• Kyungpook Mathematical Journal
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• v.49 no.4
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• pp.595-603
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• 2009

We show that for an algebraic Riccati equation $X^*B^{-1}X-T^*X-X^*T=C$, its solutions are given by X = W + BT for some solution W of $X^*B^{-1}X$ = $C+T^*BT$. To generalize this, we give an equivalent condition for $\(\array{B&W\\W*&A}\)\;{\geq}\;0$ for given positive operators B and A, by which it can be regarded as Riccati inequality $X^*B^{-1}X{\leq}A$. As an application, the harmonic mean B ! C is explicitly written even if B and C are noninvertible.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.71-111
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• 2021

We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence �� from the set of equivalent well-posed two-point boundary conditions to gl(4, ℂ). Using ��, we derive eigenconditions for the integral operator ��M for each well-posed two-point boundary condition represented by M ∈ gl(4, 8, ℂ). Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition M on Spec ��M, (2) they connect Spec ��M to Spec ����,α,k whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real λ ∉ Spec ����,α,k, there exists a real well-posed boundary condition M such that λ ∈ Spec ��M. This in particular shows that the integral operators ��M, arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to ����,α,k.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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• pp.247-256
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• 2021

Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: $$(\mathcal{P}_{\lambda})\;\{\begin{array}{lll}{\Delta}^2u-(a{\int}_{\Omega}{\mid}{\nabla}u{\mid}^2dx+b){\Delta}u+cu=f(x){\mid}u{\mid}^{-{\gamma}}-{\lambda}{\mid}u{\mid}^{p-2}u&&\text{ in }{\Omega},\\{\Delta}u=u=0&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω is a smooth bounded domain of ℝⁿ (n ≥ 5), ∆² is the biharmonic operator, and ∇u denotes the spatial gradient of u and 0 < γ < 1, λ > 0, 0 < p ≤ 2^# and a, b, c are three positive constants with a + b > 0 and f belongs to a given Lebesgue space.

• IE interfaces
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• v.22 no.2
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• pp.104-115
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• 2009

The objective of the study is to evaluate the effectiveness of Crew Resource Management (CRM) training program for operators in the Main Control Room (MCR) simulator of APR-1400 Nuclear Power Plant. The experiments were conducted for two different crews of operators performing six different emergency operating scenarios during four-week period. Each crew consisted of the five operators: senior reactor operator, safety technical advisor, reactor operator, turbine operator, and electric operator. All crews (Crew A and B) participated in the training program for the technical knowledge and skills which were required to operate the simulator of the MCR during the first week. To verify the effectiveness of the CRM training program; however, only Crew A was selected to attend the CRM training after the technical knowledge and skills training. The results of the experiments showed that the CRM training program improved the individual attitudes of Crew A significantly. Team skills of Crew A were found to be significantly better than those of Crew B. The CRM training did not have positive effects on enhancing the individual performance of Crew A; however, as compared to that of Crew B. Implication of these findings was discussed further in detail.

• Korean Journal of Mathematics
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• v.15 no.1
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• pp.51-57
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• 2007

We prove the existence of a positive solution for the system of the following nonlinear biharmonic equations with Dirichlet boundary condition $$\{{\Delta}^2u+c{\Delta}u+av^+=s_1{\phi}_1+{\epsilon}_1h_1(x)\;in\;{\Omega},\\{\Delta}^2v+c{\Delta}v+bu^+=s_2{\phi}_1+{\epsilon}_2h_2(x)\;in\;{\Omega},$$ where $u^+= max\{u,0\}$, $c{\in}R$, $s{\in}R$, ${\Delta}^2$ denotes the biharmonic operator and ${\phi}_1$ is the positive eigenfunction of the eigenvalue problem $-{\Delta}$ with Dirichlet boundary condition. Here ${\epsilon}_1$, ${\epsilon}_2$ are small numbers and $h_1(x)$, $h_2(x)$ are bounded.

• Kyungpook Mathematical Journal
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• v.50 no.2
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• pp.213-228
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• 2010

A weighted version of the geometric mean of k ($\geq\;3$) positive invertible operators is given. For operators $A_1,{\ldots},A_k$ and for nonnegative numbers ${\alpha}_1,\ldots,{\alpha}_k$ such that $\sum_\limits_{i=1}^k\;\alpha_i=1$, we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to $A_1^{\alpha_1}{\cdots}A_k^{{\alpha}_k}$ if $A_1,{\ldots},A_k$ commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.551-560
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• 2007

We prove the existence of solutions for the system of the nonlinear biharmonic equations with Dirichlet boundary condition $$\{^{-{\Delta}^2u-c{\Delta}u+{\gamma}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega},\;}_{-{\Delta}^2u-c{\Delta}u+{\delta}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega}}$$, where $u^+$ = max{u, 0}, ${\Delta}^2$ denotes the biharmonic operator and ${\phi}_1$ is the positive eigenfunction of the eigenvalue problem $-{\Delta}$ with Dirichlet boundary condition.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.1031-1047
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• 2017

In this paper, we introduce two general iterative algorithms (one implicit algorithm and other explicit algorithm) for nonexpansive mappings in a reflexive Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm. Strong convergence theorems for the sequences generated by the proposed algorithms are established.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.687-696
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• 2013

In this note, we completely characterize the reducing subspaces of $T_{{z^N_1}{z^M_2}}$ on $A^2_{\alpha}(D^2)$ where ${\alpha}$ > -1 and N, M are positive integers with $N{\neq}M$, and show that the minimal reducing subspaces of $T_{{z^N_1}{z^M_2}}$ on the unweighted Bergman space and on the weighted Bergman space are different.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.303-315
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• 2014

In this work, we construct Kantorovich type generalization of a class of linear positive operators via Riemann type q-integral. We obtain estimations for the rate of convergence by means of modulus of continuity and the elements of Lipschitz class and also investigate weighted approximation properties.