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Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pečarić and Mićić.Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and difference type, under some conditions. Also, we provide interesting inequalities for von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recovers the nonnegativity and gives a refinement for the weaker version of Fannes's inequality for only special cases.Finally, we estimate bounds for the Tsallis relative operator entropy.

In this paper, we propose the set-membership quaternion normalized least-mean-square (SM-QNLMS) algorithm. For this purpose, first, we review the quaternion least-mean-square (QLMS) algorithm, then go into the quaternion normalized least-mean-square (QNLMS) algorithm. By having the QNLMS algorithm, we propose the SM-QNLMS algorithm in order to reduce the update rate of the QNLMS algorithm and avoid updating the system parameters when there is not enough innovation in upcoming data. Moreover, the SM-QNLMS algorithm, thanks to the time-varying step-size, has higher convergence rate as compared to the QNLMS algorithm. Finally, the proposed algorithm is utilized in wind profile prediction and quaternionic adaptive beamforming. The simulation results demonstrate that the SM-QNLMS algorithm outperforms the QNLMS algorithm and it has higher convergence speed and lower update rate.

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In this paper, the set-membership affine projection (SM-AP) algorithm is utilized to censor non-informative data in big data applications. To this end, the probability distribution of the additive noise signal and the excess of the mean-squared error (EMSE) in steady-state are employed in order to estimate the threshold parameter of the single threshold SM-AP (ST-SM-AP) algorithm aiming at attaining the desired update rate. Furthermore, by defining an acceptable range for the error signal, the double threshold SM-AP (DT-SM-AP) algorithm is proposed to detect very large errors due to the irrelevant data such as outliers. The DT-SM-AP algorithm can censor non-informative and irrelevant data in big data applications, and it can improve the misalignment and convergence rate of the learning process with high computational efficiency. The simulation and numerical results corroborate the superiority of the proposed algorithms over traditional algorithms.