• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.745-763
• /
• 2017

The Borel-Serre partial compactification gives cofinite models for the proper classifying space for arithmetic lattices. Non-arithmetic lattices arise only in semisimple Lie groups of ${\mathbb{R}}$-rank one. The author generalizes the Borel-Serre partial compactification to construct cofinite models for the proper classifying space for lattices in semisimple Lie groups of ${\mathbb{R}}$-rank one by using the reduction theory of Garland and Raghunathan.

• Journal of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.563-571
• /
• 1995

The recent work of Hopkins, Kuhn and Ravenel [H-K-R] indicates the Morava K-theory, $K(n)^*(-)$, occupy an important and fundamental place in homology theory. In particular $K(n)^*(BG)$ for classifying spaces of finite groups are studied by many authors [H-K-R], [R], [T-Y 1, 2] and [Hu].

• Journal of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.179-193
• /
• 2003

The braid-permutation group is a group of welded braids which is the extension of Artin's braid groups by the symmetric groups. It is also described as a subgroup of the automorphism group of a free group. We also show that the plus-construction of the classifying space of the infinite braid-permutation group has the following two types of splittings BBP(equation omitted) B∑(equation omitted) $\times$ X, BBP(equation omitted) B $^{＋}$$\times$ Y＝S$^1$$\times$Y, where X, Y are some spaces.

• Journal of the Korean Institute of Landscape Architecture
• /
• v.33 no.2 s.109
• /
• pp.1-15
• /
• 2005

The purpose of this paper is to classify zones disrupting green spaces in city and to evaluate of their grades. The results are as follows; L There were 158 green spaces in Dalsu-gu. The 158 green spaces were classified 4 patterns and minutely classified into 9 types. The area of the 'nature park' type was turned out to be $70.1\%$ of the total area of green spaces in Dalsu-gu, then the type was considered as a important part of the green-network in Dalsu-gu. The 9 types such as 'nature park', 'river', 'neighborhood park' and so on were analysed with ecological indexes. 2. Based on the ecological indexes of 'ratio of the green space', 'features of the surrounding matrix' and 'travel distance of the wildlives' , zones disrupting green spaces were ranging widely and re-divided to 236 sectors. 3. The analysis results for classifying the grades were that grade I appeared over industrial complex and housing complex widely. On the other side, grade II and III appeared around or between nature park and neighboring park Consequently, it was necessary to consider the grade and make zones disrupting green spaces into green space for improving green network.

• Journal of the Architectural Institute of Korea Planning & Design
• /
• v.36 no.2
• /
• pp.3-11
• /
• 2020

The police substation includes work spaces for civil services, interviews, and meetings, etc, and private spaces for rest, showers and cafeterias and so on. Since a large number of rooms for each function should be installed in a relatively small building, it is important to develop an area standard for efficient space organization in consideration of the functional characteristics and usage patterns of each space. The purpose of this study is to suggest the way for improving the area standards for spaces in police substation based on the results of existing standards analysis and case study. For this objective, architectural documents of 161 police substations built after 2013 in Korea were comparatively analyzed. Sixteen of these facilities were selected for field survey and investigated how the workspace and private area were organized and used. The results of investigation showed that there were a number of problematic cases, such as spaces not installed or insufficient, spaces used for two or more functions, spaces installed even though they are not included in the standards. It was mainly due to the fact that several important spaces which had been installed in most police substations were not included in the existing standards. The ways for improvement were suggested like following four points: (1) Modifying the criteria for classifying facility size, (2) Modifying the lists of the required spaces, (3) Specifying the basis of calculation for each space in detail, and (4) Differentiating the way to organize spaces according to the facility size.

• Bulletin of the Korean Mathematical Society
• /
• v.26 no.2
• /
• pp.109-114
• /
• 1989

This paper deals with the classifying spaces of finite groups. To any finite group G we associate a space BG with the property that .pi.$_{1}$(BG)=G, .pi.$_{i}$ (BG)=0 for i>1. BG is called the classifying space of G. Consider the problem of finding a stable splitting BG= $X_{1}$$^{V}$ $X_{1}$$^{V}$..$^{V}$ $X_{n}$ localized at pp. Ideally the $X_{i}$ 's are indecomposable, thus displaying the homotopy type of BG in the simplest terms. Such a decomposition naturally splits $H^{*}$(BG). The main purpose of this paper is to give the classification theorem in stable homotopy theory for groups with periodic cohomology i.e. cyclic Sylow p-subgroups for p an odd prime and to calculate some universal stable element. In this paper, all cohomology groups are with Z/p-coefficients and p is an odd prime.prime.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.1-8
• /
• 1997

Let G be a finiteg oroup. We denote BG a classifying space of G, which a contractible universal principal G bundle EG. The stable type of BG does not determine G up to isomorphism. A simple example [due to N. Minami]is given by $Q_{4p} \times Z/2$ and $D_{2p} \times Z/4$ where ps is an odd prime, $Q_{4p} is the generalized quarternion group of order 4p and $D_{2p}$ is the dihedral group of order 2p. However the paper [6] gives us a necessary and sufficient condition for $BG_1$ and $BG_2$ to be stably equivalent localized et pp. The local stable type of BG depends on the conjegacy classes of homomorphisms from the p-groups Q into G. This classification theorem simplifies if G has a normal sylow p-subgroup. Then the stable homotopy type depends on the Weyl group of the sylow p-subgroup.

• Korean Journal of Mathematics
• /
• v.18 no.2
• /
• pp.175-183
• /
• 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 ${\mathbb{Z}}{\times}BAut^+_{\infty}$.

• Communications of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.991-1004
• /
• 2019

Firstly we consider preorders (not necessarily partial orders) on a canonical quotient of the set of the homotopy classes of continuous maps between two spaces induced by a certain equivalence relation ${\sim}_{{\varepsilon}R}$. Secondly we apply it to a classification of orientable fibrations over Y with fibre X. In the classification theorem of J. Stasheff [22] and G. Allaud [3], they use the set $[Y,\;Baut_1X]$ of homotopy classes of continuous maps from Y to $Baut_1X$, which is the classifying space for fibrations with fibre X due to A. Dold and R. Lashof [11]. In this paper we give a classification of fibrations using a preordered set (abbr., proset) structure induced by $[Y,\;Baut_1X]_{{\varepsilon}R}:=[Y,\;Baut_1X]/{\sim}_{{\varepsilon}R}$.

• Genomics & Informatics
• /
• v.17 no.4
• /
• pp.39.1-39.20
• /
• 2019

Analyzing patterns in data points embedded in linear and non-linear feature spaces is considered as one of the common research problems among different research areas, for example: data mining, machine learning, pattern recognition, and multivariate analysis. In this paper, data points are heterogeneous sets of biosequences (composite data points). A composite data point is a set of ordinary data points (e.g., set of feature vectors). We theoretically extend the derivation of the largest generalized eigenvalue-based distance metric D_ij(γ₁) in any linear and non-linear feature spaces. We prove that D_ij(γ₁) is a metric under any linear and non-linear feature transformation function. We show the sufficiency and efficiency of using the decision rule $\bar{{\delta}}_{{\Xi}i}$(i.e., mean of D_ij(γ₁)) in classification of heterogeneous sets of biosequences compared with the decision rules min_𝚵iand median_𝚵i. We analyze the impact of linear and non-linear transformation functions on classifying/clustering collections of heterogeneous sets of biosequences. The impact of the length of a sequence in a heterogeneous sequence-set generated by simulation on the classification and clustering results in linear and non-linear feature spaces is empirically shown in this paper. We propose a new concept: the limiting dispersion map of the existing clusters in heterogeneous sets of biosequences embedded in linear and nonlinear feature spaces, which is based on the limiting distribution of nucleotide compositions estimated from real data sets. Finally, the empirical conclusions and the scientific evidences are deduced from the experiments to support the theoretical side stated in this paper.