• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.825-835
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• 2024

We describe the local transition probability of a singular diagonal action on the standard non-uniform quotient of PGL₃ associated to the type 1 geodesic flow. As a consequence, we deduce the property of strongly positive recurrence.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1261-1269
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• 2023

For a fixed parametrization of a curve in an orientable two-dimensional Riemannian manifold, we introduce and investigate a new frame and curvature function. Due to the way of defining this new frame as being the time-dependent rotation in the tangent plane of the standard Frenet frame, both these new tools are called flow.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.363-368
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• 2018

The aim of this work is to derive a partial differential equation that explains the movement of vortex lines on a vortex trajectory surface in a three dimensional incompressible inviscid flow.

• The Journal of the Convergence on Culture Technology
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• v.9 no.3
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• pp.81-94
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• 2023

The study aims to compare and analyze the social network structures of Qatar Airways,s Singapore Airlines, Emirates Airlines, and ANA Airlines, recording the top 1 to 4, and Korean Air in ninth by Skytrax's airline evaluations in 2022. This study uses NodeXL, a social network analysis program, to analyze the social networks of 5 airlines, Vertex, Unique Edges, Single-Vertex Connected Components, Maximum Geodesic Distance, Average Geodesic Distance, Average Degree Centrality, Average Closeness Centrality, and Average Betweenness Centrality as indicators to compare the differences in these social networks of the airlines. As a result, Singapore's social network has a better network structure than the other airlines' social networks in terms of sharing information and transmitting resources. In addition, Qatar Airways and Singapore Airlines are superior to the other airlines in playing roles and powers of influencers who affect the flow of information and resources and the interaction within the airline's social network. The study suggests some implications to enhance the usefulness of social networks for marketing.

• The Pure and Applied Mathematics
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• v.19 no.1
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• pp.73-86
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• 2012

This is a survey on the volume entropy and its rigidity of various metric spaces. This survey is aimed to summarize recent results as well as remaining open questions and possible directions on this subject.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.427-444
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• 2011

In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ with Lie parallel normal Jacobi operator $\bar{R}_N$ and totally geodesic D and $D^{\bot}$ components of the Reeb flow.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.447-461
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• 2009

In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ satisfying the condition that the structure Jacobi operator $R_{\xi}$ commutes with the 3-structure tensors ${\phi}_i$, i = 1, 2, 3.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.737-767
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• 2016

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.855-861
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• 2019

The geodesics on the round 2-sphere $S^2$ are all simple closed curves of equal length. In 1903 Otto Zoll introduced other Riemannian surfaces with the same property. After that, his name is attached to the Riemannian manifolds whose geodesics are all simple closed curves of the same length. The question that "whether or not the set of Zoll metrics on 2-sphere $S^2$ is connected?" is still an outstanding open problem in the theory of Zoll manifolds. In the present paper, continuing the work of D. Jane for the case of the Ricci flow, we show that a naive application of some famous geometric flows does not work to answer this problem. In fact, we identify an attribute of Zoll manifolds and prove that along the geometric flows this quantity no longer reflects a Zoll metric. At the end, we will establish an alternative proof of this fact.

• Journal of Korea Water Resources Association
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• v.55 no.spc1
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• pp.1211-1222
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• 2022

Abnormal condition inevitably occurs during operation of water distribution system (WDS) and requires the isolation of certain areas using isolation valves. In general, the determination of the optimal location of isolation valves considered minimization of hydraulic failures as isolation of certain areas causes a change in hydraulic states (e.g., flow direction, velocity, pressure, etc.). Water quality failure can also be induced by changes in hydraulics, which have not been considered for isolation valve system design. Therefore, this study proposes a new isolation valve system design methodology to prevent unexpected water quality failure events. The new methodology considers flow direction change ratio (FDCR), which accounts for flow direction changes after isolation of the area, as a constraint while reliability is used as the objective function. The optimal design model has been applied to a synthetic grid network and the results are compared with the traditional design approach. Results show that considering FDCR can eliminate flow direction changes while average pressure and coefficient of variation of pressure, velocity, and hydraulic geodesic index (HGI) outperform compared to the traditional design approach. The proposed methodology is expected to be a useful approach to minimizing unexpected consequences by traditional design approaches.