• Journal of the Chungcheong Mathematical Society
• /
• v.35 no.1
• /
• pp.77-83
• /
• 2022

In this paper we investigate the topological stability of induced homeomorphisms on a hyperspace. More precisely, we show that an expansive induced homeomorphism on a hyperspace is topologically stable. We also give examples and a diagram about implications to illustrate our results.

• Journal of the Korean Society of Costume
• /
• v.60 no.5
• /
• pp.117-127
• /
• 2010

In this study, hyperspace is a result of imagination created by means of facts and fiction, represents a transfer to determination and indetermination, and means an extension to an open form. In other words, hyperspace is a high dimensional space expanded to imagination through the combination of the viewpoint on facts in this dimension and fiction. When the 2D plane surface or 3D symmetry is destroyed, or when the frame is twisted or entangled, the non-Euclidean geometry is created eventually. And when the twisting leads to transmutation and the destruction of the form reaches the extreme; this in turn became the twisting like Mbius band. Likewise, the non-Euclidean geometry is co-related to the asymmetry of the Higgs mechanism. When the 'destruction of symmetry' is considered, symmetric theory and asymmetric world can be connected. The asymmetry in turn can maintain balance by arranging the uneven weights at different distances from the shaft. Moreover, at this the concept of the upper, lower, left and right, which was included in the original form, may be crumbled down. The destruction of the symmetry is essential in order to present forecast that coincides with the phenomenon of the real world. Non-Euclidean geometry characteristic is expressed by asymmetry, twists, and deconstruction and its representative characteristic is ambiguity. The boundary between the front, back, upper, lower, inner and outer is unclear, and it is difficult and vague to pinpoint specific location. The design that does not clearly define or determine the direction of wearing costume is indeed the non-oriented design that can be worn without getting restricted by specific direction such as front and back. Non-Euclidean geometry characteristic of hyperspace have been applied to create new shapes through the modification of the substance from traditional clothing of the eastern world to modern fashion. The way of thinking in the 'hyperspace' that used to be expressed in the costumes of the east and the west in the past became the forum for unlimited creation.

• Journal of Welding and Joining
• /
• v.16 no.6
• /
• pp.86-96
• /
• 1998

This study proposes the analysis method of time series ultrasonic signal using the chaotic feature extraction for degradation extent evaluation. Features extracted from time series data using the chaotic time series signal analyze quantitatively degradation extent. For this purpose, analysis objective in this study is fractal dimension, lyapunov exponent, strange attractor on hyperspace. The lyapunov exponent is a measure of the rate at which nearby trajectories in phase space diverge. Chaotic trajectories have at least one positive lyapunov exponent. The fractal dimension appears as a metric space such as the phase space trajectory of a dynamical system. In experiment, fractal correlation) dimensions, lyapunov exponents, energy variation showed values of 2.217∼2.411, 0.097∼ 0.146, 1.601∼1.476 voltage according to degardation extent. The proposed chaotic feature extraction in this study can enhances precision ate of degradation extent evaluation from degradation extent results of the degraded materials (SA508 CL.3)

• Bulletin of the Korean Mathematical Society
• /
• v.37 no.1
• /
• pp.171-179
• /
• 2000

We prove the following: (1) For a Hausdorff space X, the hyperspace K(X) of compact subsets of X is hemicompact if and only if X is hemicompact. (2) For a regular space X, the hyperspace $C_K(X)$ of subcontinua of X is hemicompact (hemiconnected) if and only if X is hemicompact (hemiconnected). (3) For a locally compact Hausdorff space X, each open set in X is hemicompact if and only if each basic open set in the hyperspace K(X) is hemicompact. (4) For a connected, locally connected, locally compact Hausdorff space X, K(X) is hemiconnected if and only if X is hemiconnected.

• Journal of the Korean Mathematical Society
• /
• v.43 no.5
• /
• pp.1099-1114
• /
• 2006

Relations between closure-type properties of hyperspaces over a $\v{C}ech$ closure space (X, u) and covering properties of (X, u) are investigated.

• Journal of the Korean Mathematical Society
• /
• v.36 no.6
• /
• pp.1047-1059
• /
• 1999

Let $C(X)(C_{K}(X))$ denote the hyperspace of all nonempty closed connected subsets (subcontinua) of a locally compact Haus-dorff space X with the Fell topology. We prove that the following statements are equivalent: (1) X is locally connected. (2) C(X) is locally connected,. (3) C(X) is locally connected at each $E{\in}C_{k}(X).(4) C_{k}(X)$ is locally connected.

• Journal of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.309-319
• /
• 1997

We introduce $R^i$-sets and give various relations between $R^i$-sets and prove that the hyperspace of a metric continuum containing any one of the $R^i$-set also contains $R^i$ and hence is not contractible.

• Communications of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.1309-1318
• /
• 2020

In this paper we extend the concept of topological stability from continuous maps to the corresponding induced maps and prove that a continuous map is topologically stable if and only if its induced map also is topologically stable.

• Journal of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1421-1431
• /
• 2021

In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for n-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper N-expansive homeomorphisms via the topological dimension. More precisely, we show that C⁰-generically, any homeomorphism on a compact manifold is not hyper N-expansive for any N ∈ ℕ. Also we give some examples to illustrate our results.

• Journal of the Korean Society for Precision Engineering
• /
• v.15 no.12
• /
• pp.226-235
• /
• 1998

This study proposes the analysis and evaluation of method of time series of ultrasonic signal using the chaotic feature extraction for ultrasonic pattern recognition. The features are extracted from time series data for analysis of weld defects quantitatively. For this purpose, analysis objectives in this study are fractal dimension, Lyapunov exponent, and strange attractor on hyperspace. The Lyapunov exponent is a measure of rate in which phase space diverges nearby trajectories. Chaotic trajectories have at least one positive Lyapunov exponent, and the fractal dimension appears as a metric space such as the phase space trajectory of a dynamical system. In experiment, fractal(correlation) dimensions and Lyapunov exponents show the mean value of 4.663, and 0.093 relatively in case of learning, while the mean value of 4.926, and 0.090 in case of testing in slag inclusion(weld defects) are shown. Therefore, the proposed chaotic feature extraction can be enhancement of precision rate for ultrasonic pattern recognition in defecting signals of weld zone, such as slag inclusion.