出國代表作品

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)

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。

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。

本研究以攝影紀錄的方式，透過電腦進行影像分析，記錄不同刺激下美洲蜚蠊(Periplaneta americana)的觸角擺動模式，計算出各項觸角運動的參數，以瞭解光線(光刺激或光適應)、震動刺激、喝水與進食對其觸角行為的影響。我們發現在不同因子的刺激下，觸角擺動的模式具有差異，若兩種不同的刺激同時發生，蜚蠊觸角的行為亦具整合性的反應。蜚蠊於不同狀態下(如喝水或進食)，對相同的刺激有不同的反應，證明蜚蠊觸角的行為模式，受環境因子與個體狀態調節。透過掃瞄式電子顯微鏡的觀察，也發現觸角具多種感覺毛，且雌雄的感覺毛的分佈與數量具有差異。綜合以上發現，證明觸角除了為敏感的受器，亦為能反映出生理與環境狀態的動器，同時也適合進行發展檢測器的仿生學應用，用來檢測環境中物理及化學因子。The aim of this study is to investigate the different swing motion modes of antennae of American cockroach (Periplaneta americana) by computer-aided Imaging Analysis. The parameters of each swing movement were calculated in order to analyze how light (including light stimulation or light adaptation), vibration, food and drinking water may affect the antennae behavior of American cockroach. It was found that the antennae swing motion modes were significantly different under different types of stimulus. If two different types of stimulus occurred at the same time, the reactions of antennae motion may become conformable. Under different environmental conditions (such as food or water), same stimulus may result in different reactions. The antennae behavior has shown to be significantly affected by environmental conditions and individual physiological status. Through the observation with scanning electron microscope (SEM), it was found that the antennae has many types of sensilla; and the distribution and quantity of these sensilla are significant different between sexes. In conclusion, not only the antennae are considered as the sensitive receptors, but also they are the important effectors to reflect physiological status and environmental conditions. The current model is suitable for the development of specific detectors in the applications of Bionics to detect the physical and chemical factors in certain environments.

傑克船長和他的海盜們掠奪到許多箱珍寶，每箱含有數量不等的金幣及鑽石。船員們深怕傑克船長又出什麼陰謀，一致同意讓船長任選一半的箱子拿走。\r 當然，傑克不知道金幣和鑽石的價格比，為了保證可以得到一半的利益，傑克希望他拿到的金幣和鑽石都各占一半。傑克的願望會實現嗎？\r 這個問題已用高等數學證明其解，而本研究利用初等數學的方法，除證明傑克需取的最少箱子數外，同時也能更快速的算出取法。\r 類似的結果可應用至分配災區物資等情形，或任何無法轉移、獨立的資源，如各式專長的人才、多功能的機械等，期望可對更有效的分配做出貢獻。

The spiders, Neoscona nautica, often appear in groups, but individuals have their own sense of territory.They usually spin webs among branches during 6:00~ 8:00 in the evening. When building webs, they will first start with bridges and then spin Y-shaped spokes. Next, they spin meshe of net, silk frame, spokes, spirals and free-zone in order. After finishing webs, they will wait for prey on the free-zone or meshe of net. If they find something inanimate on the web, they will break the spiral attached with the inanimate object that is later removed. If the meshe of net is broken, they will fix it immediately. For them, the time to take webs back is during 2:30~ 6:00 in the morning. Most time they use the first pair and the second pair of legs to take webs back and swallow the webs. Sometimes, they break the spirals by the last pair of legs. The sequence to take webs back is : lower right section, lower middle section, lower left section, upper left section, and upper right section. At last, one thread of bridge will be left. Every early mornings they take webs back and swallow them. The next evening they rebuild webs. Possibly there are two reasons to explain why spiders eat their webs: (1).They swallow webs to get protein. (2).The web threads are easily polluted by dust and humidity and reduce stickiness. The web may also reduce the probability of capturing prey. The body length of them is not related to effective web dimensions. However, the web sizes depend on the width of web-building location. The study shows linear relation among body length, meshe of net and dimensions of free-zone. The linear relation represents that the meshe of net and free-zone have ecological or survival meaning for them. We expect that this study of Neoscona nautica can be helpful to build spider ecological database in Taiwan.簷下姬鬼蛛常成群出現，但個體卻有很強的領域性；常於下午6：00 至8：00 結網於樹枝間，結網時，先以橋絲為出發，織出一Y 形的縱絲，再由此依序織出中空網眼、絲框、縱絲、橫絲、棲息圈，網結好後，簷下姬鬼蛛則在棲息圈或網眼靜候獵物，若發現網上有非生物之異物，則將黏住異物的橫絲弄斷，再把網上的異物丟棄；若網眼被破壞，則會立即修補。收網時間為凌晨2：30 至凌晨6：00，收網時，大部分由第一、二對步足進行收網，偶爾會用最後一對步足將橫絲弄斷，一邊收網一邊將網吞食，收網的順序為：右下、中下、左下、左上、右上，最後留下一條橋絲。簷下姬鬼蛛每天清晨都會收網，並將網吃掉，翌日傍晚再重新結網，其可能原因有兩點：(1)將網吃掉以補充蛋白質。(2)蛛絲容易受灰塵、水氣之污染而減小黏性，降低獵捕功效。簷下姬鬼蛛體長與有效網面積無關，但網的大小視其結網地點寬敞程度而定。體長與網眼、棲息圈面積呈線性關係，表示網眼和棲息圈對簷下姬鬼蛛具有生態或生存意義。我們對簷下姬鬼蛛生態調查之結果，希望能幫助台灣的蜘蛛生態資料庫之建立。

我們從日常生活中的酒瓶填塞問題，延伸出圓在相守條件下產生的衍生圖面\r 積極值及相守圓排法等問題，研究過程中我們發現以下的結果：\r 一、 我們可以利用較少個相守圓的排法，以繁殖或增加的方式排出較多個相守圓\r 的衍生圖。\r 二、 當相守圓數量為2個、4個及6個時，相守圓以對角線排列，且衍生圖為正方形\r 時，面積會產生最大值。而當相守圓數量為3 個及5 個時，衍生圖面積最大\r 值會出現在相守圓排成波浪形時，但其夾角並非特殊角。\r 三、當相守圓數量為2個至6個時，相守圓以直線排列，衍生圖面積會產生最小值。\r 四、透過不同層數、個數的研究得知：當相守圓的數量為11個或14個以上時，存\r 在正三角形排法小於直線排法的情形 。而相守圓個數在10個以下時，直線排\r 法面積都叫正三角形排法小。\r 五、我們可以用質單元分割的方式討論更多圓數時，相守圓間的排列方式，使得\r 生多元間的排列變為數字間的加法排列，可以大大降低討論情形的複雜度。\r 六、我們可以用密度的概念，搭配質單元分割的方式，得到不同圓數時，以不同\r 衍生多元搭配波浪形法形成之衍生圖密度最小值。\r 七、我們可以算幾及柯西不等式來驗證特定類型的衍生圖面積極值，也可以列出\r 面積函數來分析不同類型的衍生圖面積極值。

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。

我們使用陽極氧化鋁(AAO)模板來製備銅及其氧化物的奈米線。以硫酸銅和乳酸配製電鍍液，利用氫氧化鈉水溶液(NaOH)將其pH 值調整到12，供以不同電壓，可電鍍出銅及氧化亞銅奈米線。在較高電壓下可製備出銅奈米線，而在較低電壓下可製成氧化亞銅奈米線，若使用中間電壓則能製得銅及氧化亞銅的混合態。利用x 光繞射分析儀(XRD)來分析其結晶構造、使用場發射掃描式電子顯微鏡(SEM)以得知其表面形貌。電鍍出的奈米線直徑約60 nm。奈米線的長度可藉由調整電鍍時間或電壓來控制。在製作IC 內部導線方面，銅奈米線深具開發潛能；在提升太陽能電池的轉換效率、製作可見光光觸媒方面，氧化亞銅奈米線極具前瞻性。We electrodeposited copper and cuprous oxide (Cu2O) nanowires with anodic aluminum oxide (AAO) templates. Both Cu and Cu2O nanowires could be prepared with an alkaline cupric lactate solution, which was adjusted to pH 12 using a 6 M NaOH, when supplied with different electrolytic voltages. Cu nanowires could be prepared when a higher voltage was supplied, and Cu2O nanowires could be prepared with a lower voltage. A mixture of Cu and Cu2O nanowires could be prepared with a supply of a voltage in between. X-ray diffraction (XRD) is used to determine the phase composition, and scanning electron microscopy (SEM) is employed to characterize the morphology of the nanowires. The length of nanowires can be controlled by adjusting the time spent on electrodeposition and the voltage supplied. The resultant diameter of the nanowires was about 60 nm. Cu nanowires are promising materials for making the conductive wires in IC, and Cu2O nanowires hold great promise for improving the conversion efficiency of solar cells and manufacturing visible-light photocatalyst.

在這篇報告中，我們探索了「將錯就錯的Knuth 河內塔問題」。傳統河內塔問題在電腦科學上佔有重要的地位，是一個極具內涵的模型。由於這個模型的深厚數學內涵，使其和巴斯卡三角形建立了緊密的連結，且利用這個緊密的數學連結，設計出復原任意起始狀態的良好演算法。Knuth 河內塔起因於數學家Knuth 在論文[3]中，描述傳統的河內塔問題時所發生的一次筆誤。在這個新的規則之下，我們意外發現Knuth 河內塔存在著一個和傳統河內塔平行的模型，此模型在電腦科學及數學上有著完全不同於傳統河內塔的內涵。我們的研究主要如下：(分別為內文中的四大段)(一) 結構分析。移動環所需要的次數，如何移動環並分析每一次動作所動的環，及每個環何時被動到並給出演算法。(二) 正整數的分割。所有的移動步驟將正整數做了一個新的分割(Partition)；此分割模k之後有良好的循環性質。(三) 費波那契真分數的排序。這個正整數的分割形成一張表，這張表恰好就是分子分母皆為費波那契真分數之排序。(四) 隨意亂排的Knuth 河內塔復原演算法。在Knuth 河內塔的規定下將起始狀態改變，找出良好的復原演算法，並分析。 In this project we study the "Knuth Hanoi Tower", which is motivated by a typo in a paper of Knuth. This inadvertently typo leads to a new rule of moving the discs on the Hanoi Tower (see introduction below for definition). Although seemingly similar to the traditional Hanoi-Tower problem, it turns out that under this rule the "Knuth Hanoi Tower" problem consists of amazing properties, and is totally different from the traditional one. Our study focuses on the following directions: (1) Structure analyzing: We analysis the sequences recording the disc moving and offer enumeration results and recurrsive/non-recurrsive algorithms. (2) Partition of N: The moving sequence forms a partition (a table) of N, which has an amazing congruence property. (3) The order of Fibonacci proper fraction: The row/column of the partition table is, even more amazing, exactly the order when sorting the Fibonacci proper fraction with fixed denominator/numerator. (4) The Restoration of an arbitrary initial state: We offer an efficient algorithm for restoring any initial state of discs. We hope that our study on the "Knuth Hanoi Tower" offers a simple, neat, and new example on the theory of Algorithm, Number theory and Combinatorics.