• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1287-1303
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• 2015

The Hausdorff topology induced by a fuzzy metric space under more weak assumptions is investigated in this paper. Another purpose of this paper is to obtain the Banach contraction theorem in fuzzy metric space based on a natural concept of Cauchy sequence in fuzzy metric space.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.87-94
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• 2003

We use geometric properties of Gromov-Hausdorff-convergence to present a way to construct rough but natural invariants of metric geometry.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2003.05a
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• pp.14-17
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• 2003

By Mazur's theorem, the convex hull of a relatively compact subset a Banach space is also relatively compact. But this is not true any more in the space of fuzzy numbers endowed with the Hausdorff-Skorohod metric. In this paper, we establish a necessary and sufficient condition for which the convex hull of K is also relatively compact when K is a relatively compact subset of the space F(R$\^$k/) of fuzzy numbers of R$\^$k/ endowed with the Hausdorff-Skorohod metric.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2002.05a
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• pp.277-280
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• 2002

In this paper, we characterize the Borel $\sigma$-field generated by the Hausdorff-Skorokhod metric on the space of normal and upper-semicontinuous fuzzy sets with compact support in the Ecleadean space R$\^$n/. As a result. we give a characterization of measurable fuzzy mappings .

• Journal of the Korea Society of Computer and Information
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• v.12 no.6
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• pp.147-152
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• 2007

Face detection is a crucial part of the face recognition system. It determines the performance of the whole recognition system. Hausdorff distance metric has been used in face detection and recognition with good results. It defines the distance metric based only on the geometric similarity between two sets or points. However, not only the geometry but also the local patterns around the points are available in most cases. In this paper a new Hausdorff distance measure is proposed that makes hybrid use of the similarity of the geometry and the local patterns around the points. Several experiments shows that the new method outperforms the conventional method.

• Korean Journal of Computational Design and Engineering
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• v.3 no.2
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• pp.140-144
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• 1998

We propose the area between a pair of non-self-intersecting 2-D parametric curves with same endpoints as an alternative distance metric between the curves. This metric is used when d curve is approximated with another in a simpler form to evaluate how good the approximation is. The traditional set-theoretic Hausdorff distance can he defined for any pair of curves but requires expensive calculations. Our proposed metric is not only intuitively appealing but also very easy to numerically compute. We present the numerical schemes and test it on some examples to show that our proposed metric converges in a few steps within a high accuracy.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.319-343
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• 2022

We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform h-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff h-shadowing.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.13 no.1
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• pp.83-90
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• 2013

Intervals can be used in the representation of uncertainty. In this regard, we consider monotone interval-valued set functions and the Choquet integral. This paper investigates characterizations of monotone interval-valued set functions and provides applications of the Choquet integral with respect to monotone interval-valued set functions, on the space of measurable functions with the Hausdorff metric.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1453-1462
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• 2011

In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.13-22
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• 1998

Fractals which represent many of the sets in various scien-tific fields as well as in nature is geometrically too complicate. Then we usually use Hausdorff dimension to estimate their geometrical proper-ties. But to explain the fractals from the hausdorff dimension induced by the Euclidan metric are not too sufficient. For example in digi-tal communication while encoding or decoding the fractal images we must consider not only their geometric sizes but also many other fac-tors such as colours densities and energies etc. So in this paper we define the dimension matrix of the sets by redefining the new metric.