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A class of half semilinear elliptic equations was researched on the Heisenberg group, - Δu - μu/∣x∣2 = f(x,u), where 0 ⩽ μ < (N-2/2)2. With Sobolev-type inequality, Young-type inequality and Hölder-type inequality, a solvbilivity theorem of the equations was got. [ABSTRACT FROM AUTHOR]

The article focuses on the process to obtain the maximum principle of convex functions on non-isotropic Heisenberg group through the creation of Harnack-type inequality for convex functions with iterative method on the group.

Background: Hepatic artery occlusion (HAO) after liver transplantation (LT) is a devastating complication, resulting in early graft loss and reduced overall survival. Ultrasound is an established assessment method for HAO in patients following LT, especially those with complex hepatic artery reconstruction., Aim: To investigate the ultrasound characteristics and analyze the risk factors associated with HAO in patients after LT., Methods: We retrospectively analyzed the ultrasound characteristics and the clinic risk factors associated with HAO in 400 adult LT patients who were enrolled and treated at the Third People's Hospital of Shenzhen between November 2016 and July 2022. Fourteen patients diagnosed with acute HAO (A-HAO) by surgery and fifteen diagnosed with chronic HAO (C-HAO) were included. A control group of 33 patients without HAO complications during the same period were randomly selected using a random number table. All patients underwent an ultrasonography examination. Parameters including resistance index (RI), peak systolic velocity (PSV), and portal vein velocity (PVV) were compared across the groups. Additionally, basic clinical data were collected for all patients, including gender, age, primary diagnosis, D-dimer concentration, total operation time, cold ischemia time, hot ischemia time, intraoperative blood loss and transfusion, intraoperative urine volume, infusion, model for end-stage liver disease (MELD) score, and whether complex hepatic artery reconstructions were performed. Furthermore, risk factors influencing HAO formation after LT were analyzed., Results: Compared to the non-HAO group, PVV and RI were higher in the A-HAO group, while PSV was lower. Conversely, both PSV and RI were lower in the C-HAO group compared to the non-HAO group. The proportion of patients undergoing complex hepatic artery reconstructions and the gamma-glutamyltransferase (GGT) level before occlusion were significantly higher in the A-HAO group compared to the non-HAO group. However, there were no distinct differences between the two groups in D-dimer, MELD score, pre-occlusion alanine transaminase and aspartate transaminase levels, or intraoperative conditions., Conclusion: Ultrasound features of the hepatic artery before occlusion are significantly associated with postoperative HAO development. Additionally, complex hepatic artery reconstructions, defined as revascularization of the graft requiring additional anastomosis between donor hepatic arteries, constitute a risk factor for A-HAO. Besides, abnormal pre-occlusion GGT elevation is an important biochemical indicator. Therefore, ultrasound examination serves as an important tool for screening HAO, especially in patients with the identified risk factors., Competing Interests: Conflict-of-interest statement: The authors declare no conflict of interest for this article., (©The Author(s) 2024. Published by Baishideng Publishing Group Inc. All rights reserved.)

Background: The PER1 circadian-clock gene plays an important role in the regulation of many normal physiological rhythms in vivo. It has been revealed recently that abnormal expression of PER1 correlates closely with the occurrence and development of many cancers. However, the expression and significance of PER1 in oral squamous cell carcinoma (OSCC) remains unknown. The purpose of the present study was to investigate the direct links between aberrant PER1 expression and clinicopathological features of OSCC., Methods: PER1 expression in cancerous and adjacent noncancerous tissues from 41 patients with OSCC was detected by immunohistochemical staining and real-time reverse transcriptase polymerase chain reaction, and correlations were sought with clinicopathological features in patients., Results: Expression of PER1 mRNA and protein in OSCC was significantly reduced compared with that in adjacent noncancerous tissue (P < 0.05). Expression of PER1 protein in oral phase III-IV SCC specimens was significantly lower than that in phase I-II specimens (P < 0.05), and stage T(1)-T(2) patients expressed significantly higher levels of PER1 protein than T(3)-T(4) patients (P < 0.05). Expression of PER1 in patients without lymph node metastasis was significantly higher than that in those with lymph node metastasis (P < 0.05). PER1 protein expression showed no significant correlation with patient gender and age, or with degree of tumor cell differentiation (P > 0.05)., Conclusion: Changes in PER1 expression may play an important role in the development, invasion, and metastasis of OSCC, and may also provide novel ideas and methods for investigation of the occurrence, development, and targeted treatment OSCC.