臺灣

將一個集合{1,2,...,n}分成數個非空的集合（組，區塊），稱為此集合的一個分割。如果可以找到1 ≦ a 已知無孤立點無交錯分割以Riordan 數{rn}n≥0 =1,0,1,1,3,6,15,36,... 來計數。在這篇文章中我們研究無孤立點無交錯分割的一些性質。 首先我們考慮無孤立點的無交錯分割按區塊的細分。我們得出：集合{1,2,...,n}恰含k個區塊的無孤立點的無交錯分割的個數為： 其次，我們證明bn,k和多邊形的剖分有令人訝異的關連。令dn,k是用不相交對角線將凸n 邊形分成k 塊的方法。我們用代數方法證出 bn,k = dn+2−k ,k，也給了一個新的組合證明。 最後，透過對應的方法，我們找出了七個嶄新的組合結構，這些結構都是以Riordan 數來計數。 Partition the set {1,2,...,n} into several nonempty sets (blocks) and call it a partition. If there exists 1 ≦ a It is known that the nonsingleton noncrossing partitions are counted by Riordan numbers {rn}n≥0 =1,0,1,1,3,6,15,36,... In this paper we study the properties of them. First we consider the enumeration of nonsingleton noncrossing partitions in respect to the blocks. We prove that the number of nonsingleton noncrossing partitions of {1,2,...,n} with k blocks is Then we give a connection between nonsingleton noncrossing partitions and polygon dissections. Let dn,k be the ways to dissect an n –gon with noncrossing diagonals. We prove that bn,k = dn+2−k ,k We also give a combinatorial proof. Furthermore, by way of the technic of bijection, we find 7 new combinatorial structures counted by Riordan numbers.

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.

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.

本實驗以市售本氏液與自製本氏液與葡萄糖、果糖、麥芽糖等還原醣反應，證明課本上以藍、綠、黃、橙、紅的顏色變化來顯示葡萄糖的含量多寡有誤。透過過濾觀察產物顆粒、濾液混色實驗、利用廷得耳效應以雷色光照射觀察產物顆粒大小、及溶液靜置、離心等實驗，發現藍、綠、黃、橙、紅顏色變化可能是奈米級(顆粒大小介於10-9~10-7 m)的紅色氧化亞銅，與藍色硫酸銅溶液混色結果，故不能以顏色變化來代表還原醣的多寡。若要定量葡萄糖，可由葡萄糖與本氏液反應的「顏色與反應時間關係對應圖」中，反應終點顏色(紅色)到達時間來定量葡萄糖。

從小習染之餘，對篆刻藝術的濃厚興趣及對電腦程式之熱愛，萌生將篆刻藝術與電腦 科技結合，於是應用PHP（Professional Home Page）的文字圖形（Text Image）處理功能 將中華文化之篆刻藝術透過網際網路推廣到世界各角落。本研究規劃建置了一個 FreeBSD Server 可完善處理PHP 功能的作業平台、應用PHP 的文字圖形處理功能，編 寫可在線上DIY 設計印章的程式，以及藉由Java Script 的技術在Client 端建立一個人 性化且相容性高之使用者界面網頁，達到提供多語系、多功能、依使用者偏好作調整 印章及輸出分享的功能。為使非漢語系國家能深刻體驗中華篆刻藝術，本研究同時建 立一個超過三萬筆之由英文名音譯中文名的資料庫，提供給使用者切身的服務。 Inspired by the interest of the art of seal engraving and the love for the computer programming beginning from my childhood, I combine the art of seal engraving and computer technology, and apply the Text-Image's function of PHP to promote the seal-engraving art of Chinese culture via the Internet to everywhere around the world. This project constructs a FreeBSD Server to process PHP platform perfectly, applies the Text-Image's function of PHP to write a program for online DIY pattern design, and uses Java Script to establish a human and compatible user-interface web page for clients to provide multi-language, multi-function, and being able to adjust pattern according to the user's preference and output sharing functions. This project also sets up a database of English-Chinese translation of over thirty thousand English terms to help people who are not Chinese to experience the art of seal engraving.

本研究利用佛手瓜的卷鬚探討向觸性的原理。本研究大致分為兩部份，一方面我們在卷鬚中發現了含量極為豐富的構造，此一螺旋狀構造分布於維管束中，且用雙縮?詴劑檢測後發現其含有蛋白質，且不具有運輸水分的功能；並發現此一構造的分布疏密，會影響到螺旋內側外側以及切割後片段泡溫水的彎曲方向。此外，在進行卷鬚蛋白質電泳的過程中，我們發現使用含尿素的緩衝液萃取蛋白質的效果最佳，1克的卷鬚乾重約可萃取到5毫克的蛋白質，且蛋白質總量會隨著卷鬚的成熟而遞減。利用軟體比對及質譜分析八個蛋白質點，得知此八點的蛋白質為：malate dehydrogenase, oxygen-evolving enhancer protein 1, oxyen-evolving enhancer protein 2, calreticulin, peroxidase, stromal 70 kDa heat shock-related protein, and AP2/ERF and B3 domain-containing transcription repressor。由此可知，向觸性為植物經過一連串訊號傳遞後，對外界刺激的順應。

Tea is the most widely accepted and consumed beverage worldwide due to its characteristic aroma and taste. Recent studies have provided the strong scientific basis for understanding the health promoting effects and cancer preventive actions of tea. The components of tea especial the catechins are varied with the conditions of making tea. To understand and determine the chemical composition of tea is very important. Some investigations of the parameters on the storage and making of tea were carried on in this study. The kinds of tea studied were including black tea, oolong tea, green tea, and instant tea bags. A high performance liquid chromatograph combined with UV detector and mass spectrometer used to analyze the components of tea. The results showed that the composition of tea solution is dependant of the exposure of light. One of the components of tea, (-)-epicatechin methylgallate (ECMG) was oxidized to quino form. The concentrations of (-)-catechin gallate (ECG) decreased and one of new compound (M.W. 442) produced with increasing the storage time. From the results show tea made with cold water is better than that made with hot water. The ingredients in green tea were changed faster than those in fermentative black tea and oolong tea. The components of tea can be kept unchanged for a long time at low temperature. In refrigerator, the time can be extended to overnight. 茶由於其具有特殊的芳香氣味及口味，廣泛地被世人用為飲料，近年來的研究證據顯示茶具有促進身體健康和防癌功效，但茶中成分之變化，尤其兒茶素隨茶的種類、茶沖泡保存方式及置放時間而有所不同，因此對於茶中成分及沖泡方式的認知是一門重要的課題。 本實驗主要是探討茶沖泡時間及貯存條件，對於茶液所含兒茶素變化的影響，探討的茶包含紅茶、烏龍茶、綠茶及其茶包。並使用質譜儀及高效能液相層析質譜儀配備紫外光偵測器分析茶液中之成分，實驗結果顯示成分變化速率以未發酵的綠茶較發酵的紅茶和烏龍茶快；低溫儲存時，亦可延緩茶液成分的變化，例如加蓋並存放於冰箱，則茶置放至隔夜其成分均未改變；同時結果顯示成分產生變化者，主要為兒茶素氧化，如將多酚類氫氧基經氧化變成?（quino） ，其次為斷裂再產生聚合其中綠茶以不照光變化較大。本研究並經實驗發現如果以冷開水沖泡綠茶20~30分鐘，咖啡因的溶出量雖略多於沖泡2分鐘的熱茶，但各種兒茶素的溶出量卻遠高之，尤其置放至隔夜茶液成分仍無氧化現象。

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.

很多人都有泡奶粉的習慣，卻偶爾會發現在舀完奶粉後，會有少許奶粉殘留於桌面上,仔細的觀察後發現~舀奶粉時塑膠匙上有叢狀奶粉聚集，且部分奶粉會噴射出去,若奶粉是一般的電偶，理應在排列好後維持穩定的位置，相反的，若奶粉有帶上與塑膠匙面同性的電荷，它應該不會沾附於匙面，而是被斥離。但我們的觀測結果卻同時包含以上兩種現象，我們對此感到好奇而展開研究。

流行病的傳染過程如同一個無尺度網路，但較一般無尺度網路有著更多的變數而明顯差異，因此無法直接應用一般的無尺度網路模式來描述其傳染途徑。我建立一個新模式「無尺度流行病模式」，經由比較模擬結果與疾病管制局的數據，證實此「無尺度流行病模式」是正確與確切可用，且適用於短期暴發性傳染病與長期流行病。SARS案例研究結果，顯示影響SARS疾病傳染因子的大小是：ψ>m>γ。其中降低ψ值可使SARS確定病例至5月31日止降為143人(減少確定病例190人，相當於減少死亡21人)；僅提高防疫使5=γ，亦可使確定病例減至307人(減少確定病例26人，相當於減少死亡3人)。因此強化隔離措施以減少傳染天數最為重要，且可以有效控制每日SARS新增病例，避免發生高侵襲率的現象。HIV/AIDS案例研究結果，獲知採用ψ值來進行月份模擬，則至 2005年12月HIV(+)與AIDS分別為可減少2,715與285人。而進行年度模擬結果，則至 2014年底HIV(+)與AIDS分別為可減少41,936與5,328人。無尺度流行病模式可以協助所需警戒的程度與政策決定的計畫結果。因此無尺度流行病模式在幫助政府評估社會經濟成本與健康憂慮上的有用之工具。當面臨一個全然無知的新病毒的侵襲時，如何減少死亡與傷害人數？是本研究之最終目的。因此，本研究結合了流行病、無尺度網路與灰預測，建立面對病毒侵襲，一個確切可行的對抗無尺度流行病傳染新方法，並詳細說明運作流程。\r \r \r The course of epidemic infections resembles a scale-free network. However, they are different due to more variables in the epidemic infection. Therefore, the model of scale-free networks is not enough to satisfy the reality epidemic infections. In this study, I propose a new the Scale-Free Epidemic Model. Comparison of the simulation results with Taiwan CDC report data for SARS and HIV/AIDS cases show that the Scale-Free Epidemic Model is accurate and useful. This model can be used in the short-term outbreak of infectious diseases and for the longer-term epidemics. In the SARS case study, the results show that the sequence of effect of the epidemic factors was: ψ>m>γ. The SARS confirmed cases would decrease to 143 cases (reduced 190 confirmed cases or 3 death cases) calculated to May 31, 2003, if the average infection time was reduced to two days (an optimum value of ψ). Therefore, vigorous action in isolation quarantine and treatment for SARS cases is most effective policy; the number of new cases and the attack rate would also decrease. In the HIV/AIDS case study, the simulation results of the Scale-Free Model indicates that the reduced numbers of HIV(+) and AIDS in the monthly simulation calculated to December 2005 are 2,310 and 361 and the annual simulation by December 2014 are 27,161 and 3,710. The Scale-Free Epidemic model can help determine the level of caution needed and the projected results of policy decisions. Therefore it is a useful tool in assisting the government to balance socio-economic and health concerns. The fight against a new epidemic and how to reduce the number of deaths is the main purpose of this study. So, a new method to fight against epidemics is proposed. Detailed procedures of this method are explained.