• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.143-159
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• 2013

This paper is concerned with the long time behavior for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension $n{\leq}3$. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the equations possesses a global attractor in $H^k({\Omega})^4$ ($k{\geq}0$) space.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.399-422
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• 1998

A useful method of proving the finite decidability of an equationally definable class V of algebras (i.e., variety) is to prove the decidability of the class of finite directly indecomposable members of V. The validity of this method relies on the well-known result of Feferman-Vaught: if a class K of first-order structures is decidable, then so is the class {$\prod$$_{i}$＜n/ $A_{i}$│ $A_{i}$ $\in$ X (i < n), n $\in$ $\omega$}. In this paper we show that the converse of this does not necessarily hold.d.d.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.91-97
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• 1998

In this paper we study the Weyl spectrum of weight $\alpha, \omega_\alpha(T)$, of an operator T acting on an infinite dimensional Hilbert space. Main results are as follows. Firstly, we show that the Weyll spectrum of weight $\alpha$ of a polynomially $\alpha$-compact operator is finite, and that similarity preserves polynomial $\alpha$-compactness and the $\alpha$-Weyl's theorem both. Secondly, we give a sufficient condition for an operator to be the sum of an unitary and a $\alpha$-compact operators.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.357-378
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• 2012

In this paper, we consider the convective Cahn-Hilliard equation. Based on the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors, we prove that the convective Cahn-Hilliard equation possesses a global attractor in $H^k$($k\geq0$) space, which attracts any bounded subset of $H^k({\Omega})$ in the $H^k$-norm.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.1
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• pp.81-95
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• 2007

We are concerned with the multiplicity of solutions of the nonlinear biharmonic equation with Dirichlet boundary condition, ${\Delta}^2u+c{\Delta}u=g(u)$, in ${\Omega}$, where $c{\in}R$ and ${\Delta}^2$ denotes the biharmonic operator. We show that there exists at least three solutions of the above problem under the suitable condition of g(u).

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.65-72
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• 2012

Let T > 0 be given. Let $(C[0,T],m_{\varphi})$ be the analogue of Wiener measure space, associated with the Borel proba-bility measure ${\varphi}$ on ${\mathbb{R}}$, let $(L_{2}[0,T],\tilde{\omega})$ be the centered Gaussian measure space with the correlation operator $(-\frac{d^{2}}{dx^{2}})^{-1}$ and ${\el}_2,\;\tilde{m}$ be the abstract Wiener measure space. Let U be the space of all sequence $<c_{n}>$ in ${\el}_{2}$ such that the limit $lim_{{m}{\rightarrow}\infty}\;\frac{1}{m+1}\;\sum{^{m}}{_{n=0}}\;\sum_{k=0}^{n}\;c_{k}\;cos\;\frac{k{\pi}t}{T}$ converges uniformly on [0,T] and give a set function m such that for any Borel subset G of $\el_2$, $m(\mathcal{U}\cap\;P_{0}^{-1}\;o\;P_{0}(G))\;=\tilde{m}(P_{0}^{-1}\;o\;P_{0}(G))$. The goal of this note is to study the relationship among the measures $m_{\varphi},\;\tilde{\omega},\;\tilde{m}$ and $m$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.347-355
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• 2001

Let ($X, \omega$) be a closed symplectic 4-manifold. Let a finite cyclic group G act semifreely, holomorphically on X as isometries with fixed point set $\Sigma$(may be empty) which is a 2-dimension submanifold. Then there is a smooth structure on the quotient X'=X/G such that the projection $\pi$:X$\rightarrow$X' is a Lipschitz map. Let L$\rightarrow$X be the Spin$^c$ -structure on X pulled back from a Spin$^c$-structure L'$\rightarrow$X' and b_2^$+(X')>1. If the Seiberg-Witten invariant SW(L')$\neq$0 of L' is non-zero and $L=E\bigotimesK^-1\bigotimesE$ then there is a G-invariant pseudo-holomorphic curve u:$C\rightarrowX$,/TEX> such that the image u(C) represents the fundamental class of the Poincare dual $c_1$(E). This is an equivariant version of the Taubes' Theorem.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.935-955
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• 2020

In this paper we present a measurable version of the Smale's spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.

• Journal of the Korean Institute of Intelligent Systems
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• v.19 no.6
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• pp.809-813
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• 2009

We want to show that the construct of best fuzzy tests for certain fuzzy situations of Poisson distribution. Due to Neyman and Pearson theorem, if we have ${\theta}_0$ and ${\theta}_1$ be distinct fuzzy values of ${\Omega}=\{{\theta}\;:\;{\theta}\;=\;{\theta}_0,\;{\theta}_1\}$ such that $L({\theta}_0\;:\;X)/L({\theta}_1\;:\;X)$ < k, then k is a fuzzy number. For each fuzzy random samples point $X\;{\subset}\;C$, we have most power test for fuzzy critical region C by agreement index.

• Bulletin of the Korean Chemical Society
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• v.13 no.6
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• pp.650-654
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• 1992

Absorption and fluorescence spectra are reported for four different 1 : 3 $Dy^{3+}$ : ligand systems in aqueous solution under mild alkaline pH conditions. The ligands included in this study are oxidiacetate, dipicolinate, iminodiacetate and methyliminodiacetate. The oscillator strengths for the 4f→4f multiplet-to-multiplet transitions are empirically determined from the absorption spectra and the intensity parameters ${\Omega}_{\lambda}$}(${\lambda}$ = 2, 4, 6) for the systems are also obtained by applying the Judd-Ofelt theorem to the observed oscillator strengths. The values of the intensity parameters for the systems are compared and discussed in terms of ligand structural properties to investigate how the intensity parameters can response to the minor changes in the ligand environment. In addition, the relative oscillator strengths for fluorescence are evaluated and compared to the results obtained from absorption spectra.