• Korean Journal of Mathematics
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• v.26 no.2
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• pp.167-174
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• 2018

By acting the dihedral group $D_k$ on the set of k-tuple multi-partitions, we introduce k-gonal partitions for all positive integers k. We give generating functions for these new partition functions and investigate their arithmetic properties.

• Bulletin of the Korean Chemical Society
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• v.27 no.1
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• pp.39-43
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• 2006

The relaxed torsional potential of a liquid crystalline polymer containing an ester functional group in a mesogenic unit (hereafter 12-4 oligomer) has been calculated with the ab initio self-consistent-field using 6-31G$^*$ basis set. GIAO^{13}C NMR chemical shifts also have been calculated at the B3LYP/6-31G$^*$ level of theory for each conformational structure obtained from torsional potential calculation. The results show that the phenyl ring-ester linkages are coplanar with the dihedral angle of about 0$^{\circ}$ and the ring-ring linkages in the biphenyl groups are tilted with the dihedral angle of around 43-44$^{\circ}$ in the lowest energy conformer. The biphenyl ring has a comparatively lower energy barrier of internal rotation potential in the ring-ring than that of phenyl ring-ester. The ^{13}C chemical shifts of carbonyl carbons were found to move to upfield due to $\pi$ -conjugation with phenyl ring and slightly affected about 0.5 ppm by dihedral angle of the ring-ring linkage.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.641-644
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• 2005

Let L/F be a Galois quadratic extension such that F contains a primitive n-th root of 1. Let N = L(${\alpha}^{{\frac{1}{n}}$) where ${\alpha}{\in}L{\ast}$. We show that if $N_{L/F}({\alpha})\;{\in}L^n{\cap}F$, and [N : L] = m, then $G(N/ F) {\simeq}D_m$ or generalized quaternion group whether $N_{L/F}({\alpha})\;{\in}\;F^n\;or\;{\notin}F^n$, respectively.

• Korean Journal of Crystallography
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• v.16 no.1
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• pp.1-5
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• 2005

The structure of the title compound has been determined by single-crystal X-ray diffraction work. The crystals are orthorhombic, space group $Pna2_1$ with a=11.095(3), b=21.834(7), c=12.574(4) $\AA$, and R1=0.0667. The average carbon bond length in aromaticity In the molecule is 1.386(1) $\AA$ and the average single bond length linking the central benEene ring and peripheral phenyl rings is 1.491(3) $\AA$. The average dihedral angle between the central benzene ring and each of six peripheral phenyl rings is $67.1(1)^{\circ}$ and the average dihedral angle between neighboring two phenyl rings is $55.0(1)^{\circ}$. Thus the molecule adopts a quasi-propeller configuration with approximate six-fold rotation symmetry.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.697-704
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• 2013

The function $g{\mapsto}{\zeta}^k_G(g)$ which counts the number of solutions of $x^k=g$ in a finite group G, is not necessarily a character of G. We study this function for the case of dihedral groups and generalized quaternion groups.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.167-173
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• 2023

Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ${\nu}(G)={\frac{1}{n}}$. We also classify all groups G with ${\nu}(G)={\frac{1}{2}}$. Further, we prove that if G is a solvable nonnilpotent group of even order, then ${\nu}(G){\leq}{\frac{p+3}{4p}}$, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if $\frac{G}{Z_{\infty}(G)}$ is isomorphic to the dihedral group of order 2p where Z_∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.475-486
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• 2018

We study free actions of finite nonabelian groups on 3-dimensional nilmanifolds with the first homology ${\mathbb{Z}}^2{\bigoplus}{\mathbb{Z}}_2$ which yield an orbit manifold reversing fiber orientation, up to topological conjugacy. We show that those nonabelian groups are $D_4$(the dihedral group), $Q_8$(the quaternion group), and $C_8.C_4$(the $1^{st}$ non-split extension by $C_8$ of $C_4$ acting via $C_4/C_2=C_2$).

• Bulletin of the Korean Chemical Society
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• v.18 no.1
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• pp.14-18
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• 1997

Sulconazole nitrate, C18H16Cl3N3O3S, crystallizes in monoclinic, space group C2/c, with a=14.401(1), b=8.051(1), c=34.861(2) Å, β=95.9(1)°, g=0.58 mm-1, Dc=1.523 g/cm3, Dm=1.522 g/cm3, F(000)=1888.0, and z=8. Intensities for 2460 unique reflections were measured on a CAD4 diffractometer with graphited-monochromated Mo-Kα radiation. The structure was solved by direct method and refined by full matrix least squares to a final R=0.071 for 2182 reflections (Io > 2σIo). The bond lengths and angles are comparable with the values found in the analogues imidazole derivatives. The 2,4-dichlorophenyl ring(A) and the p-chlorophenyl ring(B) are almost planar with different heights [dihedral angle 17.3°] while the imidazole ring(C) is nearly perpendicular to the two phenyl rings[dihedral angles about the two rings A, B are 110.8° and 96.1° respectively]. In order to understand the overall conformation we calculated the selected distances (l1, l2, l3) among the center of the three rings and considered the imaginary plan D[C(7), C(9) and C(16)]. The two polar group S(8) and N(19) do not have gauche conformation and l2 value (4.47 Å) is shorter than the other imidazole derivatives. One -NO3 group are hydrogen bonded the two neighbored sulconazole molecules. The molecular crystal packing is also formed by two hydrogen bondings and van der Waals forces.

• Journal of the Korean Chemical Society
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• v.24 no.1
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• pp.15-19
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• 1980

CNDO/2, EHT molecular orbital methods are used to investigate the electronic structure and reactivity of glycinato, glycine ester ligands. The results show that bidentate glycinato has a more stable structure, Gly-I with a $105.9^{\circ}$dihedral angle between ${\Delta}O_4C_3C_2$ and ${\Delta}C_3C_2N_1$ than Gly-Ⅱ. The electron inductive effects in the alkyl group substituted glycine ester ligands can also be derived from the calculation. According to the electron density, qN of ligands on the basis of CNDO/2 MO calculations, it is concluded that the stabilities are in the order of glycinato > Gly-Et-ester > Gly-i-Pr-ester > Gly-Me-ester.

• Analytical Science and Technology
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• v.36 no.4
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• pp.198-203
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• 2023

The reaction of cis-protected (Me4en)Pd(II) species with potentially multidentate bis(benzylthio)methylenepropanedioate (L) was carried out to obtain a monometallic compound, [(Me₄en)Pd(L)], in O,O'-coordination mode. The bis(benzylthio)methylene group was bent strikingly from the palladium square plane at the dihedral angle of 70.40°. The physicochemical properties of the present palladium(II) compound were fully characterized by means of infrared and nuclear magnetic resonance spectroscopy, thermogravimetric analysis, and single-crystal X-ray diffraction measurement.