一些Moire patterns 的數學性質研究
Moire 為法文，其英譯為watered， 是古代織布技術的一種應用；將印有規律條?的透明薄片重疊時，稍微移動或轉動其中的一片，會形成極大的圖形變化，稱為moire pattern本作品針對三個moire pattern 的數學式加以推導：(一)、兩張透明片各印有等間隔平行線，轉動其中一片使兩線的夾角θ，亮紋垂直距離和暗?垂直距離的比值為tanθ/2tanθ 。(二)、兩張透明片各印有輻射線，重疊後行成圓系，可由代數或幾何加以證明，利用三角函數可推導出此圓系方程式為：x2+{y-rtan[π/2-(θ-?)]}2)]}={rsec[π/2-(θ-?)]}2)]}\r \r (三)、透明片A 印有等間隔平行線，B 印有符合高斯曲線的平行線，AB 重疊時，形成一系列的高斯曲線，AB 的夾角減少時，會增大曲線的曲率，我們進一步討論曲線的曲率和平行線斜率的關係。Moire is the French word “watered” and refers to an ancient technique employed in cloth making. The moire occurs whenever two or more transparent sheets with periodic strips on them are superposed. The characteristic of moire patterns is the fact that a slight shift of sheets will create dramatic alternations in the observed patterns. In the present report, We derive the equations of three different moire patterns. First of all, take a sheet with equal spaced straight lines and placed it on top of another identical sheet. They are made to intersect and form an angle of θ. As the angle changes slightly, it produces huge changes in the spacing of moire fringes. We can derive a formula related to the interfringe distance. The ratio of bright fringes and dark fringes is tanθ/2tanθ.Secondly, two transparent sheets with radial lines on them are overlapped, forming a pattern similar to the lines of force between point charges. We can find that the pattern is a series of circle by means of algebraic and geometric proofs. And proven by trigonometric functions, we canconclude that they satisfy the equation :x2+{y-rtan[π/2-(θ-?)]}2)]}={rsec[π/2-(θ-?)]}2)]}\r Thirdly, a set of lines of equal spacing is overlapped with a second set of lines whose spacing are derived from a Gaussian curve. A series of Gaussian curves is reproduced in a moire pattern. Reducing the angle of intersection between the two figures steepen the curvature. We discussed the relation between the curvature and the slope of inclined lines.
含環胺-亞胺雙牙配基及其鎳錯合物的合成、結構鑑定，及烯烴催化聚合反應
本研究合成含環形胺—亞胺雙牙配基鎳金屬催化劑，用以催化烯烴聚合反應。將2-甲基丙醛經由溴化、胺化及亞胺化的步驟合成含胺—亞胺的雙牙配基，Et2NCMe2CH=N(2,6-iPr2C6H3) (3a)、(c-C4H8)N- CMe2CH=NPh (3b)、RNCMe2CH=N(2,6-Me2C6H3) (R = c-C4H8 3c，c-C5H10 3d)。再將配基和Ni(DME)Br2 (DME = 1,2- 二甲烷氧基乙烷) 反應， 形成鎳金屬錯合物Ni [Et2NCMe2CH=N(2,6-iPr2C6H3)]Br2 (4a)、Ni[RNCMe2CH=N (2,6-Me2C6H3)]Br2 (R = c-C4H8 4c，c-C5H10 4d)，並作結構鑑定。其中3b、3c、3d、4c、4d 均為合成的新化合物，4d 獲得X 光單晶繞射結構。
以合成之鎳錯合物作為催化劑，催化乙烯或降冰片烯(C7H10)的聚合反應，探討反應活性和高分子產物的性質。與含相同碳數的錯合物比較，在胺基具環形取代基的錯合物對乙烯的催化活性較佳，聚乙烯產物分子量較高，分子量分布範圍狹窄；其對降冰片烯的催化活性略遜於非環形者。同為環形取代基時，六環較五環者對乙烯的催化活性較差，但對降冰片烯的催化活性較佳。顯示乙烯聚合與降冰片烯聚合有不同的反應決定步驟。催化劑及配基的設計的確可以操控聚合反應及其高分子產物的性質。
The synthesis of α-amino aldeimines Et2NCMe2CH=N(2,6-iPr2C6H3) (3a), (c-C4H8)NC Me2CH=NPh (3b), RNCMe2CH=N(2,6-Me2C6H3) (R = c-C4H8 3c，c-C5H10 3d), as well as the nickel complexes Ni [Et2NCMe2C- H=N(2,6-iPr2C6H3)]Br2 (4a), Ni[RNCMe2CH=N (2,6-Me2C6H3)]Br2 (R = c-C4H8 4c，c-C5H10 4d) has been succeeded. Their structures were mainly determi ned by spectroscopy or elemental analysis. The complex 4d was characterized by X-ray crysta llographic analysis. It shows that the nickel complex has distorted tetrahedral configuration.
The catalytic reactions of ethylene or norbornene polymerization using the newly synthesized nickel complexes are studied. All catalyts show high activity toward studied olefin polymerization. Comparing the data of the catalytic ethylene polymerization for complex 4c with those of its isomer complex 4a, the former is found to results in higher activity as well as the larger molecular weight of the PE products with the narrower dispersity. On the contrary, 4c shows lower activity in the reactions of norbornene polymerization than 4a. For the cyclic amine derivatives, 4c of five-membered amino group shows better catalytic activity toward ethylene polymerization than 4d of six-membered amino group. But 4d gives better perfomance for norbornene polymerization than 4c. Such results indicate that the processes of ethylene and norbornene polymerization might have different rate-determining steps. These study confirms that the design of ligand and catalyst are crucial with respect to the control of the catalytic olefin polymerization and the properties of the polymeric products.
利用浮沈子測量液體表面張力並演示"Cheerios Cheerios effect"
密閉容器置入待測液，放入浮沉子，施加壓力，當浮沉子恰要沒入液中瞬間，因表面張力的總力達極大值且向上，外加壓力(p1)為極大值，浮沉子沒入液中；液面減壓，當浮沉子在液面正下方時，外加壓力 p2，量 p1、(p1- p2)，浮沉子的質量 m，外半徑 R，及玻璃管的體積 G V ，可求得液體表面張力。 液面再減壓，浮沉子恰要露出液面時，表面張力的總力達極大值且向下，外加壓力(p3)為極小值，量 p3、(p2- p3)，浮沈子的質量 m，外半徑 R及 G V ，應亦可求得表面張力；但實驗時浮沉子漂移到容器邊，並吸附在器壁上，因此發現浮沉子的”Cheerios effect”。 利用浮沉子和容器的相吸及相斥現象，可解釋西式早餐的小榖片放入牛奶中為何會漂移到碗緣，並支持 Vella在 2005 年 9 月份美國物理期刊(AJP)認為 Cheerios effect的成因除了由於接觸角不同外，浮力、重力、表面張力共同作用，使小榖片間有相吸、相斥現象。 The experiment apparatus is equipped with a Cartesian diver by using a glass tube with air trapped inside that floats or submerses in a closed vessel containing liquid. The external pressure may be varied with a syringe and measured with a water manometer. The maximum pressure P1 inside the vessel is measured when the diver is just about to sink, where the surface tension that acts on the diver is upward. Then the pressure P2 of the vessel is measured when the diver is just beneath the liquid surface, where no surface tension acts on the diver. Finally, the surface tension is calculated from P1, P2 and the radius of the diver, R. When the pressure inside the vessel is decreased, the diver will rise. As the diver is about to emerge from the liquid, we get the minimum pressure P3 inside the vessel, and the surface tension that acts on the diver is downward. By measuring P3, P2, and R, the magnitude of surface tension is found to be the same as above. When the diver is just about to sink into the liquid, it floats to the center of the vessel. When the diver is about to emerge from the liquid, it sticks to the wall of the vessel. This phenomenon is named the “Cheerios effect.” Our results again strongly support that the cause of the effect is due to the different contact angles between the diver and water, as well as the balance of gravity and surface tension in the case of the sinking diver, and the balance of buoyancy and surface tension in the case of rising diver as Vella claimed in his paper (AJP 73, 817 (2005)).
氣流式薄膜測厚儀
醫學上的植皮手術成功率受皮膚厚度影響，皮膚愈薄癒合速度愈快，其中以取皮厚度介於0.05mm 到0.1mm 為佳。在實驗量測時，需要經過一連串繁複的薄皮標本製作，再放到光學顯微鏡下測量，這種厚度測量方式不但耗時，又因嚴重損毀皮膚而不精確。由於使用螺旋測微器做接觸式測量會有形變的問題，因此我們想做間接接觸式的測量，所以採用氣體為媒介，做非破壞性檢測膜厚，這對於在皮膚上的施力遠小於螺旋測微器或是接觸式膜厚計。我們設計一套三頭連管線，使用空氣為媒介，儀器運作原理為在管線一端針頭非常靠近被測物時，所流出的氣體會受到被測物阻礙產生反壓使管線內的壓力上升，導致連通於另一管路的氣泡指示計壓出氣泡，當氣泡為最大氣泡時（半球形）視為達到平衡狀態。實驗時先用已知厚度且不變形的蓋玻片來當作被測物，此時可以算出針尖至蓋玻片的實際距離做為參考值。在量測軟性薄膜時，設計上採用兩側雙針頭靠近軟性被測薄膜兩側以達到氣流氣泡平衡，這時使用螺旋測微器讀取兩針尖距離，減去已知參考值的兩倍距離，即可測出未形變的軟物質厚度。本研究開發一套能測量軟性薄膜的厚度裝置，尤其在皮膚厚度測定上，不但不會直接接觸標本造成損毀，並且能夠快速地測量出厚度值，此為本儀器的最大特色。The thickness of skin graft has deterministic influences on the success of graft surgery. Experimental measurements of skin graft thickness involve complicated specimen preparation processes followed by optical microscopic examination, which are time-consuming and may incur inaccuracy due to possible damage. Here we propose a novel method using air as the media to avoid direct contact of the measured object. The physical operation relies on the following principles: When the tip of a needle connecting to a catheter system is placed close to the object to be measured, the air pumped forward from the catheter system becomes impeded by the object. The resulting backflow pressure opposing the air flow causes an increase in air pressure within the catheter and inflates the bubble connected at the other end. Balance at maximal surface tension is attained when the bubble reaches its maximum volume in hemispherical shape. In practice, a two-needle design was used, each approaching simultaneously from each side of the object. A micrometer was then used to read the distance between the two needle tips, from which the film thickness was derived, subtracting the thickness of the air layer pre-calibrated using cover glass with known thickness. The system implemented was capable of measuring thickness on soft thin films with an accuracy of ± 0.001mm. In addition to rapid measurements with high accuracy, since the pressure exerted on the skin graft is much less than in conventional calipers requiring direct contact, our method has the unique non-distorted and non-destructive advantages.
耍「薛骰」-Sicherman Dice 的探討
George Sicherman discovered that it is possible to take a couple of 6-sided dice re-labeling them with different positive integers (1,2,2,3,3,4) and (1,3,4,5,6,8) having the same probability distribution as rolling a standard pair of 6-sided dice. Such unique pair of dice is calling Sicherman dice. The secret behind the Sicherman dice can be studied by combining the powerful mathematical tool “Generating functions” with the symbolic manipulation software “Derive 6”, The same procedure may be applied to studying the possibility of the generalized Sicherman dice along the consideration of :\r (1) Adding more dice. (2) Changing the number of faces. To this end, we introduce the concept of the Sicherman Bound. For a given integer n, the number of n-sided Sicherman dice is finite. We computed manually such numbers for n?50 based on the method of “Elimination of negative terms”. Sicherman Dice 就是一對點數配置與正常骰子(6 面正立方體，點數為1到6)不同的骰子，它所拋擲出的每一種不同點數和(2,3,4...,12) 的機率恰好與一對正常的骰子相同。這種骰子是美國的Col. George Sicherman 所發現的。 Sicherman 更進一步指出：在不使用Sicherman Dice 的情形下，不可能找到一組大於或等於三顆的非正常骰子，它們拋擲出的每一種不同點數和的機率恰好與一組同數量的正常骰子相同。本研究的目標在於1. 尋求計算「Sicherman Dice 的組合和正常的骰子有相同的出現機率」的方法2. 證明Sicherman 結論的真偽及是否適用於其他正多面體(4 面/ 8 面/12 面/ 20面) 的標準骰子3. 修正Sicherman 的結論，並定義Sicherman 極限(Sicherman Limit)。在假設n面正多面體(n 為自然數， n ? 50 )存在的情形下，探討每一個正多面體的Sicherman 極限4. Sicherman Dice (Crazy Dice) 的延伸探討(1) 不同面數骰子的組合，是否可以找到面數組合相同，但點數配置不同的Crazy Dice( 如4 面與6 面的標準骰子組合，找到4 面與6 面的Crazy Dice)(2) 多個面數相同或不同骰子的組合，是否可以找到面數、個數及點數配置皆不同的Crazy Dice ( 如3 個4 面標準骰子組合， 找到2 個8 面的Crazy Dice)在研究的過程中，我發現以下的現象：(1) Sicherman Dice 的產生，是生成函數因式重新組合的結果(2) Sicherman Dice 是否存在，則視上述重新組合的結果是否有負項產生由於上述的觀察，我使用自行發展的「負項消去」法來檢驗Sicherman 結論的正確性及求得n 面正多面體其對應的Sicherman 極限。同時我也和Col. George Sicherman 取得聯繫， 討論當年他發現Sicherman Dice 的經過及其結論的限制條件，作為本研究未來發展的參考。
榕樹粗萃取液對種子萌發及生長之影響
相剋作用是指植物在代謝過程中釋放出有毒物質以抑制本身或鄰近植物種子萌發及植株發育生長的過程。一般認為，榕樹是具有相剋作用的常見植物，但很少見到相關的文獻探討。本實驗利用榕樹不同組織萃取液，以小白菜、結球白菜與阿拉伯芥作材料，探討該萃取液對上述植物發芽與生長之影響，結果顯示榕樹葉萃取液所造成之影響較顯著，不同榕樹組織的萃取液、不同濃度的萃取液對不同植物所產生的效果不盡相同，針對不同生長期的植物進行處理所造成的影響亦有差異。進一步以管柱層析法與高壓液相層析法（HPLC）分離葉萃取液，結果發現以正己烷和乙酸乙酯75：25為沖提液的條件下，於HPLC的層析圖中分離時間約8分時所流出的物質具最明顯的抑制作用。電泳剖面分析顯示處理與未處理間有些許蛋白質條帶差異，此變化蛋白可能與植物幼苗對該萃取液之耐受性有關。利用核酸微陣列(microarray)對榕樹萃取液處理之阿拉伯芥植物進行基因表現分析，發現榕樹萃取液對植物部分基因表現也有促進和抑制的情形，經由基因體與蛋白質體分析法可推論出一些受影響之相關基因。 Allelopathy was defined as a phenomenon which certain plant species, by secreting metabolites to the environment, can suppress the growth of themselves, seed germination and/or growth of other plants in the same habitats. Banyan （Ficus microcarpa L.f.） is a plant species in our campus likely to have allelopathy effect. However, documents describing such action on allelopathy were rare. In this study, we applied the crude tissue extracts including leaves, stems, roots of banyan onto the germinating seeds of Pai-Tsai （Brassica rapa L.ssp.chinensis（Rupr.（Olsson）））, Chinese cabbage （Brassica pekinensis Rupr.） and Arabidopsis（Arabidopsis thaliana, ecotype Columbia-0） to study the effects of such plant extract on the germination and seedling growth of other plants. It is found that allelopathic effects on seed germination vary among different tissues used for extract preparation. Different concentration of the extract also yield various degree of allelophathic effects. It is also noted that the application of extract onto the post-germinated sprouts has less effects on plant growth. The extract of leaves was subsequently chromatographed over silica gel using hexane/EtOH gradient solvent system and HPLC. The result showed that there was one fraction about 8 min in the chromatography of HPLC eluted with 25% EtOAc in hexane had the most inhibitory effect. SDS-PAGE analysis on the electrophoretic profiles of water soluble proteins has explored a different band pattern between the treated and non-treated sprouts. The observed band difference might provide a clue for exploring proteins which reacted differently upon the application of extract. DNA microarray analysis on the effect of banyan extract on Arabidopsis gene expression has also been employed to characterize genes responsive to the allelophathic treatment. Cross-comparison between the differential transcript and protein profiles will reveal key regulators in plants experiencing allelophathic condition. \r
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.