• Korean Journal of Mathematics
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• v.25 no.3
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• pp.437-453
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• 2017

A new class of sets, called generalized (${\Lambda}$, b)-closed sets, has been introduced and studied. Also, several properties of generalized (${\Lambda}$, b)-closed sets are investigated. Some characterizations of ${\Lambda}_b$-regular spaces and ${\Lambda}_b$-normal spaces are discussed.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1891-1908
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• 2018

In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions of the Dirichlet problem of the one-dimension Minkowski-curvature equation $$\{\(\frac{u^{\prime}}{\sqrt{1-u^{{\prime}2}}}\)^{\prime}+{\lambda}a(x)f(u)=0,\;x{\in}(0,1),\\u(0)=u(1)=0$$, where ${\lambda}$ is a positive parameter, $f{\in}C[0,{\infty})$, $a{\in}C[0,1]$. The proofs of main results are based upon the bifurcation techniques.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1471-1493
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• 2022

The goal of this paper is to establish the boundedness of bilinear Calderón-Zygmund operator BT and its commutator [b₁, b₂, BT] which is generated by b₁, b₂ ∈ BMO(ℝⁿ) (or ${\dot{\Lambda}}_{\alpha}$(ℝⁿ)) and the BT on generalized variable exponent Morrey spaces 𝓛^p(·),𝜑(ℝⁿ). Under assumption that the functions 𝜑₁ and 𝜑₂ satisfy certain conditions, the authors proved that the BT is bounded from product of spaces 𝓛^{p₁(·),𝜑₁}(ℝⁿ)×𝓛^{p₂(·),𝜑₂}(ℝⁿ) into space 𝓛^p(·),𝜑(ℝⁿ). Furthermore, the boundedness of commutator [b₁, b₂, BT] on spaces L^p(·)(ℝⁿ) and on spaces 𝓛^p(·),𝜑(ℝⁿ) is also established.

• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.961-977
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• 2022

In this paper, we consider the following Kirchhoff type equation on the whole space $$\{-(a+b{\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}\;{\mid}{\nabla}u{\mid}^2dx){\Delta}u=u^5+{\lambda}k(x)g(u),\;x{\in}{\mathbb{R}}^3,\\u{\in}{\mathcal{D}}^{1,2}({\mathbb{R}}^3),$$ where λ > 0 is a real number and k, g satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.553-563
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• 2006

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set ${\sigma}_{B{\omega}}(A)$ of all ${\lambda}{\in}\mathbb{C}$ such that $A-{\lambda}I$ is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem ${\sigma}_{B{\omega}}(A)={\sigma}(A)$\E(A), and the B-Weyl spectrum ${\sigma}_{B{\omega}}(A)$ of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalizedWeyl's theorem holds for the case where A is an algebraically ($p,\;k$)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

• Journal of the Korean Chemical Society
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• v.34 no.1
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• pp.1-9
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• 1990

Cholesteryl pentanoate $(C_{32}O_2H_{54})$ is orthorhombic, space group $P2_12_12_1$, with a = 21.930(3), b = 21.404(3), c = 6.419(5) $\AA$, Z = 4, V = 3012.8(5)$\AA$$^3$, $D_c$ = 1.04 g$cm^{-3}$, ${\lambda}(Mo\; K{\alpha}$ = 0.71069 $\AA$, $\mu$ = 0.58 $cm^{-1}$, F(000) = 1048, T = 298, R = 0.086 for 1502 unique observed reflections with I > 1.0 $\sigma$ (I). The structure was solved by direct methods and refined by cascade diagonal least-squares refinement. The C-H bond lengths and the methyl groups are fixed and refined as their ideal geometry. A comparison with other cholesteryl esters gives normal structure for the tetracyclic ring, while the tail regions of the side chain and the ester group which stands on end, show a variation from their normal values, presumably due to thermal effects. The molecules are stacked together by non-bonded van der Waals forces with the shortest intermolecular distance of 3.529 $\AA$.

• Bulletin of the Korean Chemical Society
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• v.11 no.5
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• pp.427-430
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• 1990

Cholesteryl aniline ($C_{33}H_{51}N$) is monoclinic, space group $P2_1$, with a = 9.020(3), b = 6.000(1), c = 27.130(9)${\AA},\;{\beta} = 98.22(2)^{\circ}$, Z = 2, Dc = 1.06 g/cm$^3$ and Dm = 1.04 g/cm$^3$. A diffraction data set was collected with Mo-$K_{\alpha}$ radiation (${\lambda} = 0.7107 {\AA}$) on a diffractometer with a graphite monochromator to a maximum 2${\theta}$ value of 50$^{\circ}$, by the ${\omega}-2{\theta}$ scan technique. The coordinates of the non-hydrogen atoms and their anisotropic temperature factors were refined by full-matrix least-squares methods to final R of 0.058. In cholesteryl group, bond distances were normal except in tail part, where high thermal vibration resulted in apparent shortening of the C-C distances. The crystal structure consists of bilayers of thickness $d_{001} = 27.13 {\AA}$, in each of which there is the tail to tail arrangement of molecules aligned in the unit cell with their long axes approximately parallel to the [104] axis. The two halves of the double layer are related to each other by the screw axis.