• 대한수학회보
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• 제52권5호
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• pp.1489-1493
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• 2015

For a subset $E{\subseteq}\mathbb{R}^d$ and $x{\in}\mathbb{R}^d$, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by $$dim_{H,loc}(x,E)=\lim_{r{\searrow}0}dim_H(E{\cap}B(x,r))$$, $$dim_{P,loc}(x,E)=\lim_{r{\searrow}0}dim_P(E{\cap}B(x,r))$$, where $dim_H$ and $dim_P$ denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions $f,g:\mathbb{R}^d{\rightarrow}[0,d]$ with $f{\leq}g$, it is possible to choose a set E that simultaneously has f as its local Hausdorff dimension function and g as its local packing dimension function.

• 충청수학회지
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• 제28권1호
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• pp.29-38
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• 2015

We compute the Hausdorff and packing dimensions of the cylindrical lower or upper local dimension set for a self-similar measure having different minimum and maximum of the local dimension on a self-similar set satisfying the open set condition.

• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.497-503
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• 2007

We consider a non-empty subset having same local dimension of a self-similar measure on a most generalized Cantor set. We study trans-formed lower(upper) local dimensions of an element of the subset which are local dimensions of all the self-similar measures on the most generalized Cantor set. They give better information of Hausdorff(packing) dimension of the afore-mentioned subset than those only from local dimension of a given self-similar measure.

• 한국전산응용수학회:학술대회논문집
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• 한국전산응용수학회 2003년도 KSCAM 학술발표회 프로그램 및 초록집
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• pp.12.2-12
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• 2003

미분 가능한 함수가 독립변수의 각 점에서 미분계수를 가지듯이 가장 일반화된 Cantor집합의 각 점에서 weak local dimension 을 갖는다. 이러한 weak local dimension 은 두 가지가 있는데 weak lower local dimension 과 weak upper local dimension 이 있다 weak lower local dimension 은 국소적인 의미로 perturbed Cantor 집합의 lower Cantor dimension 이고 Hausdorff dimension 과 관련이 있다. weak upper local dimension 은 국소적인 의미로 perturbed Cantor 집합의 upper Cantor dimension 이고 packing dimension 과 관련이 있다. 이때 각 점에 대응하는 유관한 측도는 quasi-self-similar measure 이며 그 점의 weak lower local dimension 이 s 이면 그 점의 s-차원 quasi-self-similar measure 의 lower local dimension 이 s 가 된다. 마찬가지로 그 점의 weak upper local dimension 이 s 이면 그 점의 s-차원 quasi-self-similar measure 의 upper local dimension 이 s 가 된다. 따라서 이와 같은 사실을 이용하면 가장 일반화된 Cantor집합의 각 점에서의 weak local dimension 을 이용하여 그 집합의 Hausdorff 또는 packing 차원의 정보를 얻을 수 있을 뿐 더러 weak local dimension 을 이용한 spectrum 을 또한 구할 수 있다. 한편 weak local dimension 과 유관한 quasi-self-similar measure 는 집합의 spectrum을 생성하며 이 spectrum 을 이루는 각 부분집합의 차원에 대하여 보다 좋은 정보를 주는 transformed dimension 과 또 다른 관련을 갖게 된다.

• 대한수학회보
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• 제41권1호
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• pp.13-18
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• 2004

We defined Cantor dimensions of a perturbed Cantor set, and investigated a relation between these dimensions and Hausdorff and packing dimensions of a perturbed Cantor set. In this paper, we introduce another expressions of the Cantor dimensions. Using these, we study some informations which can be derived from power equations induced from contraction ratios of a perturbed Cantor set to give its Hausdorff or packing dimension. This application to a deranged Cantor set gives us an estimation of its Hausdorff and packing dimensions, which is a generalization of the Cantor dimension theorem.

• 대한수학회논문집
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• 제20권4호
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• pp.695-702
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• 2005

We study the transformed measures with respect to the real parameters of a self-similar measure on a self-similar Can­tor set to give a simple proof for some result of its multifractal spectrum. A transformed measure with respect to a real parameter of a self-similar measure on a self-similar Cantor set is also a self­similar measure on the self-similar Cantor set and it gives a better information for multifractals than the original self-similar measure. A transformed measure with respect to an optimal parameter deter­mines Hausdorff and packing dimensions of a set of the points which has same local dimension for a self-similar measure. We compute the values of the transformed measures with respect to the real parameters for a set of the points which has same local dimension for a self-similar measure. Finally we investigate the magnitude of the local dimensions of a self-similar measure and give some correlation between the local dimensions.

• 대한수학회보
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• 제49권5호
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• pp.1041-1055
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• 2012

The parameter lower (upper) distribution set corresponds to the cylindrical lower or upper local dimension set for a self-similarmeasure on a self-similar set satisfying the open set condition.

• 대한수학회보
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• 제48권2호
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• pp.261-275
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• 2011

We give characterizations of the differentiability points and the non-differentiability points of the Riesz-N$\'{a}$gy-Tak$\'{a}$cs(RNT) singulr function using the distribution sets in the unit interval. Using characterizations, we show that the Hausdorff dimension of the non-differentiability points of the RNT singular function is greater than 0 and the packing dimension of the infinite derivative points of the RNT singular function is less than 1. Further the RNT singular function is nowhere differentiable in the sense of topological magnitude, which leads to that the packing dimension of the non-differentiability points of the RNT singular function is 1. Finally we show that our characterizations generalize a recent result from the ($\tau$, $\tau$ - 1)-expansion associated with the RNT singular function adding a new result for a sufficient condition for the non-differentiability points.

• Korean Journal of Mathematics
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• 제16권2호
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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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권1호
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• pp.51-61
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• 2004

We find an easier formula to compute Hausdorff and packing dimensions of a subset composing a spectral class by local dimension of a self-similar measure on a self-similar Cantor set than that of Olsen. While we cannot apply this formula to computing the dimensions of a subset composing a spectral class by local dimension of a quasi-self-similar measure on a self-similar set on the way to a perturbed Cantor set, we have a set theoretical relationship between some distribution sets. Finally we compare the behaviour of a quasi-self-similar measure on a self-similar Cantor set with that on a self-similar set on the way to a perturbed Cantor set.