給定下面範例：\r 058823529411764705882 +235294117647058823532\r =0588235294117647058823529411764705882353，\r 其等式結果與質數17 的倒數結果(1/17)有某種關聯(卻沒有一個決定性的證據)，意即\r 1/17=0.0588235294117647=\r 0.058823529411764705882352941176470588235294117647...... ( Len(17) =16 )\r \r 曾經在下列網站上發現過幾組數字(挑戰試題)，引起我們極大的興趣。\r http://www.domino.research.ibm.com/Comm/wwwr_pondernsf/challenges/March2000.html\r http://www.math.smsu.edu/~les/POW08_96.html\r \r \r The two examples that I have are 0588 2+23532=05882353 and 058823529411764705882+23529411764705882353 2=0588235294117647058823529411764705882353 These were found by the Canadian professor Alf van der Poorten, and he gave a talk on these identities in December at the west coast number theory conference. He was unspecific as to exactly where these identities were coming from, but they are connected with reciprocals of primes:1/17 = 0.0588235294117647= 0.058823529411764705882352941176470588235294117647 ΛΛ ( Len(17) = 16 ) Though not mentioning how to obtain these equations, Prof. Poorten demonstrated the relationship between the above examples and the reciprocal of the prime numbers 17 (1/17 ) without a definitive proof.

本次科展我們所探討的，是關於圓的分割問題，討論披薩被切出來的面積和及切痕的長度和等相關問題。 我們提出並且用較初等的數學方法證明了幾個與圓的分割有關的問題，包括披薩定理以及另外七個定理。 最後我們也利用GSP軟體驗證定理二、三、四分給2人時的對偶結果和定理二分給3人以上時的其他結果，並對於一般情形的推廣，作合理的猜測。

前人研究顯示結網性蜘蛛會因環境中獵物組成的改變而改變其網結構。然而，除了結構之外，構成網的絲的物理性質與蜘蛛網的捕捉效率也應有密切之關連，但目前並無人探討蜘蛛是否會因獵物而調整絲之化學及物理性質。在本研究中，我以人面蜘蛛為材料，探討：(一)其在面對不同類型之獵物時是否調整其網之結構，絲之胺基酸組成和其物理性質；(二)所產生之改變是否能促進不同類型獵物的捕食效率；以及(三)是獵物之養份或振動刺激引起這些改變。我藉著操控獵物的養份及振動刺激將人面蜘蛛分成四個處理組，分別餵以活蟋蟀(C 組)，活蒼蠅(F 組)，死蟋蟀加蒼蠅振動刺激(Cd 組)及死蒼蠅加蟋蟀振動刺激(Fd 組)。在完成食控處理後比較各組間之網結構、絲胺基酸組成、絲物理性質、對不同獵物之留置能力以及振動訊號傳遞能力。實驗結果顯示F 組之蜘蛛所結之網有較大的面積及較小的網目，並有較佳之蒼蠅留置力；然而此結構之改變卻會降低其振動訊號傳遞率。C 組之蜘蛛所結之網有較強之張力及較佳之蟋蟀留置力，而且其所產之曳絲有較高比例之alanine、glycine 及glutamine 並較佳之延伸度及韌度，以因應體型較大之獵物。上述結果顯示人面蜘蛛能調整網的結構、絲的蛋白質組成及物理性質來更有效率的捕捉不同類型的獵物。而由於上述各樣變數在C 及F 組間有顯著差異，而在獵物養份與振動訊號錯置的Cd 與Fd 組間無顯著差異，顯示此二項刺激需同時存在才能使人面蜘蛛在面對不同類型獵物時產生結網行為及絲基因表現之改變。Previous studies showed that orb-weaving spiders may adjust web structures when encountering different types of prey. In addition to web structures, silk physical properties should also be critical in affecting the prey catching efficiency of webs. However, so far no one has studied whether spiders adjust silk chemical and physical properties when encountering different types of prey. In this study, I used the giant wood spider Nephila pilipes to investigate (1) would N. pilipes adjust web structures, silk amino acid composition and silk physical properties when encountering different types of prey (2) would such changes enhance the catching efficiency of different prey and (3) were such changes generated by prey chemical composition or vibration signals? To answer these questions, I manipulated prey chemical composition and vibration signals by assigning N. pilipes into the following four feeding groups: (1) living crickets (group C), (2) living flies (group F), (3) dead cricket but with fly vibration signals (group Cd) and (4) dead flies but with cricket vibration signals (group Fd). After the feeding treatments were completed the following variables were compared between the feeding groups: web structures, silk amino acid composition, silk physical properties, retention time of different prey and vibration signal transmission rate of webs. The results showed that the webs built by spiders fed with living flies exhibited larger area, smaller mesh and longer fly retention time. However, such structural changes also reduced the vibration signal transmission ability of the web. On the other hand, webs built by spiders fed with living crickets were stiffer and retained crickets longer. Moreover, their silks exhibited higher percentages of alanine, glycine, glutamine and consequently higher elasticity and strength to cope with the larger prey. These results demonstrated that spiders can adjust web structures, silk protein composition and silk physical properties to enhance the catching efficiency of different prey. While the aforementioned variables differed significantly between spiders in C and F groups, they did not vary between spiders in Cd and Fd groups, in which groups the prey chemical composition and vibration signals were decoupled. Such result suggested that prey chemical composition and vibration signal were both needed in generating changes in web building behavior and silk gene expression in spiders.

The natural fruits of soap berry are worth-while detergent in the aspect of environmental conservation. It has been shown that the soap berry fruits can reduce the itching of head and inhibit the growth of fungi. Here we would like to test the inhibitory activity of soap berry fruits on the bacteria. This might benefit to the further application of the soap berry fruits. First,the soap berry extract was prepared and mixed with E.coli or S. aureaus. The number of bacteria was counted to realize the inhibitory activity of the soap berries. We found that the soap berry extract can inhibit the growth of E.coli and S. aureaus. Higher concentration of the soap berry extract and longer duration for the mixture make better inhibitory activities. 天然的無患子果實是一種極具環保價值的清潔劑，已知它有抑制黴菌生長的能力，可以 去頭皮屑、止頭皮癢。因此，我們想進一步探討無患子果實對於細菌生長的抑制能力，以期 了解無患子清潔劑在廣泛推廣上的應用價值。 在實驗設計上，我們自製無患子萃取液，並選擇日常生活中常見的大腸桿菌及金黃色葡 萄球菌，並然後使菌液與適量的無患子萃取液混合，計算菌液在加入無患子之前、後的細菌 個數，藉以了解無患子的殺菌力。我們發現無患子萃取液對大腸桿菌及金黃色葡萄菌有明顯 的殺菌力，且無患子濃度愈高、作用時間愈長，殺菌力愈好。

利用范式起電器將靜電導入蒸餾水、甘油、苯、環己烷、乙醇、正丙醇、正戊醇，以及HCl(aq)、NaCl(aq)、CuSO4(aq)……等不同溶液中，我們發現：非電解質溶液可以藉由極性分子的媒合作用"貯存”電子，而其中H2O效果最好，即水合電子是可以穩定存在的！但在水中若添加HCl、NaCl等電解質會提高容易導電性，以致於無貯存電子的效果。我們也發現苯雖不是極性分子，但因具有π共振系統，也能有效貯存電子。另外，從引入密度泛函數理論結合類導體可極化連續模型加以計算的結果，也能發現極性分子容易的溶劑電子親核能比非極性分子容易來的大，與實驗結果吻合。By using a Van de Graff Generator to put electrons into various solutions or solvents (such as water, alcohol, benzene, propyl alcohol, solutions of sodium chloride, etc.), we find out that nonelectrolytes can "store" electrons through hydration of polar molecules. And water has the best ability of keeping electrons among others. In other words, hydrated electrons can exist stably. However, if we add electrolyte into solvent, the solution won't store electrons because of its increasing electric conductivity. The experiment also shows that benzene can keep electrons though it is a nonpolar molecules. We infer that it is the " π resonance system " which contributes to its high ability of electron storage. In addition, by introducing the calculations of CPCM (Conductor-Like Polarizable Continuum Model), we find out that the electron affinity of polar solutions are stronger than those of nonpolar solutions. The computed results are in agreement with our experiment.

本實驗主要是研究閉管駐波的發聲原理。何謂「閉管駐波」？就是一個管子在相同長度下，用不同的力道吹，會有不同音高的聲音產生，這些音被稱為「諧音」。原管長所能發出的最低頻率稱作「第一諧音」，第二低的聲音稱作「第三諧音」，依此類推。在簫的演奏上，只要按住同樣的孔，用較大的力量吹，也同樣會發出較高的音；同樣地，在曲笛的演奏技巧上，有平吹、急吹等分別。為什麼吹越用力，音就越高呢？如果現在拿一個大吸管吹（要裝活塞），你會發現，只有在特定的位置（角度）下，才能吹出聲音。那麼，角度對於聲音也有影囉？這些現象的幕後黑手，就是在管口產生的「渦流」，渦流頻率也會隨著風速而增加；而且，渦流的頻率在特定風速下，會有特定的範圍。經由實驗可以大略歸納出，影響閉管駐波的三個主要變因，分別是「風速」、「風吹角度」、及「吹口至管口的距離」。吹得越急，風速就越快，渦流頻率越高，越易使諧音躍遷；吹的角度越小，越易產生渦流，亦易引發聲音；吹的距離越小，渦流越不?定，越易產生其他的擾動。以上就是本實驗的概略。This project is aimed to fine out how the closed tube can produce a sound. We know what harmonics are. When we hold a big straw and blow with increasing strength (the bottom should be in water), it will generate a higher sound. The high sound is called “harmonic”. The lowest sound it can make is “the first harmonic”, the second lowest sound is “the third harmonic”, and so forth. Likewise, when we press the same key on vertical bamboo flute with increasing strength, it’ll also produce a higher sound. But why do we use the strong air stream to blow the tube to cause the tone to transfer? Now let’s blow a straw flute. You will find that you need to blow in the particular position, and then the sound will be produced. So, is there any relationship between the blowing angle and the frequency? Actually, all these sound are produced by “vortex in the mouthpiece.” The frequency of vortex will increase with the wind speed. Moreover, the frequency of vortex has a range. In sum, the higher the wind speed is , the higher the frequency of the vortex is , and leads to the higher frequency of the sound. The smaller the blowing angle is, the easier the vortex will be produced; the easier the frequency will be made. The smaller the distance between the blowing angle and the frequency is, the more unstable the frequency will be. The above is the most important research in this project.

本研究探討不同的控制變因，對磁性流體薄膜在水平磁場下有序結構的影響。我們發現，外加水平磁場於磁流體薄膜時，會形成一維有序磁鍊排列，磁鍊間距除會隨著磁場增強而變小外，另外其條件值如磁場強度、流體濃度、磁增率、薄膜寬度及厚度等也影響磁鍊間距。其中磁增率及磁流體濃度增加會使磁鍊間距變小，而凹槽寬度及薄膜厚度增加會使磁鍊間距變大等現象。至於其形成磁鍊的物理作用，我們假設薄膜有三種能量交互作用，即(1)磁鍊與外加磁場間的磁能UdH。(2)兩條同方向磁化的磁鍊間相互排斥所造成的磁能Udd。(3)熱能Uther等。藉由系統能量Udh+Udd+Uther最小化，我們導出了磁鍊間距和外加磁場之間理論上的關係，並比較實驗結果確實具有相當的一致性。因這些有序結構會引發許多的光學性質，將來這些特殊的光學性質預料應可製成可調式光柵、光開關及顯示器等光電元件，使磁流體在奈米世界及光電領域裡扮演重要角色。We investigate experimentally the structure of the magnetic chains formed in the magnetic fluid thin film under the influence of the external magnetic field parallel to the film surface. A one-dimensional ordered structure forced by magnetic chains can be obtain in the magnetic fluid film. It is worth noting that the ordered structure can be manipulated by changing the control parameters such as the magnetic field, concentration of magnetic fluid, the thickness of the film, the width of the film, and the dH/dt. On the other hand, the physical mechanism of forming the ordered structure can be also studied theoretically. These magnetic chains are regarded as magnetic dipoles and three possible interactions are condisered for the energy of the system: the attracting potential energy between the magnetic chain and H(denoted by UdH),the repulsive potential energy between two magnetic chains with parallel magnetizations(denoted by Udd), and the thermal energy Uther. The relationship between the chain distance △x and the applied magnetic field H was derived by minimizing the total energy of the system with respect to the chain distance. The experimental data is consistent with the theoretical results.

The first aim of my project is to learn more about mushrooms: their lifecycle, their structures, the nutritional value of mushrooms, the medicinal values of mushrooms as well as the biological importance of mushrooms.\r The second aim of my project was to find how mushroom are cultivated in the world and specifically how they are cultivated in Namibia, where climatic conditions are rather harsh for mushrooms. Through interviews, I aimed to understand why mushroom cultivation is not widely practiced in Namibia and what are direct economic advantages of mushroom cultivation in Namibia.\r The third aim of my project was to see whether I could find cheap and economic methods of cultivating mushrooms in Namibia, a semi arid country where humidity is so scarce.

Now Artificial Neural Networks using on the basic math is fewer. This paper is to suggest the Linear equation of the basic math using the BP Artificial Neural Networks. The BP Neural Networks have power ability for learning and can approximate any function, and regularity can be found to solve the linear equation. A good sample is one of the important elements for learning of Artificial Neural Networks. Generally, the samples are a lot of amount for the resolution of Linear equation. This paper is to use the principle of two points decide one line for the samples. The experiment shows that this method curtails many samples. Furthermore we also use Artificial Neural Networks to solve the problem of point-slope form. The experiment result is very satisfactory, and it offers some idea for the basic math using Artificial Neural Networks.目前人工神經網路較少用於基礎數學方面的求解，本文針對基礎數學直線方程式提出BP 人工神經網路應用於求解直線方程式，運用其很強的學習能力、(輸入向量和其對應的目標向量來訓練網路、逼近函數)，尋求規律來求解直線方程式；而良好的樣本是人工神經網路學習的重要條件之一，一般解決直線方程式需要大量樣本，本文利用二點決定一直線的原理來解決樣本問題，實驗結果顯示，這一方法成功的縮短了可觀的學習樣本，此外我們也運用BP 人工神經網路來求解點斜式的直線方程式問題，實驗結果是可行的，並且為人工神經網路用於基礎數學提供了一些思考方向。

These days we hardly see fireflies, the Milky way, and rainbow. What’s the reason of that?\r We think the reason is related to air pollution but the concrete reason is unknown to us. Thus the purpose of this research is to see the relation between air pollution and rainbow with Aerosol. In this research we approached in some ways about why rainbow didn't appear in polluted atmosphere.

In this research we derived heating effects on anti-oxidation by optical rotation analyses. A simple and accurate Polarimeter was devised with polar screens, laser, and photo resistor; the laser was used for light source, and the photo resistor for detection. Four kind of fruits, Hyloceeus polyrhizus, California plum, kiwi and tomato were experimented by heating them in boiling water and microwave oven. Their optical rotations versus time duration were calculated. Then we used ABTS/ the H2O2/HRP analysis system from Arnao's research to obtain the heating effects on anti-oxidation by calculating the vitamin C densities versus lag time. We found that there was a tendency of increased anti-oxidation at the beginning by heating, but decreased afterwards. Better oxidation was achieved by using microwave for Hyloceeus polyrhizus, by using boiling water for kiwi and tomato, and by using both methods for California plum. The same tendency was also found in the optical rotation analyses. Therefore we successfully developed a new method to measure the antioxidant activities by the optical rotation. We can also apply this method to simulate the fruit digestion process in the stomach, which let us understand further about anti-oxidation ( or optical rotation ) versus time. 本研究利用偏振片、雷射光為光源、光敏電阻為偵測器，組裝成一個簡易且精確的旋光度計。 我們利用此旋光度計對奇異果、火龍果、聖女蕃茄、加州李子四種水果以微波、水煮及油浴三種方式來處理，求出水果的旋光度與熱處理時間的關係。再藉由Arnao 研究的ABTS/ H2O2/HRP 分析系統，以不同濃度的維生素C與延遲時間畫圖作為標準曲線來測量總抗氧化活性，並比較熱處理後抗氧化活性的變化趨勢。 透過本研究可以了解不同熱處理方式對四種水果之抗氧化活性初期皆有增加的趨勢，但隨著時間增長則抗氧化活性降低。其中奇異果與聖女蕃茄用油浴，火龍果用微波處理，加州李子則是用任何熱處理方式均可得較佳的抗氧化活性，而熱處理後在旋光度測量上也有相同的趨勢。因為此兩種方法有一致的反應趨勢，所以證明我們成功地開發出一種可以利用旋光度來測定抗氧化的新方法。 最後，應用本方法進一步模擬水果在胃部裡的消化，讓我們更可以了解在整個消化的過程中抗氧化活性(旋光度)隨時間變化的情形。

One afternoon, a photography enthusiast, Mr. Cheung was holding some photos, completely absorbed in the fond (but ancient) memories of his honeymoon in Cairo. The badly-yellowed photo prints reminded him of every sweet and romantic moment of the “good old days”. He was soon rudely awakened when Mrs. Cheung shouted, “Honey dear! I was told that high-tech is ‘in’, and the single-lens reflex camera is totally out! So stop hugging the old out-dated pictures.” Mr. Cheung fired back, “Oh, you know nothing. This is something you cannot replace at all, and my camera is a classic.” Smiling at him, Mrs. Cheung took out a new gadget, “Right, right, now calm down. It never hurts to have both worlds. Look what I’ve got you. This is called ‘Stereoscopic Camera Lens’. Its little coloured spectacles can capture everything, say, the sensational Great Pyramid, in three dimensions!” As we are asking for and relying more on technological advances, the Stereoscopic Camera lens and the Stereoscopic Magic Box are surely innovations that promise to add a new dimension to our lives. Surprisingly, the working of these two gadgets involve only some simple scientific theories such as polarization, doping and anaglyph; and employs the special property of Liquid Crystal Layer (LCD) under electric current. Stereoscopic photos can now be easily available to us. Both of our designs, the Stereoscopic Camera Lens and the Stereoscopic Magic Box, can turn something ordinary into extraordinary by making plain, 2-dimensional things ‘stand out’ before your eyes. The Lens itself is crafted with such precision that it can capture any image both faithfully and flawlessly. The Magic Box uses electric current to catalyze and enhance the 3-dimensional transformation. So each picture is vibrant and true-to-life. The Lens and the Magic Box will certainly revolutionize our vision of things around us. To enable you to have a good understanding of the theories behind the making of the Stereoscopic Camera Lens and the Stereoscopic Magic Box, we have made simple models of them with prisms, and they are in display at our booth. Anyone who is interested in these fancy gadgets should definitely come to our booth to experience a new view of the world!