• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1315-1327
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• 2011

Let M be a real hypersurface of a complex space form with almost contact metric structure (${\phi}$, ${\xi}$, ${\eta}$, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},\;{\xi}){\xi}$ is ${\xi}$-parallel. In particular, we prove that the condition ${\nabla}_{\xi}R_{\xi}=0$ characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic space when $R_{\xi}{\phi}S=R_{\xi}S{\phi}$ holds on M, where S denotes the Ricci tensor of type (1,1) on M.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.91-111
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• 2001

In this paper we prove the following : Let M be a semi-invariant submanifold with almost contact metric structure (${\phi}$, ${\xi}$, g) of codimension 3 in a complex hyperbolic space $H_{n+1}{\mathbb{C}}$. Suppose that the third fundamental form n satisfies $dn=2{\theta}{\omega}$ for a certain scalar ${\theta}({\leq}{\frac{c}{2}})$, where ${\omega}(X,\;Y)=g(X,\;{\phi}Y)$ for any vectors X and Y on M. Then M has constant eigenvalues correponding the shape operator A in the direction of the distinguished normal and the structure vector ${\xi}$ is an eigenvector of A if and only if M is locally congruent to one of the type $A_0$, $A_1$, $A_2$ or B in $H_n{\mathbb{C}}$.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.279-294
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• 2005

We study a real hypersurface M satisfying $L_{\xi}S=0\;and\;R_{\xi}S=SR_{\xi}$ in a complex hyperbolic space $H_n\mathbb{C}$, where S is the Ricci tensor of type (1,1) on M, $L_{\xi}\;and\;R_{\xi}$ denotes the operator of the Lie derivative and the Jacobi operator with respect to the structure vector field e respectively.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.185-201
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• 2009

Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ of a nonflat complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},{\xi}){\xi}$ is ${\xi}$-parallel. In this paper, we prove that the condition ${\nabla}_{\xi}R_{\xi}=0$ characterize the homogeneous real hypersurfaces of type A in a complex projective space $P_n{\mathbb{C}}$ or a complex hyperbolic space $H_n{\mathbb{C}}$ when $g({\nabla}_{\xi}{\xi},{\nabla}_{\xi}{\xi})$ is constant.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.799-817
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• 2021

In this paper, we investigate the normal and complex symmetric weighted composition operators W_𝜓,𝜑 on the Hardy space H²(𝔻). Firstly, we give the explicit conditions of weighted composition operators to be normal and complex symmetric with respect to conjugations 𝒞₁ and 𝒞₂ on H²(𝔻), respectively. Moreover, we particularly investigate the weighted composition operators W_𝜓,𝜑 on H²(𝔻) which are normal and complex symmetric with respect to conjugations 𝓙, 𝒞₁ and 𝒞₂, respectively, when 𝜑 has an interior fixed point, 𝜑 is of hyperbolic type or parabolic type.

• 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.55 no.4
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• pp.1051-1063
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• 2018

The real moment-angle complex (or, more generally, real polyhedral product) and its real toric space have recently attracted much attention in toric topology. The aim of this paper is to give two interesting remarks regarding real polyhedral products and real toric spaces. That is, we first show that Moore's conjecture holds to be true for certain real polyhedral products. In general, real polyhedral products show some drastic difference between the rational and torsion homotopy groups. Our result shows that at least in terms of the homotopy exponent at a prime this is not the case for real polyhedral products associated to a simplicial complex whose minimal missing faces are all k-simplices with $k{\geq}2$. Moreover, we also show a structural theorem for a finite group G acting simplicially on the real toric space. In other words, we show that G always contains an element of order 2, and so the order of G should be even.

• New Physics: Sae Mulli
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• v.68 no.11
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• pp.1225-1230
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• 2018

Schwinger pair production of electrons and positrons in a strong electric field is a prediction of nonperturbative quantum field theory, in which the out-vacuum is superposed of multi-particle states of the in-vacuum. Solving the Dirac or Klein-Gordon equation in the background field, though a linear wave equation, and finding the pair-production rate is a difficult or nontrivial job. The phase-integral method has recently been introduced to compute the pair production in space-dependent electric fields, and a complex analysis method has been employed to calculate the pair production in time-dependent electric fields. In this paper, we apply the complex analysis method to a Sauter-type electric field and other hyperbolic-type electric fields that vanish in the past and future and show that the Stokes phenomena in pair production occur when the time-dependent frequency for a given momentum has finite simple poles (polons) with pure imaginary residues.

• Proceedings of the Korea Association of Crystal Growth Conference
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• 1996.06b
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• pp.285-309
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• 1996

Non oxide ceramics such as nitrides of transition metals have shown significant potential for future economic impact, in diverse applications in ceramic, aerospace and electronic industries, as refractory products, abrasives and cutting tools, aircraft components, and semi-conductor substrates amid others. Combustion synthesis has become an attractive alternative to the conventional furnace technology to produce these materials cheaply, faster and at a higher level of purity. However he process os highly exothermic and manifests complex dynamics due to its strongly non-linear nature. In order to develop an understanding of this process and to study the effect of operational parameters on the final outcome, numerical modeling is necessary, which would generated essential knowledge to help scale-up the process. the model is based on a system of parabolic-hyperbolic partial differential equations representing the heat, mass and momentum conservation relations. The model also takes into account structural change due to sintering and volumetric expansion, and their effect on the transport properties of the system. The solutions of these equations exhibit steep moving spatial gradients in the form of reaction fronts, propagating in space with variable velocity, which gives rise to varying time scales. To cope with the possibility of extremely abrupt changes in the values of the solution over very short distances, adaptive mesh techniques can be applied to resolve the high activity regions by ordering grid points in appropriate places. To avoid a control volume formulation of the solution of partial differential equations, a simple orthogonal, adaptive-mesh technique is employed. This involves separate adaptation in the x and y directions. Through simple analysis and numerical examples, the adaptive mesh is shown to give significant increase in accuracy in the computations.