在此研究中，我們用類似反演變換的方法，以一個定圓創立並證明了一種新的幾何變換，稱為 「調和變換」 · 我們得到點、直線、圓與圓錐曲線經過變換的關係 ·。1 .直線可以映射成原直線或一圓錐曲線 · 2.圓可以映射成一種特殊曲線。 3 .圓錐曲線可以映射成兩條圓錐曲線或一條圓錐曲線和一直線。此外我們還發現調和變換和反演變換的特殊關係 · 最後，由於調和變換可以簡化圓錐曲線的關係，我們將調和變換應用在行星輾些的證明上，並得到了良好的結果。In this research, we use a method similar to the inversion to establish a new geometric transformation, called harmonic transformation, by a fixed circle O, we prove some of its properties. We have gotten the relationship among points. lines, circs, conies and their images: 1 .The image of a line is a conic or a line itself. 2.Thc image of a circle is a special category of curve. 3.The image of a conic with its focus at the center of O is two conies or a line and a conic. Further mote, we also find the special connection between harmonic transformation and inversion. Finally, since the harmonic transformation can simplify the conic, we apply the harmonic transformation to identify the orbit of a planet, and obtain a nice conclusion.

本研究首先確認培地茅根系具有碎形之基本特性，再進一步以方格覆蓋法計算之碎形維度來分析培地茅根系在不同時間及環境因素下的生長。主要探討碎形維度與抓地力之關係，並設計以實際根系模型來加以模擬，並發展出一可描述抓地力與碎形維度及深度關係的方程式。我們的結論為：（1） 經由方格覆蓋法之計算，培地茅此種植物，不管是整個根系或單枝根，均具有碎形基本特性，適合進一步實驗研究。（2） 碎形維度會隨著培地茅生長時間增長而增加，並且在自然光照及30℃左右會有較大值，而種植於土壤中根系發展較廣，其碎形維度比種植於沙耕中來的高。（3） 實驗結果顯示，抓地力受碎形維度及根系深度兩因素影響，而培地茅根系對土壤有較強的抓地力，推測是因為兩者根系皆又深又長，土中培地茅根碎形維度較大，接觸面積較廣，而又進一步以矽膠模型做實驗驗證。（4） 矽膠模型之目的在於減少難控制之自然變因，實驗之前，測量了根系模型與洋菜凍之基本性質，實驗結果顯示抓地力與碎形維度及根系深度皆呈正向關係，可用數學方程式加以描述。This project is mainly a research into the fractal dimension of the vetiver root system. First, we confirm the vetiver root system has the basic fractal structure by checking its self-similarity, then using box-counting method to calculate fractal dimension. We begin with a fundamental investigation into the relation between different time and environmental factors and fractal dimension. Then we move to our main point: the relation between fractal dimension and its pull-out resistance. In the next step, we make a fundamental silicon model, simulating the vetiver root system, to continue our experiments. In the end, we develop a formula that can describe the relation between its pull-out resistance, roots depth and fractal dimension. Here are our conclusions: （1） After using box-counting method to calculate fractal dimension, we discover that not only the whole vetiver root system but also a single vetiver root has the basic fractal structure. （2） Fractal dimension increases when time goes on. Also the value of fractal dimension is larger in natural sunlight and the temperature at about 30℃.The vetiver root system grows more widely in soil than those in sand. That’s why it has larger fractal dimension. （3） Data shows that its pull-out resistance is influenced by both fractal dimension and the depth of the roots. The vetiver roots, in the meantime, show greater pull-out resistance than some other plants. Thus we draw the assumption that the vetiver root system grows deep and wide, and in natural soil its fractural dimension is greater and reaches greater area. Therefore, a silicon model is constructed to further confirm the findings of the experiment.（4） The design of the silicon model is to reduce the uncontrollable variables in nature. Before starting the experiment, we measured some basic characteristics of the silicon model, including density and angle of repose. Furthermore, the experiment demonstrates that pull-out resistance and fractural dimension have a commensurate mutual relation: the stronger the pull-out resistance, the wider the fractural dimension and the deeper the root system. Thus we derive a math formula to describe this relation.

本研究主要是以一連串的地震分析，來討論台灣地區地震能量的釋放型態。 首先是求出地震的活動參數，接著再繪製地震規模的年發生率，最後是推算台灣地區 地震能量釋放時前餘震與主震所占之比重。 在地震參數上，發現地震活動度較高的月份中，在該月都發生了芮氏規模六級以上的 大地震，因其餘震發生的頻率較小震頻繁，因此造成其a值較大。 在地震規模年發生率方面，我們繪製出圖表，而最後發現台灣地區的地震仍以小震多，另外大地震則有一定的週期。 在研究的最後，了解台灣地區的地震常數之後，本研究以ESRI公司的GIS軟體ArcView8.3版疊合各種圖層，分析人口密集區及水庫的地震危害度分析，提出地圖，對大台北地區，人口稠密的幾個行政區提出警訊；另外在水庫方面，本研究則對幾個在斷層線上的水庫提出警訊。 ;This research presents a series ofearthquake analyses to discuss earthquake’s energy release type. First, we evaluated earthquake’s constant and charted the percentage of earthquake measurement every year. Finally, count the after-shocks and the major earthquake measuring 6.0 and above on the ML happening in Taiwan area. Because the after-shock frequency is more than light earthquakes, as a result it causes the “a” enumeration is large. According to the percentage of earthquake measurement happening frequency every year, we illustrate a chart. Finally we find that in Taiwan area mirror earth-quakes account for a large percentage and major earthquakes happened regularly. When we realize the constant of earthquake, the study folds a variety of layers by using ArcView 8.3 edition of GIS (Geography Information System) of ESRI co., LTD, analyzing the density-populated areas and the dams for seismic hazard to export the map. We raise a warning for the greater Taipei where some districts are densely-populated, and we also warn the dams on the line of fault of large of damage.

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.

經食道超音波 (TEE) 是診斷心血管疾病的重要方法之一，目前一般的診斷中，都是由心血管超音波影像的空間分布，藉以了解心血管的狀況或是阻塞的情況；以及用超音波都卜勒影像，觀察血液在血管中的流動。在這個研究中，我們嘗試用主動脈超音波影像圖形隨時間的變化，作為新的診斷方法，藉以定量地診斷出主動脈硬化的情形。研究過程中，我們將主動脈模擬為受到心臟之週期性壓力驅動之阻尼振盪器，其中主動脈之硬度變化即相當於阻尼因子。由心血管超音波圖形隨時間變化的分析中，測量出 “主動脈擴張到最大時” 相對於 “心電圖中的R-尖峰” 之間的延遲時間，理論上，此一延遲時間即可反應出主動脈硬化的程度。我們由10個樣品的資料中，對每個樣品進行大約50至100次心跳的分析。初步的研究成功地發現：上述之主動脈擴張延遲時間超過0.17秒時，即明確地顯示出樣品具有主動脈硬化之現象，而且此一方法也可由數據的分佈發覺心血管之其他病徵。未來仍需應用此一方法對較多樣品進行研究以確定此診斷方法之可靠性，並且由體外攝取主動脈超音波影像進行分析，藉以取代經食道超音波的方法，增加診斷的方便性以及減少病患之痛苦。 Echocardiography is a very important tool for the diagnosis of cardiovascular diseases. Important information about the intracardiac blood flow, shunt direction and cardiovascular system function can be obtained by echocardiography due to its high spatial resolution capability. However, seldom message is known about the aorta stiffness. This research investigated a new method to quantitatively analyze the aorta stiffness. The aorta was modeled as a periodic-force-driven damping oscillator, in which the stiffness of aorta was simulated as the damping factor. From the temporal analysis of the echocardiographic images, the delay time of the maximal aorta distention relative to the R-peak of electrocardiographic trace was measured to study the damping and stiffness of the aorta. The preliminary study successively found that a delay time greater than 0.17 sec could be a criterion to diagnose that the aorta is quite stiff. This method could also clearly discern some abnormal cardiac performance. A large scale study with this method should be conducted in the future.

本研究主要是探討液晶面板在不同的電壓下，對紅外光區及可見光區之穿透光譜。藉由控制外加液晶面板兩側的電壓，改變內部的電場強度，驅使液晶分子長軸方向改變(偏轉)，以達到控制穿透率之目的。施加於液晶面板兩側的電壓V大於起始電壓V0時，液晶分子長軸受電場作用與電場方向平行，減弱引導偏振光扭轉之能力，讓部分光通過偏振片。令及分別代表穿透率達到最大穿透率之10%及90%時的外加電壓，則定義「光-電開關斜率」γ為：γ =(V90-V10)/V10。透射光強度與外加電壓關係曲線則稱為「光-電開關特性曲線」。穿透率除與液晶分子之旋光程度有關，我們也做了在不同電壓下，液晶分子之穿透光譜，並討論其特性。The main idea of the project is to discuss the transmittance spectra of liquid-crystal device in the range of infrared and visual light (400~900 nm) with different electric field by changing voltage. Different biases are applied to the liquid-crystal cell, causing the axis of liquid-crystal to rotate, and the transmittances are measured. If the application of bias is greater than the threshold voltage (V0), the axis of liquid-crystal will be parallel to the electric field, and make the beam pass through polaroid. Electro-optical switching slope γ is defined as γ =(V90-V10)/V10 , where V10 and V90 are the applied voltages enabling output light signal reaches up to 10% and 90% of its maximum intensity, respectively. It is understood that transmittance depends on the optical activity of liquid-crystal cells. Besides, we will discuss the relation between wavelength and transmittance of liquid-crystal cells.

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.

隨著社會漸漸的現代化，我們需要愈來愈多的電力，然而在享用電力時，更伴隨著汗染的到來。所以討論如何有效的利用能源及探討無汗染能源，是本實驗研究的主要課題，以下即是我們的介紹。

電風扇在炙熱的夏日中為冷器以外不可或缺的生活用品。一個典型的電風扇是由架在可調高度鐵柱上的馬達及風扇構成。現行傳統電風扇有幾個缺點，一、馬達風扇(即重心)位置高，有易傾覆的缺點。二、風力分散，風速不高。三、吹出不可調室溫溫度的風。四、外觀呆板，傳統無奇。五、多噪音等。本研究根據這幾點傳統電風扇缺點，試著設計新的電風扇。首先將柱子移除使重心降至接近地面處。接著根據連續方程式(continuity equationρ1A1V1 ＝ρ2A2V2)製作圓錐狀「號角」裝置套於電風扇上以集中風力並減少噪音。最後企圖使用熱電半導體的致冷器降低輸出風的溫度。研究結果發現號角裝置有效達到預期的集中風力增加範圍目的。不幸發現以熱電半導體降溫的方式效率低且耗損過多能量所以可行性較低。雖然沒有降溫功能，本電風扇仍較傳統電風扇多出許多好處，例如加強的安全性、集中的風和強大擴充性。號角裝置可包含空氣清淨甚至音箱功能充當喇叭。Electric fans are a necessity during summer, especially in a hot and humid place such as Taiwan. A typical electric fan is primarily composed of an electric motor with fans mounted on a strut with user adjustable elevations. This setup presents several problems, including (a) the danger of an accidental tip-over due to the fan's elevated weight center (the heavy motor), (b) high wind dispersal rates, (c) room temperature winds, (d) excessive noise and (e) old, antiquated exterior appearance. This particular study attempts to alleviate all these problems. First the weight center is lowered to ground level by removing the entire strut and placing the motor close to the floor. Next by using calculations from the continuity equation(continuity equationρ1A1V1 ＝ρ2A2V2)a cone shaped “horn” device is placed over the fan in order to concentrate the wind, effectively increasing its speed while decreasing the audible noise level. Finally, an attempt was made to lower the temperature of the air in the interior of the horn by the placement of a thermoelectric semi-conductor cooling device. Our results were that the horn device worked, significantly lowered the wind dispersal rate and achieved an important objective. However, the thermoelectric cooling solution proved to be highly inefficient unless accompanied by precise wind speeds. Even without the cooling function, this new electric fan still has many distinct advantages over traditional fans such as improved safety, concentrated winds and high expandability. For instance, filters can be integrated to purify air, even an audio speaker can be made into the form of this fan.

牛樟芝(Antrodia cinnamomea)是台灣特有的原生藥用真菌，長期被民間認為具有治療癌症、腹瀉、高血壓或保肝的效果。本研究的目的是探討牛樟芝子實體乙醇萃取物對人類乳癌細胞的抑制效用，乳癌是全世界女性最常見的癌症之一，我們透過比較不同培養時期三個月(AC-3)、六個月(AC-6)及九個月(AC-9)子實體之化學指紋圖譜，並測試它們抑乳癌細胞活性之差異。結果顯示，培養三個月的牛樟芝子實體AC-3抽出成份對乳癌細胞的生長抑制作用最佳；而利用西方墨點法探討牛樟芝對細胞週期或凋亡相關蛋白質的變化，推測AC-3可影響乳癌細胞的細胞週期調控，並誘發細胞凋亡。未來我們將進一步深入研究牛樟芝子實體AC-3之抗癌作用機制，並研究對乳癌細胞轉移作用的影響，找出牛樟芝子實體中主要的抗乳癌活性成分。