臺灣

本文一開始以直線數奇偶性來探討「在平面上給定n條直線，任三條線不共點，任兩條線不平行，讓沿著任何一條直線上的n-1個交點，都剛好出現1,2,3,....,n-1 各一次」，求解的存在性。研究結果顯示直線數是偶數時有解，而直線數是奇數時無解。接著用圖論中「點著色」及「邊著色」的觀點來思考同一問題，突顯出該問題背後的數學結構。最後進一步探討「將直線替換成圓」，「有m條直線平行，任三條不共點」以及「有p條直線共點，任兩條不平行」的情況。其中最後一部分「有p條直線共點，任兩條不平行」的情況，從超圖(Hypergraph)和關聯矩陣(Incident Matrix)的觀點，得到一些有趣的結論。

異族效應可能是造成種族歧視的原因。本研究透過按鍵反應及腦波儀的形式，對本國人及高加索人兩族群進行受試。實驗中，我們給予受試者兩種族群、正反兩種方向和七種情緒的刺激材料，請受試者辨識情緒，受試完畢後再利用ANOVA、EEG lab等分析行為實驗數據及腦波圖。分析結果發現：按鍵反應方面，高加索人較會辨認情緒，也表現出較強的同理心；本國人對兩種族皆產生面部倒置效應、高加索人則僅對於本國人產生；高加索人判斷正臉情緒時會有異族效應，本國人判斷時則正反皆無。腦波儀方面，每一種情緒的異族效應是相似的，就腦內影響的區域而言，本國人面部倒反效應多由枕葉影響，高加索人則多由額葉影響，本國人異族效應由右腦影響，且僅出現在刺激後150～200毫秒，高加索人則未呈現異族效應。另外，異族效應與面部倒反效應間具有交互作用，於是在臉部辨識上的正反差異就會呈現不同結果。將上述結果進一層地推論，可得知異族效應是後天學習行為，而面部倒反效應則是受本能與後天學習共同影響而產生。 期望本研究結果未來能運用在社會議題或心理學研究上，減少異族效應可能產生的負面影響。

在實驗用共振法測量聲音在固體、液體、氣體中的駐波聲場，測量各介質中的聲速。研究超聲波在液體中的空腔效應，鋁箔在不同液體受空腔效應所破損面積與時間略成正比，並發現在水與各濃度的洗潔精水溶液中以水的破損效果最明顯。另外利用1.65MHz 高頻超聲波打入水中，因駐波使水有疏密不同產生狹縫，以雷射通過狹縫有光的繞射花紋產生，由干涉條紋可推估駐波波長。利用閃頻共振法研究光彈材料超聲波場，且發展出以肉眼觀測的裝置，由光彈材料的花紋級數與應力研究中，發現花紋級數與應力成正相關，由聲場中的花紋顏色判斷所受應力大小，並發現超聲波不僅有聲場產生並伴隨熱效應，會影響觀測花紋級數。This project began by studying the fundamental properties of acoustic waves, the relationship between its velocity, frequency and wavelength. Experiments regarding the distribution of sound waves in different mediums, and the induction of resonance in solid, liquid and gaseous materials were conducted. Results from utilizing suspending method to confirm theoretical prediction of sound velocity was accurate, and the sound wave patterns in photo-elastic materials were observed. It was also observed that an aluminum foil would be cut in an ultrasonic cleaning device. The effects of different liquids such as water and detergents on cleaning effectiveness were then experimentally determined, taking into account factors such as viscosity. From reference materials, we learned that ultrasonic waves would create Caritation in liquids. Traditionally, sound waves are expected to exhibit only longitudinal waves, yet in this study it was discovered that the residual\r stresses from resonance in photo-elastic materials also indicate the existence of transverse waves.

這份研究是關於一個「數形合一」的問題，研究的主要目的是找出同時具有兩種圖形數身分以上的數。研究結果發現雙重多角數必定存在，但個數可能有限個也可能無限多個，有些雙重中心多角數是有限多個，有些雙重中心多角數是無限多個，但令人意外的是同時是K角數和中心K角數的數卻皆是永遠是無限多個，我對其模式進行了探究並加以具體分類，並說明原因。研究過程中發現遞推關係可以大大簡化計算的步驟，可簡潔快速的求出這些數。我也證明了同時是三、四、五角數的數只有 1。此外，我將研究應用在熱門的平衡數(balancing numbers)和NSW數(NSW numbers)等上面，應用多角數的研究解決一些熱門的問題和找出了圖形上不證自明的結果。

A novel and simple method was developed to determine the activity of silver in nanometer particles more than in non-nanometer particles. The conductivity of conducting polymer, polyaniline (PANI) doped with different amount of nanometer silver particles was used to evaluated the activity of nanometer silver. In polymerization of polyaniline, hydrogen chloride solution usually used to increase the conductivity of polyaniline. When 1%(w/w) nanometer silver particles doped during the polymerization, the conductivity of polyaniline was down from 2.28 s/cm to 0.65 s/cm, then increased with increasing the amount of nanometer silver doped. The conductivity of polyaniline was changed from 2.28 s/cm to 0.47 s/cm when 3%(w/w) nanometer silver particles doped, but it is increased from 2.28 s/cm to 2.44 s/cm when was doped with 3%(w/w) micrometer silver particles. The conductivity of polyaniline changed due to the formation of silver chloride (AgCl) in doping nanometer silver. Some of the nanometer silver particles were formed to silver ion in hydrogen chloride solution for the high activity property of nanometer silver. This also can be proved from the spectra of XRD and FE-SEM. Therefore; determination the conductivity of conducting polymer by doping nanometer metal particles can be used to determine the activity of the nanometer particles. 本研究為開發一個新穎的檢測奈米金屬粒子化學活性大於非奈米金屬粒子的簡易方法。方法為利用導電高分子聚苯胺，於合成過程中添加不同濃度的奈米銀粒子，並分別偵測其成品的導電度，藉以評估奈米銀粒子的化學活性。由於聚苯胺在合成過程中通常加入鹽酸以提高其導電度，致活性較大的奈米銀粒子於氧化後，隨即與氯離子形成氯化銀的沉澱，而降低聚苯胺的導電度，如添加1﹪(w/w)奈米銀粒子的，其導電度由2.28 s/cm 降至0.65 s/cm，隨後隨著添加量的增加導電度先降後再稍回升。一般非奈米級銀粒子因氧化電位為負值，即化學活性小，而不易被氧化。由實驗結果，我們發現同樣添加3%(w/w)的奈米級銀粒子或微米級銀粒子，添加奈米級銀粒子的導電度由2.28 下降為0.47，添加微米級銀粒子的導電度卻由2.28 上升為2.44，此乃說明本方法確實足以證明奈米級金屬的化學活性的確遠大於微米級金屬，因相同條件下，微米級銀粒子未如同奈米級銀粒子一樣被氧化成銀離子。即奈米級銀粒子可以輕易的被氧化，而非奈米級銀粒子則不易被氧化。尤其也可由X 光繞射儀分析光譜圖和場發射式掃描電子顯微鏡拍攝圖證明。因此，我們可以採用添加3 %(w/w)奈米級金屬銀粒子及微米級金屬銀粒子於導電高分子的方法，並藉導電度的變化，證明奈米金屬粒子的高活潑性。

本實驗將小型塑膠圓片置於水中釋放模擬落葉的運動模式，發現圓片與水平之夾角呈現週期性變化，變化範圍會隨著時間改變。在運動幾個週期後，變化範圍會被限制，此時為穩定狀態。在水中加入鋁粉以觀察圓片運動時流場的變化，提出分流模型與最短時間原理，發現此模型可以解釋圓片的轉動機制。

由於想了解化學反應的微觀形態，我們設計微型化學反應裝置來比較巨觀(傳統型)與微觀(創新型)化學反應間的差異，並探討其實用及環保方面的問題。在顯微鏡底下，我們觀察化學反應的沉澱結晶及電解反應，嘗試以各項變因(溫度、濃度、聲波…等)來觀察其結晶的型態。我們已成功地將實驗藥品用量減少到一滴(約0.04ml)，並以微觀的角度觀察化學反應的過程。在實驗中，發現反應進行時，粒子會不斷流動，經查證後為愛因斯坦所提出的布朗運動，並且測得硫顆粒的直徑大小約4.2 ~ 6.7 微米。不同聲波所造成硫粒子的移動速率不同，而不同溫度的部份，我們發現→每增加十度硫粒子移動速率增加約兩倍。在面積4.392×10-4cm2 範圍內大約有250~300 顆硫沉澱的粒子。本實驗成功地將顯微鏡應用在化學領域上，若將此實驗推廣，可達到污染少、觀察實驗的時間短、用量少的目標。此實驗是邁向化學微觀世界，一種值得嘗試且創新的方法。In order to compare the differences between the chemical reactions of macroscopic reactor and microscopic reactor, we have designed a device of chemical reaction and researched into the problems of their environmental protections and practical aspects. Under the microscope, we observed not only their precipitating crystal compound from the chemical reaction and electrolytic reation but their types of crystal. We have successfully reduced the dose to one drop ( about 0.04ml) and observed the process of their chemical reaction from the angle of microscopic reactor. During performing the experiment, we found the particles would keep flowing while the reaction was working. It was proved as "Brown motion" introduced by Einstein. The diameter of these particles were around 4.2~6.7μm. We find that different sound waves and temperatures,the motion speeds are quite different. And the movement rate increases about two times as the sulfer particles increase 10℃ each time .Within the measure of area of 4.392×10-4cm2,there are 250~300 sulfer particles.The experiment has successfully used a microscope in the field of chemistry. If we popularize the experiment, we can reach the goal of less pollution, fewer the dose and time-saving observation. It’s an innovation to step to the world of chemical microscope world.

單淘汰賽是一種失敗一次即遭淘汰的賽制；在此假定每位選手都有一相對應的能力數值，本文主要探討在均高的單淘汰賽程表之下，若賽程安排完全依照種子安排原則(亦即最強的選手對最弱的選手、次強隊次弱….)，則對於能力越強的選手越有保障，直觀上而言能力最強的選手應有最大的奪冠機率，探討此種賽程安排是否滿足能力較強的選手有較大的勝率？因發現在某些特殊的選手能力數值分佈之下會發生次強選手勝率大於最強選手的情況，令A、B代表最強與次強選手，P(A)、P(B)代表A、B奪冠的機率，故擬定P(B)/ P(A)為參考依據，尋求P(B)/ P(A)的最大值發生處作為最極端的狀況。發現四位選手的情況下，P(B) / P(A)最大值 = 1；八位選手的情況下，P(B) / P(A)最大值=(196+98) / 343=1.0938，當選手數為2n時，P(B)/ P(A)最大值隨n的增加而遞增。 Knockout Tournament is a highly competitive system in which any player losing a game can no longer play in the tournament. Here we suppose that every player has a numerical value that corresponds to his ability. We consider a totally-seeded knockout tournament with 2n players where in the first round, the strongest player matches the weakest player, the second strongest player matches the second weakest player, and so on. We examine whether a stronger player always has a greater probability of winning the tournament. The answer is in the affirmative for n = 2. For a tournament with eight players(n = 3), the situation is much more complicated. In certain cases, the second strongest player has the greatest probability of winning the tournament. Specifically, let A and B denote the strongest and second strongest players, P(A) and P(B) their respective probability of winning the tournament. We find that the maximum value of P(B)/P(A)equals (196+98) / 343 = 1.0938. For n > 3, we have not obtained the maximum value of P(B) / P(A) . However, it can be readily seen that the maximum value of P(B) / P(A) is non-decreasing as n increases.

根據西方文獻造成狐臭的原因之一是因為Staphylococcus 將汗液中的白胺酸(leucine)代謝產生異戊酸(isovaleric)，散發出臭味，但台灣人的狐臭菌是否與西方人類似則較少被探討。 本研究透過好氧和厭氧的篩選方式篩出的優勢腋下菌種以Staphylococcus epidermidis、Staphylococcus haemolyticus、Staphylococcus lugdunensis、Micrococcus luteus、Staphylococcus aureu 為主，與國外研究結果略有不同。在Gas Chromatography (GC，氣相層析儀) 檢測結果發現以上菌種中能將白胺酸轉換成異戊酸，效果最好的菌是Staphylococcus lugdunensis，推測Staphylococcus lugdunensis為造成狐臭主因。 由結果推測，異戊酸只是臭味的一部分，未來可以藉由分析汗液成分與腋下菌的產物來找出導致狐臭的其他原因，找出更完整的腋下作用機制及更好改善狐臭的方法。

桁架(TRUSS)是以三角形為基本結構再不斷延伸的結構，其應用在生活中屢見不鮮。本實驗在探討簡單桁架結構的穩定性，其穩定性以載重比(R＝承重/自重)表示，載重比越大，穩定性越高。若以單一三角形來看桁架結構，底角越小的等腰三角形結構穩定性最好；直角三角形及不對稱的三角形都次之。其次，多三角形的簡單桁架，其構造越對稱且內含的三角形數量越多，結構越穩定，承重的載重比值(R)越大。簡單桁架內的零桿件雖不受力但對結構的穩定性卻有很大的助力。再者，不同幾何造型的桁架橋中以三角形桁架橋的載重比最大，具有以小搏大的概念，是製作桁架橋時，值得參考的桁架橋造型。