• 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.

• 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.

• 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.

• Honam Mathematical Journal
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• v.29 no.2
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• pp.205-212
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• 2007

In this paper, we give some properties of the dynamics of quadratic rational maps. Using the properties we present the algorithm for drawing the Julia sets of the quadratic rational maps. We illustrate that they are fractals by computer graphics.

• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.31-38
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• 1997

We figure out geometric properties of the Julia set $J_{\alpha}$ of cubic complex polynomial $C_{\alpha}(z)=z^3 + {\alpha}z(\alpha \epsilon \mathbb{C})$ and the smallest ellipse which surrounds $J_{\alpha}$.

• Journal of Korea Game Society
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• v.14 no.6
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• pp.109-118
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• 2014

We studied the creation of fractal images with polygonal rotation symmetry. As in Loocke's method[13] we start with IFS of affine functions that create polygonal fractals and extends the IFS by adding functions that create Julia sets instead of adding square root functions. The resulting images are rotationally symmetric and Julia set shaped. Also we can improve fractal images by modifying probabilistic IFS algorithm, and we suggest a method of deforming Julia set by changing exponent value.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.545-559
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• 2018

In this paper we mainly investigate the properties of the solutions to a type of nonhomogeneous algebraic differential equation in an angular domain. It includes the Borel directions of the solutions, the width of angular domains in which the solutions take its order and the measure of radial distributions of Julia sets of the solutions.

• Research in Mathematical Education
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• v.8 no.4
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• pp.279-291
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• 2004

Mandelbrot sets and its generalizations have been extensively studied by using the Picard iterations. The purpose of this paper is to study superior Mandelbrot sets, a new class of Mandelbrot sets by introducing the Mann iterative procedure for polynomials Q$_{c}$(z) := z$^n$ + c. We generate some superior Mandelbrot sets for different values of n ($\geq$2) and these new figures are exciting and fascinating.g.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.365-371
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• 2003

In this paper, some limit functions of the iteration of the skew product in the stable sets are investigated, which is associated with finitely generated semigroup for the class M, under some additional conditions.

• Journal of the Korean Mathematical Society
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• v.33 no.1
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• pp.47-55
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• 1996

Let $0 < \alpha < 2$ and let $T_\alpha (\theta) = \alpha tan(\theta/2)$. $T_\alpha$ has an attractive fixed point at $\theta = 0$. We denote by $C(\alpha)$ the set of points in $I = [-\pi, \pi]$ which are not attracted to $\theta = 0$ by the succesive iterations of $T_\alpha$. That is, $C(\alpha)$ is the set of points in I where the dynamics of $T_\alpha$ is chaotic.