University of IsfahanMathematics and Society2345-649310120250219Geodesic vectors of square metrics on 5- dimensional generalized symmetric spacesGeodesic vectors of square metrics on 5- dimensional generalized symmetric spaces1302843310.22108/msci.2024.139056.1609FADariushLatifiDepartment of Mathematics, University of Mohaghegh Ardabili, P.O.Box 5619911367, Ardabil, IranMiladZeinaliDepartment of Mathematics, University of Mohaghegh Ardabili, P.O.Box 5619911367, Ardabil, IranJournal Article20230928In this paper, we consider the $(\alpha, \beta)$-metric $F=\frac{(\alpha + \beta)^2}{\alpha}$ along with the function $\phi$ with the definition of $\phi(s)=1+2s+s^2$, which is known as a square metric, on 5-dimensional generalized symmetric spaces. Then we investigate and study the homogeneous geodesics of 5-dimensional generalized symmetric spaces equipped with a left invariant square metric. We also obtain and categorize the homogeneous geodesics of these spaces in some special cases, which are of type (2), (3) and (7). Also we show that for a 5-dimensional generalized symmetric space of type (2) and (7) equipped with a left invariant square metric, the geodesic vectors of $(M,F)$ are the same as the geodesic vectors of $(M, \tilde{a})$ and vice versa.In this paper, we consider the $(\alpha, \beta)$-metric $F=\frac{(\alpha + \beta)^2}{\alpha}$ along with the function $\phi$ with the definition of $\phi(s)=1+2s+s^2$, which is known as a square metric, on 5-dimensional generalized symmetric spaces. Then we investigate and study the homogeneous geodesics of 5-dimensional generalized symmetric spaces equipped with a left invariant square metric. We also obtain and categorize the homogeneous geodesics of these spaces in some special cases, which are of type (2), (3) and (7). Also we show that for a 5-dimensional generalized symmetric space of type (2) and (7) equipped with a left invariant square metric, the geodesic vectors of $(M,F)$ are the same as the geodesic vectors of $(M, \tilde{a})$ and vice versa.https://math-sci.ui.ac.ir/article_28433_3e27facdfeeb8854bfb12492e2c005ac.pdfUniversity of IsfahanMathematics and Society2345-649310120250122One Iteration method in Numerical solution of Endemic Models with arbitrarily distributed periods of infection for long timeOne Iteration method in Numerical solution of Endemic Models with arbitrarily distributed periods of infection for long time31662859010.22108/msci.2024.139744.1628FABahmanBabayar-RazlighiDepartment of Mathematics, Qom University of technology, P.O.Box 1519-37195, Qom, IranJournal Article20231124In this paper, we consider endemic models with arbitrary distributed periods of infection. Some samples of such diseases are: HIV/AIDS, rubella, influenza, and so on. Since beginning of 20th century, there are many authors that investigate about mathematical modelling, existence of solutions and stability of models. Many of these models are in the form of differential equations, integral equations, algebraic equation or hybrid of these. In this work we convert the existence model to a system of integral equations with a nonlinear algebraic equation. We solve numerically this model by an iterative process. Organization of the given algorithm is such that the problem is solved by a numerical iteration on union of short time intervals, and the process forward interval by interval. Convergence of the method is given expansively. In the numerical results section, according to the structure of the problem and by using Laplace transform, we sketch a spectrum of sample problems that have analytical solutions. Finally, we illustrate accuracy and applicability of the method by two benchmark sample problems.In this paper, we consider endemic models with arbitrary distributed periods of infection. Some samples of such diseases are: HIV/AIDS, rubella, influenza, and so on. Since beginning of 20th century, there are many authors that investigate about mathematical modelling, existence of solutions and stability of models. Many of these models are in the form of differential equations, integral equations, algebraic equation or hybrid of these. In this work we convert the existence model to a system of integral equations with a nonlinear algebraic equation. We solve numerically this model by an iterative process. Organization of the given algorithm is such that the problem is solved by a numerical iteration on union of short time intervals, and the process forward interval by interval. Convergence of the method is given expansively. In the numerical results section, according to the structure of the problem and by using Laplace transform, we sketch a spectrum of sample problems that have analytical solutions. Finally, we illustrate accuracy and applicability of the method by two benchmark sample problems.https://math-sci.ui.ac.ir/article_28590_adcc4990e829fd0061fb9b8bfc67fd93.pdfUniversity of IsfahanMathematics and Society2345-649310120250213Two proofs from Āzarkhor Ashtāz GoshnaspTwo proofs from Āzarkhor Ashtāz Goshnasp67772862910.22108/msci.2024.139764.1622FARezaKahkeshaniDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, IranJournal Article20231113One of the Iranian scientists of the fourth and fifth centuries AH is Āzarkhor Asht\={a}z Goshnasp. His name is only mentioned in the works of the prominent Iranian scientist, Ab\={u} Rayh\={a}n al-B\={i}r\={u}n\={i}. In this paper, we will explain \={A}zarkhor's two proofs on the first theorem of the book ``al-Estekhr\={a}j al-Awt\={a}r" that given by Ab\={u} Rayh\={a}n. This theorem is called ``Theorem of the broken chord" and these two proofs are the only mathematical legacy left by \={A}zarkhor.One of the Iranian scientists of the fourth and fifth centuries AH is Āzarkhor Asht\={a}z Goshnasp. His name is only mentioned in the works of the prominent Iranian scientist, Ab\={u} Rayh\={a}n al-B\={i}r\={u}n\={i}. In this paper, we will explain \={A}zarkhor's two proofs on the first theorem of the book ``al-Estekhr\={a}j al-Awt\={a}r" that given by Ab\={u} Rayh\={a}n. This theorem is called ``Theorem of the broken chord" and these two proofs are the only mathematical legacy left by \={A}zarkhor.https://math-sci.ui.ac.ir/article_28629_2b9bb0f7e92c9448c5734b947e0ade00.pdfUniversity of IsfahanMathematics and Society2345-649310120250406Graph Theory; History, Applications, and, VisionGraph Theory; History, Applications, and, Vision791032866610.22108/msci.2024.139804.1623FAMeysamTaheri DehkordiDepartment of Mathematics, University of Applied Science and Technology (UAST), Tehran, IRANJournal Article20231115Graph theory is a leading theory in mathematics, which is used in many sciences. In this article, by stating the history of this, its expansion and development, we have mentioned the course of famous problems in graph theory. Also, a practical application of this theory is stated.Graph theory is a leading theory in mathematics, which is used in many sciences. In this article, by stating the history of this, its expansion and development, we have mentioned the course of famous problems in graph theory. Also, a practical application of this theory is stated.https://math-sci.ui.ac.ir/article_28666_4a855bbf33ce553afb002fe793332b9d.pdfUniversity of IsfahanMathematics and Society2345-649310120250412New solutions to Einstein's equations to find Walker manifoldsNew solutions to Einstein's equations to find Walker manifolds1051242855010.22108/msci.2024.140664.1643FAShahroudAzamiDepartment of Pure Mathematics, Faculty of Science, Imam Khomeini International University Qazvin, IranJournal Article20240207In this paper, we investigate the Einsteinian manifolds with parallel null distribution. For this, we first obtain the equations that lead to finding the mentioned manifolds. These equations are known as Einstein's equations. Then we reduce these equations by using Lie symmetry method. These equations are known as Einstein's equations. In this method, we first obtain the generators of the symmetry algebra and then calculate the differential invariants for each of the generators and calculate the group invariant solutions of this equation. In addition to this, we also obtain the optimal system of the one-dimensional sub-algebras of these equations. This optimal system helps us to have a classification on group invariant solutions using conjugate mapping.In this paper, we investigate the Einsteinian manifolds with parallel null distribution. For this, we first obtain the equations that lead to finding the mentioned manifolds. These equations are known as Einstein's equations. Then we reduce these equations by using Lie symmetry method. These equations are known as Einstein's equations. In this method, we first obtain the generators of the symmetry algebra and then calculate the differential invariants for each of the generators and calculate the group invariant solutions of this equation. In addition to this, we also obtain the optimal system of the one-dimensional sub-algebras of these equations. This optimal system helps us to have a classification on group invariant solutions using conjugate mapping.https://math-sci.ui.ac.ir/article_28550_062b6fa81cafa081e5c2828a78285c71.pdfUniversity of IsfahanMathematics and Society2345-649310120250522Cubic semisymmetric graphs of order $ 40p $Cubic semisymmetric graphs of order $ 40p $1251442856310.22108/msci.2024.140259.1639FAMohammad RezaSalarianDeparement of Mathematics,,Faculty of Mathematical and Cumputer Science Kharazmi, ,Tehran,.IranJavanshirRezaeiStudent of ,,Faculty of Mathematical and Cumputer Science Kharazmi, ,Tehran, ,IranJournal Article20240107A simple graph $\Gamma$ is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. A simple graph $\Gamma$ is called cubic whenever it is $ 3 $-regular. An important research problem is the classification of semisymmetric cubic graphs of different orders. The purpose of this article is the classification of semisymmetric cubic graphs of order $ 40p $, where p is a prime number. We show that for $ p\ne3,31 $ such a graph Does not exist. Suppose p is a prime number, Folkman showed in\cite{fo} that there is no semisymmetric graph of order $ 2p $ or $ 2p^2 $. We show that if $\Gamma$ is a semisymmetric cubic graph of order $ 40p $, then $ p=3 $ and $\Gamma$ is isomorphic to a semisymmetric cubic graph of order $ 120 $ or $ p=31 $ and $\Gamma$ is isomorphic to the coset graph $ C(L_2 (31):\mathbb{S}_4,\mathbb{S}_4).$ Our basic tools in this research are automorphism of graphs, simple groups, solevable groups and permutation groups.A simple graph $\Gamma$ is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. A simple graph $\Gamma$ is called cubic whenever it is $ 3 $-regular. An important research problem is the classification of semisymmetric cubic graphs of different orders. The purpose of this article is the classification of semisymmetric cubic graphs of order $ 40p $, where p is a prime number. We show that for $ p\ne3,31 $ such a graph Does not exist. Suppose p is a prime number, Folkman showed in\cite{fo} that there is no semisymmetric graph of order $ 2p $ or $ 2p^2 $. We show that if $\Gamma$ is a semisymmetric cubic graph of order $ 40p $, then $ p=3 $ and $\Gamma$ is isomorphic to a semisymmetric cubic graph of order $ 120 $ or $ p=31 $ and $\Gamma$ is isomorphic to the coset graph $ C(L_2 (31):\mathbb{S}_4,\mathbb{S}_4).$ Our basic tools in this research are automorphism of graphs, simple groups, solevable groups and permutation groups.https://math-sci.ui.ac.ir/article_28563_858755db9d87b4e3493661ba115e2e32.pdf