埃及分數之固定項數分解問題
本文由‘‘分數7/17是否能表示成兩個相異的埃及分數之和’’這個問題出發，藉由簡單數論的性質以及反證法，得到一個真分數可表示成兩個相異埃及分數之和的定理檢驗法(定理1)。有了這個基礎，我們進ㄧ步推廣定理1 的結果，做出了嶄新的結果(定理2、定理3) 。此定理分別可以用來檢驗真分數表示成三個、四個相異埃及分數之和的存在性; 至於將真分數表示為5 項、6 項….k 項相異埃及分數之和的部分尚在嘗試。利用定理1、2，我們寫了兩個Matlab 軟體工具的電腦程式，使得我們可以檢驗任意真分數是否可以表示成兩項及三項的和，並可把所有的解列出來; 最後我們研究的是一個有關埃及分數的猜想(Erdos-Strauss Conjecture)問題，當分子為4，且分母為4k、4k+2、4k+3 時，猜想皆成立。對於分母為4k+1 而言，當k 為3r+1、3r+2 猜想亦成立，k=3r 且r 為奇數時也是成立的，因此目前需解決的問題只剩分母為24t+1 的情況了。值得一提的是，我們用Matlab 的程式檢驗出當分母為1014 至1014 +240000 之內的正整數時，猜想都是成立的，這已經超越了已知文獻的結果。This paper begins with the question: ‘‘Is 7/17 able to be the sum of two different Egyptian fractions?’’ to discuss the problem of Egyptian fractions. According to the complete division properties and the counter-evidence method, we get a back-check theorem which is about a true fraction can be the sum of two different Egyptian fractions (see theorem 1). Using the same method we obtain a new back-check theorem that is a fraction can be the sum of three or four different Egyptian fractions (thereom2, thereom3). Similarly, we can follow the same procedure to get the rule that a fraction can be the sum of five or six …or even more different Egyptian fractions. By the theorem1 and 2, we propose two programs written vie the Matlab software to examine that any true fraction can be the sum of two items and three items or not. Finally we focus on the Erdos-Straus Conjecture, which related about true fractions can be divided by three different Egyptian fractions. The conjecture is when the denominator is 4k, 4k+2, or 4k+3, the problem mentioned above can be solved. As for the denominator is 4k+1, then the conjecture also can be solved, as k equals to 3r+1 or 3r+2. Also, k being 3r and r is an odd number, the conjecture is satisfied. As for the case of r equals to even number, the problem has not been solved. But it is worth to mention here that we use Matlab software to examine the conjecture is agreeable as the denominator is between 1014to 1014+ 240000. This is beyond the results from the literatures.
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.
凸n 邊形等分面積線數量之分布探索
(一) 本研究首先導出ΔABC等分面積線移動所包絡出的曲線方程式，其圖形是由等分面積線段PQ(其中P、Q皆在ΔABC的周界上)的中點所構成，具有3 條曲線段(分別為3 條雙曲線之一部分)的封閉曲線，形成內文所謂的「包絡區」。利用包絡區的區隔，我們找出：1.當P 點在包絡區內，則有3 條等分面積線。2.當P 點在包絡區周界上，則有2 條等分面積線。3.當P 點曲線段的端點或在包絡區外，則有1 條等分面積線。(二) 以三角形的研究當基礎，擴展到凸n 邊形(不包含點對稱圖形)，我們發現：等分面積線數量之分布，仍然與包絡區息息相關，且1.凸2m +1邊形最多有2m +1條等分面積線。2.凸2m邊形，必發生內文所謂的「換軌」。因此，最多只有2m ?1條等分面積線。3.包絡曲線所分割出的區域，於相同區域其等分面積線數量相同，且相鄰兩區域數量差兩條。(三) 若凸n邊形有k個「換軌點」，則此n邊形過定點等分面積線至多有n ? k 條。(四) 若凸n 邊形為點對稱圖形(如正偶數邊形、平行四邊形)，則所有等分面積線皆過中心點。1) Our study got a curve equation of bisectors of a triangle. When a bisector is moving, we get three curves. They’re constructed by the midpoints of PQ. The three parts of the three curves make a closed curve which we called “the Envelope Area”. We found out:\r 1. When Point P is in the Envelope Area, we can get 3 bisectors. 2. When Point P is on the curves of the Envelope Area, we can get 2 bisectors. 3. When Point P is outside of the Envelope Area, we can get only 1 bisector. 2) Based on our study of triangles, we found that in Convex polygons(not including Point Symmetry Convex polygons), the distribution of bisectors is related to the Envelope Area. 1. We can get at most 2m +1 bisectors in a 2m +1 Convex polygon. 2. We can get at most 2m ?1 bisectors in a 2m Convex polygon, and the bisectors on the curves will “Change the Track”. 3. Envelope curve will divide a Convex polygon into several areas. The same area has the same numbers of bisectors, and the near areas have less or more 2 bisectors. 3) If a Convex polygon has k points to change the track, it will have at most n – k bisectors.\r 4) In a Point Symmetry Convex polygon (ex. Regular 2m convex polygons and parallelograms), all the bisectors will come through the center point.
心手相連的正方形
正方形兩條對角線的交點（即中心點）距四頂點等長，也與四邊等距。如果將正方形的頂點比擬成它的「手」，兩對角線的交點當成它的「心」，則兩個正方形頂點間、中心點間、或頂點與中心點間的線段相連（或重合），就如同「手」或「心」彼此相連。本文即探索當多個正方形間「心手相連」時，衍生圖形間的面積關係。而四個正方形中某幾個頂點相接（邊未重疊），恰圍出兩個三角形的圖形則是本內容討論圖形的主體架構，我們以此架構向外作出「層出不窮」的正方形，再配合中心點連接成四邊形，將推導出這些四邊形與基準正方形（Reference Square）間的面積關係。In a square, the lengths from the intersection point （center point） of two diagonal lines to the four apexes are the same, and so are they from that point to the four sides. If the apexes are “hands” and the intersection point of two diagonal lines is the “heart” of a square, the connection or overlap of two squares’ apexes and apexes, center point and center point, or apexes and center points is just like the connection of hands with hearts. In this article, hence, we are to explore the relation in area of derivative graphs formed by several squares connected “heart in hand.” When some apexes of four squares are overlain without sides overlapped, two triangles are created. And that’s the theme we are going to discuss. Furthermore, we extend the operation to infinitely overlain squares and frame out quadrangles referring to the center points of some squares. Then, the relation in areas of these overlapped squares and the Reference Square would be deduced.
Peanut Hull as an Antioxidant in Metal Coats
A study was done to determine if the antioxidants found in peanut hulls could be used\r for lessening the corrosion rate of iron. Peanut hulls were ground then divided into two\r batches, P1 and P2, then oven-dried at temperatures of 50°C and 60°C, respectively. The\r moisture content of each batch was then determined before performing methanolic extraction\r to isolate the antioxidants. Eighteen iron strips of approximately the same surface areas were\r thoroughly cleaned and weighed, then divided into six groups. The iron strips in the first five\r groups were respectively coated with pure extract from batch P1; a 1:1 mixture of P1 extract\r and turpentine; pure P2 extract; a 1:1 mixture of P2 extract and turpentine; and pure\r turpentine. No treatment was done on the sixth group. All iron strips were exposed to air to\r allow formation of rust thru atmospheric corrosion. After 12 days, the iron strip were cleaned\r and weighed; then the individual corrosion rates of the metals were determined.\r The corrosion rates of the metals treated with pure P1 extract, the P1-Turpentine, and\r the P2-Turpentine mixtures were found to be significantly lower than the corrosion rates of\r the metals without treatment, at 5% level of significance in a t-Test for independent samples.\r The average corrosion rates of all the treated metals were found to be lower than that of\r metals treated with pure turpentine, though not significantly. The corrosion rate of the metals\r coated with turpentine was not significantly less than that of untreated metals. The corrosion\r rates of the metals were also found not to be dependent with the moisture as there was no\r significant difference in the mean corrosion rates of metals treated with P1 extract and those\r treated with P2 extract, with or without turpentine.\r The project has shown that peanut hull extracts can be used to lessen the production\r of rust on the surface of the metal. Moisture content of the hulls was not found to be a factor\r in lessening the corrosion rate.
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。
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.
有毛!沒毛!哪個好!?探討石田螺及其螺殼上附生藻類與環境因子之關係
This research is about two ponds in the Ｂ park’s and the Ｄ park’s snail(Square Mystery Snail：Sinotaia quadrata) in Taipei city of Nei-hu District for research object, carry out the study of the following research proceed: 1.Discriminate the algae species that are growth on the snail shell and which is a kind of interaction with the snail; 2.The influence of the snail and algae with difference of temperature, salinity, pH value and dark ; 3. The factors affect algae growth on snail shell; 4.Use the variation of snail and algae to be a biological incator. The result manifestation: the algae that are growth on snail shell have two kinds, one is Cyanophyta and the other is Cladophora sp. The interaction between algae and snail belong to communalism, but under the condition of lacking of food, the snail then will eat the Cladophora sp. which grow on the shell of other snails. The temperature adapts aspect, upper limit of the feat existence of the snail should be low in 28℃. When over than 28℃, Cladophora sp. as the most strong, Cyanophyta is secondly, and the snail then is most poor. For the maximum tolerance of the salinity, the snail is about 4.375?, Cyanophyta is about 5.0?, Cladophora sp is then about 7.0?; Under the different salinity for the tolerance , the Cladophora sp. still the most strong, Cyanophyta is secondly, and the snail then is most poor. Under the dark environment, the speed of Cyanophyta begin to be bleaching is very fast than the Cladophora sp.. In the tolerance of pH value range: The snail is about pH=5～10, Cyanophyta is about pH=7～8, Cladophora sp. is about pH=6～8; When the pH value range is in the pH=5～8, the speed of the Cyanophyta occur changing is very fast than Cladophora sp.. The algae are growing on snail shell very different between two ponds, the main reason is water pH value dissimilarly: When pH value over than 8.5, there is no Cladophora sp. to grow on the snail shell, after the pH value to decrease, Cyanophyta then will compare early than Cladophora sp. to grow on the snail shell. Calculate by the classification of the freshwater biological incator : Two organic pollution degree of the ponds may be lain in theβ-mesosaprobic to theα-mesosaprobic, and the polluting degree of the Ｄ pond is more seriously. As for two ponds, have already faced what level of eutrophication? Belong to actually which stage of pollution grade? Not only added the classification data of floating and fixative algea in two ponds, and also according to the parts of chemistry analysis method measure of the data makes the substantial evidence, then could carry out the more accurate and thorough study in the days to come steadily studying process.本研究是以臺北市內湖區兩個綠地公園(Ｂ公園與Ｄ公園)池塘內的石田螺(Sinotaia quadrata)為研究對象，進行以下研究目的之探討：1.鑑別石田螺螺殼上藻類的種類及其與石田螺的互動關係；2.溫度、鹽度、酸鹼值及黑暗等環境因子的差異，對石田螺及螺殼上附生藻類的影響；3.影響藻類附生於石田螺螺殼上的因素；4.將石田螺及螺殼上附生藻類的變化作為監測環境因子或水質變異的指標現象。結果顯示：附生於石田螺螺殼上的藻類有藍綠藻(Cyanophyta)與剛毛藻(Cladophora sp.)兩類；與石田螺的互動關係應屬於片利共生(communalism)，但在缺乏食物的情況下，石田螺則會採食同伴殼上的剛毛藻。溫度適應方面，石田螺適宜生存的溫度上限應低於28℃，超過28℃水溫環境的耐受程度，是以剛毛藻為最強，其次是藍綠藻，而石田螺則為最差。對於環境鹽度最大耐受度方面：石田螺約為4.375??，藍綠藻約為5.0??，剛毛藻則約為7.0?；在不同鹽度環境下，鹽度的耐受程度，仍以剛毛藻為最強，其次是藍綠藻，而石田螺則是最差。在黑暗環境下，藍綠藻褪色產生白化現象的速度明顯地比剛毛藻要快了許多。在環境酸鹼值耐受的範圍方面：石田螺約在ｐＨ＝5～10 之間，藍綠藻約在ｐＨ＝7～8 之間，剛毛藻則約在ｐＨ＝6～8 之間；而酸鹼值範圍在ｐＨ＝5～8 時，藍綠藻產生變化的速度明顯地比剛毛藻還要快。而兩樣區池塘水體酸鹼值的不同，應是造成石田螺螺殼藻類附生現象差異的主要原因：當酸鹼值超過8.5 時，螺殼上就無剛毛藻附生，當酸鹼值降下後，藍綠藻則會比剛毛藻早出現在螺殼上。藉由淡水生物指標的分類推測：兩樣區池塘水體有機污染程度，可能介於β-中腐水性(β-mesosaprobic,βm)至α-中腐水性(α-mesosaprobic,αm)的範圍之間，而Ｄ池塘受污染的程度應會比Ｂ池塘還要更嚴重些。至於兩樣區池塘水體，已面臨了何種優養化的程度？究竟是屬於哪一個階段的污染等級呢？除須補充水體中浮游性及附著性藻類分類的詳細觀察資料外，仍必須參照部分水質化學分析法所測得的數據作佐證，才能在日後持續地研究過程中進行更精確及深入的探討。
化學光電池之光敏劑的開發與研究
六種自行合成出來的聯吡啶釕錯合物Ru(bpy)₃、Ru(bpy)₂(phen) 、 Ru(bpy)₂dcbpy、Ru(phen)₃、Ru(phen)₂(bpy)、Ru(phen)₂dcbpy 及商用染料N3-dye，被成功的做成光敏性太陽能電池。光電流的產生率可由IPCE (incident photon-to-current conversion efficiency) 的測量可知。此類釕錯合物可以物理吸附或化學鍵結於TiO₂奈米粒子上。IPCE 的大小可以用來探討不同吸附方式的釕錯合物轉換光電流的效率。在物理吸附上Ru(phen)₂(bpy)的效率最好。化學鍵結的以N3 Dye 最好，我們合成的錯合物以Ru(bpy)₂dcbpy 較佳。此種以TiO₂奈米結構為承載基材的太陽能光電池(Dye-Sensitized Solar Cell)，染料仍以商用染料 N3-dye 最佳。本研究發現物理吸附的Ru complexes 也可產生光電流，若能最佳化，將可簡化染料錯合物之合成。
Six ruthenium complexes, Ru(bpy)₃, Ru(bpy)₂(phen), Ru(bpy)₂dcbpy, Ru(phen)₃, Ru(phen)₂(bpy), and Ru(phen)₂dcbpy were synthesized. These Ru complexes and N3 dye have been incorporated into the dye-sensitized solar cell system. The solar energy conversion of the ruthenium complexes were measured and converted to IPCE (incident photon-to-current conversion efficiency). There complexes were either chemically bonded or physically absorbed onto the nano-sized TiO₂ particles. The IPCE were utilized to compare the photon-to-current efficiency of these Ru complexes. Among the physical-absorbed dyes, Ru(phen)₂(bpy) has the highest IPCE. For chemical-absorbed dyes, the commercial N3 dye is still the best. Among the complexes synthesized in this research that are chemical-absorbed, Ru(phen)₂dcbpy has the highest IPCE
The commercial N3 dye has the highest IPCE in the dye-sensitized TiO₂nanoparticle solar cell. We found that physically absorbed dye can convert photon to current. With better solar cell assembly, physically absorbed dye can have the same conversion efficiency as N3 dye.