• East Asian mathematical journal
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• v.35 no.1
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• pp.67-75
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• 2019

We investigate the averaging value of a random sampling of a Dirichlet series with some condition using Poisson distribution. Our result is the following: Let $L(s)={\sum}^{\infty}_{n=1}{\frac{a_n}{n^s}}$ be a Dirichlet series that converges absolutely for Re(s) > 1. If $X_t$ is an increasing random sampling with Poisson distribution and there exists a number $0<{\alpha}<{\frac{1}{2}}$ such that ${\sum}_{n{\leq}u}a_n{\ll}u^{\alpha}$, then we have $${\mathbb{E}}L(1/2+iX_t)=O(t^{\alpha}{\sqrt{{\log}t}})$$, for all sufficiently large t in ${\mathbb{R}}$. As a result, we get the behaviour of $L({\frac{1}{2}}+it)$ such that L is a Dirichlet L-function or a modular L-function, when t is sampled by the Poisson distribution.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.623-632
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• 2010

A holomorphic function F defined on the unit disc belongs to $A^{p,{\alpha}}$ (0 < p < $\infty$, 1 < ${\alpha}$ < $\infty$) if $\int\limits_U|F(z)|^p \frac{1}{1-|z|}(1+log)\frac{1}{1-|z|})^{-\alpha}$ dxdy < $\infty$. For boundedness of the composition operator defined by $C_{fg}=g{\circ}f$ mapping Blochs into $A^{p,{\alpha}$ the following (1) is a sufficient condition while (2) is a necessary condition. (1) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha}M_p(r,\lambda{\circ}f)^p\;dr$ < $\infty$ (2) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha+p}(1-r)^pM_p(r,f^#)^p\;dr$ < $\infty$.

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.569-608
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• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.

• Journal of the Korean Mathematical Society
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• v.40 no.4
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• pp.609-627
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• 2003

In this paper we describe the Einstein-Kahler metric for the Cartan-Hartogs of the first type which is the special case of the Hua domains. Firstly, we reduce the Monge-Ampere equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w) = $\midw\mid^2[det(I-ZZ^{T}]^{\frac{1}{K}}$ (see below). This differential equation can be solved to give an implicit function in Χ. Secondly, we get the estimate of the holomorphic section curvature under the complete Einstein-K$\ddot{a}$hler metric on this domain.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.543-557
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• 2017

In this paper, we investigate the notion of q-pseudoconvexity to discuss and describe some geometric characterizations of q-pseudoconvex domains ${\Omega}{\subset}{\mathbb{C}}^n$. In particular, we establish that ${\Omega}$ is q-pseudoconvex, if and only if, for every boundary point, the Levi form of the boundary is semipositive on the intersection of the holomorphic tangent space to the boundary with any (n-q+1)-dimensional subspace $E{\subset}{\mathbb{C}}^n$. Furthermore, we prove that the Kiselman's minimum principal holds true for all q-pseudoconvex domains in ${\mathbb{C}}^p{\times}{\mathbb{C}}^n$ such that each slice is a convex tube in ${\mathbb{C}}^n$.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.783-800
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• 2021

We prove the non-existence of warped product GCR-lightlike submanifolds of the type K_⊥ × _λ K_T such that K_T is a holomorphic submanifold and K_⊥ is a totally real submanifold in an indefinite Kaehler manifold $\tilde{K}$. Further, the existence of GCR-lightlike warped product submanifolds of the type K_T × _λ K_⊥ is obtained by establishing a characterization theorem in terms of the shape operator and the warping function in an indefinite Kaehler manifold. Consequently, we find some necessary and sufficient conditions for an isometrically immersed GCR-lightlike submanifold in an indefinite Kaehler manifold to be a GCR-lightlike warped product, in terms of the canonical structures f and ω. Moreover, we also derive a geometric estimate for the second fundamental form of GCR-lightlike warped product submanifolds, in terms of the Hessian of the warping function λ.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.35-54
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• 2007

Let $M_c$ = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0\;and\;c_2=c$ on a K3 surface X, where O(1) is a generic ample line bundle on X. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\geq3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq3$. This obviously implies that there is no symplectic desingularization.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.409-421
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• 2018

Let X be a complex manifold of dimension n $n{\geqslant}2$ and let ${\Omega}{\Subset}X$ be a weakly q-convex domain with smooth boundary. Assume that E is a holomorphic line bundle over X and $E^{{\otimes}m}$ is the m-times tensor product of E for positive integer m. If there exists a strongly plurisubharmonic function on a neighborhood of $b{\Omega}$, then we solve the ${\bar{\partial}}$-problem with support condition in ${\Omega}$ for forms of type (r, s), $s{\geqslant}q$ with values in $E^{{\otimes}m}$. Moreover, the solvability of the ${\bar{\partial}}_b$-problem on boundaries of weakly q-convex domains with smooth boundary in $K{\ddot{a}}hler$ manifolds are given. Furthermore, we shall establish an extension theorem for the ${\bar{\partial}}_b$-closed forms.

• Korean Journal of Mathematics
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• v.6 no.1
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• pp.67-70
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• 1998

Let ${\eta}(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$, where $q=e^{2{\pi}i{\tau}}$ and ${\tau}{\in}\mathbb{C}$. Then the transformation $$g(\tau)={\rho}\frac{\{\eta(\frac{\tau+1}{2})\eta(\frac{\tau+2}{2})\}^{16}}{\eta(\tau)^{24}}({\bar{{\rho}}{\eta}}(\frac{\tau+1}{2})^8+{\eta}(\frac{\tau+2}{2})^8)^2$$ is holomorphic for Im ${\tau}$ > 0, and has the property $$g(\tau+1)=g(\tau),\;g(-\frac{1}{\tau})={\tau}^{12}g(\tau)$$. (Theorem)

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1001-1015
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• 2010

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.