剛開始考慮平分物件時，我們從二維的多邊形部分著手，後來發現已經有人做過相關研究，並且得到類似的結論。這個部份顯現出面積平分線與其包絡曲線間的密切關係。我們將其中的方法和結果加以歸納、改善，為了更全面地研究，我們推導出一般性的包絡方程。之後當我們推廣到三維領域時，發現四面體體積平分面與之前的結論有些相似之處，平分的情況卻也更複雜，我們將推導的結果用電腦軟體呈現出來，以便更深入地了解它。最後嘗試了相當抽象的高維積平分，結果仍具有工整的對稱性，讓我們充分領略了數學之美！When considering bisecting a subject, at first we focused our attention on 2-D case, polygons. But afterwards, we found there were already some similar studies conducted by other students, which indicated the close relation between the area-bisecting lines of a polygon and their envelope. We rearranged their methods and results, and then made further improvement. Moreover, in order to study the bisecting problem entirely, we derived the general envelope equation. Then when extending the generalization to the 3-D case, we came to the conclusion that tetrahedrons’ volume-bisecting planes is similar to that in 2-D, but the circumstances are more complex. We tried to show our result with the aid of software, hoping to understand it fully. Finally, we tried to do the case in higher dimension, which is very abstract, and the result was clear-cut symmetrical. During the studying process, we had seen “the beauty of mathematics.”

我們根據物理學的彈性體振動模型發現:主動脈硬化程度可由測量主動脈相對於心臟運動的延遲時間分析得知。我們除了由樣品之超音波影像分析驗證此一觀念之外，還用一自製模型進行實驗，模擬血管厚度對延遲時間的影響，實驗結果與理論相吻合，證實了彈性體模型之可靠性。在診斷方面，此方法可用目前臨床使用的心臟超音波儀直接進行測量，使得它具有方便、普遍的優點;而且可由體外的胸前超音波掃描(TTE)進行觀測，具有非侵入性、免除受測者的不適及避免副作用，此外，能定量分析、早期診斷、鑑別度高也是此方法重要的優點。According to the elastic oscillation model of physics, we found that the aorta stiffness could be obtained by measuring the delay time of the aorta relative to the cardiac motion. The idea was confirmed by an analysis of the echocardiograph images of several samples. A home-made mechanical model was also employed to simulate the effect of cardiovascular thickness. The experimental results fitted the theory very well, verifying the feasibility of the elastic oscillation model. This measurement could be carried out with the conventional echocardiography instruments, making it convenient and common. Furthermore, the delay time could be measured with TaransThoracic Echo (TTE) instead of TransEsophadeal Echo (TEE). This non-invasion can avoid patients’ discomfort and side effect during medical process. The quantitative measurement also enables that the diagnostics can be progressed in advance.

能源危機日益嚴重，開發再生能源成為當務之急。在諸多的再生能源之中，太陽能最易於使用，且可源源不絕的不斷取得利用。但目前太陽能電池的效率始終不高。其中重要的原因是太陽本身的頻譜多為短波長，並不適合絕大部分頻譜響應較佳，多為長波長的太陽能電池。目前改善效率方式，多為改變太陽能電池的頻譜響應以配合太陽之頻譜。本研究提出反向思考的概念，藉由頻譜轉換的方法，改變照射在太陽能電池上的太陽光頻譜，以提升太陽能電池的效率。本研究利用一種珊瑚礁的螢光蛋白質（DsRed）以及人工合成的螢光染料（Cy-5）最為本研究的頻譜轉換的材料，加於低成本，目前市場佔有率較大的單晶矽太陽能電池上，經由理論與實驗的結果顯示，增加的發電效率約為3~5%，證實利用頻譜轉換的概念確實可以提升太陽能電池的效率。Energy crisis has become more and more serious in recent years, which makes recycled-energy development is a must. Among different recycled energies, solar power has two advantages, that is, easy-to-use and endless supply. However, the conventional solar cell makes poor use of the solar spectrum because the solar spectrum is mainly composed of short-wavelength, which can’t fit to most of solar cells which is more sensitive to long-wavelength. Currently, the major method to improve the efficiency is change the function of spectral response, such as concentration lens, tracking devices, and antireflection coating. Up to now, no one notices the possibility of changing solar spectrum yet. This research provides an insight into this issue. Instead of changing the function of spectral response, I changed the solar spectrum, which irradiates solar cells through spectrum conversion to improve solar cells’ efficiency. This research uses one kind of coral fluorescence proteins (DsRed) and one kind of artificial fluorescence dyes (Cy-5) as the materials of spectrum conversion. Then coat them on the low-cost and high-market-share mono-crystal-Si. According to the theories I researched and my experiments, the improvement of the efficiency is about 3~5%, which proves it is actually useful to elevate the efficiency of solar cells through spectrum conversion.

The experiment was primarily focused on studying whether the enzymes from different bacterial species collected in various hot spring areas still exhibited activities at high temperatures. A further study would be conducted on analyzing the unique characteristics universally found in the genes of selected bacteria. First, hot spring samples were collected form Peitou and Wulai, and then cultured on the PY, PTG, MFB, and TS media in the laboratory. After the broth media growing with thermophilic bacteria, a series of continuous dilution method and solid-plate spreading were applied to separate these bacterial clones. The genomic DNA of the selected bacteria was extracted and used to analyze subtilisin-like gene by polymerase chain reaction (PCR) and electrophoresis. Finally, we examined the six selected thermophilic bacteria with the enzymatic activities of fibrin- and milk-protein degradation. We successfully concluded the experiment by proving that these thermophilic bacteria still exhibited significant enzymatic activitives in the high-temperature environments. The results of this experiment can be applied in numerous fields, for example, thrombus treatment and food processing, and a more in-depth study shall warrant the due consideration. 這次的實驗，主要是研究在不同的溫泉區中所採集不同種類細菌，是否酵素在高溫下仍具活性，如果有，再進而研究它們的基因有何特別的共同之處。首先，我分別自北投和烏來採集水樣，到實驗室後再以PY、PTG、MFB 和TS 四種培養基做細菌的培養，接著再利用連續稀釋和固態塗抹來做細菌的分離。經過挑選和培養之高溫菌直接進行DNA 的抽取，並利用「聚合?鏈反應」和「電泳跑膠」技術分析其類似蛋白質分解酵素subtilisin 基因。另外，本研究同時針對所選定之6 株高溫細菌利用血栓和牛奶蛋白來測試其蛋白質分解酵素的活性。由以上實驗結果可以證明某些細菌在高溫的環境下酵素仍具活性。這次在高溫菌的實驗結論，可應用在很多地方，例如：血栓的治療、在高溫下食品處理‧‧‧等，應用相當廣泛。

林下（陰地）植物長期生活在較少陽光的地方，卻依然欣欣向榮。因此，我們針對林下植物的葉綠體囊膜電池做討論，希望可以找出發電效率更高的葉綠體囊膜電池。

This research divides into two subjects. The first subject aims at single-hole-swirls, we use single-hole-swirls to discuss water volume to the eddy current influence. The experiment obtains the relations between the aperture and the bottom area of the vessel. It also produces difference between free vortexes and forced vortexes of the eddy current. Our another discussion subject is twin-hole-swirls, which is extremely complex but actually an interesting part. Twin-hole-swirls is easy to be disturbed by the external factors. The result is different from our supposition. But from the experiment we discovered twin-hole-swirls is a mini version of twin-typhoons. Twin-hole-swirls has the same phenomenon with Fujiwhara effect. 本研究探討分為兩個主題。首先是針對單孔，我們利用單孔漩渦探討水量對漩渦的影響，並實驗求得孔徑與容器底面積的關係，以及自由渦流與強制渦流所產生之漩渦的差異。我們另一個探討的主題是雙孔，也是實驗中非常複雜但卻有趣的一部份。雙孔漩渦極易受到外在因素干擾，產生的結果與我們在實驗前的假設出入頗大，但從實驗中我們發現雙孔漩渦其實是雙颱的縮小版，雙孔漩渦與雙颱效應有著許多相符的現象。

人眼所見點光源四週的光芒（我們稱為"星星長角"），夜間瞇眼時所見向上下方向射的特別長的光芒（我們稱之為"炫光萬丈"），這二種視覺中的光學現象無法用相機拍攝，但眼睛卻能夠看見。書上及網路上均無法找到合理的解釋。因此，我們設計了數個實驗來研究這些光學現象的成因。我們發現了星星長角的成因是由光對眼睛表面刮痕的繞射和光對眼睫毛的繞射雙重原因所產生。而炫光萬丈的成因則是光線經過眼球表面淚膜層的部分反射及部分折射所造成。The purpose of the research is to verify possible reasons of some optical phenomenon in human vision, including the star shape of a dot light source and glare upward or downward when one narrowing his/her eyes. These optical phenomenon can be seen by naked eyes but can not be taken photos with a camera. We find no detailed explains which can explain the cause of glare upward or downward on the internet or in books. So we designed several experiments trying to find out the cause of these optical phenomenon . In experiments, we figured out that diffractions by notches on the surface of eyes and eyelashes are both responsible for the star shape of a dot light source. The cause of glare when narrowing eyes is that partial reflection and partial refraction of light take place between boundary of air and tear film and boundary of tear film and cornea. The appearance of glare is closely related to the thickness of the tear film . When we narrow our eyes , the tear film is thicker. The thicker the tear film is ; the longer the glare is .Also, bended extent influence the length of glare. We figured out that the more bended the tear film is ; the longer the glare is.

在做這實驗之前，我花了很長一段時間思考一個問題：我如何能得到同\r 樣大小（均勻）的水珠陣列？我在家中各個角落放置許多水瓶，看看哪\r 個地方能培養出顆粒相同的水珠？ 往往肉眼見到整齊排列的水珠，一經\r 顯微鏡觀察，可就大小不齊整了。\r 第一篇中的實驗，幾乎是「水珠日記」，記載冷凝過程；其中我特地比對\r 「穩定的水蒸氣氣流」與「擾動的水蒸氣氣流」、「水蒸氣氣流與凝結盤\r 間溫度差別」、「凝結盤密度差別」、「凝結因子的重量百分濃度差別」、「水\r 蒸氣氣流之流速差別」，並加以複合比對。希望能找到產生均勻排列的條\r 件–探討水分子的自我組裝機制（Self Mechanism of Water Droplets）。\r 這其中提出『均勻假說』： 當條件合宜時，冷凝下降的細微水珠會產生\r free vortex ring，形成整齊的渦環組合，進而產生均勻排列的細微水珠陣\r 列。\r 實驗是藉由溶液密度小於水的設計，讓水蒸氣冷凝於液面上，並且因密\r 度較大而下沉。設計的要點是：儘量減少細微水珠自冷凝後的堆疊\r （coalescence），以呈現水珠原貌。\r 在第二篇實驗中，將對渦旋比例尺修正。渦旋本身難測大小，在空氣中\r 也不易觀察，但是若由水中觀看，可藉由空氣蕊長短估算。這次更進一\r 步考慮到排水速率、水深、排水口形狀、與極值。\r \r I have tried to ask a famous math professor if he can create a formula\r describing the ordered array of water droplets。〝Then, I should study Physics\r first !〞He said。\r Condensation is the thing we live with , being found everywhere, passing\r without notice。But we never know0 when it does start？\r This experiment presented here is actually the diary of the growth of water\r droplets through condensation。Through convection and vortex ring, it\r discusses the self assembly mechanism of water droplets and peep into the\r uniformity of the size of water droplets。\r Here, vortex ring plays an important role in the self assembly mechanism of\r water droplets which is not triggered in the daily life。\r By coalescence, water droplets grow bigger, but are not round again。We used\r the polymer film as template and designed the solution lighter than water, so\r the minute droplets will sink to the bottom and layer by layer。After seconds\r we may have multilayers of ordered array。\r This is the first step in discovering the uniformity of water droplets, besides, I\r made some correction to the Vortex-Ruler。Vortex-Ruler will be useful in\r researching the flying mechanism of butterfly and dragonfly as to judge\r which one induces more vortices。

治療癌症通常以化學治療為大宗，但化療會帶來副作用，因此目前研究以溫和草藥來取代化療。在治療癌症時，癌症轉移會使病人存活率下降，因此如果找到能抑制癌症轉移的藥物，就能提高病人的存活率，而在癌症轉移機制中，癌細胞要轉移時會分泌蛋白酶，降解基底膜和細胞外間質，使癌細胞轉移出原位組織。本研究聚焦於魚腥草是否能抑制攝護腺癌細胞及胰臟癌細胞侵襲其他組織、降低基質金屬蛋白酶(matrix metalloproteinase;以下簡稱MMP)的基因量及活性及降低細胞中尿激酶型纖溶酶原激活劑(urokinase plasminogen activator ;以下簡稱uPA)表現量。根據實驗結果，魚腥草可能藉由降低MMP的基因量以抑制攝護腺癌細胞的侵襲能力及遷移能力，但抑制胰臟癌細胞MMP的活性不是藉由降低MMP的基因量，且魚腥草並不會降低兩種癌細胞的uPA表現量，但由於先前加了HCE的癌細胞轉移能力下降，因此我們認為魚腥草在不影響癌細胞的生長下，仍然可抑制兩種癌細胞的轉移能力。