We give a simple proof of the Chen inequality for the Chen invariant $\underbrace{\delta (2,...,2)}_{k\mbox{ terms}}$ of submanifolds in Riemannian space forms.

The Chen circuit system, a three-dimensional autonomous system with intriguing dynamics, has gained considerable attention due to its potential applications in diverse scientific fields. In this article, a Chen circuit-like system is defined and a comprehensive dynamics has been studied. The system exhibits chaotic dynamics, limit cycles, and bifurcations, making it a captivating subject of study. By employing numerical simulations and analytical techniques, we explore the system's stability and identify critical parameter values leading to qualitative changes. Notably, we delve Hopf bifurcations, which give rise to stability changes and the emergence of limit cycles. Furthermore, we analyze the fractal dimension of the system's attractor, providing insights into its complexity and self-similarity. Through a systematic examination of the Chen circuit-like system, we deepen our understanding of its intricate dynamics and offer valuable insights into the underlying mechanisms. The findings contribute to the field of dynamical systems and hold potential implications in areas such as chaos-based secure communications, signal processing, and nonlinear control. This work serves as a valuable reference for researchers and practitioners interested in the dynamics and bifurcation analysis of nonlinear systems.

Objective: To use the ancient and modern medical record cloud platform (V2.3.5) to organize and explore the medication patterns of the "Three Causes, Extremity, and One Disease Syndrome Formula" by Chen Wuxuan, and to summarize its therapeutic ideas. Method: Collect and organize the "San Yin Si Tian Fang" in the "San Yin Ji Yi Zheng Fang Lun", and use the modern medical record cloud platform (V2.3.5) to conduct medication frequency statistical analysis, cluster analysis, and complex network analysis. As a result, a total of 16 Chinese herbal medicines were collected in the "Three Causes, Extreme and One Disease Syndrome Formula", involving 75 traditional Chinese medicines. Statistical analysis showed that roasted licorice was the most commonly used, followed by Pinellia ternata, papaya, ginger, and Poria cocos; Medications mainly include warm drugs, sweet and pungent flavors, as well as drugs that promote lung and spleen meridians. Cluster analysis can divide the above-mentioned traditional Chinese medicine into two groups, with specific grouping as follows. Group 1: ginger, white peony, aconite, papaya, and achyranthes bidentata; Group 2: roasted licorice, poria cocos, dried ginger, pinellia ternata, and green peel. Conclusion: The characteristics of the 16 prescriptions are biased towards Xin Gan Wen, emphasizing the regulation of the spleen and stomach, which may be related to the common use of prescription by doctors in the Song Dynasty and the tendency towards warm and dry medication.

In this paper, we will use the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength ℜ=P[X>Y] based on independent adaptive progressive censored samples that were taken from the Chen distributions. An approximate confidence interval of ℜ is constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. The Bayesian estimation of ℜ is obtained based on the balanced loss function, which comes in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating ℜ using the Bayesian approach, all the unknown parameters of the Chen distributions are assumed to be independently distributed and to have informative gamma priors. Further, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm is used to compute the Bayes estimate of ℜ and the associated highest posterior density credible interval. In the end, simulation research are used to assess the general overall performance of the proposed estimators and a real data set is provided to exemplify the theoretical results.