• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1367-1382
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• 2020

We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moreover, we also provide a few rigidity results for locally conformally flat critical metrics.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1327-1340
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• 2019

We establish a monotonicity formula of V-harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for V-harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and ${\pm}holomorphic$ maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of V-harmonic maps is considered.

• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.401-418
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• 2021

For complete manifolds with α-Bach tensor (which is defined by (1.2)) flat, we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics have also been characterized by some $L^{\frac{n}{2}}$-integral inequalities. Furthermore, we also give some rigidity characterizations for constant sectional curvature.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.761-779
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• 2018

The object of the present paper is to study weakly Z symmetric spacetimes $(WZS)_4$. At first we prove that a weakly Z symmetric spacetime is a quasi-Einstein spacetime and hence a perfect fluid spacetime. Next, we consider conformally flat $(WZS)_4$ spacetimes and prove that such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field ${\rho}$. We also study $(WZS)_4$ spacetimes with divergence free conformal curvature tensor. Moreover, we characterize dust fluid and viscous fluid $(WZS)_4$ spacetimes. Finally, we construct an example of a $(WZS)_4$ spacetime.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.287-292
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• 2004

The operator $A \; {\in} \; L(H_{i})$, the Banach algebra of bounded linear operators on the complex infinite dimensional Hilbert space $\cal H_{i}$, is said to be p-hyponormal if $(A^\ast A)^P \geq (AA^\ast)^p$ for $p\; \in \; (0,1]$. Let (equation omitted) denote the completion of (equation omitted) with respect to some crossnorm. Let $I_{i}$ be the identity operator on $H_{i}$. Letting (equation omitted), where each $A_{i}$ is p-hyponormal, it is proved that the commuting n-tuple T = ($T_1$,..., $T_{n}$) satisfies Bishop's condition ($\beta$) and that if T is Weyl then there exists a non-singular commuting n-tuple S such that T = S + F for some n-tuple F of compact operators.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.679-710
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• 2021

We study the tensor product algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m), where P_k(x) is the partition algebra defined by Jones and Martin. We discuss the centralizer of this algebra and corresponding Schur-Weyl dualities and also index the inequivalent irreducible representations of the algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m) and compute their dimensions in the semisimple case. In addition, we describe the Bratteli diagrams and branching rules. Along with that, we have also constructed the RS correspondence for the tensor product of m-partition algebras which gives the bijection between the set of tensor product of m-partition diagram of P_k(n₁) ⊗ P_k(n₂) ⊗ ⋯ ⊗ P_k(n_m) and the pairs of m-vacillating tableaux of shape [λ] ∈ Γ_k^m, Γ_k^m = {[λ] = (λ₁, λ₂, …, λ_m)|λ_i ∈ Γ_k, i ∈ {1, 2, …, m}} where Γ_k = {λ_i ⊢ t|0 ≤ t ≤ k}. Also, we provide proof of the identity $(n_1n_2{\cdots}n_m)^k={\sum}_{[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ f^[λ]m_k^[λ] where m_k^[λ] is the multiplicity of the irreducible representation of $S{_{n_1}}{\times}S{_{n_2}}{\times}....{\times}S{_{n_m}}$ module indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$, where f^[λ] is the degree of the corresponding representation indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ and ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}=\{[{\lambda}]=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_m){\mid}{\lambda}_i{\in}{\Lambda}^k_{n_i},i{\in}\{1,2,{\ldots},m\}\}$ where ${\Lambda}^k_{n_i}=\{{\mu}=({\mu}_1,{\mu}_2,{\ldots},{\mu}_t){\vdash}n_i{\mid}n_i-{\mu}_1{\leq}k\}$.