• East Asian mathematical journal
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• 제39권3호
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• pp.271-278
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• 2023

We consider a 1-dimensional reaction diffusion equation with the following boundary conditions arising in a theory of the thermal explosion {-u"(t) = λf(u(t)), t ∈ (0, l), -u'(0) + C(0)u(0) = 0, u'(l) + C(l)u(l) = 0, where C : [0, ∞) → (0, ∞) is a continuous and nondecreasing function, λ > 0 is a parameter and f : [0, ∞) → (0, ∞) is a continuous function. We establish the extension of Quadrature method introduced in [8]. Using this extension, we provide numerical results for models with a typical function of f and C in a thermal explosion theory, which verify the existence, uniqueness and multiplicity results proved in [6].

• East Asian mathematical journal
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• 제39권3호
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• pp.355-367
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• 2023

Extending [14], we establish the existence of multiple positive solutions for a Schrödinger-type singular elliptic equation: $$\{{-{\Delta}u+V(x)u={\lambda}{\frac{f(u)}{u^{\beta}}},\;x{\in}{\Omega}, \atop u=0,\;x{\in}{\partial}{\Omega},$$ where 0 ∈ Ω is a bounded domain in ℝ^N, N ≥ 1, with a smooth boundary ∂Ω, β ∈ [0, 1), f ∈ C[0, ∞), V : Ω → ℝ is a bounded function and λ is a positive parameter. In particular, when f(s) > 0 on [0, σ) and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

• East Asian mathematical journal
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• 제39권4호
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• pp.419-436
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• 2023

In this study, we aimed to develop a teacher-adaptive platform model that enhances teachers' instructional activities and teaching capabilities, allowing them to conduct intended lessons effectively. The platform is designed to support various teaching and learning activities based on the instructional situation, and additionally provides teaching and learning materials, assessment questions, and results. The developed teaching and learning platform utilized Moodle, an open-source-based LMS solution that provides various tools for online learning. The platform was specifically designed for teaching conic sections in high school geometry, and it was constructed to enable teachers to deliver their intended lessons effectively.

• East Asian mathematical journal
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• 제40권3호
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• pp.295-306
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• 2024

We investigate the existence of multiple positive solutions of the following elliptic equation with a Schrödinger-type term: $$\begin{cases}-{\Delta}u+V(x)u={\lambda}f(u){\quad} x{\in}{\Omega},\\{\qquad}{\qquad}{\quad}u=0, {\qquad}\;x{\in}\partial{\Omega},\end{cases}$$, where 0 ∈ Ω is a bounded domain in ℝ^N , N ≥ 1, with a smooth boundary ∂Ω, f ∈ C[0, ∞), V ∈ L^∞(Ω) and λ is a positive parameter. In particular, when f(s) > 0 for 0 ≤ s < σ and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

• East Asian mathematical journal
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• 제40권3호
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• pp.319-328
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• 2024

We establish an existence result for a positive solution to the Schrödinger-type singular semipositone problem: $-{\Delta}u\,=\,V(x)u\,=\,{\lambda}{\frac{f(u)}{u^{\alpha}}}$ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ^N , N > 2, λ ∈ ℝ is a positive parameter, V ∈ L^∞(Ω), 0 < α < 1, f ∈ C([0, ∞), ℝ) with f(0) < 0. In particular, when ${\frac{f(s)}{s^{\alpha}}}$ is sublinear at infinity, we establish the existence of a positive solutions for λ ≫ 1. The proofs are mainly based on the sub and supersolution method. Further, we extend our existence result to infinite semipositone problems with mixed boundary conditions.

• East Asian mathematical journal
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• 제31권4호
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• pp.505-525
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• 2015

The purpose of this study is to analyze teaching method of addition and subtraction of whole number in Korea and New Zealand lower grade textbook and to get some suggestive points to develop mathematics curriculum and for a qualitative improvement of textbook. To do this, we will analyze focusing on teaching material, type and method of teaching, cases of real teaching and in the case of New Zealand, we will analyze portfolios together to see what kind of things do they deal with related to addition and subtraction. From these analyzing, the results are as follows: First, the guideline of accomplishment of group of year are stated in 2009 revised curriculum in Korea but it is rough. On the other hand, the level of accomplishment from kindergarten to high school are stated divided by eight kinds of thing in New Zealand curriculum. Second, there were common and different points in the aspect of teaching material. The common points are that both of our Korea and New Zealand are using materials related to real life intimately and the diifferent points are to use technology such as calculator and computer. They are more widely used in New Zealand than our Korea. Third, Korea had used routine method mainly but New Zealand had used method to develop creativity of learner such as to write problem corresponding to expression, posing problem corresponding to information, to complete table and find pattern and to write word problem to explain pattern and so on. Fourth, we could see special calculation strategies in the case of teaching addition and subtraction such as concept of double, compensation, various strategy based on counting of number, addition of the same number, magic square, near-double which are not finding in our mathematics textbook. Fifth, in the New Zealand textbook they had used teaching methods inducing curiosity of learner such as finding message and puzzle problem than solving given problem simply.

• East Asian mathematical journal
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• 제32권4호
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• pp.565-587
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• 2016

The purpose of this study was to analyze the understanding of the meaning of fraction division and fraction division algorithm of elementary mathematical gifted students through the process of problem posing and solving activities. For this goal, students were asked to pose more than two real-world problems with respect to the fraction division of ${\frac{3}{4}}{\div}{\frac{2}{3}}$, and to explain the validity of the operation ${\frac{3}{4}}{\div}{\frac{2}{3}}={\frac{3}{4}}{\times}{\frac{3}{2}}$ in the process of solving the posed problems. As the results, although the gifted students posed more word problems in the 'inverse of multiplication' and 'inverse of a cartesian product' situations compared to the general students and pre-service elementary teachers in the previous researches, most of them also preferred to understanding the meaning of fractional division in the 'measurement division' situation. Handling the fractional division by converting it into the division of natural numbers through reduction to a common denominator in the 'measurement division', they showed the poor understanding of the meaning of multiplication by the reciprocal of divisor in the fraction division algorithm. So we suggest following: First, instruction on fraction division based on various problem situations is necessary. Second, eliciting fractional division algorithm in partitive division situation is strongly recommended for helping students understand the meaning of the reciprocal of divisor. Third, it is necessary to incorporate real-world problem posing tasks into elementary mathematics classroom for fostering mathematical creativity as well as problem solving ability.

• East Asian mathematical journal
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• 제25권2호
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• pp.179-196
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• 2009

Let E be a uniformly convex Banach space and K a nonempty closed convex subset which is also a nonexpansive retract of E. For i = 1, 2, 3, let $T_i:K{\rightarrow}E$ be an asymptotically nonexpansive mappings with sequence ${\{k_n^{(i)}\}\subset[1,{\infty})$ such that $\sum_{n-1}^{\infty}(k_n^{(i)}-1)$ < ${\infty},\;k_{n}^{(i)}{\rightarrow}1$, as $n{\rightarrow}\infty$ and F(T)=$\bigcap_{i=3}^3F(T_i){\neq}{\phi}$ (the set of all common xed points of $T_i$, i = 1, 2, 3). Let {$a_n$},{$b_n$} and {$c_n$} are three real sequences in [0, 1] such that $\in{\leq}\;a_n,\;b_n,\;c_n\;{\leq}\;1-\in$ for $n{\in}N$ and some ${\in}{\geq}0$. Starting with arbitrary $x_1{\in}K$, define sequence {$x_n$} by setting {$$x_{n+1}=P((1-a_n)x_n+a_nT_1(PT_1)^{n-1}y_n)$$ $$y_n=P((1-b_n)x_n+a_nT_2(PT_2)^{n-1}z_n)$$ $$z_n=P((1-c_n)x_n+c_nT_3(PT_3)^{n-1}x_n)$$. Assume that one of the following conditions holds: (1) E satises the Opial property, (2) E has Frechet dierentiable norm, (3) $E^*$ has Kedec -Klee property, where $E^*$ is dual of E. Then sequence {$x_n$} converges weakly to some p${\in}$F(T).

• East Asian mathematical journal
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• 제40권2호
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• pp.183-207
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• 2024

This study is a case study that considered fractional senses and fraction operation abilities for 107 gifted students in elementary school classes. In order to find out the fractional sense, in the first question comparing the sizes of fractions 2/3 and 4/5, the students showed a variety of strategies, but the utilization rate of strategies excluding reduction to a common denominator did not exceed 50%. The second question can be solved by using the first question. It is a problem of finding two fractions by selecting four from six numbers 1, 3, 4, 5, 6, and 7 to create two fractions of which sum does not exceed 1. The percentage of correct answers to this question was about 27% (29 out of 107). Only 5 out of 29 students found answers using the first question, and the rest of the students sought answers through trial and error in various calculations. It shows that the item arrangement method from a deductive perspective has no significant effect on elementary school students. The percentage of correct answers was about 27% in the questions to find out the fraction operation ability-the question of drawing a 4/3 bar using a given 3/8-sized bar and 30.7% (23 out of 75) of the students who had wrong answers showed insufficient splitting operation. In addition, it has been shown that the operation of partitioning and iterating to form numerical senses and fractional concepts related to the fractions of the students has no significant impact.

• East Asian mathematical journal
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• 제31권4호
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• pp.483-503
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• 2015

While some studies have examined the concave and convex regular polygons respectively, very little work has been done to integrate and restructure polygon shapes. Therefore, this study aims to systematically reclassify the regular polygons on the through a comprehensive analysis of previous studies on the convex and concave regular polygons. For this study, the polygon's consistency with respect to the number of sides and angles was examined. Second, the consistency on the number of diagonals was also examined. Third, the size of the interior and exterior angels of regular polygons was investigated in order to discover the consistent properties. Fourth, the consistency concerning the area in regular polygons was inspected. Last, the consistency of the central figure number in the "k-th" regular polygons was examined. Given these examinations, this study suggests a way to create a concave regular polygon from a convex regular polygon.