波動奇蹟—皂膜與皂水共振模式之研究
本研究探討肥皂膜與肥皂水的共振現象。在肥皂膜共振方面，我們以實驗探討皂膜的共 振模式與頻率的關係；並配合共振理論模型求出薄膜厚度，再與體積密度-厚度測量方法比 較。而又發現皂膜在共振時，皂膜保存時間較平常久，故進行皂膜生命期與頻率、強度的關 係。在進行肥皂膜實驗時，肥皂水滴落在喇叭上，振動出奇特的形狀，進而探討皂水共振的 特性並嘗試建立其數學模式。 ;The research is about the resonance of soap film and soapsuds. For the resonante of soap film, we tried to find out the relation between the resonant pattern and frequency by experiment; according to the resonante model, we measured the thickness of soap film, which was compaired with volume-dencity method. We found that the life-time of the resonant soap film is longer than the normal one, so we proceeded to study the relation between the life-time, frequency, and power. We observed special resonant pattern while the soapsuds fell down on the speaker. So we studied the characteristics of resonant soapsuds, trying to make the mathematical pattern of resonant soapsuds.
誰能比我更善變???~ 釩鎵磷酸鹽的合成及其結構鑑定
This research tries to find out how changes of weight or proportion of a reactor would affect the type and structure of a crystal. In the experiment, the Hydrothermal reaction was adopted to conduct the syntheses of Vanadium-substituted Gallium Phosphates and try to pick up the single crystal to collect data with SXRD for further illustrating the crystal structure with diamond software to know well its nature. The research result finds out a new type of crystal unseen in the existing literature produced because of the weight changes of C3H10N2, therefore the weight change of a reactor may affect the structure of crystal significantly. In the future, it is expectation that comparison among structures of other Vanadium-Substituted Gallium Phosphates can be made, and the properties of this compound can be measured, and try to find out a brand new type of structure for new exploration by changing the initial reagent.
本研究係透過改變化合物所添加之反應物的比例，以了解反應物比例的改變對晶體的型態與結構的影響。實驗採用中溫水熱反應來從事金屬釩鎵磷酸鹽的合成。並挑取單晶，以X-ray 繞射儀進行數據收集，再以diamond 軟體繪出晶體的結構圖，且進一步得知其性質。研究結果發現，經改變丙二胺的量後，產生不同於文獻上的新化合物，可知改變所加反應物的比例，對於晶體結構有相當大的影響。未來可朝向與其他的釩鎵磷酸鹽做結構上的比較、對此化合物做更進一步的性質測量、改變起始試劑測試，進而得到更新穎的結構形式等部分進行探究。
吸管笛的諧音之研究與發展
吹長笛時，按同一按鍵，以大小不同的力量去吹，會引發不同頻率的泛音，而通常越用力吹，引發泛音的頻率越高，所以我們想了解為什麼越用力，泛音的頻率會高，其間的關係究竟是什麼？風經過管口會產生各種頻率的噪音，其中某些特定頻率的聲音會因為會在管內形成駐波而放大，所以我們只能聽到某些特定頻率的聲音。當風速增加時，會在管口形成渦漩逸放的紊流現象。其渦漩頻率與流速成正比（註一）。我們以塑膠管實驗。發現以特定的風速引發該基音後，繼續增加風速，當風速達某一定強度時，才會躍遷為下一個泛音的頻率。這個現象告訴我們：在一封閉管下，風速與泛音的關係並非「線性遞增」，而是越「躍遷遞增」的關係。另一個實驗測量不同管長、其諧音之頻率的關係，我們可以得知，越短的管子，因為相鄰兩泛音間頻率差較大，越不易激發更高階泛音。經由這些實驗結果，我們能夠推論：當管子越長、基音頻率越低時，諧音間頻率的差距相對越小，繪出的風速－頻率關係圖應更加顯示了風速與頻率呈正比關係。未來我們可以以閃頻器觀測紊流渦漩的產生，再變化至不同吹入角度，及各式管口造形，這些實驗能協助我們更進一步了解樂器的發聲原理，甚至開發一個以聲音頻率測量風速的儀器。註一：林婉如、張?文2006 國際科工程組佳作作品。When we press the same key and blow a flute using different strengths, we can get different overtones. Usually, the harder we blow the flute, the high the frequency we get. We want to understand why we get a higher frequency when we blow harder into the flute and to understand the relationship between them. When wind passes through the mouthpiece, many kinds of noises will be produced. Some of the frequencies will expand because they will form standing waves in the tube. Therefore, we can only hear certain frequencies. As wind speed increases, a turbulence of the vortex shedding will be formed. The frequency of the vortex shedding and wind speed will be in a direct ratio. We experiment with plastic tubes. When we increase the wind speed and get certain magnitudes, the frequency will jump to the next overtone. The phenomenon shows that the relation, in a closed tube, between wind speed and harmonics is not a linear increase but a transition increase. In another experiment, we measured the relationship between wind speed and different lengths of tubes. We can infer that the shorter the tube, the higher high-frequency harmonics can be produced. Through these experiments, we come to the conclusion that the longer the tube, the lower frequency of the fundamental tone we get and the discrepancy in frequency between harmonics is smaller. Then we make a diagram between wind speed and frequency that indicates that there is a direct ratio between wind speed and frequency. In the future, we can use “” to observe the production of turbulence. Then we can switch to different angles when we blow into flutes. Otherwise, we can experiment with different shapes of mouthpieces. These experiments can assist us to understand more how the instrument sounds. We can develop a device measuring wind speed with frequency.
液滴在我眼前蒸發
本研究的目的在觀察液滴的蒸發過程，探討影響一滴蒸發方式改變的變因。我們利用數位攝影機外加近攝鏡頭，由側面逆光拍攝液滴蒸發過程，並轉錄成電子檔利用電腦協助分析。液滴影像分隔放大投影在方格紙上，以便測量接觸角及高度的變化。結果發現蒸發情形可分為三種：Ⅰ.當接觸角大於九十度時，液滴和界面的接觸線迅速外擴，直到接觸角遞減至90度後定住不動，蒸發終了時，接觸線迅速內縮，留下縮小的實心殘跡。Ⅱ. 當接觸角小於九十度時，接觸線先些微外擴迅速外擴，之後定住不動，直到蒸發終點，液體內縮，留下環狀殘跡。Ⅲ.當接觸角小等於九十度時，接觸線始終定住不變，液滴蒸發後留下原大小的實心班跡。液滴蒸發時接觸角和高度並不是連續變化，而是交互階梯式遞減。本研究的結果可應用在探討噴漆及噴墨印表機的噴墨印刷上。The objective of this investigation is to observe the evaporation of liquid drop and find out the factors that could affect it. A digital camera with micro lens is used to film the process of the evaporation. And a close-up of the sample drop is taken with backlight. The results are analyzed with a personal computer. The magnified images of the drops are projected on grid papers for measuring the changes on contact angles, and heights of drops. We classify the evaporation of a drop into three types: Ⅰ. The contact angle is greater than 90 degrees. At first, the contact line will spread outwardly till the contact angle decrease to the 90 degrees and then be pinned until the evaporation reach its terminal. At last the contact line quickly contract inwardly, and a solid stain remain. Ⅱ. The contact angle is less than 90 degrees. The contact line will spread slightly and then be pinned. At the terminal the liquid contract inward, and a ring-like stain is formed. Ⅲ. The contact angle is equal to 90 degrees. The contact line will always be pinned, and no ring, but spot stain is left. During the evaporation, we discover that the contact angle and the height of the drop do not change continuously but alternatively decrease by steps. The results of our investigation can be applied to the aspects of paint-spraying and jet printers.
遞迥數列及渾沌現象
給定一個P∈(0,1),令k0=0, p0=p，定義k1為能使 的最小正整數k，而 ; 相同的，對於給定的kn-1, kn 為能使的最小正整數k, 。若存在kn 使得，則稱p∈ In; 若對於所有n 與kn ，，則稱p∈ I∞。如此區間(0,1)可分解成集合I1,I2,…,I∞。
國王的海市蜃樓
在夏日的午後，在炎熱的柏油路面上很容易可觀察到遠方的 路面上，出現如鏡子般的倒影，看起來彷彿前方有一灘水，但當 我們向前進一些時，倒影卻突然消失了，這個現象，一般稱為海 市蜃樓。 一般解釋海市蜃樓的成因，都是由於在上層的低溫空氣，和 在下層靠近路面的高溫空氣，因密度的不同，以致於折射率的漸 層差異，繼而產生全反射。 但我們觀察發現，地面與上層空氣的溫差，並非柏油路面上 假積水現象及倒影出現的必要條件；反而和入射光的角度、路面 的平坦程度及路面的性質有關。我們提出了粗糙面在入射光的入 射角接近90度時，可發生單向反射的模型。並由實驗來驗證假積 水現象及倒影主要的成因是「柏油路面的單向反射」而非「空氣 的折射與全反射」。 Under the scorching sunshine, we can see the reflection on the tarmac in the distance just like a water puddle on the road. And the water also reflects the people and object nearby. But, in fact, the tarmac road over there is very hot and dry. Therefore we call the phenomenon as the “false water puddle on the tarmac.” According to the textbook , the main reason for “false water puddle on the tarmac” is that the temperature difference leads to the refraction of the light and causes the phenomenon. However, from our observation, the theory still can’t explain some phenomenon, For example, the “false water puddle on the tarmac” remains to appear when the wind blows fiercely. Even with little temperature different at night the phenomenon is still obvious. Therefore, in our opinion, the temperature different of the air is not a necessary condition of “ the false water puddle on the tarmac.” We bring up the model to explain the phenomenon that when the incident angle of the light approach 90 degrees, the light will result in one-way reflection. According to the model, furthermore, we make experiments at midnight and at dawn. The result of the experiment assures us the hypothesis of the model, Consequently, we hold the ideal that the main condition of the “false water puddle on the tarmac.” Is not because of the refraction and the total reflection but because the light reflects off the road and result in the one-way reflection on the tarmac road.
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。
氧化壓力影響基因轉換表現對脂肪分化之作用
細胞脂肪分化是造成肥胖、骨質疏鬆、和糖尿病的重要前置因素。我們若要維持良好身材又想省去減肥藥的問題，那麼我們必須了解造成脂肪化的原因，才可能擁有好的預防之道。利用人類骨髓間質細胞可以分化成骨質與脂肪等細胞的特性，我們研究氧化壓力對間質細胞\r Ras基因轉換表現後骨質與脂肪分化的影響。結果發現以添加超氧根 (O2-，l5nM)形成氧化壓力，可促進正常Ras基因表現的間質細胞朝骨質分化;相反地，超氧根會促進\r Ras基因突變而不表現的細胞，朝向脂肪分化的現象。進一步研究其作可原理，發現氧化壓力可促進 RaS蛋白質啟動細胞外訊息活化酵素(ERK)，接著驅動骨質轉錄因子(CBFA1)表現，再到骨鈣蛋白質與骨結節形成。而抗氧化酵素(超氧根轉化酵素;SOD,5OOU/ml)的作用，可以抑制正常Ras基因細胞氧化壓力下骨質分化的進行;但不能防止氧化壓力促進Ras基因突變細胞，朝向脂肪分化的作用。總結而言:Ras基因的表現與否，是決定脂肪分化的關鍵切換點;也是影響氧化壓力對間質細胞朝骨質分化的樞紐。這種基因與氧化壓力互動影響骨質與脂肪分化的剖析，將有助於提醒人們:使用抗氧化劑來調節抗衰老、肥胖、和美容時，必須是在不同情況和不同基因體質的人，有所不同。
\r Human\r mesenchymal stem cells are able to differentiate into bone, muscle, cartilage or\r fat tissues. Our preliminary study with human mesenchymal cell line (HS-5) showed\r that HS-5 cells could differentiate to bone, cartilage and muscle but not fat cells\r as determined by histochemical staining of phenotypes. We have further studied the\r influence of oxidative stress on the switch between bone and fat cell differentiation.\r Results showed that oxidative stress started with exogenous superoxide, produced\r by the interaction of xanthine oxidase and hypoxanthine, promoted the differentiation\r of osteogenic lineage showing expression of osteocalcin and bone nodule formations.\r The mechanism was investigated and superoxide was found to induce ERK (extracellular\r regulated signal kinase) activation; and then the expression of osteogenic specific\r transcriptional factor (CBFA1). A plasmid containing ras-mutant (Ser 17 Asn) which\r can inactivate the expression of ERK was transfected into the HS-5 cells for studying\r the influence of oxidative stress on ras-mutated mesenchymal cells. Surprisingly,\r it was found that oxidative stress did not promote osteogenesis but it enhanced\r adipogenesis from the ras-mutated HS-5 cells. Further studies indicated that superoxide\r neither induced ERK activation nor CBFA1 expression, but it did enhance expression\r of adipogenic specific transcriptional factor (C/EBPα) and lipoprotein lipase in\r the ras-mutated mesenchvmal cells. Taken together, the study model to induce the\r bone cell differentiation from human mesenchymal stem cells may be employed to make\r bone cells for tissue engineering.