• Korean Journal of Mathematics
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• v.27 no.3
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• pp.707-722
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• 2019

For any non-negative real number ${\epsilon}_0$, we shall introduce a concept of the ${\epsilon}_0$-Cauchy sequence in a normed linear space V and also introduce a concept of the ${\epsilon}_0$-completeness in those spaces. Finally we introduce a concept of the generalized Banach spaces with these concepts.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.345-358
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• 2020

In this paper, we study the concepts of Wijsman lacunary invariant convergence, Wijsman lacunary invariant statistical convergence, Wijsman lacunary ${\mathcal{I}}_2$-invariant convergence (${\mathcal{I}}^{{\sigma}{\theta}}_{W_2}$), Wijsman lacunary ${\mathcal{I}}^*_2$-invariant convergence (${\mathcal{I}}^{\ast}^{{\sigma}{\theta}}_{W_2}$), Wijsman p-strongly lacunary invariant convergence ([W₂N_σθ]_p) of double sequence of sets and investigate the relationships among Wijsman lacunary invariant convergence, [W₂N_σθ]_p, ${\mathcal{I}}^{{\sigma}{\theta}}_{W_2}$ and ${\mathcal{I}}^{\ast}^{{\sigma}{\theta}}_{W_2}$. Also, we introduce the concepts of ${\mathcal{I}}^{{\sigma}{\theta}}_{W_2}$-Cauchy double sequence and ${\mathcal{I}}^{\ast}^{{\sigma}{\theta}}_{W_2}$-Cauchy double sequence of sets.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.403-413
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• 2007

In this paper, we introduce the notion of a norm in Hilbert algebras, and discuss some properties of Cauchy sequences.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.1017-1025
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• 2011

In this paper, we survey various notions and results related to statistical convergence of a sequence of fuzzy numbers, in which statistical convergence for fuzzy numbers was first introduced by Nuray and Savas in 1995. We will go over boundedness, convergence of sequences of fuzzy numbers, statistically convergence and statistically Cauchy sequences of fuzzy numbers, statistical limit and cluster point for sequences of fuzzy numbers, statistical mono-tonicity and boundedness of a sequence of fuzzy numbers and finally statistical limit inferior and limit inferior for the statistically bounded sequences of fuzzy numbers.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.41 no.5
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• pp.45-52
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• 2004

According tothe development of digital media technologies various algorithms for video sequence matching have been proposed to match the video sequences efficiently. A large number of video sequence matching methods have focused on frame-wise query, whereas a relatively few algorithms have been presented for video sequence matching or video shot matching. In this paper, we propose an efficientalgorithm to index the video sequences and to retrieve the sequences for video sequence query. To improve the accuracy and performance of video sequence matching, we employ the Cauchy function as a similarity measure between histograms of consecutive frames, which yields a high performance compared with conventional measures. The key frames extracted from segmented video shots can be used not only for video shot clustering but also for video sequence matching or browsing, where the key frame is defined by the frame that is significantly different from the previous fames. Several key frame extraction algorithms have been proposed, in which similar methods used for shot boundary detection were employed with proper similarity measures. In this paper, we propose the efficient algorithm to extract key frames using the cumulative Cauchy function measure and. compare its performance with that of conventional algorithms. Video sequence matching can be performed by evaluating the similarity between data sets of key frames. To improve the matching efficiency with the set of extracted key frames we employ the Cauchy function and the modified Hausdorff distance. Experimental results with several color video sequences show that the proposed method yields the high matching performance and accuracy with a low computational load compared with conventional algorithms.

• Journal of the Chungcheong Mathematical Society
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• v.11 no.1
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• pp.45-51
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• 1998

In this paper, we shall define the usual fuzzy distance between two real fuzzy points, using the usual distance between two points in $\mathbb{R}$. We introduce the fuzzy sequence in the fuzzy real line and the notion of limit of fuzzy sequence in $F_p(\mathbb{R})$, and obtain the fuzzy increasing(decreasing) sequence and fuzzy Cauchy sequence of real fuzzy points.

• Korean Journal of Mathematics
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• v.8 no.2
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• pp.139-146
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• 2000

In this paper, we discuss convergence theorem for contraction C-regularized semigroups. We establish the convergence of the sequence of generators of contraction regularized semigroups in some sense implies the convergence of the sequence of the corresponding contraction regularized semigroups. Under the assumption that R(C) is dense, we show the convergence of generators is implied by the convergence of C-resolvents of generators.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.281-286
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• 2009

In this paper, we introduce the notion of the complete fuzzy norm on a linear space. And we consider some relations between the fuzzy completeness and ordinary completeness on a linear space.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.331-339
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• 1998

We show a series of improved subseries convergence results, e.g., in a sequentially complete locally convex space X every weakly $c_0$-Cauchy series on X must be $c_0$-convergent. Thus, if X contains no copy of $c_0$, then every weakly $c_0$-Cauchy series on X must be subseries convergent.

• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.155-165
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• 2012

In this paper, we have studied some fixed point theorems on complete G-metric spaces.