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Motivation High-throughput screening (HTS) is a vital automation technology in biomedical research in both industry and academia. The well-known Z-factor has been widely used as a gatekeeper to assure assay quality in an HTS study. However, many researchers and users may not have realized that Z-factor has major issues. Results In this article, the following four major issues are explored and demonstrated so that researchers may use the Z-factor appropriately. First, the Z-factor violates the Pythagorean theorem of statistics. Second, there is no adjustment of sampling error in the application of the Z-factor for quality control (QC) in HTS studies. Third, the expectation of the sample-based Z-factor does not exist. Fourth, the thresholds in the Z-factor-based criterion lack a theoretical basis. Here, an approach to avoid these issues was proposed and new QC criteria under homoscedasticity were constructed so that researchers can choose a statistically grounded criterion for QC in the HTS studies. We implemented this approach in an R package and demonstrated its utility in multiple CRISPR/CAS9 or siRNA HTS studies. Availability and implementation The R package qcSSMDhomo is freely available from GitHub: https://github.com/Karena6688/qcSSMDhomo. The file qcSSMDhomo_1.0.0.tar.gz (for Windows) containing qcSSMDhomo is also available at Bioinformatics online. qcSSMDhomo is distributed under the GNU General Public License. Supplementary information Supplementary data are available at Bioinformatics online. [ABSTRACT FROM AUTHOR]

The corona virus disease 2019 (COVID-19) has emerged as a fatal virus. This deadly virus has taken the whole world into clutches and many people have embraced death due to this invincible bug. The death toll is rising with every tick of time. The aspiration behind this article is to discover the preventive measure that should be taken to cope with this intangible enemy. We study the prime notions of novel sort of topology accredited Pythagorean m -polar fuzzy topology along with its prime attributes. We slightly amend the well-acknowledged multi-criteria decision analysis tool TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) to befit the proposed multi-criteria group decision making (MCGDM) problem of exploring the most effective method for curing from COVID-19 employing the proposed model. [ABSTRACT FROM AUTHOR]

The terms learner, student, and mathematician are not neutral language that describe objective reality. Instead, these words, like all language, invite the speaker and a hearer to share in a common sense of the world. In this philosophical pondering on the language that researchers in mathematics use to refer to the subjects of mathematics education (i.e., learners, students, mathematicians), I begin to identify how this language is presumptive of either inequality (in the case of learner and student) or equality (in the case of mathematician). Mathematician, however, is not univocal in its meaning, instead there are two distinct ways that mathematician has been used: one as any knower and doer of mathematics and the other as people doing research in mathematics itself. I provide a genealogical discussion of mathematician, showing roots of both meanings among the ancient Greeks, with the former, broader sense rooted in the work of Aristotle while the later, more specialized sense is rooted in the Pythagorean tradition. Which legacy of the term mathematician should mathematics education researchers champion? The philosophical work of Jacques Rancière is central to the ways in which (in)equality is used here to interrogate the for-granted status of the terms and use of learner, student, and mathematician within research in mathematics education. [ABSTRACT FROM AUTHOR]

Portable innovation assumes an imperative job in each individual in the present world. Everybody utilizing a handset as their own to get the information from their vault. These advances keep people groups from their guaranteed hazard in their everyday exercises. Here in this proposed plan insert the innovations with vehicles like autos, Cars, and so forth. It enables the people groups from specific dangers to jump at the chance to meet a mishap and met in rush hour gridlock, and so on. To detect these issues by utilizing and resolved the innovation. It works dependent on the Pythagorean hypothesis from the scientific axioms, which is utilized in the spot of finding the briefest good ways from their present area and implementing the Normalized Difference Spectral Vector (NDSV). From the satellite, the sign is transmitted to the signals via mobility, with the assistance of the beneficiary the sign is gotten and transmitted through the handset to the specific portable station (MS). VANET structures a vehicle that can be worked from long division using mobile phones, the vehicle is worked using nodemcu and motor drive, and a short time later, it has used ultrasonic sensor for assessing the partition between the vehicle and when the vehicle is met with a setback it thusly sends the message to the customer that the vehicle is been met with a disaster. The detachment is resolved in ultrasonic sensor and send the division to the Arduino mega the super shows the partition in the LCD appear and thereafter, the GSM sends the message to the customer and the sign is turned on when it is descending the wiper wipes thusly and a short time later, when the opposite vehicle goes with the high bar the front light is been reduced to certain level. By utilizing the GSM (Global System Mobile Communication) innovation, the MS imparts to the suitable VANET to build up the sign and vice versa. The VANET incorporates with the IoT gadget to get and mapping the signals to mobility. Presently, the client gets the notices for the goal separation, which said what number of signs they from the source to goal and how much time takes to arrive at their goal. And furthermore, it indicated the warning while the possibility of a mishap with their course. This data is determined with the assistance of a satellite and moved to a suitable VANET. These warnings are gotten inside the client's portable for their helpful. It helps the clients from the developing issues and gives a superior reaction to spare their time. [ABSTRACT FROM AUTHOR]

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space. [ABSTRACT FROM AUTHOR]

In this article, we combine some fundamental theories of the Pythagorean 2-tuple linguistic sets (P2TLSs) with EDAS method and design the Pythagorean 2-tuple linguistic number (P2TLN) EDAS (P2TLN-EDAS) method for multiple attribute group decision making (MAGDM) issue. Firstly, the basic concepts of P2TLSs are introduced. Next, two aggregation operators of P2TLN are defined, and then the calculation steps of EDAS method are listed briefly. Furthermore, P2TLN-EDAS method is given for MAGDM problems and computing steps are proposed in detail. Finally, a computational example related to construction safety assessment is used to expound the effectiveness of the designed method. Meanwhile, we also carried out some comparative analysis between P2TLN-EDAS method and P2TLWA/P2TLWG operators and another P2TLN-TODIM method. The results show that P2TLN-EDAS method derives the same best alternative as P2TLWA, P2TLWG operators and P2TLN-TODIM method. [ABSTRACT FROM AUTHOR]

Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout's identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the different step-intervals induce to a specific set of octaves. [ABSTRACT FROM AUTHOR]

In the multi-criteria decision-making (MCDM), multi-objective optimisation by ratio analysis (MOORA) plus the full multiplicative form (MULTIMOORA) are a useful method, which has the flexibility and robustness properties. The aim of this paper is to present a more robust method of MULTIMOORA to solve the MCDM evaluations with interval-valued Pythagorean fuzzy sets (IVPFSs). More specifically, considering the mutual support relationship among the input arguments, we firstly extend the power average (PA) and power geometric (PG) operators to accommodate the interval-valued Pythagorean fuzzy environment. For this, we propose four aggregation operators, including the interval-valued Pythagorean fuzzy power average (IVPFPA), the weighted interval-valued Pythagorean fuzzy power average (WIVPFPA), the interval-valued Pythagorean fuzzy power geometric (IVPFPG) and the weighted interval-valued Pythagorean fuzzy power geometric (WIVPFPG). With the aid of WIVPFPA and WIVPFPG, we further put forward a new robustness of the ranking method, i.e., the interval-valued Pythagorean fuzzy power average-based MULTIMOORA (IVPFPA-MULTIMOORA). Due to the availability and vast diversity among open-source electronic health record (EHR) systems, choosing an appropriate system is a challenge for many health-care institutions under the interval-valued Pythagorean fuzzy environment. Based on these above-mentioned research findings, we successfully utilise the IVPFPA-MULTIMOORA method to solve the selection problem of hospital open-source EHRs systems for MedLab in Ghana. Meanwhile, the effectiveness of the proposed method is verified by some comparative analyses. [ABSTRACT FROM AUTHOR]

Complex Pythagorean fuzzy (CPF), a worthwhile generalization of Pythagorean fuzzy set, is a powerful tool to deal with two-dimensional or periodic information. In this paper, we develop two prioritized aggregation operators (AOs) under CPF environment, namely, complex Pythagorean fuzzy prioritized weighted averaging (CPFPWA) operator and complex Pythagorean fuzzy prioritized weighted geometric (CPFPWG) operator. We consider the prioritization relationship among criteria and decision makers (DMs) to make our result more accurate as in real decision making (DM) problems, the criteria and DMs have different priority level. Further, we discuss remarkable properties of our proposed AOs. Moreover, we promote the evolution of MCDM problem by investigating an algorithm in CPF environment with its flow chart. Finally, to check the superiority and validity of proposed operators, we compare the computed results with the different existing techniques. [ABSTRACT FROM AUTHOR]

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model. [ABSTRACT FROM AUTHOR]

The selection of agricultural product supplier is an important link to optimize the supply chain management of agricultural products. Due to the uncertainty factors and the lack of decision-makers' cognition, the selection of agricultural products suppliers has become a very complex and difficult work. Therefore, in order to effectively deal with these problems, this study proposes an agricultural product supplier selection algorithm based on the Pythagorean fuzzy power Bonferroni mean operator under Pythagorean fuzzy environment. In this method, first, the power operator and Bonferroni mean operator are combined and embedded in the Pythagorean fuzzy sets to build a Pythagorean fuzzy power Bonferroni mean operator. Then, a multiattribute decision-making method based on the operator proposed in this paper is proposed. Finally, an analysis of examples of agricultural product supplier selection is given to verify the rationality and effectiveness of this method. [ABSTRACT FROM AUTHOR]

In 1934, B. Berggren first discovered the surprising result that every Pythagorean triangle is the pre-product of the triangle (3, 4, 5) given as a column vector by a product of three matrices, and that every triangle is obtained in this manner exactly once and in primitive form. In this article, we show a similar result for integer triangles with an angle of 60 and 120 degrees (also known as Eisensteinian triangles). We show that any such triangle is obtained by pre-multiplication of (7, 8, 5) or (13, 15, 7) by a product arising from a set of five matrices. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets are one of the extensions of ordinary fuzzy sets and allow a larger domain to be utilized by decision makers with respect to other extensions. Pythagorean fuzzy sets have been often used as an effective tool for handling the vagueness of multi-criteria decision making problems. Aggregation operators are a useful tool in order to collect different information provided by different sources. The objective of this paper is to develop harmonic aggregation operators for trapezoidal Pythagorean fuzzy numbers. We developed trapezoidal Pythagorean fuzzy weighted harmonic mean operator, trapezoidal Pythagorean fuzzy ordered weighted harmonic mean operator, and trapezoidal Pythagorean fuzzy hybrid harmonic mean operator. We proved some theorems for the developed operators. Finally, we presented an illustrative example using the proposed aggregation operators in order to rank the alternatives. [ABSTRACT FROM AUTHOR]

We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive examples that confirm this conjecture. As a by-product, we also obtain a classification of all spherical Pythagorean triples, previously found by Smith. [ABSTRACT FROM AUTHOR]

In the context of sustainable development, building energy conservation has become the development trend of the construction industry. The selection of energy-saving building program, as a multi-criteria decision-making (MCDM) problem, has a direct influence on the actual energy-saving effect. In this paper, an integrated MCDM method combining the extended best worst method (BWM) and Weighted Aggregated Sum Product Assessment (WASPAS) method is proposed to solve the energy-saving building program selection problem under the linguistic Pythagorean fuzzy environment. The Linguistic Pythagorean fuzzy sets (LPFSs) are used to model the uncertain evaluation information of experts. The extended BWM is developed to determine the weights of criteria, while the extended WASPAS method is proposed to determine the ranking of alternatives. To validate the applicability and reliability of the proposed method, this paper presents a numerical example of the selection problem for energy-saving building programs. Some managerial insights are also given for practitioners to use the proposed method [ABSTRACT FROM AUTHOR]

The paper offers solutions to problems and proofs of claims of different nature. Although it might be possible to solve some of the tasks and prove some claims in a more elegant way, the given solutions and proofs can be used as a motivation for studying the concepts used, and also as ideas for solving similar tasks and proving other claims. [ABSTRACT FROM AUTHOR]

Interval-valued Pythagorean fuzzy set is one of the successful extensions of the interval-valued intuitionistic fuzzy set for handling the uncertainties in the data. Under this environment, in this paper we introduce the notion of induced interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (I-IVPFEOWG) operator and induced interval-valued Pythagorean fuzzy Einstein hybrid geometric (I-IVPFEHG) operator along with their some basic properties such as, idempotency, boundedness, commutatively, monotonicity. The main advantage of the induced aggregation operators is that, these operators are more suitable to aggregate the individual preference relations into a collective preference relation. Therefore these methods play a vigorous role in daily life problems. Furthermore, a method for multi-attribute group decision-making (MAGDM) problems based on these operators was developed, and the operational procedures were explained in detail. [ABSTRACT FROM AUTHOR]

For instance, newly weaned NES pups are heavier than southern elephant seal pups, by an average of 10 kg (Le Boeuf & Laws, 1994; Noren et al., 2003; but see Carlini et al., 2001). Postweaning duration and body composition changes in Southern Elephant Seal (Mirounga leonina) pups at King George Island. Energy reserve utilization in northern elephant seal (Mirounga angustirostris) pups during the postweaning fast: Size does matter. [Extracted from the article]

Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann and Clifford algebras and to the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics. [ABSTRACT FROM AUTHOR]

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincaré disk model. Our tables have m ≥ 3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU1/1,1 Z[i]. The resulting billiard map B acts on the de Sitter space x21 + x22 − x23 = 1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We compute the invariant densities of these maps, and prove the Lagrange and Galois theorems: A complex number of unit modulus has a preperiodic (purely periodic) B-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over Q(i). Each B as above is a (m − 1)-to-1 orientation-reversing covering map of the circle, a property shared by the group character T(z) = z−(m−1). We prove that there exists a homeomorphism Φ, unique up to postcomposition with elements in a dihedral group, that conjugates B with T; in particular Φ − whose prototype is the classical Minkowski question mark function − establishes a bijection between the set of points of degree ≤ 2 over Q(i) and the torsion subgroup of the circle. We provide an explicit formula for Φ, and prove that Φ is singular and Hölder continuous with exponent log(m−1) divided by the maximal periodic mean free path in the associated billiard table. [ABSTRACT FROM AUTHOR]

n this paper, we define some new operations (A@B), (A$B), (A#B), (A# B), (A * B) of q-rung orthopair fuzzy sets. Then we discuss several properties of these operations. Further, we prove necessity and possibil- ity operations of q-rung orthopair fuzzy sets. Finally, we have identified and proved several of these properties, particularly those involving the operator A - B defined as q-rung orthopair fuzzy implication with other operations. [ABSTRACT FROM AUTHOR]

Multicriteria decision-making process explicitly evaluates multiple conflicting criteria in decision making. The conventional decision-making approaches assumed that each agent is independent, but the reality is that each agent aims to maximize personal benefit which causes a negative influence on other agents' behaviors in a real-world competitive environment. In our study, we proposed an interval-valued Pythagorean prioritized operator-based game theoretical framework to mitigate the cross-influence problem. The proposed framework considers both prioritized levels among various criteria and decision makers within five stages. Notably, the interval-valued Pythagorean fuzzy sets are supposed to express the uncertainty of experts, and the game theories are applied to optimize the combination of strategies in interactive situations. Additionally, we also provided illustrative examples to address the application of our proposed framework. In summary, we provided a human-inspired framework to represent the behavior of group decision making in the interactive environment, which is potential to simulate the process of realistic humans thinking. [ABSTRACT FROM AUTHOR]

Calculation of the angle HT ht in the triangle HT ht using the law of sines. Calculation of the angle HT ht in the triangle HT ht using the law of sines. Beforehand, by applying the Pythagorean theorem to the right triangle HT ht , it is easy to deduce an expression for the length of the hypotenuse I ge i . We obtain HT ht . [Extracted from the article]

Enheduanna's temple hymns that touch on mathematics, as well as the other poem fragments appearing in this paper, allow us to hear the story of mathematics' complex beginnings in the voice of a culture that experienced it - as much as this is possible to do in a different language and several thousand years removed from that time. For example, in I Temple Hymn 8 i , dedicated to the temple of the moon god in Ur, Enheduanna's own domain (see the section "Measuring the Heavens"), she calls the moon god by both his Sumerian name, Nanna, and his Akkadian name, Suen. Returning to I Temple Hymn 42 i , we note that the first seven lines address Ezagin, Nisaba's temple in Eresh, in the second person; the last seven lines include a description of the role and the responsibilities of the "true woman", which is the goddess Nisaba, Enheduanna herself, or both - a point addressed in the concluding section of this paper. Here is the translation of the two signature lines given in [[28]]: the person who bound this tablet together is Enheduanna my king something never before created did not this one give birth to it In the next section, we begin to unravel the connections between Mesopotamian mathematics and I Temple Hymn 42 i with a discussion about numbers - their development as mathematical entities, as well as the nature of their appearance in Enheduanna's temple hymns and in particular their use in the structure of I Temple Hymn 42 i . From the point of view of architecture and construction, that is, from the point of view of activities that require mathematical skills, Enheduanna's temple hymns, particularly I Temple Hymn 1 i , reflect a number of common features of Sumerian temples that evolved from construction practices originating in deep antiquity and therefore are likely to have been a part of the ordinances of Eridu. [Extracted from the article]

From Music to Mathematics In this expanding transdisciplinary field - one that can be approached from a wide variety of perspectives - Gareth E. Roberts's work I From Music to Mathematics: Exploring the Connections i , which is intended to serve as a university textbook, offers the content of a two-semester course on music and mathematics taught by the author. Roberts's work is about finding the mathematics in certain music and discovering how we can mathematically express and explain the structure of musical ideas and language. Roberts points here to the high complexity and multiplicity of the connections between mathematics and music and also to the potential openness of mathematics to indeterminacy and choice. [Extracted from the article]

A Pythagorean fuzzy matrix is a powerful tool for describing the vague concepts more precisely. The Pythagorean fuzzy matrix based models provide more flexibility in handling the human judgment information as compared to other fuzzy models. The objective of this paper is to apply the concept of intuitionistic fuzzy matrices to Pythagorean fuzzy matrices. In this paper, we briefly introduce the Pythagorean fuzzy matrices and some theorems and examples are applied to illustrate the performance of the proposed methods. Then we define the Hamacher scalar multiplication (n:hA) and Hamacher exponentiation (A^hn) operations on Pythagorean fuzzy matrices and investigate their algebraic properties. Furthermore, we prove some properties of necessity and possibility operators on Pythagorean fuzzy matrices. [ABSTRACT FROM AUTHOR]

A continuous g‐frame is a generalization of g‐frames and continuous frames, but they behave much differently from g‐frames due to the underlying characteristic of measure spaces. Now, continuous g‐frames have been extensively studied, while continuous g‐sequences such as continuous g‐frame sequence, g‐Riesz sequences, and continuous g‐orthonormal systems have not. This paper addresses continuous g‐sequences. It is a continuation of Zhang and Li, in Numer. Func. Anal. Opt., 40 (2019), 1268‐1290, where they dealt with g‐sequences. In terms of synthesis and Gram operator methods, we in this paper characterize continuous g‐Bessel, g‐frame, and g‐Riesz sequences, respectively, and obtain the Pythagorean theorem for continuous g‐orthonormal systems. It is worth that our results are similar to the case of g‐ones, but their proofs are nontrivial. It is because the definition of continuous g‐sequences is different from that of g‐sequences due to it involving general measure space. [ABSTRACT FROM AUTHOR]

In this paper we will give q-analogues for the Pythagorean trigonometric identity sin 2 ⁡ z + cos 2 ⁡ z = 1 {\sin^{2}z+\cos^{2}z=1} in terms of Gosper's q-trigonometry. We shall also give new q-analogues for the duplicate trigonometric identity sin ⁡ (x - y) ⁢ sin ⁡ (x + y) = sin 2 ⁡ x - sin 2 ⁡ y {\sin(x-y)\sin(x+y)=\sin^{2}x-\sin^{2}y}. Moreover, we shall give a short proof for an identity of Gosper, which was also established by Mező. The main argument of our proofs is the residue theorem applied to elliptic functions. [ABSTRACT FROM AUTHOR]

The space of Kähler metrics, on the one hand, can be approximated by subspaces of algebraic metrics, while, on the other hand, it can also be enlarged to finite‐energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite‐dimensional spaces of Fubini‐Study metrics of Kähler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of Kähler potentials can be quantized. More precisely, given a Kähler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of Kähler potentials. This has a number of applications, among them a new Lidskii‐type inequality on the space of Kähler metrics, a new approach to the rooftop envelopes and Pythagorean formulas of Kähler geometry, and approximation of finite‐energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of fuzzy set, called Pythagorean linguistic sets (PLSs), to address the preferred and nonpreferred degrees of linguistic variables. Moreover, it allows decision makers to offer effectively handle uncertain information more flexible than intuitionistic linguistic sets (ILSs) when one compares two alternatives in the process of decision making. Some of the fundamental operational laws, score, accuracy, and aggregation operators are defined, and their properties are investigated. Preference relation (PR) is a useful and efficient tool for decision making that only requires the decision makers to compare two alternatives at one time. Taking the advantages of PLSs and PRs, this paper also introduces Pythagorean linguistic preference relations (PLPRs) and studies their application. We propose an approach for group decision making using group recommendations based on consistency matrices and feedback mechanism. First, the proposed method constructs the collective consistency matrix, the weight collective PRs, and the group collective PRs. Then, it constructs a consensus relation for each expert and determines the group consensus degree (GCD) for all experts. If the GCD is smaller than a predefined threshold value, then a feedback mechanism is activated to update the PLPRs. Finally, after the GCD is greater than or equal to the predefined threshold value, we calculate the arithmetic mathematical average values of the updated group collective PR to select the most appropriate alternative. [ABSTRACT FROM AUTHOR]

A Pythagorean fuzzy soft set, an extension of intuitionistic fuzzy soft set, plays an essential role to handle the vagueness in many real-life problems. We apply this concept to graph theory, and present certain new notions including, perfectly regular Pythagorean fuzzy soft graphs (PFSGs), perfectly edge-regular PFSGs and explore some of their properties. We formulate the notion of perfectly irregular PFSGs, perfectly edge-irregular PFSGs and open neighborhood degree sum (O ˆ NDS) and closed neighborhood degree sum (C ˆ NDS) of PFSGs. Finally, we discuss some decision-making problems of PFSGs. [ABSTRACT FROM AUTHOR]

In this article, we will give a geometric interpretation of certain finite arithmetic progressions. For this purpose, we will introduce the concept of the n-regular partition { P n } of a quadrilateral. [ABSTRACT FROM AUTHOR]

Despite the common knowledge that there is something "Pythagorean" about Charles Peirce's phenomenology and classification of signs there is a manifest lack of inquiries into the matter. Perhaps there is too little to go on, as Pythagoras himself did not leave us any writings to consult. Nevertheless, much of ancient Greek philosophy bears an unmistakable Pythagorean stamp, and Iamblichus' bio - graphy of Pythagoras provides us with enough to get such inquiries started. This paper examines the development of triads, beginning with the Pythagorean one (body, soul, and intellect) and proceeds to those of Immanuel Kant (Experience, Understanding, and Reason) and Peirce's compatriot and family acquaintance Pliny Earle Chase (Motivity, Spontaneity, and Rationality). The article concludes with an examination of the various triads in Peirce's early writings, especially around the time of his discovery of Chase's "Intellectual symbolism". [ABSTRACT FROM AUTHOR]

The authors describe an art-science collaboration to devise and perform a qualitatively accurate interpretation of an elliptic-hyperbolic solution to the classical astronomical "problem of three bodies," in which three point masses execute trajectories dictated by Newton's Law of Universal Gravitation. [ABSTRACT FROM AUTHOR]

This paper presents an innovative method for solving Pythagorean fuzzy (PF) multicriteria group decision-making problems with completely unknown weight information about criteria using entropy weight model, linear programming (LP) and modified technique for ordering preference by similarity to ideal solution (TOPSIS). At first, a new distance measure for PF sets is defined considering their degree of hesitancy and based on weighted Hamming distance and Hausdorff metric. To handle the fuzziness in criteria weights, PF entropy weight model is used to find the initial weights of the criteria in PF format. Following the concept of TOPSIS, an LP model is constructed on the basis of the view point that the chosen alternative should have the smallest distance from the positive ideal solution and the largest distance from the negative ideal solution. Then, the LP model is utilized to find optimal weights of the criteria. Using the newly defined distance measure, entropy weight model and LP model, TOPSIS is extended in PF environments. The existing methods are able to find criteria weights in the form of crisp values only, whereas proposed method is able to obtain those weights in PF format. Thus, the proposed method can overcome the drawback in computing criteria weight for multicriteria group decision-making in PF environments and reduce the information loss significantly. Several numerical examples are considered and solved to validate the superiority of the proposed methodology. [ABSTRACT FROM AUTHOR]

For the multi-attribute group decision-making problem in Pythagoras environment, firstly, this paper combined the Pythagorean fuzzy number and the power-average operator to creatively expand a new improved weighted support, then based on this, it proposed a Pythagorean fuzzy interaction power-average operator with improved weighted support, and discussed the properties of the operator. Then it established a decision-making method that could reflect the interaction between decision attributes under Pythagoras fuzzy background. Finally, it was applied to the evaluation of smart cities. The case study shows that the method can solve the problem of actual multi-attribute group decision-making and can be further applied to the fields of intelligent logistics, pattern recognition and artificial intelligence. [ABSTRACT FROM AUTHOR]

To contribute to the understanding of the conceptualization processes in basic education, this work intends to identify which elements of the Pythagorean Theorem conceptual field were present in a pedagogical experience, and how these characteristics indicate learning paths. The research has a qualitative approach, with features of participant research. Having the Theory of Conceptual Fields as reference, in this study we developed, applied, monitored, and investigated teaching activities through Content Analysis. The analysis showed the concepts of the conceptual field and raised considerations about the network structure of the concepts, also the various orders and intercomplementary relations between meaning, concept, and representation in the conceptualization process. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to share with colleagues about the exploration in teaching and learning mathematics. Three most commonly used mathematical theorems and their applications in undergraduate studies are presented. The theorems are the Pythagorean theorem, the fundamental theorem of calculus, and the law of large numbers. The theorems are concrete mathematical materials to motivate and engage students with diverse backgrounds in an active learning environment. For each theorem, in addition to its brief history along with hands-on activities, critical thinking components with problem solving strategies are presented. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy graph, a broadly used extension of fuzzy and intuitionistic fuzzy graph, is helpful in representing structural relationships between several objects where the relation between these objects is vague, while the Dombi operators with operational parameters have excellent flexibility. Utilizing these two concepts, this research paper proposes the novel concept of Pythagorean Dombi fuzzy graphs (PDFGs). Basically, graph terminology is employed for introducing Pythagorean fuzzy analogs of various fundamental graphical ideas using Dombi operator. Further, under Pythagorean Dombi fuzzy environment, regular, totally regular, strongly regular and biregular graphs are defined with appropriate illustration and some of their crucial properties are examined. Meanwhile, the notion of edge regularity of PDFG is also initiated with substantial characteristics. Finally, a numerical example related to evaluation of appropriate ETL software for a business intelligence project is presented to better understand PDFGs. [ABSTRACT FROM AUTHOR]