• Journal of Integrative Natural Science
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• v.10 no.2
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• pp.110-114
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• 2017

The connected and simply connected four-dimensional matrix solvable Lie group $Sol^4_1$ is the four-dimensional geometry. A crystallographic group of $Sol^4_1$ is a discrete cocompact subgroup of $Sol^4_1{\rtimes}D(4)$. In this paper, we geometrically describe the crystallographic groups of $Sol^4_1$.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.1
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• pp.17-26
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• 2021

The purpose of this paper is to determine the structure of the crystallographic group 𝚷 of the 4-dimensional solvable Lie group Sol ⁴_m,n that the translation subgroup of 𝚷, 𝚪 := 𝚷; ∩ Sol⁴_m,n, is generated by the particular elements.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1209-1251
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• 2015

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$, and more generally, the crystallographic groups of $Sol_1^4$. The maximal compact subgroup of Isom($Sol_1^4$) is $D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$. We shall exhibit an infra-solvmanifold of $Sol_1^4$ whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; the trivial group, ${\mathbb{Z}}_2$ (5 families), ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_2$ (5 families), and ${\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$ (2 families).

• Journal of Integrative Natural Science
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• v.14 no.3
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• pp.139-146
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• 2021

The purpose of this paper is to classify the torsion-free extensions 1→ℤ³→𝛱→ℤ_𝜱→1 with injective abstract kernel 𝜙 : ℤ_𝜱→Aut(ℤ³). From this classification, we handle the sufficient conditions so as to classify the crystallographic groups of Sol⁴_m,n.

• Korean Journal of Crystallography
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• v.14 no.2
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• pp.110-114
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• 2003

LiAl/sub 0.24/Mn/sub 1.76/O/sub 4-y/S/sub y/ (y=0, 0.02) were synthesized by the sol-gel method. Both structures were refined by Rietveld method, its structure refined as a cubic spinel, space group Fd-3m, a=8.17937(30) Å and 8.18331(19) Å respectively. Though it has been made a charge/discharge experiment above 20 times, there was no change of 3 V/4 V capacity degradation. It was considered that the volume change of MnO/sub 6/ octahedron induced by sulfur substitution plays a key role in keeping the 3 V/4 V capacity. The refined composition of the compound could be confirmed with the ICP analysis.

• Journal of the Korean Magnetics Society
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• v.20 no.5
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• pp.187-190
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• 2010

$Sn_{0.99}{^{57}Fe}_{0.01}O_2$ prepared by a sol-gel method, and studied by x-ray diffractometer, vibrating magnetometer, Superconducting quantum interference devices and M$\ddot{o}$ssbauer spectroscopy. the crystal structure were found to be a rutile tetragonal structure with space group $P4_2$/mnm, and oxygen deficiency are 5.6 % by Rietveld refinement. magnetization value were $M_s=1.95{\times}10^{-2}{\mu}_B/Fe$ at room temperature, and Curri-weiss temperature were and ${\theta}_{cw}$ = 18 k, measurement of VSM and SQUID, respectively. Mssbauer spectra of $Sn_{0.99}{^{57}Fe}_{0.01}O_2$ have been Sextet taken at various temperatures ranging from 4.2 K to RT, and isomer shift value $\delta$ = 0.18~0.36 mm/s of $^{57}Fe$ ion site all of the temperature range the state shows ferric.