• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.367-372
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• 2012

Taking the integrated Chebyshev-type counting function of the appropriate order, we improve the error term in Park's prime geodesic theorem for hyperbolic manifolds with cusps. The obtained estimate coincides with the best known result in the Riemann surfaces case.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1103-1107
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• 2018

Let M be a complete spacelike hypersurface in the (n + 1)-dimensional Minkowski space ${\mathbb{L}}^{n+1}$. Suppose that every unit speed curve X(s) on M satisfies ${\langle}X^{\prime\prime}(s),X^{\prime\prime}s){\rangle}{\geq}-1/r^2$ and there exists a point $p{\in}M$ such that for every unit speed geodesic X(s) of M through the point p, ${\langle}X^{\prime\prime}(s),X^{\prime\prime}s){\rangle}=-1/r^2$ holds. Then, we show that up to isometries of ${\mathbb{L}}^{n+1}$, M is the hyperbolic space $H^n(r)$.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.977-991
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• 2012

Let (M,F) and (M',F') be two foliated Riemannian manifolds with M compact. If the transversal Ricci curvature of F is nonnegative and the transversal sectional curvature of F' is nonpositive, then any transversally harmonic map ${\phi}:(M,F){\rightarrow}(M^{\prime},F^{\prime})$ is transversally totally geodesic. In addition, if the transversal Ricci curvature is positive at some point, then ${\phi}$ is transversally constant.

• Advances in biomechanics and applications
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• v.1 no.2
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• pp.85-93
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• 2014

In this paper, two procedures of enumerating the axial rotation are proposed using the unit sphere of the spherical rotation coordinate system specifying 3D rotation. If the trajectory of the movement is known, the integration of the axial component of the angular velocity plus the geometric effect equal to the enclosed area subtended by the geodesic path on the surface of the unit sphere. If the postures of the initial and final positions are known, the axial rotation is determined by the angular difference from the parallel transport along the geodesic path. The path dependency of the axial rotation of the three dimensional rigid body motion is due to the geometric effect corresponding to the closed loop discontinuity. Firstly, the closed loop discontinuity is examined for the infinitesimal region. The general closed loop discontinuity can be evaluated by the summation of those discontinuities of the infinitesimal regions forming the whole loop. This general loop discontinuity is equal to the surface area enclosed by the closed loop on the surface of the unit sphere. Using this quantification of the closed loop discontinuity of the axial rotation, the geometric effect is determined in enumerating the axial rotation. As an example, the axial rotation of the arm by the Codman's movement is evaluated, which other methods of enumerating the axial rotations failed.

• Korean Journal of Mathematics
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• v.26 no.4
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• pp.741-746
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• 2018

We study the Ihara zeta function of the dumbbell graph $D_{1,1,n}$ of type (1, 1, n) and $D_{1,2,n}$ of type (1, 2, n). Explicit formulas of the zeta functions of the graphs, their radius of convergence, and the connection with the number of closed cycles are given.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.817-827
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• 2021

In this paper, we prove that a diffeomorphism f on a normal almost contact 3-manifold M is a CRL-transformation if and only if M is an α-Sasakian manifold. Moreover, we show that a CR-loxodrome in an α-Sasakian 3-manifold is a pseudo-Hermitian magnetic curve with a strength $q={\tilde{r}}{\eta}({\gamma}^{\prime})=(r+{\alpha}-t){\eta}({\gamma}^{\prime})$ for constant 𝜂(𝛄'). A non-geodesic CR-loxodrome is a non-Legendre slant helix. Next, we prove that let M be an α-Sasakian 3-manifold such that (∇_YS)X = 0 for vector fields Y to be orthogonal to ξ, then the Ricci tensor 𝜌 satisfies 𝜌 = 2α²g. Moreover, using the CRL-transformation $\tilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\tilde{R}$, the pseudo-Ricci tensor $\tilde{\rho}$ and the torsion tensor field $\tilde{T}^t(\tilde{S}X,Y)$.