• Communications of Mathematical Education
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• v.29 no.1
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• pp.91-110
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• 2015

The purpose of this study is to investigate the understanding of pi of elementary gifted students and explore improvement direction of teaching pi. The results of this study are as follows. First, students understood insufficiently the property of approximation, constancy and infinity of pi from the fixation on 'pi = 3.14'. They mixed pi up with the approximation of pi as well. Second, they had a inclination to understand pi as algebraic formula, circumference by diameter. Third, few students understood the property of constancy and infinity of pi deeply. Lastly, the discussion activity provided the chance of finding the idea of the property of approximation of pi. In conclusion, we proposed several methods which improve the teaching of pi at elementary school.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.439-452
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• 2015

In this paper we introduce and study the new properties ($W_{\Pi}$), ($UW_{{\Pi}_a}$), ($UW_E$) and ($UW_{\Pi}$). The main goal of this paper is to study relationship between these new properties and other Weyl type theorems. Moreover, we reconsider several earlier results obtained respectively in [11], [18], [14], [1] and [13] for which we give stronger versions.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.391-401
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• 2019

A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.

• Polymer(Korea)
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• v.38 no.5
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• pp.632-639
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• 2014

2,7-Dihydroxynaphthalene bis(trimellitate anhydride) (2,7-TA) was synthesized from trimellitic anhydride chloride and 2,7-dihydroxynaphthalene. Copolyimides (Co-PI) containing ester group were synthesized from 2,7-TA, pxylylenediamine, and 2,2'-bis(trifluoromethyl)benzidine (TFB). The Co-PI films were obtained from poly(amic acid) by solution casting through thermal imidization on a glass plate. The thermal property, gas permeation, and optical transparency of the Co-PI films with various TFB monomer contents were investigated. These Co-PIs could be solution-cast into a flexible and tough film. The cast Co-PI films exhibited high optical transparency with a cut-off wavelength of 370~395 nm in UV-vis. absorption and a low yellow index value of 3.55~7.63. The thermal property of Co-PI films increased linearly with increasing TFB content. However, the oxygen permeation and optical transparency of the Co-PI films was found to worsen with increasing TFB content.

• Research in Mathematical Education
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• v.18 no.1
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• pp.1-29
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• 2014

When taught the precise definition of ${\pi}$, students may be simply asked to memorize its approximate value without developing a rigorous understanding of the underlying reason of why it is a constant. Measuring the circumferences and diameters of various circles and calculating their ratios might just represent an attempt to verify that ${\pi}$ has an approximate value of 3.14, and will not necessarily result in an adequate understanding about the constant nor formally proves that it is a constant. In this study, we aim to investigate prospective teachers' conceptual understanding of ${\pi}$, and as a constant and whether they can provide a proof of its constant property. The findings show that prospective teachers lack a holistic understanding of the constant nature of ${\pi}$, and reveal how they teach students about this property in an inappropriate approach through a proving activity. We conclude our findings with a suggestion on how to improve the situation.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1027-1040
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• 2012

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $S{\oplus}T$, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property ($gb$) and if ${\Pi}(T){\subset}{\sigma}_a(S)$, ${\PI}(S){\subset}{\sigma}_a(T)$, then $S{\oplus}T$ possesses property ($gb$) if and only if ${\sigma}_{SBF^-_+}(S{\oplus}T)={\sigma}_{SBF^-_+}(S){\cup}{\sigma}_{SBF^-_+}(T)$. Moreover, we prove that if T and S both satisfies generalized Browder's theorem, then $S{\oplus}T$ satis es generalized Browder's theorem if and only if ${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$.

• ETRI Journal
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• v.29 no.6
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• pp.838-840
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• 2007

The relation between the allowed range of variation of polarization controller wave-plates angles and the respective polarization scattering properties is investigated. It is demonstrated that a nearly uniform polarization scattering over the Poincare sphere is obtained using a concatenation of three polarization controllers with angles randomly changed between $-{\pi}$/4 and ${\pi}$/4. It is also shown that an improvement of the scattering properties is obtained if the configuration angles are allowed to change between $-{\pi}$/2 and ${\pi}$/2.

• Journal of the Korean Mathematical Society
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• v.41 no.1
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• pp.39-50
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• 2004

In [25] Taskinen shows that if $\{E_n\}_n\;and\;\{F_n\}_n$ are two sequences of Frechet spaces such that ($E_m,\;F_n$) has the BB-property for all m and n then (${\Pi}_m\;E_m,\;{\Pi}_n\;F_n$) also has the ΒΒ-property. Here we investigate when this result extends to (i) arbitrary products of Frechet spaces, (ii) countable products of DFN spaces, (iii) countable direct sums of Frechet nuclear spaces. We also look at topologies properties of ($H(U),\;\tau$) for U balanced open in a product of Frechet spaces and $\tau\;=\;{\tau}_o,\;{\tau}_{\omega}\;or\;{\tau}_{\delta}$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.925-942
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• 2016

We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing right ${\pi}$-duo as a generalization of (weakly) right duo rings. Abelian ${\pi}$-regular rings are ${\pi}$-duo, which is compared with the fact that Abelian regular rings are duo. For a right ${\pi}$-duo ring R, it is shown that every prime ideal of R is maximal if and only if R is a (strongly) ${\pi}$-regular ring with $J(R)=N_*(R)$. This result may be helpful to develop several well-known results related to pm rings (i.e., rings whose prime ideals are maximal). We also extend the right ${\pi}$-duo property to several kinds of ring which have roles in ring theory.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 1997.11a
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• pp.252-255
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• 1997

In this paper, we will propose the methodology of data analysis using the fuzzy property set model. In real world, the data can be represented with the object. $\theta$. and the property, $\pi$, and its has-property relation, P. Then, the conceptual space can be defined with the chosen properties. Each object has a unique location in the conceptual space. In Fuzzy mode, the fuzzy property, and fuzzy conceptual space can be redefined. To analyze data using the fuzzy property set model, the rough set need to be defined in the fuzzy conceptual space.