• Journal of KIISE
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• v.45 no.3
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• pp.280-287
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• 2018

Recently, the use of a recommendation system and tensor data analysis, which has high-dimensional data, is increasing, as they allow us to analyze the tensor and extract potential elements and patterns. However, due to the large size and complexity of the tensor, it needs to be decomposed in order to analyze the tensor data. While several tools are used for tensor decomposition such as rTensor, pyTensor, and MATLAB, since such tools run on a single machine, they are unable to handle large data. Also, while distributed tensor decomposition tools based on Hadoop can handle a scalable tensor, its computing speed is too slow. In this paper, we propose S-PARAFAC, which is a tensor decomposition tool based on Apache Spark, in distributed in-memory environments. We converted the PARAFAC algorithm into an Apache Spark version that enables rapid processing of tensor data. We also compared the performance of the Hadoop based tensor tool and S-PARAFAC. The result showed that S-PARAFAC is approximately 4~25 times faster than the Hadoop based tensor tool.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.163-176
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• 2016

The object of the present paper is to introduce a new curvature tensor, named generalized quasi-conformal curvature tensor which bridges conformal curvature tensor, concircular curvature tensor, projective curvature tensor and conharmonic curvature tensor. Flatness and symmetric properties of generalized quasi-conformal curvature tensor are studied in the frame of (k, ${\mu}$)-contact metric manifolds.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.255-277
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• 2024

In this paper, we introduce the notion of stress-energy tensor Q of the traceless Ricci tensor for Riemannian manifolds (Mⁿ, g), and investigate harmonicity of Riemannian curvature tensor and Weyl curvature tensor when (M, g) satisfies some geometric structure such as critical point equation or vacuum static equation for smooth functions.

• Journal of Korea Multimedia Society
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• v.22 no.4
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• pp.443-454
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• 2019

In recent years, there has been active research on a recommender system that considers three or more inputs in addition to users and goods, making it a multi-dimensional array, also known as a tensor. The main issue with using tensor is that there are a lot of missing values, making it sparse. In order to solve this, the tensor can be shrunk using the tensor decomposition algorithm into a lower dimensional array called a factor matrix. Then, the tensor is reconstructed by calculating factor matrices to fill original empty cells with predicted values. This is called tensor reconstruction. In this paper, we propose a user-based Top-K recommender system by normalized PARAFAC tensor reconstruction. This method involves factorization of a tensor into factor matrices and reconstructs the tensor again. Before decomposition, the original tensor is normalized based on each dimension to reduce overfitting. Using the real world dataset, this paper shows the processing of a large amount of data and implements a recommender system based on Apache Spark. In addition, this study has confirmed that the recommender performance is improved through normalization of the tensor.

• Honam Mathematical Journal
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• v.36 no.4
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• pp.795-803
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• 2014

In this paper, we decompose the curvature tensor (field) on the Abbena-Thurston manifold into three curvature-like tensor fields, and investigate geometric properties.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.185-195
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• 2020

This paper aims to study the 𝒲^⁎-curvature tensor on relativistic space-times. The energy-momentum tensor T of a space-time having a semi-symmetric 𝒲^⋆-curvature tensor is semi-symmetric, whereas the whereas the energy-momentum tensor T of a space-time having a divergence free 𝒲^⋆-curvature tensor is of Codazzi type. A space-time having a traceless 𝒲^⁎-curvature tensor is Einstein. A 𝒲^⁎-curvature flat space-time is Einstein. Perfect fluid space-times which admits 𝒲^⋆-curvature tensor are considered.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.149-161
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• 2019

In the present paper, we study curvature properties of ${\eta}$-Ricci solitons on para-Kenmotsu manifolds. We obtain some results of ${\eta}$-Ricci solitons on para-Kenmotsu manifolds satisfying $R({\xi},X).C=0$, $R({\xi},X).{\tilde{M}}=0$, $R({\xi},X).P=0$, $R({\xi},X).{\tilde{C}}=0$ and $R({\xi},X).H=0$, where $C,\;{\tilde{M}},\;P,\;{\tilde{C}}$ and H are a quasi-conformal curvature tensor, a M-projective curvature tensor, a pseudo-projective curvature tensor, and a concircular curvature tensor and conharmonic curvature tensor, respectively.

• Journal of KIISE
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• v.43 no.8
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• pp.910-918
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• 2016

A tensor is a multi-dimensional array that represents many data such as (user, user, time) in the social network system. A tensor generator is an important tool for multi-dimensional data mining research with various applications including simulation, multi-dimensional data modeling/understanding, and sampling/extrapolation. However, existing tensor generators cannot generate sparse tensors like real-world tensors that obey power law. In addition, they have limitations such as tensor sizes that can be processed and additional time required to upload generated tensor to distributed systems for further analysis. In this study, we propose TeT, a distributed tera-scale tensor generator to solve these problems. TeT generates sparse random tensor as well as sparse R-MAT and Kronecker tensor without any limitation on tensor sizes. In addition, a TeT-generated tensor is immediately ready for further tensor analysis on the same distributed system. The careful design of TeT facilitates nearly linear scalability on the number of machines.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.367-376
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• 2008

The main purpose of this paper is to investigate diagonal lift of tensor fields of type (1,1) from manifold to its tensor bundle of type (p, q) and to prove that when a manifold $M_n$ admits a $K\ddot{a}hlerian$ structure ($\varphi$,g), its tensor bundle of type (p,q) admits an complex structure.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.81-99
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• 2006

We investigate a curvature-like tensor defined by (3.1) in Sasakian manifold of $dimension{\geq}$ 5, and show that this tensor satisfies some properties. Especially, we determine compact Sasakian manifolds with vanishing this tensor and improve some theorems concerning contact conformal curvature tensor and spectrum of Laplacian acting on $p(0{\leq}P{\leq}2)-forms$ on the manifold by using this tensor component.