• Communications for Statistical Applications and Methods
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• 제29권6호
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• pp.721-733
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• 2022

In this paper we compare parameter estimation by Grassmann manifold optimization and sequential candidate set algorithm in a structured principal fitted component (PFC) model. The structured PFC model extends the form of the covariance matrix of a random error to relieve the limits that occur due to too simple form of the matrix. However, unlike other PFC models, structured PFC model does not have a closed form for parameter estimation in dimension reduction which signals the need of numerical computation. The numerical computation can be done through Grassmann manifold optimization and sequential candidate set algorithm. We conducted numerical studies to compare the two methods by computing the results of sequential dimension testing and trace correlation values where we can compare the performance in determining dimension and estimating the basis. We could conclude that Grassmann manifold optimization outperforms sequential candidate set algorithm in dimension determination, while sequential candidate set algorithm is better in basis estimation when conducting dimension reduction. We also applied the methods in real data which derived the same result.

• 대한수학회논문집
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• 제34권1호
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• pp.279-286
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• 2019

Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.

• 대한수학회논문집
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• 제35권1호
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• pp.279-285
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• 2020

Let Gr(k, n) be the Grassmann manifold of k-linear sub-spaces in ℂⁿ. We compute rational relative Gottlieb groups of the Plücker embedding i : Gr(2, n) → ℂP^N-1, where N = n(n - 1)/2.

• 대한수학회논문집
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• 제38권1호
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• pp.257-266
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• 2023

Let G_k,n(ℍ) for 2 ≤ k < n denote the Grassmann manifold of k-dimensional vector subspaces of ℍⁿ. In this paper, we determine the evaluation subgroups of the Plüker embedding $G{_{2,n}}({\mathbb{H}}){\hookrightarrow}{\mathbb{H}}P^{N-1}$, where $N\;=\;({n \atop 2}\)$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제5권1호
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• pp.55-63
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• 1998

A Gauss map of m-dimensional distribution on a Riemannian manifold M is called a harmonic Gauss map if it is a harmonic map from the manifold into its Grassmann bundle $G_m$(TM) of m-dimensional tangent subspace. We calculate the tension field of the Gauss map of m-dimensional distribution and especially show that the Hopf fibrations on $S^{4n＋3}$ are the harmonic Gauss map of 3-dimensional distribution.

• 대한수학회지
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• 제35권2호
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• pp.315-330
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• 1998

Chen and Piccinni [7] have classified all compact surfaces in a Euclidean space $R^{2＋p}$ with 1-type generalized Gauss map. Being motivated by this result, the purpose of this paper is to consider the Lorentz version of the classification theorem and to obtain a complete classification of space-like surfaces in indefinite Euclidean space $R_{p}$ $^{2＋p}$ with 1-type generalized Gauss map.p.

• 대한수학회지
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• 제34권1호
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• pp.23-42
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• 1997

A 6-dimensional sphere considered as a homogeneous space $G_2/SU(3)$ where $G_2$ is the group of automorphism of the octonians O. From this representation, we can define an almost comlex structure on a 6-dimensional sphere by making use of the vector cross product of the octonians. Also it is known that a homogeneous space $G_2/U(2)$ coincides with the Grassmann manifold of oriented 2-planes of a 7-dimensional Euclidean space.