نشریه ریاضی و جامعه
سال هشتم شماره 1 (بهار 1402)

Permutation patterns is a burgeoning area of research with roots in enumerative combinatorics and theoretical computer science. This article first presents a brief overview of pattern avoidance and a survey of enumeration results that are standard knowledge within the field. Then, we turn our attention to a newer optimization problem of pattern packing. We survey pattern packing results in the general case before we consider packing in a specific type of permutation that leads to a new and surprising connection with physical chemistry. Note that the original paper has published in ``Notices of the American Mathematical Society, 67, Number 7, 994-1001" and we have translated it into Farsi. This is just an extended abstract for Journal of Mathematics and Society.

This study is a descriptive study with a scientometrics approach and the method of co-word analysis which analyzes scientific outputs in the field of the cross-diffusion predator prey model on the Web of Science from 1997 to September 11, 2022. In this research, to analyze the data, Hist-Cite, Excel, Bibexcel, and Gephi software and for drawing the maps, Vos viewer software is used. In the cross-diffusion predator prey model, we investigate the structure of productions such as publications, institutions, and researchers with the high productions and citations. Also, the co-occurrence analysis related to the mentioned topic and the cooperation of countries and authors and centrality measures are discussed. The results obtained from the data analysis show that among the published works, 305 research have been published. Among the countries and the authors, China with 226 works and M. S. Fu , M. X. Wang and L. N. Guin have with 17 works respectively. M. X. Wang also has the highest number of citations with 242 local citations and 817 global citations. The journal of Nonlinear Analysis Real World Applications has published with 23 works. Among the investigated topics in the predator prey model with cross diffusion, the concepts of pattern formation, Turing instability and stability were obtained with a frequency of 49, 63 and 40, respectively. In the following, the co-word analysis of studies in this field, 6 clusters of words and concepts were identified. The cooperation map of the countries also showed that among the countries in this field, China has the largest number of works and has the highest level of the communication with other countries. The authors’ cooperation map has formed 5 clusters. Meanwhile, L. N. Guin with 10 works and 7 connections and M. X. Wang with 7 works and 4 connections have the highest number of connections. Among the top authors in the world, in terms of degree centrality, closeness centrality, and betweenness centrality respectively, S. M. Fu, L. N. Guin and W. M. Wang are ranked first.

In this paper, we prove that for every tournament matrix with nonzero spectral radius, the power iteration method converges to a nonzero eigenvector corresponding to the eigenvalue with the maximum magnitude. An application of this result for ranking the corresponding players of the matrix is also given.

Despite the many studies conducted in the field of mathematics education and especially geometry, we see that the process of teaching and learning geometry has many challenges. Students' challenges in geometry usually appear when solving problems. Two of the approaches that describe the process of solving geometric problems are called geometrical paradigms and working spaces. The purpose of this article, which is a review article, is to briefly describe these two approaches based on related studies and then show their importance in the process of teaching and learning geometry. In general, geometrical working space refers to the interaction between the three components of visualization, construction and proof. The working space in which a person is reasoning depends on the geometrical paradigm. In Geometry I, the reasoning is based on intuition and experiment, and in Geometry II, the reasoning is made by axioms, but the connection with the physical world is still maintained. Finally, in Geometry III, there’s no connection with the physical world and the reasoning is completely logical and abstract. Identifying students' geometrical paradigms and studying the working space in which they solve problems allows teachers to design their teaching according to students' understanding and also in case of difficulties in the process of teaching and learning, they can use a useful approach.

The concept map becomes meaningful with the connection between the new concepts and the existing concepts in the cognitive structure. Using the concept map, the learners combine the contents and realize the internal connection between the concepts. In this sense, statistics is important as they help people interpret important information. The purpose of the effectiveness of the concept map is to identify misconceptions in the field of mathematics education from graphic literacy. The current research method is an interpretation of two quantitative and qualitative methods. The statistical population was all preservice mathematics teachers (PMT), 18 undergraduate PMTs and 15 master's PMTs were selected as the purposive sample. The PMTs of each level were divided into three groups. The PMTs of the first group completed the concept map and then answer the questions of the written exam and the second group first answer the questions of the written exam and then completed the concept map and the third group only answer the questions of the written exam. The PMTs of group 1 and 2 did not have the opportunity to return to the exam questions they did first and correct their answers. In order to evaluate the effect of using concept maps on PMTs' learning, all PMTs of the three groups answered one question called the total exam. To analyze the data, the one-way analysis of variance test was used for each group and as an average for all groups, as well as comparing the difference of the average scores of the groups in the written exam with Tukey’s post-test method. In the end, with the help of the concept map tool, PMTs' misconceptions were identified and classified into 4 categories, failure to correctly recognize the names of graphs, incorrect drawing of statistical graphs, failure to correctly recognize the type of variable and recognition of the appropriate graph for a data set. The research findings showed that the concept map identified the misconceptions of preservice teachers with better results than the results of the written exam. PMTs can use this method to teach different subjects, especially statistics in school, and improve the teaching-learning process and strengthen meaningful learning.

The aim of this paper is to consider horizontally weakly conformal maps which have been studied in [P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, 29, The Clarendon Press, Oxford University Press, Oxford, 2003]. We first generalize the results of this book for Euclidean space with arbitrary Riemannian metrics and then we study totally umbilic and totally geodesic integral manifolds of the projection map, when regarded as conformal submersion between Euclidean spaces with arbitrary Riemannian metrics.