University of IsfahanMathematics and Society2345-649310220250406Results about crossed polysquaresResults about crossed polysquares1262877910.22108/msci.2024.141402.1656FAMohammad AliDehghanizadehDepartment of Mathematics, National University of Skills (NUS), Tehran, IranJournal Article20240507Crossed polysquares are defined by Dehghanizadeh, Davvaz and Alp. Their properties and the generalization of results from intersecting squares to intersecting polysquares have been expressed and proved by them with the help of fundamental relations.
In the following, the concept of intersected polymodules and $\Gamma$-equivalent, intersected polymodules of polygroups are introduced and some properties are obtained from it. In addition, the concept of hypermultiplying fiber and crossed polysquares in the homotopy form of kernels has been studied. These results have extended the results related to intersecting squares to intersecting polysquares. In this article, homotopy crossed polysquares are studied as homotopies of cokernel, then the image of a crossed polymodule is considered and some results are proved that show the correspondence between crossed polymodules and crossed polysquares. In the continuation of the studies, we can check the results about the crossed 2-squares and then the crossed 2-polysquares. In addition, concepts about intersecting n-squares and intersecting n-polysquares can be expanded and studied.Crossed polysquares are defined by Dehghanizadeh, Davvaz and Alp. Their properties and the generalization of results from intersecting squares to intersecting polysquares have been expressed and proved by them with the help of fundamental relations.
In the following, the concept of intersected polymodules and $\Gamma$-equivalent, intersected polymodules of polygroups are introduced and some properties are obtained from it. In addition, the concept of hypermultiplying fiber and crossed polysquares in the homotopy form of kernels has been studied. These results have extended the results related to intersecting squares to intersecting polysquares. In this article, homotopy crossed polysquares are studied as homotopies of cokernel, then the image of a crossed polymodule is considered and some results are proved that show the correspondence between crossed polymodules and crossed polysquares. In the continuation of the studies, we can check the results about the crossed 2-squares and then the crossed 2-polysquares. In addition, concepts about intersecting n-squares and intersecting n-polysquares can be expanded and studied.https://math-sci.ui.ac.ir/article_28779_cab17048f5dd7cd28415505e15a11137.pdfUniversity of IsfahanMathematics and Society2345-649310220250417Generation of symmetrical patterns using discrete dynamical systemGeneration of symmetrical patterns using discrete dynamical system27462860210.22108/msci.2024.141065.1654FAMortezaBisheh-NiasarDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, P.O.Box 8731753153,
Kashan, IranJournal Article20240407Nowadays symmetrical patterns are widely used in various industries, such as jewelry design, carpet design, patterns on wallpaper, and textile design. During the design stage, designers perform most of the work manually. Therefore, the development of methods for symmetrical pattern generation is beneficial. In this paper, we are going to present some methods for generating symmetrical patterns using the discrete dynamical system. For this, the discrete dynamical system is considered as a standard iterative method, and then the general algorithm applied for Polynomiography is used for this to generate patterns. Through phase portrait, we analyze the conditions of the existence of some conventional symmetries. Several non-standard iterative methods can be employed to create a variety of visually appealing patterns. These methods include Mann iteration, Ishikawa iteration, and S-iteration which we use them. Through numerous examples, it is demonstrated that by manipulating the parameters and coefficients, it is possible to generate beautiful symmetrical patterns that have potential artistic applications.Nowadays symmetrical patterns are widely used in various industries, such as jewelry design, carpet design, patterns on wallpaper, and textile design. During the design stage, designers perform most of the work manually. Therefore, the development of methods for symmetrical pattern generation is beneficial. In this paper, we are going to present some methods for generating symmetrical patterns using the discrete dynamical system. For this, the discrete dynamical system is considered as a standard iterative method, and then the general algorithm applied for Polynomiography is used for this to generate patterns. Through phase portrait, we analyze the conditions of the existence of some conventional symmetries. Several non-standard iterative methods can be employed to create a variety of visually appealing patterns. These methods include Mann iteration, Ishikawa iteration, and S-iteration which we use them. Through numerous examples, it is demonstrated that by manipulating the parameters and coefficients, it is possible to generate beautiful symmetrical patterns that have potential artistic applications.https://math-sci.ui.ac.ir/article_28602_c8158c03aa92c45ba481a6b3ddd47934.pdfUniversity of IsfahanMathematics and Society2345-649310220250419Approximation of symmetrically reciprocal matrices using mutations in Max AlgebraApproximation of symmetrically reciprocal matrices using mutations in Max Algebra47622859110.22108/msci.2024.140663.1644FASeyed MahmoudManjeganiDepartment of Mathematics, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, IranHojrShokooh SaljooghDepartment of Mathematics, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, IranJournal Article20240208The objective of this paper is to propose a method for constructing a transitive matrix by maximizing the mutation of a symmetrically reciprocal matrix $A$, such that the resulting matrix is closest to $A$ in terms of a relative error measure. By employing this approach, the need for calculating the maximum eigenvector is eliminated, leading to faster results. Additionally, we investigate the impact of mutations on two cases of change, specifically when a single measurement is corrected or when a new alternative is added, and analyse their effectiveness in ranking.The objective of this paper is to propose a method for constructing a transitive matrix by maximizing the mutation of a symmetrically reciprocal matrix $A$, such that the resulting matrix is closest to $A$ in terms of a relative error measure. By employing this approach, the need for calculating the maximum eigenvector is eliminated, leading to faster results. Additionally, we investigate the impact of mutations on two cases of change, specifically when a single measurement is corrected or when a new alternative is added, and analyse their effectiveness in ranking.https://math-sci.ui.ac.ir/article_28591_b60cfca932fe72ee569660a293447638.pdfUniversity of IsfahanMathematics and Society2345-649310220250421Identification of communities in social networks based on game theory with stable coalitionsIdentification of communities in social networks based on game theory with stable coalitions63832863810.22108/msci.2024.140739.1647FAMohammad ValiHairanDepartment of Mathematical Sciences, Yazd University, 89195-741, Yazd, Iran0009-0005-2915-1966Ali DelavarKhalafiDepartment of Mathematical Sciences, Yazd University, 89195-741, Yazd, IranSaeidAlikhaniDepartment of Mathematical Sciences, Yazd University, 89195-741, Yazd, IranZahraNikooeinejadDepartment of Mathematical Sciences, Yazd University, 89195-741, Yazd, IranMahdiyehHasheminezhadDepartment of Mathematical Sciences, Yazd University, 89195-741, Yazd, IranJournal Article20240310Due to the availability of more data and the increase of interactive activities in social media, the identification of overlapping associations has been considered. In this paper, a game theory-based approach to identify overlapping associations is proposed. In this method, association detection is modeled as a coalition formation game. In this game, individuals in a social network are modeled as rational actors whose goal is to improve the group's utility, which is achieved by cooperating with other players and forming coalitions. Each player can join multiple alliances, and alliances with fewer players can merge into a larger alliance as long as the joining operation is conducive to the alliance's goals. Therefore, overlapping associations can be identified simultaneously. In this article, two types of methods based on cooperative and non-cooperative game theory have been discussed. The results report is analyzed based on the comparison of association methods in the form of a diagram. It can be seen that the game group and the COFOGA method perform better association.Due to the availability of more data and the increase of interactive activities in social media, the identification of overlapping associations has been considered. In this paper, a game theory-based approach to identify overlapping associations is proposed. In this method, association detection is modeled as a coalition formation game. In this game, individuals in a social network are modeled as rational actors whose goal is to improve the group's utility, which is achieved by cooperating with other players and forming coalitions. Each player can join multiple alliances, and alliances with fewer players can merge into a larger alliance as long as the joining operation is conducive to the alliance's goals. Therefore, overlapping associations can be identified simultaneously. In this article, two types of methods based on cooperative and non-cooperative game theory have been discussed. The results report is analyzed based on the comparison of association methods in the form of a diagram. It can be seen that the game group and the COFOGA method perform better association.https://math-sci.ui.ac.ir/article_28638_c18b4d636aa4deca2b99c19e104b45a0.pdfUniversity of IsfahanMathematics and Society2345-649310220250510Artificial Intelligence: A Facilitator in Enhancing Mathematics InstructionArtificial Intelligence: A Facilitator in Enhancing Mathematics Instruction851132914810.22108/msci.2025.142998.1699FANargesYaftianDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training
UniversityReyhaneNiknamDepartment of Mathematics, of , Shahid Rajaee Teacher Training UniversityJournal Article20241009Artificial Intelligence (AI) can be regarded as a companion that enhances the pathway toward a more sustainable and intelligent educational system, facilitating a more enjoyable experience for teachers. By integrating AI technologies into educational practices, both teachers and students can benefit from a more efficient teaching-learning process. To benefit from AI capabilities in the classroom, it is essential for teachers to become familiar with AI-based tools, understand their strengths and weaknesses, and then promote the culture of proper use of these new intelligent tools in the classroom. They should guide students in using AI in the right direction. Students, as novice users of AI, need more support and guidance. Therefore, they should be taught to select the information obtained through AI according to their needs and not to accept everything AI presents easily. Instead, they should carefully evaluate the information obtained with their cognitive skills, especially critical thinking. It should be remembered that AI cannot completely replace human teachers in the classroom. In other words, an AI-supported classroom makes sense with the presence of human teachers; however, AI-based tools can complement the tireless efforts of teachers. This review article is an effort to inform teachers about the operational aspects of AI in mathematics education. In this regard, it practically introduces some AI-based tools along with several examples and presents a sample of AI applications in developing adaptive learning systems.Artificial Intelligence (AI) can be regarded as a companion that enhances the pathway toward a more sustainable and intelligent educational system, facilitating a more enjoyable experience for teachers. By integrating AI technologies into educational practices, both teachers and students can benefit from a more efficient teaching-learning process. To benefit from AI capabilities in the classroom, it is essential for teachers to become familiar with AI-based tools, understand their strengths and weaknesses, and then promote the culture of proper use of these new intelligent tools in the classroom. They should guide students in using AI in the right direction. Students, as novice users of AI, need more support and guidance. Therefore, they should be taught to select the information obtained through AI according to their needs and not to accept everything AI presents easily. Instead, they should carefully evaluate the information obtained with their cognitive skills, especially critical thinking. It should be remembered that AI cannot completely replace human teachers in the classroom. In other words, an AI-supported classroom makes sense with the presence of human teachers; however, AI-based tools can complement the tireless efforts of teachers. This review article is an effort to inform teachers about the operational aspects of AI in mathematics education. In this regard, it practically introduces some AI-based tools along with several examples and presents a sample of AI applications in developing adaptive learning systems.https://math-sci.ui.ac.ir/article_29148_5fa365fd1efe245cc9b3a7064a4b9f6c.pdfUniversity of IsfahanMathematics and Society2345-649310220250630A GAP tutorial for transformational music theoryA GAP tutorial for transformational music theory1151622943210.22108/msci.2025.139767.1621FAMehdiShamsDepartment of Statistics, Faculty of Mathematical Sciences, University of Kashan, Kashan. Kashan, I. R.
IranSeyyed AliMohammadiyehDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. IranJournal Article20231112GAP (an acronym for Groups, Algorithms, and Programming) is a computational discrete algebra system that can be applied as a practical tool for transformational music theory. In particular, this software offers theorists tools that are not easily accessible elsewhere. This tutorial demonstrates how to use GAP to generate algebraic group structures utilized by music theorists. Additionally, it explores how these structures can be applied to the study of transformational theory using familiar concepts derived from the Klumpenhouwer network and Neo-Riemannian theories.GAP (an acronym for Groups, Algorithms, and Programming) is a computational discrete algebra system that can be applied as a practical tool for transformational music theory. In particular, this software offers theorists tools that are not easily accessible elsewhere. This tutorial demonstrates how to use GAP to generate algebraic group structures utilized by music theorists. Additionally, it explores how these structures can be applied to the study of transformational theory using familiar concepts derived from the Klumpenhouwer network and Neo-Riemannian theories.https://math-sci.ui.ac.ir/article_29432_b8645ece48a7be6591b342bc54eefbfd.pdf