平面切立方體內單位立方格數極值之計算
我們先假設有一正方體及一截過正方體之平面，並設正立方體為一k*k*k 之立體。為計算平面截過之單位正立方體個數，我們必須先分別計算各層被切過之個數再將之相加，因此將各層面投影至同一平面，簡化為平面上之問題，並討論其性質／規律，計算平面截此正立方體之個數。如此，便可以一般化數學式計算平面截正立方體個數之問題。接著，用以上方法為基礎，討論各種平面切正立方體之類型，將被平面所截之單位立方體個數以電腦程式算出，觀察數字變化及其性質規則，並找出最大值發生之條件。 We initially supposed that there are a regular hexahedron consists of unitary n × n cubes and a plane which incises the regular hexahedron. To calculate the total number of the unitary cubes incised by the plane, we can first calculate them layer by layer and then sum them up. And further, we project each layer on the same plane, so the three-dimensional problem is simplified into two-dimension. By making use of the character which results from projection, we can easily calculate the number of the unitary cubes incised. Consequently, we are able to calculate them with a general equation. Afterward, we research each circumstance that the plane incises the regular hexahedron on the base of the mentioned methods. Calculate them with self-designed computer programs, and observe the regulation and change of the result. Furthermore, we can find out when it will achieve the maximum.
聽聽貝多芬作品的下一代：將碎形及基因演算法應用於數位音樂產生器
本研究整合了碎形圖形的迭代運算方法與基因交配觀念來達到音樂創新，並透過音樂和諧性判別機制來提高創新音樂的悅耳程度。利用基因觀念之交配的方法來解決長短的問題。這個方法是把原始音符輸入後，找出它們的中心點，以這個中心點為準，其他的音符按照一定比例向外延展，成為新的迭代點。再利用這些迭代點，迭代出新的音符。把製造好的音符染色體放置到交配池中，以隨機的方式在交配池中選取其中之一個染色體進行交配的動作，此二音符染色體會交換彼此的基因，產生下一代新的代表音符長短之染色體，隨後以「模仿母體判斷式」來判斷這新一代的音樂是否與母體音樂相似，藉此淘汰掉「不肖的」下一代，而若新一代與母體的相似程度高的話，它的悅耳性相信也會相對提高。最後把這些技術應用於數位音樂創作，以衍生新穎應用與創新的結果。Fractals can be produced by IFS (Iterated Function Systems). By iterative computation of many times, we can obtain the similar graphics. In my research, the methods to generate the iterative algorithms were presented. In addition, I would discuss the regularity and the content as well as the properties of those digital patterns. At last, the advanced application of fractals to digital music pieces was presented. The program took a note of several measure of music as the beginning point, and made the IFS calculations for each new note in each measure. But there was no difference in beats if you just make the IFS iteration. So I changed the beats with genetic crossover method. In this research, the expression of the DNA to each beat of note was adopted. The same way, it took a note as a beginning point. And the system obtained the new DNA from the old notes for new ones randomly. After producing the new pieces of music, I want to know if it is good to listen. So I used the algorithm that checks the simulation to the shape of mother music. If its shape is similar to the mother music, the probability that the new music is pleasing may even increase. That would make a piece of brand new music. What I want to do in this research is improve the multiformity of music and find what the relationship is of ‘good music’ and mathematical algorithms
磁流體的浪潮－磁場梯度下磁流波紋之研究
在本次實驗中，我們發現在不同厚度的磁流體薄膜中，會因本身磁性粒子結合，而呈現不同的影像圖形。隨著薄膜厚度增加，其磁性粒子會由鏈狀排列成塊狀叢集，可是一旦外加磁場後，又要全部轉向磁力線方向集結。另一個發現是將磁流體薄膜放在一個不均勻的磁場梯度中，則樣品內的磁流體粒子，不僅會隨著磁力線的方向排列移動，更會出現磁流波紋，其行進路徑是沿著垂直於磁力線的方向，向磁力線密集處移動。我們亦發現在不同的薄膜厚度及不同外加磁場下，其”磁流波紋”的波速亦會隨之改變。一般而言，樣品的厚度愈厚，或外加磁場愈大，其”磁流波紋”的波速愈快，反之則愈慢。最後，我們列出了一些磁流波紋的應用，相信是精采可期！In this experiment, we find that in different thickness of magnetic fluid different images will appear, because of the connection of magnetic particles. With the increasing of thickness the magnetic particles will change its shape from chains to blocks. But when we add external magnetic field, they will get in line one by one to the direction of magnetic line of force. We also find that we put the magnetic fluid film in the uneven magnetic gradient, the magnetic particle in the sample not only follow the direction of magnetic line of force but also show the “magnetic wave”. Its move path is perpendicular to the direction of magnetic line of force. In the different film thickness of magnetic field, the wave velocity of the “magnetic wave” will change. In generally, the thicker the sample is, or the larger the magnetic field is, the faster the wave velocity of magnetic wave is and adverse is true. At last, we list the applications of “magnetic wave”, we believe they are marvelous!
記憶學習機
一般學生對於學校課程學習的負擔，再加上對本身的自信心不足，往往導致學生自己所「背」的東西不是十分確信是否有真正記進腦中，也常常困擾著他們。因此，引發本小組想設計一個可以自我評測且可以立即得到成績的輔助學習機;在研究過程中，本小組設計記憶學習機不但可以做到使用者自我評測的功能，還可以讓使用者馬上得到檢測的結果並且使用者可以回顧先前成績欄紀錄;記憶學習機經過本小組組員的辛苦孕育下誕生，本小組便拿著「它」讓班上同學認識，雖然「它」不是同學心中的最佳主角，但是同學卻一致認定「它」是『最佳伴讀夥伴』。本作品主要功能有:1．能調整使用者所要求的測試時問。2．該作品能記憶此次測試成績，以便和下次相互比較。3．依使用者的需求選擇中文、英文、數字及遊戲模式做記憶訓練。4．能有立即性的成效及回饋。 Due to the pressure of courses and lack of confidence, many students are often not sure of what they have recited on a lesson previously, which in turn always disturbs and cut down learning confidence of students. To eradicate the obstacles they encounter, we decide to research into this topic. A memory-based learning-aided machine is designed for students to self-evaluate themselves and to get the feedback at once. In the process of development, the memory-based learning-aided machine not only lets users practice the exercise of their own, but also is able to derive the outcomes immediately and retrieve the previous records effectively and efficiently. We strive to make this learning-aided machine at its best performance as possible as we can. Besides, the memory-based learning-aided machine provides the alternatives for the users to answer the question in a way that they like. For example, users can answer questions in English mode, can choose the limit time into test themselves in a time. In addition, the system can expand its material by inputting data into the knowledge base. By our group members' lasting efforts, the learning-aided machine was created, finally. According to the result of experiment that we applied to the classmates, we conclude that the memory-based learning-aided machine is the best assistant and learning partner for the students. We would like to introduce it to classmates and hope they will be interested in using it to enhance their learning motivation and performance.
澱粉?抑制劑之研究
植物合成澱粉?抑制劑可以對抗動物的取食，國外實驗證明數種澱粉?抑制劑對害蟲防 治具有顯著效果，其中以腰豆(Phaseolus vulgaris)研究最多。我們利用5% T.C.A.進行粗萃，從台灣常見豆類中篩選出四季豆(與腰豆同種不同品系)與菜豆，對麗蠅的澱粉?具有明顯的抑制效果，對豬胰臟與黃豆澱粉?的抑制效果則小或無，此種抑制特異性深具害蟲防治的潛力。經由溫度與pH 的試驗發現粗萃中的澱粉?抑制劑成分應為蛋白質。我們以四季豆作為繼續研究的對象，將粗萃進一步純化，經由陰離子交換與膠體過濾層析，分離出單一蛋白質，經蛋 白質定序比對確認其可能為國外發表的腰豆澱粉?抑制劑—αAI-1。經由測試發現此抑制劑在 85℃時仍能抑制果蠅澱粉?，為一相當穩定的蛋白質；且抑制劑的作用受pH 值影響很大，在偏酸性環境下的效果最好，與昆蟲分泌澱粉?的部位亦為酸性環境有相當密切的關聯；且其 抑制作用具特異性，可明顯抑制果蠅、入侵紅火蟻、白蟻、蟑螂及麵包蟲等昆蟲的澱粉?活性，對人類唾液、豬胰臟、四季豆本身及黃豆澱粉?的抑制效果很小或無，值得繼續深入研究。 Plant amylase inhibitors can fight against predation from plant-eating animals. It has been reported that several amylase inhibitors have an obvious effect on pest control; among them that from Phaseolus vulgaris got the most surveyed. 5% T.C.A was employed to make crude extracts. We have screened the amylase inhibitor activities from crude extract among beans common in Taiwan. The inhibitors from both string beans (the different strain of Phaseolus vulgaris) and cowpea notably inhibited the amylases in Chrysomia megacephala, but little or no inhibition in porcine pancreas and soy bean. This specific inhibition behavior suggested strong potential in pest control. Its activity can be affected by temperature and pH suggested that amylase inhibitors in crude extracts should be proteins. String beans were chosen to be further purified from the crude extracts. A single protein was isolated after ion exchange and gel filtration chromatography. Through protein sequencing, the partial amino acid sequences were highly homologous to that ofαAI-1 from Phaseolus vulgaris, indicating it might beαAI-1. The purified protein still can inhibit the amylase from Drosophila melanogaster at 85℃, suggesting it is thermal-stable. Its activity was affected by pH and reached the peak in weak acidic environment, which might be related to the fact that amylases are secreted in acidic environment of insect’s midgut. It obviously inhibited the amylases from D. melanogaster,Solenopsis invicta, Odontotermes formosanus, Periplaneta Americana Linnaeus, and Alphitobius sp., while not to human saliva, porcine pancreas, soy bean and string beans itself. The unique pattern of inhibition activities of the purified amylase inhibitor was worthy of further anlysis.
稀子蕨的生殖策略探討
稀子蕨（Monachosorum henryi Christ）生長在台灣中海拔山區，是少數具有特殊生殖方式(不定芽)的蕨類。本研究對東眼山的稀子蕨族群進行物候調查，以了解其進行孢子繁殖及不定芽繁殖的過程，並試圖探討稀子蕨的不定芽是否可增加其族群對環境的適應性。研究結果顯示稀子蕨的孢子體及原葉體都無法在乾旱的環境下生存，常有大規模乾死的現象；而其不定芽則具有很強的耐旱性，無論是在季節普遍性及幼苖發育程度上，生殖芽都比孢子繁殖較佔優勢。而且其不定芽於旱季結束後，可迅速萌發，長出的孢子葉可隨之進行有性(孢子)及無性(不定芽)生殖，使其族群不至於在旱季中有地區性滅絕之虞。;Monachosorum henryi Christ , which exists in the middle elevations of the mountainous regions of Taiwan, is a fern with a special reproductive system seldom found in other ferns.This study investigates the phenology of Monachosorum henryi population in the Don Yang mountain region. Its purpose is to understand the sexual and asexual reproductive cycles of these ferns and to interpret whether or not the buds can increase the fitness of their population during the dry season.The results show that it is extremely difficult for the sporophyte and prothallus of Monachosorum henryi to survive in a drought. However, the buds have a much stronger drought endurance. According to this investigation, the adventitious buds of Monachosorum henryi are superior to the spores in seedling development in every season. Adventitious buds are able to germinate soon after the dry season as well as in any other season, and are able to grow spores and buds on their fronds enabling both a sexual and asexual way of proliferation. In this way the fern avoids a district extinction of their population.
見微知「駐」-水珠律動與圓駐波
It is always fascinating to see water droplet’s dancing around on a Japanese teppanyaki hotplate surface. The water usually does not evaporate immediately, but form interesting shapes, dance around and follow by evaporation of gaseous water and disappear. In this experiment, we designed a very simple experimental set-up to observe the little water droplets dancing on a heated hotplate. A homemade stainless plate and a small heater attached to the plate, and a thermal couple was assembled. With this simple setup, we observed the variation of water droplet’s shape as a function of the hotplate temperatures. The temperature of the water droplets, the duration of the water on the hotplate, and the shape number were measured. The shape formation mechanism was proposed. We found when the water droplet was subject to high heat due to the contact of the surface and the hotplate, the abrupt evaporation of the water molecules and violent vibration cause the formation of the various shapes to reach thermal equilibrium; the number of the shapes and the oscillation can be described by Laplace’s equation,Using a high-speed camera, we found the higher the temperature of the water, the more variations of the water droplet shapes can be observed. In addition, at a certain temperature range, the number of the water shapes did not change, suggesting a similar phase transformation behavior on the shape formation.
緣起： 邂逅專題研究、水珠漫舞、剪輯影片，引起我們想更進一步揭開它的神秘面紗。
緣續： 了解熱平台上水滴大小的變化及水珠基本的形狀及變化律動。
緣繫： 進一步研究水珠多變的面貌，並探討水珠的大小、溫度、停滯時間及變化規律相關機制。
緣定： 糾纏在水珠圓舞曲中有如大珠小珠落玉盤的曼妙，其中埋藏了平均圓與能量量子化的律動。
垂直水柱的成節機制探討
本研究欲探討垂直水柱遇障礙物成節的形成機制。以數位照相機、光電計時器等進行觀測。 實驗結果如下： （一）因往返水柱波速不同，而且節無波腹大幅振動現象，故節不是駐波現象。 （二）細針插入水柱表面時，當針上方超過某長度後，針下方產生Ｖ字形震波。但不論針相對水柱的速度是否超過波速，針上方都有節，故不是震波所產生的現象。 （三）根據水波槽模擬實驗，不論木條是否超過波速，木條前方均產生波紋。木條前方的水受到木條推動，往前方加速，因此顯現出波紋了。 我們認為，在水柱中所看到的節，不是震波或駐波，而是相對於木條往前傳遞的波。波源是撞擊物，改變了水柱表面的壓力，而成為波源，水柱的水因受撞擊，某個範圍內流速會小於波速，使得撞擊物前方存在波紋。This experiment uses digital camera and photoelectric timer to discuss the mechanism of causing spouts to form nodes on its surface. Because the downward wave velocity of the spout is different from that upwardand there are no significant vibrations of antinodes, standing waves are not the mechanism of causing nodes. In the experiment of inserting a needle into the spout, we found out that while the needle was inserted above a certain length of the spout, v shaped bow waves emerged. However, no matter the velocity of the needle related to the spout is over the wave velocity, there are always nodes above the needle. Therefore, bow waves are not the mechanism of causing nodes. According to the ripple tank simulating experiment, no matter whether the speed of the wooden stick is faster than the wave velocity or not, there are always waves forming in front of the wooden stick. The wooden stick pushes water in front of it and causes the water to accelerate forward. Therefore, waves appear. We think that the nodes we see on spouts are neither standing waves nor bow waves. The nodes are rather caused by the relatively moving wooden stick. The object, which impacted the spout (wooden stick), changed the pressure of the spout’s surface and became the source of wave. Because of the impact, the velocity of the water current of a certain area became slower than the wave velocity and causes nodes forming on the surface of the spout.