• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1051-1063
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• 2018

The real moment-angle complex (or, more generally, real polyhedral product) and its real toric space have recently attracted much attention in toric topology. The aim of this paper is to give two interesting remarks regarding real polyhedral products and real toric spaces. That is, we first show that Moore's conjecture holds to be true for certain real polyhedral products. In general, real polyhedral products show some drastic difference between the rational and torsion homotopy groups. Our result shows that at least in terms of the homotopy exponent at a prime this is not the case for real polyhedral products associated to a simplicial complex whose minimal missing faces are all k-simplices with $k{\geq}2$. Moreover, we also show a structural theorem for a finite group G acting simplicially on the real toric space. In other words, we show that G always contains an element of order 2, and so the order of G should be even.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1357-1366
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• 2018

It is shown that the Mayer-Vietoris sequence holds for the cohomology of complexes of Lie algebroids which are defined on simplicial complexes and satisfy the compatibility condition concerning restrictions to the faces of each simplex. The Mayer-Vietoris sequence will be obtained as a consequence of the extension lemma for piecewise smooth forms defined on complexes of Lie algebroids.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.699-705
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• 1995

We first generalize the Volodin space which Volodin constructed in order to define a new algebraic K-theory. We investigate the topological (homotopy) properties of the general Volodin space. We also provide a theorem which seems to be useful in pure homotopy theory. We prove that $V(*_\alpha G_\alpha, {G_\alpha})$ is simply connected.

• Korean Journal of Mathematics
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• v.5 no.2
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• pp.107-112
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• 1997

For a ring R with 1, the higher K-theory of Quillen is defined by the higher homotopy groups of the plus construction of the general linear group of R. On the other hand, the Volodin K-theory is defined by the higher homotopy groups of the Volodin space. In this paper we show that these two K-theories are equivalent. We show that the Volodin space is a homotopy fiber of the acyclic map from BGL(R) to its plus construction.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.175-183
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• 2010

The classical group completion theorem states that under a certain condition the homology of ${\Omega}BM$ is computed by inverting ${\pi}_0M$ in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category ${\mathcal{A}}$. The 1-morphisms of ${\mathcal{A}}$ are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of ${\mathcal{A}}$, which is shown to be homotopy equivalent to ${\mathbb{Z}}{\times}BAut^+_{\infty}$.

• Journal of Korea Multimedia Society
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• v.19 no.2
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• pp.146-154
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• 2016

This paper presents methods to analyze the topological structures of the spaces composed of patches extracted from waveform signals, which can be applied to the classification of signals. Commute time embedding is performed to transform the patch sets into the corresponding geometries, which has the properties that the embedding geometries of periodic or quasi-periodic waveforms are represented as closed curves on the low dimensional Euclidean space, while those of aperiodic signals have the shape of open curves. Persistent homology is employed to determine the topological invariants of the simplicial complexes constructed by randomly sampling the commute time embedding of the waveforms, which can be used to discriminate between the groups of waveforms topologically.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.16 no.1
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• pp.291-299
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• 2016

This paper raises a question of whether a simple periodic phenomenon is associated with the topology and provides the convincing answers to it. A variety of music instrumental sound signals are used to prove our assertion, which are embedded in Euclidean space to analyze their topologies by computing the homology groups. A commute time embedding is employed to transform segments of waveforms into the corresponding geometries, which is implemented by organizing patches according to the graph-based metric. It is shown that commute time embedding generates the intrinsic topological complexities although their geometries are varied according to the spectrums of the signals. This paper employs a persistent homology to determine the topological invariants of the simplicial complexes constructed by randomly sampling the commute time embedding of the waveforms, and discusses their applications.