University of IsfahanMathematics and Society2345-64935320201121A bound for the rate of connectionsA bound for the rate of connections1132567710.22108/msci.2021.127724.1418FARasoulAhangar MalekiMathematics Research Institute,IPM Institute For Research In Fundamental Sciences, TehranTirdadSharifMathematics Research Institute,IPM Institute For Research In Fundamental Sciences, TehranJournal Article20210308Let $K$ be a field, $R$ a standard $K$-algebra and $M$ a finitely generated $R$-module. The growth rate of $M$ is denoted by $rate_R(M)$, which is a measure of the growth rate of connections in the minimal graded free resolution of the $R$-module $M$. In this article, the behavior of this invariant under change of rings is investigated. Additionally, the growth rate for Artinian algebras is precisely determined. Moreover, an upper bound for the growth rate of the tensor product of two modules is determined based on the growth rates of each module.Let $K$ be a field, $R$ a standard $K$-algebra and $M$ a finitely generated $R$-module. The growth rate of $M$ is denoted by $rate_R(M)$, which is a measure of the growth rate of connections in the minimal graded free resolution of the $R$-module $M$. In this article, the behavior of this invariant under change of rings is investigated. Additionally, the growth rate for Artinian algebras is precisely determined. Moreover, an upper bound for the growth rate of the tensor product of two modules is determined based on the growth rates of each module.https://math-sci.ui.ac.ir/article_25677_76d88179d5aac5e3d0fe0fca4c18f2a8.pdfUniversity of IsfahanMathematics and Society2345-64935320201121A class of generalized symmetric Finsler spacesA class of generalized symmetric Finsler spaces15222580210.22108/msci.2021.128837.1435FAHamid RezaSalimi MoghaddamDepartment of Pure Mathematics, Faculty of Mathematics, University of Isfahan, IsfahanJournal Article20210523In this article, we study such Finsler spaces equipped with an $(\alpha,\beta)$-metric $F=\alpha+\beta-\beta^2/\alpha$. After examining the relationship between the s-structure of the underlying Riemannian metric and the $s$-structure of the Finsler metric under discussion, we demonstrate that every generalized symmetric space equipped with such a metric is Riemannian of Kähler type.In this article, we study such Finsler spaces equipped with an $(\alpha,\beta)$-metric $F=\alpha+\beta-\beta^2/\alpha$. After examining the relationship between the s-structure of the underlying Riemannian metric and the $s$-structure of the Finsler metric under discussion, we demonstrate that every generalized symmetric space equipped with such a metric is Riemannian of Kähler type.https://math-sci.ui.ac.ir/article_25802_472cf0c223cc3252ab38154a9ddb1354.pdfUniversity of IsfahanMathematics and Society2345-64935320201121Analysis, generalization, and application of the collatz conjectureAnalysis, generalization, and application of the collatz conjecture23372577110.22108/msci.2021.123903.1370FAMohsenFathi Aqbalagh Mustafa KhanFouman Faculty of Engineering, Campus of Technical Faculties, University of Tehran, Fouman, IranAtefehHasanzadehFouman Faculty of Engineering, Campus of Technical Faculties, University of Tehran, Fouman, IranJournal Article20200710In this article, we delve into the examination and analysis of the Collatz Conjecture. Through numerical methods and step-by-step analysis of numbers, we attempt to uncover the origin of this conjecture, which relates to an intriguing property among prime numbers. Furthermore, by identifying additional relationships among prime numbers, we generalize the Collatz Conjecture and present a comprehensive algorithm for it. Finally, we utilize this conjecture to identify and find large twin prime numbers.In this article, we delve into the examination and analysis of the Collatz Conjecture. Through numerical methods and step-by-step analysis of numbers, we attempt to uncover the origin of this conjecture, which relates to an intriguing property among prime numbers. Furthermore, by identifying additional relationships among prime numbers, we generalize the Collatz Conjecture and present a comprehensive algorithm for it. Finally, we utilize this conjecture to identify and find large twin prime numbers.https://math-sci.ui.ac.ir/article_25771_1dd868357470d971496fae8ef87658d2.pdfUniversity of IsfahanMathematics and Society2345-64935320201121Application of logic in red eye disease detectionApplication of logic in red eye disease detection39622577210.22108/msci.2021.128678.1432FAFariaNassiri-MofakhamFaculty of Computer Engineering, University of Isfahan, IranJournal Article20210512$\text{DTReD}$ is an expert system designed to diagnose and prescribe appropriate treatment for red eye diseases. This rule-based system is implemented using specialized knowledge of ophthalmology. $\text{DTReD}$ operates by analyzing responses to questions related to red eye symptoms. The information considered by this system includes any observed signs by a general physician or responses provided by the patient. Then, $\text{DTReD}$ employs established scientific knowledge and existing rules to diagnose a specific disease through forward chaining reasoning. As a result, it can prescribe suitable treatment. Additionally, $\text{DTReD}$ can provide detailed information about various characteristics of different red eye diseases to users who are non-specialists or trainers. Such an expert electronic system is beneficial in rural areas where experienced physicians are geographically less accessible or in situations where the patient cannot visit a specialist physician.$\text{DTReD}$ is an expert system designed to diagnose and prescribe appropriate treatment for red eye diseases. This rule-based system is implemented using specialized knowledge of ophthalmology. $\text{DTReD}$ operates by analyzing responses to questions related to red eye symptoms. The information considered by this system includes any observed signs by a general physician or responses provided by the patient. Then, $\text{DTReD}$ employs established scientific knowledge and existing rules to diagnose a specific disease through forward chaining reasoning. As a result, it can prescribe suitable treatment. Additionally, $\text{DTReD}$ can provide detailed information about various characteristics of different red eye diseases to users who are non-specialists or trainers. Such an expert electronic system is beneficial in rural areas where experienced physicians are geographically less accessible or in situations where the patient cannot visit a specialist physician.https://math-sci.ui.ac.ir/article_25772_378e4a282c543e0e8591e4f6f8895338.pdfUniversity of IsfahanMathematics and Society2345-64935320201121Classification of solution sets of pseudoconvex optimization problems on hadamard manifoldsClassification of solution sets of pseudoconvex optimization problems on hadamard manifolds63742581910.22108/msci.2021.127986.1421FAAliBaraniDepartment of Mathematics, Faculty of Basic Sciences, Lorestan University, Khorramabad, IranJournal Article20210404In this paper, we provide necessary and sufficient conditions for the solution sets of an optimization problem over convex subsets of Hadamard manifolds, where the objective function is Pseudoconvex. Additionally, we prove that the gradient norm of the objective function at each minimizer point is equal to a positive multiple of its norm at a specific minimizer point. The application of the obtained results is illustrated through examples.In this paper, we provide necessary and sufficient conditions for the solution sets of an optimization problem over convex subsets of Hadamard manifolds, where the objective function is Pseudoconvex. Additionally, we prove that the gradient norm of the objective function at each minimizer point is equal to a positive multiple of its norm at a specific minimizer point. The application of the obtained results is illustrated through examples.https://math-sci.ui.ac.ir/article_25819_f15fa504ab6a303eb7ebaa97666e07ca.pdfUniversity of IsfahanMathematics and Society2345-64935320201121The polynomial partition of unity method: an efficient tool in approximationThe polynomial partition of unity method: an efficient tool in approximation75982583510.22108/msci.2021.127648.1417FADavoudMirzaeiDepartment of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, Isfahan University, Isfahan, IranMohammadrezaAhmadi DaraniDepartment of Computer Science, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, IranSaeedVahdatiDepartment of Mathematics, Khansar Campus, University of Isfahan, IranJournal Article20210302In this paper, we investigate a polynomial approximation/interpolation based on the partition of unity method, and employ it as an efficient method for numerical solution of multivariate problems. First, we introduce multivariate polynomial approximations and prove their scalability properties. Then we use these properties to derive stability and convergence bounds for the proposed method. To have a stable algorithm, local approximations are computed over small subdomains and joined via the partition of unity weight functions to obtain a smooth global approximation. Finally, the global error bound is derived in terms of the error bounds of the local approximations. The idea behind this approach is to solve several small well-conditioned problems instead of solving one large ill-conditioned problem. From the computational point of view, this approach is highly efficient and applicable to a wide range of problems. As an example, the numerical solutions of differential equations are obtained using this approach. Instead of employing a background mesh (similar to finite element and finite volume methods), the unknown quantities are expressed in terms of scattered points. Hence, the given method can also be considered a so-called meshless methodIn this paper, we investigate a polynomial approximation/interpolation based on the partition of unity method, and employ it as an efficient method for numerical solution of multivariate problems. First, we introduce multivariate polynomial approximations and prove their scalability properties. Then we use these properties to derive stability and convergence bounds for the proposed method. To have a stable algorithm, local approximations are computed over small subdomains and joined via the partition of unity weight functions to obtain a smooth global approximation. Finally, the global error bound is derived in terms of the error bounds of the local approximations. The idea behind this approach is to solve several small well-conditioned problems instead of solving one large ill-conditioned problem. From the computational point of view, this approach is highly efficient and applicable to a wide range of problems. As an example, the numerical solutions of differential equations are obtained using this approach. Instead of employing a background mesh (similar to finite element and finite volume methods), the unknown quantities are expressed in terms of scattered points. Hence, the given method can also be considered a so-called meshless methodhttps://math-sci.ui.ac.ir/article_25835_70a0adf4d11c7369f5e206c7b932b79a.pdf