• 통합자연과학논문집
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• 제10권2호
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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 $Sol^4_1$.

• 대한수학회지
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• 제52권6호
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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).

• 대한수학회지
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• 제44권5호
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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.

• 충청수학회지
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• 제34권1호
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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.

• 한국결정성장학회지
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• 제12권4호
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• pp.178-183
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• 2002

용매제로서 polypropanediol을 사용한 sol-gel법으로, La의 함량을 1, 2, 5, 7 ㏖%로 변화시킨 PLZT(x/50/50) 박막을 제조하고 La 함량에 따른 유전특성의 변화를 연구하였다. 제조한 sol을 사용하여 $Pt/Ti/SiO_2$/Si 위에 10회 코팅한 후, 560~$600^{\circ}C$까지 10분간 열처리하여 600nm 두께의 박막을 얻을 수 있었다 $560^{\circ}C$ 이상에서는 조성에 관계없이 perovskite 단일 상만이 나타났고, La 함량이 증가함에 따라 입자 크기가 증가하였다. 모든 조성의 박막에서 열처리 온도가 증가함에 따라 유전상수 및 잔류분극 값이 증가되었으며 $600^{\circ}C$에서 열처리한 박막이 가장 우수한 유전 특성을 나타내었다. $600^{\circ}C$에서 열처리한 박막에서 La함량이 증가함에 따라 잔류분극, 항전계, 유전상수 모두 감소하였다.

• 한국세라믹학회:학술대회논문집
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• 한국세라믹학회 2000년도 추계총회,초청강연,특별강연,연구발표회
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• pp.180.1-180
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• 2000

• 전기화학회지
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• 제12권4호
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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.

• 한국세라믹학회지
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• 제28권1호
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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.

• 한국세라믹학회지
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• 제36권9호
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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.

• 한국자기학회:학술대회 개요집
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• 한국자기학회 2001년도 춘계연구발표회 논문개요집
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• pp.154-155
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• 2001