• Journal of Korean Society of Environmental Engineers
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• v.27 no.5
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• pp.555-561
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• 2005

A new graphical method was developed to separate two distinctive decay rate coefficients($k_1$ and $k_2$) at their respective degradable substrate fractions($S_1 and $S_2$). The mesophilic batch reactor showed $k_1$ of $0.151\;day^{-1}$ for wasted activated sludge(WAS), $0.123\;day^{-1}$ for thickened sludge(T-S), $0.248{\sim}0.358\;day^{-1}$ at S/I ratio of $1{\sim}3$ for sorghum and $0.155{\sim}0.209\;day^{-1}$ at S/I ratio $0.2{\sim}1.0$ for swine waste, whereas their long term batch decay rate coefficients($k_2$) were $0.021\;day^{-1}$, $0.001\;day^{-1}$, $0.03\;day^{-1}$ and $0.04\;day^{-1}$ respectively. At least an order of magnitude difference between $k_1$ and $k_2$ was routinely observed in the batch tests. The portion of $S_1$, which degrades with each $k_1$ appeared 71% for WAS, 39% for T-S, 90% for sorghum, and $84{\sim}91%$ at S/I ratio of $0.2{\sim}1.0$ for swine waste. Ultimate biodegradabilities of 50% for WAS, 40% of T-S, $82{\sim}92%$ for sorghum, and $81{\sim}89%$ for swine waste were observed.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.45-54
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• 2013

Let $$S^{\pm}_{(n,k)}\;:=\{(a,b,x,y){\in}\mathbb{N}^4:ax+by=n,x{\equiv}{\pm}y\;(mod\;k)\}$$. From the formula $\sum_{(a,b,x,y){\in}S^{\pm}_{(n,k)}}\;ab=4\sum_{^{m{\in}\mathbb{N}}_{m<n/k}}\;{\sigma}_1(m){\sigma}_1(n-km)+\frac{1}{6}{\sigma}_3(n)-\frac{1}{6}{\sigma}_1(n)-{\sigma}_3(\frac{n}{k})+n{\sigma}_1(\frac{n}{k})$, we find the Diophantine solutions for modulo $2^{m^{\prime}}$ and $3^{m^{\prime}}$, where $m^{\prime}{\in}\mathbb{N}$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.693-704
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• 2012

We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n{\geq}1$, we have $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1;3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.

• Journal of the Korean Chemical Society
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• v.32 no.5
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• pp.436-442
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• 1988

The $^1H-nmr$ lineshapes of $H_2O$ in the solution containing $Al^{3+}$ ion have been measured as a function of temperature and $H^+$-ion concentration. Above [$H^+$] = 0.06, the lineshape were analyzed by the uncoupled two-site exchange model. From the proton exchange rate between hexaaquaaluminium ion and bulk water as a function of H-ion concentration. These kinetic data could be fitted to a following linear rate law; that is; 1/${\tau}$ = k$_1$/12 + $k_2$[$H^+$]/6. The following proton exchange parameters were obtained; $k_1^{298}$ = 38.5s$^{-1}$ ${\{Delta}H_1^{\neq}$ = $42.9kJ mole^{-1}$ ${\{Delta}S_1^{\neq}$ = -48.6J $mole^{-1}K^{-1}$ $k_2^{298}$ = $172s^{-1}mole^{-1}$ ${\{Delta}H_2^{\neq}$ = 27.8kJ $mole^{-1}$ ${\{Delta}S_2^{\neq}$ = -90.3J $mole^{-1}K^{-1}$ These activation parameters are indicating an associative interchange, Ia, mechanism for the acid-hydrolysis of hexaaquaaluminium ion and the proton exchange between the hydration spheres of $Al^{3+}$ and $H^+$.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.693-701
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• 2006

Using Minimax principle and Linking theorem in critical point theory, we prove the existence of two nontrivial solutions for the following second order superlinear difference systems $$(P)\{{\Delta}^2x(k-1)+g(k,y(k))=0,\;k{\in}[1,\;T],\;{\Delta}^2y(k-1)+f(k,\;x(k)=0,\;k{\in}[1,\;T],\;x(0)=y(0)=0,\;x(T+1)=y(T+1)=0$$ where T is a positive integer, [1, T] is the discrete interval {1, 2,..., T}, ${\Delat}x(k)=x(k+1)-x(k)$ is the forward difference operator and ${\Delta}^2x(k)={\Delta}({\Delta}x(k))$.

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.599-607
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• 2014

In this paper, we investigate the behavior of solutions of the difference equation $$x_{n+1}=\frac{a+x_{n-1}x_{n-k}}{x_{n-1}+x_{n-k}},\;n=0,1,2,{\ldots}$$ where $k{\in}\{1,2\}$, $a{\geq}0$, and $x_{-j}$ > 0, $j=0,1,{\ldots},k$.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.739-741
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• 2009

Given $\alpha$ > 1, there exist $C(1/\alpha)$-polynomials of the form $z^n-\sum_{k=0}^{n-1}\;a_kz^k$, where $\sum_{k=0}^{n-1}\;a_k=1$, $a_{n-1}>0$ and $a_k{\geq}0$ for each k. In this paper, we obtain lower bounds for $a_{n-1}$.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.139-145
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• 1996

We show how some interesting results involving series summation and the digamma function are established by means of Riemann-Liouville operator of fractional calculus. We derive the relation $$ \frac{\Gamma(\lambda)}{\Gamma(\nu)} \sum^{\infty}_{n=1}{\frac{\Gamma(\nu+n)}{n\Gamma(\lambda+n)}_{p+2}F_{p+1}(a_1, \cdots, a_{p+1},\lambda + n; x/a)} = \sum^{\infty}_{k=0}{\frac{(a_1)_k \cdots (a_{(p+1)}{(b_1)_k \cdots (b_p)_k K!} (\frac{x}{a})^k [\psi(\lambda + k) - \psi(\lambda - \nu + k)]}, Re(\lambda) > Re(\nu) \geq 0 $$ and explain some special cases.

• Journal of the Korean Society of Food Science and Nutrition
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• v.43 no.10
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• pp.1535-1542
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• 2014

This study was conducted to investigate the physicochemical properties and biological activities of collagens with different molecular weights from Alaska pollack (Theragra chalcogramma) skin as well as their efficacies as functional materials. The molecular weights of collagens were between 1~10 kDa (below 1 kDa (AP1), 1~3 kDa (AP2), 3~10 kDa (AP3), and above 10 kDa (AP4). The protein content of AP4 (40.19 g/100 g) was the highest. Collagen contents of AP1, AP2, AP3, and AP4 were 36.43, 32.23, 19.23, and 14.89%, respectively. The free amino acid and essential amino acid contents of AP1 were higher than those of AP2, AP3, and AP4. Fourier transform infrared spectroscopy spectra of collagens with different molecular weights showed wavenumbers representing the regions of amide I, amide II, amide III, and amide A, respectively. The electron-donating ability (29.51%) and SOD-like activity (38.45%) of AP1 were higher than those of AP2, AP3, and AP4. Tyrosinase inhibition activity of AP1 improved with higher treatment concentration. The rate of inhibition of MMP-1 production in HS68 cells exposed to UVB was suppressed by treatment with AP1 (29.78%) and AP2 (26.49%) at 1 mg/mL. Furthermore, there was a strong correlation between DPPH, superoxide dismutase, tyrosinase activity, and MMP-1 inhibition rate of collagens with different molecular weights.

• Journal of the Korean Statistical Society
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• v.9 no.1
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• pp.31-38
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• 1980

The problem considered in this paper can be defiened as follows. Consider observations $x_1, x_2, \cdot, x_n$ which are assumed to come from a mixed population of the density function, $$f(x) = \sum^m_{k=1} pkf_k(x)$$ where m is the number of subpoulations and $p_k$ is the proportion of subpopulation k such that $\sum^m_{k=1} pk=1, 0