• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.711-719
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• 2018

In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring R, if its maximal right quotient ring has semisimple dimension as a right R-module, then R is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right R-modules have semisimple dimension if and only if the free right R-module ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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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.

• 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.

• Environmental Engineering Research
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• v.17 no.3
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• pp.139-144
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• 2012

In this study, the effects of starvation on floc characteristics when treating saline wastewater using a sequencing batch reactor (SBR) were investigated. The effectiveness over 5 days of starvation for aerobic and non-aerobic strategies for maintaining the physical characteristics of floc-forming sludge and the recovery period needed to regain the initial pollutant removal efficiency were investigated. Experiment results revealed that the sludge volume index (SVI) increased and the floc size and fractal dimension decreased after starvation under both aerobic and non-aerobic conditions. Sludge settleability deteriorated faster under aerobic conditions compared to non-aerobic conditions. Under non-aerobic conditions, the SBR required less time to return to its initial pollutant removal efficiency and settleability. Floc size, fractal dimension, and SVI were observed to be fairly correlated with each other. The results demonstrated that it was better to maintain the sludge under non-aerobic rather than aerobic starvation, because it adapted to, resisted starvation and had a quicker re-start afterward.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.23 no.57
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• pp.51-62
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• 2000

In this study, the psycho-physiological response of drivers was investigated in terms of EEG(Electroencephalogram), especially with the fractal dimensions computed by Chaotic algorithm. The Chaotic algorithm Is well Known to sensitively analyze the non-linear information such as brain waves. An automobile with a fully equipped data acquisition system was used to collect the data. Ten healthy subjects participated in the experiment. EEG data were collected while subjects were driving the car between Won-ju and Shin-gal J.C. on Young-Dong highway The results were presented in terms of 3-Dimensional attractor to confirm the chaotic nature of the EEG data. The correlation dimension and fractal dimension were calculated to evaluate the complexity of the brain activity as the driving duration changes. In particular, the fractal dimension indicated a difference between the driving condition and non-driving condition while other spectral variables showed inconsistent results. Based upon the fractal dimension, drivers processed the most information at the beginning of the highway driving and the amount of brain activity gradually decreased and stabilized. No particular decrease of brain activity was observed even after 100 km driving. Considering the sensitivity and consistency of the analysis by Chaotic algorithm, the fractal dimension can be a useful parameter to evaluate the psycho-physiological responses of human brain at various driving conditions.

• Steel and Composite Structures
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• v.6 no.2
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• pp.87-101
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• 2006

The deformation and dynamic behavior mechanism of submerged shell-like lattice structures with membranes are in principle of a non-conservative nature as circulatory system under hydrostatic pressure and disturbance forces of various types, existing in a marine environment. This paper deals with a characteristic analysis on quasi-periodic and chaotic behavior of a circular arch under follower forces with small disturbances. The stability region chart of the disturbed equilibrium in an excitation field was calculated numerically. Then, the periodic and chaotic behaviors of a circular arch were investigated by executing the time histories of motion, power spectrum, phase plane portraits and the Poincare section. According to the results of these studies, the state of a dynamic aspect scenario of a circular arch could be shifted from one of quasi-oscillatory motion to one of chaotic motion. Moreover, the correlation dimension of fractal dynamics was calculated corresponding to stochastic behaviors of a circular arch. This research indicates the possibility of making use of the correlation dimension as a stability index.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.785-790
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• 1998

Using spectral sequences we calculate the hightest non-vanishing index of Tor for modules of finite projective dimension.

• Communications for Statistical Applications and Methods
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• v.23 no.2
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• pp.93-103
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• 2016

In the paper, we discuss dimension reduction of predictors ${\mathbf{X}}{\in}{{\mathbb{R}}^p}$ in a regression of $Y{\mid}{\mathbf{X}}$ with a notion of sufficiency that is called sufficient dimension reduction. In sufficient dimension reduction, the original predictors ${\mathbf{X}}$ are replaced by its lower-dimensional linear projection without loss of information on selected aspects of the conditional distribution. Depending on the aspects, the central subspace, the central mean subspace and the central $k^{th}$-moment subspace are defined and investigated as primary interests. Then the relationships among the three subspaces and the changes in the three subspaces for non-singular transformation of ${\mathbf{X}}$ are studied. We discuss the two conditions to guarantee the existence of the three subspaces that constrain the marginal distribution of ${\mathbf{X}}$ and the conditional distribution of $Y{\mid}{\mathbf{X}}$. A general approach to estimate them is also introduced along with an explanation for conditions commonly assumed in most sufficient dimension reduction methodologies.

• The Journal of the Society of Korean Medicine Diagnostics
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• v.11 no.2
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• pp.84-95
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• 2007

Background and Purpose: We have studied the trends of EEG signals in the voluntary breathing condition by applying the fractal analysis. According to chaos theory, irregularity of EEG signals can result from low dimensional deterministic chaos. A principal parameter to quantify the degree of Chaotic nonlinear dynamics is correlation dimension. The aim of this study was to analyze correlation between the correlation dimension of EEG and HRV(heart rate variability). We have studied the trends of EEG signals in the voluntary breathing condition by applying the fractal analysis. Methods: EEG raw data were measured by moving windows during 15 minutes. Then, the correlation dimension(D2) was calculated by each 40-seconds-segment in 15 minutes data, totally 36 segments. 8 channels EEG study on the Fp, F, T, P was carried out in 30 subjects. Results and Conclusion: Correlation analysis of HRV was calculated with deterministic non-linear data and stochastic non-linear data. 1. Ch1(Fp1), Ch4(F3), Ch4(F4) is positive correlated with In LF. 2. Ch1(Fp1), Ch3(F3) is positive correlated with In TF.

• Journal of the Korean Society of Safety
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• v.18 no.3
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• pp.22-28
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• 2003

This study proposes the construction of chaos simulator for cutting characteristics evaluation of non-ferrous metals. Also this paper aims to find the optimal cutting conditions of diamond turning machine by measuring surface form and roughness to perform the cutting experiment of non-ferrous metals, which are aluminum, with diamond tool. As well, according to change cutting conditions such as fled rate, using diamond turning machine to perform cutting processing, by measuring cutting force and surface roughness and according to cutting conditions the aluminum about cutting properties. Trajectory changes in the attractor indicated a substantial difference in fractal characteristics. Constructed chaos simulator in this study can be used for cutting characteristics evaluation of non-ferrous metals.