• The Mathematical Education
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• v.15 no.2
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• pp.29-30
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• 1977

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.215-222
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• 2010

A T$_1$ topological space X is called an almost GP-space if every dense G$_{\delta}$-set of X has nonempty interior. The behaviour of almost GP-spaces under taking subspaces and superspaces, images and preimages and products is studied. If each dense G$_{\delta}$-set of an almost GP-space X has dense interior in X, then X is called a GID-space. In this paper, some interesting properties of GID-spaces are investigated. We will generalize some theorems that hold in almost P-spaces.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1285-1307
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• 2019

For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M/N) of semiprime submodules of M/N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval ${\uparrow}(N)^{Semp}(M)=\{P{\in}Semp(M){\mid}N{\subseteq}P\}$ and the lattice Semp (M/N) are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of M.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1341-1356
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• 2018

The author devotes this paper to defining a new class of generalized soft open sets, namely soft somewhere dense sets and to investigating its main features. With the help of examples, we illustrate the relationships between soft somewhere dense sets and some celebrated generalizations of soft open sets, and point out that the soft somewhere dense subsets of a soft hyperconnected space coincide with the non-null soft ${\beta}$-open sets. Also, we give an equivalent condition for the soft csdense sets and verify that every soft set is soft somewhere dense or soft cs-dense. We show that a collection of all soft somewhere dense subsets of a strongly soft hyperconnected space forms a soft filter on the universe set, and this collection with a non-null soft set form a soft topology on the universe set as well. Moreover, we derive some important results such as the property of being a soft somewhere dense set is a soft topological property and the finite product of soft somewhere dense sets is soft somewhere dense. In the end, we point out that the number of soft somewhere dense subsets of infinite soft topological space is infinite, and we present some results which associate soft somewhere dense sets with some soft topological concepts such as soft compact spaces and soft subspaces.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.903-921
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• 2008

A continuous linear operator $T\;:\;X{\rightarrow}X$ is called hypercyclic if there exists an $x\;{\in}\;X$ such that the orbit ${T^nx}_{n{\geq}0}$ is dense. We consider the problem: given an operator $T\;:\;X{\rightarrow}X$, hypercyclic or not, is the restriction $T|y$ to some closed invariant subspace $y{\subset}X$ hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on $H({\mathbb{C}}^d)$ (entire functions) is hypercyclic. Now, if q(D) is another such operator, p(D) maps ker q(D) invariantly (by commutativity), and we obtain a necessary and sufficient condition on p and q in order that the restriction p(D) : ker q(D) $\rightarrow$ ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of $H({\mathbb{C}}^d)$.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.259-266
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• 2024

The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to r-strongly irreducible r-ideals (endowed with Zariski topologies) of monoids, called terminal spaces. We show that terminal spaces are T₀, quasi-compact, and every nonempty irreducible closed subset has a unique generic point. We characterize rarithmetic monoids in terms of terminal spaces. Finally, we provide necessary and sufficient conditions for the subspaces of r-maximal r-ideals and r-prime r-ideals to be dense in the corresponding terminal spaces.

• Journal of Digital Convergence
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• v.20 no.2
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• pp.217-223
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• 2022

Subspace clustering for online data streams requires a large amount of memory resources as all subsets of data dimensions must be examined. In order to track the continuous change of clusters for a data stream in a finite memory space, in this paper, we propose a grid-based subspace clustering algorithm that effectively uses memory resources. Given an n-dimensional data stream, the distribution information of data items in data space is monitored by a grid-cell list. When the frequency of data items in the grid-cell list of the first level is high and it becomes a unit grid-cell, the grid-cell list of the next level is created as a child node in order to find clusters of all possible subspaces from the grid-cell. In this way, a maximum n-level grid-cell subspace tree is constructed, and a k-dimensional subspace cluster can be found at the k^th level of the subspace grid-cell tree. Through experiments, it was confirmed that the proposed method uses computing resources more efficiently by expanding only the dense space while maintaining the same accuracy as the existing method.