• Journal of the Chungcheong Mathematical Society
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• v.16 no.1
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• pp.7-14
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• 2003

Let $P_c(z)=z^n+c$ be a complex polynomial with an integer $n{\geq}2$. We derive a criterion that the critical orbit of $P_c$ escapes to infinity and investigate its applications to the degree-n bifurcation set. The intersection of the degree-n bifurcation set with the real line as well as with a typical symmetric axis is explicitly written as a function of n. A well-defined escape-time algorithm is also included for the improved construction of the degree-n bifurcation set.

• Journal of the Korean Society of Safety
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• v.13 no.4
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• pp.71-78
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• 1998

Real crack and fracture surfaces have irregularities producing zigzag contours. These irregularities are analysed by a fractal geometry which has been by a Mandelbrot. We obtained a fractal dimension which is one of the fractal characteristics. It is also estimated by an vertical section method that fractal characteristics in the fractured surfaces can be obtained as the crack grows. Moreover fractal fracture energy that corresponds to an energy release rate is shown to find relationships between fractal dimensions and crack behaviors. From these results, we concluded that a fractal characteristics analysis for a crack can be applied to a fracture mechanics.

• The Pure and Applied Mathematics
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• v.9 no.2
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• pp.113-118
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• 2002

Definition and some properties of the degree-n bifurcation set are introduced. It is proved that the interval formed by the intersection of the degree-n bifurcation set with the real line is explicitly written as a function of n. The functionality of the interval is computationally and geometrically confirmed through numerical examples. Our study extends the result of Carleson & Gamelin ［2］.

• 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.

• Proceedings of the Korean Institute of Industrial Safety Conference
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• 2002.05a
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• pp.177-183
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• 2002

인명이나 물적 재산에 많은 손실을 가져주는 기계설비 및 구조물의 파괴현상에 대한 연구는 재해 원인을 분석하고 안전대책을 수립하기 위한 측면에서 대단히 중요하며, 지금까지 많은 연구가 행하여져 오호 있다 프랙탈기하학에 대한 연구는 Mandelbrot/sup l)/에 의하여 제안되어 20년 정도의 짧은 기간임에도 불구하고 여러 분야의 자연현상을 모델화하기 위하여 다양하게 발표가 되고 있다. 프랙탈 특성은 자연현상의 불규칙한 변화를 정량적으로 나타내기 위한 프랙탈차원으로 평가된다 프랙탈차원은 파면 및 균열의 불규칙성을 정량화함으로써 균열수명을 보다 더 정확히 예측하는데 적용될 수 있다.(중략)

• Proceedings of the Korea Database Society Conference
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• 2000.11a
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• pp.116-118
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• 2000

본 논문은 만델브로트 집합의 쌍곡성분과 0＜λ＜1/e에서 초월 정함수 $E_{λ}$(z)의 Julia집합의 성질에 대한 연구이다. 만델브로트 집합의 쌍곡성분은 $P_{c}$ $^{n}$ (0)의 영점을 항상 포함하고 있고 역으로 $P_{c}$ $^{n}$ (0)의 각각의 영점은 만델브로트 집합의 한 쌍곡성분에 포함된다. 그리고 $E_{λ}$(z)의 Julia 집합이 Cantor bouquet를 포함하고 있다는 사실을 Devaney 와 Tangerman의 결과를 이용하여 설명하였다.여 설명하였다.하였다.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.3 no.3
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• pp.177-182
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• 2002

In this paper, we present a mathematical theory and algorithm consoucting some fractal sets. Among such fractal sets, the degree-n bifurcation set as well as the Julia sets is defined by extending the concept of the Mandelbrot set to the complex polynomial $Z^n$+c($c{\epsilon}C$, $n{\ge}2$). Some properties of the degree-n bifurcation set and the Julia sets have been theoretically investigated including the symmetry, periodicity, boundedness, and connectedness. An efficient algorithm constructing both the degree-n bifurcation let and the Julia sets is proposed using theoretical results. The mouse-operated software called "MANJUL" has been developed for the effective construction of the degree-n bifurcation set and the Julia sets in graphic environments with C++ programming language under the windows operating system. Simple mouse operations can construct ann magnify the degree-n bifurcation set as well af the Julia sets. They not only compute the component period but also save the images of the degree-n bifurcation set and the Julia sets to visually confirm various properties and the geometrical structure of the sets. A demonstration has verified the useful versatility of MANJUL.of MANJUL.

• Journal of the Korea Computer Graphics Society
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• v.1 no.2
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• pp.248-253
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• 1995

Mandelbrot에 의하여 제안된 Random Fractal은 현실감 있는 지형의 모델링을 가능하게 하였으며. Fournier등은 수학적으로 매우 복잡한 Fractal이론을 단순화한 중간점 분할 알고리즘(Midpoint Subdivision Algorithm)을 고안하여 다양한 형태의 지형 모델링에 매우 성공적인 결과를 얻게 되었다. 그러나, Random Fractal을 응용한 여러 종류의 알고리즘들은 이것의 특성으로 인하여, 생성되는 지형의 형태를 예측하기 어려운 단점이 있다. 따라서, 본 논문에서는 중간점 분할 알고리즘을 이용하여 사용자가 원하는 형태의 지형을 모델링할 수 있는 방법에 대하여 논하였다. 전체적인 지형의 모델링 과정을 크게 전역 제어와 지역 제어의 두 단계로 구분하여, 전역 제어 단계에서 전체 지형의 개략적인 형태를 제어하여 모델링한 후 지역 제어 단계에서의 세부적인 형태제어를 통하여 최종적으로 사용자가 원하는 형태의 지형을 모델링할 수 있는 방법을 제안하였다. 또한, GUI(Graphical User Interface)를 이용하여 전역 제어와 지역 제어에서 생성되는 전체 지형의 형태를 wire frame을 이용하여 실시간에 회전시키며 점검할 수 있도록 하여 세부적인 수정을 용이하게 하였다.

• Research in Mathematical Education
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• v.8 no.4
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• pp.261-277
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• 2004

Julia sets, their variants and generalizations have been studied extensively by using the Picard iterations. The purpose of this paper is to introduce Mann iterative procedure in the study of Julia sets. Escape criterions with respect to this process are obtained for polynomials in the complex plane. New escape criterions are significantly much superior to their corresponding cousins. Further, new algorithms are devised to compute filled Julia sets. Some beautiful and exciting figures of new filled Julia sets are included to show the power and fascination of our new venture.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.63-73
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• 2002

The governing equation locating component centers in the degree-n bifurcation set is a polynomial with a very high degree and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor(n-1). Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of computational precision digits. Many cases are studied for 2 $\leq$ n $\leq$ 25 and show a remarkably improved computation.