費馬多邊形數定理之延伸探討
本研究旨在研究費馬多邊形數定理(任意非負整數必可表成k個k邊形數的和)的一般化情況，也就是說，任意非負整數是否能表成給定的二次多項式數列中所選取的γ項和。以數學模型敘述，就是探討對一個已知的二次多項式an2+bn+c，是否可找到一正整數γ，滿足∀x∈N∪{0},∃α1,α2,…αγ,使得x=∑γi=1(aαi2+bαi+c)。 本作品主要探討若此探究模型存在，那麼數列〈an 〉的一般式an2+bn+c與γ值之間會存在什麼關係，並期望能運用一個簡潔明瞭又一般化的數學式表示。本文亦提供另一個數學模型，探討γ值與某些特殊係數a,b,c之間的關聯性。而本文探尋[a/2]n2+[b/2]n+1,a∈N,b∈Z,a+b≡0(mod 2)(此為本文主要探討的二次式)，求得此二次式所對應之γ值的方法為先令p=[2a/(a+b)]+2，再藉由所建立的模型二，求出[(p-2)/2] n2+[(4-p)/2] n的γ值，接著再用所建立的模型一來求得[a/2]n2+[b/2]n+1的γ值，進而依循此方法最後得出任意形如[a/2]n2+[b/2]n+1的二次式之γ值。
A Novel Spectroscopic-Chemical Sensor Using Photonic Crystals
Detection of harmful chemicals used in industrial complexes is crucial in order to create a safer environment for the workers. Presently, most chemical detectors used in workplaces are expensive, inefficient, and cumbersome. In order to address these deficiencies, a novel sensor was fabricated to produce a unique spectroscopic fingerprint for various toxic chemicals. The sensor was fabricated by depositing several layers of silica spheres (diameter ~250 nm) on a glass substrate using evaporation-based self assembly. As the spheres assemble to form a photonic crystal, they also create void (i.e., air) spaces in between them. Once the spheres assemble as a photonic crystal, a spectrometer was used to monitor the reflectivity. The spectrum had a high reflectivity at a specific wavelength, which is governed by the average index of refraction between the spheres and the void spaces. As a foreign chemical infiltrates into the photonic crystal, it occupies the void space, which results in an increase of the average index of refraction of the structure. Consequently, the peak wavelength of the reflectivity spectrum red-shifts, which then confirms the presence of a foreign substance. While the as-grown photonic crystal is able to detect chemicals, it is unable to differentiate between chemicals that have similar indices of refraction, such as ethanol and methanol. In order to detect chemicals with similar indices of refraction, five pieces of a single photonic crystal (i.e. five pixel device) were exposed to different silanes, which changed the surface chemistry of the silica spheres in the photonic crystal. In turn, the five pixel device was able to produce a unique chemical fingerprint for several chemicals, which can be calibrated to detect toxins in the workplace.
Studies of Hydrogen Evolution Reactions from Aluminum Foil using Waste Materials and Their Reaction Mechanism
Nowadays, the most of waste materials are incinerated and generated the toxic gases in 日本. On the other hand, the Hydrogen gas (H2) has attracted attention as clean energy due to no emissions of toxic gases. In this work, we investigated that the new hydrogen evolution system using waste materials, such as aluminum (Al) foil and lime desiccant, and also investigated their reaction mechanism. The grinded desiccant was added to Erlenmeyer flask containing 300 mL of water. After dissolution the desiccant, the Al foil was added to the solution to begin the reaction. Generated gas was determined by water displacement method. The gas components are identified by gas chromatography. We found that the waste material reaction combined with waste lime desiccant and Al foil could be used for one of the hydrogen evolution system. This reaction is depended on solubility of lime desiccant, thus mean solubility of CaO in water. The Al foil is reacted with the desiccant more than 20 times of reaction stoichiometry. The calcium ion or calcium complex ions are involved with the excess reaction of Al foil.
Sustainable Graphene Oxide Support for Ruthenium Catalysts to Improve the Efficiency of the Hydrodesulfurization of Thiophenes
沙烏地阿拉伯 is the largest oil exporter in the world. 64,000,000 tons of sulfur oxides are produced every year through the combustion of organic sulfur compounds in the oil industry. This leads to several environmentally serious problems, including air pollution. This research provides a novel strategy to utilize natural-based Graphene Oxide (GO) as a support for ruthenium (Ru/GO) to functionalize as a green catalyst for hydrodesulfurization. Physical activation of camel bone samples was carried out by carbonizing them at 500oC to produce camel bone charcoal. Modified hammer’s method was employed for GO production, followed by doping of ruthenium in a simple synthesis step. The prepared catalyst has been characterized by XRD, SEM and EDX techniques. Thiophene and 3-methylthiophene were used as model compounds in the hydrodesulfurization process. The catalytic reactions were carried out at atmospheric pressure in a continuous up-flow fixed-bed quartz reactor. The composition of the sulfur containing gaseous products was analyzed by gas chromatography. The product distribution achieved for thiophene was 3-6% butadiene and 76-77% butane. And for 3-methylthiophene, it was 32.3% of pentaned 1-pentene and 2-pentene and the selectivity percentage was 45%. Ru/GO has been found to be an excellent catalyst of thiophene and 3- methylthiophene hydrodesulfurization and is a considerable improvement when compared to the commercially available catalysts. The prepared catalyst shall potentially lead to the reduction of sulfur pollution in the future. The positive effect on the environment could be substantial.
The Polar Equation from Butterfly Sprinkler Heads
This project aims to create the polar equations from the relation of the points on the centre line of the water twisted from Butterfly sprinkler heads. The water path includes inner rim, outer rim and centre line laying in the middle of the water path is used Rhombus’s property. The diagonals are perpendicular bisectors of each other to create the centre line. Then we create the polar equation of the centre line of water that twists from 4 types of the Butterfly sprinkler heads: edge frame, curve frame, STL and STL rotary. The polar equation of outer rim and inner rim is created by adding and removing the “ f ” value ( ; is the distance between the outer rim and the centre line, and is the geometric sequence that is ) of the coefficient (a) of the polar equation respectively. The results show that the formal equation of the centre line is which can explain the different properties of Butterfly sprinkler heads. If “ f ” value is increasing the water path and the blade will be wider that affects droplets distributing thoroughtly. Furthermore the relationship between the volume of water and the radius of water distribution can be processed to find the least time that can increase the appropriate moisture level of soil.