• 대한수학회논문집
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• 제33권2호
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• pp.459-471
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• 2018

We obtain a relation between generalized Hausdorff and packing multifractal premeasures and generalized Hausdorff and packing multifractal measures. As an application, we study a general formalism for the multifractal analysis of one probability measure with respect to an other.

• 대한수학회논문집
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• 제37권4호
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• pp.1073-1097
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• 2022

In this paper, we shall be concerned with evaluation of multifractal Hausdorff measure 𝓗^q,t_𝜇 and multifractal packing measure 𝓟^q,t_𝜇 of Cartesian product sets by means of the measure of their components. This is done by investigating the density result introduced in [34]. As a consequence, we get the inequalities related to the multifractal dimension functions, proved in [35], by using a unified method for all the inequalities. Finally, we discuss the extension of our approach to studying the multifractal Hewitt-Stromberg measures of Cartesian product sets.

• Structural Engineering and Mechanics
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• 제63권6호
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• pp.751-760
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• 2017

In recent years, multifractal-based analysis methods have been widely applied in engineering. Among these methods, multifractal detrended cross-correlation analysis (MFDXA), a branch of fractal analysis, has been successfully applied in the fields of finance and biomedicine. For its great potential in reflecting the subtle characteristic among signals, a structural health monitoring (SHM) system based on MFDXA is proposed. In this system, damage assessment is conducted by exploiting the concept of multifractal theory to quantify the complexity of the vibration signal measured from a structure. According to the proposed algorithm, the damage condition is first distinguished by multifractal detrended fluctuation analysis. Subsequently, the relationship between the q-order, q-order detrended covariance, and length of segment is further explored. The dissimilarity between damaged and undamaged cases is visualized on contour diagrams, and the damage location can thus be detected using signals measured from different floors. Moreover, a damage index is proposed to efficiently enhance the SHM process. A seven-story benchmark structure, located at the National Center for Research on Earthquake Engineering (NCREE), was employed for an experimental verification to demonstrate the performance of the proposed SHM algorithm. According to the results, the damage condition and orientation could be correctly identified using the MFDXA algorithm and the proposed damage index. Since only the ambient vibration signal is required along with a set of initial reference measurements, the proposed SHM system can provide a lower cost, efficient, and reliable monitoring process.

• Korean Journal of Mathematics
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• 제28권2호
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• pp.323-341
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• 2020

In this note, we investigate the multifractal analogues of the Hewitt-Stromberg measures and dimensions in a probability space.

• Kyungpook Mathematical Journal
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• 제45권4호
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• pp.503-510
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• 2005

Let $\left{K_{\alpha}\right}_{{\alpha}{\in}{\mathbb{R}}}$ be the multifractal spectrums of a perturbed Cantor set K. We find the set of values ${\alpha}$ of nonempty set $K_{\alpha}$ by using the Birkhoff ergodic theorem. And we also show that such $K_{\alpha}$ is a fractal set in the sense of Taylor [12].

• 한국수자원학회논문집
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• 제47권9호
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• pp.839-852
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• 2014

본 연구에서는 강우의 시공간적 멀티프랙탈 특성에 기반을 둔 다운스케일링 알고리즘(RDSTMF-Rainfall Downscaling in Space-Time Multifractal Framework)을 한반도에 적용하여 그 적용성을 살펴보았다. 이를 위하여 2008년부터 2012년까지 우리나라에 호우주의보를 일으킨 8개의 이벤트에 대한 레이더강우자료를 분석하여 각 이벤트에 대한 멀티프랙탈 지수를 판별하였으며, 이에 근거하여 RDSTMF의 모수들을 산정하고 이 모수들과 시공간강우장의 평균강우량과의 관계를 도출하였다. 이 관계에 근거하여 RDSTMF를 사용하여 가상의 시공간강우장을 생성, 관측 시공간 강우장과 비교하였다. 비교 결과, RDSTMF를 사용하여 생성된 가상의 시공간 강우장은 관측 시공간 강우장의 멀티프랙지수를 3차 모멘트까지 정확히 모사함을 확인하였으며, 누적분포함수 또한 비교적 정확히 모사함을 확인하였다.

• 자원환경지질
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• 제55권3호
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• pp.231-239
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• 2022

The irregular shape of the disturbances is a fundamental issue for mining engineers at the Sidi Chennane phosphate deposit in Morocco. A precise classification of disturbed areas is therefore necessary to understand their part in the overall volume of phosphate. In this paper, we investigate the theoretical and practical aspects of studying and measuring multifractal spectrums as a defining and representative parameter for distinguishing between the phosphate deposit of a low rate of disturbances and the deposit of a high rate. An empirical multifractal approach was used by analyzing the disturbed areas through the geoelectric images of an area located in the Sidi Chennane phosphate deposit. The Generalized fractal dimension, D(q), the Singularities of strength, α(q), the local dimension, f(α) and their conjugate parameter the mass exponent, τ(q) as well as f(α)-α spectrum were the common multifractal parameters used. The results reported show wide variations of the analyzed images, indicating that the multifractal analysis is an indicator for evaluate and characterize the disturbed areas within the phosphates deposits through the studied geoelectric images. This could be the starting point for future work aimed at improving phosphate exploration planning.

• 충청수학회지
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• 제19권4호
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• pp.357-364
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• 2006

We study Hausdorff and packing dimensions of subsets of a general coding space with a generalized ultra metric from a multifractal spectrum induced by a self-similar measure on a self-similar Cantor set using a function satisfying a H${\ddot{o}}$older condition.

• Korean Journal of Mathematics
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• 제12권1호
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• pp.1-5
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• 2004

We study Hausdorff and packing dimensions of subsets of a coding space with an ultra metric from a multifractal spectrum induced by a self-similar measure on a Cantor set using a function satisfying a H$\ddot{o}$lder condition.

• 한국농공학회논문집
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• 제58권5호
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• pp.91-99
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• 2016

High resolution rainfall data at 1-hour or a finer scale are essential for reliable flood analysis and forecasting; nevertheless, many observations, forecasts, and climate projections are still given at coarse temporal resolutions. This study aims to evaluate a chaotic method for disaggregation of 6-hour rainfall data sets so as to apply operational 6-hour rainfall forecasts of the Korean Meteorological Association to flood models. We computed parameters of a state-of-the-art multiplicative random cascade model with two combinations of cascades, namely uniform splitting and diversion, using rainfall observations at Seoul station, and compared statistical performance. We additionally disaggregated 6-hour rainfall time series at 58 stations with the uniform splitting and evaluated temporal transferability of the parameters and changes in multifractal properties. Results showed that the uniform splitting outperformed the diversion in reproduction of observed statistics, and hence is better to be used for disaggregation of 6-hour rainfall forecasts. We also found that multifractal properties of rainfall observations has adequate temporal consistency with an indication of gradually increasing rainfall intensity across South Korea.