殊途同歸-格子點平面最短路徑和之探討
本研究從理想城鎮(Ideal City)街道開始，討論平面上相異n 點到某一點的最短距離和。經研究後發現：當n 為偶數時，則到相異n 點的最短距離和所形成的區域可能是一個點、一個線段或是一個矩形；當n 為奇數時，則相異n 點的最短距離和所形成的區域將會退化成一個點。此外，本研究將理想城鎮的街道換成正三角形的街道幾何平面，同樣是討論平面上相異n 點到某一點的最短距離和。經研究後發現：當n 為偶數時，則相異n 點的最短距離和所形成的區域可能為一個點、一個線段、一個四邊形、一個五邊形及一個六邊形；當n 為奇數時，相異n 點的最短距離和所形成的區域則可能為點、三角形的情況。假使考量各點重要性的比重，分別加權後再求最小點。研究發現無論在理想城鎮或正三角形幾何平面上，皆可將各點視為多個權數相同之點重疊於此點上，便可利用先前的方式求得最小點區域。透過這次的研究，可以利用n 個相異點到某一點的最短距離和實際應用在貨物運送的問題或是消防設施配置等問題。The present study was intended to start with the Ideal City and proceed to discuss the sum of the shortest distance between a point and n different points on a plane. After the discussion, it was found that if n is even, the formed region could be a point, a line segment, or a rectangle. If n is odd, then the formed region must be a mere point. Further, the current study transformed the Ideal City into the geometric plane of an equilateral triangle. Similar to the previous discussion, if n is even, the formed region could be a point, a line segment, a quadrangle, a pentagon, or a hexagon. On the other hand, if n is odd, then the formed region could be a point, or a triangle. The result of this study, which investigated the sum of the shortest distance of a certain point to n different points can be applied to the real life situation, such as transporting goods or distributing fire control facilities.
竹嵌紋病毒及其衛星核酸5'端非轉譯區與複製競爭關係之探討
RNA 病毒在複製過程中容易產生錯誤，導致其族群具中有很大的遺傳歧異度，累積的錯誤再加上選汰的壓力造成往後之變異。由於RNA 基因體之病毒變異較大，使得RNA 病毒在單一寄主上具有quasispecies 的特性，提供病毒產生新基因體的機會以適應環境或演化成新病毒。例如流行性感冒病毒與之前造成恐慌的嚴重急性呼吸道症候群病毒(severe acute respiratorysyndrome,SARS)以及禽流感病毒 (avain influenza virus) 皆為RNA 病毒，意味著RNA 病毒知不穩定性，並容易造成一些目前我們無法及時反應的危害。大部分的植物病毒又為RNA 病毒，本研究將以竹嵌紋病毒 ( Bamboo mosaic virus , BaMV )及其衛星核酸 (satellite RNA, satBaMV)為材料，進一步探討核?酸序列之變異對其族群在複製競爭上的影響。
大安水蓑衣(Hygrophila pogonocalyx)的復育對黑擬蛺蝶(Junonia iphita iphita)食性偏好??
大安水蓑衣（Hygrophila pogonocalyx）為局限分佈於臺中縣沿海溼地之稀有植物。大量境外復育栽殖後，造成近年來黑擬蛺蝶(Junonia iphita iphita)利用此種新寄主植物的比例增加。本實驗比較黑擬蛺蝶在原寄主植物臺灣馬藍(Strobilanthes formosanus)及新寄主植物大安水蓑衣上之生長環境遮蔽度、幼蟲生長發育、雌蝶產卵偏好的差異，探討大安水蓑衣復育對黑擬蛺蝶族群可能造成的影響。實驗結果發現利用大安水蓑衣之幼蟲生長發育較佳、羽化後成蟲體型較大、有效積溫常數較低。雌蝶產卵行為可能存在兩種偏好性，且子代雌蝶對寄主植物的偏好性與親代一致，不受幼蟲期取食植物影響。偏好新寄主植物之雌蝶其子代的生長發育，利用新寄主植物者顯著較利用原寄主植物者佳，顯示其對原寄主植物的適應顯著下降。本研究認為大安水蓑衣的復育結果，造成黑擬蛺蝶族群在可能共域的情況下，因為利用新的寄主植物可能已產生初步分化的現象。Hypgrophila pogonocalyx is a rare plant species which is distributed in wetlands along seashore of Taichung County. Recently, I found that the common butterfly Junonia iphita iphita recognized H. pogonocalyx as a new hostplant because of the restoration practices of H. pogonocalyx. We compared the differences of forest overstory coverage, larval growth performance and female oviposition preference between J. iphita iphita individuals exploiting the H. pogonocalyx and those utilizing the original hostplant Strobilanthes formosanus. Possible effects on J. iphita iphita by restoration of H. pogonocalyx are analyzed and discussed. It turns out that better growing performance and longer adult forewing length were found on the larvae feeding on H. pogonocalyx than those on the S. formosanus. Besides, the larvae feeding on H. pogonocalyx demonstrated lower constant value of effective accumulated temperature, suggesting that the new host may provide more energy to the larvae than the original host does. Two types of female oviposition preference seem to exist. No matter which hostplant the larvae fed on, the female adults still maintained the oviposition preference of the parental generation. Offspring of H. pogonocalyx-preferring female had better performance on this new hostplant. Our studies showed that the restoration of H. pogonocalyx might have caused primary differentiation of J. iphita iphita by using new hostplant in sympatry with the original hostplant.
Mathematics in Music
Mathematics and music are two poles of human culture. Listening to music we get into the magic world of sounds. Solving problems we are immersed in strict space of numbers and we do not reflect that the world of sounds and space of numbers have been adjoining with each other for a long time. Interrelation of mathematics and music is one of the vital topics. It hasn’t been completely opened and investigated up to now. This is the point why it draws attention of a lot of scientists and mathematicians to itself. This is the point why it draws attention of a lot of scientists and mathematicians to itself. Having considered the value of these two sciences, it seems to us that they are completely non-comparable. In fact can there be a similarity between mathematics – the queen of all sciences, a symbol of wisdom and music – the most abstract kind of art? But if you peer deeply into it you can notice that the worlds of sounds and space of numbers have been adjoining with each other for a long time. In the work I will try to establish the connection between mathematics and music and to find their common elements, to analyze pieces of music with the help of laws and concepts of mathematics to find a secret of mastery of musicians using mathematics and also to investigate the connection of music with mathematics with the “research part”. They are my own calculations and researches which are an integral part of the work. The connection of mathematic and music is caused both historically and internally in spite of the fact that mathematics is the most abstract of sciences and music is the most abstract kind of art. V. Shafutinskiy, I. Matvienko, m. Fadeev, K. Miladze, Dominik the Joker – modern composers of the XXI century – have used the golden proportion only in 4% of their pieces of music and more often in romances or children’s songs. I have revealed this fact after investigating their pieces of music of different genres. However there is a question: why does modern music attracts all of us more but the classics is being forgotten? Investigating connection between mathematics and music I had come to the conclusion that the more deeply the piece of music gives in to the mathematical analysis, to research and submits to any mathematical laws, the more harmonious and fine its sounding is, the more it excites human soul. Besides I am convinced that many important, interesting and entertaining things have not been opened in this field. We can safely continue our research of these things. I think that I have managed to lift a veil over mathematics in music, to find something common for apparently incompatible science and art. In due time English mathematician D. Silvestre called music as mathematics of feelings, and mathematics – as music of intellect. He expressed hope that each of them should receive the end from the part of the other one. In the future he expected the occurrence of a person in which Beethoven and Gauss’ greatness would unite. Terms ‘science’ and ‘art’ practically didn’t differ during far times of antiquity. And though roads of mathematics and music have gone away since then music is penetrated with mathematics and mathematics is full of poetry and music!
DNA Detection by EGFET using GaN Nanowires Gate
DNA感測器近年來蓬勃發展,應用層面包括基因工程,醫學及藥物的開發等,然而目前較常使用的感測方法,需要繁瑣耗時的標定過程,且所使用的化學藥劑對環境容易造成傷害,鑒於以上方法的不完善處,我們決定設計一套新的感測系統,此研究結合了氮化鎵奈米線(GaN Nanowires)及延伸場效電晶體(EGFET)的優點,成功的發展出創新的DNA感測系統,氮化鎵奈米線的高生物匹配性及高感測面積,能有效提高靈敏度,延伸式場效電晶體的設計,史感測器具由免標定及時感測的特性,且易於組裝及操作,我們將探針DNA(probe DNA)修是在氮化鎵奈米線作為之延伸閘極上,由於DNA在中性水溶液中帶負電,且DNA之間具有強烈的互補特性,因此當目標DNA(target DNA)與探針DNA接合,形成雙股DNA,氮化鎵奈米線(閘極)的表面電位即會有所變化,並造成FET特性的改變,藉由此性質及能成功感測DNA,研究結果顯示,此研究所發展出的DNA感測器,愈有相當高的靈敏度(10-18),相較於其他以FET技術所設計出的DNA感測器,靈敏度提升了三個數量極,此外此感測器亦具有高選擇性,即使單一鹼基對的突變也能成功辨別;-hybridization based detection techniques are widely developed due to their promising applications in genetics, medicine and drug discovery. However, current DNA detection techniques based on labels or reagents are time-consuming, environmentally-harmful and complex to implement. In this study, we have successfully demonstrated a label-free extended-gate-field-effect-transistor (EGFET) sensor utilizing a GaN-nanowires electrode with DNA probes immobilized, capable of specific DNA sequence identification. The principle behind the design is based on the change in surface potential and charge transfer after hybridization. GaN nanowires, being bio-compatible, provide direct transfer path and high surface area, thus offer an unprecedented opportunity of DNA sensing with high sensitivity. In addition, our EGFET design facilitates easy assembly and operation of DNA detection. Comparative studies on complementary and non-complementary DNA were performed to verify the specificity of the sensor. By adapting GaN nanowires structure, the assay time of DNA was shorten to within thirty minutes. Moreover, our sensor displayed an ultra-high sensitivity in the level of attoM: three orders of magnitude higher in resolution than that of other FET-based DNA detection methods.
步步為營
Two soldiers walk on a checkerboard. They can only walk one step once a time and two directions, front and left, are decided randomly. The gunshot is the column and row where a soldier is located, and one will die if he enters the gunshot area of the other. To treat the probability of winning, we first study the cases of 1×n, 2×n, 3×n, and 4×n rectangles iterately. Then we establish a general form of the probability of winning in a general n×k rectangle by using recurrence technique and generating function, respectively. Finally, we extend to the general n×m×k cuboid case to obtain the first soldier’s probability of winning.在一個長方形的棋盤中，兩士兵行走，每一次只走一步，而且上和左兩個方向是隨機的，射程範圍是所在的此行和此列，而進入他人射程範圍則死亡。探討其獲勝機率，從1×n 、2×n、3×n、4×n 矩形的情形逐步研究，並分別運用遞迴式的技巧及生成函數，導出 n×k 矩形中先走士兵獲勝機率的一般式。更進一步地，我們也獲得了n×m×k 立體空間先走士兵的獲勝機率。
再論巴斯卡三角形
本研究以b04課程中的巴斯卡三角形為研究對象，將原先巴斯卡以「1」為首、「+」為運算符號的規律三角形，改為以「-1」及「ω 」為首、「×」為運算符號，分別就其產生的新三角形作探討，發現其中似乎隱藏著原先三角形所沒有的規律性。為了更瞭解這種規律，藉由電腦軟體繪出其圖形，圖形顯示出如碎形般的複製關係，不論放大或縮小，其中的遞迴關係並未改變，頗令人好奇，因此著手研究。研究過程中對於圖形的規律性採用先臆測、接著歸納、最後給予證明的方式呈現。得到以下的結論：一、分別以數列呈現新三角形圖形的規律性。二、分別將新三角形中每一列中的某數字（如-1、ω 或ω 2 ）的個數予以通式表之。三、分別推算出新三角形第n 列第j 行的數是「1」或「?1」及「1」或「ω 」或「ω 2 」。四、相同的模式，在特定的圖形範圍中，不斷重複出現。許多研究將巴斯卡三角形中的所有數，以某數為模的餘數紀錄下，去探討其餘數在新產生的巴斯卡三角形中的分布情形；而在碎形的研究中，大部份著重如何畫出碎形。本研究著重圖形其規律性的探討，提供上述研究不同角度的詮釋與探討。 This research subject is based on Pascal’ s triangle in senior high school curriculum. The regular triangle begins with「1」and use「+」as operation. Let 「1」 be replaced with「-1」and「ω 」, the operation sign「+」be changed into「×」. I do research on the new triangle and discover the seemingly hidden regularity which doesn’t exist in the original one. To understand more about this regularity, I draw figures through the computer. The figures show the relationship of reproduction as fractal. Whether the figure is enlarged or minimized, it’s surprising curious the recursive relationship doesn’t change, so we begin to work on research. In the process of the research, we make careful observations, assumptions and deductions about the regularity of the figure. Finally, we come to some conclusions by means of giving proofs:（1）Present the regularity of the new triangle figure with progression.（2） Present such numbers as「-1」, 「ω 」, 「ω 2 」 in each row of the new triangle with formulas separately. （3）Figure out the number in the row n and in the column j of the new triangle is「1」or「-1」,and「1」or「ω 」or「ω 2 」. （4）The same model appears again and again in the specific range of figure. Many researches record Pascal’s triangle modulo certain number to explore the distribution of remainders in the new triangle. In the research of fractal, how to draw fractal is mostly focused on. The exploration of this research emphasizes the regularity of figure, offering the interpretation and exploration of researches above from different angles.
翻轉「膜」力
The starting point of this experiment is to study the structure of soap-film. By changing the height of the triangular prisms, cuboids and pentagonal prisms, I observed the patterns set by the soap within the frameworks. It is surprised that when the proportion of prism is in a specific range, the phase in the middle of the structure will overturn 90 degree and then transmitted into another kind of balance pattern. I named this process “phase transition”. According to the experiment ,we can conclude the change of film patterns within variable prisms are all applied to this regular cycle:：
We know the soap films are forever attempting to minimize their energy. It stands to reason that surface tension tend to set up the film in its minimal surface. From the point of Mathematic, each structure should have only one single balance pattern, which is set up on the base of Fermat point and this pattern should stand to the minimize of it’s energy. However, we discovered that in some specific cases, one structure can allowed two kinds of balance films-patterns to exist. In these cases, any small vibration can cause the happening of “phase transition”. To sum up, I presume some structures have two different types of balance film-patterns: one of which stands to the local minimum (in this condition the pattern’s surface area isn’t the smallest); the other stands to the absolute minimum (in this condition the pattern’s surface area is the smallest).
There is an energy valley separate local minimum from absolute minimum. The second pattern (local minimum) will appear when the structure is blocked from attaining its absolute minimum, but surface intention is not powerful enough to support the film jumping over the energy valley. In this condition, if we works on the structure (such as blowing), which would provide the film of energy to cross the valley, and then phase transition take place. Vice versa, we can also force the film to jump from absolute minimum to local minimum and phase transition will occur as well. In a word, phase transition can happen in each two way, which connects the two types of balance pattern.
This report lays stress to find out the condition of phase transition. We also analyze the structure of soap-film by its included angles and surface area in hope to go deep into the science of soap-film.
我們實驗的出發點在於研究泡膜的立體結構。藉由改變正立方柱的高，觀察其平衡薄膜形式，意外的發現當正立方柱的邊長比在某個範圍時，泡膜結構中央會瞬間90 度翻轉，形成另一種平衡型式，我們將這個過程命名為面轉變(Phase Transition)。為了進一步了解面轉變發生的相關因素，我們設計了一連串的實驗，針對正三角柱、正四角柱、正五角柱、正六角柱發生面轉變的時機和條件分析討論。此外，我們還分析了泡膜結構中膜與膜夾角的特性、最小表面積和表面能之間的相關性，對於泡膜的立體結構做了一系列深入的探討。
以阻抗匹配調整太陽能最大功率輸出之研究
In recent years, the price of the oil keeps rising continuously. As a result, the prices of the commodities are rising, too. But what does this mean? This situation stands for the resources on are becoming more and more valuable. A few months ago, I read a Weekly Business Magazine and a Scientist Magazine. They both pointed out that the resources such as oil would disappear after fifty years, and that was a horrible phenomenon. Since almost everything in our modern lives are related to oil, like automobile, motorcycle and air plane. They all need oil for its ingredient. Even plastic bag plays a part in the products of it. I really cannot imagine what it would be like if we don’t have oil after fifty years. Owing to the green house effect is becoming more severe, there are many substitution resources found, such as hydraulic power, wind power and solar energy…etc. What we’re discussing in our topic is how to enhance the power of solar energy, because for now, we all know that the solar module is very expensive and it cost a great deal of money just only one square meter, but the price of module converted from sunlight or heat energy can’t be higher. Therefore, the work is mainly to design and carry out a solar max power point track. 近年來，油價不斷的上漲，連帶著民生的物價漲幅也是越來越可觀，但…這意味著什麼 呢??這所代表的是地球上的資源可以說是越來越珍貴了，前陣子曾經看過商業週刊、科學人 雜誌…他們紛紛所指石油這種能源可能在五十年後就消失殆盡，這個可是非常可怕的結果。 鑑於溫室效應愈趨嚴重，許多的替代能源紛紛出籠，像是風力、水力和太陽能等等。我們這 次的主題是在討論如何提高太陽能效率，現在的太陽能模組我們都曉得非常的昂貴，但是由 太陽轉化成電能的效率卻是低的可以。所以本作品主要目的在於設計與實現一個太陽能最大 功率追蹤器。