• East Asian mathematical journal
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• v.37 no.4
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• pp.401-426
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• 2021

In order to propose ways to implement mathematical connection between algebra and geometry, this study reinterpreted and visualized the Khayyam's geometric method solving the cubic equations using two conic sections and the Al-Kāshi's method of constructing of angle trisection using a cubic equation. Khayyam's method is an example of a geometric solution to an algebraic problem, while Al-Kāshi's method is an example of an algebraic a solution to a geometric problem. The construction and property of conics were presented deductively by the theorem of "Stoicheia" and the Apollonius' symptoms contained in "Conics". In addition, I consider connections that emerged in the alternating process of algebra and geometry and present meaningful Implications for instruction method on mathematical connection.

• The Mathematical Education
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• v.60 no.4
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• pp.543-554
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• 2021

The dynamic geometric environment plays a positive role in solving students' geometric problems. Students can infer invariance in change through dragging, and help solve geometric problems through the analysis method. In this study, the continuous spectrum of the dynamic geometric environment can be used to solve problems of students. The continuous spectrum can be used in the 'Understand the problem' of Polya(1957)'s problem solving stage. Visually representation using continuous spectrum allows students to immediately understand the problem. The continuous spectrum can be used in the 'Devise a plan' stage. Students can define a function and explore changes visually in function values in a continuous range through continuous spectrum. Students can guess the solution of the optimization problem based on the results of their visual exploration, guess common properties through exploration activities on solutions optimized in dynamic geometries, and establish problem solving strategies based on this hypothesis. The continuous spectrum can be used in the 'Review/Extend' stage. Students can check whether their solution is equal to the solution in question through a continuous spectrum. Through this, students can look back on their thinking process. In addition, the continuous spectrum can help students guess and justify the generalized nature of a given problem. Continuous spectrum are likely to help students problem solving, so it is necessary to apply and analysis of educational effects using continuous spectrum in students' geometric learning.

• Journal of Ocean Engineering and Technology
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• v.29 no.3
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• pp.263-269
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• 2015

This paper proposes a new approach for solving the weather routing problem by dividing it into two phases with the goal of fuel saving. The problem is to decide two optimal variables: the heading angle and speed of the ship under several constraints. In the first phase, the optimal route is obtained using the Real-Time Adaptive A* algorithm with a fixed ship speed. In other words, only the heading angle is decided. The second phase is the speed scheduling phase. In this phase, the original problem, which is a nonlinear optimization problem, is converted into a geometric programming problem. By solving this geometric programming problem, which is a convex optimization problem, we can obtain an optimal speed scheduling solution very efficiently. A simple case of numerical simulation is conducted in order to validate the proposed method, and the results show that the proposed method can save fuel compared to a constant engine output voyage and constant speed voyage.

• The Mathematical Education
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• v.46 no.3
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• pp.331-349
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• 2007

This study aims to improve the teaching and learning method on the conic sections. To do that the researcher analyzed the impact of dynamic geometry software on students' problem solving of the conic sections. Students often say, "I have solved this kind of problem and remember hearing the problem solving process of it before." But they often are not able to resolve the question. Previous studies suggest that one of the reasons can be students' tendency to approach the conic sections only using algebra or analytic geometry without the geometric principle. So the researcher conducted instructions based on the geometric and historico-genetic principle on the conic sections using dynamic geometry software. The instructions were intended to find out if the experimental, intuitional, mathematic problem solving is necessary for the deductive process of solving geometric problems. To achieve the purpose of this study, the researcher video taped the instruction process and converted it to digital using the computer. What students' had said and discussed with the teacher during the classes was checked and their behavior was analyzed. That analysis was based on Branford's perspective, which included three different stage of proof; experimental, intuitive, and mathematical. The researcher got the following conclusions from this study. Firstly, students preferred their own manipulation or reconstruction to deductive mathematical explanation or proving of the problem. And they showed tendency to consider it as the mathematical truth when the problem is dealt with by their own manipulation. Secondly, the manipulation environment of dynamic geometry software help students correct their mathematical misconception, which result from their cognitive obstacles, and get correct ones. Thirdly, by using dynamic geometry software the teacher could help reduce the 'zone of proximal development' of Vigotsky.

• Journal of Korean Institute of Industrial Engineers
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• v.17 no.2
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• pp.131-142
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• 1991

The concept of criterion function is proposed as a framework for comparing the geometric and computational characteristics of various nonlinear programming approaches to linear programming such as the method of centers, Karmakar's algorithm and the gravitational method. Also, we discuss various computational issues involved in obtaining an efficient parallel implementation of these methods. Clearly, the most time consuming part in solving a linear programming problem is the direction finding procedure, where we obtain an improving direction. In most cases, finding an improving direction is equivalent to solving a simple optimization problem defined at the current feasible solution. Again, this simple optimization problem can be seen as a least squares problem, and the computational effort in solving the least squares problem is, in fact, same as the effort as in solving a system of linear equations. Hence, getting a solution to a system of linear equations fast is very important in solving a linear programming problem efficiently. For solving system of linear equations on parallel computing machines, an iterative method seems more adequate than direct methods. Therefore, we propose one possible strategy for getting an efficient parallel implementation of an iterative method for solving a system of equations and present the summary of computational experiment performed on transputer based parallel computing board installed on IBM PC.

• Journal of the Korean Geographical Society
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• v.46 no.3
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• pp.366-381
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• 2011

A competitive location problem in discrete space is computationally difficult to solve in general because of its combinatorial feature. In this paper, we address an alternative method for solving competitive location problems in discrete space, particularly employing deterministic allocation. The key point of the suggested method is to reducing the number of predefined potential facility sites associated with the size of problem by utilizing geometric concepts. The suggested method was applied to the existing broadband marketplace with increasing competition as an application. Specifically, we compared computational results and spatial configurations of two different sized problems: the problem with the original potential sites over the study area and the problem with the reduced potential sites extracted by a GIS-based geometric algorithm. The results show that the competitive location model with the reduced potential sites can be solved more efficiently, while both problems presented the same optimal locations maximizing customer capture.

• Education of Primary School Mathematics
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• v.6 no.2
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• pp.85-91
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• 2002

The purpose of this study is to analyze elementary students' understanding the relationship between perimeter and area in geometric figures. In this study, the questionaries were used. In the survey, the subjects were elementary school students in In-cheon city. They were 86 students of the fifth grade, 86 of the sixth. They were asked to solve the problems which was designed by the researcher and to describe the reasons why they answered like that. Study findings are as following; Students have misbelief about the concept of the relationship between perimeter and area in geometric figures. Therefore, 1 propose the method fur teaching about the relationship between perimeter and area in geometric figures. That is teaching via problem solving.. In teaching via problem solving, problems are valued not only as a purpose fur learning mathematics but also a primary means of doing so. For example, teachers give the problem relating the concepts of area and perimeter using a set of twenty-four square tiles. Students are challenged to determine the number of small tiles needed to make rectangle tables. Using this, students can recognize the concept of the relationship between perimeter and area in geometric figures.

• Journal for History of Mathematics
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• v.17 no.3
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• pp.93-108
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• 2004

In this article we study on various proofs of the Steiner-Lehmus theorem(any triangle that has two equal angle bisectors is isosceles). We suggest 6 geometric proofs and 3 algebraic proofs of the theorem in detail, analyze these proofs, extract related theorems, proof ideas.

• Journal of Educational Research in Mathematics
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• v.8 no.2
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• pp.605-620
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• 1998

This paper suggested a geometry learning model which relates an exploratory learning model with GSP applications, Such a model adopts GSP's capability of visualizing dynamic geometric figures and exploratory learning method's advantages of discovering properties and relations of geometric problem proving and concepts associated with geometric inferencing of students. The research was conducted for 3 middle school students by applying the proposed model for 6times at computer laboratory. The overall procedure was videotaped so that the collected data was later analyzed by qualitative methodology. The analysis indicated that the students with less than van Hiele 4 level took advantages of adoption our proposed model to gain concrete understandings of geometric principles and concepts with GSP. One of the lessons learned from this study suggested that the roles of students and a teacher who want to employ the proposed model need to change their roles respectively.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1989.04a
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• pp.33-37
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• 1989

According to the Rayleigh-Ritz approximation method, a solution can be represented as a finite series consisting of space-dependent functions, which satisfy all the geometric boundary conditions of the problem and appropriate smoothness requirement in the interior of the domain. In this paper, an efficient formulation for solving structural dynamics systems in frequency domain is presented. A general procedure called Ritz modes (or vectors) generation algorithm is used to generate the admissible functions, i.e. Ritz modes, Then, the use of direct superposition of the Ritz modes is utilized to reduce the size of the problem in spatial dimension via geometric coordinates projection. For the reduced system, the frequency domain approach is applied. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.