四等獎

「重複圖形」是本篇報告研究的問題，我們利用「方程式」建立一個尋找重複圖形，並証明其個數的方法。利用此方法得出下面的結論：1.會形成lap 2 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 2 三角形」只有1 種，即等腰直角三角形。(2)「lap 2 四邊形」只有1 種，即二邊之比為1： 且內角是45°、135°的平行四邊形。2.會形成lap 3 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 3 三角形」只有1 種，即內角為30°–60°–90°的直角三角形。3.其他的lap k 三角形：(1)任意內角為30°–60°–90°的直角三角形都是lap 3k²，其中k是正整數。(2)邊長比為1：m： 的直角三角形是lap (m²+1)k²三角形，其中m、k是正整數。 To find repeated figures, we construct a method to search them with the help of algebraic equations. Here we arrive at：1. There are only two kinds of lap 2 convex polygons, triangles and quadrilaterals. (1) The only lap 2 triangle is isogonal right-angled. (2) The only lap 2 quadrilateral is the one that contains angles 45°, 90° and two neighboring sides with the ratio 1: . 2. There are also two kinds of lap 3 convex polygons, triangles and quadrilaterals. (1) The only lap 3 triangle is the one with angles 30°, 60° and 90°. 3. Other kinds of lap k triangles are listed as following: (1) A triangle with angles 30o, 60°, 90° is a lap 3k², the k is a natural number. (2) A right-angled triangle whose ratio is 1 : m : is a lap (m2+1)k², the m and the k are natural numbers.

The purpose of this research is to create a device that is able to precisely measure small dynamic changes which cannot be recognized by the human eyes. The Vernier Caliper and the screw micrometer are common tools used to precisely measure lengths of objects. However, things which are measured by the Vernier Caliper or the screw micrometer have to be in a solid state, and the shape cannot be changed. By applying the light lever principle on Lego bricks, this research uses the LabVIEW graphical programming system to design a device which is able to automatically measure small dynamic changes. The precision of this device is higher than that of the Vernier Caliper and the screw micrometer. Moreover, this device is able to precisely detect the small dynamic changes of solids and liquids as well. Through numerous tests, the least count of the device can reach the level of 10-3cm. Also, this device has successfully measured small changes, such as the height of the liquid surface by one drop of water, the evaporation of water in one minute, and the growth of a plant in one hour. By popularizing this device, people will be able to precisely measure small dynamic changes which are difficult to be measured in a short time.

1. Purpose of the research While studying about insects for school club activity, we found that there are differences in several anatomical characters of the insects according their habitats. Especially, the unguis was different as whether the insect lives in water or on land. So we observed the structure of unguis of some insects by microscope, and physically analyze to relate with the habitat of each insect. 2. Procedures First, we read papers and books about insect morphology to study about insect's unguis. Then, we collected samples of Chironomus plumosus(larva), Neuronia regina(larva), and Carbula humerigera. After pretreatment of samples, we put them in the SEM (Scanning Electron Microscope), observed the unguis of each insect and took pictures. 3. Data First, the larva of Chironomus plumosus has prolegs with numberless hooks that has a certain arrangement and the same angle of 90º. They also has several tiny swellings around the hook. Next, the larva of Neuronia regina has pincer-like unguis which were sharp and bend, reminding the shape of a quadratic curve. Lastly, Carbula humerigera has two large, thick pincer unguis, its form same as the of Neuronia regina. The unguis are very sharp and faced towards the land. 4. Conclusions Both the aquatic insects and the terrestrial insects have structures in unguis developed to increase precision on land. Especially, the angle of unguis were all close to 90º. In addition, aquatic insects like the larvas of Chironomus plumosus or Neuronia regina have particular characters increasing friction force according to resist the flow of water.

複雜系統廣泛地存在每個人生活的周遭，儘管這些系統在表象上有明顯的差異，卻不約而同的都呈現出臨界現象。一個簡單的物理概念卻能廣泛地使用在許多不同的場合當中，這是令人始料未及的。這份報告在尋找，是什麼原因導致這許多似乎與物理沒什麼關聯的系統，都呈現出臨界現象？以及這類系統是如何演化的？臨界現象在系統的演化歷史中，又是扮演怎樣的一個角色？一開始，先介紹臨界現象在實際系統中表現的形式，在這以文字系統作為範例。接下來，進入思考的階段，猜測系統個體間存有的關聯性，提出如何影響的假設，並建立模型。從模型結果，可以了解複雜系統隨時間演化的趨勢改變，並發現臨界現象在系統中扮演的角色以及代表的意義。這份報告有三項結論：1. 臨界現象在複雜系統中呈現的形式 2. 模型結果與實際系統十分相似 3. 臨界現象代表著系統的穩定態。The theory of Scaling Law and Universality was originally used in researching Critical phenomena, and now we find it existing in our normal life. A simple physical concept can be used extensively in different courses. This study try to find out the relation between individuals which causes Critical phenomena appear in such different courses in our normal life, and figure out the effect time brings about. To begin with, we introduce the sights of Scaling Law and Universality by exploring the formation of complicated system. Facing a complicated word system as a real example, we observe an extraordinary phenomena, and find out the Universality and Scaling Law in different kinds of linguistics. In the main part, to think further, what’s the relation between the individuals that leads to Critical phenomena? We will give hypotheses and build a model to simulate complicated system from the view of Universality, trying to find out the cause of the interaction of complicated system. From this model, we could understand: When a complicated system evolves with time, the Critical phenomena will naturally occur in this system that could be considered as dynamic equilibrium. From this study we found out(1) The Universality and Scaling Law in complicated systems, here we use linguistics as an example.(2) Under the hypotheses, the simulation is almost as same as the real result of linguistics system.(3) Time plays an important part in the Critical phenomena which can be found in many different complicated systems, and Critical phenomena symbolize a stable state of such systems.

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.”

本實驗之目的在利用骨髓間質幹細胞 (Bone Marrow Mesenchymal Stem Cells, BMSCs) 培養為平台，篩選能提升BMSCs成骨分化效率的生物材料。實驗中利用osteogenic induction medium (OIM)使BMSCs朝成骨分化路徑進行，分別在第7(前期)、14(中期)、21(後期)天作Alizarin Red Stain assay(ARS assay)測定鈣沉積之效率作為評估標準，而後於平面上塗覆不同生物材料進行測試，根據實驗結果顯示，Gelatin主要在中期就可達到最高成骨分化效率，而PCL與HA到後期亦能顯著提高成骨分化效率。當進入3D立體成骨誘導試驗時，我們發現Gelatin、HA的複合材料能保持平面之特性且結合二者優點，並且更能有效提升成骨分化之效率。

給定正n邊形，於內部區域新增對角線，使得對角線不交叉且內部區域皆為三角形，則將此圖形稱為正n邊形的一個『三角剖分』。考慮正n邊形的所有三角剖分，已知其數量為卡特蘭數Catalan(n-2)。在所有三角剖分的情形中，考慮旋轉與翻轉，將同構的情形視為相同，則將所有不同構的三角剖分總數記為Dn。本文對於正n邊形的不同構三角剖分進行研究，以三種面向進行探討，首先我們以不同構三角剖分的對稱性分類，發現其和化學式CnHn+2的同分異構物有相關性；再者，以相鄰三角形的最大數量進行分類，當相鄰三角形的最大數量為n-2、n-3、n-4與n-5時，得出不同構三角剖分的計算通式；最後，以三個相鄰頂點組成的外圍三角形進行分類，將『恰包含兩個外圍三角形的不同構三角剖分』、『圖論中的毛毛蟲圖（Caterpillar）』以及『化學結構相關的Losanitsch’s triangle』進行深入探討。

葉綠素為植物體內進行光合作用關鍵物質，在能源領域內也有不少的應用。葉綠素是光合作用的鑰匙，葉綠素能夠累積光的能量並放出激發態電子，利用激發態電子的能量進行一連串複雜的反應，本研究的理論就建立在光合作用上，利用葉綠素吸收光能轉為電能的特性開發新型光電池。光電池使用金屬或非金屬材料，照光後能產生電流之材料作為發電材料。電池的製作過程非常簡單，實驗發電材料葉綠素也很容易取得，完全沒有汙染環境的疑慮，是非常乾淨的綠色能源。另外也探討葉綠素濃度對於發電效能的影響。研究使用的裝置是由兩片玻璃與矽膠片組裝而成的電池，在電池中間加入葉綠素與電解液。藉由改變電解液與有無光照來探討葉綠素在該電池中的氧化還原作用是否有明顯的變化，並推出化學反應式。結果顯示在葉綠素與電解液的配合下能夠建立出一套循環產電系統。

臺灣家庭污水污染量占總水污染的68.8%，而其中油污更是主要成份。本研究目的為篩選可分解家庭廢水中油污之微生物，並探討其降解油污的能力以及生長情形。本實驗從有油污的環境分離出三種對於家庭廢水中油污具有降解效果的菌株，經過16S ribosomal RNA gene菌種分析鑑定，得知其可能為Ralstonia sp. TFD41、Pseudomonas putida strain II-B、Sphingomonas sp. NC110。研究三種嗜油菌的最佳的生長環境條件，發現Sphingomonas sp. NC110在所有不同種類的油當中成長情形都佳於另外兩種嗜油菌。而針對Sphingomonas sp. NC110進行相關的酸鹼值及溫度測試，發現最佳生長條件為pH7以及30℃。進一步將嗜油菌放入汙水中，發現Sphingomonas sp. NC110在污水中的生長菌數最多；Ralstonia sp. TFD41的生長菌數最少，Pseudomonas putida strain II-B的生長速度最快。將嗜油菌放入100 ppm油的培養基中，以Sphingomonas sp. NC110的降解油污的能力為最佳、Ralstonia sp. TFD41的速度最快。在家庭汙水中Ralstonia sp.TFD41 + Pseudomonas putida strain II-B的生長狀況最穩定，三種嗜油菌同時培養並沒有加乘效應。未來希望能培養嗜油菌，將其放入汙水中，降解汙水中的油，以減少油對於環境的汙染。

1. Purpose: The purpose of this project was to design a theoretical tempered-air system to be integrated into an existing heating system in a home equipped with an air-to-air heat pump. This was to overcome the cost and environmental challenges of heating in cold climates as well as provide an environmentally friendly air-conditioning system in the summer at little to no cost. 2. Procedure: In the winter of 2010 an underground high-density polyethylene (HDPE) pipe (10cm diameter, and 30.5m length) was buried 2.5m underground around the footings of a new residential project. A fan with a flow of 3.4m³/min was attached to the pipe outside, and used to push air underground through the pipe and into the house. The ambient and incoming air temperatures as well as the date and time were recorded daily using a temperature probe with an error margin of ±0.1ºC throughout the summer and winter seasons of 2011. This data was then plotted and analyzed. A number of options to best extract thermal energy for potential use for heating and cooling were examined. Design components were selected to create an air chamber for an air-to-air heat pump to increase its coefficient of performance (COP). 3. Data: Graph 1 Graph 2 Winter Graph (Graph 1): The blue line represents the outdoor ambient temperature and the orange line represents the temperature of the tempered air. These are both arranged chronologically. The grey lines represent one standard deviation on either side of the incoming temperatures. The ambient temperatures vary dramatically, while the incoming temperatures remain quite stable. The lowest recorded temperature was about -9.0ºC, at which point the temperature exiting from the underground pipe was about 10.5ºC. Summer Graph (Graph 2): With dramatically varying ambient temperatures, the temperature of the incoming air consistently stays between 11.1 and 16.2ºC. The highest recorded outdoor temperature was about 30.0ºC, while the incoming temperature at that point was about 15.5ºC. This cold air was used as air conditioning.