• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.103-110
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• 2008

In this paper, we obtain the general solution and the stability of the cubic functional equation f(2x + y, 2z + w) + f(2x - y, 2z - w) = 2f(x + y, z + w) + 2f(x - y, z - w) + 12f(x, z). The cubic form $f(x,\;y)\;=\;ax^3\;+\;bx^2y\;+\;cxy^2\;+\;dy^3$ is a solution of the above functional equation.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1347-1370
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• 2016

On a closed eta-Einstein Sasakian spin manifold of dimension $2m+1{\geq}5$, $m{\equiv}0$ mod 2, we prove a new eigenvalue estimate for the Dirac operator. In dimension 5, the estimate is valid without the eta-Einstein condition. Moreover, we show that the limiting case of the estimate is attained if and only if there exists such a pair (${\varphi}_{{\frac{m}{2}}-1}$, ${\varphi}_{\frac{m}{2}}$) of spinor fields (called Sasakian duo, see Definition 2.1) that solves a special system of two differential equations.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.851-864
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• 2022

We define a kind of sectional curvature and 𝛿-invariants for statistical manifolds. For statistical submanifolds the sum of the squared mean curvature and the squared dual mean curvature is bounded below by using the 𝛿-invariant. This inequality can be considered as a generalization of the so-called Chen inequality for Riemannian submanifolds.

• East Asian mathematical journal
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• v.25 no.2
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• pp.171-177
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• 2009

Throughout this paper, we denote that R is a near-ring and G an R-group. We initiate a study of R-substructures of G, monogenic R-groups, faithful R-groups and faithful D.G. representations of near-rings. Next, we investigate some properties of monogenic R-groups, faithful monogenic R-groups and a generalization of annihilator concepts in R-groups.

• East Asian mathematical journal
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• v.31 no.4
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• pp.445-458
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• 2015

In this paper, we provide a geometrical solving method about the equiangular problem appeared in the solving process of the isoperimetric problem of polygon. In fact we deal with the following problem in the view of the productive thinking centered on the circle: Let B and G be fixed points, and let $\bar{AB}=\bar{AP_1}=\bar{DP_1}=\bar{DP_2}=\bar{FP_2}=\bar{FP_3}=\bar{HP_{n-1}}=\bar{HG}$. Then find the position of moving points $P_i(1{\leq}i{\leq}n)$ to maximize the sum of areas of the triangles that lie on the line segment $\bar{BG}$.

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1207-1235
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• 2016

Let M be a real hypersurface with constant mean curvature in a complex space form $M_n(c),c{\neq}0$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}= R({\cdot},{\xi}){\xi}$ with respect to the structure vector field ${\xi}$ is ${\phi}{\nabla}_{\xi}{\xi}$-parallel and $R_{\xi}$ commute with the structure tensor field ${\phi}$, then M is a homogeneous real hypersurface of Type A.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.575-584
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• 2011

In Shahzad and Zegeye [Nonlinear Anal. 71 (2009), no. 3-4, 838-844], the authors introduced several Ishikawa iterative schemes for xed points of multi-valued mappings in Banach spaces, and proved some strong convergence theorems by using their iterations. In their proofs of the main results, it seems reasonable and simpler to prove for the iteration {$x_n$} to be a Cauchy sequence. In this paper, we modify and improve the proofs of the main results given by Shahzad and Zegeye. Two concrete examples also are given.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.349-359
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• 2009

As students have been familiar to Internet or media which has big visual effects rather than letters, they preferred the class in computer lab to theoretical class. The class in computer lab showed all the students good reactions in concern and interest. Statistics achievements of high level students in 2008 became better than that in 2007 but those of low level students became worse than that in 2007. I analyze the reason why the achievements of low level students have become worse. The purpose of this study is to supplement of the trouble of the class in computer lab and find out a better teaching and develop mathematics education's satisfaction and qualification.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.85-91
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• 1995

For a locally compact Hausdorff space, we denote by $C_0(X)$ the Banach space of all continuous complex valued functions defined on X which vanish at infinity, equipped with the usual sup norm. In case X is compact, we write C(X) instead of $C_0(X)$. A well-known Banach-Stone theorem states that the existence of an isometry between the function spaces $C_0(X)$ and $C_0(Y)$ implies X and Y are homemorphic. D. Amir [1] and M. Cambern [2] independently generalized this theorem by proving that if $C_0(X)$ and $C_0(Y)$ are isomorphic under an isomorphism T satisfying $\left\$\mid$ T \right\$\mid$ \left\$\mid$ T^1 \right\$\mid$ < 2$, then X and Y must also be homeomorphic.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.715-726
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• 2010

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining $hs(Mat_n(R))\;=\;Mat_n(hs(R))$ for any ring R where hs(-) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.