• 한국컴퓨터정보학회논문지
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• 제29권11호
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• pp.97-105
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• 2024

화학 반응이나 경제 현상을 모델링할 때는 개체 간의 상호작용을 표현해야 한다. 상호작용의 분석이 필요한 도메인에서 그래프 뉴럴 네트워크를 활발히 적용하고 있으나, 데이터에 존재하는 위상 구조를 표현하는 과정에서 그래프 뉴럴 네트워크의 표현 능력이 충분하지 않다고 알려져 있다. 데이터에 존재하는 위상 구조를 활용하는 토폴로지 데이터 분석(TDA)과 딥러닝을 결합한 토폴로지 딥러닝(TDL)은 복잡한 관계를 내포한 데이터의 분석 및 표현에 강점이 있는데, 본 논문에서는 특히 단체 복합체(simplicial complex)와 셀 복합체(cell complex)에 주목하여 토폴로지 딥러닝의 주요 접근법을 소개하고자 한다. 본 논문에서는 단체 복합체와 셀 복합체를 이해하는 과정에서 요구되는 기본 개념과 주요 접근법을 요약 소개하여 토폴로지 딥러닝을 응용하려는 연구자에게 도움을 제공하고자 한다.

• 호남수학학술지
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• 제27권1호
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• pp.115-129
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• 2005

In this paper, we give a digital graph-theoretical approach of the study of digital images with relation to a simplicial complex. Thus, a digital graph $G_k$ with some k-adjacency in ${\mathbb{Z}}^n$ can be recognized by the simplicial complex spanned by $G_k$. Moreover, we demonstrate that a graphically $(k_0,\;k_1)$-continuous map $f:G_{k_0}{\subset}{\mathbb{Z}}^{n_0}{\rightarrow}G_{k_1}{\subset}{\mathbb{Z}}^{n_1}$ can be converted into the simplicial map $S(f):S(G_{k_0}){\rightarrow}S(G_{k_1})$ with relation to combinatorial topology. Finally, if $G_{k_0}$ is not $(k_0,\;3^{n_0}-1)$-homotopy equivalent to $SC^{n_0,4}_{3^{n_0}-1}$, a graphically $(k_0,\;k_1)$-continuous map (respectively a graphically $(k_0,\;k_1)$-isomorphisim) $f:G_{k_0}{\subset}{\mathbb{Z}}^{n_0}{\rightarrow}G_{k_1}{\subset}{\mathbb{Z}^{n_1}$ induces the group homomorphism (respectively the group isomorphisim) $S(f)_*:{\pi}_1(S(G_{k_0}),\;v_0){\rightarrow}{\pi}_1(S(G_{k_1}),\;f(v_0))$ in algebraic topology.

• 대한수학회지
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• 제57권6호
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• pp.1323-1333
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• 2020

In 2009, Borg [2] suggested a conjecture concerning the size of a t-intersecting k-uniform family of faces of an arbitrary simplicial complex. In this paper, we give a strengthening of Borg's conjecture for shifted simplicial complexes using algebraic shifting.

• 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.

• 대한수학회보
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• 제54권1호
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• pp.265-276
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• 2017

Let K be a fan-like simplicial sphere of dimension n-1 such that its associated complete fan is strongly polytopal, and let v be a vertex of K. Let K(v) be the simplicial wedge complex obtained by applying the simplicial wedge operation to K at v, and let $v_0$ and $v_1$ denote two newly created vertices of K(v). In this paper, we show that there are infinitely many strongly polytopal fans ${\Sigma}$ over such K(v)'s, different from the canonical extensions, whose projected fans ${Proj_v}_i{\Sigma}$ (i = 0, 1) are also strongly polytopal. As a consequence, it can be also shown that there are infinitely many projective toric varieties over such K(v)'s such that toric varieties over the underlying projected complexes $K_{{Proj_v}_i{\Sigma}}$ (i = 0, 1) are also projective.

• 대한수학회지
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• 제56권6호
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• pp.1613-1639
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• 2019

Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure strongly shellable complexes. Meanwhile, pure strongly shellable complexes can be characterized by the corresponding codimension one graphs. In addition, we show that the facet ideals of pure strongly shellable complexes have linear quotients.

• 호남수학학술지
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• 제18권1호
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• pp.25-31
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• 1996

• Journal of Communications and Networks
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• 제16권6호
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• pp.667-676
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• 2014

Connectivity is a crucial quality of service measure in wireless sensor networks. However, the network is always at risk of being split into several disconnected components owing to the sensor failures caused by various factors. To handle the connectivity problem, this paper introduces an in-advance mechanism to prevent network partitioning in the initial deployment phase. The approach is implemented in a distributed manner, and every node only needs to know local information of its 1-hop neighbors, which makes the approach scalable to large networks. The goal of the proposed mechanism is twofold. First, critical nodes are locally detected by the critical node detection (CND) algorithm based on the concept of maximal simplicial complex, and backups are arranged to tolerate their failures. Second, under a greedy rule, topological holes within the maximal simplicial complex as another potential risk to the network connectivity are patched step by step. Finally, we demonstrate the effectiveness of the proposed algorithm through simulation experiments.

• 대한수학회논문집
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• 제13권4호
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• pp.839-845
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• 1998

For an oriented compact, connected Seifert fibred 3-manifold M with nonempty boundary, we construct a simplicial complex using the equivalence classes of marked annulus systems and show that it is contractible.

• 응용통계연구
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• 제20권2호
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• pp.313-322
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• 2007

오염이 있는 생존자료에서 여러 가지 회귀뎁스(regression depth)를 비교 연구하였다. 중도절단 자료에서 회귀뎁스에 대한 정의는 Park과 Hwang(2003)의 반공간회귀뎁스(halfspace regression depth)와 Park(2003)의 심플리셜 회귀뎁스(simplicial regression depth)가 있다. 본 논문은 Hubert 등(2001)이 제안한 사영회귀뎁스(projection regression depth)를 생존자료에서 사용하는 방법을 제시하고 이 방법과 기존의 뎁스기반 회귀모형과의 비교를 다양한 오염 상황에서 실시하였다.