三角形與其外接錐線的生成錐線性質探討
本研究源自三角形的重心及其外接圓所構作的線段比值的古老幾何性質,我們不但推廣原命題,還創造新命題:給定△ABC與其外接錐線Γ,令直線AG, BG, CG分別交Γ 於 A', B', C' 點,再取任意k值,探討P點集合的性質。 Γ3, k={P|AA'/PA' + BB'/PB' +CC'/PC'=k} (1)Γ3,k為二次曲線系,其橢圓、拋物線、雙曲線之形態不因k值而改變,而是被外接錐線Γ所決定。 (2)發現△ABC重心 G、Γ中心O、Γ3,k 中心O3,k 的共線性及比例常數。 (3)完整劃分 Γ3,k的非退化與退化型態,並發現只有Γ3,k 為橢圓時,k 值有跳躍現象。 (4) 發現錐線Γ上取相異六點而生成兩個錐線Γ3,k、Π3,k重合的充分條件。 最後,我們以「錐線 Γ 上取一點、兩點到多點」的線性組合手法,推廣多邊形與其外接錐線的生成錐線Γn,k之性質。
Green Wastewater Treatment: Using Graphene Oxide produced from Date Pits to Degrade Organic Dyes via Novel Microwave Technique
Water treatment is considered as one of the top research priorities in Saudi Arabia. It has been reported by World Health Organization that, 50,000 people die every day from diseases caused by contaminated water. This research attempts to degrade organic pollutants present in wastewater by using Graphene Oxide synthesized from Saudi natural source. Physical activation of date pits was carried out by carbonizing the samples at different temperatures to produce active carbon. Hammer’s method was employed for the purpose of Graphene Oxide production. The resulted Graphene Oxide has been characterized using FT-Raman, XRD and SEM techniques. Methylene Blue (MB) dye was used as a model organic pollutants to examine the ability of Graphene with the aid of a microwave-system to remove such pollutants. A modified domestic MS furnace with a variable power was used to supply microwave energy. The MB solution 2.5x10-6M was mixed with 0.1gm of Graphene Oxide. The applied microwave power was ranged between 100 to 700 W and the time was set between 0 to 12 minutes. The samples were centrifuged and then filtered through a millipore filter to remove the Graphene Oxide dispersed particles. It is found that, 98% of the initial concentration of MB is removed effectively within 12 minutes under microwave power of 500 W. Chemical oxygen demand is shifted from 450 to 87 mg/L while biological oxygen demand was decreased from 270 to 12.8 mg/L which indicating the degradation of organic constituents. This method can be used for water purification from organic pollutants.