本研究中，我們將提出一些新穎結果，著重討論其在三角中的應用；同時，找出其遞迴關係式，得出三角展開式與其所對應之多項式分解式，進而討論出多種的規律性及所涵蓋的內容及推廣性質，我得到很多高中數學公式無法推導出在【4】和【8】中的漂亮公式及創新的結果，且這些等式都是由我們不太瞭解的無理數所構成的。 主要是討論我們在【7】中所得到的收穫與經驗；複數是三角、幾何、代數互動的橋樑，我是以不同的角度及嶄新的方法來綜合探討在【6】中相關的應用。提出關於正整數平方的倒數和公式更為精簡且基本的證明，將 sin−2 x 表示成級數形式的部分分式，進而應用在(a,b) = 1的機率問題上；並研究相關的等式，直接透過三角與代數來研究關於 2p 次方的倒數之求和問題，得出級數 之和的有用遞迴公式，並與最重要的常數扯上關係。 For one thing, we present diverse methods to evaluate finite trigonometric summation and related sums. Trigonometric summations over the angles equally divided on the upper half plane are investigated systematically. Several related trigonometric identities are also exhibited. What is more, we use methods of calculus, and make several surprising and unexpected transformations. A useful recursive formula for obtaining the infinite sums of even order harmonic series, infinite sums of a few even order harmonic series, which are calculated using the recursive formulas, are tabulated for easy references. Furthermore, is there any interesting results and applications？ Finally, the purpose of this paper is to develop a new proof of and related identities, but their derivations are more complicated. The following studies are completed under the instruction of the professor.

近年來重金屬污染問題日益嚴重，因此檢測地下水、工業廢水中的重金屬含量為一重要課題。本實驗即是利用錄音機中的零件－石英振盪器作為電化學分析偵測的工作電極，在0.1 M KNO3（pH = 3）溶液中利用循環伏安法以50 mV/s 的掃描速度從 – 100 到 – 700 mV 持續掃描 10 圈，來穩定電極表面。 以方波陽極剝除伏安法（SWASV）模式設定預濃縮電位– 700 mV、預濃縮時間60 秒偵測重金屬的鉛離子。掃描電位範圍由 – 700 到 – 100 mV vs. Ag/AgCl 往氧化方向掃描。偵測鉛離子濃度範圍為200－600 ppb，之後可針對各種條件作最佳化的條件探討可降低偵測濃度及實際運用於地下水及工業廢水的檢測。 Heavy metal pollution becomes very serious. Lead (Pb) determination is a continuous research interest in clinical, environmental, and industrial application. The main causes for living biological media are mutagenic, carcinogenic, and teratogenic activity. The determination of trace metal in the ground and industrial waste water will be a important project. Here, we report a simple system which using quartz crystal microbalance (QCM) (use in radio machine) for working electrode. The QCM was first equilibrated in 0.1 M KNO3（pH = 3）base electrolyte for about 1 min before electrochemical experiments. It was then pretreated by continuous scans in the window from – 100 to – 700 mV at a scan rate of 50 mV/s until a stable background current obtained. The amount of Pb2+ was detected quantitatively using square-wave anodic stripping voltammetry（SWASV）. The potential range was set from – 100 to – 700 mV in the cathodic direction for most cases. The standard addition method concentration range is from 200 to 600 ppb. Analytical conditions were systematically optimized and extended to practical applications aslo.

本研究是藉由數學手法探討；如何由一疊 36 張四種花色的撲克牌中，尋找出保證可猜中最多張花色的方法。研究過程是以在適當的猜牌時機，以邏輯推理、二進位、分析與歸納 … … 等數學原理與方法，搭配巧妙的策略運用而達到目的。 猜牌方法：先約定好猜牌規則，助手將 36 張牌背圖樣相同但非對稱的撲克牌，以旋轉牌背的方向傳達訊息。在本研究中得出「經由巧妙的猜牌方法保證可以猜中不少於 26 張花色」，並得出「當總張數趨近於無窮大時，保證可以猜中不少於 81 . 07 ％的牌，並且證出若僅使用獨立的訊息猜牌，無論任何猜牌方法皆無法猜中多於 87 . 37 ％的牌」 · 其中一個猜中多於 80 ％的例子是‘「當總張數等於 23006 張時，保證可以猜中不少於 1 8405 張牌（18405/23006 > 4/5）」 ·The study is mathematically based with reasonable explanations behind it. We are to correctly guess as many cards as possible from a deck of 36 cards, with random numbers and four different suits. We will apply mathematical methods, such as logic inference, binary system, and analytical reduction, upon right timing. Using careful arrangement of the principles and reasoning, we can reach our ultimate goal. To state guessing: Conference between the guesser and the assistant about the guessing rules, the assistant will have 36 cards with the same exact pattern on the back but not symmetrical. The pattern of the cards will be different when rotated 180o. The only communication between the two is by rotating cards. In this study, we can prove that through mathematical method, we can assure 26 or more cards can be correctly guessed. Furthermore, when the total amount of cards is close to infinity, we can assure 81.07% or more of the cards can be correctly guessed, and prove that if the cards are guessed from independent information, no more than 87.37% of the cards will be correctly guessed by any guessing methods. One of the examples, which 80% of the cards are correctly guessed, is that when the amount of the cards is 23006, 18405 or more of the cards can be correctly guessed. （18405/23006 > 4/5）

在實驗中學2007 年校內科展，參展作品《三角形中的切圓》的研究中，研究三角形內的切圓時，發現連續切圓的圓心與拋物線的軌跡有關。於是去查資料，在偶然的情況下，翻閱《平面幾何中的小花》時，接觸了「六圓定理」。因為覺得這問題非常有趣，於是便著手證明（見報告內文）。 又發現，當移動六個圓中的起始圓時，總是在某種情況下，六個圓會重合成三個圓。繼續研究其重合的狀況，發現了馬爾法蒂問題（Malfatti's Problem）的一種代數解法。 當我試著推廣六圓定理至多邊形時，發現奇數邊的多邊形似乎也有如六圓定理般圓循環的狀況，於是著手證明，但目前尚未證明成功。而偶數邊的多邊形則無類似的結果。 ;In 2007 National Experimental High School Science Exhibition, one of the exhibit works, "Inscribed Circles in Triangles", shows that the centers of the consecutive inscribed circles has something to do with the parabola's trajectory. To learn more about inscribed circles and parabolas, I referred to literature. By accident, I am faced with the problem on six circles theorem, in the book The Small Flower of Plane Geometry(平面幾何中的小花). Out of my interest in this problem, I tried to prove it. The other results are as follows: With the initial circle of six circles moved, in certain circumstances, the six circles merge into three. Further in studying this coincidence leads to an algebraic method to solve the Malfatti's Problem. Applying six circles theorem to the odd-number-sided polygons exists the same characteristic. It indicates that the inscribed circles will form a cycle. However, it hasn’t been successfully proven. The even-number-sided polygons show no similar results.

In order to facilitate a paradigm shift from traditional high volume chemical experiments to an environmentally friendly microdose experiment; I had to innovate and overcome a lot of difficulties. After six generations of experiment design I successfully reduced the volume of the combined reactants to a single drop. I utilized many recycled components to build my apparatus including a vintage computer, a simple CD disk, optical sensors and a transistor from a common computer mouse. By using a CCD monitor with an external camera feed, the single drop chemical reaction can be observed in real time and a recording of the event can be made. I chose the focus of the experiment to be the reaction of sodium thiosulfate and hydrochloric acid to create a colloidal solution of sulfonium nanoparticles. By employing a custom-made transparency target to achieve a higher precision of measurement I have also conducted deeper research into the reaction order and the rate constants. 為了將傳統高劑量的化學實驗順利的推向顯微液滴的化學實驗上，今年我們更是創新突破了很多困難的關卡。 儀器設計由第一代減量研究到第六代的減量設計，目前已能成功的運用報廢的光碟片、報廢的電腦、報廢滑鼠內的感光二極體元件及電晶體來自製設計出兩反應物各一滴溶液做自動偵測反應的記錄。透過顯微鏡加裝的CCD電子螢幕目視觀看、拍成電子影片檔由電腦播放的目標。 因此，我選定了硫代硫酸鈉溶液和鹽酸溶液反應可產生硫奈米微粒的膠態溶液作為實驗的主軸及設計可較精確定量的投影片載液面，我們也企圖對其反應速率式的級數及反應速率常數做更深入的探討。

受應力拉伸時，橡膠溫度明顯上升；縮放回原長，橡膠溫度驟降。由文獻得知橡膠內部具有特殊的鍵鍊結構，在一般的情況下，交鏈分子糾結成一團，狀態複雜；受外力拉伸時，交鏈分子依橡膠長度之增加而伸展，排列較為整齊，狀態之複雜度減小。根據熱力學第一定律，當內能變化為零，則外力作功會造成能量變化。在定溫之下，橡膠內能變化為零，當其受應力拉伸，使其內部交鏈分子排列複雜度降低，造成橡膠熵值減小，而有能量（dQ=TdS）的釋出。測量此一能量dQ 變化，即可計算出熵與狀態數之變化The temperature of rubber rises as it is stretched, its temperature comes back again while it restores to its original length. It is known that the rubber is consisted of long-chain molecules, the long-chain molecules strangle each other at normal state, however, they become more order when the rubber is stretched. Based on the 1st law of thermodynamics dU=dQ+dW, The deformation caused by applied force supplies energy to the rubber and reduce its entropy, the heat dQ (=TΔS) released by the reduction of entropy causes the temperature rise of rubber as dU=0. We report the study on the correlation of thermal properties and the molecular network in rubber, from the measurements of temperature change, the changes of entropy and the changes of states’ number were estimated.

Generally speaking in order for a exotic plant life to dominate over the local ecology, the following factors must hold true: 1. A close proximity of the environment (including geographical and climatic factors) 2. A lack of nature enemy and competitor 3. A strong reproductive and adaptability capacity. Base on the above factors, We will be using Field Observation record, Sample placement, virtual environment factor experiment and other methods to conducing our investigation. This study shown the African Touch-Me-Not（Impatiens wallerana） has a strong reproductive and adaptability capacity. Even under adverse condition the African Touch-Me-Not survive for a period of time. It can produce flowers then fruits all year round, within each fruit there is on average 45 seeds with, under the right condition, near 100% germination rate. It can also reproduce asexually via cutting. Taiwan’s warm and humid geographical condition is similar to its origin of Africa. And so far haven’t found any obvious natural enemy. In the future this plant will be likely to spread on the low to mid. level mountain region. It also can be seen that the divergence within its habitat is low. Therefore it will decrease the variety, the density and the diversification of the local eco-system. Here to advise the government’s forestry and agriculture department, to study the relevant ecological information on this species, when they are considering popularization this plant. Using the information to evaluate the impact of this plant may have on the environment, and use it as a guideline for their plant conservation policy. 一般而言，外來植物入侵本地原生植群成功的條件有：一、相似的環境（包括地理、氣候等因素）。二、缺乏天敵及競爭對手。三、旺盛的繁殖力和適應力。針對上述條件，利用野外觀察記錄、樣區設置及模擬環境因子實驗等方法，來探討非洲大鳳仙未來是否會在臺灣蔓延、擴展進而影響本土的生態系。結果顯示非洲大鳳仙有旺盛的繁殖力和適應力，?使在不利的環境下，短時間不會死亡。一年四季皆可開花結果，每粒果實平均有四十五粒種子。環境適宜時，有接近百分之百的萌發率，亦可用莖進行無性繁殖。另外，臺灣溫暖多雨的地理環境亦與其原產地非洲相似，且未發現明顯天敵，未來極有可能在中低海拔山區蔓延。調查也發現它生長的地區岐異度明顯偏低，使其他植物種類、密度減少，生物多樣性降低。因此建議政府、森林、園藝和農業工作人員在推廣時，多吸取相關物種的生態學資訊，評估其對環境可能造成之影響，以作為植物保育之重要參考。

本研究針對利吉惡地進行探討，研究此區之泥岩含水量、有機質含量、pH值、比重、可溶性陽離子含量、滲水特性和該區之植物種類，並探討坡度、水量對沖蝕率、山脊密度和溝痕形成之影響。 研究結果發現： 一、表層泥岩之含水率較高，深層泥岩最低。中層泥岩之有機質含量較高，表層與深層泥岩較低。各層泥岩pH值約8.1；無植被採樣點之 pH較高，有植被採樣點偏中性。有植被採樣點，其 Ca2+含量較高。 二、此區共發現十九種植物，其中銀合歡、相思樹屬優勢種。 三、坡度增加時，沖蝕率亦增加；水量增加時，沖蝕率、溝痕寬度也隨之增加，兩者呈高度正相關。第一區坡度較緩，山脊密度較大，第二區坡度較陡，山脊密度小，表面較平坦，溝痕較淺。Our research discusses with the contents of water in mudstone, the organic content, pH, the specific weight, the contents of dissoluble cation, dankness and the category of plants in Ligiligi Badland. Dissecting it`s slope, the abrasion of water, the density of mountain ridge and the formation of scuff mark. Outcome of our research： 1.Solum of mudstone is the dampest. Intermediate of mudstone has more organic content than others. Every bed of mudstone`s PH is 8.1. Having plants area is indifferent and having more Ca2+.2.We found nineteen categories of plants. For example Leucaena glauca and Taiwan acacia. 3.The more augmentation of gradient,the more increase of the abrasion of water and the breadth of the density and the abrasion of water too.

自組薄膜(SAMs, self-assembled monolayers)乃是具有特定官能基的化合物在不需外力作用 下自動吸附到基質表面上而自行排列成有自序規則之結構。SAMs技術因製備容易且穩定，在 應用上深具潛力。目前此方法中，以硫醇接在Au上的研究最為廣泛。本作品希望藉著SAMs 方法將硫醇分子吸附於鐵片或其他金屬表面上，以達成防鏽與抗酸目的。我們分別透過接觸 角量測研究正十二硫醇在鐵片上形成分子薄膜的可能性；利用酸與金屬產生氣體的速率研究 分子薄膜抗酸蝕的情況；測試分子薄膜的耐熱性；根據鐵片生銹時pH值變化、重量變化與[Fe2+] 含量差異來探究分子薄膜能否防鏽；最後找出適合形成分子薄膜的濃度與溫度效應。由實驗 結果我們發現SAMs薄膜確實能吸附於鐵(及鎳、銅、鋅等金屬)的表面上並增加抗酸與防鏽蝕 能力，薄膜對熱的穩定性極佳，35℃時結合效果較佳，而濃度的提高有助於SAMs的效用。 Self-assembled monolayers(SAMs) are elements which have specific functions.SAMs,ordered molecular aggregates can automatically adhere to the surface of substrate without any force.The applcation of SAMs’technique has high potential not only because ther are easy to make but also because they are stable.Exposing molecules such as alkyl thiols to an Au(0) surface is now in widespread use. In this work,we apply the thiols chemisorb onto the Fe or other metal surface to make it rust-resistant and acid-resistant.We study the following issues to find the appropriate conditions of forming monolayers in varying concentration and temperture: I. The possibility of forming n-dodecanthiol molecular monolayers on Fe surface by measuring the contact angles. II. The ability of antiacid corrodibility by comparing the rates of producing gas from acid and metal. III. The heat-resistant of molecular monolayers. IV. Whether it’s antirust by detecting the changes of pH,weight,and the concentrations of Fe2+ during the iron rust. According to the results,we conclude SAMs do adhere to the surface of Fe(and other matal like Ni,Cu,Zn), which increases the ability of antiacid and antirust.Besides they are stable to heat, have good combining effect at 35℃, and it is beneficial to the effect of SAMs through raising their concentration.

想像一下，原來固定的一個黑色大正方形(以下我們簡稱S1)四個角經過較小之正方形(以下簡稱S2)遮蓋後旋轉，為什麼會有放大縮小的感覺呢?(圖一) 我們猜測這是因為未被遮蓋部分(以下簡稱S3)的面積改變了，所造成的結果。令S1的邊長為 r，S2的一個小正方形邊長為n，k=r-n，θ為旋轉角度，我們實際算出了旋轉角度θ對S3的面積函數，