出國代表作品

The Look and Say sequence is produced by describing the appearance of the previous row. For example, start with “1,” which can be described as “one 1,” and therefore the second row is “11,” which is "two 1s," making the third row “21,” the fourth row “1211,”and so on. The main goal of this study is to work out the exact formula for this sequence, which means given the row number n, we can know at once what the n-th row is without having to start from the first row and doing the look-and-say iteration for n-1 times. Some of the methods used include dividing groups, repetition and cracks. The formula we derived speeds up the calculation and gives us a better understanding of the look and say sequence.「外觀數列」為依照外觀產生下一列的數列，第一列為「1」，第二列描述第一列「1 個1」而為「11」，第三列則描述第二列「2 個1」而為「21」，第四列「1211」，依此類推。本研究針對外觀數列的各項數學性質作研究探討，並由此推導出外觀數列的一般式，即給定第n 列就可知道該列的內容。我們運用了分組、重複性以及裂縫的方法分析數列，最後得到了其一般式，此一般式有助於運算速度的加快以及我們對數列性質的了解。

在學校科研營的教材中，有一個題目，其內容相當於:「在一列格子中 放入黑棋與白棋。白棋不可連續放置，而黑棋不受此限，請問共有幾種可能的排列方式?，在此規則下，若將格子推廣為m列n行的棋盤，那又如何呢?我們對此好奇不已。

本實驗目的在於嘗試降解傳統塑膠材料。實驗中選用極為常見的傳統塑膠，聚對苯二甲酸乙二醇ploy ethylene terephthalate (PET)，與生物可分解材料，聚乳酸poly lactic acid (PLA)，將兩者以不同比例物理性混摻，並以豬胰線脂肪酵素Lipase from porcine pancreas, TypeⅡ (PPL)進行降解，期望藉由生分解材料可被完全分解的特性牽引傳統塑膠材料進行降解。\r 本實驗共分兩階段：第一階段實驗中，將共溶劑揮發製膜進行降解，但由於溶劑法所留下的孔洞，影響了降解速率；因此，在第二階段實驗中，改以熱壓的方式製膜解決孔洞問題，並於降解後以儀器分析降解情形。結果發現：混合薄膜經降解後熔點有下降的趨勢，此可佐證確實可以生分解材料PLA牽引傳統塑膠料PET進行降解，且降解速率和結晶情形和分子鏈段纏繞程度有關，在PET與PLA濃度同為3%時降解情形最佳。

本研究所探討的主題是一道分割棋盤的題目：\r 給定一個黑白相間的mxn棋盤(設m為縱向邊，n為橫向邊)，將其分割成若干個面積大小都不等的矩形區域，且每個矩形的黑格和白格個數相等。設f(m,n)為符合上述條件的最多分割矩形的個數，則f(m,n)是多少？\r 在研究過程中，我們首先藉由對各棋盤進行分類，利用不等式的運算找出其中的關係，求得當m=n以及mn=k(k+1) 時(其中K為正整數)的f(m,n)值，並構造出其分割方法。而k(k+1)

交叉波（ cross wave ）為波峰和振源表面垂直的波，移動方向和振源表面平行，它是以表面張力為恢復力的表面張力波（ capillary wave）的一種。

Sicherman 已經找出與兩顆六面的正常骰子有相同機率分布的Sicherman 骰子，並進一步獲得與三顆六面的正常骰子有相同機率分布的骰子必為一對Sicherman 骰子與一顆六面的正常骰子之結果，我們試圖由已知的Sicherman 六面骰子的處理方法出發，透過對割圓多項式的分析來累積足夠的相關資料，以處理由兩顆四面骰子至兩顆三十面骰子，處理由三顆四面骰子至三顆三十面骰子的各種Sicherman 骰子的答案，來探索兩顆與三顆的n 面Sicherman骰子存在的充要條件與求法，並進一步將所得之結果分類，得到 ”有相同標準分解式的類型的數n，會具有相同組數的Sicherman 骰子”之猜測結果與特殊情形下的證明。 Sicherman has found out the Sicherman dice which have the same probability distribution as the normal two six-sides dice. Furthermore , he also found out a pair of Sicherman dice and a normal six-sides dice has the same result as 3 normal six-sides dice . We try to begin with the given algorithm of six-sides Sicherman dice , through the analysis of Cyclotomic Polynomials to accumulate sufficient related information then to come up with the solution from discussion of 2 four-sides dice to 2 thirty-sides dice , from 3 four-sides dice to 3 thirty-sides dice to explore the existence of necessary and sufficient condition and solution of 2 n-sides Sicherman dice and 3- sides Sicherman dice , and even to classify the results to come to a conclusion of the guessing results and proofs under special cases about “the numbers n which have the same Canonical Prime Factorization will have the same numbers of n-sides Sicherman dice.”

中文摘要 本實驗先尋求將廢棄保麗龍磺酸化為陽離子交換樹脂(本實驗稱”保麗龍膠”)的方法。將保 麗龍依：丙酮溶解→硬化→打碎→與濃硫酸共煮三小時→浸於50%硫酸溶液中→沖洗→以水 浸泡的流程，即可達再造的目的；我們測得其磺酸化比例為62.5%。再利用「碘滴定法」(浸 泡式)與「相對電壓檢測法」(流動式)，依次尋求保麗龍膠吸附金屬離子的最佳條件。其中「碘 滴定法」可有效測出銅離子濃度，但手續繁瑣；「相對電壓檢測法」最大的好處是知道保麗龍 膠何時吸附達飽和必須再生。 目前我們所知，要保麗龍膠達到吸附陽離子的最佳效能，其條件依次為：使用細粒的保 麗龍膠；低濃度的金屬離子溶液；質量愈大的保麗龍膠；低溫下較慢的金屬廢水流速及pH 值約為4.30 的銅離子廢水；鈉型的保麗龍膠吸附效能優於氫型。保麗龍膠對不同金屬離子亦 有吸附力，單位體積所含離子數愈少，初始的相對電壓會愈高；在相同莫耳濃度下，不同離 子的吸附力依次為Cr3＋＞Fe3＋＞Ni2＋＞Cu2＋＞Co2＋；分次吸附確可將金屬離子完全去除；由 吸附等溫線觀察得知，可能保麗龍膠為多孔物質，導致500ppm 以下的吸附模式無法明確判 斷，1000ppm 以上則為物理吸附模式；保麗龍膠可以再生也可被覆在砂粒上達到不錯的吸附 效能；最後，我們將吸附過金屬離子的保麗龍廢膠與硫酸鈣、紙漿及些許的石灰(質量依序為 13 克、13 克、7 克、0.04 克)混合，可製成類似紙黏土，做成造型磁鐵，廢物利用十分有趣。 Abstract The Experiment will, first of all, explore the ways to sulfonate expandable polystyrene into cation ion exchange resin (called “polystyrene rubber” hereafter in the experiment). The procedures of treating expandable polystyrene are as follows: acetone dissolve→hardening→smashing→ boiling with sulfuric acid for three hours→immersing in 50% sulfuric acid solution→washing→ immersing in water so that we may reach the goal of reconstruction. We calculate the sulfonated rate to be 62.5%. Then we make use of “Iodine Titration”(immersion method) and “Opposite Voltage”(floating method) to seek for the best conditions of adsorption the metallic ion through polystyrene rubber. The former can effectively calculate the concentration of copper ion, but the procedures are quite complex. The greatest advantage of the “Opposite Voltage” method is that we may know when the adsorption of polystyrene rubber is saturated and should be regenerated. As far as we know at present, the conditions of obtaining the best effect that polystyrene may adsorb the cation ion are as follows: fine particles of polystyrene rubber; low concentration metallic solution; polystyrene rubber of which the mass is greater; at lower temperature, slower waste water flow speed and the copper ion waste water with pH 4.30; the adsorption effect of sodium type polystyrene rubber is better than the hydrogen type. Polystyrene rubber also has adsorption effect toward different metallic ion. The less ion per cubic contains, the higher the original opposite voltage. With the same mole concentration, different ion adsorption effects may range as follows: Cr3＋＞Fe3＋＞Ni2＋＞Cu2＋＞Co2＋. The batch adsorption definitely may erase metallic ion completely. By observing the adsorption isotherm, possibly because the polystyrene rubber is a multi-apertured matter, we find that it is impossible to judge exactly the adsorption model of those metallic ion solutions of which the concentrations are below 500ppm. Those which are over 1,000ppm belong to physical adsorption models. Polystyrene may be regenerated and get an adsorption effect by coating sand particals. In the last analysis, we may make paper clay and magnets of different styles by mixing the adsorbed metallic ion polystyrene rubber with calcium sulfate, paper pulp and a little lime(the mass are respectively 13g, 13g, 7g, and 0.04g). The reuse of waste is really very interesting.

人類目前廣泛的應用奈米鑽石，奈米鑽石粒子極為微小因此很容易進入生物體內，對於生物體所產生的影響為本次的研究對象：奈米鑽石、奈米磁性粒子之生物毒性與其在生物體的分布。\r 加入奈米鑽石、奈米磁性粒子之後綠豆發芽數減少。發芽的綠豆中加入不同量的奈米鑽石、奈米磁性粒子，發現濃度越高生長速度越慢。在綠豆吸收奈米鑽石、奈米磁性粒子後，在根、莖、葉細胞內都有發現奈米鑽石、奈米磁性粒子。在麵包蟲的研究中發現，在0.25克濃度下，有些微的影響，而在0.75克和1克之影響甚大。在人類胎盤滋養層細胞中，發現加入奈米鑽石800微克以上對其有毒性。在細胞的分布，可發現細胞質及細胞核中都有奈米鑽石，當細胞內太多或作用太久時，會出現細胞凋亡的現象，此時奈米鑽石會大量聚集在細胞核，這可能是造成細胞死亡的原因之一。本研究意外發現在細胞分裂時，奈米鑽石只分布在細胞核周圍的細胞質，但是在赤道板及染色體上並沒有。

A vast amount of research has been conducted on various ant species such as Monomonum pharaonis, Formica rufa, and Lasius niger, demonstrating the coexistence of polygyny and monogyny colonies. In polygyny colony, the high heredity variability of filial generation could be expected, but what could the monogyny colony do to hold the high heredity variability in order to increase the fitness? I hope I can account the sex ratio in the colony of spiny weaver ants, Polyrhachis dives, and compare the different colony with different reproductive strategies. I also want to check by V8 that the colony can form a polygyny colony and virgin queens can mate with many males of the same colony in the artificial nest. The results suggest that in the colony of spiny weaver ants, the higher the number of the dealate queens, the lower the number of the male. In the monogyny colony, workers control the sex ratio. According to the observation, we can see the alate adult mate in the artificial nest, and multimating can be recorded for the first time. The following study was there for undertaken to examine what’s the aftereffect made by the Reproductive strategies of polygyny colony and monogyny colony.在多后聚落內，子代變異度高是可以預期的，但是單后聚落中，又如何維持遺傳的變異度以增加其對環境的適應力呢？我們希望能計算黑棘蟻聚落組成，比較不同生殖策略的聚落之各種性比，並且錄影證實黑棘蟻聚落可以為多后聚落並進行巢內自交。我們發現：黑棘蟻聚落內，蟻后數愈高，則雄蟻比例較低；單后聚落之性比決定於工蟻；而根據錄影觀察，聚落中的有翅生殖族可在人工蟻巢中完成交配，多次交配的現象首次被實地紀錄，並且佔所有交配新生蟻后的92.8%。本實驗可利用黑棘蟻來了解單后與多后聚落的生殖策略對其族群發展的影響。

此研究的目的是要設計出一套完整編輯顯現數學式、加以計算，並求出解的一套方法與成品。而這項工作的執行者，在此稱之Equatetor 。一般的數學式子，若要計算的話，普通的計算機是不足夠的。原因是它們沒有辦法表現出數學式的「原貌」，例如分號、指數、函數、根號等數學符號混在一起時的情況。於是，我便擬定了一個研究，希望設計出一套更方便且實用的方法。換句話說，我要設計出一個功能強大的工程計算機程式。其中，自然牽扯到數學式子的顯現方式（以MathML 實現），以及計算機科學的演算法及資料結構。我主要的目的有四：(1) 顯示數學式(2) 方便編輯數學式(3) 計算數學式(4) 處理可以以不同形式輸出解答的計算（如輸出分數、根號、函數解等）。研究結果中，成功地運用XML 中的MathML 與二分逼近分數等演算法及若干資料結構，達到了以下實用的幾點：(1) 結構化的數學式編輯(2) 完整地顯示數學式(3) 正確運算並輸出運算式的答案(4) 提供一般數學形式之解（非小數之解）;The object of this study is to design a method and processor which is able to edit, display a mathematical expression representing a number, calculate and output the answer. The executor of this task is called Equatetor. Normal calculators are not adequate for this kind of task. The main reason is that they can’t reveal the original expression, such as fractions, radicals, exponents or mathematic functions. Therefore, a simple and convenient method is needed. To perform the possible way of handling those tasks, a computer program has been written. Several techniques were used, such as MathML, computing algorithms, data structures, and so on. Following are main purposes: (1) Displaying mathematical expressions. (2) Editing mathematical expressions simply. (3) Calculating mathematical expressions. (4) Outputting the answers(in different expressions). And the achievements:(1) Structured methods of editing of mathematical expressions. (2) Displaying mathematical expressions completely. (3) Calculating mathematical expressions precisely. (4) Offering answers in different expressions.