• Journal of applied mathematics & informatics
• /
• v.31 no.1_2
• /
• pp.229-239
• /
• 2013

This paper studies the generalized fifth-order KdV equation with variable coefficients using Lie symmetry methods.Lie group classification with respect to the time dependent coefficients is performed. Then we get the similarity reductions using the symmetry and give some exact solutions.

• Journal of the Korean Mathematical Society
• /
• v.56 no.4
• /
• pp.917-934
• /
• 2019

In this paper, we give some classifications of almost Kenmotsu 3-manifolds under homogeneity and some symmetry conditions.

• Journal of applied mathematics & informatics
• /
• v.29 no.3_4
• /
• pp.563-572
• /
• 2011

In this paper, the Lie symmetry analysis is performed on the three mixed second-order PDEs, which arise in fluid dynamics, nonlinear wave theory and plasma physics, etc. The symmetries and similarity reductions of the equations are obtained, and the exact solutions to the equations are investigated by the dynamical system and power series methods. Then, the exact solutions to the general types of PDEs are considered through a variable transformation. At last, the symmetry and integration method is employed for reducing the nonlinear ODEs.

• Journal of the Korean Mathematical Society
• /
• v.40 no.3
• /
• pp.517-561
• /
• 2003

We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in $\mathbb{C}$. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.

• Journal of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.183-203
• /
• 1996

When dealing with a Lie group or, in general, with an analytic loop or quasigroup, its symmetry is broken by the election of the distinguished identity element.

• Journal of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1303-1336
• /
• 2023

In this paper, we study the 𝜂-parallelism of the Ricci operator of almost Kenmotsu 3-manifolds. First, we prove that an almost Kenmotsu 3-manifold M satisfying ∇_𝜉h = -2𝛼h𝜑 for some constant 𝛼 has dominantly 𝜂-parallel Ricci operator if and only if it is locally symmetric. Next, we show that if M is an H-almost Kenmotsu 3-manifold satisfying ∇_𝜉h = -2𝛼h𝜑 for a constant 𝛼, then M is a Kenmotsu 3-manifold or it is locally isomorphic to certain non-unimodular Lie group equipped with a left invariant almost Kenmotsu structure. The dominantly 𝜂-parallelism of the Ricci operator is equivalent to the local symmetry on homogeneous almost Kenmotsu 3-manifolds.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.4
• /
• pp.1249-1257
• /
• 2016

We prove that the Ricci operator S of an almost cosymplectic three-manifold M is invariant along the Reeb flow, that is, M satisfies ${\pounds}_{\xi}S=0$ if and only if M is either cosymplectic or locally isometric to the group E(1, 1) of rigid motions of Minkowski 2-space with a left invariant almost cosymplectic structure.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1345-1356
• /
• 2013

In this paper, we study rotational and helicoidal surfaces in Euclidean 3-space in terms of their Gauss map. We obtain a complete classification of these type of surfaces whose Gauss maps G satisfy $L_1G=f(G+C)$ for some constant vector $C{\in}\mathbb{E}^3$ and smooth function $f$, where $L_1$ denotes the Cheng-Yau operator.

• Analytical Science and Technology
• /
• v.5 no.1
• /
• pp.121-126
• /
• 1992

Transmission electron microscopy was used to investigate the structure of GaAs ${\Sigma}=19$, [110] tilt grain boundaries. Relative positions of Ga and As atoms in each grain on either side of the boundaries were determined by examining the dynamical coupling between HOLZ reflections and(200) beams. No inversion symmetry was present across the boundaries. These boundaries were observed to have a strong tendency to lie parallel to {331} planes. The atomic structure and lattice translation at these boundaries was studied in detail by high-resolution transmission electron microscopy(HRTEM). The boundary consists of units of 5-, 7-, and two 6-member rings.

• Bulletin of the Korean Chemical Society
• /
• v.10 no.3
• /
• pp.243-247
• /
• 1989

Two crystal structures of dehydrated $Ag^+\;and\;Ca^{2+}$ exchanged zeolite A, $Ag_2Ca_$5-A, reacting with 0.01 Torr of Cs vapor at $200^{\circ}C$ for 2 hours and 0.1 Torr of Cs vapor at $250^{\circ}C$ for 48 hours, respectively, have been determined by single crystal X-ray diffraction techniques. Their structures were solved and refined in the cubic space group Pm3m at $21(1)^{\circ}C$. The stoichiometry of first crystal was $Ag_2Ca_5$-A (a = 12.294(1)${\AA}$), indicating that Cs vapor did not react with cations in zeolite A and that of second crystal was $Ag_2Cs_{10}$-A (a = 12.166(1)${\AA}$), indicating that all $Ca^{2+}$ ions were reduced by Cs vapor and replaced by $Cs^+$ ions. Full-matrix least-squares refinements of $Ag_2Ca_5-A\;and\;Ag_2Cs_{10}$-A has converged to the final error indices, $R_1\;=\;0.041\;and\;R_2$ = 0.048 with 227 reflections, and $R_1\;=\;0.117\;an\;n\;fdd\;R_2$ = 0.120 with 167 reflections, respectively, for which I > $3{\sigma}$(I). In the structure of $Ag_2Ca_5$-A, both $Ag^+$ ions and $Ca^{2+}$ ions lie on two crystal symmetrically independent threefold axis sites on the 6-rings; $2\;Ag^+$ ions are recessed 0.33 ${\;AA}$ from the (111) planes of three O(3) oxygens and 5 $Ca^{2+}$ ions lie on the nearly center of each 6-oxygen planes. In the structure of $Ag_2Cs_{10}-A,\;Cs^+$ ions lie on the 5 different crystallographic sites. 3 $Cs^+$ ions lie at the centers of the 8-rings at sites of D4h symmetry. 6 $Cs^+$ ions lie on the threefold axes of unit cell: $4\;Cs^+$ ions are found deep in the large cavity and 2 $Cs^+$ ions are found in the sodalite cavity. One $Cs^+$ ion is found in the large cavity near a 4-ring.