• 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 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 the Korean Mathematical Society
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• v.44 no.5
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• pp.1121-1137
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• 2007

There are 7 types of 4-dimensional solvable Lie groups of the form ${\mathbb{R}^3}\;{\times}_{\varphi}\;{\mathbb{R}}$ which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups $({\mathbb{R}^4},\;Nil^3\;{\times}\;{\mathbb{R}\;and\;Nil^4)$ are well known to have lattices. All the compact forms modeled on the remaining four solvable groups $Sol^3\;{\times}\;{\mathbb{R}},\;Sol_0^4,\;Sol_0^'4\;and\;Sol_{\lambda}^4$ are characterized: (1) $Sol^3\;{\times}\;{\mathbb{R}}$ has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, ${\mathbb{Z}}_2\;or\;{\mathbb{Z}}_4$. (2) Only some of $Sol_{\lambda}^4$, called $Sol_{m,n}^4$, have lattices with no non-trivial infra-solvmanifolds. (3) $Sol_0^{'4}$ does not have a lattice nor a compact form. (4) $Sol_0^4$ does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on $Sol_0^4$. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.

• 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 Crystal Growth and Crystal Technology
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• v.12 no.4
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• pp.178-183
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• 2002

The PLZT (x/50/50, x = 1, 2, 5, 7) thin films have been prepared by a sol-gel method using polypropanediol as a solvent, and their dielectric properties have been investigated. The prepared sol was coated 10 times on $Pt/Ti/SiO2_2$/Si wafer. After post-annealing at 560~$600^{\circ}C$ for 10 min, the 600 nm-thick PLZT (x/50/50) thin films were formed with pure perovskite phase. Grain size of the PLZT (x/50/50) thin films was increased with increasing the amount of La. For all the compositions, dielectric properties such as dielectric constant and remnant polarization were enhanced with increasing annealing temperatures. As the amount of La was increased, the remnant polarization, coercive field and dielectric constant of the PLZT (x/50/50) thin films fired at $600^{\circ}C$ were decreased.

• Proceedings of the Korean Ceranic Society Conference
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• 2000.10a
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• pp.180.1-180
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• 2000

• Journal of the Korean Electrochemical Society
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• v.12 no.4
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• pp.335-341
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• 2009

Organic-inorganic hybrid sol-gel matrices were used as hosts for chloro (5, 10, 15, 20-tetraphenylporphyrinato) cobalt (III) (Co[TPP]Cl), a known ionophore for nitrite. The sol-gel precursor was prepared by the reaction of (3-isocyanopropyl) triethoxysilane with 1,4-butanediol. An appropriate amount of the anion-exchanger, tridodecylmethylammonium chloride (TDMAC) and the plasticizer, tributylphosphate (DBP) were used as membrane additives. On mixing with an acidic catalyst, the sol-state precursors slowly gelled, yielding a membrane in which the active components, Co[TPP]Cl and TDMAC, were encapsulated. The performances of the sol-gel membrane-based electrodes were compared to those of Co[TPP]Cl-based poly(vinyl chloride) (PVC) membrane electrodes. Membranes with a molar ratio of Co[TPP]Cl: TDMAC (1 : 0.1) showed reasonable response slopes toward nitrite. The response slopes were typically 53 mV/decade between $10^{-5.4}$ and $10^{-1.0}\;M$. Selectivities toward nitrite over hydrophilic and small anions such as chloride were somewhat inferior to those observed with PVC-based membranes, but selectivities over lipophilic anions were quite similar. Reduced asymmetry potentials due to protein adsorption were found to occur with the sol-gel matrix relative to PVC-based films when the sensors were employed as a detector in flow-through configuration.

• Journal of the Korean Ceramic Society
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• v.28 no.1
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• pp.60-66
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• 1991

The polymeric alumina sol was prepared by partial hydrolysis of aluminum sec-butoxide reacted with acetylacetone and its characteristics was investigated. The effects of alcohol solvent, acetylacetone, and acid concentration to the sol were investigated. FT-IR and Al27-MASNMR were used to study hydrolysis and polymerization reaction of aluminum complex. Synthesized sol showed the characteristics of polymeric alumina sol. To make a clear polymeric alumina sol, the optimum contents of acetylacetone, H2O and alcohol solvent were 0.4-0.6 mole, 0.25-1.25 mole, 3-5 mole per one mole alkoxide respectively. As a result of the Al27-MASNMR analysis, it was noted that hexa-penta-coordinated Al were main structure.

• Journal of the Korean Ceramic Society
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• v.36 no.9
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• pp.893-900
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• 1999

Silica films were prepared on Si single crystal substrates by sol-gel process using TEOS as starting materials. Films were fabricated by a spin coating technique. Sol solutions were prepared by varying the compositions of CH3OH, H2O and DMF with fixed molar ratio of TEOS=1, HCl=0.05(mol). Wetting behavior viscosity of solutions gelation time thickness of films and cracking behavior were investigated with the various solution compositions. Wetting behaviors of solutions depended on the solution compositions mixing method and mixing rate. The optimum composition of sol was TEOS : DMF ; CH3OH: H2O :HCl=1:2:4:4:0.05(mol) and the mixing rate of solution was optimized at 1 ml/min. Viscosity of solutions were controlled by choosing a reaction time(elapsed time after mixing) at a room temperature so that we could get up to 800nm thick film The surface roughness was getting poor when thickness of films was thicker than 500nm. Thickness of coated films were increased with decreasing amount of CH3OH. The best surface roughness was obtained at the content of CH3OH 4 mol. The shortest gelation time was obtained with the content of CH3OH 8 mol. Crack-free filkms were fabricated when sintered at 500$^{\circ}C$ for 1 hr with heating rate of 0.6$^{\circ}C$/min.

• Proceedings of the Korean Magnestics Society Conference
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• 2001.04a
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• pp.154-155
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• 2001