• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.105-110
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• 2013

A logarithmic composition inequality in Besov spaces is derived which generalizes Vishik's inequality: ${\parallel}f{\circ}g{\parallel}_{B^s_{p,1}}{\leq}(1+{\log}({\parallel}{\nabla}g{\parallel}_{L^{\infty}}{\parallel}{\nabla}g^{-1}{\parallel}_{L^{\infty}})){\parallel}f{\parallel}_{B^s_{p,1}}$, where $g$ is a volume-preserving diffeomorphism on ${\mathbb{R}}^n$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.1
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• pp.17-29
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• 2005

In this paper we consider finite element methods for nonlinear hyperbolic integro-differential problems defined in ${\Omega}\;{\subset}\;R^d(d\;{\leq}\;4)$. A new initial approximation of $u_t(0)$ is taken. Optimal order error estimates in $L_p$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are demonstrated as well.

• Journal of the Korean Mathematical Society
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• v.36 no.6
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• pp.1133-1143
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• 1999

Chaterji strengthened version of a theorem for martin-gales which is a generalization of a theorem of Marcinkiewicz proving that if $X_n$ is a sequence of independent, identically distributed random variables with $E{\mid}X_n{\mid}^p\;<\;{\infty}$, 0 < P < 2 and $EX_1\;=\;1{\leq}\;p\;<\;2$ then $n^{-1/p}{\sum^n}_{i=1}X_i\;\rightarrow\;0$ a,s, and in $L^p$. In this paper, we probe a version of law of large numbers for double arrays. If ${X_{ij}}$ is a double sequence of random variables with $E{\mid}X_{11}\mid^log^+\mid X_{11}\mid^p\;<\infty$, 0 < P <2, then $lim_{m{\vee}n{\rightarrow}\infty}\frac{{\sum^m}_{i=1}{\sum^n}_{j=1}(X_{ij-a_{ij}}}{(mn)^\frac{1}{p}}\;=0$ a.s. and in $L^p$, where $a_{ij}$ = 0 if 0 < p < 1, and $a_{ij}\;=\;E[X_{ij}\midF_[ij}]$ if $1{\leq}p{\leq}2$, which is a generalization of Etemadi's marcinkiewicz-type SLLN for double arrays. this also generalize earlier results of Smythe, and Gut for double arrays of i.i.d. r.v's.

• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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• pp.35-43
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• 2009

We say that an isometric immersed hypersurface x : $M^n\;{\rightarrow}\;{\mathbb{R}}^{n+1}$ is of $L_k$-finite type ($L_k$-f.t.) if $x\;=\;{\sum}^p_{i=0}x_i$ for some positive integer p < $\infty$, $x_i$ : $M{\rightarrow}{\mathbb{R}}^{n+1}$ is smooth and $L_kx_i={\lambda}_ix_i$, ${\lambda}_i\;{\in}\;{\mathbb{R}}$, $0{\leq}i{\leq}p$, $L_kf=trP_k\;{\circ}\;{\nabla}^2f$ for $f\;{\in}\'C^{\infty}(M)$, where $P_k$ is the kth Newton transformation, ${\nabla}^2f$ is the Hessian of f, $L_kx\;=\;(L_kx^1,\;{\ldots},\;L_kx^{n+1})$, $x=(x^1,\;{\ldots},\;x^{n+1})$. In this article we study the following(hyper)surfaces in ${\mathbb{R}}^{n+1}$ from the view point of $L_1$-finiteness type: totally umbilic ones, generalized cylinders $S^m(r){\times}{\mathbb{R}}^{n-m}$, ruled surfaces in ${\mathbb{R}}^{n+1}$ and some revolution surfaces in ${\mathbb{R}}^3$.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.185-220
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• 2016

We study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator defined on the half plane. Next, we establish a principle of maximum for the singular partial differential operator $${\Delta}_{\alpha}={\frac{{\partial}^2}{{\partial}r^2}+{\frac{2{\alpha}+1}{r}{\frac{\partial}{{\partial}r}}+{\frac{{\partial}^2}{{\partial}x^2}}+{\frac{{\partial}^2}{{\partial}t^2}}};\;(r,x,t){\in}]0,+{\infty}[{\times}{\mathbb{R}}{\times}]0,+{\infty}[$$. Later, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every $p{\in}]1,+{\infty}[$, there exists a positive constant $C_p$ such that for every $f{\in}L^p(d{\nu}_{\alpha})$, $${\frac{1}{C_p}}{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}{\leqslant}{\parallel}g(f){\parallel}_{p,{\nu}_{\alpha}}{\leqslant}C_p{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}$$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.2
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• pp.55-64
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• 2005

In this paper, generalized difference methods(GDM) for one-dimensional viscoelastic problems are proposed and analyzed. The new initial values are given in the generalized difference scheme, so we obtain optimal error estimates in $L^p$ and $W^{1,p}(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}(2\;{\leq}\;p\;{\leq}\;{\infty})$ between the GDM solution and the generalized Ritz-Volterra projection of the exact solution.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.7 no.2
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• pp.35-49
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• 2003

In this paper, finite volume element methods for nonlinear parabolic integrodifferential problems are proposed and analyzed. The optimal error estimates in $L^p\;and\;W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ are obtained. The main results in this paper perfect the theory of FVE methods.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1029-1036
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• 1997

Let ${A_t)}_{t>0}$ be a dilation group given by $A_t = exp(-P log t)$, where P is a real $n \times n$ matrix whose eigenvalues has strictly positive real part. Let $\nu$ be the trace of P and $P^*$ denote the adjoint of pp. Suppose that $K$ is a function defined on $R^n$ such that $$\mid$K(x)$\mid$ \leq k($\mid$x$\mid$_Q)$ for a bounded and decreasing function $k(t) on R_+$ satisfying $k \diamond $\mid$\cdot$\mid$_Q \in \cup_{\varepsilon >0}L^1((1 + $\mid$x$\mid$)^\varepsilon dx)$ where $Q = \int_{0}^{\infty} exp(-tP^*) exp(-tP)$ dt and the norm $$\mid$\cdot$\mid$_Q$ stands for $$\mid$x$\mid$_Q = \sqrt{}, x \in R^n$. For $f \in L^1(R^n)$, define $mf(x) = sup_{t>0}$\mid$K_t * f(x)$\mid$$ where $K_t(X) = t^{-\nu}K(A_{1/t}^* x)$. Then we show that $m$ is a bounded operator of $L^1(R^n) into L^{1, \infty}(R^n)$.

• KSBB Journal
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• v.15 no.2
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• pp.134-138
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• 2000

Addition of carbon and nitrogen source to an alcohol distillery wastewater was tried to increase the cell concentration of a b baker's yeast cultivated in that wastewater. Carbon was found to be primary limiting nutrient and nitrogen secondary limiting o one. Glucose addition increased the cell concentration 1.3 times higher than no addition, and both glucose and $(NH_4)_2S0_4$ a addition did 5.8 times. A fed-batch cultivation by the automatic addition of glucose and ammonium was executed. Added g glu$\infty$se was automatically controlled to low concentration by a method using DO as control parameter. Ammonium was a automatically added as NH40H used as pH $\infty$ntrol agent after initiating glucose addition. By this simple cultivation method t the cell concentration $\infty$내d be efficiently increased from 2.6g/L to 12.0g/L, and maximum specific growth rate and biomass y yield to glu$\infty$se were $0.18hr^{-1}$ and about 0.54g/g respectively. By increasing cell concentration, COD of the wastewater m media could be additionally reduced by about 22%.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.417-426
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• 2011

Let ${\mu}$ be a finite positive Borel measure on the unit ball $B{\subset}{\mathbb{C}}^n$ and ${\nu}$ be the Euclidean volume measure such that ${\nu}(B)=1$. For the unit sphere $S=\{z:{\mid}z{\mid}=1\}$, ${\sigma}$ is the rotation-invariant measure on S such that ${\sigma}(S) =1$. Let ${\mathcal{P}}[f]$ be the Poisson-$Szeg{\ddot{o}}$ integral of f and $\tilde{\mu}$ be the Berezin transform of ${\mu}$. In this paper, we show that if there is a constant M > 0 such that ${\int_B}{\mid}{\mathcal{P}}[f](z){\mid}^pd{\mu}(z){\leq}M{\int_B}{\mid}{\mathcal{P}}[f](z){\mid}^pd{\nu}(z)$ for all $f{\in}L^p(\sigma)$, then ${\parallel}{\tilde{\mu}}{\parallel}_{\infty}{\equiv}{\sup}_{z{\in}B}{\mid}{\tilde{\mu}}(z){\mid}<{\infty}$, and we show that if ${\parallel}{\tilde{\mu}{\parallel}_{\infty}<{\infty}$, then ${\int_B}{\mid}{\mathcal{P}}[f](z){\mid}^pd{\mu}(z){\leq}C{\mid}{\mid}{\tilde{\mu}}{\mid}{\mid}_{\infty}{\int_S}{\mid}f(\zeta){\mid}^pd{\sigma}(\zeta)$ for some constant C.