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.

本研究主要以一連串地震分析，來討論台灣地區地震能量釋放的型態。首先求出地震活動參數，接著再繪製地震規模的年發生率，最後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。另外本研究發現，在大地震來臨之前，地震A 值會升高，應可視為其A 值升高之因是由於前震的關係。在地震規模年發生率方面，繪製出圖表，最後發現台灣地區地震仍以小震多，另外大地震則有一定週期。在研究最後，了解台灣地區地震常數之後，本研究更以ESRI 的GIS 軟體ArcView9.0 版輸出台灣地震目錄，並從中了解台灣的震測儀器發展。This research presents a series of earthquake 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 9.0 edition of GIS (Geography Information System) of ESRI , knowing the earthquake in Taiwan.

Most football managers are not aware of the need for analysis of soccer data, which is one of the dynamic sports. In this study, we developed a statistical model with performance indicators and score indicators of individual soccer player based on various event data including dynamic features such as goal, assist, pass, etc. In this study, the correlation between the dependent variables and the explanatory variables, and each explanatory variable was confirmed through the correlation analysis to solve the problem of multiple communicability from the regression model and to analyze the statistically significant preliminary model. In addition, we analyzed the correlation between individual rating of the players and the data recorded in the soccer games, and found that there has been a problem with the rating of the soccer players evaluated by the reporters and the soccer statistics site. To solve this problem, we developed a model that best fit the performance indicators of individual soccer player using the linear regression model and the beta regression model. The performance index model of the athletes was developed by comparing the R-squared value and the mean absolute percentage error of two models, the linear regression model and the beta regression model, and found out the beta regression model is better model to use. By using the estimated regression coefficients of the regression model we made new PI model. Score Index, which is the attractive point of soccer, was developed by comparing Poisson regression model and negative binomial regression model based on AIC value, and the one using negative binomial regression model was found to be better. Through the model developed by this study, it is possible to collect the event data recorded by individual athletes for each soccer game, and obtain the PI & SI index which are the athlete performance index models. This allows us to evaluate each team's players objectively, analyze the team's deficiencies, and provide tools to find players, who can fill in the missing positions of the teams. This study can also be utilized to grasp the performance of athlete in real time by simulating the resultative model.

本研究目的在發展以水活性值作為食品水分含量和水溶液濃度估算之快速量測工具。對檢體之多點30℃水活性與水含量作圖，結合四階多項式數學模式模擬等溫吸濕曲線方程式(Moisture Sorption Isotherms)。以水活性實測值內插換算，獲取食品水分含量。對檢體之多點水活性與濃度作圖，結合多項式數學模式模擬水活性濃度曲線方程式。以水活性實測值內插換算，獲取水溶液濃度。結果顯示，隨機收集的食品檢體實測值與以模擬曲線內插而得之水分含量值偏差多不超過3%，實測值落點皆分布於模擬曲線附近，表示以此擬合曲線推算含水率確實可行。同時，水活性濃度模擬曲線準確度佳，不只適用於稀薄溶液，大範圍濃度均適用。

It’s very important to control and monitor gases that are produced by industrial applications in different values and kinds because they can cause environmental pollution and health problems. Photonic Crystal Fiber (PCF), which is a different kind of optical fiber, is a new alternative for gas sensors due to their small sample volumes, low transmission losses and high flexibility properties. PCF’s are silica-glass fibers, made by periodic sequence of hollows along the fiber. By filling these hollows with optical liquids or gases very sensitive sensors can be made. In this project, we aimed to design a sensitive sensor by filling the hollows with proper gases and liquids in the solid core PCF. For these applications ethanol, methanol, toluene vapors and carbon dioxide was used. And to observe carbon dioxide’s effects ionic liquid (EMIMBF4), which carbon dioxide can dissolve in, was filled then the experiments were repeated. It was observed that the transmission of light in PCF changed depending on the refractive index of the gas that was filled. With this change, it was understood that there were another gas besides the usual containments of air. Our system could measure the absorbtion peak of toluen so it can be used as a toluen dedector and when ionic liquid filled the fiber, two steps that occur in the spectra of corbon dioxide so it can also be used as a carbon dioxide dedector. The system was customized as a carbon doxide sensor in a cost-efficient and portable way. Our system can be specialized and easiliy used with right light source, which is efficient to see the absorbtion peaks, and proper liquids to dedect intended gas. Making a carbon dioxide sensor by filling PCF with ionic liquid was never attempted before. Also the lack of carbon dioxide sensor studies supports the originality of our project. That’s why we think our project will contribute very importancies to the existing literature.

Modern capacitive touch input and proximity sensing technologies are rigid and limited to flat substrates making it impossible to apply them onto objects with irregular geometries like textiles or car handles. Furthermore, the high cost restricts the applications to small surfaces and cannot be scaled up to be applied on large surfaces such as walls. Therefore, a paint-on scheme would broaden the applications of capacitive touch input and proximity sensing devices. Paintable capacitive sensors are an emerging technology hindered by the high cost and lackluster properties of conductive paints. Existing conductive paints utilize expensive filler materials such as silver and gold to achieve high conductivity but suffer from low surface area. High surface area is critical for capacitive proximity sensors to detect objects from far distances and for overall sensitivity. Carbonaceous alternatives using micronized graphite exhibit low conductivity, require high loadings and most disintegrate when in contact with water. Multilayer graphene nanoplatelets are investigated for their high conductivity, high surface area, low cost, flexibility and eco friendliness. A waterproof acrylic latex is combined with multilayer graphene and dispersed via bath sonication. The optimal time of sonication and optimal graphene loading is determined through systematic testing. An Arduino Uno is loaded with a CapSense library and the graphene based paint is utilized as the interface to sense both touch and proximity.

Gangrene is the death of tissue of certain parts of the body. In Indonesia, people who suffer from Diabetes will also often suffer from Gangrene, which usually affects a patient’s feet. The medication for it is not affordable for everybody. I have chosen the Horseshoe Crab, simply because it’s known widely in Indonesian and can be found easily. Also, due to the fact that, among all crustaceans, the Horseshoe Crab contains the highest levels of Chitin. By using the Chitin found in the shell of the Horseshoe Crab, I shall endeavor to heal the Gangrene of Diabetic Patients.

本作品為環球數學城市競賽的考題之推廣（ International Mathematics Tournament of the Towns, Senior A-Level Paper, Fall 2009, No. 7 ），然而此題目本身比較接近Scientific American （ Feb 1979 ）中 Martin Gardner 的文章 The Rotating Table 所提出的問題。 而此問題的多邊形版已被數學家解決(Ted Lewis & Stephen Willard, 1980, The Rotating Table,Mathematics Magazine, 53,Page 174-175.)。然而此份作品採用簡潔的初等數學歸納法證明了原本的多邊形情形，且專注於探討條件改變為m=n-1時的多邊形情形。 m=n-1時的情形是前人所沒有做出結果的，且m=n-1的情形遠遠比m=n 的情況來得複雜多變，其中估計上下界所使用的許多組合技巧相當特別。然而其中當n為3 和5的倍數時有比較特殊的情況，這一個部分僅能給出目前得出的k值上界。而其他情況則能夠找出所有的k值。

1 Purpose of the Research Nanocrystal thin films exhibit many useful properties, including electrochromicity and superconductivity. When synthesised via Molecular Beam Epitaxy (MBE), selection of substrate, specifically knowledge of crystal orientation, is critical. Reflection High Energy Electron Diffraction (RHEED) is an in situ crystal characterisation method highly compatible with MBE. This study explores a new method of RHEED analysis to determine crystal orientation. 2 Procedure/Theoretical Framework RHEED characterization is the incidence of a beam of high-energy electrons at a low angle with respect to the sample surface. Electrons diffract, and interfere to form patterns on the detector. Traditionally, studies of RHEED analyse one static image as a representation of the surface structure, or observations of RHEED patterns over time. The approach to RHEED analysis in this study exploits changes in RHEED patterns given a rotating substrate. Having specific rotational symmetries along different axes, crystal structures can be differentiated by determining rotational symmetry through RHEED. Electrons scatter upon incidence with crystal planes within the crystal to form Kikuchi lines on the RHEED detector (Fig. 2). The orientation of crystal with respect to incident electron beam affects the Kikuchi line patterns. If the crystal is rotated, crystal planes change orientation, and electrons would diffract from crystal planes in different directions. As such, as the crystal is rotated, the Kikuchi lines move. When the degree of rotation of the crystal corresponds to the rotational symmetry of the crystal (Fig. 1), the Kikuchi lines return to their original position. As crystals with different crystal plane orientations exhibit different orders of symmetries, analyzing the Kikuchi line patterns of the crystal at different degrees of rotation can reveal the rotational symmetry and consequently crystal plane orientation of a crystal. 3 Data/Experimental Testing In order to assess the practical viability of the proposed method, experiments were conducted on SrTiO3 (001), (110), and (111). SrTiO3 exists as a typical perovskite structure (Fig. 3), often used in the synthesis of superconductors via MBE. 3.1 Methodology RHEED images of each sample were taken at 0◦, 60◦, 90◦ and 180◦. Curves were fit to each Kikuchi line observed in the image (Fig. 4). These Kikuchi line approximations are compared by superimposing the curves traced and qualitatively assessing the degree of similarity between the Kikuchi lines of 2 images, to verify the order of symmetry and crystal orientation of the crystal. In the images of the superimposed Kikuchi lines illustrated in Fig. 5, there is similarity between the Kikuchi lines when only when the sample has been rotated by an angle corresponding its degree of symmetry. 4 Conclusions This study offers a method to determine the crystal orientation of thin film through determining the degree of rotational symmetry of the sample, by observation of Kikuchi lines in the RHEED pattern as the sample is rotated. Experimental data was analyzed qualitatively to verify the viability of this theoretical method in practice. This method could be extended to analyze the symmetry of other crystal structures. As it does not require information on the machine settings or usage of complex functions to produce a reliable output, this method is fast and straightforward, opening doors to more streamlined RHEED analysis.

本研究主要探討傾角對面轉變的影響與面轉變的原因。當柱體由肥皂水中拉起時，泡膜圖形可分為中央膜平行與垂直底面的形式，兩種形式因高的變化而互相轉換的過程稱為面轉變。三到六角柱傾角越大，面轉變時的高越大，反之亦然；六角柱可以面轉變，且在40°到45°之間有臨界角度存在。本研究以力與能量的角度解釋面轉變．泡膜藉由改變面積以達到最低的能量、維持穩定狀態，因此本研究計算、比較不同形式的泡膜面積，以解釋面轉變。我們同時發現三角錐與四角錐都不會發生面轉換；三角錐的實驗值多大於理論值，四角錐的實驗值則多小於理論值。

我們發現當CdS 與ZnS 重量比以2:1 時，能在Na2SO3 及Na2S 的混合溶液中，照射日光產生氫氣，並能有最佳的產率38.18(ml/hr*g)，比一般直接照紫外光的效果更好。不只善用自然界的太陽能，還可以生成最乾淨"綠色"的能源－氫氣。我們以 CdCO3 和ZnCO3 做為產氫觸媒的原料，其為黃色的粉末，並以AA 原子吸收光譜儀求得樣品溶液中測得CdS 與ZnS 重量比為２：１。 把產生的觸媒放入以 Na2S 和Na2SO3 混合而成的溶液中，以鋁薄紙包覆以防受光實驗提早進行。把實驗裝置放在日光下照射，以一端管口所接的細量管得失產生氫氣的體積，進而換算產率。 我們的實驗裝置為密閉式的，並無循環裝置，故產氫量有限。未來的發展將是朝著大型開放式的循環系統，使氫氣能源源不絕的產生。 The production of molecular hydrogen upon sunlight illumination of the mixture of CdS and ZnS suspensions in the weight ratio 2:1 in Na2SO3 / Na2S solution have been observed. The best rate of evolution of hydrogen is 38.18(ml/hr*g), which has a higher efficiency than upon UV illumination. It not only make a great use of the solar energy, but also produce the cleanest “green” energy resource─hydrogen.

尿酸氧化酶參與嘌呤代謝機制，然人類尿酸氧化酶基因已退化，易使過量尿酸沉積於關節造成痛風，因此微生物源尿酸氧化酶具極高研發價值。 本研究針對微生物源酵素進行基因體探勘，篩選出抗輻射奇異球菌(Deinococcus radiodurans)及耐熱雙球桿菌(Thermobispora bispora)源尿酸氧化酶基因，取得尿酸氧化酶重組蛋白質進行酵素活性測定、熱穩定分析、蛋白質結構解析、酵素對金屬離子耐受性分析與尿酸檢測應用。抗輻射奇異球菌源酵素於30℃催化速率較過去報導之酵素佳，而耐熱雙球桿菌源酵素於70℃催化效率與熱穩定性仍佳。解析耐熱雙球桿菌源酵素結構發現羧基端具特異性構型，推測與熱穩定性有關。本研究的兩種尿酸氧化酶，可分別作為降解尿酸的速效及緩解型藥物，並開發成快速檢測尿酸的簡易方法。