• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.19-23
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• 1995

We will estimate the lower bound of the first nonzero Neumann and Dirichlet eigenvalue of Laplacian equation on compact Riemannian manifold M with boundary. In case that the boundary of M has positive second fundamental form elements, Ly-Yau[3] gave the lower bound of the first nonzero neumann eigenvalue $\eta_1$. In case that the second fundamental form elements of $\partial$M is bounded below by negative constant, Roger Chen[4] investigated the lower bound of $\eta_1$. In [1], [2], we obtained the lower bound of the first nonzero Neumann eigenvalue is estimated under the condtion that the second fundamental form elements of boundary is bounded below by zero. Moreover, I realize that "the interior rolling $\varepsilon$ - ball condition" is not necessary when the first Dirichlet eigenvalue was estimated in [1].ed in [1].

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.603-624
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• 2015

We first extend the notions of weighted curvatures, including the weighted flag curvature and the weighted Ricci curvature, for a Finsler manifold with given volume form. Then we establish some basic comparison theorems for Finsler manifolds with various weighted curvature bounds. As applications, we obtain some McKean type theorems for the first eigenvalue of Finsler manifolds, some results on weighted curvature and fundamental group for Finsler manifolds, as well as an estimation of Gromov simplicial norms for reversible Finsler manifolds.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.507-512
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• 1996

A totally umbilic submanifold of a pseudo-Riemanian manifold is a submanifold whose first fundamental form and second fundamental form are proportiona. An ordinary hypersphere $S^n(r)$ of an affine (n + 1)-space of the Euclidean space $E^m$ is the best known example of totally umbilic submanifolds of $E^m$.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.49-60
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• 2007

In this article, the concept of rigidity of smooth surfaces in the three dimensional Euclidean space which naturally arises in elementary geometry is introduced, and the natural process of the development of rigidity theory for compact surfaces and its generalizations are investigated.

• Honam Mathematical Journal
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• v.39 no.4
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• pp.465-473
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• 2017

In this paper, we define a totally umbilic hypersurface of first order and show that a totally umbilic hypersurface of first order in an Einstein manifold has a parallel second fundamental form. Furthermore we prove that a complete, simply connected and totally umbilic hypersurface of first order in a space of constant curvature is a Riemannian product of Einstein manifolds. Finally we show a proper example which is a totally umbilic hypersurface of first order but not a totally umbilic hypersurface.

• Transactions of the Korean Society of Machine Tool Engineers
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• v.13 no.6
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• pp.107-113
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• 2004

This paper presents the pillar section optimization technique considering the reliability of the vehicle body structure consisted of complicated thin-walled panels. The response surface method is utilized to obtain the response surface models that describe the approximate performance functions representing the system characteristics on the section properties of the pillar and on the mass and the natural frequencies of the vehicle B.I.W. The reliability-based design optimization on the pillar sections Is performed and compared with the conventional deterministic optimization. The FORM is applied for the reliability analysis of the vehicle body structure. The developed optimization system is applied to the pillar section design considering the fundamental natural frequencies of passenger car body structure. By applying the proposed RBDO technique, it can be possible to optimize the pillar sections considering the reliability that engineers require.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.253-266
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• 2011

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.

• Journal of computational fluids engineering
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• v.2 no.1
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• pp.13-20
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• 1997

A fundamental study for the numerical scheme to simulate unsteady nonlinear waves by solving Euler equations is presented. First a conservation form and a non-conservation form of the Euler equations with a free surface fitted coordinate system are compared. Next, a time splitting fractional step method and an alternating direction implicit(ADI) method for the time integration are compared. For the comparative study, flow calculations around a bottom bump in a channel and a NACA 0012 hydrofoil in a flume are performed. The results show that the ADI method with a third order upwind differencing scheme is very efficient in reducing the computing time with keeping the accuracy, And, there is no distinct difference between two expression forms except that the non-conservative form shows faster wave propagating velocity than the conservation form. Some results are compared with experiments and show good agreement.

• Smart Structures and Systems
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• v.21 no.6
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• pp.715-725
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• 2018

Truss-Z (TZ) is an Extremely Modular System (EMS). Such systems allow for creation of structurally sound free-form structures, are comprised of as few types of modules as possible, and are not constrained by a regular tessellation of space. Their objective is to create spatial structures in given environments connecting given terminals without self-intersections and obstacle-intersections. TZ is a skeletal modular system for creating free-form pedestrian ramps and ramp networks. The previous research on TZ focused on global discrete geometric optimization of the spatial configuration of modules. This paper reports on the first attempts at structural optimization of the module for a single-branch TZ. The internal topology and the sizing of module beams are subject to optimization. An important challenge is that the module is to be universal: it must be designed for the worst case scenario, as defined by the module position within a TZ branch and the geometric configuration of the branch itself. There are four variations of each module, and the number of unique TZ configurations grows exponentially with the branch length. The aim is to obtain minimum-mass modules with the von Mises equivalent stress constrained under certain design load. The resulting modules are further evaluated also in terms of the typical structural criterion of compliance.

• Journal for History of Mathematics
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• v.10 no.1
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• pp.19-32
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• 1997

This paper deals with the relationship between the order structure and topological structure in the historical point of view. We first investigate how the order structure has developed along with the set theory and logic in the second half of the nineteenth century. After the general topology has emerged in the beginning of the twentieth century, two disciplines of the order theory and topology give each other a great deal of effect for their development via various dualities, compactifications by maximal filter spaces and Alexandroff's specialization order, which form eventually a fundamental setting for the development of the category theory or functor theory.