• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.525-535
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• 2015

In this paper, we study on spinors with two hyperbolic components. Firstly, we express the hyperbolic spinor representation of a spacelike curve dened on an oriented (spacelike or time-like) surface in Minkowski space ${\mathbb{R}}^3_1$. Then, we obtain the relation between the hyperbolic spinor representation of the Frenet frame of the spacelike curve on oriented surface and Darboux frame of the surface on the same points. Finally, we give one example about these hyperbolic spinors.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.549-555
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• 2021

We show the solvability of the ordinary differential equations describing curves on sphere or hyperbolic space. Making use of the geometric properties of the equations, we derive explicit solution formulae.

• 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 $H^n(r)$.

• Honam Mathematical Journal
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• v.45 no.3
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• pp.513-541
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• 2023

In this paper, we calculate Frenet frames, Frenet derivative formulas, curvatures, arc lengths, geodesic curvatures according to the Lorentzian 3-space ℝ³₁, Lorentzian sphere 𝕊²₁ and hyperbolic sphere ℍ²₀ of the spherical indicatrix curves of the spacelike Salkowski curve with the timelike principal normal in ℝ³₁ and draw the graphs of these indicatrix curves on the spheres.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.159-175
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• 2017

We solve the $Bj{\ddot{o}}rling$ problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve ${\gamma}$ and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to ${\gamma}$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains ${\gamma}$ and the unit normal of which on ${\gamma}$ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1339-1351
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• 2015

In this paper, using pseudo-spherical Frenet frame of pseudo-spherical curves in hyperbolic space, we define the notion of the structure functions on the non-developable ruled surfaces with timelike ruling. Then we obtain the properties of the structure functions and a complete classification of the non-developable ruled surfaces with timelike ruling in Minkowski 3-space by the theories of the structure functions.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.957-964
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• 2014

In this paper, we prove that there is no branch point in the Lorentz space (M, d) which is the limit space of a sequence {($M_{\alpha},d_{\alpha}$)} of compact globally hyperbolic interpolating spacetimes with $C^{\pm}_{\alpha}$-properties and curvature bounded below. Using this, we also obtain that every maximal timelike geodesic in the limit space (M, d) can be expressed as the limit curve of a sequence of maximal timelike geodesics in {($M_{\alpha},d_{\alpha}$)}. Finally, we show that the limit space (M, d) satisfies a timelike triangle comparison property which is analogous to the case of Alexandrov curvature bounds in length spaces.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.733-743
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• 2012

For the "Hopf bundle" $S^1{\rightarrow}S^{2n,1}{\rightarrow}\mathbb{C}H^n$, horizontal lifts of simple closed curves are studied. Let ${\gamma}$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along ${\gamma}$ is given by $$V({\gamma})=e^{{\lambda}A({\gamma})i}$$ where $A({\gamma})$ is the area of the region on the surface $S$ surrounded by ${\gamma}$; ${\lambda}=1/2$ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group $\mathbb{R}{\rightarrow}\mathcal{H}^{2n+1}{\rightarrow}\mathbb{C}^n$.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1299-1320
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• 2023

In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hyper-cones whose centers lie on a non-null curve with non-null Frenet vector fields in E⁴₁ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E⁴₁ by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.

• Geomechanics and Engineering
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• v.29 no.6
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• pp.615-626
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• 2022

The soft clays are widely distributed, and one of the prominent engineering problems is the creep behavior. In order to predict the creep deformation of soft clays in an easier and more acceptable way, a simple creep constitutive model has been proposed in this paper. Firstly, the triaxial creep test data indicated that, the strain-time (𝜀-t) curve showing in the 𝜀-lgt space can be divided into two lines with different slopes, and the time referring to the demarcation point is named as t_EOP. Thereafter, the strain increments occurred after the time t_EOP are totally assumed to be the creep components, and the elastic and plastic strains had occurred before t_EOP. A hyperbolic equation expressing the relationship between creep volumetric strain, stress and time is proposed, with several triaxial creep test data of soft clays verifying the applicability. Additionally, the creep flow law is suggested to be similar with the plastic flow law of the modified Cam-Clay model, and the proposed volumetric strain equation is used to deduced the scaling factor for creep strains. Therefore, a creep constitutive model is thereby established, and verified by successfully predicting the creep principal strains of triaxial specimens.