• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2004.04a
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• pp.510-514
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• 2004

In this paper, we introduce Fuzzy Beppo Levi's Theorem in which we use the supremum instead of addition in the expression of Beppo Levi's Theorem. That holds under the conditions which are continuity of t-seminorm ┬and the fuzzy additivity of a fuzzy measure g.

• Journal of the Korean Mathematical Society
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• v.51 no.5
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• pp.881-895
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• 2014

In a recent paper [10] we introduced the notion of Levi harmonic map f from an almost contact semi-Riemannian manifold (M, ${\varphi}$, ${\xi}$, ${\eta}$, g) into a semi-Riemannian manifold $M^{\prime}$. In particular, we compute the tension field ${\tau}_H(f)$ for a CR map f between two almost contact semi-Riemannian manifolds satisfying the so-called ${\varphi}$-condition, where $H=Ker({\eta})$ is the Levi distribution. In the present paper we show that the condition (A) of Rawnsley [17] is related to the ${\varphi}$-condition. Then, we compute the tension field ${\tau}_H(f)$ for a CR map between two arbitrary almost contact semi-Riemannian manifolds, and we study the concept of Levi pluriharmonicity. Moreover, we study the harmonicity on quasicosymplectic manifolds.

• Journal of the Korean Mathematical Society
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• v.37 no.4
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• pp.503-519
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• 2000

It is well-known that an analytic generic CR submainfold M of codimension m in Cn+m is locally transformed by a biholomorphic mapping to a plane Cn$\times$Rm ⊂ Cn$\times$Cm whenever the Levi form L on M vanishes identically. We obtain such a normalizing biholomorphic mapping of M in terms of the defining function of M. Then it is verified without Frobenius theorem that M is locally foliated into complex manifolds of dimension n.

• Journal for History of Mathematics
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• v.28 no.2
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• pp.103-115
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• 2015

Contemporary differential geometry owes much to the theory of connections on the bundles over manifolds. In this paper, following the work of Gauss on surfaces in 3 dimensional space and the work of Riemann on the curvature tensors on general n dimensional Riemannian manifolds, we will investigate how differential geometry had been developed from mid 19th century to early 20th century through lives and mathematical works of Christoffel, Ricci-Curbastro and Levi-Civita. Christoffel coined the Christoffel symbol and Ricci used the Christoffel symbol to define the notion of covariant derivative. Levi-Civita completed the theory of absolute differential calculus with Ricci and discovered geometric meaning of covariant derivative as parallel transport.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.169-178
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• 2019

Let $M^{2n+1}$, $n{\geq}1$, be a smooth manifold with a pseudoconvex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets $M={\mathcal{S}}_0{\supset}{\mathcal{S}}_1{\supset}{\cdots}{\supset}{\mathcal{S}}_n$, where $S_q$ is the set of points where the Levi-form has nullity ${\geq}q$. We prove that ${\mathcal{S}}{_q}^{\prime}s$ are locally given as common zero sets of the coefficients $A_j$, $j=0,1,{\ldots},q-1$, of the characteristic polynomial of the Levi-form. Some sufficient conditions for local existence of complex submanifolds are presented in terms of the coefficients $A_j$.

• Animal Systematics, Evolution and Diversity
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• v.5 no.1
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• pp.33-58
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• 1989

The identified Ceratinomorpha consist of 41 species, 21 genera and 12 families. Among them, two species, Clathria mosulpia and Haliclona ulreungia, were new species and the following species were new to Korea: Ophlitaspongia pennata california De Laubenfels, 1936, Desmacella rosea Fristedt, 1887, Clathria dayi Levi, 1963, Clathria parva Levi, 1963, Ax-ociella cylindrica Hallman, 1920, Axocielita calla (De Laubenfels, 1934), Myxilla sigmatifera ( (Levi, 1963), Haliclona perlucida (Griessinger, 1971), Petrosia nigricans Lindgren, 1897, G Gel/ius arcoferus Vosmaer, 1885, Reniera ventillabrum Fristedt, 1887, Reniera pigmentifera D Dendy, 1905, and Coelosphaera physa (Schmidt, 1875)

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.781-794
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• 2021

Let Ω be a smoothly bounded pseudoconvex domain in ℂⁿ and assume that the (n - 2)-eigenvalues of the Levi-form are comparable in a neighborhood of z₀ ∈ bΩ. Also, assume that there is a smooth 1-dimensional analytic variety V whose order of contact with bΩ at z₀ is equal to 𝜂 and 𝚫_n-2(z₀) < ∞. We show that the maximal gain in Hölder regularity for solutions of the ${\bar{\partial}}$-equation is at most ${\frac{1}{\eta}}$.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.425-437
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• 2002

Let $\Omega$ be a smoothly bounded pseudoconvex domain in $C^{n}$ and let $z^{0}$ $\in$b$\Omega$ a point of finite type. We also assume that the Levi form of b$\Omega$ is comparable in a neighborhood of $z^{0}$ . Then we get precise estimates of the Bergman kernel function, $K_{\Omega}$(z, w), and its derivatives in a neighborhood of $z^{0}$ . .

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.91-102
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• 2002

Let M and N be CR manifolds with nondegenerate Levi forms of hypersurface type of dimension 2m ＋ 1 and 2n ＋ 1, respectively, where 1 $\leq$ m $\leq$ n. Let f : M longrightarrow N be a CR mapping. Under a generic assumption we construct a complete system of finite order for the infinitesimal deformations of f. In particular, we prove the space of infinitesimal deformations of f forms a finite dimensional Lie algebra.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1075-1085
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• 2020

We investigate the relation between the notion of e-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor P : Mod-R → Spec(Mod-R), where Spec(Mod-R) denotes the spectral category of Mod-R, and the localization functor with respect to the singular torsion theory.