長方體中切割正立方體之研究
在1940 年代，Bouwkamp 提出一系列有關如何將矩形切割成若干個正方形的研究報告，但是如何找出正方形個數最少的方法仍是長久以來懸而未決的問題。在本研究報告中，首先引進「四角切割」的方法，並結合輾轉相除法的概念，來研究矩形的切割問題。我們的方法能大幅度降低正方形的個數，也適合做為此問題的上界函數。有關如何在長方體中切割出正立方體的組合，我們也將輾轉相除法的概念延伸到三維空間，進而建立所切割出最少個正立體數的一個上界模式。此外，藉由四角切割概念的延伸，我們也發現這個上界亦可再予修正。In 1940’s, Bouwkamp proposed the study of dissecting squares from rectangles. Among the study, the problem of the least number of dissected squares has been open for decades. In this project, we first propose a corner dissection method, associated with the famous Euclidean algorithm. By reducing nearly three fourths of the number dissected by the primitive Euclidian algorithm, our method indeed establish a suitable upper bound of the minimal number of dissected squares from the given rectangles Meanwhile, the Euclidean algorithm has also been considered to dissect the cubes from cuboids. We analyze the fundamental properties of the method and establish a prototype of upper bound function for the minimal number of dissected cubes. Moreover, the method of corner dissection has also been implemented for some cuboids, which also exhibits the acceptable improvement being a suitable upper bound.
新網路模式對抗無尺度流行病傳染A New Network Model to Fight Against Scale-Free Epidemi
流行病的傳染如同一個無尺度網路，但有一些特殊特性在發展一套新傳播模式時，是需要被詳加考慮的。我採用時間位移(t 與ti 分別以t?(ti?1)與ti?(ti?1)來取代)至無尺度網路模式中，再引入一個非連續強力函數H(t; ti?1, L+ti?1)來描述流行病傳播的特定時段與強度，並重新定義機率p 為無效傳染率。之後，我建立了新模式「無尺度流行病模式」－SFE-1與SFE-2。模擬六種病原的傳染途徑，結果證實SFE-1與SFE-2模式是正確與確切可用。案例研究結果，顯示傳染強度H可為固定值或為變數；p可以是一固定值、雙固定值或為新增病例的函數。更進一步解析美國AIDS病例在不同族群與行為上的差異，獲知亞裔/太平洋裔與印第安人/阿拉斯加人的H值低於其他族群，其原因可能是由於小的族群具有較高的接觸所致。異性性接觸的H值低於其他，顯示性交易是傳播HIV的主要途徑。SFE-1與SFE-2模式也可被用在流行病的預測上，因為SFE-2使用已知值而非估算值，所以SFE-2模擬結果較佳；但是SFE-1更可以明確提供一個流行病在失控或控制下的預測結果。無尺度流行病模式可以協助所需警戒的程度與政策決定的計畫結果。因此在幫助政府評估社會經濟成本與健康憂慮上是一個有用的工具。所以我提出一個確切可行的對抗無尺度流行病傳染新方法，並詳細說明運作流程。The course of epidemics resembles a scale-free network, but some specific elements should be considered in developing a new model. I introduced a time shifting (replacing t and ti by t?(ti?1) and ti?(ti?1)) and a discontinuous forcing function H(t; ti?1, L+ti?1) into the scale-free network model to fit the specific period and intensity of the infection, and redefined the probability p as an invalid infection rate. Then I proposed the new Scale-Free Epidemic Models, SFE-1 and SFE-2. The simulation results of six types of epidemic transmission showed that the SFE models were accurate and useful. In the case studies, the results showed that H were constant or variable, and p were a fixed constant, a dual constant, or a function of new addition cases in the epidemic periods. The further studies for comparisons of the difference races/ethnics and the difference transmission category of AIDS cases in USA were analyzed. The H value for Asian/Pacific and Indian/Alaska Native race were lower than others, it may be due to small clusters with constant high contact rates. The H value for heterosexual contact was lower than the others, indicating that whoredom was the main transmission for HIV. Both SFE models can be used to predict epidemics, SFE-2 is better than SFE-1 due to SFE-2 using given indices and not conjectured values, but SFE-1 can more clearly suggest results of epidemics when under control or not. SFE models can help the government to determine the level of caution needed and the projected results of policy decisions. They are useful tools in assisting to balance socio-economic and health concerns. I hereby propose a new method to fight against epidemics with detailed procedures of using the SFE models.
重複圖形
「重複圖形」是本篇報告研究的問題，我們利用「方程式」建立一個尋找重複圖形，並証明其個數的方法。利用此方法得出下面的結論：1.會形成lap 2 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 2 三角形」只有1 種，即等腰直角三角形。(2)「lap 2 四邊形」只有1 種，即二邊之比為1： 且內角是45°、135°的平行四邊形。2.會形成lap 3 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 3 三角形」只有1 種，即內角為30°–60°–90°的直角三角形。3.其他的lap k 三角形：(1)任意內角為30°–60°–90°的直角三角形都是lap 3k²，其中k是正整數。(2)邊長比為1：m： 的直角三角形是lap (m²+1)k²三角形，其中m、k是正整數。
To find repeated figures, we construct a method to search them with the help of algebraic equations. Here we arrive at：1. There are only two kinds of lap 2 convex polygons, triangles and quadrilaterals. (1) The only lap 2 triangle is isogonal right-angled. (2) The only lap 2 quadrilateral is the one that contains angles 45°, 90° and two neighboring sides with the ratio 1: . 2. There are also two kinds of lap 3 convex polygons, triangles and quadrilaterals. (1) The only lap 3 triangle is the one with angles 30°, 60° and 90°. 3. Other kinds of lap k triangles are listed as following: (1) A triangle with angles 30o, 60°, 90° is a lap 3k², the k is a natural number. (2) A right-angled triangle whose ratio is 1 : m : is a lap (m2+1)k², the m and the k are natural numbers.
數列生成遞迴
這個題目是源自2003年的TRML思考賽的題目，原題目並不難，它只有用到簡單的排列方法，主要是討論 an 、bn 兩種數字的排列，其中 an 為滿足下列所有條件之N位數A的個數。
I. A中每一個數字為1或2
II. A中至少有相鄰的兩數字是1
而 bn 表示滿足下列所有條件的N位數B的個數
I. B中每一個數字為0或1
II. B中至少有相鄰的兩數字是1
以及探討an 、bn 與費氏數列cn之關係，其中 cn = cn-1 + cn-2 ，n≧3 ,c1=1, c2=2 。
其中 an 如果改成考慮為一數列，其值不變；而 bn 如果改為數列，那麼就不需要考慮0不能為首位數字的情況。如此，讓人聯想到一個用生成函數解的題目「一個N項數列，其中每一項只能是0或1或2，其中0和2永不能相鄰，求這個數列個數的一般式。」，因此，我們嘗試將這個題目改變它的要求繼續做下去，發現其中有某些規則，例如：不只是原來的11相鄰，甚至是排列其它種方式，都可能從其遞迴式看出它排列的意義，甚至這種排列數是可以用遞迴式求出來的。這提供了我們另一種求數字排列的方法，也是我們覺得有趣的地方。
在過程中我們初步得到以下結論：
This solution is according to power contest of 2003 TRML. It is composed of two number arrangements, an , bn .
First, suppose an is the total number conforming to the following rules.
I. Each number is 1 or 2 in A.
II. There is a couple of (11) in A at least.
Then, suppose bn is the total number conforming to the following conditions.
I. Each number is 0 or 1 in B.
II. There is a couple of (11) in B at least.
Furthermore , we give the thought to the relation among an ， bn ，and cn (Fibonacci Sequence).
By the way, if an is changed to a sequence, and the result is the same. But if bn is to arrange number, we have to give thought to the fact that the first number can’t be zero. If it is a sequence, we don’t have to consider it.
The problem belongs to combinatorics. After we do this problem, we find not only original question but also other permutation can be understood by its formula. The problem provides us with other means to solve permutation and combination question. Then, we get the conclusion as follows：
「膠」流電-黏度及外加電壓對電解質溶液離子暫穩態通道之影響
在本次實驗中，我發現膠狀電解質溶液中的帶電離子，會因為離子團的熱運動，和電偶極的庫倫吸引力 (electric dipole) 的交互作用下，使溶液的I-V curve (電流-電壓曲線)，具有類似磁滯曲線(Hysteresis curve) 的效果；而膠狀溶液之濃度越高，電解起始點的對應I-V 值也越大。此外，白金電極與銅箔電極的距離若改變，也會使溶液的I-V curve 變的不一樣。另一方面，我也發現，在給予膠狀電解質溶液一緩慢外加的電壓或衝擊電壓並持續維持此一定額外加電壓時，會因為該溶液的黏度持續增高、帶電離子濃度增高且反應不斷變化下，而使該溶液的對應電壓，形成一重複出現「先降-後升-再降」的震盪現象，且電壓值節節升高。最後，我利用掃描式電子顯微鏡（SEM）及能量分散光譜儀（EDS）觀察銅箔電極之表面變化並分析其上之化學組成，藉此嘗試解釋上述這些有趣的現象。In this experiment, with the interaction of the heating action of ionic atmosphere and electric dipole, I find that ions in the gel make the I-V curve in the colloid electrolyte liquor show up with the effect similar to Hysteresis curve. The higher concentration of the colloidal solution, the bigger value of I-V at the initial electrolysis reaction was found. Furthermore, the shape of I-V curve is dependent on the distance between platinum electrode and cupper electrode. On the other hand, I find that when I apply a gradual extra-voltage or a fast extra-voltage to the colloidal electrolyte solutions and then maintain to a fixed value, this will make a unique ‘two peaks’ state oscillation of corresponding voltage. The reason is owing to the climbing viscosity and ion concentration in the solution. With the methods of scanning electron microscope (SEM) and energy dispersive spectrometer (EDS), I observe the change and analyze the components of chemicals on the surface of the cupper electrodes. Finally, I present the interesting results and try to interpret these phenomena.
太陽能發電環境評估與追蹤器探討
本研究首先探討台灣各地的日照時數與世界重要都市的比較，發現台灣南部日照時數皆超過2000 小時，適合發展太陽能。接著，?了增加陽光的能量密度而加設弗瑞奈透鏡，雖然能順利的使照度放大三百餘倍，但歲日照角度的影響甚鉅，?了克服角度的問題，我們決定開發自製的追蹤器來改善角度的問題。太陽能板需要改變仰角跟傾角（雙軸調整），由光感應器判斷及自動控制程式，判斷隨時辰與季節變化的太陽角度。當搭配奈米塗料、弗瑞奈透鏡與追蹤器，總輸出功率可增加約50%。太陽電池表面玻璃會阻擋藍紫光的吸收，但本研究在太陽能板上塗佈奈米塗料，發現能增加短波長的吸收；經實驗後奈米等級表面具有自潔效應，可防止灰塵雨滴的堆積影響光線吸收，具有開發價值。This project first compares Taiwanese locations with other places in the world on average daylight times. It was discovered that southern Taiwan has the longest average daylight time all over 2000 hour sand therefore most ripe for solar power development. To increase the energy density of solar Fresnel lenses were incorporated. Although this has the advantage of magnifying illumination by three hundred percent, the alignment angle for the solar panel will have a significant impact on performance. We then designed and built a automated tracking device with illumination sensors to control the elevation and inclination of the solar panel which adjusts the angle according to environmental conditions such as time of day and season. When solar cell collocate Nano coating, Fresnel lens, tracking device, its power can promote almost150%. The glasses on the solar cell will interfere solar cell absorbing blue and purple light, but we lay on a Nano coating and we find Nano coating can improve solar cell to absorb short wave; and surface o Nano have lotus effect, it can prevent dust and rain effecting solar cell absorb lights, and it is worth developing .
Is the fruit safe?-吊白塊的簡易自製試劑
吊白塊是一種在現切水果中常見的食品添加物，它可使剛切的水果不易被氧化，並同時具有漂白的效果，但此種添加物會對人體造成許多疾病。本研究針對吊白塊作嘗試性的初級檢驗，選用一般常見的氧化劑和染料，自行研發簡易的檢驗方法，且進一步製作安定性佳且攜帶方便的試紙。本實驗結果發現，由衛生局提供的「藍吊試劑」本身不甚穩定，且顏色變化不明顯；在自製檢驗試劑方面，效果最佳的是過錳酸鉀，濃度可測至0.0005M，且反應相當快速，唯試液容易與水果表面的Fe(II)離子反應；孔雀綠和晶紅酸等染料效果亦佳，且變色相當明顯，但反應時間較長。Rongalit is a bleaching agent commonly used as a food additive. It can prevent fresh fruits to be oxidized (without color-changed), especially when they were cut for sale. However, as for this additive, it is not good on health and is necessary to be detected. The test-paper currently used, the so-called “blue-test paper”, can be obtained from the Department of Health (Taipei). However, its stability is poor; the color change is not clear when it reacts with Rongalit. For this reason, I developed simple methods for detecting Rongalit by using various oxidizers and dyes. A test-paper, with better stability and easily for carry, was successfully developed. The findings show that the use of KMnO4 on the homemade test-paper provides the best result. The reaction time is short and the limit of detection can be improved to 5 × 10-4 M. The color changes were also clear when malachite green and fuchsin acid were used, but the reaction times were longer.
高分子複合材料的性質、製作與分解
Synthesis of Polymer Material and its Decomposing Processes Because plastic cannot be decomposed naturally by itself, therefore, additives needed to be added to facile the decomposing process. Let us choose one common material: thermoformed Nylon 66. During the formation process, addition of glucose powder and monosaccharide polymerized will result in yielding the products of methyl cellulose, soluble starch and agar powder. Observe whether adding additives would allow changes to occur structurally, or would the elasticity be improved when exist in a linear state, or even it would form a better pH resistance property. According to the experiments, when Nylon 66 contains methyl cellulose, it can sustain the highest tension. Its coefficient of elasticity is 2 times as large as the original one. In terms of the data, we can also observe that when Nylon66 contains soluble starch, it has the lowest ability to sustain tension. Besides, it has the lowest coefficient of elasticity. And when Nylon 66 contains cellulose, it has the highest rate in the process of decomposing.As we look at the surface of polymers under 400 diameters, we can observe that the Nylon 66 with agar powder has some filiferous substance. But we have not confirmed what the matter is. 由於塑膠不能在自然情況下順利分解，所以我們在塑膠中添加其他成分使塑膠可以較易 分解。我們選定常見的塑膠—熱塑性的耐綸-66。在聚合物的製作過程中添加葡萄糖、澱粉、 洋菜粉末以及甲基纖維素，並觀察加入添加物的塑膠在結構上是否有變化？其塑膠在線型時 之張力是否有增強？耐酸鹼性是否有變化？由實驗結果我們可以得知含有甲基纖維素之耐綸 -66 所能承受之張力強度最高，且其彈性係數也比無添加物之耐綸-66 高出近2 倍；而含可溶 性澱粉之耐綸-66 所能承受之張力最小，且彈性係數也最低。此外，進行生物分解的實驗可發 現，含葡萄糖的耐綸-66 分解的速率最快。使用400 倍的光學顯微鏡可發現含有洋菜粉末的耐 綸-66 表面與其他耐綸-66 複合材料差異較大，值得進一步研究。