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In this paper, we consider the estimation algorithm of tensor response on vector covariate regression model. Based on projection theory of tensor and the idea of randomized algorithm for tensor decomposition, three new algorithms named SHOLRR, RHOLRR and RSHOLRR are proposed under the low‐rank Tucker decomposition and some theoretical analyses for two randomized algorithms are also provided. To explore the nonlinear relationship between tensor response and vector covariate, we develop the KRSHOLRR algorithm based on kernel trick and RSHOLRR algorithm. Our proposed algorithms can not only guarantee high estimation accuracy but also have the advantage of fast computing speed, especially for higher‐order tensor response. Through extensive synthesized data analyses and applications to two real datasets, we demonstrate the outperformance of our proposed algorithms over the stat‐of‐art. [ABSTRACT FROM AUTHOR]

Inspired by the definition of tensor-tensor product and tensor tubal rank, a randomized singular value decomposition of tensor is presented in this paper. Based on tensor singular value decomposition (t-SVD) and tensor randomized singular value decomposition (t-RSVD), we obtain two efficient algorithms to solve tensor completion problem. We also propose the adaptive rank method to adjust the tubal rank of tensor. The main advantage of the random projection-based t-RSVD is to cut down the computing time in consideration of large-scale problems. In the optimization process, the alternating minimization algorithm is employed to solve the tensor completion problem. Finally, numerical experiment results indicate that the t-RSVD is competitive and consumes less time than the t-SVD. The efficiency and feasibility of our methods are illustrated by the image and video recovery. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of tensor equations. Some explicit solutions to these equations are obtained. They are extensions of some results given by Lancaster. At the end of the article, these results are also employed to prove Lyapunov and Stein stability theorems for tensors. [ABSTRACT FROM AUTHOR]

Mercury (Hg 2+ ) contamination in water is associated with potential toxicity to human health and ecosystems. Many research studies have been ongoing to develop new materials for the remediation of Hg 2+ pollution in water. In this study, a novel thiol- and amino-containing fibrous adsorbent was prepared by grafting 2-amino-5-mercapto-1,3,4-thiadiazol (AMTD) onto PAN fiber through a microwave-assisted method. The synthesized functional fiber was characterized by FTIR, SEM, and elemental analysis. Adsorption tests depicted that for mercury uptake, PAN MW -AMTD fiber exhibited enhanced adsorption capacity compared with other fibrous adsorbents and selective adsorption feature under the interference of other metal ions, including Pb 2+ , Cu 2+ , Cd 2+ , and Zn 2+ . The influence of pH on the adsorption process was investigated and the effect of temperature revealed that the adsorption sorption process was endothermic and the adsorption performance of PAN MW -AMTD was elevated with the increase of temperature. Kinetic studies of PAN MW -AMTD fiber followed the pseudo-second-order and the adsorption isotherm of Hg 2+ was well fitted by Sips and Langmuir equations, given the maximum adsorption amount of 332.9 mg/g. XPS results suggested that a synergetic coordination effect of sulfur and nitrogen in functional fiber with mercury took responsibility for the adsorption mechanism in the uptake process. In addition, the prepared PAN MW -AMTD fiber could easily be regenerated with 0.1 M HCl for five times without significant reduction of mercury removal efficiency. Thus, this study will facilitate the research on novel functional material for the removal of mercury from water., (Copyright © 2021 Elsevier Ltd. All rights reserved.)