• Korean Journal of Mathematics
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• v.8 no.1
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• pp.97-102
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• 2000

In this paper, we study the first and second formulas of energy functions on the Teichm$\ddot{u}$ller spaces and prove the relation between index and nullity of Jacobi operator.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.585-593
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• 1996

We make the definition of hermitian K-theory for nonunital rings which makes as many senses as possible. We next show that the excision property in rational hermitian K-theory implies the nullity of rational $H B^-$-homology which is the antisymmetric part of Bar homology.

• THE INTERNATIONAL COMMERCE & LAW REVIEW
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• v.69
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• pp.699-720
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• 2016

In the past 10 years, there have been lots of misuses of the trademark system in China. For example, some Chinese companies have registered same or similar oversea's well-known trademark as a prior rights holder, and oversea's companies lost a chance to register their own trademarks or commence cases to acquire their own trademark determination in China. So Chinese government revised Chinese Trademark Law in 2014 to remedy these mistakes. Article 30 is intended to crack down on preemptive registration and compensate for the possible unfair consequences resulted from the principle of prior registration. Under the principle of prior registration, only where the unregistered trademarks of prior use have certain influence, and where the applicant of latter applied trademark knows or should know the prior trademark and the applicant has the bad faith of obtaining unjustified interests from goodwill of such unregistered marks, it shall be curbed by Article 30. Furthermore, trademark oppositions could be filed by anybody previously. Under the revised Trademark Law Article 44, oppositions based on absolute grounds can still be filed by anyone, but oppositions based on other available grounds can only be filed by a prior rights holder or a materially-interested party with undefined but similar to the standing requirement for filing nullities under Article 41 of the old law, and likely intended to cover trademark licensees and successors.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.797-820
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• 2019

For every tree T, we introduce a topological invariant, called the T-cross-index, for connected graphs. The T-cross-index of a graph is a non-negative integer or infinity according to whether T is a tree basis of the graph or not. It is shown how this cross-index is independent of the other topological invariants of connected graphs, such as the Euler characteristic, the crossing number and the genus.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.213-219
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• 2014

We prove that a semi-symmetric 3-dimensional gradient Ricci soliton is locally isometric to a space form $\mathbb{S}^3$, $\mathbb{H}^3$, $\mathbb{R}^3$ (Gaussian soliton); or a product space $\mathbb{R}{\times}\mathbb{S}^2$, $\mathbb{R}{\times}\mathbb{H}^2$, where the potential function depends only on the nullity.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.191-203
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• 2021

Let M be an almost Kenmotsu manifold of dimension 2n + 1 having non-vanishing ��-sectional curvature such that trℓ > -2n - 2. We prove that any second order parallel tensor on M is a constant multiple of the associated metric tensor and obtained some consequences of this. Vector fields keeping curvature tensor invariant are characterized on M.

• Korean Journal of Mathematics
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• v.29 no.4
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• pp.801-810
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• 2021

This paper deals with the study of N (k)-quasi Einstein manifolds that satisfies the certain curvature conditions 𝒞_*·𝒞_* = 0, 𝓢·𝒞_* = 0 and ${\mathcal{R}}{\cdot}{\mathcal{C}}_*=f{\tilde{Q}}(g,\;{\mathcal{C}}_*)$, where 𝒞_*, 𝓢 and 𝓡 denotes the quasi-conformal curvature tensor, Ricci tensor and the curvature tensor respectively. Finally, we construct an example of N (k)-quasi Einstein manifold.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.333-345
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• 2022

We study *-Ricci solitons on non-cosymplectic (κ, µ)-acs (almost cosymplectic) manifolds M. We find *-solitons that are steady, and such that both the scalar curvature and the divergence of the potential field is negative. Further, we study concurrent, concircular, torse forming and torqued vector fields on M admitting Ricci and *-Ricci solitons. Also, we provide some examples.

• Honam Mathematical Journal
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• v.43 no.4
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• pp.571-586
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• 2021

The present paper deals with the study of generalized quasi-conformal curvature tensor inside the setting of (𝑘, 𝜇)'-almost Kenmotsu manifold with respect to 𝜂-Ricci soliton. Certain consequences of these curvature tensor on such manifold are likewise displayed. Finally, we illustrate some examples based on this study.

• Honam Mathematical Journal
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• v.43 no.4
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• pp.655-665
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• 2021

In the present paper, we study the geometric properties of the invariant submanifold of an almost Kenmotsu structure whose Riemannian curvature tensor has (𝜅, 𝜇, 𝜈)-nullity distribution. In this connection, the necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu (𝜅, 𝜇, 𝜈)-space to be totally geodesic under the behavior of functions 𝜅, 𝜇, and 𝜈.