• 대한수학회보
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• 제50권1호
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• pp.25-36
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• 2013

By using some existence theorems of maximal elements for a family of set-valued mappings involving a better admissible set-valued mapping under noncompact setting of FC-spaces, we present some non-empty intersection theorems for a family $\{G_i\}_{i{\in}I}$ in product FC-spaces. Then, as applications, some new existence theorems of equilibrium for a system of generalized vector equilibrium problems are proved in product FC-spaces. Our results improve and generalize some recent results.

• Bulletin of the Korean Chemical Society
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• 제15권5호
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• pp.379-386
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• 1994

Reactions of 1',2-bis(diphenylphosphino)-l-(N,N-dimethylaminomethyl)ferrocene (FcNPP) with $M(CO)_6$ (M=Cr, Mo, W) in the presence of TMNO (Trimethylamine oxide) in a stoichiometric ratio of 1 : 1.5 : 3.5 produced a series of Gp 6 metal carbonyl derivatives with a variety of coordination modes: M(CO)$_4({\eta}^2$-FcNPP-P,P) (M=Cr, Mo, W), $M(CO)_5({\eta}^1-FcNPP-P) (M=Mo, W)\;,\; M_2(CO)_9({\eta}^1\;,\;{\eta}^2-FcNPP-P,P,N) (M=Cr, Mo)\;,\;M_2(CO)_{10}({\eta}^1\;,\;{\eta}^1-FcNPP-P,P) (M=Cr, Mo, W)\;, and\;W(CO)_4({\eta}^2-FcNPP(O)-P,N)$. All these complexes were characterized by microanalytical and spectroscopic techniques. In one case, the structure of W(CO)$_4({\eta}^2$-FcNPP(O)-P,N) was determined by X-ray crystallography. Crystals are monocinic, space group P$2_{1/C}$, with a=10.147(2), b=19.902(3), c=19.821(4) ${\AA},\;{\beta}=96.88(2)^{\circ},\;V=3974(l){\AA}^3$, Z=4, and $D_{calc}=1.64 g cm^{-3}$. The geometry around the central tungsten metal is a distorted octahedron, with the nitrogen and phosphorus atoms being cis to each other. Some of these complexes exhibited catalytic activities in the allylic oxidation and epoxidation of cholesterly acetate. Other oxidation products were also formed with the different chemical yields and product distribution depending upon the ligand and the central metal.