• Title, Summary, Keyword: n-dimensional volume

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A Design of Teaching Unit to Foster Secondary Pre-service Teachers' Mathematising Ability: Inquiry into n-volume of n-simplex (예비중등교사의 수학화 능력을 신장하기 위한 교수단원의 설계: n-단체(simplex)의 n-부피 탐구)

  • Kim Jin-Hwan;Park Kyo-Sik
    • School Mathematics
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    • v.8 no.1
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    • pp.27-43
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    • 2006
  • The objective of this paper is to design teaching units to foster secondary pre-service teachers' mathematising abilities. In these teaching units we focus on generalizing area of a 2-dimensional triangle and volume of a 3-dimensional tetrahedron to n-volume of n-simplex In this process of generalizing, principle of the permanence of equivalent forms and Cavalieri's principle are applied. To find n-volume of n-simplex, we define n-orthogonal triangular prism, and inquire into n-volume of it. And we find n-volume of n-simplex by using vectors and determinants. Through these teaching units, secondary pre-service teachers can understand and inquire into n-simplex which is generalized from a triangle and a tetrahedron, and n-volume of n-simplex which is generalized from area of a triangle and volume of a tetrahedron. They can also promote natural connection between school mathematics and academic mathematics.

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A CHARACTERIZATION OF ELLIPTIC HYPERBOLOIDS

  • Kim, Dong-Soo;Son, Booseon
    • Honam Mathematical Journal
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    • v.35 no.1
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    • pp.37-49
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    • 2013
  • Consider a non-degenerate open convex cone C with vertex the origin in the $n$2-dimensional Euclidean space $E^n$. We study volume properties of strictly convex hypersurfaces in the cone C. As a result, for example, if the volume of the region of an elliptic cone C cut off by the tangent hyperplane P of M at $p$ is independent of the point $p{\in}M$, then it is shown that the hypersurface M is part of an elliptic hyperboloid.

THE HARDY TYPE INEQUALITY ON METRIC MEASURE SPACES

  • Du, Feng;Mao, Jing;Wang, Qiaoling;Wu, Chuanxi
    • Journal of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1359-1380
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    • 2018
  • In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.

Computing Rotational Swept Volumes of Polyhedral Objects (다면체의 회전 스웹터 볼륨 계산 방법)

  • 백낙훈;신성용
    • Korean Journal of Computational Design and Engineering
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    • v.4 no.2
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    • pp.162-171
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    • 1999
  • Plane sweep plays an important role in computational geometry. This paper shows that an extension of topological plane sweep to three-dimensional space can calculate the volume swept by rotating a solid polyhedral object about a fixed axis. Analyzing the characteristics of rotational swept volumes, we present an incremental algorithm based on the three-dimensional topological sweep technique. Our solution shows the time bound of O(n²·2?+T?), where n is the number of vertices in the original object and T? is time for handling face cycles. Here, α(n) is the inverse of Ackermann's function.

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3-Dimensional Representation of Heart by Thresholding in EBT Images (EBT 영상에서 임계치 설정법에 의한 심장의 3차원 표현)

  • Won, C.H.;Koo, S.M.;Kim, M.N.;Cho, J.H.
    • Proceedings of the KOSOMBE Conference
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    • v.1997 no.11
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    • pp.533-536
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    • 1997
  • In this paper, we visualized 3-dimensional volume of heart using volume method by thresholding in EBT slices data. Volume rendering is the method that acquire the color by casting a pixel ray to volume data. The gray level of heart region is so high that we decide heart region by thresholding method. When a pixel ray is cast to volume data, the region that is higher than threshold value becomes heart region. We effectively rendered the heart volume and showed the 3-dimensional heart volume.

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Three dimensional numerical simulations for non-breaking solitary wave interacting with a group of slender vertical cylinders

  • Mo, Weihua;Liu, Philip L.F.
    • International Journal of Naval Architecture and Ocean Engineering
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    • v.1 no.1
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    • pp.20-28
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    • 2009
  • In thus paper we validate a numerical model for wave-structure interaction by comparing numerical results with laboratory data. The numerical model is based on the Navier-Stokes (N-S) equations for an incompressible fluid. The N-S equations are solved by a two-step projection finite volume scheme and the free surface displacements are tracked by the volume of fluid (VOF) method The numerical model is used to simulate solitary waves and their interaction with a group of slender vertical piles. Numerical results are compared with the laboratory data and very good agreement is observed for the time history of free surface displacement, fluid particle velocity and wave force. The agreement for dynamic pressure on the cylinder is less satisfactory, which is primarily caused by instrument errors.

WEAKLY EINSTEIN CRITICAL POINT EQUATION

  • Hwang, Seungsu;Yun, Gabjin
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.1087-1094
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    • 2016
  • On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

RELATIVE ISOPERIMETRIC INEQUALITY FOR MINIMAL SUBMANIFOLDS IN SPACE FORMS

  • Seo, Keomkyo
    • Korean Journal of Mathematics
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    • v.18 no.2
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    • pp.195-200
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    • 2010
  • Let C be a closed convex set in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$. Assume that ${\Sigma}$ is an n-dimensional compact minimal submanifold outside C such that ${\Sigma}$ is orthogonal to ${\partial}C$ along ${\partial}{\Sigma}{\cap}{\partial}C$ and ${\partial}{\Sigma}$ lies on a geodesic sphere centered at a fixed point $p{\in}{\partial}{\Sigma}{\cap}{\partial}C$ and that r is the distance in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ from p. We make use of a modified volume $M_p({\Sigma})$ of ${\Sigma}$ and obtain a sharp relative isoperimetric inequality $$\frac{1}{2}n^n{\omega}_nM_p({\Sigma})^{n-1}{\leq}Vol({\partial}{\Sigma}{\sim}{\partial}C)^n$$, where ${\omega}_n$ is the volume of a unit ball in ${\mathbb{R}}^n$ Equality holds if and only if ${\Sigma}$ is a totally geodesic half ball centered at p.

Correlation between the 2-Dimensional Extent of Orbital Defects and the 3-Dimensional Volume of Herniated Orbital Content in Patients with Isolated Orbital Wall Fractures

  • Cha, Jong Hyun;Moon, Myeong Ho;Lee, Yong Hae;Koh, In Chang;Kim, Kyu Nam;Kim, Chang Gyun;Kim, Hoon
    • Archives of Plastic Surgery
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    • v.44 no.1
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    • pp.26-33
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    • 2017
  • Background The purpose of this study was to assess the correlation between the 2-dimensional (2D) extent of orbital defects and the 3-dimensional (3D) volume of herniated orbital content in patients with an orbital wall fracture. Methods This retrospective study was based on the medical records and radiologic data of 60 patients from January 2014 to June 2016 for a unilateral isolated orbital wall fracture. They were classified into 2 groups depending on whether the fracture involved the inferior wall (group I, n=30) or the medial wall (group M, n=30). The 2D area of the orbital defect was calculated using the conventional formula. The 2D extent of the orbital defect and the 3D volume of herniated orbital content were measured with 3D image processing software. Statistical analysis was performed to evaluate the correlations between the 2D and 3D parameters. Results Varying degrees of positive correlation were found between the 2D extent of the orbital defects and the 3D herniated orbital volume in both groups (Pearson correlation coefficient, 0.568-0.788; $R^2=32.2%-62.1%$). Conclusions Both the calculated and measured 2D extent of the orbital defects showed a positive correlation with the 3D herniated orbital volume in orbital wall fractures. However, a relatively large volume of herniation (>$0.9cm^3$) occurred not infrequently despite the presence of a small orbital defect (<$1.9cm^2$). Therefore, estimating the 3D volume of the herniated content in addition to the 2D orbital defect would be helpful for determining whether surgery is indicated and ensuring adequate surgical outcomes.