蜘蛛數
We understood the definition and meaning of spider number by reading〝Wonders of Numbers〞. It interested us so much. So, we took further step to study the situation of extreme value when the gap sometimes lie on the line and sometimes on the circle or even on both. That is to say, we explored the relation between spider number and the gap when the spider number is maximum or minimum. New research for the application of spider number involves several directions. First, we design a new game called〝Stepping Land Mine〞with the rule of spider number. Give you a net with several hidden gaps, trying to find the right positions of gaps. Second is the further result for a different type of net about regular n-polygon. Third is a tactic for a net with destroying of the strategy points. In this situation, the gaps amount on the circle and on the line are fixed. At the same time, consider the situation of circles and lines designing the tactic of placing the gaps to attain the maximum of the destructive effect. 在本文中我們定義一個蜘蛛網上的蜘蛛數，若在蜘蛛網中加入缺口後，會影響蜘蛛數的大小。我們探討蜘蛛網上的缺口，該如何分配才能夠得到蜘蛛數的極值(最大值及最小值)。先觀察一直線和圓上缺口如何放置蜘蛛數有極值，再探討許多條直線及圓上的情況，進而推展至許多同心圓及通過圓心的許多條放射線的缺口，該如何放置，蜘蛛數才會有極值發生。
重複圖形
「重複圖形」是本篇報告研究的問題，我們利用「方程式」建立一個尋找重複圖形，並証明其個數的方法。利用此方法得出下面的結論：1.會形成lap 2 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 2 三角形」只有1 種，即等腰直角三角形。(2)「lap 2 四邊形」只有1 種，即二邊之比為1： 且內角是45°、135°的平行四邊形。2.會形成lap 3 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 3 三角形」只有1 種，即內角為30°–60°–90°的直角三角形。3.其他的lap k 三角形：(1)任意內角為30°–60°–90°的直角三角形都是lap 3k²，其中k是正整數。(2)邊長比為1：m： 的直角三角形是lap (m²+1)k²三角形，其中m、k是正整數。
To find repeated figures, we construct a method to search them with the help of algebraic equations. Here we arrive at：1. There are only two kinds of lap 2 convex polygons, triangles and quadrilaterals. (1) The only lap 2 triangle is isogonal right-angled. (2) The only lap 2 quadrilateral is the one that contains angles 45°, 90° and two neighboring sides with the ratio 1: . 2. There are also two kinds of lap 3 convex polygons, triangles and quadrilaterals. (1) The only lap 3 triangle is the one with angles 30°, 60° and 90°. 3. Other kinds of lap k triangles are listed as following: (1) A triangle with angles 30o, 60°, 90° is a lap 3k², the k is a natural number. (2) A right-angled triangle whose ratio is 1 : m : is a lap (m2+1)k², the m and the k are natural numbers.
完全圖立方乘積之最小控制
完全圖Kn是指一個圖中有n個點，且任意一個點都跟其它的點有邊相連。兩個圖G和H的卡氏乘積G□H的點集V(G□H)＝{(g,h)| g∈V(G),h∈V(H)}，兩個點(g1,h1)和(g2,h2)有邊相連若且為若g1=g2 且h1~h2，或g1~g2且h1=h2。
三個完全圖Ka、Kb、Kc 的立方乘積是指Ka□Kb□Kc。一個圖G中的一點v所連的其它點稱為這個點v的鄰居，也就是N(v)={x | x~v}。一個點集S中的所有點的鄰居的聯集稱為這個點集的鄰居，也就是N(S)=∪v∈S N(v)。如果一個點集S和它的鄰居N(S)包含了一個圖G的所有的點，也就是S∪N(S)＝V(G)稱這個點集S是這個圖G的一個控制集。我們把圖G的所有控制集中點數最少的稱為最小控制集，並定最小控制集的點數為最小控制數γ(G)，也就是γ(G)＝min { | S |, S是G的控制}。
本文的目的在於研究完全圖立方乘積的最小控制，也就是要給γ(Ka□Kb□Kc)一個上界。特別當 a = b = c = n時，γ(Ka□Kb□Kc) = 。
A complete graph Kn is a graph with n vertices, which any vertex is adjacency to every other vertices. The Cartesian product of two graph G and H which is denoted G□H is define as follow: the vertex set V(G□H)＝{(g,h)| g∈V(G),h∈V(H)}，and two vertices (g1,h1) and (g2,h2) is adjacent if and only if g1=g2 and h1~h2 or g1~g2 and h1=h2. The Cartesian product of three complete graph Ka,Kb,Kc is Ka□Kb□Kc,which is the same with (Ka□Kb)□Kc.
In a graph G, the neighbor of a vertex v N(v) is the set of the vertices adjacent to the vertex v, that is N(v)={x | x~v}。 The neighbor of a vertex set S is N(S), which is the union of the neighbors of vertex v over S, that is N(S)=∪v∈SN(v). For a graph G, if a vertex set S unions its neighbor N(S) equal to the vertex set of G, that is S∪N(S)＝V(G), we say that S is a dominating set of G. The domination number of a graph G will be denoted as γ(G), which is the minimum size of all dominating set of G..
We give an upper bound to γ(Ka□Kb□Kc). And when a=b=c, γ(Ka□Kb□Kc) ≦
Self Assembly Mechanism of Water Droplets
這是一系列關於水蒸氣冷凝為極細微水珠的實驗。其中可以歸納為三大部分，第一部分是基礎實驗，將水蒸氣導引至親水性介面上，觀察冷凝水珠的結構。雖然看似簡單平常，但是卻發現：不同溫度的水蒸氣，其冷凝最初始的細微珠粒，尺寸相同；爾後溫度高者，堆疊速率較大，以至於最後同時呈現的水珠大小不一，尺寸不同！
第二部分，是針對冷凝水珠自我組裝機制的探討。實驗是將水蒸氣導引至密度小於1的高分子溶液上，並藉由揮發性溶劑快速揮發，將水珠粒「分層保留」以便更深入了解「解構」後的水珠群聚機制。在這組實驗中得到兩張有趣的圖片：
在討論時，我是從對流機制切入，嘗試解構上面兩張圖。
第三部分的實驗，是將水蒸氣導引到磁場及靜電場上，觀察冷凝的機構。這部分呈現出來的結果，推翻了一般「水分子為電中性應該在電場與磁場中不受影響？」刻板觀念，實驗呈現水分子：不但在電磁場上不易長大同時也有固定的散佈模式(assembly pattern)。同時也觀察到：水分子在正電場形成的凝結水珠較為均勻，在負電場則表現出較大親水性特質。這部分的實驗對日後研究細胞膜上水分子通道應有助益。
I have tried to ask a famous math professor if he can create a formula describing the ordered array of water droplets. “Then, I should study Physics first!” He said. Condensation is the thing we live with, being found everywhere, passing without notice. But we never know when it dose start? By coalescence, water droplets grow bigger, but are not round again. We used the polymer film as template and designed the solution lighter than water, so the minute droplets will sink to the bottom and layer by layer. After seconds we may have multilayers of ordered array. This experiment presented here is actually the diary of the growth of water droplets through condensation, upon volatile fluid, magnetic field and electric field. Through convection, it discusses the self assembly mechanism of water droplets and peep into the uniformity of the size of water droplets. In this experiment, convection and magneto-electric force did play important roles in the self assembly mechanism of water droplets. The topic is mostly concerned as we are surrounded by magneto-electric waves in today’s world. This is the first step in discovering the homogeneous state of water droplets, providing insights into the self assembly mechanism of water droplets with nano sizes.
四面體體積平分面的包絡方程探討
剛開始考慮平分物件時，我們從二維的多邊形部分著手，後來發現已經有人做過相關研究，並且得到類似的結論。這個部份顯現出面積平分線與其包絡曲線間的密切關係。我們將其中的方法和結果加以歸納、改善，為了更全面地研究，我們推導出一般性的包絡方程。之後當我們推廣到三維領域時，發現四面體體積平分面與之前的結論有些相似之處，平分的情況卻也更複雜，我們將推導的結果用電腦軟體呈現出來，以便更深入地了解它。最後嘗試了相當抽象的高維積平分，結果仍具有工整的對稱性，讓我們充分領略了數學之美！When considering bisecting a subject, at first we focused our attention on 2-D case, polygons. But afterwards, we found there were already some similar studies conducted by other students, which indicated the close relation between the area-bisecting lines of a polygon and their envelope. We rearranged their methods and results, and then made further improvement. Moreover, in order to study the bisecting problem entirely, we derived the general envelope equation. Then when extending the generalization to the 3-D case, we came to the conclusion that tetrahedrons’ volume-bisecting planes is similar to that in 2-D, but the circumstances are more complex. We tried to show our result with the aid of software, hoping to understand it fully. Finally, we tried to do the case in higher dimension, which is very abstract, and the result was clear-cut symmetrical. During the studying process, we had seen “the beauty of mathematics.”
「膠」流電-黏度及外加電壓對電解質溶液離子暫穩態通道之影響
在本次實驗中，我發現膠狀電解質溶液中的帶電離子，會因為離子團的熱運動，和電偶極的庫倫吸引力 (electric dipole) 的交互作用下，使溶液的I-V curve (電流-電壓曲線)，具有類似磁滯曲線(Hysteresis curve) 的效果；而膠狀溶液之濃度越高，電解起始點的對應I-V 值也越大。此外，白金電極與銅箔電極的距離若改變，也會使溶液的I-V curve 變的不一樣。另一方面，我也發現，在給予膠狀電解質溶液一緩慢外加的電壓或衝擊電壓並持續維持此一定額外加電壓時，會因為該溶液的黏度持續增高、帶電離子濃度增高且反應不斷變化下，而使該溶液的對應電壓，形成一重複出現「先降-後升-再降」的震盪現象，且電壓值節節升高。最後，我利用掃描式電子顯微鏡（SEM）及能量分散光譜儀（EDS）觀察銅箔電極之表面變化並分析其上之化學組成，藉此嘗試解釋上述這些有趣的現象。In this experiment, with the interaction of the heating action of ionic atmosphere and electric dipole, I find that ions in the gel make the I-V curve in the colloid electrolyte liquor show up with the effect similar to Hysteresis curve. The higher concentration of the colloidal solution, the bigger value of I-V at the initial electrolysis reaction was found. Furthermore, the shape of I-V curve is dependent on the distance between platinum electrode and cupper electrode. On the other hand, I find that when I apply a gradual extra-voltage or a fast extra-voltage to the colloidal electrolyte solutions and then maintain to a fixed value, this will make a unique ‘two peaks’ state oscillation of corresponding voltage. The reason is owing to the climbing viscosity and ion concentration in the solution. With the methods of scanning electron microscope (SEM) and energy dispersive spectrometer (EDS), I observe the change and analyze the components of chemicals on the surface of the cupper electrodes. Finally, I present the interesting results and try to interpret these phenomena.
新型空氣清淨燈具之研究與開發
本研究主要的目的是在開發同時具有空氣清淨與照明的兩種燈具。其中桌燈是基於自然對流原理，利用燈泡發熱讓氣流通過燈具上方的濾網達到過濾功能，為了尋求過濾效果與照度兼顧的最佳值，本研究並提出比較因子的概念。在吊燈方面，除了運用自然對流原理之外，還更進一步利用太陽能驅動風扇，進行強制對流，強化過濾的效果，使得本研究成果更趨於完善。 由實驗結果可得知，桌燈在四星期長期測試條件之下，其過濾效果增進率分別為39.1, 40.8與 40.1%。在吊燈四週長期實驗的結果方面，螺旋與 100W 鎢絲燈泡在自然對流的過濾效果增進率分別為49.1%與 51.4%，而100W鎢絲燈強制對流方面過濾效果增進率則為60.2%。由整個研究結果可以發現，本燈具對於空氣清淨有極佳的效果，在不增加額外耗能條件之下，能增加燈具的散熱效果與延長壽命，同時又具備空氣清淨效果，對環境空氣品質具有相當的貢獻。 The purpose of this study is to develop a novel lamp with both the functions of air-cleaning and lighting. One of it is the desk light. Basing on free convection principle, it makes the air run through the filter on the top of the lamp by its heat in order to attain the aim of air cleaning. To find the optimum value of both cleaning effect and illumination, we advanced the compare factor. The other is the droplight, though it is based on the same principle, we use the solar energy as its power to drive the fan. So that the effect of the filter can be augmented and the result of this research approach perfect. According to the experimental result, in the four-week experiment with desk light, the enhanced efficiency of filter is 39.1%, 40.8% and 40.1% respectively. On the way of droplight with four-week experiment, the enhanced efficiency of filter is 49.1% and 51.4% with helix and tungsten(100W) lamp under the condition of free convection; the enhanced efficiency of filter is 60.2% with tungsten(100W) lamp under forced convection. All these results of the research shows that the novel lamp has great performance on air cleaning and much better effect of heat sink without extra consuming of energy, also the lifespan of the lamp can be extended. Furthermore, it is capable of air cleaning and contributes to the quality of environmental air.
流體碰撞物體所產生的波形之研究及應用
當流體由圓管流下，在碰撞到物體後水流會產生類似駐波的形狀。為瞭解此現象的產生機制，及影響此現象的變因，我們改變流體的表面張力、流速及與碰撞物體間的距離，以探討各變因對波形所產生的影響，進而研究此現象的成因。由實驗結果發現波形會因流速加快、擋板距離增加、表面張力減少而有波長變短的趨勢，且可以用表面張力波的理論解釋。由理論推導的結果，可測量液體表面張力。由於圓球狀的外型使表面積增大，可增加液體之散熱的面積，因此可應用在水冷系統方面。A phenomenon similar to the standing wave, which occurs when a slow-velocity fluid jet collides with an obstacle, was observed. Because the free surface profile was observed to be stable, the phenomenon was not considered as standing wave. To understand the mechanism of this phenomenon and the factors that can affect the free surface profile, the surface tension of the fluid, jet velocity and the distance between the exit of the tube and the obstacle are varied to study their influences on the free surface profile. According to our experiment, the wave length is shortened when the jet velocity or the distance between the tube and the obstacle increases or when the surface tension decreases. The tendency of the investigated phenomenon can be explained by the capillary wave theory. Based on Bernoulli’s principle, continuity principle and surface tension\r equation, an ODE (ordinary differential equation) could be formulated. By using numerical method to solve this ODE, we predict the free surface profile which could match the experimental photo well. The tendency of the phenomenon can also be explained by the ODE. In order to measure the surface tension of the fluid, we wish to minimize the experiment apparatus. To enhance our assumption we use laser to locate the individual particle that we add in the fluid and calculate the velocity field of the flow jet.
利用雙雷射精密測定準靜物的極微小變位
To-be static objects, such as bridges, volcanoes, seldom move ordinarily but have mini displacement only under special conditions, like flood or earthquakes. Therefore, how to measure their mini displacement has never become fully popular with scientists’ research. Then, beginning with “ Optical Lever Theorem”, through a series of speculation and discussion, I decide to use laser ray as light source to perform an experiment ------- trying to find objects with mini displacement in our daily lives, such as revolving electric fans, engine-opening motorcycles, shaken trees, testing their magnifying effect first. Next, I try to use the control-experiment method to find out the magnifying relation and formula of the rotation angle of the plane mirror and the displacement quantity of light focus. As to the measure of mini displacement on objects, I utilize the pillar mirror as a reflection plane to research the magnifying relation of reflection light focus and original displacement quantity. The image made from the light focus of pillar mirror’s reflection, however, isn’t so perfect that I have to use a special plastic-made light-concentrating mirror, which is also called “ Fresnel Lens ”, to focalize the light for easy observation. Besides, I find out the “ function graph ” of the mini displacement quantity on to-be static objects and the displacement quantity of reflection light focus. At last, I try to build up a “ Bridge Alarm System ” of Optical Lever Theorem. 準靜物如橋樑、火山，由於平常不輕易移動，只有在特殊情況下（如洪水、地震）時，才會發生位移的現象。因此，其微小變位如何測量，一向是科學界較少探討的題目。於是，先由光學槓桿原理著手，經過一番思考、探討，決定採用雷射光作為光源，並作了第一個實驗－－找生活中具極微小變位的東西，如轉動中的電扇、引擎發動的摩托車、被搖動的樹木等，先測試其放大效果。接著就試圖用控制變因的方法，找出平面鏡旋轉角度和光點平移量的放大關係和公式。至於物體微小的平移量之測量，則是利用柱面鏡作為反射面，來探討反射光點的位移與本來的平移量之放大關係。然而，柱面鏡的反射光點成像並不理想，於是用一種特製的塑膠集光鏡，又稱Fresnel Lens，將光點集中以利觀測，並且找出準靜物之微小平移量及反射光點的位移量的函數圖形。最後，嘗試建立一套光學槓桿式的「橋樑預警系統」。