台灣稀有水生植物蓴菜生長型態構造觀察成分分析研究
本研究針對台灣產水生植物，蓴菜之構造與生長環境、蓴菜對腸胃道常見致病細菌之抑菌效果以及主要成分暨化合物分析。由本研究結果得知，崙埤湖內之稀有浮葉型水生植物蓴菜，其生長環境為無汙染之乾淨偏酸性水源，最適合生長之生深為50-160 ㎝；水溫則為22-25℃；而蓴菜之地下根莖對表皮金黃葡萄球菌（Staphylococcus aureus）具有輕度之抑菌效果，經由分離純化得知為BS-1：沒食子酸（Gallic acid）；另外，由蓴菜之葉片分離出十種成分分別為BS-2 （Kaempferol-7-O-Glucosids）、BS-3 （Quercetin-7-O-glucosids）、BS-4（5,8,4’-Trihydroxyflavone-7-O-glucosids）、BS-5 （3,5,8,3’4’-Pentahydroxy flavone）、BS-6（Vitamin E: d-Tocopherol）、BS-7 （Glyceride）、BS-8 （Phenolic A）、BS-9（Quercetin）、BS-10（Kaempferol）、BS-11（Phenolic B）。其中發現BS-8 對神經膠腫瘤細胞株有18.42%之抑癌效果，另外，BS-2、BS-3、BS-5、BS-10、BS-11 等成分，呈現良好之美白作用。This investigation is to analyze Brasenia schreberi Gmel., a native rare floating water plant in Taiwan, focusing on the plant’ s structure, its growth environment and, most importantly, the effect of chemical compounds it produces on restraining the common pathogenic bacteria in human stomach. The result indicates that the most suitable growth environment for Brasenia schreberi Gmel. is in slightly acid, pollution-free water such as that in the lake Lung Pi in northern Taiwan. The ideal water depth for its growth is 50-160 cm, and the water temperature is 22-25°C. The impractical BS-1 (Gallic acid) extracted from the izome of Brasenia schreberi Gmel. by separation and purification has a light effect on restraining Staphylococcus aureus, a bacteria in the stomach. From the epidermis of the blade of Brasenia schreberi Gmel., ten other ingredients are also isolated, including BS-2 (Kaempferol-7-O-glucosids), BS-3 (Quercetin-7-O-glucosids), BS-4 (5,8,4’-Trihydroxyflavone-7-O-glucosids), BS-5 (3,5,8,3’,4’-Pentahydroxyflavone), BS-6 (Vitamin E: d-Tocopherol ), BS-7 ( Glyceride ), BS-8 (Phenolic A ), BS-9 (Quercetin), BS-10 (Kaempferol),and BS-11 (Phenolic B). BS-8 is found to resist cancer C6 ( Glioma ) by 18.42%, while BS-2,BS-3, BS-5, BS-10, and BS-11 show an outstanding effect on skin-whitening.
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。
培地茅根系碎形維度及抗拉力
本研究首先確認培地茅根系具有碎形之基本特性，再進一步以方格覆蓋法計算之碎形維度來分析培地茅根系在不同時間及環境因素下的生長。主要探討碎形維度與抓地力之關係，並設計以實際根系模型來加以模擬，並發展出一可描述抓地力與碎形維度及深度關係的方程式。我們的結論為：（1） 經由方格覆蓋法之計算，培地茅此種植物，不管是整個根系或單枝根，均具有碎形基本特性，適合進一步實驗研究。（2） 碎形維度會隨著培地茅生長時間增長而增加，並且在自然光照及30℃左右會有較大值，而種植於土壤中根系發展較廣，其碎形維度比種植於沙耕中來的高。（3） 實驗結果顯示，抓地力受碎形維度及根系深度兩因素影響，而培地茅根系對土壤有較強的抓地力，推測是因為兩者根系皆又深又長，土中培地茅根碎形維度較大，接觸面積較廣，而又進一步以矽膠模型做實驗驗證。（4） 矽膠模型之目的在於減少難控制之自然變因，實驗之前，測量了根系模型與洋菜凍之基本性質，實驗結果顯示抓地力與碎形維度及根系深度皆呈正向關係，可用數學方程式加以描述。This project is mainly a research into the fractal dimension of the vetiver root system. First, we confirm the vetiver root system has the basic fractal structure by checking its self-similarity, then using box-counting method to calculate fractal dimension. We begin with a fundamental investigation into the relation between different time and environmental factors and fractal dimension. Then we move to our main point: the relation between fractal dimension and its pull-out resistance. In the next step, we make a fundamental silicon model, simulating the vetiver root system, to continue our experiments. In the end, we develop a formula that can describe the relation between its pull-out resistance, roots depth and fractal dimension. Here are our conclusions: （1） After using box-counting method to calculate fractal dimension, we discover that not only the whole vetiver root system but also a single vetiver root has the basic fractal structure. （2） Fractal dimension increases when time goes on. Also the value of fractal dimension is larger in natural sunlight and the temperature at about 30℃.The vetiver root system grows more widely in soil than those in sand. That’s why it has larger fractal dimension. （3） Data shows that its pull-out resistance is influenced by both fractal dimension and the depth of the roots. The vetiver roots, in the meantime, show greater pull-out resistance than some other plants. Thus we draw the assumption that the vetiver root system grows deep and wide, and in natural soil its fractural dimension is greater and reaches greater area. Therefore, a silicon model is constructed to further confirm the findings of the experiment.（4） The design of the silicon model is to reduce the uncontrollable variables in nature. Before starting the experiment, we measured some basic characteristics of the silicon model, including density and angle of repose. Furthermore, the experiment demonstrates that pull-out resistance and fractural dimension have a commensurate mutual relation: the stronger the pull-out resistance, the wider the fractural dimension and the deeper the root system. Thus we derive a math formula to describe this relation.
對抗無尺度流行病傳染之新方法
流行病的傳染過程如同一個無尺度網路，但較一般無尺度網路有著更多的變數而明顯差異，因此無法直接應用一般的無尺度網路模式來描述其傳染途徑。我建立一個新模式「無尺度流行病模式」，經由比較模擬結果與疾病管制局的數據，證實此「無尺度流行病模式」是正確與確切可用，且適用於短期暴發性傳染病與長期流行病。SARS案例研究結果，顯示影響SARS疾病傳染因子的大小是：ψ>m>γ。其中降低ψ值可使SARS確定病例至5月31日止降為143人(減少確定病例190人，相當於減少死亡21人)；僅提高防疫使5=γ，亦可使確定病例減至307人(減少確定病例26人，相當於減少死亡3人)。因此強化隔離措施以減少傳染天數最為重要，且可以有效控制每日SARS新增病例，避免發生高侵襲率的現象。HIV/AIDS案例研究結果，獲知採用ψ值來進行月份模擬，則至 2005年12月HIV(+)與AIDS分別為可減少2,715與285人。而進行年度模擬結果，則至 2014年底HIV(+)與AIDS分別為可減少41,936與5,328人。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。因此無尺度流行病模式在幫助政府評估社會經濟成本與健康憂慮上的有用之工具。當面臨一個全然無知的新病毒的侵襲時，如何減少死亡與傷害人數？是本研究之最終目的。因此，本研究結合了流行病、無尺度網路與灰預測，建立面對病毒侵襲，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。\r \r \r The course of epidemic infections resembles a scale-free network. However, they are different due to more variables in the epidemic infection. Therefore, the model of scale-free networks is not enough to satisfy the reality epidemic infections. In this study, I propose a new the Scale-Free Epidemic Model. Comparison of the simulation results with Taiwan CDC report data for SARS and HIV/AIDS cases show that the Scale-Free Epidemic Model is accurate and useful. This model can be used in the short-term outbreak of infectious diseases and for the longer-term epidemics. In the SARS case study, the results show that the sequence of effect of the epidemic factors was: ψ>m>γ. The SARS confirmed cases would decrease to 143 cases (reduced 190 confirmed cases or 3 death cases) calculated to May 31, 2003, if the average infection time was reduced to two days (an optimum value of ψ). Therefore, vigorous action in isolation quarantine and treatment for SARS cases is most effective policy; the number of new cases and the attack rate would also decrease. In the HIV/AIDS case study, the simulation results of the Scale-Free Model indicates that the reduced numbers of HIV(+) and AIDS in the monthly simulation calculated to December 2005 are 2,310 and 361 and the annual simulation by December 2014 are 27,161 and 3,710. The Scale-Free Epidemic model can help determine the level of caution needed and the projected results of policy decisions. Therefore it is a useful tool in assisting the government to balance socio-economic and health concerns. The fight against a new epidemic and how to reduce the number of deaths is the main purpose of this study. So, a new method to fight against epidemics is proposed. Detailed procedures of this method are explained.
六圓定理
在實驗中學2007 年校內科展，參展作品《三角形中的切圓》的研究中，研究三角形內的切圓時，發現連續切圓的圓心與拋物線的軌跡有關。於是去查資料，在偶然的情況下，翻閱《平面幾何中的小花》時，接觸了「六圓定理」。因為覺得這問題非常有趣，於是便著手證明（見報告內文）。 又發現，當移動六個圓中的起始圓時，總是在某種情況下，六個圓會重合成三個圓。繼續研究其重合的狀況，發現了馬爾法蒂問題（Malfatti's Problem）的一種代數解法。 當我試著推廣六圓定理至多邊形時，發現奇數邊的多邊形似乎也有如六圓定理般圓循環的狀況，於是著手證明，但目前尚未證明成功。而偶數邊的多邊形則無類似的結果。 ;In 2007 National Experimental High School Science Exhibition, one of the exhibit works, "Inscribed Circles in Triangles", shows that the centers of the consecutive inscribed circles has something to do with the parabola's trajectory. To learn more about inscribed circles and parabolas, I referred to literature. By accident, I am faced with the problem on six circles theorem, in the book The Small Flower of Plane Geometry(平面幾何中的小花). Out of my interest in this problem, I tried to prove it. The other results are as follows: With the initial circle of six circles moved, in certain circumstances, the six circles merge into three. Further in studying this coincidence leads to an algebraic method to solve the Malfatti's Problem. Applying six circles theorem to the odd-number-sided polygons exists the same characteristic. It indicates that the inscribed circles will form a cycle. However, it hasn’t been successfully proven. The even-number-sided polygons show no similar results.
線蟲補捉菌Arthrobotrys musiformis 黏液相關基因之選殖與功能界定
線蟲捕捉菌Arthrobotrys musiformis 是一種可經線蟲誘導產生捕捉網來捕捉線蟲的真菌，本實驗即針對A. musiformis 的捕捉網黏液相關基因：Manosyltransferase（AH73）, β-1,3-glucan transferase（AH102）, fimbrin（AH121）及mannose-specific lectin precursor（AH338）進行選殖與功能界定，希望建立這方面的研究基礎，將來能應用在松材線蟲的生物防治上。首先我們大量培養A. musiformis，萃取菌絲體的DNA；接著進行聚合?連鎖反應 (Polymerase Chain Reaction，PCR) ，利用專一性引子對 (primer) 大量增幅AH73、AH102、AH121 及AH338之基因片段；增幅後的產物經過純化、選殖，定序並進行分析比對，確認增幅之序列無誤後，以 Digoxigenin (DIG) 標示當為探針，篩檢A. musiformis 的Fosmid Library﹔目前已成功選殖出AH73 之可能基因，完成AH73 之探針製備，並以其篩檢A. musiformis 的Fosmid Library﹔呈雜合正反應之選殖株 (clones) 將以散彈槍方法（shotgun）定序，作序列組合，探索相關的基因；接下來用 Rapid Amplification of cDNA Ends（RACE) 做出互補DNA (complementary DNA , cDNA) 全長度後；最後建構基因缺失株，驗證此基因所調控的生理以及生化機能。 Nematode trapping fungus Arthrobotrys musiformis can capture nematodes by producing adhesive nets when nematodes go through. Many kinds of nematodes, including pine wood nematode (Bursaphelencus xylophilus), can be captured. Pine wood nematode causes serious pine wood disease. Therefore, A. musiformis has the potential of biocontrol in pine wood nematode. Our research focused on adhesion and adhesive relevant genes of A. musiformis ：Manosyltransferase (AH73), β-1,3-glucan transferase (AH102), fimbrin (AH121), and mannose-specific lectin precursor (AH338). We try to clone these genes and carry out functional analysis. In order to achieve this goal, we used specific primers derived from previously obtained complementary DNA (cDNA), by Polymerase Chain Reaction (PCR) to amplify these genes and gained adequate quantity of genomic DNA products. After sequencing and verifying of the identity of the genomic DNA, we use Digoxigenin （DIG） to label them and use them as probes to screen the constructed A. musiformis Fosmid Library. Currently, the Southern colony hybridization is undergoing. The positive Fosmid clones against the specific probes will be sequenced completely by shotgun library to monitor the existence of adhesion related gene cluster. After working out the full length cDNA of these genes, we will use them to construct replacement vectors to knockout the adhesion related genes, creating mutants and further verify their functions through genotype or phenotype bioassay.
從有限三角和公式研究偶次調和級數之遞迴公式及其相關等式之推廣與應用
本研究中，我們將提出一些新穎結果，著重討論其在三角中的應用；同時，找出其遞迴關係式，得出三角展開式與其所對應之多項式分解式，進而討論出多種的規律性及所涵蓋的內容及推廣性質，我得到很多高中數學公式無法推導出在【4】和【8】中的漂亮公式及創新的結果，且這些等式都是由我們不太瞭解的無理數所構成的。
主要是討論我們在【7】中所得到的收穫與經驗；複數是三角、幾何、代數互動的橋樑，我是以不同的角度及嶄新的方法來綜合探討在【6】中相關的應用。提出關於正整數平方的倒數和公式更為精簡且基本的證明，將 sin−2 x 表示成級數形式的部分分式，進而應用在(a,b) = 1的機率問題上；並研究相關的等式，直接透過三角與代數來研究關於 2p 次方的倒數之求和問題，得出級數 之和的有用遞迴公式，並與最重要的常數扯上關係。
For one thing, we present diverse methods to evaluate finite trigonometric summation and related sums. Trigonometric summations over the angles equally divided on the upper half plane are investigated systematically. Several related trigonometric identities are also exhibited.
What is more, we use methods of calculus, and make several surprising and unexpected transformations. A useful recursive formula for obtaining the infinite sums of even order harmonic series, infinite sums of a few even order harmonic series, which are calculated using the recursive formulas, are tabulated for easy references. Furthermore, is there any interesting results and applications？
Finally, the purpose of this paper is to develop a new proof of and related identities, but their derivations are more complicated. The following studies are completed under the instruction of the professor.
生生不息-正五邊形的繁衍法則
This study was to explore the nature of two basic constitutes of the regular pentagon,With these two constitutes, the regular pentagon could be multiplied into any times. We used four multiplication methods (m2 = 2m1 + n1 、n2 = m1 + n1 、m2= k2m1 、n2= k2n1、a2 = a1 + 1、a2 = a1 + ) to show how the regular pentagon could enlarge and to verify that the enlarged regular pentagons derived from computer did exist. By integrating these four multiplication methods, we were able to arrange regular pentagon of any length of side, and evidenced the equation was
( If the side length of a regular pentagon is a form of m,n is the number of A,B respectively )
We further proved that the first multiplication method could be developed into a new modified method, which could divide a regular pentagon with a given side length into a combination of A and B. But only when the x and y of side length of a regular pentagon could be divided by a natural number, k, and made x/k into an item of the Fibonacci Sequence and y/k a successive item.
When we tried to verify if any regular pentagon could be constituted by other smaller regular pentagons, we also found that it was un-dividable only if the length of pentagon side were ( the number of A, B were the 2n and 2n-1 item of Lucas Sequence). Otherwise, any regular pentagon might be able to be constituted by other smaller regular pentagons.
本研究是以正五邊形的兩個基本組成元素(B)作為討論對象，利用此二元素可以將正五邊形做任意倍數的放大。我們共使用4種繁殖法則(m2 = 2m1 + n1 、n2 = m1 + n1 、m2= k2m1 、n2= k2n1、a2 = a1 + 1、a2 = a1 + ) 來說明正五邊形的放大情形，並利用此4 種繁殖法驗證電腦運算出的放大圖形確實存在。利用這4 種繁殖法則的改良與整合，已達到能排出任意邊長之正五邊形的目標，並能計算並證明出其通式為。
(若正五邊形的邊長為形式,m、n代表、的個數)
更特別的是，我們能用第一繁殖法反推出一種方法，將給定邊長的正五邊形利用簡單的切割方式分成由A、B 組合成的形式，但只有正五邊形邊長之x、y 值可同除以任一自然數k 而使 x/k 為費波那契數列之一項且 y/k 為其後一項者才可以使用。
將此想法推廣至一個正五邊形能否由比他小的其他五邊形組合而成時，我們也發現當正五邊形之邊長為時（其A、B 個數為盧卡斯數列之第2n,2n-1 項），不可分解，否則應該皆可將一個正五邊形分解成比它小的其他五邊形組合（我們也可以利用這些質形檢驗出其他正五邊形是否也為質形）。但其分解形式，不只一種，而我們推測只用兩種較小的正五邊形就能達成，我們期待能找出一或多種分解方法，能將正五邊形分解成標準的分解形式。
佛手瓜卷鬚之向觸性及其參與蛋白質之探討
本研究利用佛手瓜的卷鬚探討向觸性的原理。本研究大致分為兩部份，一方面我們在卷鬚中發現了含量極為豐富的構造，此一螺旋狀構造分布於維管束中，且用雙縮?詴劑檢測後發現其含有蛋白質，且不具有運輸水分的功能；並發現此一構造的分布疏密，會影響到螺旋內側外側以及切割後片段泡溫水的彎曲方向。此外，在進行卷鬚蛋白質電泳的過程中，我們發現使用含尿素的緩衝液萃取蛋白質的效果最佳，1克的卷鬚乾重約可萃取到5毫克的蛋白質，且蛋白質總量會隨著卷鬚的成熟而遞減。利用軟體比對及質譜分析八個蛋白質點，得知此八點的蛋白質為：malate dehydrogenase, oxygen-evolving enhancer protein 1, oxyen-evolving enhancer protein 2, calreticulin, peroxidase, stromal 70 kDa heat shock-related protein, and AP2/ERF and B3 domain-containing transcription repressor。由此可知，向觸性為植物經過一連串訊號傳遞後，對外界刺激的順應。