• The Pure and Applied Mathematics
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• v.28 no.3
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• pp.235-246
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• 2021

In this study, a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions, is considered.The results of Rogosinskis lemma and Jacks lemma have been utilized to derive novel inequalities. Also, these inequalities have been strengthened by considering the critical points which are different from zero.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.411-417
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• 2009

The Urysohn's lemma which is crucial tool for the study of the metrization problem is proved in the sense of set-theoretic concept, namely, by the idempotent relation defined on a given topology.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.989-993
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• 2022

In this paper, we prove Pascal's hexagon theorem by elementary tools only. We follow the well-known route to prove the theorem by Bezóut's theorem, explaining all the details in elementary argument. In particular, we prove a toy version of Study's lemma.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.2
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• pp.67-77
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• 2007

This paper is concerned with the proof of so-called Bramble-Hilbert Lemma. We present that $Poincar{\acute{e}}'s$ inequality in [3] implies one of results of Morrey which is crucial in the proof. In this point of view, we recognize that removing the average term in $Poincar{\acute{e}}'s$ inequality fulfills a crucial role in the proof of Bramble-Hilbert Lemma. It is accomplished by adding some polynomial of degree one less than the degree of the Sobolev space in the outset. So, the condition annihilating the set of polynomials $P_{k-1}$ of degree k - 1 is required necessarily in Bramble-Hilbert Lemma.

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.501-510
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• 2021

A uniform study towards normality is provided for topological spaces. Following Császár, 𝛄-normality and 𝛄(𝜃)-normality are introduced and investigated. For 𝛄 ∈ 𝚪₁₃, 𝛄-normality is found to satisfy Urysohn's lemma and provide partition of unity. Several existing variants of normality such as 𝜃-normality, 𝚫-normality etc. are shown to be particular cases of 𝛄(𝜃)-normality. In this process, 𝛄-regularity and 𝛄(𝜃)-regularity are introduced and studied. Several important characterizations of all these notions are provided.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1577-1596
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• 2017

We study contraction of points on ${\mathbb{P}}^1({\bar{\mathbb{Q}}})$ with certain control on local ramification indices, with application to the unramified curve correspondence problem initiated by Bogomolov and Tschinkel.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.3
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• pp.393-400
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• 2012

In this paper, we study the process of McShane delta integrals on time scales and discuss the relation between McShane delta integral and Henstock delta integral. We also prove the mono- tone convergence theorem, Fatou's Lemma and the dominated con- vergence theorems for the McShane delta integral.

• East Asian mathematical journal
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• v.32 no.3
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• pp.377-385
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• 2016

Global attractor is a basic concept to study the long-time behavior of solutions of the various equations. This paper is investigated with the existence of a global attractor for the beam equation $$u_{tt}+{\Delta}^2u-{\nabla}{\cdot}\{{\sigma}({\mid}{\nabla}u{\mid}^2){\nabla}u\}+f(u)+a(x)g(u_t)=h,$$ using multipliers technique and Nakao's Lemma.

• Journal of the Korean Mathematical Society
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• v.46 no.2
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• pp.363-447
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• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

• The Pure and Applied Mathematics
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• v.16 no.4
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• pp.405-416
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• 2009

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.