• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.41-63
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• 2024

In this paper, we study complete Riemannian immersions into a semi-Riemannian warped product obeying suitable curvature constraints. Under appropriate differential inequalities involving higher order mean curvatures, we establish rigidity and nonexistence results concerning these immersions. Applications to the cases that the ambient space is either an Einstein manifold, a steady state type spacetime or a pseudo-hyperbolic space are given, and a particular investigation of entire graphs constructed over the fiber of the ambient space is also made. Our approach is based on a parabolicity criterion related to a linearized differential operator which is a divergence-type operator and can be regarded as a natural extension of the standard Laplacian.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.915-927
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• 2000

This paper is concerned with degenerate Volterra equations Mu(t) + ∫(sub)0(sup)t k(t-s) Lu(s)ds = f(t) in Banach spaces both in the hyperbolic case, and the parabolic one. The key assumption is played by the representation of the underlying space X as a direct sum X = N(T) + R(T), where T is the bounded linear operator T = ML(sup)-1. Hyperbolicity means that the part T of T in R(T) is an abstract potential operator, i.e., -T(sup)-1 generates a C(sub)0-semigroup, and parabolicity means that -T(sup)-1 generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel k($.$) is degenerated or singular at t=0 using the results of Pruss[8] on analytic resolvents. Finally, we consider the case where $\lambda$ is a pole for ($\lambda$L + M)(sup)-1.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1201-1207
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• 2013

Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and ${\int}_M{\mid}A{\mid}^2\;dv={\infty}$, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of $L^2$ harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.

• Korean Journal of Mathematics
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• v.18 no.4
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• pp.369-380
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• 2010

We give a criterion for vertex extremal length parabolicity of locally finite planar graphs, and use it to show that a disk triangulation graph is circle packing parabolic if and only if its immediate finer graphs are circle packing parabolic.

• East Asian mathematical journal
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• v.37 no.3
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• pp.307-317
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• 2021

In this paper, we deal with a problem to determine the type of automorphisms of the unit disc in ℂ in terms of intrinsic geometry. We will characterize the hyperbolicity and parabolicity of automorphism by the distance function of the Poincaré metric.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.73-95
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• 1999

We prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincar inequality and the finite covering condition at infinity on each end, then every positive harmonic function on the manifold is asymptotically constant at infinity on each end. This result is a direct generalization of those of Yau and of Li and Tam.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1577-1586
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• 2018

In this paper, we prove that if every bounded ${\mathcal{A}}$-harmonic function on a complete Riemannian manifold M is asymptotically constant at infinity of p-nonparabolic ends of M, then each bounded ${\mathcal{A}}$-harmonic function is uniquely determined by the values at infinity of p-nonparabolic ends of M, where ${\mathcal{A}}$ is a nonlinear elliptic operator of type p on M. Furthermore, in this case, every bounded ${\mathcal{A}}$-harmonic function on M has finite energy.

• Proceedings of the KIEE Conference
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• 1992.07b
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• pp.813-815
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• 1992

Accurate Capacitance-Voltage characteristics of Metal-Insulator-Semiconductor (MIS) devices in narrow band-gap semiconductors are presented. The unique band structure of narrow band-gap semiconductors is taken into account such as non-parabolicity and degeneracy. Compensated and partially ionized impurities either in the bulk or the space charge region are also considered. HgCdTe is a defect semiconductor, so this approach is very important for characterization and analysis of MIS devices.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.43 no.3
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• pp.420-431
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• 1994

The HgCdTe material, which is II-VI compound semiconductor, is important materials for the fabrication of the infrared detectros. To suggest the model of accurate MIS C-V calculation for narrow band gap semiconductors such as HgCdTe, non-parabolicity from k.p theory and degeneracy effect are considered. And partially ionized effect and compensation effect which are material's properties are also considerd. Especially, degenerated material C-V characteristics from Fermi-Dirac statistics and exact charge theory are presented to get more accurate analysis of the experimental results. Also the comparison with calculation results between the general MIS theory from Boltzmann appoximation method and this model which is considered the narrow band gap semiconductor properties, show that this model is more useful theory to determination of accurate low and high frequency C-V characteristics.

• Bulletin of the Korean Chemical Society
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• v.20 no.1
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• pp.35-41
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• 1999

Stability characteristics of hyperbolic reaction-diffusion equations with a reversible Brusselator model are investigated as an extension of the previous work. Intensive stability analysis is performed for three important parameters, Nrd, β and Dx, where Nrd is the reaction-diffusion number which is a measure of hyperbolicity, β is a measure of reversibility of autocatalytic reaction and Dx is a diffusion coefficient of intermediate X. Especially, the dependence on Nrd of stability exhibits some interesting features, such as hyperbolicity in the small Nrd region and parabolicity in the large Nrd region. The hyperbolic reaction-diffusion equations are solved numerically by a spectral method which is modified and adjusted to hyperbolic partial differential equations. The numerical method gives good accuracy and efficiency even in a stiff region in the case of small Nrd, and it can be extended to a two-dimensional system. Four types of solution, spatially homogeneous, spatially oscillatory, spatio-temporally oscillatory and chaotic can be obtained. Entropy productions for reaction are also calculated to get some crucial information related to the bifurcation of the system. At the bifurcation point, entropy production changes discontinuously and it shows that different structures of the system have different modes in the dissipative process required to maintain the structure of the system. But it appears that magnitude of entropy production in each structure give no important information related for states of system itself.