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.
Synthesis of Nanocomposite Nanocellulose From Durio zibethinus L. and TiO2 NPs as Potential Food Packaging Antibacterial (E. coli Wild Type and Resistance)
According to the 印尼n Association of Olefin Aromatic and Plastic Industries/INAPLAS, 2019 national plastic consumption still relies on plastic packaging at 65% and surprisingly, around 60% of plastic waste is absorbed by the food and beverage industry. The waste has been widely sought to be environmentally friendly, one of which is by developing biodegradable packaging. The purpose of this research is to make durian peel cellulose nanocomposites impregnated with TiO2 NPs, to form antibacterial properties against E. coli wild type and resistance. In this research, there are research methods consisting of nanocomposite synthesis, PSA test, FTIR, physical characteristics test and resistance test. The results analyzed that the nanocomposite nanocellulose-TiO2 NPs was successfully made using a 1:1 ratio and had a particle size of 458.7 nm based on the PSA test, which is classified as a nano size. The success of nanocomposite synthesis was proven by the results of FTIR analysis, which showed the formation of 698.65cm-1 and 1633.99cm-1 spectra, indicating the peak of TiO2 NPs and O-H functional groups on TiO2 NPs, as well as 1028.98cm-1 and 1158.42cm-1 showing C-O and C-O-C bonds in cellulose. The antibacterial test performed showed no significant activity in disc diffusion and well diffusion tests against E. coli wild type and resistance. This is potentially caused by inhomogeneous particle size variation. Physical characteristics test showed that the tensile strength test (0.075 > 0.0125 MPa) Durio Nano-Pack is superior to styrofoam, but the compressive strength test (0.125 > 0.875 MPa) shows the opposite. In this study, nanocomposite has a potential innovation that provides good mechanical properties and has a dual function mechanically as bio-based food packaging and chemically as antibacterial. Further research is needed to improve the particle size homogeneity of nanocomposites, modify the impregnation method, so that it has the potential to develop multifunctional materials that excel in various applications.
Proposal for the Restoration of Fire-Damaged Soil Using Water-Soluble Aromatic Compounds Derived from Soil Actinomycetes
The following issues associated with soil affected by wildfires were identified: First, there was a significant decline in the populations of anaerobic and aerobic soil bacteria, which play a critical role in the decomposition and cycling of organic matter. This decline resulted in reduced water retention capacity and porosity of the soil, leading to poor moisture retention and increased evaporation compared to unaffected soil. Moreover, the organic matter content in the soil was significantly depleted, inhibiting plant growth. Additionally, there was a notable proliferation of methane-producing bacteria, which contribute to the greenhouse effect. It was further observed that fire-damaged soils exhibit limited natural recovery, even over prolonged periods. .An investigation into the underlying causes of these problems revealed that actinomycetes, the primary microorganisms responsible for producing watersoluble aromatic compounds in soil, are particularly sensitive to heat compared to other bacterial species. Research demonstrated that the population and diversity of actinomycetes are significantly diminished in soils exposed to wildfires. To mitigate these issues, water-soluble aromatic compounds produced by actinomycetes were extracted and introduced into wildfire-affected soil. This intervention promoted the restoration of actinomycetes populations, enabling their normal growth in the affected soil. Consequently, various wildfire-induced soil problems were effectively resolved. These outcomes were confirmed through the study...Key Words: Actinomycetes, anaerobic and aerobic soil bacteria, methane-producing bacteria, wildfires, water-soluble aromatic compounds.
Utilizing Sparse Optimal Linear Feedback Control to Design Targeted Therapeutic Strategies for Enhancing Gut Microbiome Stability
According to the 2024 American Cancer Risk Survey, one in 24 individuals is at high risk of developing colon cancer. This condition is linked to gut microbiome instability. Consequently, there is a pressing need for a more effective and precise approach to maintaining gut microbiome stability, which this research aims to solve by finding the most crucial bacteria species in maintaining the stability of the gut microbiome through the application of Optimal Linear Feedback Control. Two of its variants being applied in this research are Sparsity Promoting Linear Quadratic Regulator (LQRSP) with a variety range of (0.05, 44.58, and 49.84) and Linear Quadratic Regulator (LQR) ( = 0) along with other supporting methods; Controllability Gramian and Network Theory (graph analysis). The finding in this research shows that bacteria species Bacteroides hydrogenotrophica, Bacteroides uniformis, Bacteroides vulgaris, Bacteroides thetaiotaomicron, Escherichia lenta, and Dorea formicigenerans have an important role for preventing and medicating a variety of gut-related diseases. This conclusion is reinforced by the analysis conducted using the Controllability Gramian, displaying five of the chosen bacteria with the highest controllability index, which demonstrates that the system can be effectively controlled. This finding suggests a potential for enhancing therapeutic strategies, rendering them more precise and systematic. To gain deeper insights into the relationship between each bacteria and the rationale behind the selection of these bacteria by LQRSP, this study also employs network theory, which successfully elucidates the choice of Bacteroides uniformis despite its low controllability index. Additionally, to further validate the efficacy of these bacteria, the research develops a simulation that compares the controlled system with the uncontrolled system, utilizing two types of disturbances. The results indicate a significant difference in robustness against disturbances between the controlled and uncontrolled systems. The findings from this research can be used as a foundation for a more efficient and systematic intervention strategy findings. By researching gut microbiome composition regulation using a mathematical approach, it opens new opportunities for new method discoveries aiming to increase the health of the gut microbiome which is beneficial for the medical field and prevention of gut related diseases.