• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.79-95
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• 2023

In this note, we define a Berger type deformed Sasaki metric as a natural metric on the second tangent bundle of a manifold by means of a biparacomplex structure. First, we obtain the Levi-Civita connection of this metric. Secondly, we get the curvature tensor, sectional curvature, and scalar curvature. Afterwards, we obtain some formulas characterizing the geodesics with respect to the metric on the second tangent bundle. Finally, we present the harmonicity conditions for some maps.

• Journal for History of Mathematics
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• v.24 no.3
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• pp.1-12
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• 2011

It is well known that the sexagesimal cycle(干支) has been playing very important role in ordinary human affairs including astrology and almanacs and the arts of divination(術數). Yin-Yang school related the cycle with the sixty four hexagrams and the system of five notes(五音) and twelve pitch-pipes(十二律), and the processes to relate them are called respectively NaJia(納甲) and NaYin(納音) and quoted in Shen Kuo's Meng qi bi tan(夢溪筆談, 1095). Yang Hui obtained the process NaYin in the context of mathematics. In this paper we show that Yang Hui introduced the concept and notion of functions and then using congruences and the composite of functions, he could succeed to describe perfectly the process in his Xu gu zhai qi suan fa(續古摘奇算法, 1275). We also note that his concept and notion of functions are the earliest ones in the history of mathematics.

• The Mathematical Education
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• v.45 no.1 s.112
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• pp.75-96
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• 2006

The purpose of this study was to analyze how elementary pre-service teachers learned the pedagogical content knowledge of mathematics and to understand the challenges and difficulties that they experienced in the course of student teaching. A qualitative case study provided an in-depth description of the whole three weeks of student teaching process. Four pre-service teachers and two mentor teachers participated in this study. Multiple data collection techniques were used; classroom observations, in-depth interviews, document analysis, and researcher's field notes. The results of this study showed how pre-service teachers learn PCK of mathematics in designing mathematics lessons, understanding mathematics learners and delivering mathematics lessons and what are the difficulties and challenges they experienced. Finally this study discussed about some suggestions to pre-service program and future research.

• Journal of Educational Research in Mathematics
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• v.12 no.2
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• pp.213-228
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• 2002

We know that curriculum is, first of all, related to teaching materials, namely, contents. Therefore, when we think of mathematics curriculum, we must take account of characteristic of mathematics. Vygotsky has studied the development of scientific concepts and everyday concepts. According to Vygotsky, scientific concepts grow down through spontaneous concepts; spontaneous concepts grow upward through scientific concepts. And mathematics is a representative of subjects dealing with scientific or theoretical concept. Therefore, his study provides scientific basis for mathematics curriculum design. In this context, Davydov notes that everyday concepts are developed through empirical abstraction, while scientific concepts require a theoretical abstraction. And Davydov constructed the curriculum materials for the teaching of number concept. Davydov's curriculum is an example of reflecting Vygotsky' theoretical view and his view about the types of abstraction. In particular, it represents mathematical characteristic of a 'science' by introducing number concept through quantitative relationship and use of signs. In conclusion, stance mathematical concepts have scientific characteristic, mathematics curriculum reflects this characteristic.

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.831-841
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• 1998

We shall discuss on some vanishing theorems of harmonic sections of a Riemannian vector bundle over a compact Riemannian manifold with boundary. In relating the results of H. Donnelly - P. Li ([4]), for special case of harmonic forms satisfying absolute or relative boundary problem, our results improve the vanishing results of T. Takahashi ([9]).

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.235-243
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• 2009

In this paper we will investigate the superstability of the generalized d'Alembert type functional equations ${\sum}^m_{i=1}f(x+{{\sigma}^i}(y))$ = kg(x)f(y) and ${\sum}^m_{i=1}f(x+{{\sigma}^i}(y))$ = kf(x)g(y).

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.179-189
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• 2011

The object of the present paper is to obtain a more general condition for univalence of analytic functions in the open unit disk U. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1255-1265
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• 2009

We show that the Riemannian geometry of a tangent sphere bundle of a Riemannian manifold (M, g) of constant radius $\gamma$ reduces essentially to the one of unit tangent sphere bundle of a Riemannian manifold equipped with the respective induced Sasaki metrics. Further, we provide some applications of this theorem on the $\eta$-Einstein tangent sphere bundles and certain related topics to the tangent sphere bundles.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.867-875
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• 1998

In this paper we prove that any continuous function on a bounded closed interval of can be approximated by the superposition of a bounded sigmoidal function with a fixed weight. In addition we show that any continuous function over $\mathbb{R}$ which vanishes at infinity can be approximated by the superposition f a bounded sigmoidal function with a weighted norm. Our proof is constructive.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.821-827
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• 2012

In this paper, we prove that for all odd prime powers $q$ there exist MDS(maximum distance separable) self-dual codes over $\mathbb{F}_{q^2}$ for all even lengths up to $q+1$. Additionally, we prove that there exist MDS self-dual codes of length four over $\mathbb{F}_q$ for all $q$ > 2, $q{\neq}5$.