• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.121-126
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• 1985

D.K. Harrison [5] has shown that if R and S are fields of characteristic different from 2, then two Witt rings W(R) and W(S) are isomorphic if and only if W(R)/I(R)$^{3}$ and W(S)/I(S)$^{3}$ are isomorphic where I(R) and I(S) denote the fundamental ideals of W(R) and W(S) respectively. In [1], J.K. Arason and A. Pfister proved a corresponding result when the characteristics of R and S are 2, and, in [9], K.I. Mandelberg proved the result when R and S are commutative semi-local rings having 2 a unit. In this paper, we prove the result when R and S are 2-fold full rings. 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 nondegenerate (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)=(.phi.(x+y)-.phi.(x)-.phi.(y))/2 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 U(R). 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 [ $a_{11}$, $a_{22}$] for the space.the space.e.e.e.

• Kyungpook Mathematical Journal
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• v.13 no.2
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• pp.235-240
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• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.701-710
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• 1994

Suppose ${X_n}$ is a Markov process taking values in some arbitrary space $(S, \varphi)$ with n-stemp transition probability $$ P^{(n)}(x, B) = Prob(X_n \in B$\mid$X_0 = x), x \in X, B \in \varphi.$$ We shall call a Markov process with transition probabilities $P{(n)}(x, B)$ $\phi$-irreducible for some non-trivial $\sigma$-finite measure $\phi$ on $\varphi$ if whenever $\phi(B) > 0$, $$ \sum^{\infty}_{n=1}{2^{-n}P^{(n)}}(x, B) > 0, for every x \in S.$$ A non-trivial $\sigma$-finite measure $\pi$ on $\varphi$ is called invariant for ${X_n}$ if $$ \int{P(x, B)\pi(dx) = \pi(B)}, B \in \varphi $$.

• Bulletin of the Korean Chemical Society
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• v.23 no.6
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• pp.845-851
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• 2002

The synthesis of the intramolecular donor - stabilized silyl and germyl complexes of the type ($Cab^c.p) MMe_2X$ (2a:M=Si, X=Cl;2b;M= Ge, X=Cl;2e;M=Si,X=H) was achieved by the reaction of $LiCab^c,p$ (1) with $Me_2SiClX$ and $Me_2GeCl_2$ respectively. The intramolecular M←P interacion in 2a-2c is provided by $^1H$, $13^C.$, $31^P$ and $29^Si$ NMR spectroscopy. The salt elimination reactions of dichlorotetramethyldisilane and -digermane with 1 afforded the $bis(\sigma-carboranylphosphino)disilane$ and disgermane [$(Cab^C.P)MMe_2]_2(4a;M$ = Si;4b: M=Ge). The oxidative addition reaction of 4a-4b with $pd_2(dba)_3CHCl_3afforded$ the bis(silyl)-and bis(germyl)-palladium complexes. The chloro-bridged dipalladium complexes were obtained by the reaction of 2a-2b with $pd_2(dba)_3CHCl_3$ The crystal structures of 5a and 7b were determined by X-ray structural studies.

• Journal of the Korean Magnetics Society
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• v.3 no.2
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• pp.130-134
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• 1993

The magnetic properties of the amorphous $Fe_{73.5-X}Co_{X}Cu_{1}Nb_{3}Si_{13.5}B_{9}(x=2,\;4)$ alloys, fabricated by a single roll rapid quenching technique and annealed at $400~650^{\circ}C$, have been investigated. The optimum annealing temperature is $550^{\circ}C$ for the amorphous $Fe_{71.5}Co_{2}Cu_{1}Nb_{3}Si_{13.5}B_{9}$ alloy. The properties of the nanocrystalline $Fe_{71.5}Co_{2}Cu_{1}Nb_{3}Si_{13.5}B_{9}$ alloy show the relative permeability of $1.1{\times}10^{4}$ and the coercive force of 0.22 Oe at 1 kHz. When annealed at $600^{\circ}C$, the nanocrystalline $Fe_{69.5}Co_{4}Cu_{1}Nb_{3}Si_{13.5)B_{9}$ alloy shows the relative permeability of $1.0{\times}10^{4}$ and the coercive force of 0.19 Oe at 1 kHz. From the X-ray measurement, it is found that the remarkably improved soft magnetic properties are the effect of the formation of $\alpha$-Fe(Si) grain. By the results of FMR exper-imeIlt, the optimum annealing condition is just below temperature which the peak-to-peak line width of FMR spectrum increase rapidly.

• Kyungpook Mathematical Journal
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• v.49 no.1
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• pp.167-181
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• 2009

Based on Ricatti equation $XA^{-1}X=B$ for two (positive invertible) operators A and B which has the geometric mean $A{\sharp}B$ as its solution, we consider a cubic equation $X(A{\sharp}B)^{-1}X(A{\sharp}B)^{-1}X=C$ for A, B and C. The solution X = $(A{\sharp}B){\sharp}_{\frac{1}{3}}C$ is a candidate of the geometric mean of the three operators. However, this solution is not invariant under permutation unlike the geometric mean of two operators. To supply the lack of the property, we adopt a limiting process due to Ando-Li-Mathias. We define reasonable geometric means of k operators for all integers $k{\geq}2$ by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.

• Journal of the Korean Magnetics Society
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• v.7 no.6
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• pp.293-298
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• 1997

$Fe_5Si_Xb_{5-x}$ (x=0, 1, 2, 3) powders were prepared by mechanical alloying, and their phases and magnetic properties were investigated by using XRD, TEM, Mossbauer spectroscopy and VSM. Starting elements are incorporated into $\alpha$-Fe in the early stage of mechanical alloying, and the stable phases are formed as the milling proceeds. After the annealing at 80$0^{\circ}C$ for 2 hours, it is found that the FeB and $Fe_2B$ phases coexist for the $Fe_5B_5$(x=0) composition. By substituting Si for B, the formation of $Fe_2B$ phase is restricted and the $Fe_5SiB_2$, $Fe_2Si_{0.4}B_{0.6}$ and paramagnetic phase begin to appear. The FeB phase has wide range of hyperfine magnetic field because it is not fully crystallized on the annealing at 800 $^{\circ}C$. On the contrary, others have good crystalline phases and show well-defined hyperfine magnetic field. Magnetic saturation is highest for the $Fe_5B_5$ composition where the amount of the $Fe_2B$ phase in large.

• Journal of Magnetics
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• v.14 no.4
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• pp.150-154
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• 2009

The effect of Ga and, Nb addition on the kinetics and mechanism of the disproportionation of a Nd-Fe-B alloy were investigated by isothermal thermopiezic analysis (TPA) using $Nd_{12.5}Fe_{(81.1-(x+y))}B_{6.4}Ga_xNb_y$ (x=0 and 0.3, y= 0 and 0.2) alloys. The addition of Ga and Nb retarded the disproportionation kinetics of the Nd-Fe-B alloy significantly, and increased the activation energy of the disproportionation reaction. The disproportionation kinetics of the $Nd_{12.5}Fe_{(81.1-(x+y))}B_{6.4}Ga_xNb_y$ alloys measured under an initial hydrogen pressure of 0.02 MPa were fitted to a parabolic rate law. This suggested that during the disproportionation of $Nd_{12.5}Fe_{(81.1-(x+y))}B_{6.4}Ga_xNb_y$ alloys with an initial hydrogen pressure of 0.02 MPa, a continuous disproportionation product is formed and the overall reaction rate is limited by the diffusion of hydrogen atoms (or ions).

• Magazine of the Korean Society of Agricultural Engineers
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• v.15 no.4
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• pp.3160-3169
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• 1973

Rainfall, evaporation, and permeability of water are the most important factors in determining the demand of water. The Daegu area has only a meteorologi observatory and there is not sufficient data for adapting the advanced method for derivation of the estimated of evaporation in the Daegu area. However, by using available data, the writer devoted his great effort in deriving the most reasonable formula applicable to the Daegu area and it is adaptable for various purposes such as industry and estimation of groundwater etc. The data used in this study was the monthly amount of evaporation of the Daegu area for the past 13 years(1960 to 1970). A year can be divided into two groups by relative degrees of evaporation in this area: the first group (less evaporation) is January, February, March, October, November, and December, and the second (more evaporation) is April, May, June, July, August, and September. The amount of evaporation of the two groups were statistically treated by the theory of probability for derivation of estimated formula of evaporation. The formula derved is believed to fully consider. The characteristic hydrological environment of this area as the following shows: log(x＋3)=0.8963＋0.1125$\xi$..........(4, 5, 6, 7, 8, 9 month) log(x-0.7)=0.2051＋0.3023$\xi$..........(1, 2, 3, 10, 11, 12 month) This study obtained the above formula of probability of the monthly evaporation of this area by using the relation: $F_(x)=\frac{1}{{\surd}{\pi}}\int\limits_{-\infty}^{\xi}e^{-\xi2}d{\xi}\;{\xi}=alog_{\alpha}({\frac{x_0+b'}{x_0+b})\;(-b<x<{\infty})$ $$log(x_0＋b)=0.80961$ $$\frac{1}{a}=\sqrt{\frac{2N}{N-1}}\;Sx=0.1125$$ $$b=\frac{1}{m}\sum\limits_{i-I}^{m}b_s=3.14$$ $$S_x=\sqrt{\frac{1}{N}\sum\limits_{i-I}^{N}\{log(x_i+b)\}^2-\{log(x_i+b)\}^2}=0.0791$$ (4, 5, 6, 7, 8, 9 month) This formula may be advantageously applied to estimation of evaporation in the Daegu area. Notation for general terms has been denoted by following: $W_(x)$: probability of occurance. $$W_(x)=\int_x^{\infty}f(x)dx$$ P : probability $$P=\frac{N!}{t!(N-t)}{F_i^{N-{\pi}}(1-F_i)^l$$ $$F_{\eta}:\; Thomas\;plot\;F_{\eta}=(1-\frac{n}{N+1})$$ $X_l\;X_i$: maximun, minimum value of total number of sample size(other notation for general terms was used as needed)

• Journal of the Korean Ceramic Society
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• v.25 no.6
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• pp.577-584
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• 1988

Gd1Ba2Cu3O9-x has been found to be a high Tc superconductor with a transitiion onset at 91K and zero resistance achieved at 87K. The structure as determined from x-ray diffraction is orthorhombic, with lattice constants b=3.842$\pm$0.002$\AA$, b=3.895$\pm$0.003$\AA$, and c=11.684$\pm$0.007$\AA$. The structural similarities between the Gd1Ba2Cu3O9-x compound and the well-studied single phase perovskite, Y1Ba2Cu3O9-x are discussed. A correlation between the observed x-ray spectrum and the effect of oxygen deficiencies in several of the unit cell planes is also discussed.