• Journal of Forest and Environmental Science
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• v.15 no.1
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• pp.107-116
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• 1999

For the study on the variation of Pinus densiflora S. et Z. natural populations. 13 stands were selected in Kangwondo region in 1994. They were located in Hyoja-Dong, Chunchon-Si [plot 1]; Sinnam-Myun, lnje-Gun [plot 2]; Buk-Myun, lnje-Gun [plot 3]; Seo-Myun, Yangyang-Gun [plot 4 and 5]; Sonyang-Myun, Yangyang-Gun [plot 6]; Toseong-Myun, Goseong-Gun [plot 7]; Synrim-Myun, Wonju-Si [plot 8]; Dunnae-Myun, Hoengsong-Gun [plot 9]; Jinbu-Myun, Pyungchang-Gun [plot 10]; Sogumgang, Kangneung-Si [plot 11]; Seo-Myun, Ulchin-Gun [plot 12]; and Mitan- Myun, Pyungchang-Gun [plot 13]. The morphological characteristics of needles and cones were investigated for twenty individual trees from each population. The results are summarized as follows: 1. Among needle characteristics, length, the number of serration. and the number of stoma row were significantly different among populations and between individuals within a population. They were 69.3~91.9mm, 25.1~28.7 per 0.5cm, and 4.1~6.2 rows for abaxial side and 2.9~4.6 rows for adaxial side, respectively. 2. Among cone characters surveyed. length and diameter were significantly different among populations and between individuals within a population. They were 31.1~43.7mm and 20.0~24.1mm, respectively. 3. A highly positive correlation was observed between the number of stoma row of abaxial side and that of adaxial side. and between length and diameter of cone. But the number of serration was negatively correlated with all traits. 4. The distance from seashore was positively correlated with needle length and the number of serration, but negatively correlated with the number of stoma row and cone length. However, the altitude was positively correlated with the number of serration, but negatively correlated with the number of stoma row and cone length. 5. Cluster analysis using average linkage method for needle and cone characteristics showed two groups to Euclidean distance 1.0. They were group I consisting of plots 1, 8, 12, and 13 and group II of plots 2, 3, 4, 5, 6, 7, 9, 10, and 11. However, group II was divided again to Euclidean distance 0.8, that is a group including plots 2, 3, 9, and 11 and the other group comprising plots 4, 5, 6, 7, and 10.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1485-1500
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• 2021

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an (m + 2)-dimensional space and spherical curves in an (m + 1)-dimensional space.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.905-913
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• 2023

Suppose that M is a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space 𝔼ⁿ⁺¹ with the origin o in its convex side and with the outward unit normal N. For a fixed point p ∈ M and a positive constant t, we put 𝚽_t the hyperplane parallel to the tangent hyperplane 𝚽 at p and passing through the point q = p - tN(p). We consider the region cut from M by the parallel hyperplane 𝚽_t, and denote by I_p(t) the (n + 1)-dimensional volume of the convex hull of the region and the origin o. Then Schneider's characterization theorem for ellipsoids states that among centrally symmetric, strictly convex and closed surfaces in the 3-dimensional Euclidean space 𝔼³, the ellipsoids are the only ones satisfying I_p(t) = 𝜙(p)t, where 𝜙 is a function defined on M. Recently, the characterization theorem was extended to centrally symmetric, strictly convex and closed hypersurfaces in 𝔼ⁿ⁺¹ satisfying for a constant 𝛽, I_p(t) = 𝜙(p)t^𝛽. In this paper, we study the volume I_p(t) of a strictly convex and complete hypersurface in 𝔼ⁿ⁺¹ with the origin o in its convex side. As a result, first of all we extend the characterization theorem to strictly convex and closed (not necessarily centrally symmetric) hypersurfaces in 𝔼ⁿ⁺¹ satisfying I_p(t) = 𝜙(p)t^𝛽. After that we generalize the characterization theorem to strictly convex and complete (not necessarily closed) hypersurfaces in 𝔼ⁿ⁺¹ satisfying I_p(t) = 𝜙(p)t^𝛽.