Li-Yau Inequality and Liouville Property to a Semilinear Heat Equation on Riemannian Manifolds

This work deals with the Entire solutions of a nonlinear equation. The first part of this paper is devoted to investigation of the Liouville property on compact manifolds, which extends a result by Castorina-Mantegazza [4] for positive f. Secondly, we will turn to non-compact manifolds and prove a Liouville theorem under the assumptions of boundedness of the Ricci curvature from below, diffeomorphism of M with R^N and sub-criticality of p defined below. Finally, we also present simplified proofs of Yau's theorem for harmonic function and Gidas-Spruck's theorem for elliptic semilinear equation. Our proofs are based on Li-Yau type estimation for nonlinear equations

Black hole shadow, photon ring and lensing ring in the cold dark matter halo

The Schwarzschild black hole in the cold dark matter (CDM) halo is deeply studied, and the radiation laws of the thin accretion disk near the black hole is discussed and summarized. The light orbits around the black hole are calculated. According to the number of times that the light from the north pole passes through the equatorial plane of the black hole, the light orbits can be divided into three categories. At the same time, the innermost stable circular orbit of the black hole is also calculated. By analyzing the calculation results of radiation intensity of three kinds of radiation which emitted by thin accretion disk on the equatorial plane of black hole and observed at infinity away from the north pole of black hole, the radiation laws of the thin accretion disk of Schwarzschild black hole suitable for the background of the CDM halo is obtained.Comment: 21pages. arXiv admin note: text overlap with arXiv:2105.15073 by other author

Tamoxifen inhibits astrocytic JAK2/STAT3 pathway in rats with traumatic spinal injury

Purpose: To investigate the reducing effect of Morinda citrifolia capsules (MCC) on total cholesterol (TC) and low-density lipoprotein cholesterol (LDL-C) in hypercholesterolemia patients.Methods: This study was a randomized double-blind placebo-controlled clinical trial, with 60 subjects placed in two groups, viz, experimental (MCC group) and placebo (P group). The first group received two capsules of MCC (each capsule contains 500 mg extract) while the second group received two capsules of placebo (comprised of 500 mg fillers) 3 times daily, for 14 days for both groups. Overnight fasting cubiti venous blood (3 mL) was taken from each subject each time measurements were carried out. TC and LDL-C were measured by spectrophotometric assay using an automated analyzer.Results: The results show that there was significant decrease in TC and LDL-C on day 14, compared to control (P group). Reduction in TC and LDL-C was 13.8 and 15.5 %, respectively. Decrease in TC and LDL-C levels was influenced by factors, such as age, BMI, exercise, diet, and smoking habits. In MCC group, the capsules significantly decreased TC levels (p < 0.05).Conclusion: The results suggest that 1 g MCC, given orally thrice daily, significantly reduces TC and LDL-C levels in patients with hypercholesterolemia.Keywords: Hypercholesterolemia, Cholesterol levels, Morinda citrifoli

Hexaaqua­nickel(II) 4,4′-(1,2-dihy­droxy­ethane-1,2-di­yl)dibenzoate monohydrate

In the title compound, [Ni(H2O)6](C16H12O6)·H2O, the NiII cation is located on a mirror plane and is coordinated by six water mol­ecules, two of which are also located on the mirror plane, in a distorted octa­hedral geometry. The 4,4′-(1,2-dihy­droxy­ethane-1,2-di­yl)dibenzoate anion is centrosymmetric with the mid-point of the central ethane C—C bond located on an inversion center. The uncoordinated water mol­ecule is located on a mirror plane. Extensive O—H⋯O hydrogen bonding is present in the crystal structure

Limited medial osteochondral lesions of the talus associated with chronic ankle instability do not impact the results of endoscopic modified Broström ligament repair

Background The arthroscopic modified Brostrom procedure, with repair of the anterior talofibular ligament and extensor retinaculum, produces good functional outcomes in patients with chronic lateral ankle instability (CLAI). CLAI can be associated with osteochondral lesions of the talus (OLTs). It remains unclear whether associated limited OLTs affect clinical outcomes in such patients. Methods This retrospective cohort study included 92 CLAI patients with and without OLTs undergoing an all-inside arthroscopic modified Brostrom procedure from June 2016 to May 2019. The patients were divided into non-lesion group (n = 32) and lesion group (n = 60) according to whether CLAI was associated or not with OLTs. All the osteochondral lesions less than 15 mm in diameter were managed with bone marrow stimulation techniques (arthroscopic microfracture) at the time of the arthroscopic modified Brostrom procedure. The Visual Analogue Scale (VAS) scores, American Orthopedic Foot and Ankle Society (AOFAS) scores, Karlsson Ankle Function Score (KAFS), Anterior Talar Translation (ATT), Active Joint Position Sense (AJPS), and the rate of return to sports were compared in both groups. Results Increase in all the functional scores (VAS, AOFAS, KAFS, ATT, and AJPS) in both groups was, respectively, recorded 1 year and 2 years after surgery. At the 1-year and 2-year follow-up, there was no significant difference in the VAS, AOFAS, KAFS, ATT, and AJPS scores between the non-lesion and lesion groups. Conclusion In patients with CLAI who underwent an arthroscopic modified Brostrom procedure, the presence of limited OLTs (less than 15 mm in diameter), which required arthroscopic microfracture, did not exert any influence on outcome