• Title, Summary, Keyword: unit (4m+3)-sphere

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SCALAR CURVATURE OF CONTACT THREE CR-SUBMANIFOLDS IN A UNIT (4m + 3)-SPHERE

  • Kim, Hyang-Sook;Pak, Jin-Suk
    • 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.

ON CONTACT THREE CR SUBMANIFOLDS OF A (4m + 3)-DIMENSIONAL UNIT SPHERE

  • Kwon, Jung-Hwan;Pak, Jin--Suk
    • 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.

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CONTACT THREE CR-SUBMANIFOLDS OF A (4m + 3)-DIMENSIONAL UNIT SPHERE

  • Kim, Hyang-Sook;Kim, Young-Mi;Kwon, Jung-Hwan;Pak, Jin-Suk
    • 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 $r^{2}_{1}+r^{2}_{2}=1$.

HYPERSURFACES IN THE UNIT SPHERE WITH SOME CURVATURE CONDITIONS

  • Park, Joon-Sang
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.641-648
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    • 1994
  • Let M be a minimally immersed closed hypersurface in $S^{n+1}$, II the second fundamental form and $S = \Vert II \Vert^2$. It is well known that if $0 \leq S \leq n$, then $S \equiv 0$ or $S \equiv n$ and totally geodesic hypersheres and Clifford tori are the only possible minimal hypersurfaces with $S \equiv 0$ or $S \equiv n$ ([6], [2]). From these results, Chern suggested some questions on the study of compact minimal hypersurfaces on the sphere with S =constant: what are the next possible values of S to n, and does in the ambient sphere\ulcorner By the way, S is defined extrinsically but, in fact, it is an intrinsic invariant for the minimal hypersurface, i.e., S = n(n-1) - R, where R is the scalar, curvature of M. Some partial answers have been obtained for dim M = 3: Assuming $M^3 \subset S^4$ is closed and minimal with S =constant, de Almeida and Brito [1] proved that if $R \geq 0$ (or equivalently $S \leq 6$), then S = 0, 3 or 6, Peng and Terng ([5]) proved that if M has 3 distint principal curvatures, then S = 6, and in [3] Chang showed that if there exists a point which has two distinct principal curvatures, then S = 3. Hence the problem for dim M = 3 is completely done. For higher dimensional cases, not much has been known and these problems seem to be very hard without imposing some more conditions on M.

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GRADIENT EINSTEIN-TYPE CONTACT METRIC MANIFOLDS

  • Kumara, Huchchappa Aruna;Venkatesha, Venkatesha
    • Communications of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.639-651
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    • 2020
  • Consider a gradient Einstein-type metric in the setting of K-contact manifolds and (κ, µ)-contact manifolds. First, it is proved that, if a complete K-contact manifold admits a gradient Einstein-type metric, then M is compact, Einstein, Sasakian and isometric to the unit sphere 𝕊2n+1. Next, it is proved that, if a non-Sasakian (κ, µ)-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension 3, and for higher dimension, M is locally isometric to the product of a Euclidean space 𝔼n+1 and a sphere 𝕊n(4) of constant curvature +4.

The solid angle estimation of acetabular coverage of the femoral head using 3D method (입체각 측정을 통한 대퇴골두에 대한 관골구 coverage 측정)

  • Choi, K.H.;Kim, M.C.;Lim, C.T.;Kim, S.I.
    • Proceedings of the KOSOMBE Conference
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    • v.1997 no.05
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    • pp.123-126
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    • 1997
  • We present a method for the estimation of 3D solid angle assessment of the acetabular coverage of the femoral head in 3D space. At first, femoral head and acetabulum is segmented from the original CT scan images. The slice thickness is 1.5mm and the number of slices is usually 30-40 to cover the entire acetabulum. The superior half of the femoral head is modeled as part of a sphere. Thus, the axial cross sections of the upper half of the femoral head are also modeled as circles. A set of points from each outline image of femoral head is fitted recursively into a circle by minimizing root-mean-square (RMS) error. With these fitted circles, a center point of the femoral head model is evaluated. This is a reference point for calculating the solid angle of the acetabular inner surface. Next, the tangent lines connecting from a set of points of the acetabular edge to the center of the fitted sphere are obtained. The lines pass through the unit sphere whose center is the same as that of the femoral head. With the points on the unit sphere, we calculate area and estimate the solid angle. Based on this solid angle, the deformity of the acetabulum is analyzed. In case of normal subject, the solid angle is about 4.3 (rad) and acetabular coverage is 68%.

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A Study on the Planning for PC Use Space in Residences - Focus on the Unit Plan of Apartments According to the Size - (주거 내 PC 사용공간 계획에 관한 연구)

  • Jeong, Byeol-Ah;Kim, Bong-Ae
    • Journal of the Korean housing association
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    • v.18 no.3
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    • pp.51-61
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    • 2007
  • This study is aimed to provide a guide for PC use space in apartment. To achieve this, a design about PC use space which was considered adequate location, area, form was suggested. As methods of the study, survey research method and case study. The survey was taken by five hundred-eighty-nine people who used PCs in apartment from May 2005 to July 2005 in JeJu Special Self-governing Province for an analysis of data. The results of the survey were showed PC use space situation and furnitures layout patterns. The case was classified by unit plans of size, and it showed design example of PC use space. In types of $61{\sim}69$ square meters ($m^2$), PC use space was located in dead-space in the residence with minimum scale, because of $61{\sim}69$ square meters types are smaller than the other types. In types $100{\sim}129$ square meters, PC use space which was divided with a partition panel was located in the livingroom expanding the room space to the balcony. In types of 130 square meters, PC use space was located in the room of sons or daughters which was arranged with other furniture. The adequate area of PC use space is $1.4{\sim}2.2$ square meters, and it should take into account furniture size and human active sphere. And PC use space of form should consider traffic line, personal space and privacy. Besides it is required of ergonomic design in chair, table in its scale. And it is necessary development of systematic furniture for PC use space.

The Consideration about Heavy Metal Contamination of Room and Worker in a Workshop (공작실에서 실내 및 작업종사자의 중금속 오염도에 관한 고찰)

  • Kim Jeong-Ho;Kim Gha-Jung;Kim Sung-Ki;Bea Suk-Hwan
    • The Journal of Korean Society for Radiation Therapy
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    • v.17 no.2
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    • pp.87-94
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    • 2005
  • Purpose : Heavy metal use when producing the block from the workshop. At this time, production of heavy metal dust and fume gives risk in human. This like heavy metal to improve seriousness through measurement and analysis. And by the quest in solution is purpose of this thesis. Materials and Methods : Organization is Inductively Coupled Plasma Atomic Emission Spectrometer, and the object is Deajeon city 4 workshops in university hospital radiation oncology (Bismuth, Lead, Tin and cadmium). Method is the ppb the pumping it does at unit, comparison analysis. And the Calculation heavy metal standard level in air through heavy metal standard level in body and blood, so Heavy metal temporary standard set. Results : Subterranean existence room air quality the administration laws appointed Lead and Cadmium's exposure recommend that it is $3{\mu}g/m^3\;and\;2{\mu}g/m^3$. And Bismuth and Tin decides $7{\mu}g/m^3\;and\;6{\mu}g/m^3$ through standard level in air heavy metal and standard level in body and blood. Heavy metal measurement level of workshops in 4 university hospital Daejeon city compares with work existence and nonexistence. On work nonexistence almost measurement level is below the recommend level. But work existence case express high level. Also consequently in composition ratio of the block is continuous with the detection ratio. Conclusion : Worker's heavy metal contamination imbrued serious for solution founds basic part. In hospital may operation on local air exhauster and periodical efficiency check, protector offer, et al. And worker have a correct understanding part of heavy metal contamination, and have continuous interest, health control. Finally, learned society sphere administer to establishment standard level and periodical measurement. And it founds basic solution plan of periodical special health checkup.

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