• 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 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 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 $65.53SiO_2{\cdot}25.00Al_2O_3{\cdot}5.00La_2O_3{\cdot}2.07ZrO_2{\cdot}0.70MgO{\cdot}1.70SrO$.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.257-264
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• 1992

Since Alfsen and Effors [1] introduced the notion of an M-ideal, many authors [3,6,9,12] have worked on the problem of finding those Banach spaces X and Y for which K(X,Y), the space of all compact linear operators from X to Y, is an M-ideal in L(X,Y), the space of all bounded linear operators from X to Y. The M-ideal property of K(X,Y) in L(X,Y) gives some informations on X,Y and K(X,Y). If K(X) (=K(X,X)) is an M-ideal in L(X) (=L(X,X)), then X has the metric compact approximation property [5] and X is an M-ideal in $X^{**}$ [10]. If X is reflexive and K(X) is an M-ideal in L(X), then K(X)$^{**}$ is isometrically isomorphic to L(X)[5]. A weaker notion is a semi M-ideal. Studies on Banach spaces X and Y for which K(X,Y) is a semi M-ideal in L(X,Y) were done by Lima [9, 10].

• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.103-113
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• 1991

Some properties of a path space and the fundamental group ${\sigma}(X,x_0,G)$ of a topological transformation group (X, G, ${\pi}$) are described. It is shown that ${\sigma}(X,x_0,H)$ is a normal subgroup of ${\sigma}(X,x_0,G)$ if H is a normal subgroup of G ; Let (X, G, ${\pi}$) be a transformation group with the open action property. If every identification map $p:{\Sigma}(X,x,G)\;{\longrightarrow}\;{\sigma}(X,x,G)$ is open for each $x{\in}X$, then ${\lambda}$ induces a homeomorphism between the fundamental groups ${\sigma}(X,x_0,G)$ and ${\sigma}(X,y_0,G)$ where ${\lambda}$ is a path from $x_0$ to $y_0$ in X ; The space ${\sigma}(X,x_0,G)$ is an H-space if the identification map $p:{\Sigma}(X,x_0,G)\;{\longrightarrow}\;{\sigma}(X,x_0,G)$ is open in a topological transformation group (X, G, ${\pi}$).

• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.221-229
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• 2013

We show that if X is a space such that ${\beta}QF(X)=QF({\beta}X)$ and each stable $Z(X)^{\sharp}$-ultrafilter has the countable intersection property, then there is a homeomorphism $h_X:vQF(X){\rightarrow}QF(vX)$ with $r_X={\Phi}_{vX}{\circ}h_X$. Moreover, if ${\beta}QF(X)=QF({\beta}X)$ and $vE(X)=E(vX)$ or $v{\Lambda}(X)={\Lambda}(vX)$, then $vQF(X)=QF(vX)$.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.181-189
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• 2011

Let G be a group and X be a nonempty set and H be a subgroup of G. For a given ${\phi}_G\;:\;G{\times}X{\rightarrow}X$, a group action of G on X, we define ${\phi}_H\;:\;H{\times}X{\rightarrow}X$, a subgroup action of H on X, by ${\phi}_H(h,x)={\phi}_G(h,x)$ for all $(h,x){\in}H{\times}X$. In this paper, by considering a subgroup action of H on X, we have some results as follows: (1) If H,K are two normal subgroups of G such that $H{\subseteq}K{\subseteq}G$, then for any $x{\in}X$ ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}_H}(x)$) = ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}_K}(x)$) = ($orb_{{\phi}_K}(x)\;:\;orb_{{\phi}_H}(x)$); additionally, in case of $K{\cap}stab_{{\phi}_G}(x)$ = {1}, if ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}H}(x)$) and ($orb_{{\phi}_K}(x)\;:\;orb_{{\phi}_H}(x)$) are both finite, then ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}_H}(x)$) is finite; (2) If H is a cyclic subgroup of G and $stab_{{\phi}_H}(x){\neq}$ {1} for some $x{\in}X$, then $orb_{{\phi}_H}(x)$ is finite.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.1-11
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• 2019

Let R be a prime ring with involution of the second kind and centre Z(R). Suppose R admits a generalized derivation $F:R{\rightarrow}R$ associated with a derivation $d:R{\rightarrow}R$. The purpose of this paper is to study the commutativity of a prime ring R satisfying any one of the following identities: (i) $F(x){\circ}x^*{\in}Z(R)$ (ii) $F([x,x^*]){\pm}x{\circ}x^*{\in}Z(R)$ (iii) $F(x{\circ}x^*){\pm}[x,x^*]{\in}Z(R)$ (iv) $F(x){\circ}d(x^*){\pm}x{\circ}x^*{\in}Z(R)$ (v) $[F(x),d(x^*)]{\pm}x{\circ}x^*{\in}Z(R)$ (vi) $F(x){\pm}x{\circ}x^*{\in}Z(R)$ (vii) $F(x){\pm}[x,x^*]{\in}Z(R)$ (viii) $[F(x),x^*]{\mp}F(x){\circ}x^*{\in}Z(R)$ (ix) $F(x{\circ}x^*){\in}Z(R)$ for all $x{\in}R$.

• Korean Journal of Mathematics
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• v.32 no.3
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• pp.533-544
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• 2024

Given a groupoid (X, *) and a real-valued function d : X → R, a new (derived) function Φ(X, *)(d) is defined as [Φ(X, *)(d)](x, y) := d(x * y) + d(y * x) and thus Φ(X, *) : R^X → R^X2 as well, where R is the set of real numbers. The mapping Φ(X, *) is an R-linear transformation also. Properties of groupoids (X, *), functions d : X → R, and linear transformations Φ(X, *) interact in interesting ways as explored in this paper. Because of the great number of such possible interactions the results obtained are of necessity limited. Nevertheless, interesting results are obtained. E.g., if (X, *, 0) is a groupoid such that x * y = 0 = y * x if and only if x = y, which includes the class of all d/BCK-algebras, then (X, *) is *-metrizable, i.e., Φ(X, *)(d) : X² → X is a metric on X for some d : X → R.