• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.593-607
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• 2021

The aim of the present paper is to establish an intrinsic generalization of Cartan connection in Finsler geometry. This connection is called the vertical recurrent Finsler connection. An intrinsic proof of the existence and uniqueness theorem for such connection is investigated. Moreover, it is shown that for such connection, the associated semi-spray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. Explicit intrinsic expression relating this connection and Cartan connection is deduced. We also investigate some applications concerning the fundamental geometric objects associated with this connection. Finally, three important results concerning the curvature tensors associated to a special vertical recurrent Finsler connection are studied.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.363-388
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• 2022

In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. For illustration of our methods a constructive proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.

• Journal of the Korean School Mathematics Society
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• v.10 no.2
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• pp.207-219
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• 2007

Students have a hard time with a formal proof, which is one of most important part in mathematics education. They were taught the proof with algebraic visual materials using technology and specialized visual materials. But, they experienced the difficulty in justifying due to the lack of experimental recognition with the representation using technology. The specialized visual materials limited the extension of mathematics thinking of students because it worked only for the case that is fixed. In order to solve this type of problem, we made algebraic visual materials for 9th graders using technology and generalized visual materials so that students experience for themselves to help them to experience experimental justification, thus we recognized that they were improved in enhancing mathematical thinking.

• Nuclear Engineering and Technology
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• v.55 no.6
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• pp.2018-2025
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• 2023

In our previous study, we proposed an integrated PG-PET-based imaging method to increase the prediction accuracy for patient dose distributions. The purpose of the present study is to experimentally validate the feasibility of the PG-PET system. Based on the detector geometry optimized in the previous study, we constructed a dual-head PG-PET system consisting of a 16 × 16 GAGG scintillator and KETEK SiPM arrays, BaSO₄ reflectors, and an 8 × 8 parallel-hole tungsten collimator. The performance of this system as equipped with a proof of principle, we measured the PG and positron emission (PE) distributions from a 3 × 6 × 10 cm³ PMMA phantom for a 45 MeV proton beam. The measured depth was about 17 mm and the expected depth was 16 mm in the computation simulation under the same conditions as the measurements. In the comparison result, we can find a 1 mm difference between computation simulation and measurement. In this study, our results show the feasibility of the PG-PET system for in-vivo range verification. However, further study should be followed with the consideration of the typical measurement conditions in the clinic application.

• Communications of Mathematical Education
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• v.26 no.1
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• pp.85-108
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• 2012

The comprehensive implication in justification activity that includes the proof in the elementary school level where the logical and formative verification is hard to come has to be instructed. Therefore, this study has set the following issues. First, what is the mathematical justification type shown in the Number and Operations, and Geometry? Second, what are the errors shown by students in the justification process? In order to solve these research issues, the test was implemented on 62 third grade elementary school students in D City and analyzed the mathematical justification type. The research result could be summarized as follows. First, in solving the justification type test for the number and operations, students evenly used the empirical justification type and the analytical justification type. Second, in the geometry, the ratio of the empirical justification was shown to be higher than the analytical justification, and it had a difference from the number and operations that evenly disclosed the ratio of the empirical justification and the analytical justification. And third, as a result of analyzing the errors of students occurring during the justification process, it was shown to show in the order of the error of omitting the problem solving process, error of concept and principle, error in understanding the questions, and technical error. Therefore, it is prudent to provide substantial justification experiences to students. And, since it is difficult to correct the erroneous concept and mistaken principle once it is accepted as familiar content that it is required to find out the principle accepted in error or mistake and re-instruct to correct it.

• Journal of Educational Research in Mathematics
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• v.23 no.2
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• pp.173-192
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• 2013

It may be replacement proofs with understanding and explaining geometrical properties that was a remarkable change in school geometry of 2009 revised national curriculum for mathematics. That comes from the difficulties which students have experienced in learning proofs. This study focuses on one of those difficulties which are caused by the forms of proofs: using letters for designating some sides or angles in writing proofs and understanding some long sentences of proofs. To overcome it, this study aims to investigate the applicability of Byrne's method which uses coloured diagrams instead of letters. For this purpose, the proofs of three geometrical properties were taught to middle school students by Byrne's visual method using the original source, dynamic representations, and the teacher's manual drawing, respectively. Consequently, the applicability of Byrne's method was discussed based on its strengths and its weaknesses by analysing the results of students' worksheets and interviews and their teacher's interview. This analysis shows that Byrne's method may be helpful for students' understanding of given geometrical proofs rather than writing proofs.

• School Mathematics
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• v.11 no.1
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• pp.55-70
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• 2009

The purpose of this study is to implement Lakatos method in the teaching and learning of geometry for middle school students. In his landmark book , Lakatos suggested the following instructional approach: an initial conjecture was produced, attempts were made to prove the conjecture, the proofs were repeatedly refuted by counterexamples, and finally more improved conjectures and refined proofs were suggested. In the study, students were selected from the high achieving students who participated in the special mathematics and science program offered by the city council of Seoul. The students were given a contradictory geometric proposition, and expected to find the cause of the fallacy. The students successfully identified the fallacy following the Lakatos method. In this process they also set up a primitive conjecture and this conjecture was justified by the proof and refutation method. Some implications were drawn from the result of the study.

• East Asian mathematical journal
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• v.32 no.4
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• pp.483-500
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• 2016

Ancient Greek mathematician Euclid left three books about mathematics. It's 'The elements', 'The data', 'On divisions of figure'. This study is based on the analysis of Euclid's 'On divisions of figure'. 'On divisions of figure' is a book about the construction of the shape. Because, there are thirty six proposition in 'On divisions of figure', among them 30 proposition are for the construction. In this study, based on the 'On divisions of figure' we explore the direction for construction education. The results were as follows. First, the proposition of 'On divisions of figure' shall include the following information. It is a 'proposition presented', 'heuristic approach to the construction process', 'specifically drawn presenting', 'proof process'. Therefore, the content of textbooks needs a qualitative improvement in this way. Second, a conceptual basis of 'On divisions of figure' is 'The elements'. 'The elements' includes the construction propositions 25%. However, the geometric constructions contents in middle school area is only 3%. Therefore, it is necessary to expand the learning of construction in the our country mathematics curriculum.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.795-834
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• 2016

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^{\Omega}$ ($D^2$, ${\partial}D^2$) of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism Cal : $Diff^{\Omega}$ ($D^1$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to a homomorphism ${\bar{Cal}}$ : Hameo($D^2$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to that of the vanishing of the basic phase function $f_{\underline{F}}$, a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian ${\underline{F}}$ on $S^2$ that is obtained via the natural embedding $D^2{\hookrightarrow}S^2$. Here Hameo($D^2$, ${\partial}D^2$) is the group of Hamiltonian homeomorphisms introduced by $M{\ddot{u}}ller$ and the author [18]. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on $D^2$ via a study of the associated Hamiton-Jacobi equation.

• Journal of Institute of Control, Robotics and Systems
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• v.12 no.8
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• pp.729-738
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• 2006

In recent days, navigation technology becomes more important as location based service (LBS) such as E911 and telematics are considered as attractive business fields. Commercial LBS requires that navigation system should be inexpensive and available anytime and anywhere - indoors and outdoors. If we consider these requirements, it is out of question that GPS is the most favorite system in the world. However, GPS has a serious problem. The one is that GPS does not operate indoors well. This is because GPS satellites are about 20,000km above the ground so that indoor signals are too weak to be tracked in GPS receiver. And the other is that vertical accuracy is less than horizontal accuracy, because of GPS satellites' geometry. To solve these problems, many researches have been done around the world since 1990s. This paper is also one of them and we will introduce an excellent solution by use of pseudolite. Pseudolite is a kind of signal generator, which transmits GPS-like signal. So it is same as GPS satellite in ground. In this paper, we will propose the integrated navigation system of GPS and pseudolite and show the flight test results using RC airplane to proof our navigation system. As a result, we could improve the vertical accuracy of airplane into the horizontal accuracy.