• Journal for History of Mathematics
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• v.31 no.6
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• pp.315-337
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• 2018

Pythagorean thoerem exists in several equivalent forms in the Euclidean plane, that is, the Hilbert plane which in addition satisfies the parallel axiom. In this article, we investigate the truthness and mutual relationships of those propositions in various non-Hilbert planes which satisfy the parallel axiom and all the Hilbert axioms except the SAS axiom.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.865-879
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• 2022

Let (M^m, g) be an m-dimensional Riemannian manifold. In this paper, we introduce a new class of metric on (M^m, g), obtained by a non-conformal deformation of the metric g. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. In the last section we characterizes some class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when (M^m, g) is an Euclidean space.

• KCI Concrete Journal
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• v.14 no.1
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• pp.30-35
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• 2002

The fractal geometry is a non-Euclidean geometry which describes the naturally irregular or fragmented shapes, so that it can be applied to fracture behavior of materials to investigate the fracture process. Fractal curves have a characteristic that represents a self-similarity as an invariant based on the fractal dimension. This fractal geometry was applied to the crack growth of cementitious composites in order to correlate the fracture behavior to microstructures of cementitious composites. The purpose of this study was to find relationships between fractal dimensions and fracture energy. Fracture test was carried out in order to investigate the fracture behavior of plain and fiber reinforced cement composites. The load-CMOD curve and fracture energy of the beams were observed under the three point loading system. The crack profiles were obtained by the image processing system. Box counting method was used to determine the fractal dimension, D$_{f}$. It was known that the linear correlation exists between fractal dimension and fracture energy of the cement composites. The implications of the fractal nature for the crack growth behavior on the fracture energy, G$_{f}$ is apparent.ent.

• Proceedings of the Korea Concrete Institute Conference
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• 2001.05a
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• pp.333-338
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• 2001

Fractal geometry is a non-Euclidean geometry which has been developed to quantitative analysis irregular or fractional shapes. Fractal dimension of irregular surface has fractal values ranging from 2 to 3 and of irregular line profile has fractal values ranging from 1 to 2. In this paper, quantitative analysis of crack growth patterns during the fracture processing of fiber-reinforced cement composites based on fractal geometry. The fracture behaviors of fiber reinforced mortar beams subjected to three-point loading in flexure. The beams all had a single notch depth, but varing volume fractions of polypropylene, cellulose fibers. The crack growth behaviors, as observed through the image processing system, and the box counting method was used to determine the fractal dimension, Df. The results showed that the linear correlation exists between fractal dimension and fracture energy of the fiber reinforced cement mortar.

• Communications for Statistical Applications and Methods
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• v.22 no.4
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• pp.321-331
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• 2015

The K-means clustering algorithm is a popular and widely used method for clustering. For covariance matrices, we consider a geodesic clustering algorithm based on the K-means clustering framework in consideration of symmetric positive definite matrices as a Riemannian (non-Euclidean) manifold. This paper considers a geodesic clustering algorithm for data consisting of symmetric positive definite (SPD) matrices, utilizing the Riemannian geometric structure for SPD matrices and the idea of a K-means clustering algorithm. A K-means clustering algorithm is divided into two main steps for which we need a dissimilarity measure between two matrix data points and a way of computing centroids for observations in clusters. In order to use the Riemannian structure, we adopt the geodesic distance and the intrinsic mean for symmetric positive definite matrices. We demonstrate our proposed method through simulations as well as application to real financial data.

• Journal of Korea Technical Association of The Pulp and Paper Industry
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• v.47 no.4
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• pp.177-186
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• 2015

Until Mandelbrot introduced the concept of fractal geometry and fractal dimension in early 1970s, it has been generally considered that the geometry of nature should be too complex and irregular to describe analytically or mathematically. Here fractal dimension indicates a non-integer number such as 0.5, 1.5, or 2.5 instead of only integers used in the traditional Euclidean geometry, i.e., 0 for point, 1 for line, 2 for area, and 3 for volume. Since his pioneering work on fractal geometry, the geometry of nature has been found fractal. Mandelbrot introduced the concept of fractal geometry. For example, fractal geometry has been found in mountains, coastlines, clouds, lightning, earthquakes, turbulence, trees and plants. Even human organs are found to be fractal. This suggests that the fractal geometry should be the law for Nature rather than the exception. Fractal geometry has a hierarchical structure consisting of the elements having the same shape, but the different sizes from the largest to the smallest. Thus, fractal geometry can be characterized by the similarity and hierarchical structure. A process requires driving energy to proceed. Otherwise, the process would stop. A hierarchical structure is considered ideal to generate such driving force. This explains why natural process or phenomena such as lightning, thunderstorm, earth quakes, and turbulence has fractal geometry. It would not be surprising to find that even the human organs such as the brain, the lung, and the circulatory system have fractal geometry. Until now, a normal frequency distribution (or Gaussian frequency distribution) has been commonly used to describe frequencies of an object. However, a log-normal frequency distribution has been most frequently found in natural phenomena and chemical processes such as corrosion and coagulation. It can be mathematically shown that if an object has a log-normal frequency distribution, it has fractal geometry. In other words, these two go hand in hand. Lastly, applying fractal principles is discussed, focusing on pulp and paper industry. The principles should be applicable to characterizing surface roughness, particle size distributions, and formation. They should be also applicable to wet-end chemistry for ideal mixing, felt and fabric design for papermaking process, dewatering, drying, creping, and post-converting such as laminating, embossing, and printing.

• Journal of the Korea Concrete Institute
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• v.13 no.2
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• pp.146-152
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• 2001

The fractal geometry is a non-Euclidean geometry which discribes the naturally irregular or fragmented shaps, so that it can be applied to fracture behavior of materials to investigate the fracture process. Fractal curves have a characteristic that represents a self-similarity as an invariant based on the fractal dimension. This fractal geometry was applied to the crack growth of cementitious composites in order to correlate the fracture behavior to microstructures of cemposite composites. The purpose of this study was to find relationships between fractal dimensions and fracture energy. Fracture test was carried out in order to investigate the fracture behavior of plain and fiber reinforced cement composites. The load-CMOD curve and fracture energy of the beams were observed under the three point loading system. The crack profiles were obtained by the image processing system. Box counting method was used to determine the fractal dimension, D$_{f}$. It was known that the linear correlation exists between fractal dimension and fracture energy of the cement composites. The implications of the fractal nature for the crack growth behavior on the fracture energy, G$_{f}$ is appearent.ent.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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• pp.57-66
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• 2003

The various deformation spaces associated with maximal geometric structures on closed oriented 3-manifolds was studied in [2], leaving out the geometry of $\mathbb{R}^3$. In this paper, we study the Weil spaces and Teichm$\ddot{u}$ller spaces of non-orientable 3-dimensional flat Riemannian manifolds. In particular, we find the Teichm$\ddot{u}$ller spaces are homeomorphic to the Euclidean spaces $\mathbb{R}^4$ or $\mathbb{R}^3$ depending on the holonomy group $\mathbb{Z}_2$ or $\mathbb{Z}_2{\times}\mathbb{Z}_2$ respectively.

• Journal of Digital Contents Society
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• v.9 no.1
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• pp.109-116
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• 2008

The variation of real phenomena and shape of nature in our world is so complicated that some mathematical tools using the traditional geometric methods based on the Euclidean geometry and analytical differential method may be irrelevant or insufficient in some problems. Recently, to deal with these circumstances, one can use the fractal geometric method. As another measures, in this paper we introduce the non-integral order derivative function for the analytical method and construct to facilitate their calculation.

• Journal for History of Mathematics
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• v.23 no.4
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• pp.17-30
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• 2010

The aim of this paper is to classify mathematical revolutions by discussing the concept of revolution, and to suggest criteria to judge mathematical revolutions. I examine the relation between the types and the criteria of mathematical revolutions, and explore what types of revolutions several instances of changes in mathematical knowledge are.