• East Asian mathematical journal
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• v.22 no.2
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• pp.131-149
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• 2006

In this paper we introduce the concept of $\gamma$-preopen sets in a topological space together with its corresponding $\gamma$-preclosure and $\gamma$-preinterior operators and a new class of topology $\tau_{{\gamma}p}$ which is generated by the class of $\gamma$-preopen sets. Also we introduce $\gamma$-pre $T_i$ spaces(i=0, $\frac{1}{2}$, 1, 2) and study some of its properties and we proved that if $\gamma$ is a regular operation, then$(X,\;{\tau}_{{\gamma}p})$ is a $\gamma$-pre $T\frac{1}{2}$ space. Finally we introduce $(\gamma,\;\beta)$-precontinuous mappings and study some of its properties.

• Kyungpook Mathematical Journal
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• v.52 no.3
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• pp.265-269
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• 2012

In this paper, we prove the following theorem which gives a characterization of $M_1$-spaces by g-function.

• The Mathematical Education
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• v.21 no.1
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• pp.19-21
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• 1982

In this paper, we define the z-S-closed spaces using the notions of zero-sets and S-closed spaces introduced by T. Thompson, and investigate some properties of these spaces. We also obtain the following results. If a space X is z-S-closed, then every cover of z-regular semiopen sets has a finite proximate subcover. A z-extremally disconnected z-QHC space is z-S-closed, z-S-closed is contagious.

• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.17-22
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• 1985

B.M. Munshi and D.S. Bassan defined and developed the concept of super continuity in [5]. The concept has been investigated further by I. L. Reilly and M. K. Vamanamurthy in [6] where super continuity is characterized in terms of the semi-regularization topology. Super continuity is related to the concepts of .delta.-continuity and strong .theta.-continuity developed by T. Noiri in [7]. The purpose of this note is to derive relationships between super continuity and other strong continuity conditions and to develop additional properties of super continuous functions. Super continuity implies continuity, but the converse implication is false [5]. Super continuity is strictly between strong .theta.-continuity and .delta.-continuity and strictly between complete continuity and .delta.-continuity. The symbols X and Y will denote topological spaces with no separation axioms assumed unless explicity stated. The closure and interior of a subset U of a space X will be denoted by Cl(U) and Int(U) respectively and U is said to be regular open (resp. regular closed) if U=Int[Cl(U) (resp. U=Cl(Int(U)]. If necessary, a subscript will be added to denote the space in which the closure or interior is taken.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.725-742
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• 1996

In this paper we give some sharp expressions of the weakly convergent sequence coefficient WCS(X) of a Banach space X. They are used to prove fixed point theorems for involution mappings T from a weakly compact convex subset C of a Banach space X with WCS(X) > 1 into itself which $T^2$ are both of asymptotically nonexpansive type and weakly asymptotically regular on C. We also show that if X satisfies the semi-Opial property, then every nonexpansive mapping $T : C \to C$ has a fixed point. Further, some questions for asymtotically nonexpansive mappings are raised.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.135-149
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• 2011

Using (r, s)-preopen sets [14] and pre-${\omega}_t$-closures [6], a new kind of covering property $P^t_{({\omega}_r,s)}$-closedness is introduced in a bitopological space and several characterizations via filter bases, nets and grills [30] along with various properties of such concept are investigated. Two new types of cluster sets, namely pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets and (r, s)t-${\theta}_f$-precluster sets of functions and multifunctions between two bitopological spaces are introduced. Several properties of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets are investigated and using the degeneracy of such cluster sets, some new characterizations of some separation axioms in topological spaces or in bitopological spaces are obtained. A sufficient condition for $P^t_{({\omega}_r,s)}$-closedness has also been established in terms of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets.

• Journal of the Korean Mathematical Society
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• v.35 no.1
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• pp.115-126
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• 1998

We study the optimal control problem of system governed by retarded functional differential $$ x'(t) = A_0 x(t) + A_1 x(t - h) + \\ulcorner\ulcorner\ulcorner_{-h}^{0} a(s)A_2 x(t + s)ds + B_0 u(t) $$ in Hilbert space H. After the fundamental facts of retarded system and the description of condition so called a weak backward uniqueness property are established, the technically important maximal principle and the bang-bang principle are given. its corresponding linear system.

• The Mathematical Education
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• v.16 no.2
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• pp.22-26
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• 1978

It is shown that a map having an extension to an open map between the Alex-androff base compactifications of its domain and range has a unique such extension. J.S. Wasileski has introduced the Alexandroff base compactifications of Hausdorff spaces endowed with Alexandroff bases. We introduce a definition of morphism between such spaces to obtain a category which we denote by ABC. We prove that the Alexandroff base compactification on objects can be extended to a functor on ABC and that the compact objects give an epireflective subcategory of ABC. For each topological space X there exists a completely regular space $\alpha$X and a surjective continuous function $\alpha$$_{x}$ : Xlongrightarrow$\alpha$X such that for each completely regular space Z and g$\in$C (X, Z) there exists a unique g$\in$C($\alpha$X, 2) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\alpha$$_{x}$, $\alpha$X) is called a completely regularization of X. Let TOP be the category of topological spaces and continuous functions and let CREG be the category of completely regular spaces and continuous functions. The functor $\alpha$ : TOPlongrightarrowCREG is a completely regular reflection functor. For each topological space X there exists a compact Hausdorff space $\beta$X and a dense continuous function $\beta$x : Xlongrightarrow$\beta$X such that for each compact Hausdorff space K and g$\in$C (X, K) there exists a uniqueg$\in$C($\beta$X, K) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\beta$$_{x}$, $\beta$X) is called a Stone-Cech compactification of X. Let COMPT$_2$ be the category of compact Hausdorff spaces and continuous functions. The functor $\beta$ : TOPlongrightarrowCOMPT$_2$ is a compact reflection functor. For each topological space X there exists a realcompact space (equation omitted) and a dense continuous function (equation omitted) such that for each realcompact space Z and g$\in$C(X, 2) there exists a unique g$\in$C (equation omitted) with g=g$^{\circ}$(equation omitted). Such a pair (equation omitted) is called a Hewitt's realcompactification of X. Let RCOM be the category of realcompact spaces and continuous functions. The functor (equation omitted) : TOPlongrightarrowRCOM is a realcompact refection functor. In [2], D. Harris established the existence of a category of spaces and maps on which the Wallman compactification is an epirefiective functor. H. L. Bentley and S. A. Naimpally [1] generalized the result of Harris concerning the functorial properties of the Wallman compactification of a T$_1$-space. J. S. Wasileski [5] constructed a new compactification called Alexandroff base compactification. In order to fix our notations and for the sake of convenience. we begin with recalling reflection and Alexandroff base compactification.

• Kyungpook Mathematical Journal
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• v.51 no.3
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• pp.323-338
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• 2011

In this paper, we introduce ($$\tilde{g}$$, s)-continuous functions between topological spaces, study some of its basic properties and discuss its relationships with other topological functions.

• Journal of the Korea Safety Management & Science
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• v.17 no.1
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• pp.53-66
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• 2015

This study classified the study subjects as apartment residents and managers, and conducted the descriptive statistics of research questions related to the effectiveness of the lightweight partition walls and the evacuation space. This study also conducted T-test and Anova to find the different perceptions of the lightweight partition walls and the evacuation space, and the different perceptions of related laws according to the characteristics of the residents and managers. The findings and suggestions were as follows. Firstly, strengthening of the Senior Fire Safety Manager system to prevent the insolvency of fire safety management due to the affiliate. Secondly, implementing enhanced penalties in accordance with not implementing residents education, in order to prevent not smoothly delivering manager trained education to residents. Thirdly, adjusting hands-on training time according to the gender of the managers to supplement women's relatively low perceptions of lightweight partition. Fourthly, mandating evacuation guidance stickers on the evacuation space for residents to remove obstacles in the evacuation space. Finally, establishing regular residents education system in order to improve the awareness of the related laws for residents residing over 16th floors.