• Biomaterials and Biomechanics in Bioengineering
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• v.1 no.1
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• pp.1-12
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• 2014

Polymer multilayered hydrogels were prepared on a titanium alloy (Ti) substrate using a layer-by-layer (LBL) process to load a cell growth factor. Two water-soluble polymers were used to fabricate the multilayered hydrogels, a phospholipid polymer with both N, N-dimethylaminoethyl methacrylate (DMAEMA) units and 4-vinylphenylboronic acid (VPBA) units [poly(MPC-co-DMAEMA-co-VPBA) (PMDV)], and the polysaccharide alginate (ALG). PMDV interacted with ALG through a selective reaction between the VPBA units in PMDV and the hydroxyl groups in ALG and through electrostatic interactions between the DMAEMA units in PMDA and the anionic carboxyl groups in ALG. First, the Ti substrate was covered with photoreactive poly vinyl alcohol, and then the Ti alloy was alternately immersed in the respective polymer solutions to form the PMDV/ALG multilayered hydrogels. In this multilayered hydrogel, vascular endothelial growth factor (VEGF) was introduced in different layers during the LbL process under mild conditions. Release of VEGF from the multilayered hydrogels was dependent on the location; however, release continued for 2 weeks. Endothelial cells adhered to the hydrogel and proliferated, and these corresponded to the VEGF release profile from the hydrogel. We concluded that multilayered hydrogels composed of PMDV and ALG could be loaded with cell growth factors that have high activity and can control cell functions. Therefore, this system provides a cell function controllable substrate based on the controlled release of biologically active proteins.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.421-428
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• 2009

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this paper, we showed the following : Let $\mathcal{L}$ be a subspace lattice acting on a Hilbert space $\mathcal{H}$ and let $X_i$ and $Y_i$ be operators in B($\mathcal{H}$) for i = 1, 2, ${\cdots}$. Let $P_i$ be the projection onto $\overline{rangeX_i}$ for all i = 1, 2, ${\cdots}$. If $P_kE$ = $EP_k$ for some k in $\mathbb{N}$ and all E in $\mathcal{L}$, then the following are equivalent: (1) $sup\;\{{\frac{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}:f{\in}H,n{\in}{\mathbb{N}},E{\in}\mathcal{L}}\}$ < ${\infty}$ range $\overline{rangeY_k}\;=\;\overline{rangeX_k}\;=\;\mathcal{H}$, and < $X_kf,\;X_kg$ >=< $Y_kf,\;Y_kg$ > for some k in $\mathbb{N}$ and for all f and g in $\mathcal{H}$. (2) There exists an operator A in Alg$\mathcal{L}$ such that $AX_i$ = $Y_i$ for i = 1, 2, ${\cdots}$ and AA$^*$ = I = A$^*$A.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.329-334
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• 2008

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, the following is proved: Let $\mathcal{L}$ be a subspace lattice on $\mathcal{H}$ and let X and Y be operators acting on a Hilbert space H. Let P be the projection onto the $\overline{rangeX}$. If PE = EP for each E ${\in}$ $\mathcal{L}$, then the following are equivalent: (1) sup ${{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}}:f{\in}\mathcal{H},\;E{\in}\mathcal{L}}$ < ${\infty},\;\overline{rangeY}\;{\subset}\;\overline{rangeX}$, and there is a bounded operator T acting on $\mathcal{H}$ such that < Xf, Tg >=< Yf, Xg >, < Tf, Tg >=< Yf, Yg > for all f and gin $\mathcal{H}$ and $T^*h$ = 0 for h ${\in}\;{\overline{rangeX}}^{\perp}$. (2) There is a normal operator A in AlgL such that AX = Y and Ag = 0 for all g in range ${\overline{rangeX}}^{\perp}$.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.981-986
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• 2011

Given vectors x and y in a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equations $Tx_i=y_i$, for i = 1, 2, ${\cdots}$, n. In this paper the following is proved : Let $\cal{L}$ be a subspace lattice on a Hilbert space $\cal{H}$. Let x and y be vectors in $\cal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE=EP_x$ for each $E{\in}\cal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$\cal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and $A=A^*$. (2) sup $sup\;\{\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}\;:\;E\;{\in}\;{\cal{L}}\}$ < ${\infty}$, $y\;{\in}\;sp(x)$ and < x, y >=< y, x >.

• Korean Journal of Environmental Biology
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• v.37 no.4
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• pp.741-748
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• 2019

As the occurrence of harmful algal blooms (HABs) have become severe in precious water resources, the development of efficient harmful algae treatment methods is considering as an important environmental issue for sustainable conservation of water resources. To treat HABs in water resources, various conventional physical and chemical methods have been utilized and showed treatment efficiency, However, these methods can lead to discharging of cyanotoxins into the water bodies by chemical or physical algal cell lysis or destruction. Thus, to overcome this limitation, the development of safe HABs treatment methods is required. In the present study, adsorption technology was investigated for the removal of harmful algal species, Microcystis aeruginosa from aqueous phases. Industrial waste biomass, Corynebacterium glutamicum biomass was valorized as biosorbent (PEI-modified alginate/biomass composite fiber; PEI-AlgBF) for M. aeruginosa through immobilization with alginate matrix and cationic polymer (polyethylenimine; PEI) coating. The functional groups characteristic of PEI-Alg was determined using FT-IR analysis. By adsorption process used PEI-AlgBF, 52 and 67% of M. aeruginosa could be removed under the initial density of M. aeruginosa 200×10⁴ cells mL^-1 and 50×10⁴ cells mL^-1, respectively. As the increasing surface area of PEI-AlgBF, the removal efficiency was increased. In addition, we could find that adsorptive removal of M. aeruginosa has occurred without any M. aeruginosa cell lysis and destruction.

• Honam Mathematical Journal
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• v.33 no.1
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• pp.115-120
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• 2011

Given vectors x and y in a separable complex Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following : Let Alg$\cal{L}$ be a tridiagonal algebra on a separable complex Hilbert space H and let x = ($x_i$) and y = ($y_i$) be vectors in H. Then the following are equivalent: (1) There exists an invertible operator A = ($a_{kj}$) in Alg$\cal{L}$ such that Ax = y. (2) There exist bounded sequences $\{{\alpha}_n\}$ and $\{{{\beta}}_n\}$ in $\mathbb{C}$ such that for all $k\in\mathbb{N}$, ${\alpha}_{2k-1}\neq0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=\frac{\alpha_{2k}}{{\alpha}_{2k-1}\alpha_{2k+1}}$ and $$y_1={\alpha}_1x_1+{\alpha}_2x_2$$ $$y_{2k}={\alpha}_{4k-1}x_{2k}$$ $$y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}x_{2k+2}$$.

• Honam Mathematical Journal
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• v.29 no.1
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• pp.55-60
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• 2007

Given operators X and Y acting on a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we showed the following : Let $\cal{L}$ be a subspace lattice acting on a Hilbert space $\cal{H}$ and let X and Y be operators in $\cal{B}(\cal{H})$. Let P be the projection onto $\bar{rangeX}$. If FE = EF for every $E\in\cal{L}$, then the following are equivalent: (1) $sup\{{{\parallel}E^{\perp}Yf\parallel\atop \parallel{E}^{\perp}Xf\parallel}\;:\;f{\in}\cal{H},\;E\in\cal{L}\}\$ < $\infty$, $\bar{range\;Y}\subset\bar{range\;X}$, and < Xf, Yg >=< Yf,Xg > for any f and g in $\cal{H}$. (2) There exists a self-adjoint operator A in Alg$\cal{L}$ such that AX = Y.

• Bulletin of the Korean Mathematical Society
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• v.39 no.3
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• pp.423-430
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• 2002

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_{}i$ = $Y_{i}$ for i/ = 1,2,…, n. In this article, we obtained the following : Let X ＝ ($x_{i\sigma(i)}$ and Y ＝ ($y_{ij}$ be operators in B(H) such that $X_{i\sigma(i)}\neq\;0$ for all i. Then the following statements are equivalent. (1) There exists an operator A in Alg L such that AX = Y, every E in L reduces A and A is a self-adjoint operator. (2) sup ${\frac{\parallel{\sum^n}_{i=1}E_iYf_i\parallel}{\parallel{\sum^n}_{i=1}E_iXf_i\parallel}n\;\epsilon\;N,E_i\;\epsilon\;L and f_i\;\epsilon\;H}$ < $\infty$ and $x_{i,\sigma(i)}y_{i,\sigma(i)}$ is real for all i = 1,2, ....

• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.237-244
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• 2018

Let ${\mathcal{H}}$ be an infinite dimensional separable Hilbert space with a fixed orthonormal base $\{e_1,e_2,{\cdots}\}$. Let L be the subspace lattice generated by the subspaces $\{[e_1],[e_1,e_2],[e_1,e_2,e_3],{\cdots}\}$ and let $Alg{\mathcal{L}}$ be the algebra of bounded operators which leave invariant all projections in ${\mathcal{L}}$. Let p and q be natural numbers (p < q). Let ${\mathcal{A}}$ be a linear manifold in $Alg{\mathcal{L}}$ such that $T_{(p,q)}=0$ for all T in ${\mathcal{A}}$. If ${\mathcal{A}}$ is a Lie ideal, then $T_{(p,p)}=T_{(p+1,p+1)}={\cdots}=T_{(q,q)}$ and $T_{(i,j)}=0$, $p{\eqslantless}i{\eqslantless}q$ and i < $j{\eqslantless}q$ for all T in ${\mathcal{A}}$.

• The Pure and Applied Mathematics
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• v.14 no.3
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• pp.161-166
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• 2007

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_i=Y_i$, for i = 1,2,...,n. In this article, we showed the following: Let L, be a subspace lattice on a Hilbert space H and let X and Y be operators in B(H). Then the following are equivalent: (1) $$sup\{\frac{{\parallel}E^{\bot}Yf{\parallel}}{{\overline}{\parallel}E^{\bot}Xf{\parallel}}\;:\;f{\epsilon}H,\;E{\epsilon}L}\}\;<\;{\infty},\;sup\{\frac{{\parallel}Xf{\parallel}}{{\overline}{\parallel}Yf{\parallel}}\;:\;f{\epsilon}H\}\;<\;{\infty}$$ and $\bar{range\;X}=H=\bar{range\;Y}$. (2) There exists an invertible operator A in AlgL such that AX=Y.