Gougu Rule for the right triangles is the Chinese Pythagorean theorem. In the late age of the Chosun Dynasty, mathematicians of Chosun pioneered the study of the Chinese Nine Chapters and other advanced mathematical problems as well as the Easternism in spite of the various difficulties after the Imchinoeran(임진왜란), Chungyuchairan(정유재란) and Byungchahoran(병자호란) The technologies of the addition and addition twice are the methods of the solution of the problems in the right triangles. This paper is intended to introduce some problems using these methods of solution.

The article is to understand the meaning of numbers on the cuboctahedron which was used in the United Shilla and to find the way to apply it into our current school curriculum. We want to reinterpret the meaning of tile numbers on this cuboctahedron. Also, we introduce this meaning to Symbols and Expressions in the 7th-grade math and expand it to Operations of Polynomials.

In this paper, we investigate the origin of Euclid's fifth postulate. For this we analyze the Euclid's proof of the Pythagorean theorem, so form a hypothesis "The Euclid's fifth postulate originated from the Pythagorean theorem." And we test our hypothesis by some historical evidences.evidences.

Infinity is very important concept in mathematics. In history of mathematics, potential infinity concept conflicts with actual infinity concept for a long time. It is reason that actual infinity concept causes difficulty in our perceptions. This phenomenon is called epistemological obstacle by Brousseau. So, in this paper, we examine the infinity in terms of mathematical didactics. First, we examine the history of development of infinity and reveal the similarity between the history of debate about infinity and episternological obstacle of students. Next, we investigate obstacle of students about infinity and the contents of curriculum which treat the infinity Finally, we suggest the methods for overcoming obstacle in learning of infinity concept.

In this article, We explained the historical origin and significance of decimal fraction, and draw some educational implications based on that. In general, it is accepted that decimal fraction was first invented by a Belgian man, Simon Stevin(1548-1620). In short, the idea of infinite decimal fraction refers to the ratio of the whole quantity to a unit. Stevin's idea of decimal fraction is significant for the history of mathematics in that it broke through the limit of Greek mathematics which separated discrete quantity from continuous quantity, and number from magnitude, and it became the origin of modern number concept. H. Eves chose the invention of decimal fraction as one of the "Great moments of mathematics."The method of teaching decimal fraction in our school mathematics tends to emphasize the computational aspect of decimal fraction too much and ignore the conceptual aspect of it. In teaching decimal fraction, like all the other areas of mathematics, the conceptual aspect should be emphasized as much as the computational aspect.al aspect.

In this paper, we consider relationship of the monastic mathematics and the arts of the middle ages. Because mathematics and arts are effects of the spirit of the ages. Here, we concern with the Rome and the religion of Christ. Next, we think of the bible and Christian doctrine and then compare with Christian arts and arts of the customs of the middle ages. The middle ages is the period of women's inequality according to the feudal system. So we investigate the correlation of the christianity, arts of the middle ages, women's inequality and monastic mathematics which is worthless in mathematics history.

Though the concept of unique factorization was formulated in tile 19th century, Euclid already had considered the prime factorization of natural numbers, so called tile fundamental theorem of arithmetic. The unique factorization of algebraic integers was a crucial problem in solving elliptic equations and the Fermat Last Problem in tile 19th century On the other hand the unique factorization of the formal power series ring were a critical problem in the past century. Unique factorization is one of the idealistic condition in computation and prime elements and prime ideals are vital ingredients in thinking and solving problems.

The recent surge of interest in the more technical aspects of nonparametric density estimation and nonparametric regression estimation has brought the subject into public view. In this paper, we investigate the general concept of the nonparametric density estimation, the nonparametric regression estimation and its performance criteria.

The RSA cryptosystem is most commonly used for providing privacy and ensuring authenticity of digital data. 1'his system is based on the difficulty of integer factoring. Many attacks had been done, but none of them devastating. They mostly illustrate the dangers of improper use of RSA. Improper use implies many aspects, but here we imply the misuse of the parameters of RSA. Specially, sizes of parameters give strong effects on the efficiency and the security of the system. Parameters are also related each other. We analyze the relation of them. Recently many researchers are interested in side-channel attacks. We also investigate partial key exposure attacks, which was motivated by side-channel attacks. If a fraction of tile secret key bits is revealed, the private key will be reconstructed. We also study mathematical background of these attacks, solving modular multivariate polynomial equations.

In this paper, we deal with partially conformal geodesic transformations in Kahler geometry by using Fermi coordinates when tile submanifold is a geodesic sphere. We derive the necessary and sufficient condition for tile existence of such transformation in terms of the Jacobi operator and its derivative.