The Maximum Area of N-gons within the Intersection Region of Two Congruent Circles
At the 61st National High School Science Fair of Taiwan, the first-rank paper "The Study of the Largest Area of Inscribed Triangle within the Intersection of two circles" was presented. The authors discussed several properties of maximum area of inscribed triangles within intersection regions of two congruent circles. They only claim their results but without providing a rigorous proof. However, we give a proof by showing the convergence of the iteration of finding the largest height. Subsequently, we offer new methods to approach the problems such as the trigonometric identities, Jensen's Inequality to prove the maximum area of triangles and quadrangles within the intersection region of two congruent circles. Finally, we determined the maximum area for the case of n-gons. We conducted further research and discussion on this issue. In the future, we hope to prove why the maximum area of n-gons within the intersection region of two congruent circles occurs when there are two points on the intersection points of the two circles. We aim similar problems in the three-dimensional space, namely the maximum volume of tetrahedron within the intersection of two unit spheres.
橡實代謝物與飛鼠小腸內生菌共發酵產物之應用
This study primarily investigates the components of food residues in the stomach of flying squirrels and the metabolism of the intestinal bacteria Floricoccus tropicus. Using nuclear magnetic resonance (NMR) hydrogen spectrum analysis, the signals of long-chain fatty acids were detected in both n-hexane and 75% ethanol extracts of flying squirrel gastric residues and acorns, suggesting that acorns may be a primary food source for the flying squirrels. The study also identified Floricoccus tropicus, one of the lactic acid bacteria, from the intestines of flying squirrels and discovered its role in the metabolism of fatty acids in acorns. Results showed that polyunsaturated fatty acids significantly decreased during fermentation, indicating that they were converted into short-chain fatty acids with anti-inflammatory properties. In the antibacterial activity experiments, the acorn grease showed no inhibitory effects before fermentation, but after fermentation, the acorn grease exhibited inhibitory effects against E. coli. Furthermore, in anti-inflammatory tests, fermented acorn grease samples significantly suppressed the production of NO and TNF-α in LPSinduced RAW 264.7 cells, with greater inhibition at higher concentrations. In conclusion, the lactic acid bacteria Floricoccus tropicus was found to metabolize fatty acids of acorns into compounds with antibacterial and anti-inflammatory effects.