When you draw a wet finger around the edge of a half filled wine glass, a sweet musical sound comes forth. The pitch of this sound is directly correlated to the amount of liquid in the glass- the higher the height of the liquid is, the lower the frequency is. It means that the shorter the air column in the glass is, the lower the frequency is. This phenomenon differs from the variance in pitch in a wind instrument. In a wind instrument such as a flute, the shorter the air column in its chamber is, the higher the resulting pitches are. In order to study the wine glass phenomenon, we used a piezoelectric crystal loudspeaker connected to an oscilloscope. We recorded the resulting data by using a digital video recorder to capture the images of the waveform of sound, and than analyzed the waveform by using the computer. Our conclusions are as follows: 1. The frequency of sound thus produced was the same whether we draw our finger around the rim, or we strike the glass rim. The higher the height of the liquid is, the lower the frequency is. But the frequencies vary when we strike the glass and when we blow on the edge. 2. When we used a glass without liquid in it, the frequency emitted when we drew our finger around the edge, this frequency varied inversely as the cube root of their weights. 3. In a glass with liquid, the emitted frequency did not have any correlation to the weight of the contents. By taking two identically filled glasses and placing in each a solid object of the same size but different weight, we were able to see that there was no change in the frequency emitted between the two glasses as long as the height of the liquid remained constant. 4. According to “The Flying Circus of Physics”, if we tap the side of a glass of beer, because of the air bubbles in the beer, the frequency emitted will be lower than that from a glass of pure water. This is according to the book, because the speed of sound is lower in air than in water, therefore the speed of sound in an air-water mixture would be lower than in pure water. The resonant frequencies of the mixture will also be lower. However, in our experiment, we discovered that\r when the glass contained air bubbles, the frequency emitted higher. Our explanation is that the sound emitted since the rim of the glass oscillated transversely, the frequency depends only on the retard of the rim and that the frequency is independent of the speed of sound. The intention of this research is to clarify the many misconceptions of this interesting phenomenon.以溼的手指在玻璃酒杯邊緣摩擦，會有悅耳的聲音，而且頻率會隨著內裝液體減少（空氣柱變長）而變高，這種變化與管樂器隨空氣柱的變長而音調變低不同，為了研究它的原因，我們利用壓電晶片喇叭連接到示波器上，並且利用數位錄影機錄下示波器上的訊號，再以電腦分析出瞬間的頻率，結果發現：一、摩玻璃杯與敲玻璃杯，杯所發出之頻率相同，都是所裝液體愈多發出之頻率愈低。但敲玻璃管與吹玻璃管所發出之頻率不同。二、不裝液體之高腳杯，摩擦時所發出之頻率與重量之立方根成反比。（與鐘相同）\r 三、裝液體之高腳杯發出之頻率，不再與總重量有關，而是與液體之高度有關，保持液體高度不變，即使在杯子中央加入不同重量之固體，杯子振動頻率還是不變。若改裝不同密度之液體，則密度愈大頻率愈低。四、在“The Flying Circus of Physics”書中提到輕敲裝有啤酒之杯時，會因杯中含有氣泡而聽到較低之音調，書中解釋是”空氣中之音速低於水中之音速，混有空氣之水中音速變低，其共振頻率也會降低。”但我們的實驗結果是有氣泡時頻率反而高。我們的解釋是杯子所發出之聲音是由於杯面之振動也就是杯壁的橫向振盪，振盪頻率與液體對杯壁之阻尼有關，但與液中聲速無關，密度愈大之液體阻尼愈大。有氣泡時接觸杯壁之液體變少，阻尼較少所以頻率高。希望本研究能使大多數人對這有趣之現象不再有誤解。

我們以電腦程式控制喇叭的聲音頻率，並利用保利龍球的跳動來觀測音叉及金屬片的共振(鳴)現象：、實驗發現音叉、金屬片都有多個共振頻率的情形，而共振頻率彼此之間呈簡單的數學比例關係。、我們利用操縱變因的方式，研究了長度、寬度、厚度、溫度等因素對共振(鳴)頻率的影響。發現隨著長度、寬度減少，共振頻率反而增高;隨著溫度上升，共振頻率反而下降。、移動保利龍球和金屬片的接觸位置，可以測得各點的振動強度，進而繪成振動的駐波波形。縱向(y)上的駐波波形是由多個振動模式的混合組成，這實驗結果可由混波的理論數值作圖印証。研究振動波形，可以得到平面波速的資訊。研究顯示，不同材料有不同的波速。

In the fast-paced environment of a hospital intensive care unit (ICU), good doctor-patient communication is essential. However, medical conditions and devices often inhibit a patient’s ability to speak, write or type. Current assistive communication devices are either prohibitively expensive or cumbersome and time-consuming, creating a gap in communication during a patient’s first days in the ICU. This project applies computer vision to develop a low-cost software solution that bridges this gap by enabling patients to generate words with eye movements. In the system, a webcam acquires an image of the patient, and an image processing algorithm classifies patient’s gaze as pointing in one of eight directions. Each direction corresponds to an option on a graphical menu presented to the patient on the computer’s display. The patient can use the menu to select a preformed phrase from a list of common phrases. Patients desiring to express more complex ideas can type custom words using the menu as an ambiguous keyboard (similar to a phone keypad). In either case, the patient-generated text will be displayed on screen and read aloud through the computer’s audio system. The only hardware requirements are an existing computer and a $6 webcam. The program can process and respond to an image in 148ms. A new user can be trained in approximately 10 minutes, and after training can type a simple phrase such as “hello world” in 40 seconds. While further testing and improvement is required before the system will be ready for implementation, the project shows promise as a low-cost solution to ICU communication.

我們藉由電腦模擬來研究宇宙微波背景輻射中之Sunyaev-Zel’dovich 效應，以探討星系團及宇宙的一些根本性質。重要的發現有： 以上的結果，將可在短期的未來直接應用在許多期待中的觀測結果上，以揭開星團的總質量、質量密度、以及宇宙中的黑暗能量等神祕面紗。 We study the important properties of the galaxy clusters and our universe by using numerical simulations for the Sunyaev-Zel’dovich effect in the Cosmic Microwave Background. We found that: These results can be applied to the observations in the near future, in order to reveal the total mass of clusters, their mass density profile, and the dark energy of our universe.

放洗澡水時，偶然間見到旋轉中的液面呈現下凹現象，做化學實驗調配藥品攪拌溶液時，亦可看到液面下凹的現象，十分有趣。而數學正學到拋物線的我們，靈機一動，心想：「這下凹的液面，會不會是「拋物面」呢？ 因此我們將整個容器旋轉，觀察其中之液體所形成之凹面，並設計實驗及利用電腦計算、繪圖，二者配合，加以比較、證明。若我們能在實驗室中以簡易的方式將拋物面製造成功，並設法產生鏡面，即可利用此結果，自製各式的光學儀器。

本研究在不改變傳統汽車電動後視鏡機械結構下，設計出一可自動調整後視鏡視角的智慧型汽車後視鏡。本研究首先以真值表研究電動後視鏡的機電特性，並藉以設計出一可以與原先人工控制後視鏡並存之驅動電路。微電腦單晶片被使用來控制後視鏡之轉向時間及轉向條件。後視鏡之轉角偵測乃利用方向盤帶動齒輪，藉由齒輪間的傳動並利用位置編碼器將轉角資訊轉換成為電訊號，並將訊號傳送至單晶片來控制後視鏡之轉向時間。研究結果顯示，一可隨方向盤轉動而自動改變後視鏡視角之汽車後視鏡確能減少開車時視角的因轉向而減少的問題。

本文由‘‘分數7/17是否能表示成兩個相異的埃及分數之和’’這個問題出發，藉由簡單數論的性質以及反證法，得到一個真分數可表示成兩個相異埃及分數之和的定理檢驗法(定理1)。有了這個基礎，我們進ㄧ步推廣定理1 的結果，做出了嶄新的結果(定理2、定理3) 。此定理分別可以用來檢驗真分數表示成三個、四個相異埃及分數之和的存在性; 至於將真分數表示為5 項、6 項….k 項相異埃及分數之和的部分尚在嘗試。利用定理1、2，我們寫了兩個Matlab 軟體工具的電腦程式，使得我們可以檢驗任意真分數是否可以表示成兩項及三項的和，並可把所有的解列出來; 最後我們研究的是一個有關埃及分數的猜想(Erdos-Strauss Conjecture)問題，當分子為4，且分母為4k、4k+2、4k+3 時，猜想皆成立。對於分母為4k+1 而言，當k 為3r+1、3r+2 猜想亦成立，k=3r 且r 為奇數時也是成立的，因此目前需解決的問題只剩分母為24t+1 的情況了。值得一提的是，我們用Matlab 的程式檢驗出當分母為1014 至1014 +240000 之內的正整數時，猜想都是成立的，這已經超越了已知文獻的結果。This paper begins with the question: ‘‘Is 7/17 able to be the sum of two different Egyptian fractions?’’ to discuss the problem of Egyptian fractions. According to the complete division properties and the counter-evidence method, we get a back-check theorem which is about a true fraction can be the sum of two different Egyptian fractions (see theorem 1). Using the same method we obtain a new back-check theorem that is a fraction can be the sum of three or four different Egyptian fractions (thereom2, thereom3). Similarly, we can follow the same procedure to get the rule that a fraction can be the sum of five or six …or even more different Egyptian fractions. By the theorem1 and 2, we propose two programs written vie the Matlab software to examine that any true fraction can be the sum of two items and three items or not. Finally we focus on the Erdos-Straus Conjecture, which related about true fractions can be divided by three different Egyptian fractions. The conjecture is when the denominator is 4k, 4k+2, or 4k+3, the problem mentioned above can be solved. As for the denominator is 4k+1, then the conjecture also can be solved, as k equals to 3r+1 or 3r+2. Also, k being 3r and r is an odd number, the conjecture is satisfied. As for the case of r equals to even number, the problem has not been solved. But it is worth to mention here that we use Matlab software to examine the conjecture is agreeable as the denominator is between 1014to 1014+ 240000. This is beyond the results from the literatures.

We study the effect of dark energy on the evolution of cosmic structure in a scenario where the dark energy is treated as free particles and thus can be localized. By theoretical derivation and numerical simulations, we found that: 1. The dark energy particles gain kinetic energy from a moving dark matter particle through gravitational interaction. Due to energy conservation, the dark matter particle will slow down with time Ek(t) = Ek0 - 9 × 10-5[|1+3w|ρDE]1.92t where Ek(t) is the kinetic energy of the dark matter particle,Ek0 is its initial kinetic energy, w is the coefficient of equation of state for dark energy, ρDE is the mean energy density of dark energy, and t is the time. 2. The formation history and structure of galaxy clusters are different in the presence of localized dark energy. The more the localized dark energy, the earlier the formation of the cluster core. In addition, the kinetic energy Ek(R) as a function of R will be different if the ρDE is different. Thus we can compare the observed Ek(R) of clusters with our results to deduce the ρDE in our universe. The results here can be applied to the observations in the near future. 我們探討宇宙結構演化受到可局部叢集之黑暗能量粒子的影響。藉由理論推導及電腦模擬，我們發現： 一、黑暗能量粒子會透過重力交互作而從運動中的黑暗物質粒子獲得力學能。因力學能守恆，黑暗物質粒子的速率會減慢，滿足 Ek(t) = Ek0 - 9 × 10-5[|1+3w|ρDE]1.92t 其中Ek(t) 為黑暗物質粒子的動能，Ek0 為其初始動能，w 為狀態方程式係數，ρDE 為黑暗能量的平均密度，t 為時間。 二、星系團的形成過程及結構，會因可局部叢集之黑暗能量的存在而改變。黑暗能量越多時，星系團的核心會越早形成。而且動能 Ek(R) 隨著至星系中心距離R 的變化，會因 ρDE 的不同而不同，因此可以將量測到的 Ek(R) 和這裡的結果比對，推導出宇宙中的 ρDE 。 這些研究成果將可直接應用在未來的觀測結果上。

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.

This study was to explore the nature of two basic constitutes of the regular pentagon,With these two constitutes, the regular pentagon could be multiplied into any times. We used four multiplication methods (m2 = 2m1 + n1 、n2 = m1 + n1 、m2= k2m1 、n2= k2n1、a2 = a1 + 1、a2 = a1 + ) to show how the regular pentagon could enlarge and to verify that the enlarged regular pentagons derived from computer did exist. By integrating these four multiplication methods, we were able to arrange regular pentagon of any length of side, and evidenced the equation was ( If the side length of a regular pentagon is a form of m,n is the number of A,B respectively ) We further proved that the first multiplication method could be developed into a new modified method, which could divide a regular pentagon with a given side length into a combination of A and B. But only when the x and y of side length of a regular pentagon could be divided by a natural number, k, and made x/k into an item of the Fibonacci Sequence and y/k a successive item. When we tried to verify if any regular pentagon could be constituted by other smaller regular pentagons, we also found that it was un-dividable only if the length of pentagon side were ( the number of A, B were the 2n and 2n-1 item of Lucas Sequence). Otherwise, any regular pentagon might be able to be constituted by other smaller regular pentagons. 本研究是以正五邊形的兩個基本組成元素(B)作為討論對象，利用此二元素可以將正五邊形做任意倍數的放大。我們共使用4種繁殖法則(m2 = 2m1 + n1 、n2 = m1 + n1 、m2= k2m1 、n2= k2n1、a2 = a1 + 1、a2 = a1 + ) 來說明正五邊形的放大情形，並利用此4 種繁殖法驗證電腦運算出的放大圖形確實存在。利用這4 種繁殖法則的改良與整合，已達到能排出任意邊長之正五邊形的目標，並能計算並證明出其通式為。 (若正五邊形的邊長為形式,m、n代表、的個數) 更特別的是，我們能用第一繁殖法反推出一種方法，將給定邊長的正五邊形利用簡單的切割方式分成由A、B 組合成的形式，但只有正五邊形邊長之x、y 值可同除以任一自然數k 而使 x/k 為費波那契數列之一項且 y/k 為其後一項者才可以使用。 將此想法推廣至一個正五邊形能否由比他小的其他五邊形組合而成時，我們也發現當正五邊形之邊長為時（其A、B 個數為盧卡斯數列之第2n,2n-1 項），不可分解，否則應該皆可將一個正五邊形分解成比它小的其他五邊形組合（我們也可以利用這些質形檢驗出其他正五邊形是否也為質形）。但其分解形式，不只一種，而我們推測只用兩種較小的正五邊形就能達成，我們期待能找出一或多種分解方法，能將正五邊形分解成標準的分解形式。

課堂上介紹到彩虹形成的原理是透過水珠的色散、折射與反射，而我們平時可以看到的虹和霓分別是反射一次與兩次所造成，本研究便欲探討更高階的彩虹。我們開發出電腦模擬程式，由「光線追蹤法」繪出一到十階的彩虹，研究光在水珠內反射次數、光的入射角與總偏向角之關係，發現反射次數越多，入射角要越大才會產生總偏向角的最小極限。此外我們也計算出每一條光線在各階彩虹中的能量，以及各階射出光當中每單位角度的光線數目與每單位角度的能量，結果發現能量最大、光線數目最多及總偏向角最小這三者的方向相同。並由以上數據的分析，成功設計出能呈現第三道彩虹的實驗。推廣研究：電腦模擬圖形中高階彩虹內部反射線所形成的包絡線。

Hamiltonian和Hiroimono路徑問題都有百年以上的歷史，且沒有直接解決方法，可能因為是題目本身的條件在討論時尚不足解決問題，現階段只能用電腦去跑。