• Journal of the Korean Mathematical Society
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• v.41 no.5
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• pp.933-944
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• 2004

Packing dimension of a set is an upper bound for the packing dimensions of measures on the set. Recently the packing dimension of statistically self-similar Cantor set, which has uniform distributions for contraction ratios, was shown to be its Hausdorff dimension. We study the method to find an upper bound of packing dimensions and the upper Renyi dimensions of measures on a statistically quasi-self-similar Cantor set (its packing dimension is still unknown) which has non-uniform distributions of contraction ratios. As results, in some statistically quasi-self-similar Cantor set we show that every probability measure on it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely and it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely for almost all probability measure on it.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.241-246
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• 2007

We investigate the relation between Hausdorff(packing) measure and lower(packing) Cantor measure on a deranged Cantor set. If the infimum of some distortion of contraction ratios is positive, then Hausdorff(packing) measure and lower(packing) Cantor measure of a deranged Cantor set are equivalent except for some singular behavior for packing measure case. It is a generalization of already known result on the perturbed Cantor set.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.745-754
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• 2016

In a graph G = (V, E), a non-empty set $S{\subseteq}V$ is said to be an open packing set if no two vertices of S have a common neighbour in G. The maximum cardinality of an open packing set is called the open packing number and is denoted by ${\rho}^{\circ}$. In this paper, we examine the effect of ${\rho}^{\circ}$ when G is modified by deleting a vertex.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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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.

• Journal of Navigation and Port Research
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• v.30 no.7
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• pp.561-566
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• 2006

In this paper, we propose a new ship scheduling set packing model considering limited risk or variance. The set packing model is used in many applications, such as vehicle routing, crew scheduling, ship scheduling, cutting stock and so on. As long as the ship scheduling is concerned, there exits many unknown external factors such as machine breakdown, climate change and transportation cost fluctuation. However, existing ship scheduling models have not considered those factors apparently. We use a quadratic set packing model to limit the variance of expected cost of ship scheduling problems under stochastic spot rates. Set problems are NP-complete, and additional quadratic constraint makes the problems much harder. We implement Kelley's cutting plane method to replace the hard quadratic constraint by many linear constrains and use branch-and-bound algorithm to get the optimal integral solution. Some meaningful computational results and comments are provided.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.397-403
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• 2004

Cutler showed some duality results about Hausdorff and packing dimensions of a measure on a compact set in Euclidean space if its s-dimensional Hausdorff measure or packing measure is positive. We show that the similar results in a perturbed Cantor set hold according to its quasi s-dimensional Hausdorff measure or packing measure and we find concrete measures in this case while Cutler showed the existence of such measures. Finally under some strong condition, we give a concrete measure whose Hausdorff and packing dimensions are the same as those of the perturbed Cantor set without the condition that it has positive s-dimensional Hausdorff or packing measures.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1173-1183
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• 2017

We give some sufficient conditions for the null and infinite derivatives of the $Riesz-N{\acute{a}}gy-Tak{\acute{a}}cs$ (RNT) singular function. Using these conditions, we show that the Hausdorff dimension of the set of the infinite derivative points of the RNT singular function coincides with its packing dimension which is positive and less than 1 while the Hausdorff dimension of the non-differentiability set of the RNT singular function does not coincide with its packing dimension 1.

• International Journal of Highway Engineering
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• v.15 no.5
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• pp.31-45
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• 2013

PURPOSES : Determination of particle packing model variables that can be used for formulation of new DEM based particle packing model by examining existing particle packing models METHODS : Existing particle packing models are thoroughly examined by analytical reformulation and sensitivity analysis in order to set up DEM based new particle packing model and to determine its variables. All model equations considered in this examination are represented with consistent expressions and are compared to each others to find mathematical and conceptual similarity in expressions. RESULTS : From the examination of existing models, it is observed that the models are very similar in their shapes although the derivation of the models may be different. As well, it is observed that variables used in some existing models are comprehensive enough to estimate particle packing but not applicable to DEM simulation. CONCLUSIONS : A set of variables that can be used in DEM based particle packing model is determined.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.447-454
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• 2004

We study the Hausdorff and packing dimensions of subsets composing multifractal spectrum generated by a quasi-self-similar probability measure on a perturbed Cantor set.