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The Bracing Rectangular Frameworks in STEAM Financial Engineering Education Program for Elementary School Students

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The Bracing Rectangular Frameworks in STEAM Financial Engineering Education Program for Elementary School Students International Journal of Pure and Applied Mathematics Volume 118 No. 19 2018, 2169-2182 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu The Bracing Rectangular Frameworks in STEAM Financial Engineering Education Program for Elementary School Students Keunbae Choi1 and Namje Park2,3 1Dept. of Math. Education, Teachers College, Jeju National University [email protected] 2Department of Computer Education, Teachers College, Jeju National University [email protected] 3Elementary Education Research Institute, Jeju National University January 23, 2018 Abstract Background/Objectives: As the financial market be- comes more and more complicated and specialized due to development of IT technologies and globalization of the fi- nancial environment, financial engineering experts will come into the spotlight with the ability to analyze the situation of the financial market, to prospect the future, and analyze it scientifically and statistically based on mathematical un- derstanding and analysis ability. In this paper, we provide an algorithm to determine the rigidity of braced rectangular frameworks consisting of rectangular array of girder beams and riveted joints. Methods/Statistical analysis: The STEAM educa- tion program suggested in this paper will improve the logical 1 2169 International Journal of Pure and Applied Mathematics Special Issue decision-making ability required to analyze and estimate the uncertain future accurately with scientific thinking, as well as to acquire the basic knowledge in the economic or mathe- matical area deemed necessary for the financial engineering experts. For the purpose of a coding program, we study the matrices induced by given rectangular frameworks. Findings: We have a necessary and sufficient condition to determine the rigidity of a rectangular framework. Also, we investigate an algorithm and python code of the problem. Key Words : Rectangular framework, bipartite graph, connected graph, disconnected graph, matrix, python. 1 Introduction The new STEAM (Science, Technology, Engineering, Art, and Math) program suggested in this paper aims at helping students to have interest in financial engineering experts and to design their career creatively through the project on future promising career. The program was designed to help teachers and students understand the jobs and capabilities required for financial engineering experts through direction and execution of the financial engineering expert project. The STEAM program suggested in this paper will improve the logical decision-making ability required to analyze and estimate the uncertain future accurately with scientific thinking, as well as to acquire the basic knowledge in the economic or mathematical area deemed necessary for the financial engineering experts. With occurrence of the recent global economic crisis, this paper intends to provide the program which enables the students to acquire cre- ative thinking strategies required to handle this crisis wisely. Ulti- mately, the program will provide the students with the chance to have emotional experience through designing of financial products and communication for mutual evaluation. Many buildings are maintained by steel frameworks consisting of rectangular array of girder beams and welded or riveted joints. This is especially the case when designing high-rise buildings. However, for many reasons, these structures are treated as planar structures with pin-joints rather than rigid welds when joining the beams to- gether. The simplest form (see. Figure 1) is a rectangle consisting of four beams and four pin-joints. This structure is unstable because 2 2170 International Journal of Pure and Applied Mathematics Special Issue it can be easily deformed under sufficiently high loads as Figure 1. For the stability of the structure, it must be braced. Figure 1. Simplest rectangular framework6 In the case of larger structure containing many rectangular cells, it is possible to ensure the rigidity by attaching support rods to all the rectangular cells, but it is costly as shown in figure 2. Figure 2. Rectangular framework Actually, it is possible to ensure the rigidity of rectangular frameworks by providing the support rods in a part of the rect- angles. Then you can ask a natural question about what is the most economical way to ensure rigidity. To do this, you need to 3 2171 International Journal of Pure and Applied Mathematics Special Issue know the ways to install the supports to ensure rigidity and what 1 7 is the least costly − . In this article, we will study a necessary and sufficient condition to determine the rigidity of a rectangular framework. Also, we investigate an algorithm and python code of the problem. 2 Rectangular frameworks and bipar- tite graph The rigidity of rectangular frameworks with bracing is corresponds 1 7 to the connectivity of its bipartite graph − . Here, our bipartite graph is a graph; one set of vertices corresponds to the rows of the rectangular framework, the other set of vertices corresponds to the columns of the framework, and an edge joins a row-vertex and column-vertex if the cell in the corresponding row and column is 1 7 braced. In fact, it is well known that − A rigid bracing of rectangular frameworks corresponds to con- • nected bipartite graphs (figure 3). A non-rigid bracing of rectangular frameworks corresponds • to disconnected bipartite graphs (figure 4). A minimum rigid bracing of rectangular frameworks corresponds to spanning trees (figure 5) Figure 3. Rigid rectangular framework and bipartite graph 4 2172 International Journal of Pure and Applied Mathematics Special Issue Figure 4. Non-rigid rectangular framework and bipartite graph Figure 5. Rigid rectangular framework and spanning tree 5 2173 International Journal of Pure and Applied Mathematics Special Issue 3 Rectangular frameworks and their ma- trix In this section, for the purpose of a coding program, we study the matrices induced by given rectangular frameworks as shown in figure 6. Definition5. Let T be a rectangular framework and let MT = [aij], where aij = 1 if there is a braced cell in ith row and jth column in T, otherwise aij = 0. And let ET be the matrix deleted the rows of MT that the sum of entries is less than equal to 1. For example, Figure 6. The matrices correspond to a rectangular framework. Definition. Two row vectors rk and rl of ET (MT ) is r-connectable if there exists row vectors ri1 , ri2 , ....., riw of ET (MT ) such that r .r = 0, r .r = 0, ...., r .r = 0 k i1 6 i1 i2 6 iw il 6 where is an inner product of two row vectors of ET (MT ) . In figure 6, the connected components of M are r1, r2, r3 and r4 . T { } { } The matrix ET has only one connected component. 1. Theoretical results. The following theorem is a modification of Theorem 105. 6 2174 International Journal of Pure and Applied Mathematics Special Issue Theorem 1. A Rectangular framework T is rigid if and only if (1) All row vectors of MT is non-zero. (2) Every row of ET is r-connectable. (3) The number of non-zero column vectors of ET is equal to the number of columns of MT . Proof. Suppose that T is rigid. Then clearly (1) is satisfied. Also, the bipartite graph GT corresponding to T is connected. No- tice that zero row vectors in MT do not influence the connectedness of ET . Since a row vector ri of MT in which the only one entry is 1 is connected only one column cj in GT as shown in figure 7. That is to say, the degree of ri is 1. Hence the deleting of ri does not influence the connectedness of the remaining rows and also the row ri with degree 1 does not influence the connectedness of any two columns. Figure 7. The row with degree 1 and the connectedness of bipartite graph Thus (2) and (3) are satisfied. Conversely, we assume that the condition (1), (2) and (3) are satisfied. By (2), any two rows ri and rk of ET is connectable in GT . By (3), any cj (j=1, 2,...n) is connected to a row of ET inGT . By the condition (1), any the row of MT does not contained in ET is connected to a column in GT . Thus the bipartite graph GT is connected, and hence T is rigid8-13. Corollary 23. Rectangular framework T is rigid if and only if (1) All column vectors of MT is not zero. 7 2175 International Journal of Pure and Applied Mathematics Special Issue (2) Every row of MT is r-connectable Theorem 3. A Rectangular framework T is rigid as minimum bracings if and only if (1) The number of non-zero column vectors of ET is equal to the number of columns of MT . (2) The graph GE induced by ET is a spanning tree. Proof. Suppose that T is rigid as minimum bracing. Then by Theorem 1, the condition (1) is satisfied and also the graph GE t induced by ET = [r1r2...rk] is connected. If there exists a cycle in GE, then one edge deleting in the cycle does not influence the connectedness of GE. Let ri be the changed row of ET (corre- sponding to one edge deleting in the cycle), in which some entry 1 of the row is changed with 0, and hence one extra beam in T is deleted. Let (ri) is the changed row from ri and let S be the rect- angular framework as one extra beam is deleted in T. The matrix t ES = [r1r2...(ri)...rk] induced by S has a r connected component r1, r2, ..., (ri), ...rk and the number of non-zero columns is equal to the{ number of columns} of T. Thus S is rigid. This is a contradiction to the fact T is minimum rigid bracing of rectangular framework. Therefore (2) is satisfied.
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