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The multi-step inertial randomized Kaczmarz (MIRK) method is an iterative method for solving large-scale linear systems. In this paper, we enhance the MIRK method by incorporating the greedy probability criterion, coupled with the introduction of a tighter threshold parameter for this criterion. We prove that the proposed greedy MIRK (GMIRK) method enjoys an improved deterministic linear convergence compared to both the MIRK method and the greedy randomized Kaczmarz method. Furthermore, we exhibit that the multi-step inertial extrapolation approach can be geometrically interpreted as an orthogonal projection method, and establish its relationship with the sketch-and-project method in Gower and Richtárik (SIAM J Matrix Anal Appl 36(4):1660–1690, 2015) and the oblique projection technique in Li et al. (Results Appl. Math. 16:100342, 2022). Numerical experiments are provided to confirm our results. [ABSTRACT FROM AUTHOR]

Prostate cancer (PC) is the most common malignancy in male reproductive system. Sennoside A (SA) is an anthraquinone active ingredient extracted from Rheum officinale Baill., which exerts anti-tumor activity on different tumors. In the present study, the toxicity of SA on PC3 and DU 145 cells was detected via CCK-8. The effects of SA on growth, apoptosis, and autophagy were determined through CCK-8, Hoechst stain, flow cytometry, western blot, and immunofluorescence examinations. An in vivo experiment was performed in xenografted mice with intraperitoneal introduction of 10 mg/kg SA and validated via TUNEL, immunohistochemistry and western blot. The results showed that SA inhibited the cell viability with a IC50 value of 52.36 and 67.48 µM in DU 145 and PC3 cells respectively, and enhanced the apoptosis of PC3 and DU 145 cells. Additionally, SA elevated the relative LC3B expression, and the relative protein expression of LC3II/LC3I and Beclin-1, but diminished the P62 protein expression. The relative protein level of p-PI3K/PI3K, p-AKT/AKT and p-mTOR/mTOR was reduced with SA treatment, which was verified by the 740 Y-P application. The 740 Y-P treatments also restored the SA-induced the cell viability, apoptosis rate and relative LC3B expression. Meanwhile, SA inhibited the growth of PC cell and the relative protein level of PI3K/AKT/mTOR axis in vivo. Taken together, SA regulated the proliferation, apoptosis and autophagy via inactivating the PI3K/AKT/mTOR axis in PC. [ABSTRACT FROM AUTHOR]

We investigate the randomized Kaczmarz method that adaptively updates the stepsize using readily available information for solving inconsistent linear systems. A novel geometric interpretation is provided which shows that the proposed method can be viewed as an orthogonal projection method in some sense. We prove that this method converges linearly in expectation to the unique minimum Euclidean norm least-squares solution of the linear system, and provide a tight upper bound for the convergence of the proposed method. Numerical experiments are also given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]