• 호남수학학술지
• /
• 제38권4호
• /
• pp.753-782
• /
• 2016

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

• 한국연초학회지
• /
• 제16권1호
• /
• pp.3-13
• /
• 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.

• 대한환경공학회지
• /
• 제32권1호
• /
• pp.33-46
• /
• 2010

본 연구는 광분해 산화공정으로 난분해성 물질인 N-Nitrosodimethylamine (NDMA)인 제거 및 부산물 생성 특성을 파악하기 위한 3개의 독립변수 (자외선 강도($X_1:\;1.5{\sim}4.5\;mW/cm^2$, 초기 NDMA 농도($X_2:\;100{\sim}300\;uM$), pH(X3:3~9))와 4개의 종속변수(NDMA 제거율($Y_1$), dimethylamine (DMA) 생성농도($Y_2$), dimethylformamide (DMF) 생성농도($Y_3$) 및 $NO_2$-N 생성농도($Y_4$))로 구성된 박스-벤켄 설계를 이용한 실험계획을 적용시켜 예측 모델과 광분해 산화 최적조건을 수립하였다. 실험결과 2시간 광분해 후 NDMA는 거의 완전히 제거되었으며 DMA, DMF와 $NO_2$-N은 NDMA 광분해와 동시에 부산물로 생성되었다. 광분해 최적의 조건을 얻기 위해 정준분석을 수행하여 최적 점 (반응값, 독립변수 조건)과 예측반응모델을 수립한 결과, 다음과 같은 결과를 얻었다 ($Y_1=117+21X_1-0.3X_2-17.2X_3+{2.43X_1}^2+{0.001X_2}^2+{3.2X_3}^2-0.08X_1X_2-1.6X_1X_3-0.05X_2X_3$ ($R^2$ = 96%, Adjusted $R^2$ = 88%)와 99.3% ($X_1:\;4.5\;mW/cm^2$, $X_2:\;190\;uM$, $X_3:\;3.2$), $Y_2=-101+18.5X_1+0.4X_2+21X_3-{3.3X_1}^2-{0.01X_2}^2-{1.5X_3}^2-0.01X_1X_2-0.07X_1X_3-0.01X_2X_3$ ($R^2$= 99.4%, 수정 $R^2$ = 95.7%)와 35.2 uM ($X_1:\;3\;mW/cm^2$, $X_2:\;220\;uM$, $X_3:\;6.3$), $Y_3=-6.2+0.2X_1+0.02X_2+2X_3-{0.26X_1}^2-{0.01X_2}^2-{0.2X_3}^2-0.004X_1X_2+0.1X_1X_3-0.02X_2X_3$ ($R^2$= 98%, 수정 $R^2$ = 94.4%)와 3.7 uM ($X_1:\;4.5\;mW/cm^2$, $X_2:\;290\;uM$, $X_3:\;6.2$), $Y_4=-25+12.2X_1+0.15X_2+7.8X_3+{1.1X_1}^2+{0.001X_2}^2-{0.34X_3}^2+0.01X_1X_2+0.08X_1X_3-3.4X_2X_3$ ($R^2$= 98.5%, 수정 $R^2$ = 95.7%)와 74.5 uM ($X_1:\;4.5\;mW/cm^2$, $X_2:\;220\;uM$, $X_3:\;3.1$). 반응표면분석법 중 하나인 박스-벤켄법은 UV 광분해에 의한 NDMA 분해 및 부산물 생성에 대한 통계학적 및 수학적인 결과 및 최적의 운전조건을 제시하였다. 예측모델의 검정을 통하여 박스-벤켄법은 매우 높은 신뢰성을 보였다.

• 한국세라믹학회지
• /
• 제48권2호
• /
• pp.127-133
• /
• 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 $65.53SiO_2{\cdot}25.00Al_2O_3{\cdot}5.00La_2O_3{\cdot}2.07ZrO_2{\cdot}0.70MgO{\cdot}1.70SrO$.

• 대한수학회보
• /
• 제29권2호
• /
• pp.257-264
• /
• 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].

• 충청수학회지
• /
• 제4권1호
• /
• pp.103-113
• /
• 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, ${\pi}$).

• 충청수학회지
• /
• 제26권1호
• /
• pp.221-229
• /
• 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
• /
• 제19권2호
• /
• pp.181-189
• /
• 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
• /
• 제59권1호
• /
• pp.1-11
• /
• 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 $x{\in}R$.

• 한국잠사곤충학회지
• /
• 제9권
• /
• pp.21-25
• /
• 1969

상엽의 수확고를 측정하기 위하여 상엽의 수량과 높음 상관관계가 있는 형질중 상전에서 쉽게 측정할수 있는 기조장(X$_1$), 기조직경(X$_2$), 엽수(X$_3$), 엽면적(X$_4$)의 4 개형질을 측정하여 이들 형질의 수량에 영향하는 가중치를 다중회분방정식에 의하여 계출하여 수량을 측정할수 있도록 여러가지 식을 유도하였다. 1. 기조장(X$_1$)과 기조직경(X$_2$)을 측정하여 수량을 측정하기 위하여는 개량서반에 었어서는 y$_1$v$_1$＝-115.760＋0.068X$_1$＋165.756X$_2$(g) 일지뢰에 있어서는 y$_1$v$_2$＝-221.500＋1.768X$_1$＋38.152X$_2$(g) 노상에 있어서는 y$_1$v$_3$＝-253.826-0.116X$_1$＋289.507X$_2$(g) 수원상 4호에 있어서는 y$_1$v$_4$＝ -157.559＋1.063X$_1$＋106.088X$_2$(g)의 식에 의해서 기조장(X$_1$)과 기조직경(X$_2$)의 측정치를 대입하면 수량을 견적할수 있다. 2. 기조장(X$_1$), 기조직경(X$_2$), 엽수(X$_3$)의 3 개형질을 측정하여 수량을 견적하는 데는 각품종별로 각각 y$_{7}$v$_1$＝-118.478-0.665X$_1$＋184.445X$_2$＋2.346X$_3$ y$_{7}$v$_2$＝-217.432＋2.062X$_1$＋35.668X$_2$-1.058X$_3$ y$_{7}$v$_3$＝-206. 249-0.739X$_1$＋268.08X$_2$＋2.770X$_3$ y$_{7}$v$_4$＝-153.383＋0.009X$_1$＋2.024X$_2$＋0.171X$_3$ 의 식에 의하여 수량을 견적할수 있다. 3. 기조장(X$_1$), 기조직경(X$_2$), 엽수(X$_3$), 엽면적(X$_4$)의 4개형질을 측정하고 수량을 견적하기 위하여는 각품종별로 각각 y$_{11}$v$_1$＝82. 567-1.283X$_1$＋15.501X$_2$＋0.640X$_3$＋3.511X$_4$ y$_{11}$v$_2$＝136.411＋0.311X$_1$＋1.921X$_2$-0. 217X$_3$＋0.214X$_4$ y$_{11}$v$_3$＝150.2Z7-0.139X$_1$＋11.788X$_2$＋0.143X$_3$＋0.381X$_4$ y$_{11}$v$_4$＝160.850＋0.323X$_1$＋66.076X$_2$-0.794X$_3$＋2..614X$_4$등의 식에 의하여 수량을 견적할수 있다.의하여 수량을 견적할수 있다.