這篇報告要探討下列的「轉沙發的問題」是否有解？有一個長方形的沙發，如圖一，若要求每次只能以「四個頂點逆時針或順時針連續旋轉90度」的方式轉動，請問當長寬具備何種關係時，沙發經數次轉動後，剛好可以「轉」到相鄰的位置，如圖一，而且沙發坐人的正面方向仍保持不變呢？ 我們把原問題看成「平面座標上長方形旋轉的數學問題」，再利用「平面座標、三角函數、複數、複數的極式表示及向量」等數學工具，導出符合題目要求的方程式，最後證出當長與寬的比值為正實數時，有下列的結果： 1.當長與寬比值為無理數時，此問題無解。 2.當長與寬比值是最簡分數時，若分子為奇數，此問題無解。 3.當長與寬比值是最簡分數時，若分子為偶數，分母為奇數，此問題有解。 4.在有解的情況下，我們可以找出特定轉法的最小值。 5.當長與寬比值是最簡分數時，若分子為偶數，分母為奇數，沙發可轉至A點座標為(αp,0) 的位置，其中 α∈Z，且沙發坐人的正面方向保持不變。 6.當長與寬比值是最簡分數時，若分子為奇數，分母為偶數，沙發可轉至A點座標為(0,βq) 的位置，其中 β∈Z，且沙發坐人的正面方向保持不變。 7當的長與寬比值為正實數時，可將沙發轉至A點的座標為(2αp + 2βq,2γp + 2qω)的位置，其中 α,β,γ,ω∈Z，且沙發坐人的正面方向保持不變。 In this paper we discuss the solution of rotating sofa problem as follows : The condition is : Merely allow to rotate the sofa several times by rotating 90 degrees clockwise or counterclockwise around the vertex. (maybe A, B, C, or D in Fig. 1) The question is : What’s the relationship between the length and the width of the sofa, if we request the sofa translated next to the original position with direction unchanged. (as shown in Fig. 1 with A’B’C’D’). We take this problem as a mathematical one of rotating a rectangle in plane coordinates. Then we derive the desired equations by using the tools of plane coordinates, trigonometric functions, complex number, polar form of complex number, and vector. Finally, we prove that： 1. When the ratio of length and width is irrational, the problem has no solution. 2. When the length of sofa is odd in the ratio of length and width, the problem has no solution. 3. When the ratio of length and width is even, the problem has solutions. 4. When the solutions exist , we can find the minimum of the number of rotations. 5. When the ratio of length and width is an irreducible fraction, which has the even numerator and the odd denominator, the sofa can be rotated to the coordinate (αp,0)(α∈Z)which is the new position of A and keep the original position with direction unchanged. 6. When the ratio of length and width is an irreducible fraction, which has the odd numerator and the even denominator, the sofa can be rotated to the coordinate (0,βq)(β∈Z) which is the new position of A and keep the original position with direction unchanged. 7. When the ratio of length and width is a real positive number, the sofa can be rotated to the coordinate (2αp + 2βq,2γp + 2qω)(α,β,γ,ω∈Z)which is the new position of A and keep the original position with direction unchanged.

At present Malaysia is the largest exporter of palm oil in the international market. In the process of extraction of palm oil from oil palm fruit, biomass materials such as palm empty fruit bunch (EFB) and palm pressed fibre (PPF) are generated as waste products. Natural fibres reinforced cement-based materials have gain increasing application in residential housing components. One of the natural fibres considered is oil palm empty fruit bunches (EFB) fibres which offer advantages such as availability, renewability, low cost and the established technology to extract the fibres. This study investigates the properties of cement board incorporated with large amount of oil palm EFB fibres Among the tests conducted was compressive strength, density, water absorption and thickness swelling tests. It was found that high EFB fibres content lead to lower strength and higher absorption . The results also indicate that high EFB fibres contents reduced the self-weight of the blocks and the resulting blocks can be classified as lightweight cement blocks suitable to be used as lightweight walling materials. Our research is to study the production of cement board using Empty fruit bunch(EFB)These board were made from empty fruit bunch, cement and water. Two chemical are added is aluminium sulphate and sodium silicate. Cement : EFB mixture by weight was 2.5:1, 2.75:1 and 3.0:1 used to produces a cement board.

我們發現當CdS 與ZnS 重量比以2:1 時，能在Na2SO3 及Na2S 的混合溶液中，照射日光產生氫氣，並能有最佳的產率38.18(ml/hr*g)，比一般直接照紫外光的效果更好。不只善用自然界的太陽能，還可以生成最乾淨"綠色"的能源－氫氣。我們以 CdCO3 和ZnCO3 做為產氫觸媒的原料，其為黃色的粉末，並以AA 原子吸收光譜儀求得樣品溶液中測得CdS 與ZnS 重量比為２：１。 把產生的觸媒放入以 Na2S 和Na2SO3 混合而成的溶液中，以鋁薄紙包覆以防受光實驗提早進行。把實驗裝置放在日光下照射，以一端管口所接的細量管得失產生氫氣的體積，進而換算產率。 我們的實驗裝置為密閉式的，並無循環裝置，故產氫量有限。未來的發展將是朝著大型開放式的循環系統，使氫氣能源源不絕的產生。 The production of molecular hydrogen upon sunlight illumination of the mixture of CdS and ZnS suspensions in the weight ratio 2:1 in Na2SO3 / Na2S solution have been observed. The best rate of evolution of hydrogen is 38.18(ml/hr*g), which has a higher efficiency than upon UV illumination. It not only make a great use of the solar energy, but also produce the cleanest “green” energy resource─hydrogen.

The purpose of my project was to prove if the seeds of invasive plants are spread throughout the Peace region in hay harvested by local farmers. Invasive plants are so dangerous because they adapt to their environment fast and some of them can cause harm to both animals and humans. Within a year they can completely wipe the ecosystem out of native plants. Many invasive plants are either poisonous or can affect the systems of the animals body’s. I collected 22 hay samples from local farms around the North Peace region. I conducted a survey to help collect information about the samples. A pretest was conducted to determine which of two homemade sifters (one with three layers of decreasing size wire mesh, and one with 6 layers of materials with larger size holes) was the best for sifting through hay. I was able to determine that the larger sifter was easier for sifting hay. I sifted through all 22 samples with the larger sifter. I individually looked through each layer and removed what I believed to be seeds. Each seed was individually bagged, labeled, and photographed through a microscope. After each seed was photographed they were planted to help identify the seeds. I contacted a seed specialist. I was able to send him the images of my seeds. He helped me to identify my findings. I removed a total of 5568 potential seeds in my 22 samples. Out of all the potential seeds found 628 seeds were invasive. All invasive plants identified either cause pain to animals or they easily over take the native plants.

本實驗主要探討變頻的意義，利用廢棄風扇設計轉子和線圈互感實驗中發現轉子上渦電流存在的事實。台灣交流電的頻率為60Hz，故四極馬達極速為30rps在實驗中觀察到，本實驗觀察到轉速越大則線圈和轉子上渦電流互感值越大並導致線圈電流變小，進而降低線圈電流熱效應佔輸入電能比例。在分析轉子處消耗能量實驗中發現如果轉速越接近極速則磁力在轉子切線方向分力做功比例會減少，且大部分輸入電能轉變成渦電流熱能，線圈與轉子間的平均磁力和轉子切平面的方向夾角(θ值)會決定線圈熱效應及渦電流熱效應佔輸入電能比例，實驗中發現最佳轉速(最佳θ值)下運轉馬達會達到最佳能源效益比，實驗結果顯示本次實驗採用馬達在轉速約25(rps)時能源效益比最佳。

我們的目的在於完成一架能攀爬樓梯的「仿生獸」Strandbeest註。研究則專注在連桿系統上，並發現了腳尖的軌跡與系統所承載的阻力，分別肇因於連桿長度的比例與重心位置相對于曲柄的力矩有關，又分別影響著仿生獸是否能克服地形與產生足夠強大的前進力量。最後，我們再探討了足底的摩擦力和四足間軸距的學理，並使用3D列印調整連桿長度設計的便利性，完成一架可以克服地形的「仿生獸」。（註：所謂「仿生獸」，是由一位荷蘭科學家泰奧·楊森所發明的機械體，利用風力與力學原理制動，可以躲避障礙物或模仿生物的一些行為。）

本文所探討的是給定一個 n×n 的棋盤及 n^2個兩面棋(一面為黑色，一面為白色)，若規定其中一個棋子翻面時，則與此棋相鄰的所有棋子亦須跟著翻面，而我們想探討在此規定下的所有棋局是否皆可被翻成同一面。因此我們將每一個 n×n 的棋局對應到一個矩陣，且翻棋的過程則對應到矩陣二進位的加法。利用此思考模式我們可以將此遊戲問題轉換成是解聯立方程組與判別矩陣是否可逆的問題，最後並借助數學軟體 Mathematics 4求其解。

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.

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.

In this project, we defined a mid-tangent point with respect to a fixed point X and a tangent at a point Y on a planar curve C as a point on the tangent that is equidistant from X and Y. We studied the locus of mid-tangent points of conic sections. We found that the locus of mid-tangent points of most conic sections are non-linear curves. However, we observed and proved by using Euclidean geometry that the locus of mid-tangent points of circles are straight lines. The mapping defined by mid-tangent points was studied further. The similarity between a mid-tangent mapping and a stereographic projection was displayed as a one – to – one correspondence function. We also extended the concept of mid-tangent points to three dimensional space and found that the similarity with the stereographic projection was retained in higher dimensions. Finally, we studied the locus of mid-tangent points of a sphere to create a mapping of the sphere to a plane.

We all do it – walk along with a cup in hand, and carelessly spill it. While it’s usually more annoying than anything else, it happens to affect almost all of us, and little is done to minimise the likelihood of it occurring. So my aim was to explain the physics behind why we spill drinks when we walk, and to investigate how we can minimise the likelihood of this occurring. I broke this investigation into two distinct parts, explaining the system of the cup, and explaining the effect of walking. From initial observations, it was clear that the cup was a resonating system. Like any resonating system, the cup has a natural frequency. When the cup is oscillated – moved back and forth – at near this frequency, the size of the liquid oscillations is very large. This is because the acceleration is in phase with the motion of the liquid, so in each cycle maximum energy is input into the system. In my investigation I experimentally measured this natural frequency, and created a mathematical model to explain this frequency. It was also found that as the size of liquid oscillations in the cup increases, so does distortion of the fluid surface, possibly enabling spilling. To systematically analyse the effect of walking, I had subjects walk on a treadmill, so walking surface and speed were controlled. However, I also needed an accurate way of measuring the motion of a carried cup. Firstly, I tried to use video analysis; however I found this far too imprecise for measuring small changes in velocity of a cup. In the end I used a smartphone to record the acceleration of a carried cup, as acceleration is what causes the movement of liquid in a cup. This allowed surprisingly accurate measurements to be made, and allowed both the size and frequency of the acceleration to be recorded. In order to relate the system of the cup and the oscillation provided whilst walking I conducted a qualitative experiment into the effect of stride frequency on the likelihood of spilling. When stride frequency was very close to the natural frequency of the cup, spilling occurred almost instantly, while it did not occur if stride frequency was much higher or lower. In the end, my research showed that to minimise the likelihood of spilling your drink walk slowly, use a narrow cup, focus on walking smoothly, and fill the cup well below the rim. Despite this, some people happen to be much smoother cup carriers than others, likely due to their individual biomechanics. And, if you really don’t want to spill your drink, you can always use a lid.

Immunoglobulin G is a glycoprotein structured molecule that is produced by the immune system and protects organism from harmful effects of antigens. Ig G amount in the blood plasma is an appropriate indicator of; infection, cancer, diabetes, cardiovascular diseases, Alzheimer and other autoimmune diseases. Besides, purification of Ig G used in the treatment of these diseases from naturel sources is carried out at high costs on the World market. It is hard to obtain Ig G in high amounts and without any decomposes, that’s why it is important to develop new systems that will help to recognize and purify Ig G antibody. In this project, my purpose was; recognizing Ig G antibody with efficient, high amounted, fast, easily, with less toxicity, economically and purifying Ig G in high ratios from its natural sources. For this purpose p(HEMA-EDMA) membranes are synthesized with free radical photo polymerization method and characterized according to SEM images, swelling behaviors FTIR analysis and elemental analysis. In order to adsorb Ig G to polymeric membranes; polymeric membranes are activated with silanization agent (IMEO) and derivatized with Con A which is a lectin affinity ligand. In the SEM results it is examined that membranes are in spherical structures. Highest swelling value is determined as 224.8%.Binding of IMEO was demonstrated with FTIR and Elemental Analysis. Optimum conditions for Ig G adsorption to membranes are; 1.5 mg/ml initial Ig G concentration, 30 minutes of adsorption time, pH 4 citrate buffer 37 0C and without any different ion strength. Optimum adsorption capacity is determined as 253.8 mg/cm2 and it is also determined that this value is 7 times higher than nonspecific Ig G adsorption to p(HEMA-EDMA) membranes. Ig G adsorption-desorption cycles (5 times) proved that product is reusable without losing its adsorption capacity. According to the electrophoresis, Ig G could be desorbed in pure form without any denaturation to its structure.