• 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 $y^2=x^3+b$.

• Communications for Statistical Applications and Methods
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• v.9 no.2
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• pp.555-563
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• 2002

This paper discusses cubic equations in general saddlepoint approximations. Exact roots are found for various cases by trigonometric identities, the root which is appropriate for the general saddlepoint approximations is selected and discussed, and the defective cases in which the general saddlepoint approximations cannot be used are found.

• Journal of Korea Multimedia Society
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• v.22 no.9
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• pp.1029-1035
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• 2019

In this paper, a tentative Kth order Goldschmidt floating point number Nth root algorithm for K order convergence rate in one iteration is proposed by applying Taylor series to the Goldschmidt square root algorithm. Using the proposed algorithm, Nth root and Nth inverse root can be computed from iterative multiplications without division. It also predicts the error of the algorithm iteration. It iterates until the predicted error becomes smaller than the specified value. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a floating point number Nth root unit.

• IEMEK Journal of Embedded Systems and Applications
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• v.13 no.1
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• pp.45-51
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• 2018

In this paper, a tentative Kth order Newton-Raphson's floating point number Nth root algorithm for K order convergence rate in one iteration is proposed by applying Taylor series to the Newton-Raphson root algorithm. Using the proposed algorithm, $F^{-1/N}$ and $F^{-(N-1)/N}$ can be computed from iterative multiplications without division. It also predicts the error of the algorithm iteration and iterates only until the predicted error becomes smaller than the specified value. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a floating point number Nth root unit.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.123-132
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• 2021

Let K be a number field and L a finite abelian extension of K. Let E be an elliptic curve defined over K. The restriction of scalars ResKLE decomposes (up to isogeny) into abelian varieties over K $$Res^L_KE{\sim}{\bigoplus_{F{\in}S}}A_F,$$ where S is the set of cyclic extensions of K in L. It is known that if L is a quadratic extension, then AL is the quadratic twist of E. In this paper, we consider the case that K is a number field containing a primitive third root of unity, $L=K({\sqrt[3]{D}})$ is the cyclic cubic extension of K for some D ∈ K×/(K×)3, E = Ea : y2 = x3 + a is an elliptic curve with j-invariant 0 defined over K, and EaD : y2 = x3 + aD2 is the cubic twist of Ea. In this case, we prove AL is isogenous over K to $E_a^D{\times}E_a^{D^2}$ and a property of the Selmer rank of AL, which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.

• Textile Coloration and Finishing
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• v.2 no.2
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• pp.20-32
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• 1990

In this paper, a new method by Cubic Spline to analyze Munsell Value Function is proposed. The values calculated by this method are compared with ones by Judd's Polynomial and Cube Root Functions, etc. For performing these computation algorithms have been developed.

• Communications of Mathematical Education
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• v.30 no.4
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• pp.469-497
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• 2016

In this research, we develop a systematic learning the materials on constructibility of cubic roots. We propose two sets of materials: one is based on concepts of field, vector space, minimal polynomial in abstract algebra, another based on properties of cubic roots in elementary algebra. We assess the validity, applicability, defects and merits of developed materials through prospective teachers, in-service teachers, and professionals. It could be expected that materials be used for advanced secondary students, mathematics majoring college students and mathematics teachers. Furthermore, we may expect the materials be useful for understanding and solving the (un)constructibility problems.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.10C
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• pp.614-620
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• 2011

RSA, the most commonly used public-key cryptosystem, is based on the difficulty of factoring very large integers. The fastest known factoring algorithm is the Number Field Sieve(NFS). NFS first chooses two polynomials having common root modulo N and consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root, of which the most time consuming step is the Sieving step. However, in recent years, the importance of the Polynomial Selection step has been studied widely, because one can save a lot of time and memory in sieving and matrix step if one chooses optimal polynomial for NFS. One of the ideal ways of choosing sieving polynomial is to choose two polynomials with same degree. Montgomery proposed the method of selecting two (nonlinear) quadratic sieving polynomials. We proposed two cubic polynomials using 5-term geometric progression.

• Communications of Mathematical Education
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• v.22 no.4
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• pp.505-514
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• 2008

In this paper we study on derivation of formulas for roots of quadratic equation, cubic equation, and quartic equation through method analogy. Our argument is based on the norm form of polynomial. We also present some mathematical content knowledge related with main discussion of this article.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.217-227
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• 2009

Assuming that a given nonlinear function f : $\mathbf{R}{\rightarrow}\mathbf{R}$ has a zero $\alpha$with integer multiplicity $m{\geq}1$ and is sufficiently smooth in a small neighborhood of $\alpha$, we define extended leap-frogging Newton's method. We investigate the order of convergence and the asymptotic error constant of the proposed method as a function of multiplicity m. Numerical experiments for various test functions show a satisfactory agreement with the theory presented in this paper and are throughly verified via Mathematica programming with its high-precision computability.