• Communications of the Korean Mathematical Society
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• v.33 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.213-230
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• 2019

We construct new metric outer measures (multifractal analogues of the Hewitt-Stromberg measure) $H^{q,t}_{\mu}$ and $P^{q,t}_{\mu}$ lying between the multifractal Hausdorff measure ${\mathcal{H}}^{q,t}_{\mu}$ and the multifractal packing measure ${\mathcal{P}}^{q,t}_{\mu}$. We set up a necessary and sufficient condition for which multifractal Hausdorff and packing measures are equivalent to the new ones. Also, we focus our study on some regularities for these given measures. In particular, we try to formulate a new version of Olsen's density theorem when ${\mu}$ satisfies the doubling condition. As an application, we extend the density theorem given in [3].

• Communications of the Korean Mathematical Society
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• v.37 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.549-554
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• 2008

We study the multifractal spectrum of two dimensionally indexed classes whose members are distribution sets of a self-similar attractor in the unit interval.

• Korean Journal of Mathematics
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• v.28 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.229-241
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• 1999

The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $\sigma$-multifractal from $\sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $\sigma$-multifractal. We show that the non-differentiable function on the $\sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $\sigma$-multifractal spaces.

• Structural Engineering and Mechanics
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• v.63 no.6
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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.

• Economic and Environmental Geology
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• v.55 no.3
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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.

• Korean Journal of Mathematics
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• v.26 no.2
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• pp.271-283
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• 2018

In this paper, we generelize the Olsen's density theorem to any measurable set, allowing us to extend the main results of H.K. Baek in (Proc. Indian Acad. Sci. (Math. Sci.) Vol. 118, (2008), pp. 273-279.). In particular, we tried through these results to improve the decomposition theorem of Besicovitch's type for the regularities of multifractal Hausdorff measure and packing measure.

• Kyungpook Mathematical Journal
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• v.45 no.4
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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].