• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.259-268
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• 2013

In this paper, we define the Banach-valued strong $M_{\alpha}$-integral and study the primitive of the strong $M_{\alpha}$-integral in terms of the $M_{\alpha}$-variational measures. We also prove that every function of bounded variation is a multiplier for the strong $M_{\alpha}$-integral.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.763-769
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• 2012

In this paper, we define the strong $M_{\alpha}$-integral of Banach-valued functions and investigate some properties of the strong $M_{\alpha}$-integral.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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• pp.41-46
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• 2003

In this paper, we investigate the $SP_{\alpha}$-integral and the $M_{\alpha}$-integral defined on an interval of the n-dimensional Euclidean space $\mathbb{R}^n$. In particular, we show that these two integrals are equivalent.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.415-421
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• 2022

This paper presents a fundamental theorem of calculus, an integration by parts formula and a version of equiintegrability convergence theorem for the M_α-integral using the M_α-strong Lusin condition. In the convergence theorem, to be able to relax the condition of being point-wise convergent everywhere to point-wise convergent almost everywhere, the uniform M_α-strong Lusin condition was imposed.

• The Bulletin of The Korean Astronomical Society
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• v.42 no.1
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• pp.28.1-28.1
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• 2017

To investigate AGN outflows as a tracer of AGN feedback on the host galaxies, we perform integral-field spectroscopy of 20 type 2 AGNs at z<0.1 using the Magellan/IMACS and the VLT/VIMOS. The observed objects are luminous AGNs with the [O III] luminosity >$10^{41.5}erg/s$, and exhibit strong outflow signatures in the [O III] kinematics. We obtain the maps of the narrow and broad components of [O III] and $H{\alpha}$ lines by decomposing the emission-line profile. The broad components in both [O III] and $H{\alpha}$ represent the non-gravitational kinematics, (i.e., gas outflows), while the narrow components represent the gravitational kinematics (i.e., rotational disks), especially in $H{\alpha}$. By using the spatially integrated spectra within the flux-weighted size of the narrow-line region, we estimate the outflow energetics. The ionized gas mass is $(1.0-38.5){\times}10^5M_{\odot}$, and the mean mass outflow rate is $4.6{\pm}4.3M_{\odot}/yr$, which is a factor of ~260 higher than the mean mass accretion rate $0.02{\pm}0.01M_{\odot}/yr$. The mean energy injection rate is $0.8{\pm}0.6%$ of the AGN bolometric luminosity Lbol, while the mean momentum flux is $(5.4{\pm}3.6){\times}L_{bol}/c$, except for two most kinematically energetic AGNs. The estimated energetics are consistent with the expectations for energy-conserving outflows from AGNs, yet we do not find any supporting evidence of instantaneous star-formation quenching due to the outflows.

• The Bulletin of The Korean Astronomical Society
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• v.40 no.2
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• pp.32.3-33
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• 2015

There exist scaling relations that link the mass of supermassive black holes with both the velocity dispersion and the mass of the central stellar cusp of their host galaxies. This implies that these two components grow in tandem. Feedback from actively accreting supermassive black holes (AGN), in the form of multi-phase gas outflows, has been argued to be the agent of this co-evolution. Here we employ the powerful GMOS integral field spectroscopy unit on the 8.2m Gemini-North telescope to investigate ionized gas outflows of luminous Type 2 AGN in the local Universe (z<0.1). Our sample of 6 galaxies is drawn from the Sloan Digital Sky Survey (SDSS) and was selected based on their [OIII] dust-corrected luminosity (>1042 erg/s) and signatures of outflows in the [OIII] line profile of their spatially integrated SDSS spectra. These are arguably the best candidates to explore AGN feedback in action since they are < 1% of a large local type 2 AGN SDSS sample selected based on their [OIII] kinematics. We combine a careful spectral decomposition of the [OIII] and $H{\alpha}$ line profiles with spatial information on ~0.5kpc scales to understand the outflow kinematics and energetics in these objects. We find clear evidence for strong outflows in [OIII] and occasionally $H{\alpha}$ that are clearly driven by the ionizing radiation of the AGN. We kinematically and spatially decompose outflowing and rotating ionized gas components. We find [OIII] to be a better tracer of AGN outflows, while $H{\alpha}$ appears to be strongly affected by both stellar rotation and outflows induced by ongoing star formation. The observed kinematics and spatial distribution of the ionized gas imply a large opening angle for the outflow. Finally, we find the projected outflow velocity to decrease as a function of distance, while its dispersion shows a more complex structure with a potentially initially increasing trend (out to 0.5-1kpc distances).