• 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 $y^2=x^3+b$.

• International Journal of Control, Automation, and Systems
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• v.5 no.4
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• pp.471-478
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• 2007

In this paper, an optimal Wiener-Hopf decoupling controller formula is obtained which is expressed in terms of rational matrices, thereby readily allowing the use of state-space algorithms. To this end, the characterization formula for the class of all realizable decoupling controller is formulated in terms of rational functions. The class of all stabilizing and decoupling controllers is parametrized via the free diagonal matrices and the optimal decoupling controller is determined from these free matrices.

• Journal of the Korean Society of Hazard Mitigation
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• v.7 no.4
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• pp.67-74
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• 2007

Rational method has been widely used to calculate peak runoff drainage design or small watershed because of simplicity and convenience. Runoff coefficient(C) is the most important parameter in the rational method which varies according to rainfall intensity, return period, rainfall duration time and soil characteristics. In practice, constant which is value of C in rational formula has been used from the table, originally based on ASCE. These table value does not consider the upper conditions of the depending factors, hence peak runoff calculation could be in correct. Therefore to calculate C in this paper we have devised an improved formula, considering relationship with rainfall duration, return period and CN of NRCS method. This formula is considered to be more reliable and helpful to the hydrologists and engineers to predict correct peak runoff.

• Journal of the Korean Chemical Society
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• v.13 no.1
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• pp.56-61
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• 1969

By the chemical procedure and infra-red spectroscopy, the characteristics of humic acid were studied. The results were as follows: 1. Molecular weight. 5,200. 2. Molecular formula. $C_{240}H_{250}O_{120}N_{1O}$. 3. Rational formula. 4. Confirmation of the accurate structure of humic acid is beyond us nowadays. The structure is speculated that it may be a kind of condensed polymer of many benzene kerns in which above mentioned various functional groups are attached. Also some part of the large quantities of oxygen would be furan type carbonyl and aliphatic ethereal forms.

• Kyungpook Mathematical Journal
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• v.52 no.3
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• pp.347-357
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• 2012

When Newton's method, or Halley's method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

• Korean Journal of Mathematics
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• v.30 no.1
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• pp.43-51
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• 2022

The aim of this paper is to obtain a Radau type quadrature formula for a rational interpolation process satisfying the almost quasi Hermite Fejér interpolatory conditions on the zeros of Chebyshev Markov sine fraction on [-1, 1).

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.919-947
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• 2009

In this paper, we give a recursive formula for the Jones polynomial of a 2-bridge knot or link with Conway normal form C($-2n_1$, $2n_2$, $-2n_3$, ..., $(-1)_r2n_r$) in terms of $n_1$, $n_2$, ..., $n_r$. As applications, we also give a recursive formula for the Jones polynomial of a 3-periodic link $L^{(3)}$ with rational quotient L = C(2, $n_1$, -2, $n_2$, ..., $n_r$, $(-1)^r2$) for any nonzero integers $n_1$, $n_2$, ..., $n_r$ and give a formula for the span of the Jones polynomial of $L^{(3)}$ in terms of $n_1$, $n_2$, ..., $n_r$ with $n_i{\neq}{\pm}1$ for all i=1, 2, ..., r.

• 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 $\infty$.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.29 no.1
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• pp.67-75
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• 2016

In this paper, a pressure-time history curve of blast load and Conwep model are presented, and a simplified blast load formula is suggested. Generally, a blast load are applied as a pressure-time history curve, and it is calculated by blast load formula such as Conwep model. The Conwep model which is used in most of the blast analysis is quiet difficult to calculate because of its complex process. Therefore, a simplified formula is proposed to calculate blast load by simple rational expressions and to make a simplified pressure-time history curve. In this process, a curve fitting method was used to find the simple rational expressions. The calculation results of the simplified formula have an error of less than 1% in comparison with the Conwep model. And, blast analyses using finite elements method are accomplished with the Conwep model and simplified formula for verification.

• Honam Mathematical Journal
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• v.43 no.1
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• pp.88-99
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• 2021

The purpose of this paper is to assign a movable frame to an arbitrary point of a rational Bézier curve on the 2-sphere S2 in Euclidean 3-space R3 to provide a better understanding of the geometry of the curve. Especially, we obtain the formula of geodesic curvature for a quadratic rational Bézier curve that allows a curve to be characterized on the surface. Moreover, we give some important results and relations for the Darboux frame and geodesic curvature of a such curve. Then, in specific case, given characterizations for the quadratic rational Bézier curve are illustrated on a unit 2-sphere.