• Communications of the Korean Mathematical Society
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• v.34 no.1
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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.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.569-579
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• 2014

We consider a condition under which the projectivization $P(E^k)$ of a complex k-bundle $E^k{\rightarrow}M$ over an even-dimensional manifold M can have the hard Lefschetz property, affected by [10]. It depends strongly on the rank k of the bundle $E^k$. Our approach is purely algebraic by using rational Sullivan minimal models [5]. We will give some examples.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.835-840
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• 2021

In this paper, we compute the rational evaluation subgroup of the Hopf fibration S²ⁿ⁺¹ ↪ ℂP(n). We show that, for the Sullivan model 𝜙 : A → B, where A and B are the minimal Sullivan models of ℂP(n) and S²ⁿ⁺¹ respectively, the evaluation subgroup G_n(A, B; 𝜙) and the relative evaluation subgroup G^rel_n (A, B; 𝜙) of 𝜙 are generated by single elements.

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.131-139
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• 2023

Let V_k,n (ℂ) denote the complex Steifel and Gr_k,n (ℂ) the Grassmann manifolds for 1 ≤ k < n. In this paper, we compute, in terms of the Sullivan minimal models, the evaluation subgroups and, more generally, the relative evaluation subgroups of the fibration p : V_k,k+n (ℂ) → Gr_k,k+n (ℂ). In particular, we prove that G_* (Gr_k,k+n (ℂ), V_k,k+n (ℂ) ; p) is isomorphic to G^rel_* (Gr_k,k+n (ℂ), V_k,k+n (ℂ) ; p) ⊕ G_* (V_k,k+n (ℂ)).

• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.643-648
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• 2023

Let X be a simply connected rationally elliptic space such that H²(X; ℚ) ≠ 0. In this paper, we show that if H²ⁿ(X^[2n-2]; ℚ) = 0 or if π_2n(X²ⁿ) ⊗ ℚ = 0 for all n, then X is an F₀-space.

• Anxiety and mood
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• v.1 no.1
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• pp.14-17
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• 2005

By the influence of the descriptive approach of DSM-III, the anxiety became the same thing as the anxiety disorder to the clinicians. This unfortunate result sacrificed psychodynamic model of symptom formations and simplified the anxiety as one of the disease entity not as the overdetermined symptoms. These phenomenon awakened the psychoanalytic interest which was in sleep. Freud was the first major articulator of the basic significance of anxiety in human behavior. He attributed the particular quality of the anxiety experience to the trauma of birth, and subsequently to the fear of castration. Such classification of the anxiety according to the psychosexual development is helpful for the clinicians in understanding the origin of anxiety which the patient shows during the psychotherapy. The other analytical view of interpersonal psychoanalysis came from Sullivan. A large part of his therapy is taken up with recognizing and correcting parataxic distortions that interfere with realistic self-appraisal of events and of oneself in relation to others. Perhaps no explanation is the 'most basic' explanation for human anxiety. Anxiety is a multifaceted entity consisting of aspects of realm of discourse. Existential anxiety is inescapable in Western culture but it can be transcended by the cultivation of mind in Eastern culture. The analysts need to stay attuned to their own propensities for anxiety and must permit their own experiences with anxiety to be the grist for the psychotherapeutic mill.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.313-318
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• 1998

In this paper we prove two independent theorems concerning formality of a nilmanifold and a differential graded algebra using the well-known theorem of Deligne-Griffiths-Morgan-Sullivan. We first give a rational homotopy theoretic proof to the statement that a nilmanifold is formal if and only if it is a torus. And then we study some conditions with which formality of one dga implies formality of the other in an extension of dga's.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.445-449
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• 2021

We define the relative self-closeness number N��(g) of a map g : X → Y, which is a generalization of the self-closeness number N��(X) of a connected CW complex X defined by Choi and Lee [1]. Then we compare N��(p) with N��(X) for a fibration $X{\rightarrow}E{\rightarrow\limits^p}Y$. Furthermore we obtain its rationalized result.

• The Korean Journal of Applied Statistics
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• v.30 no.3
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• pp.475-486
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• 2017

This paper estimates the health life expectancies for Korean people based on a sample cohort database collected through objective measurements by the National Health Insurance Service. Health life expectancy is estimated using the single-state approach of Sullivan (1971). The 9-order correction factor method of Greville (1945) and Brass-logit model of Brass (1971) are also adopted for unobserved or incompletely observed age-specific morbidity and mortality. Based on the mortality and morbidity estimated from sample cohort DB, men and women in Korea are expected to live a 'healthy life' for 61 and 60 years in 2013, respectively, whereas life expectancies of men and women are 80 and 87, respectively. We also estimate certain disease-free life expectancies for each of genders, income levels, and types of insurance from 2003 to 2013 in Korea. We found that there exists an inequality of healthy life expectancy in Korea for different genders, income levels, and types of insurance.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.991-1004
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• 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 $[Y,\;Baut_1X]_{{\varepsilon}R}:=[Y,\;Baut_1X]/{\sim}_{{\varepsilon}R}$.