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The main purpose of this paper is to analyze the mean-field stochastic differential equations driven by $ G $-Brownian motion (mean-field $ G $-SDEs). We first investigate the existence and uniqueness of the solution for the mean-field $ G $-SDEs by utilizing the Banach fixed point theorem. Moreover, by the method of coupling by change of measures, the Harnack and log-Harnack inequalities for mean-field $ G $-SDEs with additive noise are established. [ABSTRACT FROM AUTHOR]

Sufficient and necessary conditions are presented for the comparison theorem of path dependent G-SDEs. Different from the corresponding study in path independent G-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case. [ABSTRACT FROM AUTHOR]

A one‐pot approach for the asymmetric synthesis of C2‐tetrasubstituted indolin‐3‐ones from 2‐substituted indoles was developed via merging transition metal catalysis with organocatalysis. This strategy involves two processes, including CuI catalyzed oxidative dearomatization of 2‐substituted indoles using O2 as green oxidant, and followed by an proline‐promoted asymmetric Mannich reaction with ketones or aldehydes. A series of C2‐tetrasubstituted indolin‐3‐ones were obtained in 35–86% yields, 2:1‐>20:1 dr and 48–99% ee. Moreover, the synthetic 2‐tetrasubstituted indolin‐3‐ones could be easily transformed into 1H‐[1,3] oxazino [3,4‐a]indol‐5(3H)‐ones via a [4+1] cyclization process. In addition, the synthetic compound 3 s show certain antibacterial activity against S. aureus ATCC25923 and multi‐drug resistance bacterial strain of S. aureus (20151027077) and its MIC values up to 8 μg/mL and 16 μg/mL, respectively. [ABSTRACT FROM AUTHOR]

A [4+3] annulation of 2‐ylideneoxindoles with crotonate‐derived sulfur ylides has been developed. A series of oxepino[3,2‐b]indoles were prepared in moderate to excellent yields (62‐93%) under mild conditions. Moreover, the synthetic oxepino[3,2‐b] indoles can be further transformed into more complex cyclopropa[5,6]oxepino[3,2‐b]indoles via a [2+1] cyclopropanation. In addition, the synthetic compounds show certain antiproliferative activity against K562 and MCF‐7 cells, and its IC50 values for these two kinds of tumor cells up to 5.40±0.88 μM and 18.41±0.50 μM, respectively. [ABSTRACT FROM AUTHOR]

In this paper, Wang's Harnack and shift Harnack inequality for a class of stochastic differential equations driven by G-Brownian motion are established. The results generalize the ones in the linear expectation setting. Moreover, some applications are also given. [ABSTRACT FROM AUTHOR]

The first KI-catalyzed direct acyloxylation of indolin-3-ones with carboxylic acids has been developed using 30% aq. H2O2 as a green oxidant at room temperature. Through this strategy, various C2-acyloxy indolin-3-ones were obtained in up to 96% yield. Moreover, the C2-acyloxy indolin-3-ones can serve as versatile intermediates for the synthesis of nucleophilic 2-monoarylated indolin-3-ones and 3-ylidene indoles. [ABSTRACT FROM AUTHOR]