• Journal of the Korean Mathematical Society
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• v.39 no.6
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• pp.881-898
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• 2002

Our aim in this paper is to introduce a generalization of some notions in homological algebra. We define the concepts of chain U-complex, U-homology, chain (U, U')-map, chain (U, U')-homotopy and $\mu$-functor. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma, Snake Lemma, Connecting Homomorphism and Exact Triangle.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.577-587
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• 2002

Let A be a commutative ring and M an Artinian .A-module. Let $\sigma$ be a torsion radical functor and (T, F) it's corresponding partition of Spec(A) In [1] the concept of Cohen-Macauly modules was generalized . In this paper we shall define $\sigma$-co-Cohen-Macaulay (abbr. $\sigma$-co-CM). Indeed this is one of the aims of this paper, we obtain some satisfactory properties of such modules. An-other aim of this paper is to generalize the concept of cograde by using the left derived functor $U^{\alpha}$$_{I}$(-) of the $\alpha$-adic completion functor, where a is contained in Jacobson radical of A.A.

• 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.