• 호남수학학술지
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• 제30권1호
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• pp.137-146
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• 2008

Many fractal objects observed in reality are characterized by some irregularities or complexities in their features. These properties can be measured and analyzed by means of fractal dimension. However, in many cases, the calculation of this value may not be so easy to utilize in applications. In this respect, we have treated a formal method to estimate the dimension of fractal curves.

• 한국생산제조학회지
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• 제7권5호
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• pp.7-14
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• 1998

In this paper, Fractal analysis applied to evaluate machined surface profile. The spectrum method was used to calculate fractal dimension of generated surface profiles by Weierstrass-Mandelbrot fractal function. To avoid estimation errors by low frequency characteristics of FFT, the Maximum Entropy Method (MEM) was examined. We suggest a new criterion to define the MEM order m. MEM power spectrum with our criterion is proved to be advantageous by the comparison with the experimental results.

• 대한수학회보
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• 제46권1호
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• pp.1-10
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• 2009

Main purpose of this paper is to define an elliptic analogue of the Hardy sums. Some results, which are related to elliptic analogue of the Hardy sums, are given.

• The Journal of the Acoustical Society of Korea
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• 제17권1E호
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• pp.36-43
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• 1998

A fuzzy logic filter is constructed from a set of fuzzy IF-THEN rules which change adaptively to minimize some criterion function as new information becomes available. This paper generalizes the fuzzy logic filter and it's adaptive filtering algorithm to include complex parameters and complex signals. Using the complex Stone-Weierstrass theorem, we prove that linear combinations of the fuzzy basis functions are capable of uniformly approximating and complex continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, a complex orthogonal least-squares (COLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs. Also, we propose an adaptive algorithm based on LMS which adjust simultaneously filter parameters and the parameter of the membership function which characterize the fuzzy concepts in the IF-THEN rules. The modeling of a nonlinear communications channel based on a complex fuzzy is used to demonstrate the effectiveness of these algorithm.

• 한국통신학회논문지
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• 제21권1호
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• pp.251-268
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• 1996

In this paper, a higher-order feedforward neural network called ridge polynomial network (RPN) which shows good approximation capability for nonlnear continuous functions defined on compact subsets in multi-dimensional Euclidean spaces, is presented. This network provides more efficient and regular structure as compared to ordinary higher-order feedforward networks based on Gabor-Kolmogrov polynomial expansions, while maintating their fast learning property. the ridge polynomial network is a generalization of the pi-sigma network (PSN) and uses a specialform of ridge polynomials. It is shown that any multivariate polynomial can be exactly represented in this form, and thus realized by a RPN. The approximation capability of the RPNs for arbitrary continuous functions is shown by this representation theorem and the classical weierstrass polynomial approximation theorem. The RPN provides a natural mechanism for incremental function approximation based on learning algorithm of the PSN. Simulation results on several applications such as multivariate function approximation and pattern classification assert nonlinear approximation capability of the RPN.

• 대한수학회보
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• 제50권4호
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• pp.1389-1413
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• 2013

In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $\mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $\mathcal{C}_k(m,n;N)$ ($k=1,2,3,{\cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${\wp}$-function, its derivative and certain linear combination of Eisenstein series is established.

• 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.

• 한국수학사학회지
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• 제23권3호
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• pp.57-73
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• 2010

초월수의 연구는 2000년 이상 수학자들을 괴롭혀 왔던 고대 그리스의 기하학 문제의 하나인 원적문제가 불가능하다는 것을 보여줌으로써 수학사의 중요한 분야임을 입증하였다. Liouville은 1844년에 처음으로 구체적인 초월수의 예를 제시하였고, 칸토어는 1874년에 초월수의 존재성을 증명하였다. Louville 정리는 많은 초월수를 만들어 낼 뿐 아니라 초월수의 존재성을 증명하는데 이용할 수 있다. 1873년에 Hermite가 자연로그의 밑수 e가 초월수임을 보이고, 1882년에 Lindemann이 원주율 $\pi$가 초월수임 증명하였다. 1934년에 Gelfond와 Schneider는 각각 힐버트의 7번째 문제에 대한 서로 다른 완전한 해를 찾았다. 1966년에 Baker는 Gelfond-Schneider 정리의 일반화된 결과를 증명하였다. 이 연구의 목적은 초월수의 개념과 발달과정을 살피고, 미해결 문제를 제시하여 초월수의 연구가 촉진되도록 후학들에게 연구 동기를 부여하고자 한다.

• 한국지능시스템학회논문지
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• 제14권2호
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• pp.178-183
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• 2004

본 논문에서는 기존의 계층 퍼지 시스템에서 입력 변수로 사용되던 이전 계층의 출력값을 퍼지 규칙의 전건부에서는 사용하지 않고 후건부에서만 사용하는 계층 퍼지 시스템을 제안하였다. 또한 제안된 계층 퍼지 시스템을 구성할 때에 싱글톤 퍼지화기와 평균 중심법 비퍼지화기를 사용하는 경우에는 시스템 입력 변수의 멤버쉽 함수가 주어진 컴팩트 도메인 내에서 완전하기만 하다면, 임의의 연속 함수에 대하여 그에 해당하는 제안된 형태의 계층 퍼지 시스템이 존재한다는 것을 수학적으로 증명하였다.

• 대한수학회논문집
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• 제18권1호
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• pp.105-115
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• 2003

It is well known that there exists a regular branched covering map from T$^2$ onto $\={C}$ iff the ramification indices are (2,2,2,2), (2,4,4), (2,3,6) and (3,3,3). In this paper we construct (count-ably many) chaotic homeomorphisms induced by hyperbolic toral automorphism and regular branched covering map corresponding to the ramification indices (2,2,2,2). And we also gave an example which shows that the above construction of a chaotic map is not true in general if the ramification indices is (2,4,4) and also show that there are no chaotic homeomorphisms induced by hyperbolic toral automorphism and regular branched covering map corresponding to the ramification indices (2,3,6) and (3,3,3).