University of IsfahanMathematics and Society2345-64939220240822Bollobás lemma: analysis, application, algorithmBollobás lemma: analysis, application, algorithm1182814010.22108/msci.2024.139150.1606FAMohsenAlambardar MeybodiDepartment of Applied Mathematics and Computer Science, University of Isfahan,AfshanHashemiLinköping University, Department of Computer and Information ScienceJournal Article20230922Many of the theorems that were developed by researchers in the 1960s and 1970s in combinations and graph theory as mathematical theorems, are still used today in designing algorithms and solving new problems. The purpose of this article is to show that useful things happened in the past that we should not forget and use them with a creative eye. Bollobás Lemma \cite{bollobas}, which was proposed in the 1970s, is a well-known problem in combinatorics. In this problem, we have a family of sets $A_1, A_2,\ldots,A_m$ each with size a and another family of sets $B_1, B_2,\ldots,B_m$ each with size b. The goal is to find the maximum size m, the number of sets so that for each index i we have $A_i \cap B_i = \emptyset $ and also $A_i \cap B_j \neq \emptyset$ where $(i \neq j).$ The Bollobás lemma expresses the upper bound for the maximum number of these sets as $m\leq \binom{a+b}{b}$. In this paper, after stating the versions of the lemma and the existing proof for this lemma, we present another probability-based proof for the Bollobás Lemma, and then with a different look at this combinatorial problem, we investigate the interesting applications of this lemma in graph theory problems and parameterized algorithms.Many of the theorems that were developed by researchers in the 1960s and 1970s in combinations and graph theory as mathematical theorems, are still used today in designing algorithms and solving new problems. The purpose of this article is to show that useful things happened in the past that we should not forget and use them with a creative eye. Bollobás Lemma \cite{bollobas}, which was proposed in the 1970s, is a well-known problem in combinatorics. In this problem, we have a family of sets $A_1, A_2,\ldots,A_m$ each with size a and another family of sets $B_1, B_2,\ldots,B_m$ each with size b. The goal is to find the maximum size m, the number of sets so that for each index i we have $A_i \cap B_i = \emptyset $ and also $A_i \cap B_j \neq \emptyset$ where $(i \neq j).$ The Bollobás lemma expresses the upper bound for the maximum number of these sets as $m\leq \binom{a+b}{b}$. In this paper, after stating the versions of the lemma and the existing proof for this lemma, we present another probability-based proof for the Bollobás Lemma, and then with a different look at this combinatorial problem, we investigate the interesting applications of this lemma in graph theory problems and parameterized algorithms.https://math-sci.ui.ac.ir/article_28140_20845725fa4c1229fd8131026bb48d5b.pdfUniversity of IsfahanMathematics and Society2345-64939220240822Employability of mathematics, statistics, and computer science graduates: introduction to skill coursesEmployability of mathematics, statistics, and computer science graduates: introduction to skill courses19302826210.22108/msci.2024.138322.1591FAEhsanMovahedniaDepartment of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, IranJournal Article20230707The purpose of this article is to investigate job opportunities for graduates of mathematics, statistics and computer science. By introducing skill courses related to these fields, suggestions are made to improve the employment status of graduates. These proposals provide the opportunity for students to increase their preparation for entering the labor market by acquiring some technical and specialized skills appropriate to their field. These skills can be a valuable advantage for graduates in the labor market. Moreover, learning such skills can pave the way for nurturing their creativity and entrepreneurial opportunities.The purpose of this article is to investigate job opportunities for graduates of mathematics, statistics and computer science. By introducing skill courses related to these fields, suggestions are made to improve the employment status of graduates. These proposals provide the opportunity for students to increase their preparation for entering the labor market by acquiring some technical and specialized skills appropriate to their field. These skills can be a valuable advantage for graduates in the labor market. Moreover, learning such skills can pave the way for nurturing their creativity and entrepreneurial opportunities.https://math-sci.ui.ac.ir/article_28262_2da46c8936f7df80a3d3d2760e9c768c.pdfUniversity of IsfahanMathematics and Society2345-64939220240528Music and linear algebraMusic and linear algebra31472822910.22108/msci.2024.139462.1616FAMarziyehFelahatDepartment of Basic Sciences, Bozorgmehr University of Qaenat, Qaen, IranJavadTayyebiDepartment of Industrial Engineering, Birjand University of Technology, Birjand, IranJournal Article20231018Sound, with all its manifestations, presents itself as a form of energy with properties and capabilities that remain partially shrouded in mystery. The art of composing music, the artful assemblage and fusion of sounds, still lingers in a realm of obscurity, fraught with complexities. The techniques wielded by composers often stem from an intrinsic gift that defies replication or sharing. However, akin to other art forms, music can be deciphered through the lenses of mathematics and geometry. Through the utilization of linear algebra, a paramount mathematical discipline, a practical and efficient framework can be established for the comprehension and manipulation of musical compositions. Within this discourse, we delve into the essence of music and its fundamental principles, subsequently exploring the application of mathematical concepts; linear algebra in particular; in the analysis of music. The findings put forth in this manuscript hold the potential to enlighten researchers and enthusiasts within the realm of music and mathematical sciences, enhance comprehension and knowledge, refine and enhance analytical methodologies and musical procedures across various facets of the music industry, and foster a deeper bond between the realms of mathematics and artistic expression.Sound, with all its manifestations, presents itself as a form of energy with properties and capabilities that remain partially shrouded in mystery. The art of composing music, the artful assemblage and fusion of sounds, still lingers in a realm of obscurity, fraught with complexities. The techniques wielded by composers often stem from an intrinsic gift that defies replication or sharing. However, akin to other art forms, music can be deciphered through the lenses of mathematics and geometry. Through the utilization of linear algebra, a paramount mathematical discipline, a practical and efficient framework can be established for the comprehension and manipulation of musical compositions. Within this discourse, we delve into the essence of music and its fundamental principles, subsequently exploring the application of mathematical concepts; linear algebra in particular; in the analysis of music. The findings put forth in this manuscript hold the potential to enlighten researchers and enthusiasts within the realm of music and mathematical sciences, enhance comprehension and knowledge, refine and enhance analytical methodologies and musical procedures across various facets of the music industry, and foster a deeper bond between the realms of mathematics and artistic expression.https://math-sci.ui.ac.ir/article_28229_012e6997c4dece5ba9a2357bd87d37c5.pdfUniversity of IsfahanMathematics and Society2345-64939220240822Mathematics, an irreplaceable achievement with no expiration dateMathematics, an irreplaceable achievement with no expiration date49692852210.22108/msci.2024.140207.1635FAMaryamAlipourDepartment of Mathematics-University of Sistan and BaluchestanOmidZabetiDepartment of Mathematics- University of Sistan and BaluchestanJournal Article20231225Undoubtedly, what introduces mathematics as the queen of human sciences and distinguishes it from all sciences is the existence of the essence of freedom in meaning and Its meaning is In other words, the role of mathematics in the universe is the discovery of tools and features of the secrets of the natural world, which is important in two ways: 1. It confirms and interprets all scientific theories. 2. It has a deep, profound and basic view of all issues. In this article, we try to look at the irreplaceable applications of mathematics from another angle. How mathematics has a tremendous impact on our lives moment by moment and leads us towards the true meaning of life. In fact, we try to express and interpret the applications of the irreplaceable role of mathematics in intellectual, analytical and skill abilities by relying on general mathematics materials, which is considered a basic course in most scientific fields. The main goal of this article is concrete promotion of mathematics.Undoubtedly, what introduces mathematics as the queen of human sciences and distinguishes it from all sciences is the existence of the essence of freedom in meaning and Its meaning is In other words, the role of mathematics in the universe is the discovery of tools and features of the secrets of the natural world, which is important in two ways: 1. It confirms and interprets all scientific theories. 2. It has a deep, profound and basic view of all issues. In this article, we try to look at the irreplaceable applications of mathematics from another angle. How mathematics has a tremendous impact on our lives moment by moment and leads us towards the true meaning of life. In fact, we try to express and interpret the applications of the irreplaceable role of mathematics in intellectual, analytical and skill abilities by relying on general mathematics materials, which is considered a basic course in most scientific fields. The main goal of this article is concrete promotion of mathematics.https://math-sci.ui.ac.ir/article_28522_e2b6798dc356d9a88562a4240b7970a0.pdfUniversity of IsfahanMathematics and Society2345-64939220240602Soliton and its application in the study of nonlinear dynamics of acoustic waves in multi-particle fluidSoliton and its application in the study of nonlinear dynamics of acoustic waves in multi-particle fluid71852828810.22108/msci.2024.139913.1629FANoraNassiri-MofakhamFuel Cycle Research School, Nuclear Science and Technology Research Institute Tehran, IranJournal Article20231125Solitons have many applications in pure and applied mathematics, especially in the fields such as nonlinear differential equations, Lie algebra, and algebraic geometry. Solitons are ubiquitous in nature and have many applications in nonlinear dynamics. The discovery of solitons made it possible to obtain analytical solutions for nonlinear differential equations. Among them Korteweg-de Vries equation is the most familiar one, which is the most canonical nonlinear wave equation. In an electrostatic fluid that is partially ionized, the collision of ions with each other and also with electrons cause the propagation and coupling of ionic electrostatic waves to behave nonlinear. The equations describing the dynamics of acoustic waves in this fluid are transformed into the Korteweg-de Vries equation by the reduction perturbation method, which analytical solutions are in the form of solitary waves, and their amplitude and speed depend on the properties of the fluid medium. A brief history of solitons and their applications are given. Then, the nonlinear propagation of the traveling ion electrostatic wave in a multi-particle fluid is obtained by taking into account the production-loss rate of ions. The analytical and numerical solution of the wave equations is obtained. The results show that the wave dispersion curves have bifurcation points and only high-frequency waves are propagated as solitary waves with complex modes. The results of this study can be used in the observation of ion acoustic waves in space plasmas and the presence of dust grains in laboratory plasma.Solitons have many applications in pure and applied mathematics, especially in the fields such as nonlinear differential equations, Lie algebra, and algebraic geometry. Solitons are ubiquitous in nature and have many applications in nonlinear dynamics. The discovery of solitons made it possible to obtain analytical solutions for nonlinear differential equations. Among them Korteweg-de Vries equation is the most familiar one, which is the most canonical nonlinear wave equation. In an electrostatic fluid that is partially ionized, the collision of ions with each other and also with electrons cause the propagation and coupling of ionic electrostatic waves to behave nonlinear. The equations describing the dynamics of acoustic waves in this fluid are transformed into the Korteweg-de Vries equation by the reduction perturbation method, which analytical solutions are in the form of solitary waves, and their amplitude and speed depend on the properties of the fluid medium. A brief history of solitons and their applications are given. Then, the nonlinear propagation of the traveling ion electrostatic wave in a multi-particle fluid is obtained by taking into account the production-loss rate of ions. The analytical and numerical solution of the wave equations is obtained. The results show that the wave dispersion curves have bifurcation points and only high-frequency waves are propagated as solitary waves with complex modes. The results of this study can be used in the observation of ion acoustic waves in space plasmas and the presence of dust grains in laboratory plasma.https://math-sci.ui.ac.ir/article_28288_83ed0311ae1067d94d175db906f3899a.pdfUniversity of IsfahanMathematics and Society2345-64939220240610Generalized derivations on certain Banach algebrasGeneralized derivations on certain Banach algebras871052828710.22108/msci.2024.140007.1630FAMehdiNematiDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.AliEbrahimzadeh EsfahaniDepartment of Mathematical Sciences,
Isfahan University of Technology,
Isfahan 84156-83111, IranSimaSoltani RenaniDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, IranSeyed MahmoudManjeganiDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, IranJournal Article20231203In this paper, we apply the well-known results concerning derivations and generalized derivations of commutative Banach algebras and of prime rings to certain Banach algebras that are neither commutative Banach algebras nor prime rings. For example, we investigate the truth of Singer-Wermer conjecture and Posner's second theorem for this class of Banach algebras.In this paper, we apply the well-known results concerning derivations and generalized derivations of commutative Banach algebras and of prime rings to certain Banach algebras that are neither commutative Banach algebras nor prime rings. For example, we investigate the truth of Singer-Wermer conjecture and Posner's second theorem for this class of Banach algebras.https://math-sci.ui.ac.ir/article_28287_35bf96e6641655026d3e398c875e83fd.pdf