• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.619-626
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• 2006

Let m be any positive integer, R be a ring with identity, $M_m(R)$ be the matrix ring of all m by m matrices eve. R and $G_m(R)$ be the multiplicative group of all n by n nonsingular matrices in $M_m(R)$. In this pape., the following are investigated: (1) for any pairwise coprime ideals ${I_1,\;I_2,\;...,\;I_n}$ in a ring R, $M_m(R/(I_1{\cap}I_2{\cap}...{\cap}I_n))$ is isomorphic to $M_m(R/I_1){\times}M_m(R/I_2){\times}...{\times}M_m(R/I_n);$ and $G_m(R/I_1){\cap}I_2{\cap}...{\cap}I_n))$ is isomorphic to $G_m(R/I_1){\times}G_m(R/I_2){\times}...{\times}G_m(R/I_n);$ (2) In particular, if R is a finite ring with identity, then the order of $G_m(R)$ can be computed.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.3
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• pp.349-363
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• 2019

Let ${\mathbb{H}}^5$ be the 5-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we calculate the volumes of geodesic balls in ${\mathbb{H}}^5$. Let $B_e(R)$ be the geodesic ball with center e (the identity of ${\mathbb{H}}^5$) and radius R in ${\mathbb{H}}^5$. Then, the volume of $B_e(R)$ is given by $${\hfill{12}}Vol(B_e(R))\\{={\frac{4{\pi}^2}{6!}}{\left(p_1(R)+p_4(R){\sin}\;R+p_5(R){\cos}\;R+p_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^R}{\frac{{\sin}\;t}{t}}dt\right.}\\{\left.{\hfill{65}}{+q_4(R){\sin}(2R)+q_5(R){\cos}(2R)+q_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^{2R}}{\frac{{\sin}\;t}{t}}dt}\right)}$$ where $p_n$ and $q_n$ are polynomials with degree n.

• Bulletin of the Korean Chemical Society
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• v.6 no.2
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• pp.119-120
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• 1985

Stereospecific coordination of racemic 2,3-diaminobutane has been observed in the reaction with platinum(Ⅱ) complexes of optically active 6,6'-dimethyl-2,2'-diaminobiphenyl. The reaction between [Pt (R-dmdabp) Cl$_{2}$] (R-dmdabp is R-6,6'-dimethyl-2,2'-diaminobiphenyl) and unresolved bn (bn is 2,3-diaminobutane) has yielded [Pt(R-dmdabp)-(R-bn)] Cl$_{2}$ only, while the reaction of [Pt(S-dmdabp)Cl$_{2}$] with unresolvd bn has yielded [Pt(S-dmdabp) (S-bn)]Cl$_{2}$ only. On the other hand, the standard [Pt(R-dmdabp) (R-bn)] Cl$_{2}$ complex has been independently prepared from the reaction of [Pt(R-dmdabp)Cl$_{2}$] with R-bn, and the standard [Pt(S-dmdabp) (S-bn)] Cl$_{2}$ from the reaction of [Pt(S-dmdabp)Cl$_{2}$] with S-bn. The stereospecific behavior of the racemic 2,3-diamino-butane is thus confirmed from the comparison of these Pt(Ⅱ) complexes prepared using racemic bn with the standard Pt(Ⅱ) complexes prepared using R-bn or S-bn.

• The Journal of the Petrological Society of Korea
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• v.25 no.4
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• pp.311-324
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• 2016

The characteristics of the rock cleavage in Jurassic granite from Geochang were analysed. The evaluation for three quarrying planes and three rock cleavages was performed using the parameters such as (1) reduction ratio between the value of spacing and the value of length, (2) microcrack spacing frequency(N), (3) total spacing($1mm{\geq}$), (4) exponential constant(a), (5) magnitude of exponent(${\lambda}$), (6) mean spacing($S_{mean}$), (7) difference value($S_{mean}-S_{median}$) between mean spacing and median spacing($S_{median}$) and (8) density of spacing. Especially the close dependence between the above spacing parameters and the parameters from the spacing-cumulative frequency diagrams was derived. The discrimination factors representing three quarrying planes and three rock cleavages were acquired through these mutual contrast. The analysis results of the research are summarized as follows. First, the reduction ratios of frequency(N), mean value, median value, the above difference value($S_{mean}-S_{median}$) and density for three rock cleavages are in orders of G(grain, (G1 + G2)/2) < H(hardway, (H1 + H2)/2) < R(rift, (R1 + R2)/2), H < G $\ll$ R, H < G $\ll$ R, H < G < R and H < G $\ll$ R. The values of the above five parameters for three planes show the various orders of R'(rift plane) $\ll$ H'(hardway plane) < G'(grain plane), R' $\ll$ G' < H', R' < H' < G', R' < G' < H' and R' $\ll$ H' < G', respectively. Second, the values of (I) parameters(2, 3, 4 and 5) and (II) parameters(6, 7 and 8) are in orders of (I) H < G < R and (II) R < G < H. On the contrary, the values of the above two groups(I~II) of parameters for three planes show reverse orders. Third, to review the overall characteristics of the arrangement among the six diagrams, these diagrams show an order of R2 < R1 < G2 < G1 < H2 < H1 from the related chart. In other words, above six diagrams can be summarized in order of rift(R1 + R2) < grain(G1 + G2) < hardway(H1 + H2). These results indicate a relative magnitude of rock cleavage related to microcrack spacing. Especially, two parameters for each diagram, the above difference value($S_{mean}-S_{median}$) and mean spacing, could provide advanced information for prediction the order of arrangement among the diagrams. Finally, the general chart for three planes and three rock cleavages were made. From the related chart, three exponential straight lines for three rock cleavages show an order of R(R1 + R2) < G(G1 + G2) < H(H1 + H2). On the contrary, three lines for three planes show an order of H'(R2 + G2) < G'(R1 + H2) < R'(G1 + H1). Consequently, correlation of the mutually reverse order between three planes and three rock cleavages can be drawn from the related chart.

• Bulletin of the Korean Chemical Society
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• v.5 no.6
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• pp.237-240
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• 1984

2,2'-Diaminobiphenyl platinum (II) complexes of optically active trans-1,2-diaminocyclohexane and 1,2-diaminopropane, [Pt(S,S-chxn)(dabp)]Cl2, [Pt(R,R-chxn)(dabp)]$C_{12}$, [Pt(S-pn)(dabp)]$C_{l2}$, and [Pt(R-pn)(dabp)]Cl2, where S,S-chxn and R,R-chxn are, respectively, S and R isomers of trans-1,2-diaminocyclohexane, and S-pn and R-pn are, respectively, S and R isomers of 1,2-diaminopropane, and dabp the 2,2'-diaminobipheny, have been prepared. The dabp ligand has been found to take the $=delta$ conformation in the S,S-chxn and S-pn platinum (II) complexes, while it takes the $\lambda$ conformation in the R,R-chxn and R-pn platinum (II) complexes.

• Journal of the Korean Chemical Society
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• v.41 no.3
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• pp.150-156
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• 1997

A precursor of bulgecinine, (4S,5R)-1-acetyl-2-formyl-5-benzyloxymethyl-4-pyrrolidinol (15) has been synthesized from diacetone-D-glucose. Barton deoxygenation, conversion to an L-sugar and displacement with $N_3^-$ at C-5, and one-pot reductive cyclization at C-2 produced (6R)-6-Ο-benzyloxymethyl-(3R)-3-methoxy-2-oxa-5-azabicyclo-[2,2,1]heptane(13), a key intermediate for bulgecinine. N-Acetylation and acid hydrolysis of 13 furnished a precursor of bulgecinine, (2S,4S,5R)-pyrrolidinol derivative 15 and its (2R,4S,5R)-diastereomer.

• Communications for Statistical Applications and Methods
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• v.2 no.1
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• pp.32-47
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• 1995

The coefficient of determination R/sup 2/, as the proprtation of by explained by a set of independent variavles x/sub 1/, x/sub 2, .cdots., x/sub k/ through a linear regression model, is a very useful tool in linear regression analysis. Suppose R/sup 2//sub yx/ is the coefficient of determination when y is regressed only on x/sub i/ alone. If the independent variables are correlaated, the sum, R/sup 2//sub {yx/sub 1/}/ +R/sup 2//sub {yx/sub 2/}/+.cdots.R/sup 2//sub {yx/sub k/}/, is not equal to R/sup 2/sub {yx/sub 1/x/sub 2/.cots.x/sub k/}/, where R/sup 2//sub {yx/sub 1/x/sub 2/.cdots.x/sub k/}/ is the coefficient of determination when y is regressed simultaneously on x/sub 1/, x/sub 2/,.cdots., x/sub k/. In this paper it is discussed that under what conditions the sum is greater than, equal to, or less than R/sup 2//sub {yx/sub 1/x/sub 2/.cdots.x/sub k/}/, and then the proofs for these conditions are given. Also illustrated examples are provided. In addition, we will discuss about inequality between R/sup 2//sub {yx/sub 1/x/sub 2/.cdots.x/sub k/}/ and the sum, R/sup 2//sub {yx/sub 1/}/+R/sup 2//sub {yx/sub 2/}/+.cdots.+R/sup 2//sub {yx/sub k/}/.

• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.11-15
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• 1985

Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.

• Journal of Environmental Science International
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• v.9 no.3
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• pp.223-228
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• 2000

The purpose of this study is to evaluate the purification capacity of pollutants by column test with the tidal flat sediment. Sediment materials were taken from Chunjangdae tidal flat which located in Chungnam Seochungun. The column tests were conducted on four conditions(R1 : raw sewage filtered by G2 filter, R2 : sterilized sewage after filtered by GF/C filter, R3 : R2 /filtered(membrane) seawater (1:1), R4 : R2 /filtered(membrane) seawater(1:2)). The results of this study may be summarized as followed ; The removed COD by column tests were increased according to increasing the quantity of sewage. During the column tests of 580min, the total removed ammonia nitrogen were 90.1mg for R1, 81.0mg for R2, 27.6mg for R3 and 4.1mg for R4. The result was similar to COD experiment. During the 580min, the total removed total-phosphates were 3.4mg for R1, 4.2mg for R2, 5.6mg for R3 and 2.0mg for R4. The removal efficiency of Pb and Cd for R3 and R4 reactor were higher than R1 and R2 reactor. The remove of heavy metal by the column test was high in sample with seawater. But in the initial 20min, the adsorbed Pb and Cd showed about 3% of the total adsorbed Pb and Cd during 580min.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.20 no.2
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• pp.730-740
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• 1996

Boiling heat transfer coefficients of pure refrigerants (R22, R32, R134a, R125, R290, and R600a) and refrigerant mixtures (R32/Rl34a, R290/ R600a, and R32/R125) are measured experimentally and compared with Chen's correlation. The test section is a seamless stainless steel tube with inner diameter of 7.7mm and uniformly heated by applying electric current directly to the tube. Heat fluxes range from 10 to 30kW$^2$. Mass fluxes are set to 424 ~ 742kg/m$^{2}$s for R22, R32, R134a, R32/R134a, and R32/Rl25 ; 265 ~ 583kg/m$^{2}$s for R290, R600a, and R290/R600a. Heat transfer coefficients depend strongly on heat flux at a low quality region and become independent as quality increases. Convective boiling term in the Chen's correlation predicts experimental data of the pure refrigerants fairly well (relative error of 12.1% for the data of quality over 0.2). The correlation for pure substances overpredicts the heat transfer coefficients for nonazeotropic refrigerant mixtures.