• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.855-866
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• 2020

For two essentially bounded Lebesgue measurable functions 𝜙 and ξ on unit circle 𝕋, we attempt to study properties of operators $S^k_{\mathcal{M}({\phi},{\xi})=S^k_{T_{\phi}}+S^k_{H_{\xi}}$ on H²(𝕋) (k ≥ 2), where $S^k_{T_{\phi}}$ is a k^th-order slant Toeplitz operator with symbol 𝜙 and $S^k_{H_{\xi}}$ is a k^th-order slant Hankel operator with symbol ξ. The spectral properties of operators S^k_{𝓜(𝜙,𝜙)} (or simply S^k_𝓜(𝜙)) are investigated on H²(𝕋). More precisely, it is proved that for k = 2, the Coburn's type theorem holds for S^k_𝓜(𝜙). The conditions under which operators S^k_𝓜(𝜙) commute are also explored.

• Communications of Mathematical Education
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• v.24 no.3
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• pp.731-744
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• 2010

Nowadays in school mathematics, the skill and method for solving problems are often emphasized in preference to the theoretical principles of mathematics. Students pay attention to how to make an equation mechanically before even understanding the meaning of the given problem. Furthermore they do not get to really know about the principle or theorem that were used to solve the problem, or the meaning of the answer that they have obtained. In contemporary textbooks the conic section such as circle, ellipse, parabola and hyperbola are introduced as the cross section of a cone. But they do not mention how conic section are connected with the quadratic equation or how these curves are related mutually. Students learn the quadratic equations of the conic sections introduced geometrically and are used to manipulating it algebraically through finding a focal point, vertex, and directrix of the cross section of a cone. But they are not familiar with relating these equations with the cross section of a cone. In this paper, we try to understand the quadratic curves better through the analysis of the discussion made in the process of the discovery and eventual development of the conic section and then seek for way to improve the teaching and learning methods of quadratic curves.

• Journal of the Korean School Mathematics Society
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• v.21 no.4
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• pp.401-418
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• 2018

The current geometric and vector textbooks focus on the mechanical activities of finding focus, corner, etc. through elliptic equations. In this paper, we propose a process in which analogy and analytical methods are used in reversible activities of focusing from a given elliptic graph without a coordinate plane. The exploratory tool was used as Geogebra. At first, students tried to find the focus of the ellipse by randomly constructing the major a is and the minor a is in the given ellipse. However, we have experienced a method of constructing the circle of symmetry and analyzed this principle and deduced it to the ellipse. As a result, we could construct the center, long a is and short a is of the ellipse. Then, using the analytical method, the focus formula was recognized as the Pythagorean theorem, and the ellipse's focus was constructed by using the original drawing. Therefore, it is confirmed that analogy and analytical method can positively affect the elliptical focus.