Mathematics and Society
Year:2023 | Volume:8 | Issue:1|Papers Count:6

(The above abstract has been extracted by the translator from the original article (L. Pudwell, From Permutation Patterns to the Periodic Table, Notices of the American Mathematical Society, 67 994–1001.))Abstract: 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.  1. IntroductionLet $S_k $ be the set of all permutations on $[k]=\{1, 2,\ldots,k\}$. Given $\pi \in S_k$ and $\rho \in S_l$, we say that $\pi$ contains $\rho$ as a pattern if there exist $1\leq i_1 The definition of pattern containment may be made more visual by considering the plot of $\pi$. In particular, for $\pi=\pi_1\pi_2\cdots \pi_k\in S_k$, the plot of $\pi$ is the graph of the points $(i‎,‎\pi_i)$ in the Cartesian plane. Of particular interest are the sets $S_k(\rho)=\{ \pi \in S_k \vert \pi ~ avoids~ \rho\}$. For example, ‎\[S_4(123)=\{1432‎, ‎2143‎, ‎2413‎, ‎2431‎, ‎3142‎, ‎3214‎, ‎3241,‎ 3412‎, ‎3421‎, ‎4132‎, ‎4213‎, ‎4231‎, ‎4312‎, ‎4321\}‎\]and $\pi= 43512\in S_5(123)$‎ since there is no increasing subsequence of length 3 in $\pi$. 2. Main ResultsMuch of the existing literature in permutation patterns studies the quantity $s_k(\rho)=\vert S_k(\rho)\vert$ for various patterns $\rho$. Starting with the simplest case, it is immediate that $s_k(1)=0$ if $k\geq 1$ since each digit of a nonempty permutation is a copy of the pattern 1. We also have that $s_k(\rho)=\vert S_k(\rho)\vert$ for $k\geq 0$, since the unique permutation of length $𝑘$ avoiding 12 (resp., 21) is $J_k$ (resp., $I_k$). For more information, please refer to the original paper.  Rather than focusing on packing in all permutations, the author of the original paper in the rest of the paper focus on packing patterns into permutations with extra restrictions. This family of packing problems will provide a new link between permutations and physical chemistry.Definition 2.1. Permutation $\pi$ is an alternating permutation if ‎\[\pi_1 <\pi_2>\pi_3<\pi_4\cdots.\] Alternating permutations are also known as zig-zag permutations or up-down permutations.Theorem 2.2. The maximum number of copies of $123$ in an alternating permutation of length $k$ is given by ‎\[ν(123‎, ‎\widehat{I_k})=\lbrace\dfrac{(k-2)(k^2-4k+6)}{6}     k‎~is ~ even‎,\dfrac{(k-1)(k-2)(k-3)}{6}         k~is~‎ odd‎.\rbrace ‎‎\]‎3. ConclusionThis connection between pattern packing and physical chemistry is striking even to long-time permutation patterns researchers. Similarly, the quasi-polynomial sequence obtained for $\nu(123‎, ‎\widehat{I_k})$ had no previous interpretation in the literature other than as a sequence of atomic numbers. What, if any, chemical interpretation is there for $\nu(123‎, ‎\widehat{I_k})$ when $k>10?$ What other chemical or physical structures can be described in terms of pattern packing or pattern avoidance? Are there other combinatorial structures that give alternate ways to generate the sequences of atomic numbers of particular groups of chemical elements? The variety of applications of permutation patterns has grown tremendously in recent decades, and modeling electron orbitals can now be added to the list.

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. 1. IntroductionThe maturity, dynamism, and innovativeness of various fields of science can be measured by the research activities of that field. The manifestation of these activities may be in various forms. Scientometrics draws a map of knowledge by processing, extracting, and sorting information and provides the possibility of analysis, navigation and display of knowledge. One of the most used methods for drawing and analyzing the structure of knowledge in different fields is the co-occurrence of words, or in other words, the relationship between the words used in different parts of documents. 2. Main ResaltsThe results obtained from the data analysis show that among the published works, 305 research have been published. Among the topics, the predator-prey model with cross-diffusion is ranked first with a frequency of 150 and the predator-prey model with a frequency of 125 have the highest repetition rate. Among the countries, China has been ranked first with 226 works. After that, India with 35 works and the United States with 18 works are in second and third place. It's remarked that Iran with 2 works has taken the 26th rank. Among the authors, 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. After that, P. Y. H. Pang with 146 local citations and 461 global citations is in second and Z. Jin with 75 local citations and 247 global citations has taken the 3rd rank. Among the institutions, Lanzhou University and Shaanxi University have the most number of works, with 14 works. The Journal of Nonlinear Analysis Real World Applications has published 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.The highest co-word is related to cross-diffusion and the predator-prey model, which was observed in 68 studies. Cross-diffusion and pattern formation are in the second place, and the third place is occupied by cross-diffusion and Turing instability.In the following, co-word analysis of studies in this field, 6 clusters of words and concepts were identified. The keywords of cluster one have been proposed for the topics of pattern formation. Cluster two is about the existence of solutions and in cluster three, the stability of solutions is formed. In cluster four, the cross-diffusion model is investigated. Cluster five examines instabilities and it is placed in the cluster of six reduction methods of Lyapunov-Schmidt.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 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 world's top authors, respectively, S. M. Fu, W. M. Wang and L. Zhang have the highest ranks in terms of degree centrality.In closeness centrality, L. N. Guin, S. Ghorai and V. N. Biktashev won the highest ranks respectively. In betweenness centrality, W. M. Wang, S. M. Fu and L. Zhang are ranked first to third respectively. 3. Summary of ProofsThe trend of topics done in the study of the cross-diffusion predator-prey model has been the following. At first, the researchers investigated the existence and boundedness of solutions in these models. Then they discussed stability, global stability, and instability of solutions. After that, Turing instability, pattern formation, and time delay were carried out. Finally, Hopf bifurcation, Turing bifurcation and Hopf-Turing bifurcation were analyzed.

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. 1. IntroductionThe power iteration method is a simple numerical method for finding a corresponding eigenvector of a dominant eigenvalue of a matrix, i.e., an eigenvalue with the largest absolute value. Given a matrix $A$, this method starts as the first vector with an initial guess for an eigenvector of a dominant eigenvalue of $A$. For $n>1$, the $n$-th vector in the sequence is obtained by multiplying the $(n-1)$-th vector on the left by $A$. The process continues until either the sequence converges to the desired eigenvector, or it is clear that the sequence is not convergent. The power iteration method is very useful, but it is not convergent in general. A tournament matrix is a square matrix $A$ whose entries are $0$ or $1$ such that $A+A^t=J-I$, where $I$ is the identity matrix and $J$ is a matirx with all entries equal to $1$. In this paper, we show that for every tournament matrix with a nonzero spectral radius, the power iteration method for finding the non-negative eigenvector corresponding to the dominant eigenvalue is convergent. As an application, we will rank the corresponding players of the tournament matrix. 2. Main ResaltsThe spectral radius of a square matrix $A$ is defined as the maximum absolute value of its eigenvalues and is denoted by $\rho(A)$. Also, for a vector $R=(r_1,\cdots,r_n)\in\mathbb{R}$, we use the notation $\| R\|_1=|r_1|+\cdots+|r_n|$.Theorem 2.1.Let $A$ be a tournament matrix.Then $\rho(A)$ is an eigenvalue of $A$ with the geometric multiplicity one. Moreover, there exists a unique eigenvector $R$ with nonnegative entries sush that $\| R\|_1=1$. Definition 2.2.Let $A$ be a tournamnet matrix. The eigenvalue $R$ in Therorem 2.1 is called the generalized Perron eigenvector of $A$. Definition 2.3.Let $A$ be a tournamnet matrix of order $n$ with $\rho(A)\neq0$.For every positive integer $k$, let \[v_k=\frac{A^kv_{0}}{\| A^kv_{0}\|_1},\]where\[v_0=(\frac{1}{n},\ldots,\frac{1}{n})^t\in\mathbb{R}^n.\]We say that $A$ satisfies the power condition if the sequence $\{v_k\}$ converges to the generalized Perron eigenvector of $A$. Theorem 2.4.Let $A$ be a tournament matrix of order $n$. If $\rho(A)\neq0$, then $A$ satisfies the power condition. Let $A=(a_{ij})$ be a tournament matrix. Then $A$ may be considered as the matrix of a {\it round robin tournamnet}, i.e., a tournament for which every player plays exactly one match against each of the other players. We label the players as $1,2,\cdots,n$. Then $a_{ij}=1$ if and only if player $i$ defeats player $j$. Now, for $X=(1,\cdots,1)^t\in\mathbb{R}^n$, the $i$-th component of the vector $AX$ is the number of wins for player $i$. The product $a_{ij}a_{jk}$ is nonzero if and only if player $i$ defeats player $j$ and player $j$ defeats player $k$. This suggests that player $i$ defeats {\it indirectly} player $j$. The number of such wins can be computed by the vector $A^2X$. Similarly, the product $a_{ij}a_{jk}a_{ks}$ is nonzero exactly when player $i$ defeats player $j$, player $j$ defeats player $k$ and player $k$ defeats player $s$, i.e., again player $i$ defeats indirectly player $j$. Hence, one can count these wins by the matrix $A^3X$. Continuing this argument, we get the vector $(A+A^2+\cdots)X$ which counts all of these direct and indirecet wins. However, this vector may deverge to infinity. To settle this problem, one can normalize the vectors as follows: use the vector\[v_0=\frac{1}{n}X=(\frac{1}{n},\ldots,\frac{1}{n})^t,\]instead of $X$. For every positive number $k$, set\[w_k=\frac{(A+\cdots+A^k)v_0}{\| (A+\cdots+A^k)v_0\|_1}.\]Note taht $\| w_k \|_1=1$ for every nonnegative integer $k$. The following result shows that the sequence $\{w_k\}$ can be used for our ranking problem.Theorem 2.5.Let $A$ be a tournament matrix. If $\rho(A)\neq0$, then the sequence $\{w_k\}$ converges to the generalized Perron eigenvector of $A$. 3. Summary of ProofsTheorem 2.1 follows from [4, (8.3.1)] and [6. (3.1)]. To proof Theorem 2.4, we need some preliminaries as follows: Definition 3.1.A square matrix $P$ is called a {\it permutation matrix} if its rows are obtained by permuting the rows of the identity matrix. Lemma 3.2.A tournament matrix $A$ of order $n$ with $\rho(A)\neq0$ satisfies the power condition if and only if $PAP^t$ satisfies the power condition for every permutation matrix $P$ of order $n$. Proposition 3.3.Let $A$ be a tournament matirx with $\rho(A)\neq0$. If the algebraic multiplicity of $\rho(A)$ is $1$, then $A$ satisfies the power condition. Proposition 3.4.Let $A=\left(\begin{smallmatrix}B&J\\0&C\end{smallmatrix}\right)$ be a tournament matrix of order $n$, where $B$ is a square matirx of nonzero order, $C$ is a nonzero matrix of order $n-m$ and $J$ is an $m\times (n-m)$ matrix whose entries all are $1$. If $\rho(B)=\rho(C)\neq0$ and $B$ and $C$ satisfy the power condition, then so is $A$. Proof of Theorem 2.4.We use induction on the algebraic multiplicity of $\rho(A)$. If the algebraic multiplicity of $\rho(A)$ is $1$ the result follows from Proposition 3.3. Suppose that every tournamnet matrix $T$ with $\rho(T)\neq0$ and the algebraic multiplicity $l$ satisfies the power condition. Assume that the algebraic multiplicity of $\rho(A)$ is $l+1$. Then there exists a permutation matrix $P$ such that $PAP^t$ has a decomposition $PAP^t=\left(\begin{smallmatrix}B&J\\0&C\end{smallmatrix}\right)$ where $B$ is a square matrix of nonzero order $m$ satisfying $\rho(B)=\rho(A)$ with the algebraic multiplicity $1$, $C$ is a square matrix of nonzero order $n-m$ satisfying $\rho(B)=\rho(A)$ with the algebraic multiplicity $l$ and $J$ is an $m\times (n-m)$ matrix whose entries are all $1$. By the hypothesis, $B$ and $C$ satisfy the power condition. Hence, by by Proposition 3.4, $PAP^t$ satisfies the power condition. By Lemma 3.2, the matrix $A$ satisfies the power condition, proving the result. Proof of Theorem 2.5.For a nonnegative number $k$, let $a_k=A^kv_0$, $b_k=\| A^kv_0\|_1$. Since the entries of $A^kv_0$ are nonzero for all $k$, we have\[b_1+\cdots+b_k=\| Av_0\|_1+\cdots+\| A^kv_0\|_1=\| (A+\cdots+A^k)v_0\|_1.\]Hence,\[w_k=\frac{a_1+\cdots+a_k}{b_1+\cdots+b_k}.\]By Theorem 2.4, the sequence $\{\frac{a_k}{b_k}\}$ converges to the generalized Perron vector of $A$. Hence, so does the sequence $\{w_k\}$, thanks to Stolz-Ces\`aro Theorem (see [10, p. 181]).

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.1. IntroductionLearning geometry causes many challenges for students. Many of these difficulties appear when facing geometric problems and choosing the right strategies for solving or proving them. In general, the way students deal with geometric problems and the strategies they use to solve or prove them, reveals important information about their attitude toward geometry. The geometrical paradigms and working spaces describe the students’ way of looking at problems and also the strategies they use in solving them [3, 4].2. Main ResultsParadigm means all the beliefs held by the members of a community. Learners and teachers with similar paradigms can easily communicate with each other, and when they have different paradigms, many difficulties and misconceptions occur \cite{5}. For example, in the process of proving a claim, it is sometimes allowed to use drawing, but sometimes it is not acceptable and providing more detailed reasonings is needed. Therefore, in different situations and problems, learners use different paradigms and we can't say one paradigm is more accurate than the others. In general, three different geometrical paradigms are introduced by Houdement and Kuzniak [3]. In Geometry I, learners use perception, experiment and connection with the physical world to solve a geometric problem. It is related to reality. So the backward and forward movement between the model and reality is allowed to prove the geometric assertions. In Geometry II, the objects are not material. Definitions and axioms are necessary to define and create the objects, but in this paradigm, they are close to intuition. At last, in Geometry III, the system of axioms has no relation with reality. It is independent of any application of the objects of the real world. This paradigm is mainly used in university courses and it doesn’t exist in school geometry. A common educational misconception happens when students and teachers are not in the same paradigm. The passage from one type of Geometry to another is a complex phenomenon. Because it’s a change of theory and can be considered an educational evolution. At least, two transitions happen that are not the same. The first transition (from Geometry I to Geometry II) deals with the nature of the objects and the space. The second one (from Geometry II to Geometry III) concerns the system of axioms and it leads to a more complex process. During elementary school, the first transition must happen and teachers can think about how to prepare students for Geometry II [2,3,5,6]. In order to identify the students' geometrical paradigm, it is necessary to examine the strategies they use in problem solving. In fact, the geometrical working space in which they solve problems should be identified. If we consider mathematics as an activity that is done by the human brain, we can find out how learners have a geometrical paradigm. When experts solve geometric problems, they go back and forth between paradigms. A geometrical working space is a place that is organized to explain the process of solving geometric problems. It illustrates the structure of the complex situation in which the problem solver acts. It involves two planes which are called the epistemological and the cognitive planes. In the epistemological plane, there are three elements. In fact, learners use three components which are the theoretical system of references, the real space, and artifacts to solve a geometric problem. These components are not sufficient to define the meaning of the geometrical working space clearly. Because it strongly depends on its users too. So the cognitive plane was introduced to describe the cognitive activity of each user which consists of visualization, construction and proof. The process of linking the epistemological plane and the cognitive plane is part of geometrical work. In fact, problem solvers use more than one component to reach the correct response to a problem and a set of these actions represents their geometrical working space [4,5,6,9]. The variety of geometrical working spaces depends on the way users synthesize the cognitive and epistemological planes to solve geometric problems. It also depends on the cognitive abilities of each user too. In fact, being an expert or a beginner in solving problems affects the structure of geometrical working space [4].3. Summary of Proofs/ConclusionMany of the difficulties in the process of teaching and learning geometry are due to the difference in the paradigms and working spaces of students and teachers. Several factors influence the students' geometrical paradigm and the working space in which they solve problems, among which the role of the teachers and the textbooks are the most prominent. In fact, the educational system of each country can determine the type of preferred paradigm for each educational level according to the goals of the curriculum. So teachers should be familiar with these approaches and in addition to being aware of them, they can guide the students toward a suitable working space [4,6].

AbstractThe 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.   1. IntroductionStatistical graphics are important because they help people interpret a lot of important information. View statistical graphics as a powerful tool, giving people a summary of quantitative data and even categorical data. Humans interpret various findings about sports games, medical reports, health statistics, etc [3]. with the help of Statistical graphics; Therefore, Statistical graphics play a very important role in our human lives. In this research, concept map is used as a valuable tool to evaluate PMTs' understanding of statistical literacy and to identify PMTs' misconceptionss of this concept. A concept map leads to meaningful learning by making connections between new concepts and existing concepts in the cognitive structure. By using the concept map, the learners combine many contents and realize the internal connection between the concepts. The purpose of the research is the effectiveness of the use of concept maps on identifying the misconceptions of PMTs about graphic literacy [19]. The present research method is a combination of quantitative and qualitative methods. The statistical population was all PMTs, 18 undergraduate PMTs and 15 master's PMTs in secondary school were selected as a 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. In the second stage, all PMTs in all three groups answered one question called the total exam. In the third stage, regarding the total exam, deeper investigations were conducted using the results of the interviews conducted with PMTs and qualitative analysis methods. 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 identify the type of variable and recognition of the appropriate graph for a data set. 2. Main ResaltsThe results indicate that using concept maps as an evaluation tool improves PMTs' learning and identifies their misconceptions. The use of concept maps in evaluation is supported by the idea of changing approaches in evaluation in order to use different tools in addition to written exams. The research findings showed that the concept map identified the misconceptions of PMTs with better results than the results of the written exam. The first misconception identified in the research was about the lack of recognition of different charts such as curve graph, histogram and bar graph. The PMTs did not know the characteristics of these charts correctly and were confused in distinguishing between charts and their drawing. Some PMTs used a bar graph instead of a histogram. Most of the PMTs had problems in drawing the histogram and did not use the right and left limits of the graph for drawing. But this misconception was not clearly visible in the concept map and it was revealed in the exam. Therefore, it cannot be said that concept maps show us comprehensive evidence of people's knowledge. Therefore, it seems that the concept map cannot completely replace the written exam. Some PMTs had confusion and misconception in identifying the variable type of each of the statistical charts. A number of PMTs considered bar graphs to be related to discrete variables or did not correctly identify the type of histogram graph variable. These misconceptions could be recognized and evaluated with the help of concept map. The results of the present study are in line with the findings of Reyhani [32] and Novak's [19] studies in the science course. It is also in line with the results of Barwells [44], Williams [40] and Smita's [45] research in mathematics course, in the sense that concept maps are used in the assessment of mathematics course, it helps PMTs in the field of graphic literacy in recognizing the misconceptions. 3. ConclusionThe results of this study show that teaching through a concept map can make the learning process more meaningful. Therefore, evaluation through concept map gives us a better possibility to identify misconceptions and the level of understanding of the audience, so that PMTs can improve the teaching-learning process and strengthen meaningful learning. PMTs can use this method to teach different subjects, especially statistics in school. We need more research in this field. Issues such as how to use concept maps in teaching other concepts of mathematics and statistics in secondary and even higher education can be considered in future researches. It can also be suggested that PMTs can use different methods of presenting pre-prepared concept maps as an educational strategy in their future teaching in schools at different stages of education. In this context, PMTs can also be encouraged to use it as a teaching-learning strategy by preparing concept maps of the subject matter.

Abstract 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. 1. IntroductionThe aim of this paper is to consider horizontally weakly conformal maps which have been studied in [3]. Horizontally weakly conformal maps are generalization of Riemannian submersions in a sense that at the point where $ d\psi_x\neq 0 $, where $ d\psi_x$ denotes the differential of the map $ \psi: (M^n,g)\rightarrow (\overline{M}^m,h)$, $ d\psi_x $ preserves horizontal angels [6]. This is equivalent to the existence of a function $ \Lambda $ on $M$ such that $ \left_h=\Lambda(x)\left_g $, for any horizontal vectors $ X,Y $. In section 2 of this paper, we study horizontally weakly conformal maps from a Riemannian manifold into an Euclidean space equipped with an arbitrary Riemannian metric and obtain some generalizations of some results of [3] for Euclidean spaces with the canonical metric. We also give some results regarding the relationship between these maps and two dimensional Riemannian manifolds. Finally, we study the horizontal and vertical distributions of the projection map between the Euclidean spaces equipped with an arbitrary Riemannian metrics, under the condition of conformality, and obtain some generalizations of the results in [3] which have been proved there in the setting of the Euclidean spaces equipped with their canonical metric. 2. Main ResultsDefinition 2.1. [3]  Let $ \psi: (M^n,g)\rightarrow (\overline{M}^m,h)$ be a smooth map from Riemannian manifold $ (M,g) $ into Riemannian manifold $ (\overline{M},h) $. The map $ \psi $ is called horizontally weakly conformal when     * $d\psi_x=0 $, or$   * the linear transformation $ d\psi_x:T_xM\rightarrow T_{\psi(x)} \overline{M} $ is surjective and there is a number $ \Lambda(x) $, which is called square       dilation, such that for any $ X,Y\in \mathcal{H}_x $, $$\left_h=\Lambda(x)\left_g,$$     where $\mathcal{H}_x=\{\ker d\psi_x\}^\bot $ is the horizontal space.Lemma 2.2. Let $ \psi=(\psi^1,\ldots,\psi^m)$ be a smooth map from Riemannian manifold $ (M,g) $ into the Euclidean space $ \mathbb{R}^m $ (with or without a Riemannian metric). Then $ \ker d\psi=\cap_{i=1}^m\ker {d\psi}^i $ and $ \mathcal{H}_{\psi}=\mathcal{H}_{\psi^1}+\cdots+\mathcal{H}_{\psi^m} $ which for every $ i=1,\ldots,m $, $\mathcal{H}_{\psi^i}=\left<\nabla \psi^i\right> $. In particular, if $ \psi $ is a submersion then $ \mathcal{H}_{\psi}=\mathcal{H}_{\psi^1}\oplus\cdots\oplus\mathcal{H}_{\psi^m} $. Theorem 2.3. Let $ \psi=(\psi^1,\ldots,\psi^m)$ be a smooth map from Riemannian manifold $ (M,g) $ into Riemannian manifold $ (\mathbb{R}^m,h) $. Then $ \psi $ is a horizontally weakly conformal map with the square dilation $ \Lambda $ if and only if for any $ i,j=1,\ldots,m $, $\left<\nabla \psi^i,\nabla\psi^j\right>_{g}=\Lambda (h^{ij}\circ \psi)$. Proof. By Lemma 2.2 and for any $ i,j=1,\ldots,m $, we have $$\left_{h}=\Lambda \left<\nabla \psi^i,\nabla\psi^j\right>_{g},$$ and so$$\sum_{k,l=1}^md\psi^k(\nabla\psi^i)(h_{kl}\circ\psi) d\psi^l(\nabla\psi^j) =\sum_{k,l=1}^m\left<\nabla \psi^i,\nabla\psi^k\right>_{g}(h_{kl}\circ\psi) \left<\nabla \psi^l,\nabla\psi^j\right>_{g}$$ $$=\Lambda \left<\nabla \psi^i,\nabla\psi^j\right>_{g},$$ which yields the conclusion. Theorem 2.4. Let $ \psi=(\psi^1,\ldots,\psi^m)$ be a smooth map from Riemannian manifold $ (M,g) $ into Riemannian manifold $ (\mathbb{R}^m,fh_{can}) $, where $ f $ is a positive smooth function on the Euclidean space $ \mathbb{R}^m $. Then $ \psi $ is a horizontally weakly conformal map with the square dilation $ \Lambda $ if and only if for any $ i,j=1,\ldots,m $, $ \Lambda\delta_{ij}=(f\circ\psi)\left<\nabla \psi^i,\nabla \psi^j\right>_g $. Corollary 2.5. [3]   Let $ \psi=(\psi^1,\ldots,\psi^m)$ be a smooth map from Riemannian manifold $ (M,g) $ into the Euclidean space $ \mathbb{R}^m $. Then, $ \psi $ is a horizontally weakly conformal map with square dilation $ \Lambda $ if and only if for any $ i,j=1,\ldots,m $, $ \Lambda\delta_{ij}=\left<\nabla \psi^i,\nabla \psi^j\right>_g $. Theorem 2.6. [3]  Composition of two horizontally weakly conformal maps $ \psi: (M^n,g)\rightarrow (\overline{M}^m,h)$ and $ \varphi: (\overline{M}^m,h) rightarrow (\overline{\overline{M}}^k,l)$ with square dilations $ \Lambda:M\rightarrow[0,\infty) $ and $ \overline{\Lambda}:\overline{M}\rightarrow[0,\infty) $, is a horizontally weakly conformal map $ \varphi\circ\psi $ with square dilation $ \Lambda (\overline{\Lambda}\circ\psi):M\rightarrow[0,\infty) $. Two dimensional Riemannian manifolds have locally isothermal coordinates. Therefore by use of Theorems 2.4 and 2.6, we get the following result. Theorem 2.7. Consider $ \psi: (M^n,g)\rightarrow (\overline{M}^2,h)$ is a smooth map from Riemannian manifold $ (M,g) $ into two dimensional Riemannian manifold $ (\overline{M},h) $. Then $ \psi $ is a horizontally weakly conformal map if and only if for any local isothermal coordinate $ z $ (write $ z $ in a complex form) on $ \overline{M}^2 $, $ \nabla z $ is isotropic, that is $ \left<\nabla z,\nabla z\right>_g=0 $. Corollary 2.8. [3]  A smooth map from Riemannian manifold $ (M,g) $ into a Riemann surface $ \overline{M}^2 $, is a horizontally weakly conformal map if and only if for any local complex coordinate $ z $ on $ \overline{M}^2 $, $ \nabla z $ is isotropic. Lemma 2.9. Suppose $ (M^n,g) $ is a smooth Riemannian manifold and $(x^1,\ldots,x^m,x^{m+1},\ldots,x^n) $ be its local coordinate and consider distributions$ \mathcal{H}=\left<\nabla x^1,\cdots,\nabla x^m\right> $ and $ \mathcal{V}=\left<\frac{\partial}{\partial x^{m+1}},\cdots,\frac{\partial}{\partial x^{n}}\right> $. Then    * $ \mathcal{H}=\mathcal{V}^\perp $,    * The distribution $ \mathcal{H} $ is integrable if and only if for any $ i,j=1,\ldots,m $, $ r=m+1,\ldots,n $,$$ \sum_{A,B=1}^{n}\left(g^{iB}g^{jA}-g^{jB}g^{iA}\right)\dfrac{\partial g_{Ar}}{\partial x^B}=0, $$    * The distribution $ \mathcal{V} $ is integrable,    * The maximal integral manifolds of $ \mathcal{V} $ are totally geodesic if and only if for any $ i=1,\ldots,m $, $ r,s=m+1,\ldots,n $,$$\sum_{j=1}^{m}T^{ij}{\Gamma_{rs}^j}=0,$$       where $ \{\Gamma_{rs}^j \}$ are Christoffel symbols of Levi-Civita connection $ \nabla $ and $ (T_{ij})=(g^{ij}) $,    * The maximal integral manifolds of $ \mathcal{V} $ are totally umbilic if and only if for any $ i=1,\ldots,m $, $ r,s=m+1,\ldots,n $,$$\sum_{j=1}^{m}T^{ij}\Big({\Gamma_{rs}^j-\dfrac{g_{rs}}{n-m}\sum_{t,v=m+1}^{n}L^{tv}\Gamma_{tv}^j}\Big)=0,$$     where $ (L_{tv})=(g_{tv}) $. Theorem 2.10. Consider the projection map $ \psi:(\mathbb{R}^n,g)\rightarrow (\mathbb{R}^m,h) $, $ \psi(x^1,\ldots,x^m,x^{m+1},\ldots,x^n) =(x^1,\ldots,x^m) $. In any point $ \mathbf{x}\in\mathbb{R}^n $, vertical space $\mathcal{V}_{\mathbf{x}} $ is spanned by $ \{\frac{\partial}{\partial x^{m+1}},\cdots,\frac{\partial}{\partial x^{n}}\}$ and horizontal space $ \mathcal{H}_{\mathbf{x}} $ is spanned by $ \{\overset{g}{\nabla} x^1,\cdots,\overset{g}{\nabla} x^m\} $. Then $ \psi $ is a conformal submersion with the square dilation $ \Lambda $ if and only if for any $ i,j=1,\ldots,m $, $ g^{ij}=\Lambda (h^{ij}\circ \psi) $, and so    * $ \mathcal{H}=\mathcal{V}^\perp $,    * The distribution $ \mathcal{H} $ is integrable if and only if for any $ i,j=1,\ldots,m $, $ r=m+1,\ldots,n $,  $$\Lambda^2\sum_{k,l=1}^{m}\left((h^{il}h^{jk}-h^{jl}h^{ik})\circ\psi\right)~\dfrac{\partial g_{kr}}{\partial x^l}$$$$+\Lambda\sum_{s=m+1,\ldots,n,~l=1,\ldots,m}\left((h^{il}\circ\psi)~g^{js}-(h^{jl}\circ\psi)~g^{is}\right)\dfrac{\partial g_{sr}}{\partial x^l}$$$$+\Lambda\sum_{s=m+1,\ldots,n,~l=1,\ldots,m}\left(g^{is}(h^{jl}\circ\psi)-g^{js}(h^{il}\circ\psi)\right)\dfrac{\partial g_{lr}}{\partial x^s}$$$$+\sum_{t,s=m+1}^{n}\left(g^{is}g^{jt}-g^{js}g^{it}\right)\dfrac{\partial g_{tr}}{\partial x^s}=0,$$   * The distribution $ \mathcal{V} $ is integrable and its maximal integral manifolds are $ (n-m) $-dimensional planes which are fibers of $ \psi $,       * The plane fibers are totally geodesic if and only if for any $ i=1,\ldots,m $, $ r,s=m+1,\ldots,n $,$$\sum_{j=1}^{m}(h_{ij}\circ\psi){\Gamma_{rs}^j}=0,$$     where $ \{\Gamma_{rs}^j \}$ are Christoffel symbols of Levi-Civita connection $\overset{g}{\nabla} $,   * The plane fibers are totally umbilic if and only if for any $ i=1,\ldots,m $, $ r,s=m+1,\ldots,n $,$$\sum_{j=1}^{m}(h_{ij}\circ\psi)\Big({\Gamma_{rs}^j-\dfrac{g_{rs}}{n-m}\sum_{t,v=m+1}^{n}L^{tv}\Gamma_{tv}^j}\Big)=0,$$     where $ (L_{tv})=(g_{tv}) $.  Corollary 2.11. [3]  Consider the orthogonal projection $ \psi:\mathbb{R}^n\rightarrow \mathbb{R}^m $, $ \psi(x^1,\ldots,x^m,x^{m+1},\ldots,x^n)  (x^1,\ldots,x^m) $. In any point $ \mathbf{x}\in\mathbb{R}^n $, vertical space $\mathcal{V}_{\mathbf{x}} $ is spanned by $ \{\frac{\partial}{\partial x^{m+1}},\cdots,\frac{\partial}{\partial x^{n}}\}$ and horizontal space $ \mathcal{H}_{\mathbf{x}} $ is spanned by $ \{\frac{\partial}{\partial x^{1}},\cdots,\frac{\partial}{\partial x^{m}}\}$, and the map $ \psi $ is a conformal submersion with square dilation one. Distributions $ \mathcal{V} $ and $ \mathcal{H} $, are orthogonal to each other, integrable and their maximal integral manifolds are totally geodesic, and respectively are $ (n-m )$-dimensional and $ m$-dimensional planes. Theorem 2.12.Consider the double covering map $ \psi:(\mathbb{S}^n,g)\rightarrow(\mathbb{R}P^n,h_{std} )$, $ \psi(p)=[p] $, $ p\in \mathbb{S}^n$, where $ g $ is a Riemannian metric on $ \mathbb{S}^n $ and $ h_{std} $ is the standard Riemannian metric of $ \mathbb{R}P^n $. Then $ \psi $ is a conformal submersion if and only if $ g $ and the standard Riemannian metric of $ \mathbb{S}^n $ are conformally equivalent. 3. ConclusionsIn this paper, we considered the horizontally weakly conformal maps which have been studied in [3]. We generalized the results of \cite{baird2003harmonic} for Euclidean space with arbitrary Riemannian metrics and then we studied totally umbilic and totally geodesic integral manifolds of the projection map, when regarded as conformal submersion between Euclidean spaces with arbitrary Riemannian metrics.