• 대한수학회논문집
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• 제9권3호
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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.

• 대한수학회논문집
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• 제35권2호
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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.

• 호남수학학술지
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• 제33권3호
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• pp.321-333
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• 2011

During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors. The functional determinant for the n-dimensional sphere $S^n$ with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on $S^n$ (n = 8, 9) by mainly using ceratin known closed-form evaluations of series involving Zeta function.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제4권2호
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• pp.137-144
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• 1997

In this paper we consider covering problems in spherical geometry. Let $cov_q{S_1}^n$ be the smallest radius of q equal metric balls that cover n-dimensional unit sphere ${S_1}^n$. We show that $cov_q{S_1}^n\;=\;\frac{\pi}{2}\;for\;2\leq\;q\leq\;n+1$ and $\pi-arccos(\frac{-1}{n+1})$ for q = n ＋ 2. The configuration of centers of balls realizing $cov_q{S_1}^n$ are established, simultaneously. Moreover, some properties of $cov_{q}$X for the compact metric space X, in general, are proved.

• 한국구조물진단유지관리공학회 논문집
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• 제18권3호
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• pp.160-166
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• 2014

본 연구에서는 건축물의 외피를 통해 손실되는 냉난방 에너지를 감소시킬 수 있는 방안으로써, 구조용 콘크리트에 단열성능을 향상시킬 수 있는 할로마이크로스피어 (Hollow Micro Sphere, 이하 HMS)를 사용하였다. 중공구조의 마이크로 크기의 입자인 HMS를 사용한 콘크리트를 실험한 결과는 아래와 같다. HMS의 사용으로 인해 슬럼프에 감소가 나타나 고성능감수제의 사용이 필요할 것으로 판단된다. 공기량은 HMS의 치환율이 증가할수록 감소하는 경향을 보였으며, 압축강도는 HMS의 계면부착력이 형성되지 않는 것에 기인하여 치환율 증가에 따라 감소하는 것으로 나타났다. 열전도율과 단위용적질량은 치환율이 증가할수록 감소하였다. 열전도율은 보통콘크리트 약 30.0~46.5% 감소한 것으로 나타났다.

• 충청수학회지
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• 제22권1호
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• pp.89-99
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• 2009

Let $\mu$ be a finite positive Borel measure on the unit ball $B{\subset}\mathbb{C}^n$ and $\nu$ be the Euclidean volume measure such that ${\nu}(B)=1$. For the unit sphere $S=\{z:{\mid}z{\mid}=1\}$, $\sigma$ is the rotation-invariant measure on S such that ${\sigma}(S)=1$. Let $\mathcal{P}[f]$ be the invariant Poisson integral of f. We will show that there is a constant M > 0 such that $\int_B{\mid}{\mathcal{P}}[f](z){\mid}^{p}d{\mu}(z){\leq}M\;{\int}_B{\mid}{\mathcal{P}}[f](z)^pd{\nu}(z)$ for all $f{\in}L^p({\sigma})$ if and only if ${\parallel}{\mu}{\parallel_r}\;=\;sup_{z{\in}B}\;\frac{\mu(E(z,r))}{\nu(E(z,r))}\;<\;\infty$.

• 한국건축시공학회:학술대회논문집
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• 한국건축시공학회 2011년도 춘계 학술논문 발표대회 1부
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• pp.131-132
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• 2011

This study analyzes the yearly-best delivered size range of truck mixer based on the specified mix, Water to Binder Ratio, aggregate proportion and unit amount with statistical method targeting on Kyeongi and Kyeongsang province and compares with the similar materials of Japan to propose as a basic standard for the quality control of mixer truck. As a result, in case of the Water to Binder Ratio of these areas, it is higher than Japan's due to the excessive safety rate reflecting the changes of differential value impact and unit amount, and the unit amount's standard deviation is very large by reflecting the changes of the amount used and chemical admixture susceptibility. In case of aggregate proportion, the frequency rate is about 50%, which is very similar value with Japan's one.

• 대한수학회보
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• 제39권1호
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• pp.89-95
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• 2002

In this paper, we obtain an integral formula relating the measure of great spheres $S^{n-2}$ and arc length of a curve on the unit sphere $S^{n-1$ As an application of the formula, we develop a geometric inequality for a spherical curve and prone generalized version of Fenchel＇s theorem in $S^n$.

• 대한수학회보
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• 제41권1호
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• pp.117-123
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• 2004

It has been conjectured that, on a compact orient able manifold M, a critical point of the total scalar curvature functional restricted the space of unit volume metrics of constant scalar curvature is Einstein. In this paper we show that if a manifold is a 3-dimensional warped product, then (M, g) cannot be a critical point unless it is isometric to the standard sphere.

• 대한수학회보
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• 제26권2호
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• pp.119-126
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• 1989

Throughout this paper, X will always denote a Banach space over the complex numbers C, and L(X) will denote the Banach algebra of all continuous linear operators on X. Operator will always mean continuous linear operator. An n-tuple of operators T$_{1}$,..,T$_{n}$ on X will be denoted by over ^ T=(T$_{1}$,..,T$_{n}$ ). Let L$^{n}$ (X) be the set of all n-tuples of operators on X. X' will denote the dual space of X, S(X) its unit sphere and .PI.(X) the subset of X*X' defined by .PI.(X)={(x,f).mem.X*X': ∥x∥=∥f∥=f(x)=1}.