• Honam Mathematical Journal
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• v.30 no.1
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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.

• Journal of the Korean Society of Manufacturing Technology Engineers
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• v.7 no.5
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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.

• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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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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• v.17 no.1E
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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.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.21 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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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.

• Journal for History of Mathematics
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• v.28 no.2
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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.

• Journal for History of Mathematics
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• v.23 no.3
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• pp.57-73
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• 2010

Transcendental numbers are important in the history of mathematics because their study provided that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was insoluble. Liouville established in 1844 that transcendental numbers exist. In 1874, Cantor published his first proof of the existence of transcendentals in article [10]. Louville's theorem basically can be used to prove the existence of Transcendental number as well as produce a class of transcendental numbers. The number e was proved to be transcendental by Hermite in 1873, and $\pi$ by Lindemann in 1882. In 1934, Gelfond published a complete solution to the entire seventh problem of Hilbert. Within six weeks, Schneider found another independent solution. In 1966, A. Baker established the generalization of the Gelfond-Schneider theorem. He proved that any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental. This study aims to examine the concept and development of transcendental numbers and to present students with its open problems promoting a research on it any further.

• Journal of the Korean Institute of Intelligent Systems
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• v.14 no.2
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• pp.178-183
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• 2004

This paper presents a class of hierarchical fuzzy systems where previous layer outputs are used not in IF-parts, but only in THEN -parts of the fuzzy rules of the current layer. The existence of the proposed hierarchical fuzzy system which approximates a given real continuous function on a compact set is proven if complete fuzzy sets are used in the IF-parts of the fuzzy rules with singleton fuzzifier and center average defuzzifier.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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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).