• Journal of applied mathematics & informatics
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• 제10권1_2호
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• pp.119-130
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• 2002

An extended Jacobin elliptic function method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations(PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation that Jacobin elliptic functions satisfy and use its solutions to replace Jacobin elliptic functions in Jacobin elliptic function method. It is interesting that many other methods are special cases of our method. Some illustrative equations are investigated by this means.

• 대한수학회보
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• 제37권4호
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• pp.675-683
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• 2000

As a by-product of [4], we give algebraic integers of certain values of quotients of Weierstrass $\delta'(\tau),\delta'(\tau)$-functions. We also show that special values of elliptic functions are transcendental numbers.

• 대한수학회지
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• 제50권4호
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• pp.847-864
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• 2013

By a change of variables we obtain new $y$-coordinates of elliptic curves. Utilizing these $y$-coordinates as meromorphic modular functions, together with the elliptic modular function, we generate the fields of meromorphic modular functions. Furthermore, by means of the special values of the $y$-coordinates, we construct the ray class fields over imaginary quadratic fields as well as normal bases of these ray class fields.

• 한국수학사학회지
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• 제28권2호
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• pp.85-102
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• 2015

Elliptic curves are a common theme among various fields of mathematics, such as number theory, algebraic geometry, complex analysis, cryptography, and mathematical physics. In the history of elliptic curves, we can find number theoretic problems on the one hand, and complex function theoretic ones on the other. The elliptic curve theory is a synthesis of those two indeed. As an overview of the history of elliptic curves, we survey the Diophantine equations of 3rd degree and the congruent number problem as some of number theoretic trails of elliptic curves. We discuss elliptic integrals and elliptic functions, from which we get a glimpse of idea where the name 'elliptic curve' came from. We explain how the solution of Diophantine equations of 3rd degree and elliptic functions are related. Finally we outline the BSD conjecture, one of the 7 millennium problems proposed by the Clay Math Institute, as an important problem concerning elliptic curves.

• 대한전자공학회논문지
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• 제20권5호
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• pp.37-42
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• 1983

Elliptic 함수는 허수 축상에 영점을 가지며 통과영역과 차단영역에서 같은 파상을 갖는데 결과적으로 elliptic 함수는 천이영역을 최소로함으로써 감도면에서 최적인 상태를 만들 수 있으나, 시간 영역에서의 특성은 Chebyshev 함수나 Butterworth 함수 응답 특성보다 떨어진다. 본 논문은 통과영역 감쇠율, 차단 주파수 ωs 등과 같이 여러 요소를 변화시켰을 때의 효과를 분석하기 위해서 단위계단 응답은 임펄스 응답을 분석해 보았다. 그 결과 다음과 같은 독특한 특성이 나타났는데 elliptic 필터 함수의 계단 응답은 높은 고유주파구에서 오버슈우트와 언더슈우트가 크고 빠르게 발생했고 초기값은 ωs가 증가하므로 감소하나 원점에서 시작하지는 않았다. Butterworth나 Chebyshev의 경우와는 다르게 임펄스 응답은 원점에서 시작하지 않는 것도 있으며 위와 같은 독특한 특성을 알아보도록 8개의 특성곡선을 제시하였다.

• 대한수학회논문집
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• 제33권4호
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• pp.1285-1301
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• 2018

This paper is devoted to the study of $Tur{\acute{a}}n$-type inequalities for some well-known special functions such as Gauss hypergeometric functions, generalized complete elliptic integrals and confluent hypergeometric functions which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality. We also apply these inequalities for some sample of interest such as incomplete beta function, incomplete gamma function, elliptic integrals and modified Bessel functions to obtain their corresponding $Tur{\acute{a}}n$-type inequalities.

• 대한수학회지
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• 제61권4호
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• pp.761-778
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• 2024

A separable minimal surface is represented by the form of f(x) + g(y) + h(z) = 0, where f, g and h are real-valued functions of x, y and z, respectively. We provide exact equations for separable minimal surfaces with elliptic functions that are singly, doubly and triply periodic minimal surfaces and completely classify all them. In particular, parameters in the separable minimal surfaces change the shape of the surfaces, such as fundamental periods and its limit behavior, within the form f(x) + g(y) + h(z) = 0.

• Advances in nano research
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• 제9권1호
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• pp.47-58
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• 2020

The present paper deals with analyzing nonlinear forced vibrational behaviors of nonlocal multi-phase piezo-magnetic beam rested on elastic substrate and subjected to an excitation of elliptic type. The applied elliptic force may be presented as a Fourier series expansion of Jacobi elliptic functions. The considered multi-phase smart material is based on a composition of piezoelectric and magnetic constituents with desirable percentages. Additionally, the equilibrium equations of nanobeam with piezo-magnetic properties are derived utilizing Hamilton's principle and von-Kármán geometric nonlinearity. Then, an exact solution based on Jacobi elliptic functions has been provided to obtain nonlinear vibrational frequencies. It is found that nonlinear vibrational behaviors of the nanobeam are dependent on the magnitudes of induced electrical voltages, magnetic field intensity, elliptic modulus, force magnitude and elastic substrate parameters.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제21권3호
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• pp.167-180
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• 2017

According to the fact that the low convergence level of the complete elliptic integral of the first kind for the modulus which having values approach to one. In this paper we propose novelty of the complete elliptic integral which having new infinite series that consists of new modulus introduced as own modulus function. We apply scheme of iteration by substituting the common modulus with own modulus function into the new infinite series. We obtained so many new exact formulas of the complete elliptic integral derived from this method correspond to the number of iterations. On the other hand, it has been also obtained a lot of new transformation functions with the corresponding own modulus functions. The calculation results show that the enhancement of the number of significant figures of the new infinite series of the complete elliptic integral of the first kind corresponds to the level of quadratic convergence.

• Korean Journal of Mathematics
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• 제7권2호
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• pp.159-166
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• 1999

For classical elliptic pseudodifferential operators $A({\lambda})$ of order $m$ > 0 with parameter ${\lambda}$ of weight ${\chi}$ > 0, it is known that $logDet_{\theta}A({\lambda})$ admits an asymptotic expansion as ${\theta}{\rightarrow}+{\infty}$. In this paper we show, with some assumptions, that the coefficients of ${\lambda}^-{\frac{n}{\chi}}$ can be expressed by the values of zeta functions at 0 for some elliptic ${\psi}$DO's on $M{\times}S^1{\times}{\cdots}{\times}S^1$ multiplied by $\frac{m}{c_{n-1}}$.