• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.97-101
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• 1995

In [6], the author studied the property of the Szeg kernel and had a result that if $\Omega$ is a smoothly bounded domain in C and the Szeg kernel associated with $\Omega$ is rational, then any proper holomorphic map from $\Omega$ to the unit disc U is rational. It leads to the study of the proper rational map of $\Omega$ to U. In this note, first we simplify the proof of the above result and prove an existence theorem of a proper rational map. Before we proceed to state our result, we must recall some preliminary facts.(omitted)

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.807-814
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• 1996

Let $G = Z_2$. Let X be any compact connected orientable nonsingular real algebraic variety of dim X = k = odd with the trivial G action, and let Y be the unit sphere $S^{2n-k}$ with the antipodal action of G. Then we prove that any G invariant entire rational map $f : x \times Y \to S^{2n}$ is G homotopically trivial. We apply this result to prove that any entire rational map $g : X \times RP^{2n-k} \to S^{2n}$ is homotopically trivial.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1309-1320
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• 2023

Let f : X → Y be a map between simply connected CW-complexes of finite type with X finite. In this paper, we prove that the rational cohomology of mapping spaces map(X, Y ; f) contains a polynomial algebra over a generator of degree N, where N = max{i, π_i(Y)⊗ℚ ≠ 0} is an even number. Moreover, we are interested in determining the rational homotopy type of map(𝕊ⁿ, ℂP^m; f) and we deduce its rational cohomology as a consequence. The paper ends with a brief discussion about the realization problem of mapping spaces.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1317-1332
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• 2018

In this paper, we introduce the D-ratio of a rational map $f:{\mathbb{P}}^n{\dashrightarrow}{\mathbb{P}}^n$, defined over ${\bar{\mathbb{Q}}}$, whose indeterminacy locus is contained in a hyperplane H on ${\mathbb{P}}^n$. The D-ratio r(f; ${\bar{V}}$) characterizes endomorphisms and provides a useful height inequality on ${\mathbb{P}}^n({\bar{\mathbb{Q}}}){\backslash}H$. We also provide a dynamical application: preperiodic points of dynamical systems of small D-ratio are of bounded height.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.215-223
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• 2009

The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1-type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine helicoidal surface of polynomial kind with pointwise 1-type Gauss map of the first kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a characterization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.949-961
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• 1995

Let f be a smooth G map from a nonsingular real algebraic G variety to an equivariant Grassmann variety. We use some G vector bundle theory to find a necessary and sufficient condition to approximate f by an entire rational G map. As an application we algebraically approximate a smooth G map between G spheres when G is an abelian group.

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1171-1187
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• 2011

Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly regular pair. In this paper, we provide the same improvement for a jointly regular family: let h : ${\mathbb{P}}_{\mathbb{Q}}^n{\rightarrow}{{\mathbb{R}}$ be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is de ned in [10] and let {$f_1,{\ldots},f_k|f_l:\mathbb{A}^n{\rightarrow}\mathbb{A}^n$} bbe finite set of polynomial maps which is defined over a number field K. If the intersection of the indeterminacy loci of $f_1,{\ldots},f_k$ is empty, then there is a constant C such that $ \sum\limits_{l=1}^k\frac{1}{def\;f_\iota}h(f_\iota(P))>(1+\frac{1}{r})f(P)-C$ for all $P{\in}\mathbb{A}^n$ where r= $max_{\iota=1},{\ldots},k(r(f_l))$.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.567-576
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• 2009

We classify and characterize the rational helicoidal surfaces in a three-dimensional Minkowski space satisfying pointwise 1-type like problem on the Gauss map.

• The Pure and Applied Mathematics
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• v.19 no.3
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• pp.229-237
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• 2012

In this article, we study generalized slant cylindrical surfaces (GSCS's) with pointwise 1-type Gauss map of the first and second kinds. Our main results state that the right circular cones are the only rational kind GSCS's with pointwise 1-type Gauss map of the second kind.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.259-267
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• 2022

We use L_∞ models to compute the rational homotopy type of the mapping space of the component of the natural inclusion i_n,k : ℂPⁿ ↪ ℂP^n+k between complex projective spaces and show that it has the rational homotopy type of a product of odd dimensional spheres and a complex projective space. We also characterize the mapping aut₁ ℂPⁿ → map(ℂPⁿ, ℂP^n+k; i_n,k) and the resulting G-sequence.