• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.941-947
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• 2007

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let ${\mu}0$ be the least eigenvalue of the Laplacian acting on $L^2-functions$ on M. We show that if $Ric^M{\ge}-{\mu}0$ at all $x{\in}M$ and either $Ric^M>-{\mu}0$ at some point x0 or Vol(M) is infinite, then every harmonic morphism ${\phi}:M{\to}N$ of finite energy is constant.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.213-228
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• 2022

In the present paper, we aim to study Yamabe soliton and Riemann soliton on Lorentzian para-Sasakian manifold. First, we proved, if the scalar curvature of an 𝜂-Einstein Lorentzian para-Sasakian manifold M is constant, then either 𝜏 = n(n-1) or, 𝜏 = n-1. Also we constructed an example to justify this. Next, it is proved that, if a three dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton for V is an infinitesimal contact transformation and tr 𝜑 is constant, then the soliton is expanding. Also we proved that, suppose a 3-dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton, if tr 𝜑 is constant and scalar curvature 𝜏 is harmonic (i.e., ∆𝜏 = 0), then the soliton constant λ is always greater than zero with either 𝜏 = 2, or 𝜏 = 6, or λ = 6. Finally, we proved that, if an 𝜂-Einstein Lorentzian para-Sasakian manifold M represents a Riemann soliton for the potential vector field V has constant divergence then either, M is of constant curvature 1 or, V is a strict infinitesimal contact transformation.

• Honam Mathematical Journal
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• v.38 no.2
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• pp.385-391
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• 2016

We consider Jacobi filds as the first derivatives for ${\varepsilon}$, the energy of harmonic extensions, in a given manifold. In this paper we see that the Jacobi fild is bounded by the given boundary map. Here we give no restriction concerned with the curvature for the given manifold.

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.649-650
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• 2022

In this erratum, we offer a correction to [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435-1449]. Theorem 1 in the original paper has three assertions (i)-(iii), but we add (iv) after having clarified the argument.

• Geomechanics and Engineering
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• v.19 no.5
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• pp.369-381
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• 2019

The objective of this paper is to study the two dimensional deformation in transversely isotropic thermoelastic medium without energy dissipation due to time harmonic sources using new modified couple stress theory, a continuum theory capable to predict the size effects at micro/nano scale. The couple stress constitutive relationships have been introduced for transversely isotropic thermoelastic medium, in which the curvature tensor is asymmetric and the couple stress moment tensor is symmetric. Fourier transform technique is applied to obtain the solutions of the governing equations. Assuming the deformation to be harmonically time-dependent, the transformed solution is obtained in the frequency domain. The application of a time harmonic concentrated and distributed sources have been considered to show the utility of the solution obtained. The displacement components, stress components, temperature change and couple stress are obtained in the transformed domain. A numerical inversion technique has been used to obtain the solutions in the physical domain. The effects of angular frequency are depicted graphically on the resulted quantities.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.233-236
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• 1992

Let f:M.rarw.N be a smooth map between Rioemannian manifolds M and N. If f maps geodesics of M to geodesics of N, f is called totally geodesic. As is well known, totally geodesic maps are harmonic and the image f(M) of a totally geodesic map f:M.rarw. N is an immersed totally geodesic submanifold of N (cf. .cint. 6.3 of [W]). We are interested in the following question: When is a harmonic map f:M .rarw. N with rank .leq. 1 everywhere on M totally geodesic\ulcorner In other words, when is the image of a harmonic map f:M .rarw. N with rank .leq. 1 everywhere on M geodesics of N\ulcorner In this note, we give some sufficient conditions on curvatures of M. It is interesting that no curvature assumptions on target manifolds are necessary in Theorems 1 and 2. Some properties of totally geodesic maps are also given in Theorem 3. We think our Theorem 3 is somewhat unusual in view of the following classical theorem of Eells and Sampson (see pp.124 of [ES]).

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.2 s.95
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• pp.123-128
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• 2005

Forced vibration of a thin circular ring with a concentrated harmonic force is analyzed when the ring is free and has only the in-plane motion. Using the unit doublet function for external force, the governing equation is obtained and is solved by the use of Laplace transform. The exact solutions of displacement components and bending moment are obtained. In order to verify the solutions of analysis, finite element analysis is performed and the results shows good agreement. Then, frequency response curves for displacement and bending moment are obtained. In deriving the governing equations and the solutions, nondimensional parameter of the exciting frequency and the magnitude of exciting force are extracted. As the displacement components are obtained, the remaining bending strain, slope, curvature, shear force, etc. can also be derived. With the results of this work, the responses of a free ring excited on multiple points with different frequencies can also be obtained easily by superposition.

• International journal of advanced smart convergence
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• v.4 no.2
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• pp.163-169
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• 2015

This study proposes a sports simulation device with various harmonics generation. The proposed system is composed of 6 degrees of freedom simulator devices and three types of sports simulation such as walking, snowboard, and jet-ski. In this research, every joint movement is designed with a crank-and-slider mechanism, which is efficient for generating continuous curvature smoothly. Contrary to the conventional spatial simulator with linear actuators, harmonics generation and its spatial combinations become the crucial issue in this research. The harmonic pattern in each joint is modelled for generating smooth curvatures that are also superposed for achieving overall motions. In addition, the targeted motions of sports simulations have different physical factors of periodic gait motion, frictionless surface, and buoyant effects, which are respectively designed by integrating three dimensional graphics information.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2091-2106
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• 2017

The authors investigate f-biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature, and derive that if $\int_{M}f^p{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$, $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}du{\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, the authors also get that if u satisfies some integral conditions, then it is minimal. These results give an affirmative partial answer to conjecture 4 (generalized Chen's conjecture for f-biharmonic submanifolds).

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.424-429
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• 2005

In this paper, the response characteristics of one to one resonance on the rectangular cantilever beam in which basic harmonic excitations are applied by nonlinear coupled differential integral equations are studied. This equations have 3-dimensional non-linearity of nonlinear inertia and nonlinear curvature. Galerkin and multi scale methods are used for theoretical approach to one to one internal resonance. Nonlinear response characteristics of 1st, 2nd, 3rd modes are measured from the experiment for basic harmonic excitation. From the experimental result, geometrical terms of nonlinearity display light spring effect and these terms play an important role in the response characteristics of low frequency modes. Dynamic behaviors in the out of plane are also studied.