臺灣

設Γ為一平面曲線而 P 為一定點 , 自P 向Γ所有的切線作對稱點，則所有對稱點所成的圖形Γ1 稱為曲線Γ對定點P 的double pedal curve , Γ1 對定點P 的double pedal curve Γ2 稱為曲線Γ對定點P 的2-th double pedal curve , Γ2 對定點P 的double pedal curve Γ3 稱為曲線 Γ對定點P 的3-th double pedal curve ,…… 。以下是本文主要的結果：結論A：當Γ為一圓形而P 為圓上一點時 , 計算其n−th double pedal curve 的方程式。結論B：當Γ為任意平滑的參數曲線而P 為任意一點時 , Γ的 double pedal curve 的切線性質。結論C：當Γ為任意平滑的參數曲線而P 為(0,0)時, 計算其n−th double pedal curve 的方程式。 Given a plane curve Γand a fixed point P ,the locus of the reflection of P about the tangent to the curveΓis called the double pedal curve of Γwith respect to P.We denote Γ1 as the double pedal curve of Γwith respect to P, Γ2 as the double pedal curve of Γ1 with respect to P , Γ3 as the double pedal curve of Γ2 with respect to P ,and so on , we call Γn the n-th double pedal curve of Γwith respect to P. If Γ is a circle, and P is a point on the circle, we got the parametric equation of the n−th double pedal curve of Γ with respect to P. And, for any parametric plane curve Γ; we got the method to draw the tangent of the double pedal curve of Γ.

實驗發現，在高溫並排鋸條之齒面上可以形成懸浮水珠，而水珠具有爬坡的現象。當鋸條齒面溫度達四百度且鋸條傾斜角低於二十度時，水滴均可向上爬坡，特別的是，角度在十二度內，大小水滴的爬升速度皆有隨著角度提高而變快的趨勢，且質量較小的水珠爬升速率較快。 水珠之所以能向上爬坡與其表面受熱快速蒸發和齒廓曲線有關。首先，水珠因受熱在周圍形成一層很薄的蒸氣層，阻隔熱的傳遞，使得在鋸齒面上可以形成懸浮水珠，持續一段時間而不會立即蒸發；而水珠爬坡的現象，肇因於蒸氣壓力和蒸發所產生的反向動量，配合不對稱的齒廓曲線，造成水珠會受到一個沿斜坡向上之淨力，進而可以爬坡。綜合前述的原因，我們可以合理的解釋各項實驗結果。

本文藉由模糊數學理論所提的分類方式，來設計一套對於DNA序列的分類方法，並利用了40筆人工已分完的樣本，分別作為測試及學習樣本。\r 發現可以順利的將每筆DNA序列中的鹼基每3個一組轉成胺基酸序列，再利用模糊分類方式將所有胺基酸序列進行分類，最後並利用原先滾動方式的環狀排列方式，來對同一筆鹼基的3組胺基酸資料進行檢核，發現可以有效提高分類的正確性。\r 之後我們再利用182筆自然樣本進行檢檢，也發現模糊分類方式亦可正確完成此次分類結果。

磁粉探傷過程包含兩個重要的物理現象，其一是磁力線於工作瑕疵處的漏磁現象而形成邊緣磁場，其二是鐵粉顆粒受邊緣磁場的影響而向工作瑕疵處附近聚集現象分別反應出磁場在通過不同介質時所遵循的折射原哩，以及磁場分佈對鐵粉顆粒產生的磁力原理。本研究以電磁通電產生靜磁場，並利用兩電磁鐵間的氣隙來模擬工件瑕疵，因電磁鐵的磁導係數遠大於空氣之磁導係數而造成漏磁場方向機與漏磁面垂直，形成一單純的邊界條件使得邊緣磁通密度的解析解可直接利用馬克斯威爾方程式求得。我們亦導出空氣中的磁通分佈對微小的鐵粉顆粒所產生的磁力公式，發現鐵粉顆粒受靜磁力的大小與該顆粒的體積、磁通密度與磁通密度之梯度成正比，而其方向則與磁通密度之梯度一致，此結論與磁粉探傷過程中，鐵粉向工件瑕疵處聚集的現象吻合。實驗設計採用螢光粉混合鐵粉以獲致明顯的鐵粉顆粒運動軌跡，用數位錄影機紀錄後再擷取影像圖檔判讀其位置與時間之關係，進而反算鐵粉顆粒之位置與所受之靜磁力的關係，以定量的方式證實所推導的邊緣磁場分佈公式以及磁力公式。Two important physical phenomena are observed in the practice of magnetic particle inspection (MPI). The first one is that leakage flux is present in the defect area of the work-piece under inspection. The second one is that magnetic particles aggregate in the vicinity of the defect. These phenomena manifest the theory of flux refraction, which occurs in the intersection area of two different magnetic materials, and the theory of magneto-static force, which is experienced by the iron powder in a magnetic field distribution. Two electromagnets, made of cast steel, are aligned together such that the leakage flux in the air gap forms a fringing field distribution. It is this magnetic field distribution that simulates a defect area in a magnetized magnetic work-piece. Since the permeability of cast steel is far larger than that of air, the direction of the fringing field at the surface of the electromagnets is almost perpendicular to the surface. Such a simple geometry renders an analytical solution to the Maxwell’s equations. The magnetic force of the magneto-static field exerting on the magnetic particle, an iron powder in this case, can be derived by using the principle of virtual displacement. We obtain a formula of magnetic force, whose direction coincides with the gradient of the magnetic flux density and whose magnitude is proportional to the magnitude of the particle volume, the magnetic flux density and its gradient. This formula also agrees with the observation in MPI that the magnetic particles aggregate in the vicinity of the defect.

因應咖啡消費量與日俱增，產生大量廢棄咖啡渣的問題，我們想找出咖啡渣的再利用價值。本研究先以不同溶劑、體積、溫度浸泡咖啡渣，將濾液經蒸餾設備濃縮取得咖啡渣油，找出抗紫外線效果最佳的條件；再比較不同沖煮方式、烘焙程度的咖啡渣，萃取出的咖啡渣油抗紫外線效果；最後將最佳抗紫外線效果的咖啡渣油與市售防曬品，照射紫外燈與塗抹在豬皮上比較。實驗結果顯示，咖啡渣油具有良好的抗紫外線效果，冷泡淺焙咖啡渣100 g，以150 ml丙酮，常溫浸泡，所萃取出的咖啡渣油抗紫外線效果最佳。利用「小畫家」軟體比較豬皮照射前後顏色差異，發現咖啡渣油防曬效果與市售防曬品PA+++效果接近。希望未來能應用在各式防曬用品，達到綠色環保的效益。

費曼曾說：There is plenty of room at the bottom。喬治亞理工大學的Mostafa El-Sayed 教授發表的癌細胞辨識、與科學月刊報導『台大抗煞一號』引發我們對膠體金粒子的興趣。膠體的性質主要是由界達電位 (zeta potential)決定。參考台科大、成大、中山…等超音波應用研究，提出改良篩選物理法製造之膠體金粒子的儀器設計與製作。經沉降過濾可達平均粒徑 100 nm；而離心式篩選機與超音波管式篩選機可達平均粒徑30 nm。篩選後的膠體粒子以電容原理調控膠體金粒子之界達電位 (zeta potential)，成功地從-30 mV 提升至-59 mV，並發展成電容超音波界達電位控制儀(Capacitor Ultrasonic Zeta Potential Controller)。以膠體金粒子與蛋白質鍵結量來測試調控界達電位的效果，發現蛋白質鍵結量之增加曲線與界達電位的增加曲線的增加趨勢相似；此功能的發現對於生物科技方面的應用應會有很大的幫助。透過界達電位控制系統，本研究達到費曼先生所期望的「在原子或分子的尺度上來加工材料和製造設備」。“There is plenty room at the bottom.” The words of Mr. Feynman are the beginning of nano technology. Mostafa El-Sayed, a professor of Georgia Institute Technology, identified cancer cells through nano gold-antibody complex. So, our study focuses on the zeta potential of colloidal gold particles. At first, the filtering method and equipments were developed. The theories were based on the ultrasonic studies of universities such as National Taiwan University of Science and Technology. Then the colloidal gold’s sizes were filtered to100 nm through settling. At last, by using Continual-Filtering Centrifuge (CoCe.) and Tube Well Mass (TW-MS), the mean particles sizes can be filtered to 30 nm. The most important results are: Zeta potential of the gold colloid was controlled with Capacitor Ultrasonic Zeta Potential Controller. The zeta potential can be raised from -30 mV up to -59 mV, which is -20 mV higher than the conventional pH-changing way. The function of zeta potential to protein binding quantity was tested. The increasing curves of zeta potential and protein binding quantity were similar. This property would be a significance of biotechnology. Thourgh Capacitor Ultrasonic Zeta Potential Control system, the zeta potential’s limitation of gold colliod, which is produced by SANSS (Submerged Arc Nanoparticles Synthesis System), can be controled in a wilder range. The study which is focused on nano-scale, like the wish of Mr. Feynman – “To manufacture material and produce equipment in atom and molecular scale”.

一顆純水水滴落在水面上通常會立刻與水面結合，但當液滴以固定高度朝向有界面活性劑的液面釋放時，這顆水滴會較在沒有添加界面活性劑的水面上時生存更久。根據統計結果，不同釋放高度和界面活性劑的有無及水滴大小均會影響水滴的生存期。當液面垂直振動或是被風吹動時，因為水滴和液面之間的氣墊會不斷地被更新，結合的時間可以被延長。當液面垂直振動時，可觀察到漂浮水滴在液面上產生上下振動、在2點間來回運動(束縛)、行進、跳動等運動行為，在運動期間水滴本身也會伴隨著振盪變形。當水滴在振動液面產生行進所對應的加速度門檻值，在特定頻率下會出現極小值，此時水滴與水浴會形成共振現象。當兩顆漂浮液滴在垂直振動的液面互相靠近時，除了會有結合、彈開的現象外，兩者間也會產生互繞質心轉動及排斥的現象。漂浮水滴在振動液面上的多種運動現象很類似Couder’s團隊針對在油面上跳動油滴(walker)所做的研究，但兩者間仍然存在著差異。

由於台灣山區溪流短小陡急，土石流災害嚴重，透過式防砂壩攔阻工法為現\r 今之趨勢，而帄面透水柵具有一般透過式壩的優點，不但能將土石流轉化為水砂\r 流，還可以減低土石流衝擊力造成的損壞及改善上游儲砂空間不足的問題。本研\r 究採用改良式帄面柵，在下游處增設分流河道，可改善分離出之細顆粒土砂水與\r 大礫石再度結合之危險。此工法於2003 年引進台灣後，尚未廣泛應用，主要原因\r 為缺乏設計之經驗式，因此本研究針對透水柵的柵棒長度（L/ Dmax）、棒淨間距\r （b/ Dmax）、柵面架設方式、柵面篩分角度（θ）等多項重要因子進行室內渠槽\r 試驗，最後提出土砂篩分比與攔阻率的趨勢方程式，設計時以總攔阻率（R）高為\r 原則，輔以篩分比（S）與貯砂率（R1）高，即可有良好之成效，期望能作為國內\r 外現場工程施做時之參考，結果如下所示。

Thermal expansion exists in our daily life. However, thermal expansion is generally too slight to be seen by naked eyes. Therefore, in the present project, a dilatometer was assembled to enhance better sensitivity toward thermal expansion. Hopefully the self-assembled dilatometer could contribute to teaching purpose.The structure of our 4th generation dilatometer is showed below. Using an ‘L’ square to hang up the metal stick and a rolling needle with a mirror to reflect the laser light are the critical parts of this equipment. By using this special reflection mechanism, the slight expansion of a metal stick caused by heat can be enlarged to a large scale. This special mechanism is where our creativity laid. Measuring in millimeter (mm), the measurement precision of the equipment can be extended to 0.0001 decimal. Our dilatometer was used to measure the expansion of various metal sticks caused by the temperature changes. Results were drawn from analysis of the data: 1) The average relative deflection was within 1.0~1.8%; 2) The relative deviation of linear thermal expansion coefficient was within –1.2~-4.4%. 物質熱漲冷縮的特性普遍存在於我們的生活環境中，但因其變化量相對微小，一般並不容易直接觀察，爲了進一步研究這課題，我們組裝偵測熱膨脹的儀器，並希望儀器的靈敏度高，能推廣為教學器材，經過我們不斷努力與改良，終於有了令人愉悅的成果。 自製第四代熱膨脹儀的結構如圖，設計「角尺懸吊金屬棒」與「滾針及鏡面反射」是儀器的重要部份，利用滾針旋轉及鏡面反射雷射光，加乘放大熱膨脹的微量變化，這是我們主要的創意，以公厘（mm）為單位，儀器的精確值到小數第四位。 利用自製的熱膨脹儀，探討金屬熱膨脹的影響因素。分析實驗所得數據，平均相對偏差在1.0~1.8﹪，而線膨脹係數的相對誤差約-1.2~-4.4﹪。

Refer to Figure 1. Suppose ABCD is a trapezoid and . Passing the intersection M of and we construct a parallel line intersecting and at E and F, respectively. We obtain that , and then we can generalize the result. 如(圖一)，若ABCD為梯形且，過 和交點M 分別作平行線交、 於E 、F ，可得 的關係，再加以推廣。