• East Asian mathematical journal
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• v.28 no.5
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• pp.533-541
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• 2012

A common fixed point result of Gregus type for subcompatible mappings defined on a complete linear metric space is obtained. The considered underlying space is generalized from Banach space to complete linear metric spaces, which include Banach space and complete metrizable locally convex spaces. Invariant approximation results have also been determined as its application.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.93-102
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• 1995

In [6], we introduce a hyperbolic homeomorphism on a compact metrizable space and show that a hyperbolic homeomorphism is topologically stable. The purpose of this paper is to study a necessary and sufficient condition for a homeomorphism to be hyperbolic. We get the following theorem.

• Communications of the Korean Mathematical Society
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• v.16 no.2
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• pp.277-285
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• 2001

The purpose of this paper is to give a proof of a generalized convex-valued selection theorem which is given by weakening a Banach space to a completely metrizable locally convex topological vector space, i.e., a Frechet space. We also develop the properties of upper semi-continuous singlevalued mapping to those of upper semi-continuous multivalued mappings. These properties wil be applied in our further consideraations of selection theorems.

• Journal of the Korean Mathematical Society
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• v.35 no.1
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• pp.23-76
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• 1998

We prove that a non-empty separable metrizable space X admits a totally bounded metric for which the metric dimension of X in Assouad's sense equals the topological dimension of X, which leads to a characterization for the latter. We also give a characterization based on this Assouad dimension for the demension (embedding dimension) of a compact set in a Euclidean space. We discuss Assouad dimension and these results in connection with porous sets and measures with the doubling property. The elementary properties of Assouad dimension are proved in an appendix.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.585-602
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• 2019

We characterize Gelfand-continuous functions from a Tychonoff space X into an Arens-Michael algebra A. Then we define several algebras of such functions, and investigate them as topological algebras. Finally, we provide a class of examples of (metrizable) commutative unital complete Arens-Michael algebras A and locally compact spaces X for which all these algebras differ from each other.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1157-1175
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• 2018

For a switched system with constraint on switching sequences, which is also called a subshift action, on a metric space not necessarily compact, two kinds of topological entropies, average topological entropy and maximal topological entropy, are introduced. Then we give some properties of those topological entropies and estimate the bounds of them for some special systems, such as subshift actions generated by finite smooth maps on p-dimensional Riemannian manifold and by a family of surjective endomorphisms on a compact metrizable group. In particular, for linear switched systems on ${\mathbb{R}}^p$, we obtain a better upper bound, by joint spectral radius, which is sharper than that by Wang et al. in [42,43].

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.195-201
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• 1988

In general relativity, analyzing causality is central to the study of black holes, to cosmology, and to each of the major recent mathematical theorems. By causality we refer to the general question of which points in a space-time can be joined by causal curves; relativistically which events can influence (be influenced by) a given event. Various causality conditions have been developed for space-times of the problems associated with examples of causality violations (2, 4). Causally continuous space-times were defined by Hawking and Sachs (5). Budic and Sachs (3) established causal completion. A metrizable topology on the causal completion of a causally continuous space-time was studied by Beem(1). Recently the region of space-time where causal continuity is violated was studied by Ishikawa (6) and Vyas and Akolia (8). In this paper we show characterization for reflectingness in terms of continuity of set valued functions. We investigate some properties of the region related to a causally continuous space-time where distinguishingness is violated, and characterize the chronology condition in terms of distinguishing-violated region.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.571-583
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• 2012

Let $G$ be a metrizable, ${\sigma}$-compact locally compact abelian group with a compact open subgroup. In this paper we define the Gramian and the dual Gramian operators for shift invariant subspaces of $L^2(G)$ and we use them to characterize shift generated dual frames for shift in- variant spaces, which forms a frame for a subspace of $L^2(G)$. We present necessary and sufficient conditions for which standard dual is a unique SG-dual frame of type I and type II.

• Journal of the Korean Mathematical Society
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• v.49 no.1
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• pp.165-185
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• 2012

In this paper, the topological entropy and measure-theoretic entropy for nonautonomous dynamical systems are studied. Some properties of these entropies are given and the relation between them is discussed. Moreover, the bounds of them for several particular nonautonomous systems, such as affine transformations on metrizable groups (especially on the torus) and smooth maps on Riemannian manifolds, are obtained.