• Korean Journal of Mathematics
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• v.16 no.2
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• pp.157-163
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• 2008

Using informations of subsets of divergence points and the relation between members of spectral classes, we give another complete decomposition of spectral classes generated by lower(upper) local dimensions of a self-similar measure on a self-similar Cantor set with full information of their dimensions. We note that it is a complete refinement of the earlier complete decomposition of the spectral classes. Further we study the packing dimension of some uncountable union of distribution sets.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.469-475
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• 2004

We generalize S. Ikeda's results for perturbed cantor sets showing how we get the dimensions for deformed self-similar sets.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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• pp.231-236
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• 2010

Using the properties of the concave function, we show that the Hausdorff dimension of the product $C_{\frac{a+b}{2},\frac{a+b}{2}}{\times}C_{\frac{a+b}{2},\frac{a+b}{2}}$ of the same symmetric Cantor sets is greater than that of the product $C_{a,b}{\times}C_{a,b}$ of the same anti-symmetric Cantor sets. Further, for $1/e^2$ < a, b < 1/2, we also show that the dimension of the product $C_{a,a}{\times}C_{b,b}$ of the different symmetric Cantor sets is greater than that of the product $C_{\frac{a+b}{2},\frac{a+b}{2}}{\times}C_{\frac{a+b}{2},\frac{a+b}{2}}$ of the same symmetric Cantor sets using the concavity. Finally we give a concrete example showing that the latter argument does not hold for all 0 < a, b < 1/2.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.827-835
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• 2005

As a tool of measuring the irregularity of curve, fractal dimensions can be used. For an irregular function, fractional calculus are more available. However, to know its fractional differentiability which is related to its complexity is complicated one. In this paper, variants of the Hausdorff dimension and the packing dimension as well as the derivative order are defined and the relations between them are investigated so that the differentiability of fractal curve can be explained through its complexity.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.731-749
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• 2005

In this paper, we study the two dimensional g-Navier­Stokes equations on some unbounded domain ${\Omega}\;{\subset}\;R^2$. We prove the existence of the global attractor for the two dimensional g-Navier­Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.245-250
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• 2009

We consider a code function from the unit interval which has a generalized dyadic expansion into a coding space which has an associated ultra metric. The code function is not a bi-Lipschitz map but a dimension-preserving map in the sense that the Hausdorff and packing dimensions of any subset in the unit interval and its image under the code function coincide respectively.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2003.09a
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• pp.372-375
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• 2003

The incidence of blindness resulting from diabetic retinopathy has significantly increased despite the intervention of insulin to control diabetes mellitus. Early signs are microaneurysms, exudates, intraretinal hemorrhages, cotton wool patches, microvascular abnormalities, and venous beading. Advanced stages include neovascularization, fibrous formations, preretinal and vitreous microhemorrhages, and retinal detachment. Microaneurysm count is important because it is an indicator of retinopathy progression. The purpose of this paper is to apply data mining to detect diabetic retinopathy patterns in routine fundus fluorescein angiography. Early symptoms are of principal interest and therefore the emphasis is on detecting microaneurysms rather than vessel tortuosity. The analysis does not involve image-recognition algorithms. Instead, mathematical filtering isolates microaneurysms, microhemorrhages, and exudates as objects of disconnected sets. A neural network is trained on their distribution to return fractal dimension. Hausdorff and box counting dimensions grade progression of the disease. The field is acquired on fluorescein angiography with resolution superior to color ophthalmoscopy, or on patterns produced by physical or mathematical simulations that model viscous fingering of water with additives percolated through porous media. A mathematical filter and neural network perform the screening process thereby eliminating the time consuming operation of determining fractal set dimension in every case.

• 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.