• East Asian mathematical journal
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• v.31 no.4
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• pp.459-481
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• 2015

In this study, we investigated whether the theorem is established even if we replace a 'square' element in the Euclidean proof of the Pythagorean theorem with different figures. At this time, we used different figures as equilateral, isosceles triangle, (mutant) a right triangle, a rectangle, a parallelogram, and any similar figures. Pythagorean theorem implies a relationship between the three sides of a right triangle. However, the procedure of Euclidean proof is discussed in relation between the areas of the square, which each edge is the length of each side of a right triangle. In this study, according to the attached figures, we found that the Pythagorean theorem appears in the following three cases, that is, the relationship between the sides, the relationship between the areas, and one case that do not appear in the previous two cases directly. In addition, we recognized the efficiency of Euclidean proof attached the square. This proving activity requires a mathematical process, and a generalization of this process is a good material that can experience the diversity and rigor at the same time.

• School Mathematics
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• v.10 no.4
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• pp.573-581
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• 2008

School geometry takes various approaches such as deductive, analytic, and vector methods. Especially, the mathematical connections between these methods are closely related to the mathematical connections between geometry and algebra. This article analysed the geometric consequences of vector algebra from the viewpoint of teacher's subject-matter knowledge and investigated the connections between the geometric proof and the algebraic proof with vector and inner product.

• The Mathematical Education
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• v.45 no.3 s.114
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• pp.367-373
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• 2006

This study suggests that it is necessary to prove that the values of three areas of a triangle, which are obtained by the multiplication of the respective base and its corresponding height, are the same. It also seeks to deeply understand the meaning of Area formula of triangles by exploring some questions raised in the analysis of the proof. Area formula of triangles expresses the invariance of congruence and additivity on one hand, and the uniqueness of parallel line, one of the characteristics of Euclidean geometry, on the other. This discussion can be applied to introducing and developing exploratory learning on area in that it revisits the ordinary thinking on area.

• East Asian mathematical journal
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• v.39 no.4
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• pp.479-492
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• 2023

Although Euclidean plane geometry is implemented in the middle school course, there are three major problems in high school space geometry that can be intuitively taken for granted or misinterpreted as circular arguments. In order to solve this problem, this study proved three major problems using sets, Euclidean plane geometry, and parallel line postulates. This corresponds to a logical sequence and has mathematical and mathematical educational values. Furthermore, it will be possible to configure spatial geometry using sets, and by giving legitimacy to non-Euclidean spatial geometry, it will open the possibility of future research.

• Journal for History of Mathematics
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• v.12 no.2
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• pp.25-39
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• 1999

In Euclidean plane geometry, areas of polygons can be computed through a finite process of cutting and pasting. The Hilbert's third problem is that a theory of volume can not be based on the idea of cutting and pasting. This problem was solved by Dehn a few months after it was posed. The purpose of this article is not only to study Hilbert's third Problem and its proof but also to provide basis for the secondary school mathematics.

• Communications of Mathematical Education
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• v.20 no.4 s.28
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• pp.621-634
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• 2006

In this study we describe the characteristics of solving geometry problems related with the ratio of segments using the principle of the lever and the center of gravity, compare and analyze this problem solving method with the traditional Euclidean proof method and the analytic method.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.897-921
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• 2018

Given bounded pseudoconvex domains in 2-dimensional complex Euclidean space, we derive analytical and geometric conditions which guarantee the Diederich-$Forn{\ae}ss$ index is 1. The analytical condition is independent of strongly pseudoconvex points and extends $Forn{\ae}ss$-Herbig's theorem in 2007. The geometric condition reveals the index reflects topological properties of boundary. The proof uses an idea including differential equations and geometric analysis to find the optimal defining function. We also give a precise domain of which the Diederich-$Forn{\ae}ss$ index is 1. The index of this domain can not be verified by formerly known theorems.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.269-277
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• 2008

Let $f=(f_n)$ be a nonnegative submartingale such that ${\parallel}f{\parallel}{\infty}{\leq}1\;and\;g=(g_n)$ be a martingale, adapted to the same filtration, satisfying $${\mid}d_{gn}{\mid}{\leq}{\mid}df_n{\mid},\;n=0,\;1,\;2,\;....$$ The paper contains the proof of the sharp inequality $$\limits^{sup}_ n\;\mathbb{E}{\Phi}({\mid}g_n{\mid}){\leq}{\Phi}(1)$$ for a class of convex increasing functions ${\Phi}\;on\;[0,\;{\infty}]$, satisfying certain growth condition. As an application, we show a continuous-time version for stochastic integrals and a related estimate for smooth functions on Euclidean domain.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.521-528
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• 2007

We present a new proof of the characterization theorem for Minkowski Pythagorean-hodograph curves in the Minkowski spaces $\mathbf{R}^{n+1,m}$. For an polynomial curves $\mathbf{s}(t)=(x_1(t),...,\;x_{n+m}(t))$, we also find Minkowski Pythagorean-hodograph curves $\mathbf{r}(t)=(x_0(t),\;x_1(t),...,\;x_{n+m}(t))$. In case m=0, Minkowski Pythagorean-hodograph curves become Pythagorean-hodograph curves in the Euclidean spaces $\mathbf{R}^{n+1}$ and Theorems in this paper hold for these Pythagorean-hodograph curves.

• (The)Study of the Eastern Classic
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• no.35
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• pp.275-295
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• 2009

The aim of this paper is to reject Graham's interpretation of bi (必) and xianzhi (先知) of Later Mohists' Mojing ("墨經") as logical necessity and a priori knowledge respectively. Graham's interpretations of them are based on his beliefs that Mojing distinguishes lun (論), the art of description from bian (辯), the art of inference in the Mohist disciplines and that the latter art should be seen as such a rigorous proof as Euclidean geometry even though it is not a Western formal logic. His beliefs also start from his distinguishing 'knowledge of names' from 'knowledge of conjunction of names and objects' according to the objects of knowledge. In my reading, the art of description and the art of inference, however, can't be sharply distinguished each other in Mojing and bi and xianzhi should be taken as suggesting both a normative necessity and an empirical necessity. A normative necessity is derived from 'normative theory of definition' which comes form the theory of rectification of names in China. The normative theory of definition, unlike the descriptive theory of definition, defines terms normatively rather than descriptively. For example, although such a definition of father, 'father is beneficient', has the form of being descriptive, but it actually is prescriptive and therefore means 'father should be beneficient'. Through this normative theory of definition, empirical knowledge, as long as it is a knowledge, is seen as necessary and so can't be wrong. To conclude, for Mohists an empirical knowledge is always a basis of an inferential knowledge or a priori knowledge, so Mohists' a priori knowledge is not really a fundamental knowledge and its necessity therefore is nothing but both a normative necessity and an empirical necessity.