• 대한수학회보
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• 제55권4호
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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.

• 대한수학회논문집
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• 제33권4호
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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.

• 대한수학회논문집
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• 제10권3호
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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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• 제5권2호
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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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• 제18권2호
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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 ${\mathbb{Z}}{\times}BAut^+_{\infty}$.

• 한국멀티미디어학회논문지
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• 제19권2호
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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.

• 한국인터넷방송통신학회논문지
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• 제16권1호
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• pp.291-299
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• 2016

본 논문에서는 임의의 주기적인 현상이나 특성은 위상구조와 밀접한 관련이 있음을 추론하고 이를 실험적으로 확인한다. 실험대상으로 주기적 특성이 있는 다양한 악기음을 선택하여 이를 유클리드 공간에 임베딩하고 이로부터 호몰로지 군을 계산하여 위상특성을 분석한다. 이를 위하여, 파형신호에서 추출한 패치모음을 패치 그래프로 구성한 다음, 대표적인 다양체 학습 방식인 통근시간 임베딩 기법을 이용하여 기하구조로 변환한다. 스펙트럼이 시간에 따라 가변적인 파형신호를 통근시간 임베딩할 때, 그에 따라 생성되는 기하구조는 변화하지만 그 신호 고유의 내재된 위상구조는 거의 변하지 않는다. 본 논문에서는 임베딩 데이터의 일부를 표본화하여 단순 복합체를 구성한 다음 이로부터 호몰로지를 계산하여 임베딩 기하구조의 위상특성을 분석하고, 이의 활용방안을 논의한다.