• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1109-1127
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• 2021

We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃₁ and 𝜃₂. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.137-155
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• 2021

In this paper, first, we prove some common fixed point theorems for the generalized contraction condition under newly defined modified simulation function which generalize and include many results in the literature. Second, we give two numerical examples with graphical representations for verifying the proposed results. Third, we discuss and study a set of common fixed point theorems for two pairs (finite families) of self-mappings. Finally, we give some applications of our results in discrete and functional fractional economic systems.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.797-836
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• 2024

We consider the following strongly damped wave equation on ℝ³ with memory u_tt - αΔu_t - βΔu + λu - ∫₀^∞ κ'(s)∆u(t - s)ds + f(x, u) + g(x, u_t) = h, where a quite general memory kernel and the nonlinearity f exhibit a critical growth. Existence, uniqueness and continuous dependence results are provided as well as the existence of regular global and exponential attractors of finite fractal dimension.

• The Journal of the Korea Contents Association
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• v.17 no.1
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• pp.465-474
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• 2017

This study aimed to identify the trends of related research as a basic study for the development of creative convergent formative education program of natural structure concept. This study sought to identify research trends present in previous studies. The nature structure of the preliminary research was limited to the three concepts: (i) fractal ; (ii) kinetic and (iii) biomimicry. In this study, the trends of domestic research in the last 10 years related to the concept of natural structure were analyzed using academic research information service. It was found that, to date, little research has been conducted on the three concepts across education fields. In relation to the fractal concept, previous research has focused on mathematics. This preliminary study sought to review the abovementioned three concepts or the development of a modeling education program. It should be significant, if an education program adopted unlimited modeling principles to understand the innate features of the nature structure. However, very few education programs have adopted the three concepts of the nature structure. Future studies would seek to review international research trends based on the three concepts of the nature structure and combine the results on international research trends with the results on domestic research trends found in this study.

• Korean Institute of Interior Design Journal
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• v.19 no.2
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• pp.90-98
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• 2010

The concepts of laws like regularity or persistence or recurrence those are discovered in nature, became the essential elements in speculative philosophy, study and scientific technology. Western civilization was spread out by these natural laws. As this background, this study is aimed to research the theories of natural laws and the development of geometry as the descriptive tools and the development aspects of the concepts of space. According to Whitehead's four theories on the natural laws, the result of this study that aimed like that as follows. First, the theories on the immanence and imposition of the natural laws were the predominant ideas from ancient Greek to before the scientific revolution, the theory on the simple description like the positivism made the Newton-Cartesian mechanism and an absolutist world view. The theory on the conventional interpretation made the organicism and relativism world view according to non-Euclidean geometry. Second, the geometrical composition of ancient Greek architecture was an aesthetics that represented the immanence of natural laws. Third, in the basic symbol of medieval times, the numeral symbol was the frame of thought and was an important principal of architecture. Fourth, during the Renaissance, architecture was regarded as mathematics that made the order of universe to visible things and the geometry was regarded as an important architectural principal. Fifth, according to the non-Euclidean geometry, it was possible to present the natural phenomena and the universe. Sixth, topology made to lapse the division of traditional floor, wall and ceiling in contemporary architecture and made to build the continuous space. Seventy, the new nature was explained by fractal concepts not by Euclidean shapes, fractal presented that the essence of nature had not mechanical and linear characteristic but organic and non-linear characteristic.

• Computers and Concrete
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• v.32 no.2
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• pp.149-163
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• 2023

One of the main design parameters traditionally utilized in projects of geotechnical engineering is the uniaxial compressive strength. The present paper employed three artificial intelligence methods, i.e., the stochastic fractal search (SFS), the multi-verse optimization (MVO), and the vortex search algorithm (VSA), in order to determine the compressive strength of concrete (CSC). For the same reason, 1030 concrete specimens were subjected to compressive strength tests. According to the obtained laboratory results, the fly ash, cement, water, slag, coarse aggregates, fine aggregates, and SP were subjected to tests as the input parameters of the model in order to decide the optimum input configuration for the estimation of the compressive strength. The performance was evaluated by employing three criteria, i.e., the root mean square error (RMSE), mean absolute error (MAE), and the determination coefficient (R²). The evaluation of the error criteria and the determination coefficient obtained from the above three techniques indicates that the SFS-MLP technique outperformed the MVO-MLP and VSA-MLP methods. The developed artificial neural network models exhibit higher amounts of errors and lower correlation coefficients in comparison with other models. Nonetheless, the use of the stochastic fractal search algorithm has resulted in considerable enhancement in precision and accuracy of the evaluations conducted through the artificial neural network and has enhanced its performance. According to the results, the utilized SFS-MLP technique showed a better performance in the estimation of the compressive strength of concrete (R²=0.99932 and 0.99942, and RMSE=0.32611 and 0.24922). The novelty of our study is the use of a large dataset composed of 1030 entries and optimization of the learning scheme of the neural prediction model via a data distribution of a 20:80 testing-to-training ratio.

• Journal of the Daesoon Academy of Sciences
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• v.32
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• pp.137-173
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• 2019

Modern scientists are trying to find the basic unit of order, fractal geometry, in the complex systems of the universe. Fractal is a term often used in mathematics or physics, it is appropriate as a principle to explain why some models of ultimate reality are represented as multifaceted. Fractals are already widely used in the field of computer graphics and as a commercial principle in the world of science. In this paper, using observations from fractal geometry, I present the embodiment of ultimate reality as understood in Daesoon Thought. There are various models of ultimate reality such as Dao (道, the way), Sangje (上帝, supreme god), Sinmyeong (神明, Gods), Mugeuk (無極, limitlessness), Taegeuk (太極, the Great Ultimate), and Cheonji (天地, heaven and earth) all of which exist in Daesoon Thought, and these concepts are mutually interrelated. In other words, by revealing the fact that ultimate reality is embodied within fractal geometry, it can be shown that concordance and transformation of various models of ultimate reality are supported by modern science. But when the major religions of the world were divided along lines of personality (personal gods) and non-personality (impersonal deities), most religions came to assume that ultimate reality was either transcendental or personal, and they could not postulate a relationship between God and humanity as Yin Yang (陰陽) fractals (Holon). In addition, religions, which assume ultimate reality as an intrinsic and impersonal being, are somewhat different in terms of their degree of Holon realization - all parts and whole restitution. Daesoon Thought most directly states that gods (deities) and human beings are in a relationship of Yin Yang fractals. In essence, "deities are Yin, and humanity is Yang" and furthermore, "human beings are divine beings." Additionally, in the Daesoon Thought, these models of ultimate reality are presented through various concepts from various viewpoints, and they are revealed as mutually interrelated concepts. As such, point of view regarding the universe wherein Holarchy becomes a models in a key idea within perennial philosophy. According to a universalized view of religious phenomena, perennial philosophy was adopted by the world's great spiritual teachers, thinkers, philosophers, and scientists. From this viewpoint, when ultimate reality coincides, human beings and God are no longer different. In other words, the veracity of the theory of ultimate reality that has appeared in Daesoon Thought can find support in both modern science and perennial philosophy.

• Journal of the Korean Mathematical Society
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• v.41 no.6
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• pp.945-955
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• 2004

In this paper, we establish a number system in $R^n$ which arises from a Haar wavelet basis in connection with decompositions of certain Cuntz algebra representations on $L^2$( $R^n$). Number systems in $R^n$ are also of independent interest [9]. We study radix-representations of $\chi$ $\in$ $R^n$: $\chi$:$\alpha$$_{ι}$ $\alpha$$_{ι-1}$…$\alpha$$_1$$\alpha$$_{0}$ ㆍ$\alpha$$_{-1}$ $\alpha$$_{-2}$ … as $\chi$= $M^{ι}$$\alpha$$_{ι}$ $\alpha$＋…M$\alpha$$_1$＋$\alpha$$_{0}$ ＋ $M^{-1}$ $\alpha$$_{-1}$ ＋ $M^{-2}$ $\alpha$$_{-2}$ ＋… where each $\alpha$$_{k}$ $\in$ D, and D is some specified digit set. Our analysis uses iteration techniques of a number-theoretic flavor. The view-point is a dual one which we term fractals in the large vs. fractals in the small,illustrating the number theory of integral lattice points vs. fractions.s vs. fractions.

• Proceedings of the KIEE Conference
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• 2004.11a
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• pp.251-254
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• 2004

The aging diagnosis technique using partial discharge detection method detects partial discharge signals cause of power equipment failuer and able to forecast the aging state of insulation system through analysis algorithm, in this paper accumulates HFPD signal during constant scheduled cycles to build HFPD pattern and then analyzes HFPD pattern using statistical parameters and ${\Delta}F$ pattern. The 3D pattern is composed of detected signal frequency, amplitude and repeated number and the FRPDA(frequency resolved partial discharge analysis) technique is used in 3D pattern construction. The ${\Delta}F$ pattern shows variation characteristics of amplitude gradient of consecutive HFPD signal Pulses and able to classify discharge types-internal discharge, surface discharge and coronal discharge etc. Fractal mathematics applied to ${\Delta}F$ pattern quantification and neural networks is used in aging diagnostic algorithm.