• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.209-224
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• 2013

The problem of finding rational or integral points of an elliptic curve basically boils down to solving a cubic equation. We look closely at the cubic formula of Cardano to find a criterion for a cubic polynomial to have a rational or integral roots. Also we show that existence of a rational root of a cubic polynomial implies existence of a solution for certain Diophantine equation. As an application we find some integral solutions of some special type for $y^2=x^3+b$.

• Honam Mathematical Journal
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• v.39 no.2
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• pp.161-174
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• 2017

The objective of this manuscript is to continue the study of fixed point theory in dislocated metric spaces, introduced by Hitzler et al. [12]. Concretely, we apply the concept of dislocated metric spaces and obtain theorems asserting the existence of common fixed points for a pair of mappings satisfying new generalized rational contractions in such spaces.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.753-771
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• 2015

A complete invariant defined for (closed connected orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself can be reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In a previous work, a characteristic lattice point invariant defined for the 3-manifolds was constructed by using an embedding of the prime links into the set of lattice points. In this paper, a characteristic rational invariant defined for the 3-manifolds called the characteristic genus defined for the 3-manifolds is constructed by using an embedding of a set of lattice points called the PDelta set into the set of rational numbers. The characteristic genus defined for the 3-manifolds is also compared with the Heegaard genus, the bridge genus and the braid genus defined for the 3-manifolds. By using this characteristic rational invariant defined for the 3-manifolds, a smooth real function with the definition interval (-1, 1) called the characteristic genus function is constructed as a characteristic invariant defined for the 3-manifolds.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1635-1646
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• 2008

Schanuel's formula describes the distribution of rational points on projective space. In this paper we will extend it to algebraic points of bounded degree in the case of ${\mathbb{P}}^1$. The estimate formula will also give an explicit error term which is quite small relative to the leading term. It will also lead to a quasi-asymptotic formula for the number of points of bounded degree on ${\mathbb{P}}^1$ according as the height bound goes to $\infty$.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.569-579
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• 2019

Cusp (parabolic) points in the extended modular group ${\bar{\Gamma}}$ are basically the images of infinity under the group elements. This implies that the cusp points of ${\bar{\Gamma}}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q{\cup}\{{\infty}\}$.The Farey graph F is the graph whose set of vertices is $Q_{\infty}$ and whose edges join each pair of Farey neighbours. Each rational number x has an integer continued fraction expansion (ICF) $x=[b_1,{\cdots},b_n]$. We get a path from ${\infty}$ to x in F as $<{\infty},C_1,{\cdots},C_n>$ for each ICF. In this study, we investigate relationships between Fibonacci numbers, Farey graph, extended modular group and ICF. Also, we give a computer program that computes the geodesics, block forms and matrix represantations.

• Korean Journal of Remote Sensing
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• v.34 no.3
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• pp.565-578
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• 2018

In this paper, we investigate the feasibility of modelling image strips, instead of individual scenes, that have been acquired from the same orbital pass through the process of bundle adjustments. Under this approach, First, a rational function model (RFM) of the strip image is generated from the RFMs of individual images, such that the entire strip of images can be treated as a single image. Correction parameters are calculated through bundle adjustments between strip images. For the experiment, we used two stereo strips. Each strip image consists of three KOMPSAT-3A scenes. Experimental results show that it was possible to improve the initial model by using the control points located in a specific region of the strip. We showed that absolute orientation with moderate accuracy of 2 m errors were achieved from 12 ground control points for the three-image strips. The test results indicate that bundle adjustment of strip images could be more efficient than bundle adjustments of the individual scenes.

• Korean Journal of Computational Design and Engineering
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• v.4 no.4
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• pp.350-354
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• 1999

The problem of the computation of derivatives arises in various applications of rational Bezier curves. These applications sometimes require the computation of derivative on numerous points. Therefore, many researches have dealt with the representation for the computation of derivatives with the small computation error. This paper compares the performances of the representations for the derivative of rational Bezier curves in the performances. The performance is measured as computation requirements at the pre-processing stage and at the computation stage based on the theoretical derivation of computational bound as well as the experimental verification. Based on this measurement, this paper discusses which representation is preferable in different situations.

• Journal of Family Resource Management and Policy Review
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• v.11 no.1
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• pp.1-20
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• 2007

This study identified financial management patterns of college students, and compared socioeconomic characteristics among different groups of financial management patterns. Also, the study examined the level of financial knowledge of college students, and compared it among the groups of financial management patterns. Data fur this study were from a questionnaire completed by 4-year college students (n=364), and were analyzed by factor analysis, cluster analysis, chi-square test, and ANOVA. The findings of this study were as follows: First, the financial management patterns were categorized by four groups: rational management group, future-oriented group, active management group, and present-oriented group. Secondly, younger students were more likely to be in the present-oriented group, while older students were likely to be in the future-oriented or active management group. Male students were likely to be the active managers, but female were likely to be the rational managers. Students' income was higher for future-oriented or active management groups, and their part-time jobs and their experiences of financial education were also significant variables. Thirdly, the average score of college students' financial knowledge was 49.9 on a 100 point basis. The part of financial assets and investment had only 47 points. The group of rational managers and active managers received higher points than the other groups.

• Journal of Korea Multimedia Society
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• v.5 no.4
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• pp.386-392
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• 2002

In this paper, we propose a novel blocking effect reduction method based on a rational B-spline. The blocking effect results from independent coding of each image block and becomes highly visible especially coded at very low bit rates. The proposed approach adopts a rational open uniform B-spline curve that used to produce a smooth curve through a set of control points. The pixels on the block boundary are treated as control points, and the weight values, which decide the shape of curve, are determined differentially by considering the distance the position of the pixels and that of the block boundary. The simulation results show that the proposed method has excellent performance for all pattern of the blocking effect with less computational complexity.

• The Mathematical Education
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• v.60 no.1
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• pp.21-39
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• 2021

The aim of this study was to deduce implications of the growth of mathematics teachers' specialty for effective instruction about the formulae of exponentiation with rational exponents by analyzing teachers' understanding of the definition of rational exponent. In order to accomplish the aim, this study ascertained the nature of the definition of rational exponent through examining previous literature and established specific research questions with reference to the results of the examination. A questionnaire regarding the nature of the definition was developed in order to solve the questions and was taken to 50 in-service high school teachers. By analysing the data from the written responses by the teachers, this study delineated four characteristics of the teachers' understanding with regard to the definition of rational exponent. Firstly, the teachers did not explicitly use the condition of the bases with rational exponents while proving f'(x)=rxr-1. Secondly, few teachers explained the reason why the bases with rational exponents must be positive. Thirdly, there were some teachers who misunderstood the formulae of exponentiation with rational exponents. Lastly, the majority of teachers thought that $(-8)^{\frac{1}{3}}$ equals to -2. Additionally, several issues were discussed related to teacher education for enhancing teachers' knowledge about the definition, features of effective instruction on the formulae of exponentiation and improvement points to explanation methods about the definition and formulae on the current high school textbooks.