• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.351-381
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• 2021

In this paper we introduce the notion of strong Galois H-progenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C. We prove that the set of isomorphism classes of strong Galois H-progenerator objects is a subgroup of the group of strong Galois H-objects introduced in [3]. Moreover, we show that strong Galois H-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if H is finite, we find exact sequences of Picard groups related with invertible left H-(quasi)modules and an isomorphism Pic(HMod) ≅ Pic(C)⊕G(H∗) where Pic(HMod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H∗) the group of group-like morphisms of the dual of H.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1203-1227
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• 2022

In this paper we develop the homological properties of the Gorenstein (𝓛, 𝓐)-flat R-modules 𝓖𝓕_{(𝓕(R),𝓐)} proposed by Gillespie, where the class 𝓐 ⊆ Mod(R^op) sometimes corresponds to a duality pair (𝓛, 𝓐). We study the weak global and finitistic dimensions that come with the class 𝓖𝓕_{(𝓕(R),𝓐)} and show that over a (𝓛, 𝓐)-Gorenstein ring, the functor - ⊗_R - is left balanced over Mod(R^op) × Mod(R) by the classes 𝓖𝓕_{(𝓕(R^op),𝓐)} × 𝓖𝓕_{(𝓕(R),𝓐)}. When the duality pair is (𝓕(R), 𝓕𝓟Inj(R^op)) we recover the G. Yang's result over a Ding-Chen ring, and we see that is new for (Lev(R), AC(R^op)) among others.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.137-148
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• 2023

We show the existence of inductive limit in the category of C^*-ternary rings. It is proved that the inductive limit of C^*-ternary rings commutes with the functor 𝓐 in the sense that if (M_n, ϕ_n) is an inductive system of C^*-ternary rings, then $\lim_{\rightarrow}$ 𝓐(M_n) = 𝓐$(\lim_{\rightarrow}\;M_{n})$. Some local properties (such as nuclearity, exactness and simplicity) of inductive limit of C^*-ternary rings have been investigated. Finally we obtain $\lim_{\rightarrow}\;M_{n}^{**}$ = $(\lim_{\rightarrow}\;M_{n})^{**}$.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.409-425
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• 2024

The category of pairs is the category whose objects are maps between two based spaces and morphisms are pair-maps from one object to another object. To study the self-homotopy equivalences in the category of pairs, we use covariant functors from the category of pairs to the group category whose objects are groups and morphisms are group homomorphisms. We introduce specific subgroups of groups of self-pair homotopy equivalences and put these groups together into certain sequences. We investigate properties of these sequences, in particular, the exactness and split. We apply the results to two special functors, homotopy and homology functors and determine the suggested several subgroups of groups of self-pair homotopy equivalences.

• International Journal of Advanced Culture Technology
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• v.5 no.4
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• pp.1-9
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• 2017

This study of Cho Ji - Hoon's Poem "Falling Flowers" was attempted to find the mechanism of poetic healing and utilize it for literary therapy. In this study, I examined how Cho Ji-Hoon's poem "Falling Flowers" encoded crying. Especially, we focused on the organic relationship of each layer represented by poem and put emotional codes on the layer of functor and argument. The results are as follow. It represents the Separation Layer of 1-3strophes, 4-6strophes constitute the Time Layer, and 7-9strophes the Sadness Layer. This poem proceeds the encoding of the sentence in which the crying of cuckoo in the 1-3strophes transforms into the crying of the poetic narrator in the last 9strophe. The relation of emotional layers in this poem is in the same function relations as "(1-3strophes) ${\subset}$ (4-6strophes) ${\subset}$ (7-9strophes)". Since these functional relations consist of the encoding of sadness, encrypts emotion signals of sadness as "U+U+U" becomes "UUU". 1-3strophes' U is the cry of the cuckoo, and U of the 4-6strophes is blood cry. Therefore, "UUU" is the blood cry of poetic narrator. This Cho Ji-Hoon's poem has a Han(恨) at its base. So, as Cho Ji-Hoon's poem "Falling Flowers" is uttered, the poetic mechanism of U, the code of sadness, is amplified. Then we get caught up in the emotions we want to cry. The poetic catharsis of "crying" is providing the effect of literary therapy. In the future, it will be possible to develop a more effective literary therapy technique by developing a literary therapy program like this poetic structure.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1483-1504
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• 2017

We consider weak solutions of the instationary Navier-Stokes system in a smooth bounded domain ${\Omega}{\subset}{\mathbb{R}}^3$ with initial value $u_0{\in}L^2_{\sigma}({\Omega})$. It is known that a weak solution is a local strong solution in the sense of Serrin if $u_0$ satisfies the optimal initial value condition $u_0{\in}B^{-1+3/q}_{q,s_q}$ with Serrin exponents $s_q$ > 2, q > 3 such that ${\frac{2}{s_q}}+{\frac{3}{q}}=1$. This result has recently been generalized by the authors to weighted Serrin conditions such that u is contained in the weighted Serrin class ${{\int}_0^T}({\tau}^{\alpha}{\parallel}u({\tau}){\parallel}_q)^s$ $d{\tau}$ < ${\infty}$ with ${\frac{2}{s}}+{\frac{3}{q}}=1-2{\alpha}$, 0 < ${\alpha}$ < ${\frac{1}{2}}$. This regularity is guaranteed if and only if $u_0$ is contained in the Besov space $B^{-1+3/q}_{q,s}$. In this article we consider the limit case of initial values in the Besov space $B^{-1+3/q}_{q,{\infty}}$ and in its subspace ${{\circ}\atop{B}}^{-1+3/q}_{q,{\infty}}$ based on the continuous interpolation functor. Special emphasis is put on questions of uniqueness within the class of weak solutions.