• 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.

• The Transactions of the Korea Information Processing Society
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• v.4 no.2
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• pp.596-605
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• 1997

The synamic systrm emplohing the self-squared function, , $f(Z)=z^2+c$, provides the Mandelbrot set which classifies constants c using the divergence of the sequence starting from the origin.To speed-up the generation of Mandelbrot set images, two approaches , called as the divide-and-conquer technique and the triangular boundary tracing technique, have been developed.However , the divede-and-conquer technique genrates sequences of some pixels that so not affect graphical representations of the Mandelbrot set.The triangular boundary tracing tech-mique does noot represent some 8-connected components of the Mandelbrost set.In this paper, we prorose a new ;method which solves the 8-connectivity problem of triangular boundary tracing technique.This algorithm considers the divergence for only pixels which are essential to the graphical repressentation of the Mandelbrot set.It also foves good representations for 8-connected components like hairly structures.

• 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 Korean Society of Industrial and Systems Engineering
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• v.19 no.37
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• pp.233-240
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• 1996

Julia set, Fatou set와 Mandelbrot set 가 컴퓨터에 의하여 도형화된 후부터 혼돈 역학체계 (chaotic dynamical system)에 대한 연구가 모든 학계에 비상한 관심을 모으고 있으며 특히 수학자들에 의하여 많은 연구가 이루어지고 있다. 또한 혼돈 역학체계를 기초로 하여 컴퓨터 그래픽스를 이용한 후랙탈(fractal)들의 매혹적인 시각적 표현으로 인하여 최근들어 과학자들 뿐 아니라 일반대중의 후랙탈에 대한 관심이 매우 높아지고 있다. 후랙탈이란 말은 라틴어 fractus(부서진 상태를 뜻함)에서 유래되었으며 1975년 Mandelbrot가 수학 및 자연계의 비정규적 패턴들에 대한 체계적 고찰을 담은 자신의 에세이의 표제를 주기 위해서 만들었다(〔6〕). 후랙탈을 기술하는데 있어서 가장 중요한 양은 차원(dimension)으로, 예컨데 Cantor 1/3 집합은 길이 1인 선분으로부터 시작하야 매 단계마다 모든 선분들의 가운데 1/3을 잘라내는 것을 무한히 반복함으로써 얻어지는데 이 집합의 Lebesgue measure는 0이지만 후랙탈 차원은 log2/log3 로 정수차원이 아닌 실수차원을 갖으며 또한 Cantor 1/3집합은 연속이 아니면서 점도 선도 아닌 집합인 것이다. 이 논문에서는 Box counting dimension 과 Hausdorff dimension에 대한 몇 가지 정의를 하고 정리 2.6, 정리2.7 및 정리 3.3을 증명함으로써 어떤 성질을 갖는 후랙탈의 가장 중요한 양인 후랙탈 차원에 대하여 논의 하고자 한다.

• Communications of Mathematical Education
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• v.18 no.1 s.18
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• pp.239-248
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• 2004

본 논문에서는 맨델브로트(Mandelbrot) 집합의 개념을 3차의 복소 다항식 z^3$+c 에 확장시켜 3차 분기집합을 정의하고, 이 집합의 2-주기 성분의 경계선 방정식과 관련 기하학적 성질을 고등학교 및 대학에서 다루는 미적분학 관점에서 분석하고자 한다. 복소수, 삼각함수, 매개함수, 함수의 극값, 미분 및 적분 등의 기초 이론을 활용하여 2-주기 성분의 경계선 방정식을 매개함수로 표시하고, 경계선의 내부 면적, 둘레 길이, 무게중심 등을 이론적으로 기술한다. 수학 소프트웨어인 매스매티카(Mathematica)를 활용하여 2-주기성분의 작도 및 기하학적 성질에 관한 수치 해석적 결과를 제시하고자 한다.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.7 no.6
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• pp.1421-1424
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• 2006

By extending the Mandelbrot set for the complex polynomial $$M={c\in C\;:\; _{k\rightarrow\infty}^{lim}P_c^k(0)\;{\neq}\;{\infty}$$ we define the degree-n bifurcation set. In this paper, we formulate the boundary equation of a period-2 component on the main component in the degree-9 bifurcation set by parameterizing its image. We establish an algorithm constructing a period-2 component in the degree-9 bifurcation set and the typical implementations show the satisfactory result with Mathematica codes grounded on the analysis.

• Proceedings of the KAIS Fall Conference
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• 2002.05a
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• pp.115-117
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• 2002

In this paper, 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 $z^{n}{\;}+{\;}c(c{\;}\in{\;}C,{\;}n{\;}\geq{\;}2)$. Some properties of the degree-n bifurcation set and the Julia sets have been theoretically investigated including the symmetry, periodicity, boundedness, connectedness and the bifurcation points as well as the governing equation for the component centers. An efficient algorithm constructing both the degree-n bifurcation set and the Julia sets is proposed using theoretical results. The mouse-operated software calico "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 and magnify the degree-n bifurcation set as well as the Julia sets. They not only compute the component period, bifurcation points and component centers 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.

• The KIPS Transactions:PartA
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• v.19A no.3
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• pp.129-138
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• 2012

In distributed memory architectures like clusters, each node stores a portion of data. How data is distributed across nodes influences the performance of such systems. The data distribution scheme is the strategy to distribute data across nodes and realize parallel data processing. Due to various reasons such as maintenance, scale up, upgrade, etc., the performance of nodes in a cluster can often become non-identical. In such clusters, data distribution without considering performance cannot efficiently distribute data on nodes. In this paper, we propose a new data distribution scheme based on the number of cores in nodes. We use the number of cores as the performance factor. In our data distribution scheme, each node is allocated an amount of data proportional to the number of cores in it. We implement our data distribution scheme using the Chapel language. To show our data distribution is effective in reducing the execution time of parallel applications, we implement Mandelbrot Set and ${\pi}$-Calculation programs with our data distribution scheme, and compare the execution times on a cluster. Based on experimental results on clusters of 8-core and 16-core nodes, we demonstrate that data distribution based on the number of cores can contribute to a reduction in the execution times of parallel programs on clusters.