• Honam Mathematical Journal
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• v.38 no.4
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• pp.753-782
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• 2016

In this paper we deduce the number of representations of a positive integer n by each of the six triangular forms as $${\frac{1}{2}}x_1(x_1+1)+{\frac{3}{2}}x_2(x_2+1)+{\frac{3}{2}}x_3(x_3+1)+{\frac{3}{2}}x_4(x_4+1),\\{\frac{1}{2}}x_1(x_1+1)+{\frac{1}{2}}x_2(x_2+1)+{\frac{3}{2}}x_3(x_3+1)+{\frac{3}{2}}x_4(x_4+1),\\{\frac{1}{2}}x_1(x_1+1)+{\frac{1}{2}}x_2(x_2+1)+{\frac{1}{2}}x_3(x_3+1)+{\frac{3}{2}}x_4(x_4+1),\\x_1(x_1+1)+x_2(x_2+1)+{\frac{3}{2}}x_3(x_3+1)+{\frac{3}{2}}x_4(x_4+1),\\x_1(x_1+1)+{\frac{3}{2}}x_2(x_2+1)+{\frac{3}{2}}x_3(x_3+1)+3x_4(x_4+1),\\{\frac{1}{2}}x_1(x_1+1)+{\frac{1}{2}}x_2(x_2+1)+3x_3(x_3+1)+3x_4(x_4+1).$$

• Journal of Korean Society of Environmental Engineers
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• v.32 no.1
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• pp.33-46
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• 2010

We investigated and estimated at the characteristics of decomposition and by-products of N-Nitrosodimethylamine (NDMA) using a design of experiment (DOE) based on the Box-Behken design in an UV process, and also the main factors (variables) with UV intensity($X_2$) (range: $1.5{\sim}4.5\;mW/cm^2$), NDMA concentration ($X_2$) (range: 100~300 uM) and pH ($X_2$) (rang: 3~9) which consisted of 3 levels in each factor and 4 responses ($Y_1$ (% of NDMA removal), $Y_2$ (dimethylamine (DMA) reformation (uM)), $Y_3$ (dimethylformamide (DMF) reformation (uM), $Y_4$ ($NO_2$-N reformation (uM)) were set up to estimate the prediction model and the optimization conditions. The results of prediction model and optimization point using the canonical analysis in order to obtain the optimal operation conditions were $Y_1$ [% of NDMA removal] = $117+21X_1-0.3X_2-17.2X_3+{2.43X_1}^2+{0.001X_2}^2+{3.2X_3}^2-0.08X_1X_2-1.6X_1X_3-0.05X_2X_3$ ($R^2$= 96%, Adjusted $R^2$ = 88%) and 99.3% ($X_1:\;4.5\;mW/cm^2$, $X_2:\;190\;uM$, $X_3:\;3.2$), $Y_2$ [DMA conc] = $-101+18.5X_1+0.4X_2+21X_3-{3.3X_1}^2-{0.01X_2}^2-{1.5X_3}^2-0.01X_1X_2+0.07X_1X_3-0.01X_2X_3$ ($R^2$= 99.4%, 수정 $R^2$ = 95.7%) and 35.2 uM ($X_1$: 3 $mW/cm^2$, $X_2$: 220 uM, $X_3$: 6.3), $Y_3$ [DMF conc] = $-6.2+0.2X_1+0.02X_2+2X_3-0.26X_1^2-0.01X_2^2-0.2X_3^2-0.004X_1X_2+0.1X_1X_3-0.02X_2X_3$ ($R^2$= 98%, Adjusted $R^2$ = 94.4%) and 3.7 uM ($X_1:\;4.5\;$mW/cm^2$, $X_2:\;290\;uM$, $X_3:\;6.2$) and $Y_4$ [$NO_2$-N conc] = $-25+12.2X_1+0.15X_2+7.8X_3+{1.1X_1}^2+{0.001X_2}^2-{0.34X_3}^2+0.01X_1X_2+0.08X_1X_3-3.4X_2X_3$ ($R^2$= 98.5%, Adjusted $R^2$ = 95.7%) and 74.5 uM ($X_1:\;4.5\;mW/cm^2$, $X_2:\;220\;uM$, $X_3:\;3.1$). This study has demonstrated that the response surface methodology and the Box-Behnken statistical experiment design can provide statistically reliable results for decomposition and by-products of NDMA by the UV photolysis and also for determination of optimum conditions. Predictions obtained from the response functions were in good agreement with the experimental results indicating the reliability of the methodology used.

• Journal of the Korean Ceramic Society
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• v.48 no.2
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• pp.127-133
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• 2011

Non-Alkali multicomponent $La_2O_3-Al_2O_3-SiO_2$ glasses has been designed and analyzed on the basis of a mixture design experiment with constraints. Fitted models for thermal expansion coefficient, glass transition temperature, Young's modulus, Shear modulus and density are as follows: ${\alpha}(/^{\circ}C)=8.41{\times}10^{-8}x_1+5.72{\times}10^{-7}x_2+2.13{\times}10^{-7}x_3+1.09{\times}10^{-7}x_4+1.10{\times}10^{-7}x_5+1.15{\times}10^{-7}x_6+2.72{\times}10^{-8}x_7+2.41{\times}10^{-7}x_8-1.08{\times}10^{-8}x_1x_2+4.28{\times}10^{-8}x_3x_7-2.02{\times}10^{-8}x_3x_8-1.60{\times}10^{-8}x_4x_5-2.71{\times}10^{-9}x_4x_8-2.19{\times}10^{-8}x_5x_6-3.89{\times}10^{-8}x_5x_7$ $T_g(^{\circ}C)=7.36x_1+15.35x_2+20.14x_3+8.97x_4+13.85x_5+4.22x_6+28.21x_7-1.44x_8-0.84x_2x_3-0.45x_2x_5-1.64x_2x_7+0.93x_3x_8-1.04x_5x_8-0.48x_6x_8$ $E(GPa)=2.04x_1+14.26x_2-1.22x_3-0.80x_4-2.26x_5-1.67x_6-1.27x_7+3.63x_8-0.24x_1x_2-0.07x_2x_8+0.14x_3x_6-0.68x_3x_8+0.29x_4x_5+1.28x_5x_8$ $G(GPa)=0.35x_1+1.78x_2+1.35x_3+1.87x_4+9.72x_5+29.16x_6-0.99x_7+3.60x_8-0.48x_1x_6-0.50x_2x_5+0.08x_3x_7-0.66x_3x_8+0.94x_5x_8$ ${\rho}(g/cm^3)=0.09x_1+0.51x_2-4.94{\times}10^{-3}x_3-0.03x_4+0.45x_5-0.07x_6-0.10x_7+0.07x_8-9.60{\times}10^{-3}x_1x_2-8.20{\times}10^{-3}x_1x_5+2.17{\times}10^{-3}x_3x_7-0.03x_3x_8+0.05x_5x_8$ The optimal glass composition similar to the thermal expansion coefficient of Si based on these fitted models is $65.53SiO_2{\cdot}25.00Al_2O_3{\cdot}5.00La_2O_3{\cdot}2.07ZrO_2{\cdot}0.70MgO{\cdot}1.70SrO$.

• Journal of Sericultural and Entomological Science
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• v.9
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• pp.21-25
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• 1969

Various formulae for estimation of leaf production in mulberry trees were investigated and obtained. Four varieties of mulberry trees were used as the materials, and four characters. namely branch length (X, 1). branch diameter (X, 2). leaf number per branch (X, 3), and leaf area per branch (X, 4). were studies. The formulae to eatimate the leaf yield of mulberry trees are as follows: 1. Y$_1$v$_1$＝-115.760＋0.068X$_1$＋165.756X$_2$ Y$_1$v$_2$＝-221.500＋1.768X$_1$＋38.152X$_2$ Y$_1$v$_3$＝-253.826-0.116X$_1$＋289.507X$_2$ Y$_1$v$_4$＝ -157.559＋1.063X$_1$＋106.088X$_2$ where Y$_1$v$_1$, Y$_1$v$_2$, Y$_1$v$_3$, Y$_1$v$_4$, are showed the estimated yield of the each variety, namely Gaeryang souban, Ilchirye, Nosang. and Suwon Sang No. 4, respectively. X$_1$ and X$_2$ denote the measured values of branch length and branch diameter, respectively. 2. Y$\sub$7/v$_1$＝-118.478-0.665X$_1$＋184.445X$_2$＋2.346X$_3$ Y$\sub$7/v$_2$＝-217.432＋2.062X$_1$＋35.668X$_2$-1.058X$_3$ Y$\sub$7/v$_3$＝-206. 249-0.739X$_1$＋268.08X$_2$＋2.770X$_3$ Y$\sub$7/v$_4$＝-153.383＋0.009X$_1$＋2.024X$_2$＋0.171X$_3$where Y$\sub$7/v$_1$, Y$\sub$7/v$_2$, Y$\sub$7/v$_3$, Y$\sub$7/v$_4$, are the estimated yield of the each variety, namely Gaeryang. Souban, Ilichirye, Nosang, and Suwon Sang No. 4, respectively. X$_1$, X$_2$, X$_3$, denote the measured values of each character. branch length, branch diameter and leaf number per branch, respectively. 3. Y$\sub$11/v$_1$＝82. 567-1.283X$_1$＋15.501X$_2$＋0.640X$_3$＋3.511X$_4$ Y$\sub$11/v$_2$＝136.411＋0.311X$_1$＋1.921X$_2$-0. 217X$_3$＋0.214X$_4$ Y$\sub$11/v$_3$＝150.2Z7-0.139X$_1$＋11.788X$_2$＋0.143X$_3$＋0.381X$_4$ Y$\sub$11/v$_4$＝160.850＋0.323X$_1$＋66.076X$_2$-0.794X$_3$＋2..614X$_4$ where Y$\sub$11/v$_1$, Y$\sub$11/v$_2$, Y$\sub$11/v$_3$, Y$\sub$11/v$_4$, are the estimated yield values of four varieties, and X$_1$, X$_2$, X$_3$, X$_4$ denote the measured values of four characters. namely branch length, branch diameter. leaf number per branch and leaf area per branch. respectively. The estimation method of mulberry leaf yield by measurement of some characters, branch length. branch diameter. leaf number per branch and leaf area per branch. could be the better method to determine the leaf yield of mulberry trees without destroying the leaves and without weighting the leaves of mulberry trees than the other methods.

• Journal of the Korean Society of Tobacco Science
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• v.16 no.1
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• pp.3-13
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• 1994

To minimize the time ordinarily spent in mono filter cigarette design, we studied the relationship between major seven independant variables ; filament(X1) and total denier(X2), porosity of the aller plug wrap(X3), filter length(X4), Porosity of the tip paper(X5) and cigarette paper(X6) and net weight of the reference cut tobacco(X7). Ninty trial numbers were obtained as a results of using rotatable central composite design and it is analyzed by the multiple regression analysis with stepwise in SAS/pc under restricted conditions. That is, UPD (Y1) = 82.96 - 3.80X1 + 2.50X2 - 3.29X3 - 3.15X5 - 0.83X22 + 1.88X5X6 - 1.38 X5X7(R2: 0.63), EPD(Y2) : 120.91 - 5.70X1 + 3.60X2 + 4.23X4 - 0.93X6 + 4.06X7 (R2=0.84), TVR(Y3) = 49.70 - 0.78X1 + 3.60X3 + 2.00X4 + 4.20X5 - 0.93X6 + 2.64X7 - 1.07X1X2 + 1.0IX1 X3 + 1.05X2X6 + 0.45X22 - 0.64X42 + 1.29X4X6 - 0.97X4X7 - 1.28X5X6 + 1.53X5X7 + 1.39X6X7(R2=0.65), and EVR(Y4) : 3.24-0.21X3-0.20X4 -0.24X5+0.67X6+0.26X4X7 (R2=0.55), where EPD : encapsulated pressure drop, VPD : unencapsulated pressure drop, TVR ; tip ventilation rate, and En : envelope ventilation rate. All variables in the model are significant at the 0.05 level.

• Journal of Sericultural and Entomological Science
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• v.10
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• pp.35-40
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• 1969

Various formulae for estimation of spring leaf production in mulberry trees were calculated and obtained. Four varieties of mulberry trees were used as the materials, and four characters, namely branch length (X$_1$), node number (X$_2$), branch diameter (X$_3$) and branch number per stock (X$_4$) were studied. The formulae to estimate the leaf yield of spring mulberry trees are as follows: 1. $Y_1$v$_1$= -26.8939＋50.3950X$_1$＋1.1403X$_2$ $Y_1$v$_2$= -372.1091＋116.6371X$_1$＋0.1984X$_2$ $Y_1$v$_3$= 149.8203＋90.5125X$_1$-0.9775X$_2$ $Y_1$v$_4$= 108, 1496＋59.4533X$_1$＋1.4965X$_2$ Where $Y_1$v$_1$, $Y_1$v$_2$, $Y_1$v$_3$, $Y_1$v$_4$, are showed the estimated yield of the each variety, namely Gaeryang Seuban, Ilchirye, Nosang, and Suwon Sang No. 4, respectively. X$_1$ and X$_2$ denote the measured values of branch length and node number, respectively. 2. $Y_{7}$v$_1$= -54.4411＋32.9869c1.1127X$_2$＋21.7600X$_3$ $Y_{7}$v$_2$= -494.1480-1.8756X$_1$＋0.9788X$_2$＋110.0039X$_3$ $Y_{7}$v$_3$= 143.2836＋29.1779X$_1$＋0.1644X$_2$＋48.4135X$_3$ $Y_{7}$v$_4$= 1243.2549＋1.9454X$_1$＋2.7118X$_2$-75.6669X$_3$ Where $Y_{7}$v$_1$, $Y_{7}$v$_2$, $Y_{7}$v$_3$, $Y_{7}$v$_4$, are the estimated yield of the each variety, namely Gaeryang-Seuban, Ilchirye, Nosang, Suwon Sang No 4, respectively. X$_1$, X$_2$, X$_3$ denote the measured values of each character, branch length, node number, branch diameter and branch number per stock, respectively. 3. $Y_{11}$v$_1$=233.4780＋74.3713X$_1$＋1.2912X$_2$＋39.0420X$_3$-148.9300X$_4$ $Y_{11}$v$_2$=-317.0150＋15.l524X$_1$＋1.0861X$_2$＋156.7973X$_3$-148.3742X$_4$ $Y_{11}$v$_3$=178.7011＋29.8664X$_1$-0.2562X$_2$＋102.4632X$_3$-83.2693X$_4$ $Y_{11}$v$_4$= 264.0062＋47.7742X$_1$＋2.6996X$_2$＋92.8882X$_3$-192.3464X$_4$ Where $Y_{11}$v$_1$, $Y_{11}$v$_2$, $Y_{11}$v$_3$, $Y_{11}$v$_4$, are the estimated yield values of four varieties, and X$_1$, X$_2$, X$_3$, X$_4$, denote the measured values of four characters, namely branch length, node number, branch diameter and branch number per stock, respectively. The estimation method of mulberry spring leaf yield by measurement of some characters, in autumn the year before, could be the better method to determine the leaf yield of mulberry trees without destroying the leaves and without weighting the leaves of mulberry trees than the other methods.

• Korean Journal of Materials Research
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• v.15 no.2
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• pp.112-114
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• 2005

Dispersion stability of the $Sm_xCe_{1-x}O_{2-2/x}$ nanoparticles, which was produced by hydrothermal process, was studied in aqueous suspension using ESA (Eletrokinetic Sonic Amplitude). The average particle size of the synthesized $Sm_xCe_{1-x}O_{2-2/x}$ at nanoparticles was about $5{\pm}2nm$. The dispersion and rheological behavior of the $Sm_xCe_{1-x}O_{2-2/x}$ nanoparticles aqueous suspension was investigated using $NH_4OH\;and\;HNO_3$ as a disperse agent. The colloidal stability of aqueous suspensions with $Sm_xCe_{1-x}O_{2-2/x}$ nanoparticles at different pH values has been investigated by means of zeta potential, average particle size, and the distribution of synthesized $Sm_xCe_{1-x}O_{2-2/x}$ nanoparticles. The isoelectric point of the $Sm_xCe_{1-x}O_{2-2/x}$ nanoparticles was at pH around 11 and the value of zeta potential was at its maximum near pH 6.5.

• Applied Chemistry for Engineering
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• v.3 no.4
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• pp.678-685
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• 1992

The effects of functional monomers on the pressure sensitive adhesive proporties were studied. Acrylic acid and other monomers were copolymerized by radical solution polymerization and their properties were measured. The desirability function methodology was applied to obtain optimum pressure sensitive adhesive properties. Acrylic acid showed more effective than acrylamide on peel strength increase. On the other hand acrylamide showed more effective than acrylic acid on tack decrease. The optimum monomer ratio of the acrylic pressure sensitive adhesive recipe containing n-butylacrylate 81.7 mole%, acrylic acid 8.0 mole%, acrylamide 2.1 mole% and vinylacetate 8.2 mole% was obtained to result from the statistical analysis with the desirability function methodology. The estimated regression equation of desirability function(D) is as follows: $D=.857+.072X_1-.114X_2-.027X_3-.126X_1{^2}-.046X_1{\cdot}X_2-.063X_1{\cdot}X_3-.152X_2{^2}+.027X2{\cdot}X_3-.120X_3{^2}$ $X_1$:coded acylic acid, $X_2$:coded acylamide, $X_3$:coded vinylacetate

• Honam Mathematical Journal
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• v.34 no.1
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• pp.45-54
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• 2012

In this paper, we prove stabilities of multiplicative functional equations in three variables such as $r(\frac{x+y+z}{3})-r(x+y+z)$=$\frac{2r(\frac{x+y}{2})r(\frac{y+z}{2})r(\frac{z+x}{2})}{r(\frac{x+y}{2})r(\frac{y+z}{2})+r(\frac{y+z}{2})r(\frac{z+x}{2})+r(\frac{z+x}{2})r(\frac{x+y}{2})}$ and $r(\frac{x+y+z}{3})+r(x+y+z)$=$\frac{4r(\frac{x+y}{2})r(\frac{y+z}{2})r(\frac{z+x}{2})}{r(\frac{x+y}{2})r(\frac{y+z}{2})+r(\frac{y+z}{2})r(\frac{z+x}{2})+r(\frac{z+x}{2})r(\frac{x+y}{2})}$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.529-572
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• 2012

A basis of a subspace of $S_4({\Gamma}_0(79))$ is given and the formulas for the number of representations of positive integers by some direct sums of the quadratic forms $x^2_1+x_1x_2+20x^2_2$, $4x^2_1{\pm}x_1x_2+5x^2_2$, $2x^2_1{\pm}x_1x_2+10x^2_2$ are determined.