• 충청수학회지
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• 제20권1호
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• pp.37-45
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• 2007

Let X, Y be vector spaces. In this paper, we investigate the generalized Hyers-Ulam-Rassias stability problem for a cubic function $f:X{\rightarrow}Y$ satisfies $$r^3f(\frac{\Sigma_{j=1}^{n-1}x_j+2x_n}{r})+r^3f(\frac{\Sigma_{j=1}^{n-1}x_j-2x_n}{r})+8\sum_{j=1}^{n-1}f(x_j)=2f{\sum_{j=1}^{n-1}}x_j)+4{\sum_{j=1}^{n-1}}(f(x_j+x_n)+f(x_j-x_n))$$ for all $x_1,{\cdots},x_n{\in}X$.

• 충청수학회지
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• 제32권3호
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• pp.295-307
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• 2019

In this paper, we investigate Hyers-Ulam-Rassias stability of the functional equation $$f(x+ky)-{\frac{k^2+k}{2}}f(x+y)+(k^2-1)f(x)-{\frac{k^2-k}{2}}f(x-y)\\{\hfill{67}}-f(ky)+{\frac{k^2+k}{2}}f(y)+{\frac{k^2-k}{2}}f(-y)=0.$$

• International Journal of Internet, Broadcasting and Communication
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• 제9권3호
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• pp.1-8
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• 2017

Recently, the development of the head mounted display (HMD) device has attracted a great deal of attention to the actual contents. Especially, Augmented Reality (AR), which is a mixture of actual information and virtual world information, is focused on. AR HMD is able to interact by arranging virtual objects in real space through spatial recognition using depth camera. In order to naturally mix virtual space with real space, it is necessary to develop a technology for realizing spatial mapping information with high accuracy. The purpose of this paper is to evaluate the optimal configuration of augmented reality application program by realizing accurate spatial mapping information when mapping a real space and an object placement environment using HoloLens. To do this, we changed the spatial mapping information in real space to three levels, which are the number of meshes used in cubic meters to scan step by step. After that, it was compared with the 3D model obtained by changing the actual space and mesh number. Experimental result shows that the higher the number of meshes used in cubic meters, the higher the accuracy between real space and spatial mapping. This paper is expected to be applied to augmented reality application programs that require scanning of highly mapped spatial mapping information.

• 대한수학회보
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• 제53권4호
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• pp.959-970
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• 2016

In this paper, we generalize the stability for an n-dimensional cubic functional equation in Banach space to set-valued dynamics. Let $n{\geq}2$ be an integer. We define the n-dimensional cubic set-valued functional equation given by $$f(2{{\sum}_{i=1}^{n-1}}x_i+x_n){\oplus}f(2{{\sum}_{i=1}^{n-1}}x_i-x_n){\oplus}4{{\sum}_{i=1}^{n-1}}f(x_i)\\=16f({{\sum}_{i=1}^{n-1}}x_i){\oplus}2{{\sum}_{i=1}^{n-1}}(f(x_i+x_n){\oplus}f(x_i-x_n)).$$ We first prove that the solution of the n-dimensional cubic set-valued functional equation is actually the cubic set-valued mapping in [6]. We prove the Hyers-Ulam stability for the set-valued functional equation.

• 대한수학회보
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• 제45권4호
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• pp.739-748
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• 2008

In this paper, we obtain the general solution of a generalized cubic functional equation, the Hyers-Ulam-Rassias stability, and the stability by using the alternative fixed point for a generalized cubic functional equation $$4f(\sum_{j=1}^{n-1}\;x_j\;+\;mx_n)\;+\;4f(\sum_{j=1}^{n-1}\;x_j+mx_n\;x_j\;-\;mx_n}\;+\;m^2\sum_{j=1}^{n-1}\;(f(2x_j)\;=\;8f(\sum_{j=1}^{n-1}\;x_j)\;+\;4m^2{\sum_{j=1}^{n-1}}\;\(f(x_j+x_n)\;+\;f(x_j-x_n)\)$$ for a positive integer $m\;{\geq}\;1$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제19권4호
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• pp.383-396
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• 2012

In this paper, we investigate a fuzzy version of stability for the functional equation $$f(x+2y)-3f(x+y)+3f(x)-f(x-y)-3f(y)+3f(-y)=0$$ in the sense of M. Mirmostafaee and M. S. Moslehian.

• East Asian mathematical journal
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• 제35권5호
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• pp.559-568
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• 2019

In this paper, we investigate the stability problems for a functional equation f(x + 2y)+f(x - 2y) - 4f(x + y) - 4f(x - y) + 6f(x) - 2f(2y) + 12f(y) - 4f(-y) = 0 by using the fixed point theory in the sense of L. C˘adariu and V. Radu.

• 충청수학회지
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• 제19권2호
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• pp.189-196
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• 2006

The generalized Hyers-Ulam stability problems of the cubic functional equation f(x + y + z) + f(x + y - z) + 2f(x - y) + 4f(y) = f(x - y + z) + f(x - y - z) +2f(x + y) + 2f(y + z) + 2f(y - z) shall be treated under the approximately odd condition and the behavior of the cubic mappings and the additive mappings shall be investigated. The generalized Hyers-Ulam stability problem for functional equations had been posed by Th.M. Rassias and J. Tabor [7] in 1992.

• 대한수학회보
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• 제55권1호
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• pp.267-296
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• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 1994년도 추계학술대회 논문집
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• pp.298-302
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• 1994

A three dimensional FE mesh generation scheme based on mapping approach is proposed in this study. A volume in Euclcdian space is represented by composite hyperpatches which are piecewise cubic functions with parameters u,v,w. A key idea in the proposed approach is that sampled grid data points only on the boundary surfaces are needed for the shape representation. Inner points which are necessary of form a hyperpatch are internally generated by Coons patches. This approach is most appropriate for the shapes which are compositions of hexahedron-like shapes and also severely curved.