's Theorem. This paper also includes proof of the Andersen's Theorem, as well as that of the Andersen's Lemma, which are both attributed to Jens Kruce Andersen. [ABSTRACT FROM AUTHOR]

In this work, we characterize the Pythagorean orthogonality of bounded linear operators using the concept of norm attainment set and disjoint support in Hilbert spaces. [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, I address how music theory, generally, and the case of the octave more specifically, played an instrumental and under-appreciated role in the development of seventeenth-century mechanization of sensory perception. In particular, I argue that investigation into the causes of the octave, Aristotle's chosen example of a formal cause, and other musical intervals like it served to undermine the Neo-Aristotelian framework, while also providing key evidence in support of a mechanistic account of sensory perception. I begin by discussing how the case of the octave was understood in Scholastic Neo-Aristotelian philosophy and how this framework was applied in sixteenth-century music theory. Following this, I show how developments in music theory and acoustics challenged the status of formal causes and real qualities in this Neo-Aristotelian framework. Finally, I show how these developments provided a much needed detailed, empirically supported example of the mechanistic-style metaphysics of sensory perception that was prevalent in the Early Modern period, focusing primarily on Descartes. [ABSTRACT FROM AUTHOR]

In this paper, we present the Pythagorean neutrosophic interval valued fuzzy soft set. This is a generalization of the Pythagorean interval valued fuzzy soft set as well as the neutrosophic interval valued fuzzy soft set. It is discussed in this paper how an aggregated operation is used to aggregate the decision matrix of PNIVS. There are a number of extensions to the normosophic fuzzy soft sets that involve the use of multi-criteria decisionmaking. The aim of this study is to develop a score function based on aggregating TOPSIS methods in order to find ideal solutions for PNIVS that have both positive and negative values. The purpose of this study is to identify the optimal alternative under closeness conditions. It gives us the opportunity to interact with two real life problems, such as the production of ten different types of motorbikes by an automobile company. According to this set of parameters, a motorbike is determined by the fuel tank capacity, better styling, a better price, more mileage, durable, and other factors that determine how a customer can choose which bike to buy. [ABSTRACT FROM AUTHOR]

In contrast to the Pythagorean interval valued fuzzy soft set and the neutrosophic interval valued fuzzy soft set, the Pythagorean neutrosophic interval valued fuzzy soft set is a generalization of these sets. We discuss aggregating PyNIVFS decision matrixes by using aggregated operations. The VIKOR method, which is an extension of neutrosophic fuzzy soft sets, is a powerful method for evaluating multi-criteria group decision making. The score function in this approach is based on the aggregation of the VIKOR method to a PyNIVFSpositive and negative solution. Optimal alternatives are introduced under closeness. An investment company plans to purchase some shares of the top five companies in the stock exchange to invest some money on. In order to minimize this factor, they decided to invest some of their cash in percentages of 30 dollars, 25 dollars, 20 dollars, 15 dollars and 10 dollars in accordance with the top five ranked companies in order to minimize the effect of this factor. [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]

As an important way to cultivate talents, school-enterprise cooperation is highly praised by the educational and economic circles of all countries. Vocational education spans enterprises and schools, and is an education of cross-border innovation. Vocational schools should make the people they cultivate meet the needs of the market and enterprises, and carry out in-depth school-enterprise cooperation is one of the effective ways. The school-enterprise cooperation between vocational colleges and enterprises makes vocational education more targeted and practical, and plays an important role in the training of skilled talents. It is not only an inevitable requirement for enterprise development and participation in competition, but also an important direction of China's vocational education reform, and is the fundamental outlet for the development of China's vocational education. The performance evaluation of school-enterprise cooperation in vocational colleges is classical multiple-attribute group decision-making (MAGDM) issues. Recently, the TODIM and VIKOR method has been used to solve MAGDM issues. The 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) are used as a tool for characterizing uncertain information during the performance evaluation of school-enterprise cooperation in vocational colleges. In this manuscript, we design the 2-tuple linguistic Pythagorean fuzzy TODIM-VIKOR(2TLPF-TODIM-VIKOR) method to solve the MAGDM under 2TLPFSs. In the end, a numerical case study for performance evaluation of school-enterprise cooperation in vocational colleges is given to validate the proposed method. [ABSTRACT FROM AUTHOR]

By the telescoping of the partial sums, the desired series converges to Graph HT ht . Prove Graph HT ht . Write Graph HT ht for the sum of the entries of vector I v i . I Proposed by Colin Defant, Princeton University, Princeton, NJ, and Pakawut Jiradilok, Massachusetts Institute of Technology, Cambridge, MA. i Let Graph HT ht be a simple polygon that is a union of triangles chosen from the infinite plane tiling by equilateral triangles of side length 1. [Extracted from the article]

A new orthogonality type in normed linear spaces, which is based on invariant inner products, is introduced. It is shown that this orthogonality has properties of existence, uniqueness, and homogeneity in general normed linear spaces. New characterizations of inner product spaces are obtained by studying properties of this orthogonality and its relation to other orthogonality types, including Birkhoff orthogonality, isosceles orthogonality, and Pythagorean orthogonality. Since this orthogonality is not additive in general, sufficient conditions for its local additivity are presented. Finally, a new geometric constant i(X) is introduced to characterize the difference between this orthogonality and isosceles orthogonality quantitatively. It is shown that 0 ≤ i (X) ≤ 1 holds in any normed linear space, i (X) = 0 iff the underlying space is an inner product space, and that the upper bound 1 is not attainable. [ABSTRACT FROM AUTHOR]

We introduce a new generalization of -congruent numbers by defining a rational -parallelogram envelope. It turns out that the existence of a rational -parallelogram envelope associated to an n is connected to the property of n being a -congruent number. Then, we study some problems related to the rational -parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of Ochiai, where only the case = π / 2 was considered. Moreover, we answer open questions raised by Ochiai in the more general setting of an arbitrary Pythagorean angle. [ABSTRACT FROM AUTHOR]

Objectives: Gauss-Markov model, the model with condition equations and Gauss-Helmert model are three classical adjustment models. Gauss-Helmert model can be seen as the combined form of the other two models. They are usually distinguished by the number of unknown parameters. However, this classification method cannot reflect the essential relationship between them. In addition, the connections between Gauss-Markov model and the conditional model has been intensively investigated, while that of Gauss-Helmert model has been ignored. Methods: Some equivalent objective functions of the adjust⁃ ment of the least-square in the context of Gauss-Markov model and the conditional model are proposed based on the projection theory and the dual optimization technique. Based on the theory of matrix analysis, this paper realizes the conversion between Gauss-Markov model and Gauss-Helmert model according to partial orthogonality of coefficient matrices, thus completing the structure of these models and revealing the mathematical relationships. Results: The least-square adjustment within Gauss-Markov model and the model with condition equations are explained geometrically by employing the linear space and projector. The equivalence of the objective functions of these two models are also expounded. Combining the Pythagorean theorem and duality theory, we extend them to four equivalent objective functions and illustrate the construction principle. The transformation between Gauss-Markov model and Gauss-Helmert model are achieved by the so-called partial-orthogonality. Conclusions: The geometric reasoning of the least-squares adjustment is analyzed once more. Two objection functions are newly proposed, and the geometric relation and connotation between objective functions are revealed. Gauss-Helmert model is not only the combined model in terms of the model appearance, these three adjustment models can actually be mutually trans⁃ formed. As a result, the understanding of the mathematical nature of adjustment models and the corresponding estimation in this paper has theoretical and practical value. [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]

Pythagorean neutrosophic sets (PNS) have been recognized as a highly effective mechanism for managing situations characterized by indeterminacy and inconsistency within decision-making procedures. This paper delves into the examination of algebraic operations performed on PNS, thereby expanding their scope of application, and enhancing their utility. In this study, we put forth a set of algebraic operations that can be applied to PNS. These operations encompass addition, multiplication, scalar multiplication, and power. These operations facilitate the efficient manipulation and combination of PNS, thereby enhancing decision-making in scenarios characterized by uncertainty and vagueness. To demonstrate the efficacy of these operations, we will present several illustrative examples accompanied by corroborating proofs. The introduction of algebraic operations enhances the capabilities of PNS, thereby creating opportunities for their practical application. [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 research proposes multicriteria decision-making (MCDM)-based real-time Mesenchymal stem cells (MSC) transfusion framework. The testing phase of the methodology denotes the ability to stick to plastic surfaces, the upregulation and downregulation of certain surface protein markers, and lastly, the ability to differentiate into various cell types. First, two scenarios of an enhanced dataset based on a medical perspective were created in the development phase to produce varying levels of emergency. Second, for real-time monitoring ofCOVID-19 patients with different emergency levels (i.e.,mild, moderate, severe, and critical), an automated triage algorithm based on a formal medical guideline is proposed, taking into account the improvement and deterioration procedures fromone level to the next. For this strategy, Einstein aggregation information under the Pythagorean probabilistic hesitant fuzzy environment (PyPHFE) is developed. Einstein operations on PyPHFE such as Einstein sum, product, scalar multiplication, and their properties are investigated. Then, several Pythagorean probabilistic hesitant fuzzy Einstein aggregation operators, namely the Pythagorean probabilistic hesitant fuzzy weighted average (PyPHFWA) operator, Pythagorean probabilistic hesitant fuzzy Einstein weighted geometric (PyPHFEWG) operator, Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted average (PyPHFEOWA) operator, Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted geometric (PyPHFEOWG) operator, Pythagorean probabilistic hesitant fuzzy Einstein hybrid average (PyPHFEHA) operator and Pythagorean probabilistic hesitant fuzzy Einstein hybrid geometric (PyPHFEHG) operator are investigated. All the above-mentioned operators are helpful in design the algorithm to tackle uncertainty in decision making problems. In last, a numerical case study of decision making is presented to demonstrate the applicability and validity of the proposed technique. Besides, the comparison of the existing and the proposed technique is established to show the effectiveness and validity of the established technique. [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]

In this paper, we introduce spherical neutrosophic matrices (SNMs) as generalization of intuitionistic fuzzy matrices, Pythagorean fuzzy matrices, picture fuzzy matrices and spherical fuzzy matrices. Some algebraic operations such as max-min, min-max, complement, algebraic sum and algebraic product are defined in SNMs and investigated. Further, scalar multiplication (nA) and exponentiation (An ) operations of a spherical neutrosophic matrix A using algebraic operations are constructed, and their desirable properties are proved. Finally, define a new operation(@) on spherical neutrosophic matrices and discuss distributive law in the case where the operations of ⊕, ⊗, ∧ and ∨ are combined each other. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets assist in handling more uncertain and vague data than fuzzy sets and intuitionistic fuzzy sets. The notion of cubic pythagorean fuzzy sets is defined by combining interval valued pythagorean fuzzy sets and pythagorean fuzzy sets In this paper, based on the notion of cubic Pythagorean fuzzy sets we initiate a new theory called cubic pythagorean fuzzy linear spaces. Inspired by the notion of Cubic linear spaces we also present P(resp. R)-union and P(resp. R) intersection of cubic pythagorean fuzzy linear spaces. The concept of internal(resp. external) cubic pythagorean fuzzy linear spaces and its properties are examined. [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.)

What is it that makes us judge two proofs of the same theorem to be the same or different? This is not an idle question: one central aspect of judging mathematics is the novelty of the mathematics presented. This is important everywhere, from the peer-review system, to assigning international prestige, to funding agencies' grant decisions. It even matters to some extent in examinations, to avoid accusations of collusion. Surprisingly, philosophers of mathematics have not paid the question of novelty much attention. In this Article, we will consider the appealing conjecture that the main ideas that make up the proof, the essence of a proof, can indeed be identified and that very different styles of proofs can share common main ideas. Further, that a particular theorem can be proved using quite different, independent main ideas. As a means of exploring whether this is plausible, we will present a number of novel proofs of the following theorem. [ABSTRACT FROM AUTHOR]

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.)