Search

Plato used mathematics extensively in his account of the cosmos in the Timaeus, but as he did not use equations, but did use geometry, harmony and according to some, numerology, it has not been clear how or to what effect he used mathematics. This paper argues that the relationship between mathematics and cosmology is not atemporally evident and that Plato's use of mathematics was an open and rational possibility in his context, though that sort of use of mathematics has subsequently been superseded as science has progressed. I argue that there is a philosophically and historically meaningful space between 'primitive' or unreflective uses of mathematics and the modern conception of how mathematics relates to cosmology. Plato's use of mathematics in the Timaeus enabled the cosmos to be as good as it could be, allowed the demiurge a rational choice (of which planetary orbits and which atomic shapes to instantiate) and allowed Timaeus to give an account of the cosmos (where if the demiurge did not have such a rational choice he would not have been able to do so). I also argue that within this space it is both meaningful and important to differentiate between Pythagorean and Platonic uses of number and that we need to reject the idea of 'Pythagorean/Platonic number mysticism'. Plato's use of number in the Timaeus was not mystical even though it does not match modern usage. [ABSTRACT FROM AUTHOR]

To eliminate racist prejudices, it is necessary to identify the root cause(s) of racism. American slavery preceded racism, and it was closely associated with genocide. Accordingly, we seek the unique cause of the unique event of genocide + slavery. This was initially justified by religious prejudice, rather than colour prejudice. This religious justification was weakened when many Blacks converted to Christianity, after the trans-Atlantic slave trade. The curse of Kam, using quick visual cues to characterize Blacks as inferior Christians, was inadequate. Hence, the church fell back on an ancient trick of using false history as secular justification for Christian superiority. This trick had resulted in a false history of science during the Crusades when scientific knowledge in translated Arabic texts was indiscriminately attributed to the early Greeks, without evidence. This false history enabled belief in religious superiority to mutate into a secular belief in White superiority. After colonialism, and the Aryan race conjecture, the belief in White superiority further mutated into a belief in Western civilizational superiority, openly propagated today by colonial education. Hence, to eliminate racist prejudice, it is necessary to engage simultaneously with the allied prejudices about Christian/White/Western superiority, based on the same false history of science.

This paper deals with 1) angle trisection, 2) Bhaskara’s first proof, and 3) Pythagorean theorem. The purpose of this paper is threefold. First, to show a new, direct method of trisecting the 90deg angle using unmarked straight edge and compass; secondly, to show Bhaskara’s first proof of the Pythagorean theorem (c2 = a2 + b2) as embedded in this new, direct trisection of the 90deg angle; lastly, to show the derivation of the Pythagorean theorem from this trisection of the 90deg angle. This paper employs the direct dissection method. It concludes by presenting four points: a) the concept of trisectability as distinct from concept of constructability; b) the trisection of the 90deg angle as really a new, different method; c) Bhaskara’s first proof of the Pythagorean theorem as truly embedded in this trisection of the 90deg angle and; d) another way of deriving Pythagorean theorem from this trisection of the 90deg angle.

Spatial utilization of pasture landscapes by grazing animals is often heterogeneous and driven by complex environmental and physiological interactions between plants and animals. Synergistic energy pathways can be modeled using mechanistic relationships describing resource distribution, animal behavior, metabolic energy rate, and pasture utilization using biometric sensors. The objective of this study was to demonstrate the ability to measure the energy landscape by mapping fine-scale contributions to animal energy expenditure, nutrient acquisition, growth, and overall economic profitability. We utilized animal trajectories and location fixes recorded by virtual fencing collars (Vence) daily body weight (BW) using precision livestock scales (Smartscale, C-Lock Inc.) and oxygen consumption (GreenFeeds, C-Lock Inc.) on steers (n = 127), managed as part of a broader project to evaluate environmental synergies in extensive grazing systems. Steers were allocated to one of six native grass pastures assigned to either rotational or continuous grazing strategies, and one of three stocking rates (low, moderate, high) in a 2 x 3 factorial arrangement. Space utilization was quantified as the frequency of fixes located within each cell of a 5 x 5-meter Grid and the distance between grid cells and water was calculated using the Pythagorean theorem. Movement behavior was quantified using Program R to calculate step length, the distance between two temporally continuous fix points utilizing the Pythagorean Theorem, and turn angles calculated as the cosine of the pre- and pro-ceeding step lengths where larger values indicated greater deviation from a straight trajectory. A mixed linear regression model was fitted using the lmer function in the lme4 package. Space utilization decreased with increased distance from water (P < 0.05), with rotational grazing strategies increasing concentration around water sources (P < 0.05). This indicates an opportunity to utilize precision livestock technology to facilitate pasture management and increase space utilization in grazing cattle. [ABSTRACT FROM AUTHOR]

This article discusses the work of Thābit ibn Qurra, a mathematician and astronomer from the Islamic world who made significant contributions to the development of mathematics and science. Thābit ibn Qurra provided an extension of the Pythagorean theorem for general triangles and offered an original proof of the theorem using rearrangement and juxtaposition. The article presents an extension of Thābit ibn Qurra's method to any triangle, demonstrating how the theorem can be applied to triangles with obtuse or acute angles. The article concludes with a theorem that states the relationship between the lengths of the sides of a triangle and the scalar projection of one side onto another. [Extracted from the article]

Artificial intelligence (AI)-enabled energy efficiency in buildings leverages AI to optimize energy consumption, enhancing sustainability and aiding decision-makers in selecting cost-effective and environmentally friendly solutions. Pythagorean fuzzy Z ̃ number sets, capable of handling high uncertainty, play a crucial role in representing fuzzy information in decision-making. This research introduces novel operational laws and weighted aggregation operators, focusing on neutral addition and scalar multiplication. Comprehensive investigations into the properties of these laws lead to groundbreaking aggregation operators tailored for Pythagorean fuzzy Z ̃ number information. These include weighted, ordered weighted, and hybrid neutral averaging aggregation operators, along with a novel approach for calculating attribute weights. Relationships and characteristics of these operators are explored, and an algorithm for solving multiple attribute group decision-making problems is provided. A practical example illustrates the approach's superiority, supported by comparative results. • The advanced Pythagorean fuzzy Z-number-based neutrality aggregation model is given. • Fundamental operational guidelines for neutrality AOs for PFZNs are introduced. • The theoretic algorithm to tackle MAGDM problems in the PFZN environment is proposed. • The problem of AI-enabled energy efficiency is addressed from the MAGDM aspect. • Guidance for experts when evaluating energy-efficient building solutions is offered. [ABSTRACT FROM AUTHOR]

Background: In recent years, smaller-sized (diameter < 2.5 cm) meningiomas are diagnosed due to increased cranial imaging. Symptomatic meningiomas need to be removed surgically. Therefore, it is extremely important to locate the lesion exactly to tailor the craniotomy especially if the neuro-navigation system is not available. Many hospitals located in the underdeveloped countries cannot afford the high costs of neuro-navigation equipment. Hence, it is relevant to discover low-cost associated and effective methods for lesion localization for surgery. Methods: The use of localization markers in advance can help to acquire preoperative CT images of the patients to create and calculate a three-dimensional (3D) virtual graph using a computer. With the 3D graph, spatial distance of the tumor from the markers is calculated and the tumor location projected on the scalp by the Triangle Pythagorean theorem. This enables precise localization of intracranial microlesions preoperatively. Results: The location of the tumor was consistent with that of the pre-operative virtual image, and the craniotomy was exact. The patient was discharged 3 days later without any neurological deficits. Conclusions: This method is simple and reliable, inexpensive, and accurate in the location of small-sized lesions, which can partially compensate for the lack of neuro-navigation and is suitable for widespread application in hospitals in developing countries. [ABSTRACT FROM AUTHOR]

But where popular mathematics's closest kin, popular physics, deals with the nature of concrete, if distant, things (the nature of space and time, the structure of the universe), the stuff of mathematics and philosophy is intangible abstractions. The best-known attempt I know is David Foster Wallace's I Everything and More i [[13]], a "pop-technical" introduction, as he puts it, to the mathematics and philosophy of the infinite. Even though Wallace spent time as a doctoral philosophy student at Harvard and had "developed a jones for mathematics" in high school [[14], p. 68], he was neither a trained mathematician nor a philosopher. Wallace assured the reader that it is "a piece of pop technical writing" and claimed that his own math background didn't go much beyond high school. [Extracted from the article]

Pythagorean fuzzy set (PFS) is a significant soft computing tool for tackling embedded fuzziness in decision-making. Many computing methods have been studied to facilitate the application of PFS in modeling practical problems, among which the concept of correlation coefficient is very important. This article proposes some novel methods of computing correlation between PFSs via the three characteristic parameters of PFS by incorporating the ideas of Pythagorean fuzzy deviation, variance, and covariance. These novel methods evaluate the magnitude of relationship, show the potency of correlation between the PFSs, and also indicate whether the PFSs are related in either negative or positive sense. The proposed techniques are substantiated together with some theoretical results and numerically validated to be superior in terms of reliability and accuracy compared to some similar existing techniques. Decision-making processes involving pattern recognition and career placement problems are determined using the proposed techniques. [ABSTRACT FROM AUTHOR]

This article contributes to the advancement and evolution of outranking decision-making methodologies, with a novel essay on the ELimination and Choice Translating REality (ELECTRE) family of methods. Its primary target is to unfold the constituents and expound the implementation of the ELECTRE II method for group decision making in complex Pythagorean fuzzy framework. This results in the complex Pythagorean fuzzy ELECTRE II method. By inception, it is intrinsically superior to models using one-dimensional data. It is designed to perform the pairwise comparisons of the alternatives using the core notions of concordance, discordance and indifferent sets, which is then followed by the construction of complex Pythagorean fuzzy concordance and discordance matrices. Further, the strong and weak outranking relations are developed by the comparison of concordance and discordance indices with the concordance and discordance levels. Later, the forward, reverse and average rankings of the alternatives are computed by the dint of strong and weak outranking graphs. This methodology is supported by a case study for the selection of wastewater treatment process, and by a numerical example for the selection of the best cloud solution for a big data project. Its consistency is confirmed by an effectiveness test and comparison analysis with the Pythagorean fuzzy ELECTRE II and complex Pythagorean fuzzy ELECTRE I methods. [ABSTRACT FROM AUTHOR]

Pythagorean Transportation Problem (PTP) is applied on Capacity and Requirement of commodities taken from one source to the different destinations. In this paper Decagonal Pythagorean Fuzzy Numbers using Transportation problem by ranking method and Centroid Ranking Technique and Proposed Ranking Method.. The transportation cost can be minimized by using of Proposed Ranking Method under Score function Method. The procedure is illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

A dynamic uncertain causality graph (DUCG) is a probabilistic graphical model for knowledge representation and reasoning, which has been widely used in many areas, such as probabilistic safety assessment, medical diagnosis, and fault diagnosis. However, the convention DUCG model fails to model experts' knowledge precisely because knowledge parameters were crisp numbers or fuzzy numbers. In reality, domain experts tend to use linguistic terms to express their judgements due to professional limitations and information deficiency. To overcome the shortcomings of DUCGs, this article proposes a new type of DUCG model by integrating Pythagorean uncertain linguistic sets (PULSs) and the evaluation based on the distance from average solution (EDAS) method. In particular, experts express knowledge parameters in the form of the PULSs, which can depict the uncertainty and vagueness of expert knowledge. Furthermore, this model gathers the evaluations of experts on knowledge parameters and handles conflicting opinions among them. Moreover, a reasoning algorithm based on the EDAS method is proposed to improve the reliability and intelligence of expert systems. Lastly, an industrial example concerning the root cause analysis of abnormal aluminum electrolysis cell condition is provided to demonstrate the proposed DUCG model. [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.)

In this article, the concept of parametric interval valued Pythagorean number (PIVPN) has been introduced, which is an extended version of Pythagorean number (PN). Here, a new score and accuracy function have been innovated in the PIVPN environment along with the De-Pythagorean value concept. The new tool and techniques have been fruitfully applied to two realistic problems, namely the networking critical path model (CPM) problem and the multicriteria group decision making problem (MCGDM) problem. In order to solve the MCGDM problem, we have prepared Parametric Interval valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO) and Parametric Interval valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO) operator in PIVPN environment. Finally, sensitivity analysis and industrious comprehensive numerical simulations have been performed to identify the reliability, efficiency, and usefulness of this novel work. In this article, we have shown that PIVPNs are a more well-organized representation to grip a real-life problem, and they can handle inconsistent conditions in a better compatible way in comparison to the other existing methods. [ABSTRACT FROM AUTHOR]

A sustainable supply chain (SSC) is vital for company's sustainability success, so it is imperative to identify and prioritize SSC's design requirements (DRs) for better SSC planning. For customer-centric markets, the customer requirements (CRs) need to be integrated into SSC's DRs. This paper thus proposes a customer-centric approach based on Analytic Network Process (ANP), Quality Function Deployment (QFD), Grey Relational Analysis (GRA), and Pythagorean Fuzzy Set (PFS) to rank SSC's DRs, considering CRs and information ambiguity. The PFS is combined with ANP, QFD, and GRA to better handle uncertainty in the SSC. The Pythagorean fuzzy ANP is applied to analyze the correlations among the sustainable CRs and determine the corresponding weights. The sustainable CRs are transformed into the DRs using the Pythagorean fuzzy QFD. The relationships among the resulting DRs are analyzed through Pythagorean fuzzy GRA to prioritize DRs. The approach is validated through a case study. The results obtained in this paper shows that the proposed method is efficient to prioritize DRs of SSC with the consideration of sustainable CRs under uncertain environment. The novelties of proposed method are that it not only offers a customer-oriented SSC planning method through the integration of ANP, QFD and GRA, but also can reflect the uncertain information with a broader membership representation space via PFSs. Based on the proposed method, the decision-maker can conduct comprehensive analysis to prioritize DRs and design appropriate SSC to fulfill CRs under uncertain environment. [ABSTRACT FROM AUTHOR]

We bound the Pythagoras number of a real projective subvariety: the smallest positive integer r such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most r squares. Enhancing existing methods, we exhibit three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant of a projective subvariety called the quadratic persistence. Defined by projecting away from points, this numerical invariant is closely related to the linear syzygies of the variety. In addition, we classify the projective subvarieties of maximal and almost-maximal quadratic persistence, and determine their Pythagoras numbers. [ABSTRACT FROM AUTHOR]

We first propose the normal Pythagorean neutrosophic set (NPNS) in this paper,which synthesizes the distribution of the incompleteness, indeterminacy, and inconsistency of the Pythagorean neutrosophic set (PNS) and normal fuzzy number. We also define some properties of NPNS. For solving the decision-making problem of the nonstrictly independent and interacting attributes, two kinds of NPNS Choquet integral operators are proposed. First, the NPNS Choquet integral average (NPNSCIA) operator and the NPNS Choquet integral geometric (NPNSCIG) operator are proposed. Then, their calculating formulas are derived, their properties are discussed, and an approach for solving the interacting multi-attribute decision making based on the NPNS is studied. Finally, the two kinds of operators are applied to validate the stability of the new method. [ABSTRACT FROM AUTHOR]

Hence HT ht is constant. We might write this as HT ht . Each smaller triangle has the same angles as HT ht , so the same HT ht works for all three triangles. We can divide HT ht into two smaller triangles HT ht and HT ht by dropping the altitude perpendicular to I c i . [Extracted from the article]

We establish a generalization of the Pythagorean theorem with a proof using Ptolemy's theorem. [ABSTRACT FROM AUTHOR]

Its base has length HT ht . By similar triangles, HT ht . The images of segments orthogonal to the major axis, like HT ht and HT ht , do not change length. In Figure 5, left, HT ht is the polar of the point at infinity on HT ht , and vice versa. [Extracted from the article]

The aim of this paper is to develop many characterizations of Pentapartitioned Neutrosophic Pythagorean (PNP) strongly irresolvable spaces and its properties is also studied. Several characterizations of Pentapartitioned Neutrosophic Pythagorean strongly irresolvable spaces are investigated in this study. Also examined are the conditions under which Pentapartitioned Neutrosophic Pythagorean strongly irresolvable spaces become Pentapartitioned Neutrosophic Pythagorean first category spaces, Pentapartitioned Neutrosophic Pythagorean Baire spaces, and Pentapartitioned Baire spaces. [ABSTRACT FROM AUTHOR]

Any decision-making method involves some degree of uncertainity. To deal with the uncertain environment of collective decision making, various tools and modern techniques have been proposed. Among those techniques, Pythagorean Fuzzy sets are one of the most recently using approach for dealing with imprecision. These sets enhance intuitionistic fuzzy sets with a wider range of applications, which is the reason for looking into their applicability in dealing with the problem of career placements. In this paper, we examine at the concept of Pythagorean Fuzzy sets and some definitions about score and accuracy functions of fuzzy sets. Some Pythagorean Fuzzy set properties are discussed. The concept of a relation is established in a Pythagorean Fuzzy set termed as Pythagorean Fuzzy Relation, which is accompanied by numerical examples. Finally, a career placement decision-making process on the proposed Pythagorean fuzzy relation called min-max-min composition can be used to determine the appropriate suitability of careers to the applicants on the basis of skills of academic performance. [ABSTRACT FROM AUTHOR]

In this study, a cutting stock problem is addressed to determine the width/length of the wooden boards and select lumber in standard lengths for cutting a cable spool. A nonlinear mathematical model is introduced using Pythagoras' theorem. The aim is to minimize the total length of lumber used and equivalently the total amount of wood wasted. To reduce the computational burden, the mathematical model is decomposed into two submodels for sizing and cutting decisions, and a two‐stage decomposition algorithm is proposed for solving the submodels subsequently. A simulated annealing metaheuristic combining the first‐fit decreasing and increasing techniques (SA‐FFD/I) is proposed to show the computational efficiency of the decomposition approach. The savings on the total length of lumber used and the total amount of wood wasted in production are achieved by the decomposition algorithm, which is 8% and 86.4% on average compared to the SA‐FFD/I heuristic. Accordingly, a numerical analysis is conducted on a real case to assess how capacity load and demand pattern scenarios impact the solution. The ratio between the total amount of wood waste and the total length of lumber does not exceed 2.54% for a weekly planning horizon. [ABSTRACT FROM AUTHOR]

This paper is devoted to introduce a novel concept known as restricted neutrosophic set (RNS) as another subclass of neutrosophic set (NS). The purpose of introducing the notion of RNS is to give a new mathematical theory that is more promising and purposeful than the existing fuzzy-centric theories to solve the uncertainty based real-world problems in a lucid manner. From decision-makers point of view, the new mathematical tool can be viewed as a direct extension of Pythagorean neutrosophic set (PNS). The PNS has its own inherent limitation for which the decision-makers can't answer a certain type of problem. For example, in a certain problem, if we consider the degree of truth-membership () A Tx=0.8, degree of indeterminate-membership () 0.9 A I x =, and the degree of falsitymembership () A Fx=0.8, then it gives an absurd result under PNS. To remove such kind of absurdity, there is a demand to introduce another superior set-theoretical concept that provides more information for the decisionmakers. This gives rise to the introduction of RNS. In RNS, we choose any value belongs to [0,1] for the three membership degrees so that their product always limited to 1. So, the beauty of RNS is that it can accommodate more information within small range with relaxed membership values i.e under RNS we can consider the maximum membership triplet as () 1,1,1. Undoubtedly the RNS gives more compact set-theoretical model to describe imprecise knowledge with ease. Finally, a decision-making approach based algorithm is introduced and applied to solve medical diagnosis problem. [ABSTRACT FROM AUTHOR]

The Pythagorean fuzzy soft set (PFSS) is the most influential and operative tool for maneuvering compared to the Pythagorean fuzzy set (PFS), which can accommodate the parameterization of alternatives. It is also a generalized form of intuitionistic fuzzy soft sets (IFSS), which delivers healthier and more exact valuations in the decision-making (DM) procedure. The primary purpose is to extend and propose ideas related to Einstein's ordered weighted geometric aggregation operator from fuzzy structure to PFSS structure. The core objective of this work is to present a PFSS aggregation operator, such as the Pythagorean fuzzy soft Einstein-ordered weighted geometric (PFSEOWG) operator. In addition, the basic properties of the proposed operator are introduced, such as idempotency, boundedness, and homogeneity. Moreover, a DM method based on a developed operator has been presented to solve the multiattribute group decision-making (MAGDM) problem. A real-life application of the anticipated method has been offered for a capitalist to choose the most delicate business to finance his money. Finally, a brief comparative analysis with some current methods demonstrates the proposed approach's effectiveness and reliability. [ABSTRACT FROM AUTHOR]

COVID-19 poses many challenges for hospitals around the world. Each country attempts to solve the problems in its hospitals using different methods. In Turkey, two pandemic hospitals were built in İstanbul, the most crowded province. In addition, some hospitals were designated as pandemic hospitals. This study focuses on the methods used for site selection for a pandemic hospital in Atakum, a district of Samsun City, Turkey. As a solution to the problem, initially, spatial analysis was performed using GIS to produce maps based on seven criteria obtained from the insight of an expert team. Analytic hierarchy process (AHP) augmented by interval-valued Pythagorean fuzzy numbers (PFNs) was then used to determine weights for the criteria. Distance to transportation network was the most important criterion influencing the selection process and the least significant one was the distance to fire stations. Based on the criteria weights, and five rules specified by the expert team, 13 suitable locations for a pandemic hospital were determined using GIS. The technique for order preference by similarity to ideal solution (TOPSIS) method was used to determine the final ranking of 13 alternative locations (A1–A13). A10 was identified as the most appropriate site and A11 as the least appropriate site for a pandemic hospital. Finally, sensitivity analysis was performed to investigate how changes in weight values of the criteria affect the ranking of the alternatives. [ABSTRACT FROM AUTHOR]

The present paper contains a comparative study between two non-deterministic approaches to the Delphi method: one based on Pythagorean fuzzy numbers, and the other based on the cloud model. From the empirical evidence obtained in an experimental investigation (N = 56 experts), related to the academic results forecast, it is observed that both approaches produce relatively similar results, even close to the results of an ARIMA test based on series chronological. The study reveals that both methods differ substantially in the way they input, process and output information. Advantages and disadvantages are analyzed, which must be addressed in each case. It is concluded that both approaches are viable, as alternative resources in unusual situations such as the COVID-19 pandemic, where the use of diachronic information is not always completely justified. [ABSTRACT FROM AUTHOR]

Research's goal is to identify and connect measurements and patterns in coconut palm planting types constructed by a Mexican farmer. The study was theoretically based on Ethnomathe-matics and was developed under a qualitative-ethnographic methodology in three stages: first, a participant observation was carried out to achieve familiarization with the peasant. Second, to collect the data, semi-structured interviews were implemented, and third, an ethnomathematical thematic analysis of the data was followed. The results show that the farmer builds two sowing systems (real frame and three bobbins) using non-conventional measures such as step and string, and, as a conventional unit of measurement, the meter. In addition, he applied trigonometric ratios and the Pythagorean theorem to find the height of the triangle, used in the triangular planting system, and implement it as a measurement standard. This work has mathematical potential for the teaching and learning of mathematical topics connected to the sociocultural context of students and teachers. [ABSTRACT FROM AUTHOR]

The overarching structures like intuitionistic fuzzy sets, Pythagorean fuzzy sets, m-polar fuzzy sets, and neutrosophic fuzzy sets etc. have their own inadequacies and impediments. These models are unable to do work because of their impediments in many real life situations. To overcome these deficiencies, in this paper, we introduce a set entitled Pythagorean m-polar fuzzy neutrosophic set (PmFNS), as a hybrid model of Pythagorean fuzzy set, m-polar fuzzy set and single-valued neutrosophic set. We define some notions related to PmFNS with the help of illustrations. We also present some concept of Pythagorean m-polar fuzzy neutrosophic topology alongside its leading characteristics. We render two applications of PmFNS of scarcity of water and uplifting economy ruined due to COVID-19 using TOPSIS. [ABSTRACT FROM AUTHOR]

In this paper, the authors give a comparison version of Pythagorean theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds). [ABSTRACT FROM AUTHOR]

In this study, we investigate spacelike Pythagorean normal (PN) surfaces with normal N = e1 in Minkowski 3-space. We obtain cubic, quartic and quintic spacelike PN surfaces by means of bivariate polynomials with split quaternion coefficients. In addition, maximal spacelike PN surfaces are also examined in the Minkowski 3-space. Moreover, we give some examples of the cubic, quartic and quintic spacelike PN surfaces and illustrated them. [ABSTRACT FROM AUTHOR]

This article presents a systematic review of research on language and multilingualism in mathematics education published in the South African journal Pythagoras from 1994 to 2021. This time frame was chosen as the year 1994 marked the acknowledgement of 11 official languages in the new democratic South Africa (including 9 indigenous languages), compared to only Afrikaans and English during the apartheid era. The review considers emergent themes in the included articles and examines what the articles reveal about mathematics education in South Africa. In addition to other findings, our corpus of literature indicates that research in this field of mathematics has been mostly undertaken in under-resourced schools and that research under the theme of multilingual education is at the forefront of research in South African mathematics education, while research on language policy needs more attention. Research on multilingual education in our corpus of literature also reveals great awareness of the value of seeing language as a resource, as well as the benefits that accrue when learners' home languages are taken into account in mathematics teaching and learning. The gaps in research in the field of language and multilingualism in the teaching and learning of mathematics are also noted and recommendations for future research are made. [ABSTRACT FROM AUTHOR]

We consider a modified Pythagorean identity which yields an analogue of the geometric mean theorem for obtuse triangles. Several consequences of this identity are also examined, including: the associated orthic triangle being isosceles, the nine-point center lying on a side, and the Kosnita point coinciding with a vertex. [ABSTRACT FROM AUTHOR]

This study aims to present the board game Rythmomachia as described by the mathematician and philosopher Francesco Barozzi in a manual published in Venice in 1572. Both the rules of the game and their relation to the Pythagorean tradition are described. This work is part of a research project sponsored by A.G. Leventis Foundation in Cyprus and is implemented by the Associate Professor of the Department of Byzantine and Modern Greek Studies, Irene Papadaki and the Emeritus Professor of the Department of Social Sciences and Education, Athanasios Gagatsis. [ABSTRACT FROM AUTHOR]

Copyright of Divulgaciones Matematicas is the property of Universidad del Zulia, Facultad Experimental de Ciencias 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.)