• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.373-391
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• 2007

We study an (n+3)($n\;{\geq}\;7-dimensional$ real submanifold of a (4m+3)-unit sphere $S^{4m+3}$ with Sasakian 3-structure induced from the canonical quaternionic $K\"{a}hler$ structure of quaternionic (m+1)-number space $Q^{m+1}$, and especially determine contact three CR-submanifolds with (p-1) contact three CR-dimension under the equality conditions given in (4.1), where p = 4m - n denotes the codimension of the submanifold. Also we provide necessary conditions concerning sectional curvature in order that a compact contact three CR-submanifold of (p-1) contact three CR-dimension in $S^{4m+3}$ is the model space $S^{4n_1+3}(r_1){\times}S^{4n_2+3}(r_2)$ for some portion $(n_1,\;n_2)$ of (n-3)/4 and some $r_1,\;r_2$ with $r^{2}_{1}+r^{2}_{2}=1$.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.561-577
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• 1998

We study (n＋3)-dimensional contact three CR submanifolds of a Riemannian manifold with Sasakian three structure and investigate some characterizations of $S^{4r＋3}$(a) $\times$ $S^{4s＋3}$(b) ($a^2$＋ $b^2$=1, 4(r ＋ s) = n - 3) as a contact three CR sub manifold of a (4m＋3)-dimensional unit sphere.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.585-600
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• 2011

In this paper we derive an integral formula on an (n + 3)-dimensional, compact, minimal contact three CR-submanifold M of (p-1) contact three CR-dimension immersed in a unit (4m+3)-sphere $S^{4m+3}$. Using this integral formula, we give a sufficient condition concerning the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

• Journal of Korea Foundry Society
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• v.20 no.2
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• pp.104-109
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• 2000

Three different white cast irons alloyed with Cr, V, Mo and W were prepared in order to study their abrasion wear behavior in as-cast and heat-treated conditions. The specimens were produced using a 15㎏-capacity high frequency induction furnace. Melts were super-heated to $1600^{\circ}C$, and poured at $1550^{\circ}C$ into Y-block pepset molds. Three combinations of the alloying elements were selected so as to obtain the different types of carbides : 3%C-10%Cr-5%Mo-5%W(alloy No. 1: $M_7C_3$ and $M_6C$), 3%C -10%V-5%Mo-5%W(alloy No. 2: MC and $M_2C$) and 3%C-17%Cr-3%V(alloy No. 3: $M_7C_3$ only). A scratching type abrasion test was carried out in the states of as-cast(AS), homogenizing(AH), air-hardening(AHF) and tempering(AHFT). First of all, the as-cast specimens were homogenized at $950^{\circ}C$ for 5h under the vacuum atmosphere. Then, they were austenitized at $1050^{\circ}C$ for 2h and followed by air-hardening in air. The air-hardened specimens were tempered at $300^{\circ}C$ for 3h. 1 ㎏ load was applied in order to contact the specimen with abrading wheel which was wound by 120 mesh SiC paper. The wear loss of the test piece(dimension: $50{\times}50{\times}5$ mm) was measured after one cycle of wear test and this procedure was repeated up to 8 cycles. In all the specimens, the abrasion wear loss was found to decrease in the order of AH, AS, AHFT and AHF states. Abrasion wear loss was lowest in the alloy No.2 and highest in the alloy No.1 except for the as-cast and homogenized condition in which the alloy No.3 showed the highest abrasion wear loss. The lowest abrasion wear loss of the alloy No.2 could be attributed to the fact that it contained primary and eutectic MC carbides, and eutectic $M_2C$ carbide with extremely high hardness. The matrix of each specimen was fully pearlitic in the as-cast state but it was transformed to martensite, tempered martensite and austenite depending upon the type of heat-treatment. From these results, it becomes clear that MC carbide is a significant phase to improve the abrasion wear resistance.