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In this paper, we present two S-type Z-eigenvalue inclusion sets involved with a nonempty proper subset S of N for general tensors. It is shown that the new sets are tighter than those provided by Wang et al. (Discrete Contin. Dyn. Syst., Ser. B 22(1):187-198, 2017). Furthermore, we obtain upper bounds for the spectral radius of weakly symmetric nonnegative tensors, which are sharper than existing results.

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The effects of substitution of micron size powder by nanometer size powder in magnetorheological (MR) fluids are investigated in this study. Three MR fluid samples containing iron powder with 45% weight fraction in a carrier fluid were made by Materials Modification Inc. The difference among these three fluids is size of the magnetic particles. The first MR fluid sample contained only micron size iron powder with lO^m particle size. In the second sample, 5% micron iron was substituted with nano powders having 30~40nm mean diameter, while the third sample had 37.5% micron powder and 7.5% nano powder. Rheological tests were conducted on the three samples using a parallel disk rheometer. Highest yield stress was observed in the second MR fluid sample containing 40% micron and 5% nano iron powder. By replacing only 5% micron iron powder with nanoparticles, we achieved substantial increment in yield stress. However, when nano powder content is increased to 7.5%, the yield stress decreases and is lower than that in the all micron MR fluid. Thus, by doping a reasonable percent of nano iron powder in the MR fluid, a substantial change in the rheological characteristics is obtainable. Further investigations of effects of nano iron powder in MR fluids for higher weight fraction MR fluids will be carried out in future.

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