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We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schröcker and Šír 2023). The third method generalizes the dual approach of (Pottmann 1995) from planar to spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples. • Three methods to compute rational curves with rational arc length function. • Compact quaternionic formulations, extending formulas for rational PH curves. • Comparison and critical discussion of merits of each approach. • Extensive examples relating our findings to existing literature. [ABSTRACT FROM AUTHOR]

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 paper a new definition for Menger probabilistic inner product spaces is presented. In the new setting a Menger probabilistic inner product space can naturally become a Menger probabilistic normed space, and a classical inner product space can be considered as a special case of Menger probabilistic inner product spaces. An example is given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in probability. By virtue of this definition, the topological structure is discussed and some elementary properties are described in terms of the families of semi-inner products. Also, some convergence theorems and probabilistic Pythagorean theorem are given. As applications, a new fixed point theorem for nonlinear contractions in Menger probabilistic inner product spaces is established. [ABSTRACT FROM AUTHOR]

In the textile industry, selecting the optimal fabric is crucial in the design and production process. The multiple-criteria decision-making (MCDM) technique has been applied to address fabric selection problems. Because the sine trigonometry function maintains the periodicity and symmetry of the origin in nature, it satisfies the preferences of experts over multiple parameters. Considering the advantages of the sine trigonometry function, we introduce sine trigonometry Pythagorean fuzzy sets (ST-PFSs) and weighted averaging (ST-PFWA) and geometric (ST-PFWG) aggregation operators (AOs). Additionally, we propose a novel method based on ST-PFWA and ST-PFWG AOs to solve two different types of fabric selection MCDM problems. The first problem is provided by Pythagorean fuzzy numbers (PFNs), and we apply the proposed method to address the issue and conduct a comparative analysis with other PFS AOs. The second problem is offered with crisp values, and we apply a PFS linguistic term transform system to convert the values into PFNs. Then, we apply ST-PFWA and ST-PFWG AOs to settle the problem and perform a comparative analysis among different PFS linguistic terms. The complete ranking results illustrate that our proposed methodology is more efficient and reliable than previous approaches and can be used in other textile fields. [ABSTRACT FROM AUTHOR]

For 0 ≤ α ≤ 1, let Hα(x, y) be the convex weighted harmonic mean of x and y. We establish differential subordination implications of the form Hα(p(z), p(z)Θ(z) + zp0 (z)Φ(z)) ≺ h(z) ⇒ p(z) ≺ h(z), where Φ, Θ are analytic functions and h is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence. [ABSTRACT FROM AUTHOR]

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]

It seems that in the time of Euclid, very few examples of triples that satisfy the Pythagorean equality not of the form ( I p i , I q i , I r i ) or HT ht could have been known. Indeed, if I BD i and I CD i are of lengths I k i and I l i , then from the proportionalities HT ht and HT ht , it follows that HT ht and HT ht . Therefore, it was proved as a proposition of "geometric algebra."[2] Euclid's approach to the Pythagorean theorem is analyzed in the first three sections below: "Pythagorean Triples", "A Most Lucid Demonstration", and "Proposition VI.31 HT ht ." Proclus does not claim that Euclid proved VI.31 HT ht but that he insisted on a more general theorem, that is, VI.31. [Extracted from the article]

Copyright of Gazi University Journal of Gazi Educational Faculty (GUJGEF) is the property of Gazi University Journal of Gazi Educational Faculty (GUJGEF) 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 discuss the zoom in and zoom out approach to mathematics teacher education (ZIZO), which is intentionally designed to foster commognitive conflicts through the exploration of historical sources and the engagement with collective discussions. ZIZO is structured in three interconnected moments: firstly, participants are invited to immerse in primary sources, as a means of shaking their comfort zones regarding legitimate mathematical procedures and criteria; secondly, they engage in plenary discussions on the historical contextualisation of mathematical ideas, supported by the study of secondary sources; finally, they return to the source to unveil specific aspects of historical practices. This approach is part of a broader proposal, which builds upon participants' collective experiences to stimulate the development of a problematising stance toward knowledge production and teaching of mathematics. We exemplify ZIZO by a case in which the study of books I and II of Euclid's Elements was used to problematise usual approaches to the teaching of Pythagoras' theorem and areas in Brazilian secondary school. Our findings suggest that the engagement with Euclidean practices led teachers to problematise their views on the teaching of areas as geometrical magnitudes and as measurements, as well as the pedagogical uses of manipulable materials. These results highlight ZIZO's potential for teacher education, especially regarding the problematisation of views on what mathematics is and how it is learned and taught. [ABSTRACT FROM AUTHOR]

In this study, the mathematical thinking processes on geometry of 6th, 7th and 8th grade students who have the same geometric thinking levels were examined. Two students from each grade level were selected for the study. The geometric thinking levels of the students were determined as the third level. In addition, the algebraic thinking levels of the students were discussed. The three worlds of mathematics was used in the research. The mathematical thinking processes of the students were examined in terms of the embodied world, the proceptual world and the formal world. The Pythagorean Theorem was chosen as the geometry subject. Two-stage semi-structured interviews were conducted with the students. In the first part of the interviews, the verbal expression of the Pythagorean Theorem was directed to the students. In the second part, an activity was presented for them to discover the theorem in a real-life situation. As a result, while the students had difficulty in explaining the theorem in the verbal expression, they were able to express it more easily in a real-life situation. 7th and 8th grade students were more successful than 6th grade students in demonstrating the processes of the three worlds of mathematics. [ABSTRACT FROM AUTHOR]

This study aims to determine the effect of the cognitive conflict approach on the elimination of misconceptions in square root numbers. For this purpose, this study was conducted with 8th-grade students of a secondary school in a region of Turkey. A three tiered diagnostic test was used to determine the impact of the 5-step cognitive conflict approach in the study. This test was used as a pre-test before applications and the same test was used as a post-test after applications in the same class. The test results of the students were divided into 4 groups called "misconception", "lack of knowledge", "lack of confidence", "scientific knowledge", and these groups were also divided into categories (A, B, ...). 5-step cognitive conflict approach made some transformations between categories. The greatest transformation was the transformation of misconceptions into scientific knowledge. However, it was seen that some misconceptions (B category) continued after the applications. In light of these findings, it is suggested that teachers pay attention to the mathematical language and mathematical representations they use during instruction. Moreover, the topic of root numbers should definitely be explained using the Pythagorean Theorem, and the guidebooks should be revised accordingly. [ABSTRACT FROM AUTHOR]

Each of the Pythagorean means corresponds to the centroid of a region in the Cartesian plane. We show how this insight leads to a short proof of a result that generalizes the HM-GM-AM inequality. [ABSTRACT FROM AUTHOR]

In the past few decades polynomial curves with Pythagorean hodograph (PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining, and robotics. This work deals with classes of PH curves built upon exponential-polynomial spaces (EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the Bézier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau–like algorithm and two recently proposed methods: Woźny and Chudy’s algorithm and the dynamic evaluation procedure by Yang and Hong. [ABSTRACT FROM AUTHOR]

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can describe both interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs) and Pythagorean fuzzy (PF) sets (PFSs) simultaneously for addressing information ambiguities. Since an aggregation operator (AO) is a significant mathematical approach in decision-making (DM) problems, this article presents some novel Bonferroni mean (BM) and weighted Bonferroni mean averaging operators between CPF-numbers (CPFNs) for aggregating the different preferences of the decision-makers. Then, using the proposed AOs, we develop a DM approach under the CPF environment and demonstrated it with a numerical example. Furthermore, a comparative study between the proposed and existing methods has been performed to demonstrate the practicality and efficiency of the proposed DM approach. [ABSTRACT FROM AUTHOR]

In this article, we present the concept of Pythagorean fuzzy threshold graphs (FTGs) which are generalizations of intuitionistic FTGs. We show that Pythagorean FTGs do not contain Pythagorean fuzzy alternating (FA) 4 - cycle. Pythagorean FTGs can be constructed by consecutively adding an isolated node or a dominating node. We propose that Pythagorean FTGs are Pythagorean fuzzy split graphs (SGs). Also threshold dimension and threshold partition number of Pythagorean fuzzy graphs (PFGs) and some basic theorems on threshold dimension and threshold partition number have been presented. Finally, we discuss the implementations of Pythagorean FTGs in the gas supply allocation problem and traffic flow problem. By means of implementations, we observe that Pythagorean FTGs are applicable to a greater extent than intuitionistic FTGs to deal with uncertainty and vagueness. [ABSTRACT FROM AUTHOR]

This study aims to describe learning media to facilitate students in constructing an understanding of the Pythagorean theorem. This study describes a learning media product in the form of a puzzle called Pupphy (Pythagorean Proving Puzzle). Pupphy is a puzzle proving the Pythagorean theorem on the Pythagorean theorem material for grade VIII students in junior high school. This media is different from the others because it is made from environmentally friendly materials that are easily found in the surrounding environment and can also be made by students themselves. This research uses a qualitative approach that uses narrative data and aims to describe a medium that can be used in learning mathematics using Pythagorean material as an innovative learning method. Pupphy is packaged in an interactive animation form that is equipped with an evaluation of the understanding of the Pythagorean theorem so that it allows students to carry out individual evaluations of competency achievements related to the Pythagorean theorem. The structure of the material in Pupphy consists of an explanation of the elements of a right triangle, an animation of the discovery of the Pythagorean theorem formula, and the application of this formula to right triangles. to find out the feasibility of a test conducted by media experts and material experts and also conducted on students. [ABSTRACT FROM AUTHOR]

In daily life, decision-making (DM) problem is a complicated work related to uncertainties and vagueness. To overcome these imprecisions, many fuzzy sets and theories have been presented by different scholars. Probabilistic models are the communal models proposed for the management of uncertainties. On the other hand, if these uncertainties are not probabilistic in nature, then other models such as fuzzy linguistic and fuzzy logic are developed. Here, a new approach known as the complex Pythagorean fuzzy Maclaurin symmetric mean (CPFMSM) operator is used to handle these uncertainties in DM issues. This complex Pythagorean fuzzy set (CPFS) is a modified form of the Pythagorean fuzzy set (PFS) and of the complex intuitionistic fuzzy set (CIFS). The aggregation operators have the ability to combine different sources of information. Therefore, an aggregation operator known as the MSM operator is utilized under the complex Pythagorean fuzzy (CPF) environment to extend the theory and applications of traditional MSM. For this purpose, we devised new operators known as CPFMSM and CPF dual Maclaurin symmetric mean (CPFDMSM) to aggregate CPF data. To evaluate an emergency program, the MAGDM approach is used, which is based on the newly introduced operators. Furthermore, the viability and applicability of the propounded method are certified by a detailed analysis with the other approaches researched in the past. [ABSTRACT FROM AUTHOR]

Continuing inductively, we note that the probability that Graph HT ht is Graph HT ht where we have used the fact that there are 6 ways to order I u i , I v i , and I w i . [Extracted from the article]

The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group W relative to its normal subgroup S by defining a PFSG of W S . At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved. [ABSTRACT FROM AUTHOR]

This paper revisits the topic of Pythagorean triples with a different perspective. While numerous methods were explored to generate Pythagorean triples, none of them is whole in terms of producing all of the triples without repetitions. Indeed, many current methods focus on producing primitive triples but there does not exist multiple triples. We explores a new formation of Primitive Pythagorean triple for an even and odd shorter leg, which helps to investigate multiple primitive triples and some conjectures based on graphical representation. We genuinely hope researchers, students, and instructors of mathematics will enjoy this article and search for new directions. [ABSTRACT FROM AUTHOR]

Cache replacement policy (CRP) in content-centric network (CCN) can reduce cache redundancy, optimize cache utility, and improve network performance. When assessing the CRPs in CCN, it is often full of great uncertainty. Set pair analysis (SPA) is a pioneering uncertainty theory, which consists of three components of the connection number (CN), and overlaps with Pythagorean fuzzy set. The goal of this article is to concentrate on multi-criteria group decision-making (MCGDM) method under Pythagorean fuzzy SPA environment. For it, some revised Pythagorean fuzzy aggregation operators are given for integrating multiple group information as one. Then, the CN has been constructed, and the distance measure between CNs is proposed. Later, the CN score function based distance measure is introduced for dealing with the comparison issue. In addition, the combined weight determining method by fusing the subjective weight preference and objective weight technology (water-filling theory) is shown. Subsequently, the PRSRV MCGDM approach fusing with aggregation operator, distance measure, score function and combined weight is developed for evaluating CRPs. In the end, a comparison with some existing approaches indicates that the developed approach has strong data adaptability. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Interval valued q- Rung Orthopair Hesitant fuzzy sets (IVq-ROHFS) with motivation of Interval valued pythagorean Hesitant fuzzy sets [44] as a new concept. Then, we give some basic operations as complement, union, intersection, addition, scalar multiplication, scalar power. Also, we combine to (IVq-ROHFS) and choquet integral together with aggregating operators and develop to Interval valued q- rung orthopair hesitant fuzzy Choquet averaging operator (IVq-ROHCA) and Interval valued q- rung orthopair hesitant fuzzy Choquet geometric operator (IVq-ROHCG). Then, we offer to indicate soft approach of proposed IVq-ROHCA and IVq-ROHCG an example adopted from Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS (IVIHCI) [43]. The obtained results are agreement with IVIHCI but presented IVq-ROHCA and IVq-ROHCG have more advantages than existing structures as Interval-valued intuitionistic hesitant fuzzy sets (IVIHFS), interval-valued Pythagorean Hesitant fuzzy sets (IVPHFS) owing to reasons changing according to need, requirement, prefer of decision makers and moreover because of IVIHFS and IVPHFS are special cases of IVq-ROHFS. It is open from comparative analysis that while some of offered approaches like IVIHFS, IVPHFS etc. are giving no solution for some values, our operators present to needed results. [ABSTRACT FROM AUTHOR]

The Greek philosopher Pythagoras swam in math, leading devotees to believe in the "harmony of the spheres", a relation of the vibrations and distance of celestial bodies from earth to string length-ratios. This idea, that numerical ratios are present in both math and music, was taken by Johannes Kepler and further extended into geometry. A poster in the back of my high school math classroom announced MATH Is EVERYWHERE. [Extracted from the article]

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.

Wepresent a proof of Apollonius's theorem via Heron's formula. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article published in the International Journal of Fuzzy Systems. It states that certain tables and a step in the original publication contained incorrect data, and provides the correct data in the correction notice. The corrected tables present the results of a mathematical operator called 2TLPFWA, showing values for lambda and omega combinations for different variables. The purpose and context of these calculations are not clear from the given text. The document also includes a table and calculations related to attribute solutions, providing values for different attributes and their corresponding solutions for two different scenarios. The document is authored by Yangjingyu Zhang, Qiang Cai, Guiwu Wei, Hongjun Wang, and Cun Wei. [Extracted from the article]

We give a proof of Heron's formula using the Pythagorean theorem, via the medial triangle. [ABSTRACT FROM AUTHOR]

A new design of a triple band perfect metamaterial absorber based on Pythagorean fractal geometry is proposed and analyzed for terahertz sensing applications. The proposed design showed an enhanced sensing performance and achieved three intensive peaks at 33.93, 36.27, and 38.39 THz, corresponding to the absorptivity of 98.5%, 99.3%, and 99.6%, respectively. Due to the symmetrical nature of the recommended design, the structure exhibited the characteristics of independency on the incident wave angles. Furthermore, a parametric study was performed to show the effects of the change in substrate type, resonator material, and substrate thickness on the absorption spectrum. At a fixed analyte thickness (0.5 μm), the resonance frequency of the design was found to be sensitive to the refractive index of the surrounding medium. The proposed design presented three ultra-sensitive responses of 1730, 1590, and 2050 GHz/RIU with the figure of merit (FoM) of 3.20, 1.54, and 4.28, respectively, when the refractive index was changed from 1.0 to 1.4. Additionally, the metamaterial sensor showed a sensitivity of 1230, 2270, and 1580 GHz/μm at the three resonance frequencies, respectively, when it was utilized for the detection of thickness variation at a fixed analyte refractive index (RI) of 1.4. As long as the RI of the biomedical samples is between 1.3 and 1.4, the proposed sensor can be used for biomedical applications. [ABSTRACT FROM AUTHOR]

The German attack on France in May 1940 sent Alfred into a profound depression, but there were happier occasions such as the birth of a great-grandson, Frido Mann, on July 31, 1940, though Alfred and Hedwig never saw him. Hedwig liked her, and corresponded with her until her death, long after Alfred had lost interest.) The letters also contain some epistolary flirtation by Hedwig, and invitations to Harden to visit her ("Alfred wouldn't mind"), though she makes it clear that she doesn't seriously expect that he will come. It is curious that the life of Alfred Pringsheim (1850-1941) has attracted less attention from mathematicians than from literary and musical scholars. In 1939, their strict curfew was extended; that same year, Alfred got his new identity card, which, not content with a "J" stamp, also added the extra name Israel, which beginning in 1939 was given to all Jewish men with non-Jewish names, with the requirement that he henceforth sign himself Alfred Israel Pringsheim [[11], Figure 40]. [Extracted from the article]

In extreme cases, evacuation difficulties could cause casualties in a closed layout during an emergency. The existing emergency routes are designed based on the shortest path to the nearest egress in a vacant layout. This research aims to design an emergency route plan in a realistic closed layout with interior arrangements and crowds as static and dynamic impediments that cause movement divergence and misdirection that affect evacuation time. This research proposes an automated emergency route design using Cellular Automata (CA) based pedestrian simulation in a realistic layout. The simulation was integrated with the Pythagorean Theorem (PT) and Dijkstra's Algorithm (DA) to imitate human exitfinding behaviour during evacuation. The results showed that PT is viable in layouts with static obstacles, requiring 20% less travel distances and evacuation time than DA with similar experiment sets. However, the DA approach results have become on par with the PT in a layout with dynamic and static obstacles. DA outperforms PT in densely populated regions, while PT outperforms DA in less populated regions. (Received in March 2022, accepted in July 2022. This paper was with the authors 1 month for 1 revision.) [ABSTRACT FROM AUTHOR]

Copyright of Logos: A Journal, of Religion, Philosophy Comparative Cultural Studies & Art (08687692) is the property of Logos 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 introduces the generalized q-rung neutrosophic soft set (GqRNSSS) theory and its use to solve actual problems. We also define a few operations that make use of the GqRNSSS. The GqRNSSS is constructed by generalizing both the Pythagorean neutrosophic soft set (PyNSSS) and Pythagorean fuzzy soft set (PyFSS). We give a method for agricultural output that is based on the proposed similarity measure of GqRNSSS. If two GqRNSSS are compared, it can be determined whether or not a person produces good agricultural output. We support a strategy for dealing with the decision-making (DM) problem that makes use of the generalized q-rung soft set model. In this article, we discuss the application of a similarity measure between two GqRNSSS in agricultural output. Show how they can be successfully applied to challenges with uncertainty. [ABSTRACT FROM AUTHOR]

For the multiattribute group decision-making problem in an interval Pythagorean fuzzy environment, the existing experts and scholars have extended the weighted average (WA), ordered weighted average (OWA), generalized ordered weighted average (GOWA), weighted ordered weighted average (WOWA), and other operators to interval fuzzy environment, while the research on the application and promotion of interval Pythagorean fuzzy with generalized weighted ordered weighted average (GWOWA) operator has not been carried out, GWOWA operator not only retains the advantages of WOWA operator but also introduces artificial variables, which increases the ability of decision-makers to control the aggregation of fuzzy information. Therefore, the GWOWA operator model based on interval Pythagorean fuzzy sets is constructed. First, it is proved that interval Pythagorean fuzzy generalized weighted average operator (IVPFGWA) and interval Pythagorean fuzzy generalized ordered weighted average operator (IVPFGOWA) are special cases of IVPFGWOWA operator, and their idempotence, monotonicity, and boundedness are proved; second, a group decision-making method based on interval Pythagorean fuzzy GWOWA operator is presented. Finally, an example is given to illustrate the effectiveness and scientificity of this method. It is found that the interval Pythagorean fuzzy decision-making method of the GWOWA operator alleviates the loss of information in the decision-making process to a great extent. At the same time, with the increase in the value of artificial variables, the gap between the best scheme and other schemes continues to increase, making the decision-making results more obvious, scientific, and accurate. [ABSTRACT FROM AUTHOR]

The concept of linear Diophantine fuzzy set (LDFS) is a new mathematical tool for optimization, soft computing, and decision analysis. The aim of this article is to extend the notion of graph theory towards LDFSs. We initiate the idea of linear Diophantine fuzzy graph (LDF-graph) as a generalization of certain theoretical concepts including, q-rung orthopair fuzzy graph, Pythagorean fuzzy graph, and intuitionistic fuzzy graph. We extend certain properties of crisp graph theory towards LDF-graph including, composition, join, and union of LDF-graphs. We elucidate these operations with various illustrations. We analyze some interesting results that the composition of two LDF-graphs is a LDF-graph, cartesian product of two LDF-graphs is a LDF-graph, and the join of two LDF-graphs is a LDF-graph. We describe the idea of homomorphisms for LDF-graphs. We observe the equivalence relation via an isomorphism between LDF-graphs. Some significant results related to complement of LDF-graph are also investigated. Lastly, an algorithm based on LDFSs and LDF-relations is proposed for decision-making problems. A numerical example of medical diagnosis application is presented based on proposed approach. [ABSTRACT FROM AUTHOR]

Bonferroni means (BM) operator is the extended form of the arithmetic mean operator, used for simplifying non-dominant and non-feasible problems diagnosed in genuine life scenarios. A lot of aggregation operators are the specific parts of the BM operators under the consideration of different values of parameters which are the main parts of the BM operators. In the presence of the BM operator and a very well-known conception in the scenario of fuzzy set, called complex Pythagorean fuzzy (CPF) setting, the objective of this scenario is to diagnose the CPF power BM (CPFPBM) operator and utilize their beneficial results with important properties. Moreover, a multi-attribute decision-making (MADM) technique is evaluated in the presence of invented operators for CPF settings. In the last of this study, we diagnosed the superiority and efficiency of the invented works with the help of sensitive analysis and graphical illustrations to enhance the gap of the research works. [ABSTRACT FROM AUTHOR]

This paper is devoted to discussing the reverse triple I method based on the Pythagorean fuzzy set (PFS). We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The reverse triple I methods for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including the robustness, continuity and reversibility. Finally, a practical problem is provided to illustrate the effectiveness of the reverse triple I method for PFMP in decision-making problems. The advantages of the new method over existing methods are also expounded. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so the novel methods are more logical, can better deal with the uncertain problems in complex decision-making environments and can completely reflect the decision-making opinions of decision-makers. [ABSTRACT FROM AUTHOR]

In psychology and sociology fields, the information is either dependent or independent and sometimes requires combinations of both. The existing structures like the fuzzy set, intuitionistic fuzzy set, and Pythagorean set have limitations when the information requires combinations of dependent and independent components. To overcome these limitations, we introduce the concept of a simplified intuitionistic neutrosophic hypersoft set. We present some properties of the correlation coefficient, weighted correlation coefficient, and aggregation operators on a simplified intuitionistic hypersoft set. Finally, we develop an algorithm and illustrate with a case study for identifying the leader; who can bring changes to society in the socio-political context. [ABSTRACT FROM AUTHOR]

In this paper, after a brief survey of the discovery of the Pythagorean theorem in ancient civilisations, we present eight historical proofs of this theorem in detail: the proofs of Euclid and Ptolemy, three proofs by Thabit ibn Qurra, the proof attributed to Leonardo da Vinci, U.S. president James A. Garfield’s proof, and Albert Einstein’s boyhood proof. [ABSTRACT FROM AUTHOR]

This paper is devoted to the research of full implication triple I method under Pythagorean fuzzy environment. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The full implication triple I method for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, the properties of full implication triple I method of PFMP and PFMT models including the robustness, continuity and reversibility are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in medical diagnosis. The advantages of the new method are also explained. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so they can more accurately and completely express decision information. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new set called Pythagorean neutrosophic beta-open set with this concept, and we introduce interior and closure of Pythagorean neutrosophic beta-open set in a Pythagorean neutrosophic topological spaces by utilizing beta-open set and we introduce the chii, ii, iiibeta-spaces and di, ii, iiibeta-spaces from the pair of distinct points and we have derived the necessary and sufficient conditions by utilizing beta-open sets. We also go through some containment relations for interiors and closures of beta-open sets and studied some of their characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of fuzzy Neutrosophic set called n-Cylindrical fuzzy Neutrosophic set (n-CyFNS), with I as independent neutrosophic component. The n-CyFNS can be claimed as the largest extension of fuzzy sets. In n-CyFNS, the degree of positive, neutral and negative membership functions are satisfying the condition, 0≤ βA(x) ≤1 and 0≤ αA n(x) + γA n(x) ≤ 1, n>1, is an integer. Also the distance between two n-CyFNS and its properties are also defined. Along with basic operations on n-CyFNSs, we put forward two concepts-Neutrosophic affinity degree & Neutrosophic similarity index which is used to compare and correlate n-CyFNSs respectively. A comparison is made in the n-CyFNS environment using the existing correlation measures to check its reliability. [ABSTRACT FROM AUTHOR]

The objective of any decision-making method in a plithogenic environment is to make an optimal ranking of the alternatives subjected to core essential criteria. The preference for plithogenic decision-making methods is gaining momentum in recent times as plithogenic representations are more comprehensive and efficient in handling uncertain and imprecise decision-making data. In this paper, a plithogenic CRITIC-MAIRCA decision-making model is developed and applied to decisionmaking on livestock feeding stuff. A total of 20 alternatives under three major feed categories of green fodder, subsidiary fodder and concentrate feed are ranked using MAIRCA (The Multi Attributive Ideal-Real Comparative Analysis) method and the criterion weights are determined using the method of CRITIC (CRiteria Importance Through Inter-criteria Correlation). The final results of the plithogenic ranking are compared with fuzzy and crisp ranking methods and it is observed that the plithogenic CRITIC-MAIRCA method is highly efficient in making a feasible ranking. [ABSTRACT FROM AUTHOR]

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