• 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 Korean Mathematical Society
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• v.47 no.2
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• pp.299-309
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• 2010

The Weyl group of a compact connected Lie group is a reflection group. If such Lie groups are locally isomorphic, the representations of the Weyl groups are rationally equivalent. They need not however be equivalent as integral representations. Turning to the invariant theory, the rational cohomology of a classifying space is a ring of invariants, which is a polynomial ring. In the modular case, we will ask if rings of invariants are polynomial algebras, and if each of them can be realized as the mod p cohomology of a space, particularly for dihedral groups.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.397-410
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• 2002

In this paper, some inequalities for vector-valued maximal functions over locally compact Vienkin groups are obtained.

• Journal of the Korean Mathematical Society
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• v.14 no.2
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• pp.213-215
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• 1978

• Bulletin of the Korean Mathematical Society
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• v.13 no.2
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• pp.85-92
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• 1976

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.371-381
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• 2021

In this paper, we attempt to obtain sufficient conditions for the existence of tight wavelet frame sets in locally compact abelian groups. The condition is generated by modulating a collection of characteristic functions that correspond to a generalized shift-invariant system via the Fourier transform. We present two approaches (for stationary and non-stationary wavelets) to construct the scaling function for L²(G) and, using the scaling function, we construct an orthonormal wavelet basis for L²(G). We propose an open problem related to the extension principle for Riesz wavelets in locally compact abelian groups.

• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.47-52
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• 1989

Methods of Riemannian geometry has played an important role in the study of compact transformation groups. Every effective action of a compact Lie group on a differential manifold leaves a Riemannian metric invariant and the study of such actions reduces to the one involving the group of isometries of a Riemannian metric on the manifold which is, a priori, a Lie group under the compact open topology. Once an action of a compact Lie group is given an invariant metric is easily constructed by the averaging method and the Lie group is naturally imbedded in the group of isometries as a Lie subgroup. But usually this invariant metric has more symmetries than those given by the original action. Therefore the first question one may ask is when one can find a Riemannian metric so that the given action coincides with the action of the full group of isometries. This seems to be a difficult question to answer which depends very much on the orbit structure and the group itself. In this paper we give a sufficient condition that a subgroup action of a compact Lie group has an invariant metric which is not invariant under the full action of the group and figure out some aspects of the action and the orbit structure regarding the invariant Riemannian metric. In fact, according to our results, this is possible if there is a larger transformation group, containing the oringnal action and either having larger orbit somewhere or having exactly the same orbit structure but with an orbit on which a Riemannian metric is ivariant under the orginal action of the group and not under that of the larger one. Recently R. Saerens and W. Zame showed that every compact Lie group can be realized as the full group of isometries of Riemannian metric. [SZ] This answers a question closely related to ours but the situation turns out to be quite different in the two problems.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.149-161
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• 2002

We study the torsions in the integral homology of the double loop space of the compact simple Lie groups by determining the higher Bockstein actions on the homology of those spaces through the Bockstein lemma and computing the Bockstein spectral sequence.

• Journal of Korean Neurosurgical Society
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• v.67 no.4
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• pp.431-441
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• 2024

Objective : Gamma Knife radiosurgery (GKRS) is an effective and noninvasive treatment for high-risk arteriovenous malformations (AVMs). Since differences in GKRS outcomes by nidus type are unknown, this study evaluated GKRS feasibility and safety in patients with brain AVMs. Methods : This single-center retrospective study included patients with AVM who underwent GKRS between 2008 and 2021. Patients were divided into compact- and diffuse-type groups according to nidus characteristics. We excluded patients who performed GKRS and did not follow-up evaluation with magnetic resonance imaging or digital subtraction angiography within 36 months from the study. We used univariate and multivariate analyses to characterize associations of nidus type with obliteration rate and GKRS-related complications. Results : We enrolled 154 patients (mean age, 32.14±17.17 years; mean post-GKRS follow-up, 52.10±33.67 months) of whom 131 (85.1%) had compact- and 23 (14.9%) diffuse-type nidus AVMs. Of all AVMs, 89 (57.8%) were unruptured, and 65 (42.2%) had ruptured. The mean Spetzler-Martin AVM grades were 2.03±0.95 and 3.39±1.23 for the compact- and diffuse-type groups, respectively (p<0.001). During the follow-up period, AVM-related hemorrhages occurred in four individuals (2.6%), three of whom had compact nidi. Substantial radiation-induced changes and cyst formation were observed in 21 (13.6%) and one patient (0.6%), respectively. The AVM complete obliteration rate was 46.1% across both groups. Post-GKRS complication and complete obliteration rates were not significantly different between nidus types. For diffuse-type nidus AVMs, larger AVM size and volume (p<0.001), lower radiation dose (p<0.001), eloquent area location (p=0.015), and higher Spetzler-Martin grade (p<0.001) were observed. Conclusion : GKRS is a safe and feasible treatment for brain AVMs characterized by both diffuse- and compact-type nidi.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.1-13
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• 1999

Let G be the 3-dimensional Heisenberg group. A discrete subgroup of Isom(G), acting freely on G with non-compact quotient, must be isomorphic to either 1, Z, Z2 or the fundamental group of the Klein bottle. We classify all discrete representations of such groups into Isom(G) up to affine conjugacy. This yields an affine calssification of 3-dimensional non-compact infra-nilmanifolds.