未得獎作品

這個研究起源於一個平分圓的問題：在平面上2n +1個點(n∈N)，其中任三點不共線，任四點不共圓，任取三點可以畫出唯一的圓，若一半的點在圓內，一半的點在圓外，則此圓為平分圓，Federico Ardila 教授在America Monthly 111 期[2]中發表了一篇論文，證明平分圓的個數為n2個。我們研究的目的是：如果將圓改成拋物線，則平分拋物線的個數是否為一定值? 若為定值，則為多少個? 我們的研究題目是：平面上2n +1個在正常位置上的點(n∈N)，平分拋物線的個數為何? 我們將研究的主要結果分述如下： 一、證明在平面上2n +1個點(n∈N)，平分拋物線個數為定值。 二、證明在平面上2n +1個點(n∈N)，平分拋物線個數為n2個。 接著推廣至：若平分拋物線改成(a ∨ b)拋物線，則個數為何? 我們將研究的主要結果分述如下： 一、證明在平面上2n +1個點(n∈N)，(a ∨ b)拋物線個數為定值。 二、證明在平面上2n +1個點(n∈N)，(a ∨ b)拋物線個數為2(ab + a + b +1)個。 This study originated from a question of “The Number of Halving Circles＂: Setting 2n +1 points in the plane is in general position if no three of the points are collinear and no four are concyclic. We call a circle halving with respect to those 2n +1 points if it has three points of those 2n +1 points on its circumference, n −1 points in its interior, and n −1 in its exterior. Then we call this circle “Halving Circle.＂ Professor Federico Ardila issued a paper in the America Monthly 111 [2]. The goal of that paper is to prove the following fact: any set of 2n +1 points in general position in the plane has exactly n2 halving circles. The purpose we make the study of is: If we turn circles into parabolas, how many Halving Parabolas are there? The title we make the study of is: Setting 2n +1 points in the plane (n∈N) , how many Halving Parabolas are there? We show our main effect below: 1. Proving that 2n +1 points in the plane (n∈N) , the number of Halving Parabolas is constant. 2. Proving that 2n +1 points in the plane (n∈N) , the number of Halving Parabolas is n2 . Spread: If we turn Halving Parabolas into (a ∨ b) Parabolas, how many (a ∨ b) Parabolas are there? We show our main effect below: 1. Proving that 2n +1 points in the plane (n∈N) , the number of (a ∨ b) Parabolas is constant. 2. Proving that 2n +1 points in the plane (n∈N) , the number of (a ∨ b) Parabolas is 2(ab + a + b +1) .

Positive –A Study on a Nanoscale Revision Method and Sensitivity Evaluation This paper took a pyramid and a fixed point as the reference level. It was the intention of our team to establish and prove a new hardness value revising method that is to be used in the deflection of indentation of nano indentor. Such new method was named Material Surface Nanoscale Hardness Revision Method with which we re-measured various materials and error sensitivity of hardness values. We obtained the following conclusions:(1). This paper revision modification method have a highly precision. (2). When the round tip or plane tip was closed to ideal indentor tip, the contact areas during indentation process were close, not demonstrating significant difference. (3). The indentation triangle created when loading effort P was similar with the one left on the sample material when unloading the effort; thus, even though the sink-in and pile up effects due to the mechanical properties of sample material caused the differentiation of side lengths and two indentation areas, the angles of two indentation areas was the same. (4). When the effort was loaded by the tip onto the sample material, if the tip had a certain deflection ψ or rotation ω, the indentation triangle left on the sample material was still significant. (5). In the observation of the indentation triangle left on the sample material, when the triangle cannot become a regular triangle, it meant that there is a deflection or rotation happening to the tip and a further revision of the deflected angle ψ or rotated angle ω is required. (6). The hardness value revision method under indentation deflection situation had the best effect on the projected area revision; the second was on the indentation volume revision and than on the indentation contact area revision. (7). The hardness error sensitivity of hardness value revision method under indentation deflection situation had the best effect on the projected area; the second was on the contact area and than on the indentation volume revision. (8). The method proposed by this study was proved by the silica and aluminum single crystal indentation results and is thus able to be applied to the engineering in the nanoscale measurement of metal materials to obtain more precise data.不偏不倚-奈米級修準方法與敏感度評估之研究在這篇研究報告中，以一個三角錐和一個定點為基準，本團隊建立並證明一個新的在奈米硬度測試儀壓痕偏斜情況下，硬度值的修正方法，取名材料表面奈米硬度修正方法。在新的材料表面奈米硬度修正方法下，重新檢測各種材料及硬度誤差敏感度，得到許多好的結論：（1）本研究之修準方法具有高度精確性。（2）利用圓球尖端或平面尖端的方法近似理想壓頭時其壓痕過程中之接 觸面積相近，並無明顯差異。（3）作用力P 施加(Loading)時之壓痕三角形與卸載時(unloading)殘留於測試材料上之壓痕三角形係屬於相似形；因此，即使各該三角形之邊長因為該測試材料本身的機械性質所產生的滲入(sink-in)與堆放(pile-up)的效應而造成作用力施加與卸載時，壓痕面積上的差異。不過，該兩壓痕面積的角度卻是一致的。（4）當該作用力隨著該壓頭施加於測試材料時，若該壓頭產生某一程度的偏斜ψ 或旋轉ω 時，該殘留於測試材料上之壓痕三角形仍然具有代表性。（5）藉由觀察該殘留於測試材料上之壓痕三角形，當該三角形無法成為一正三角形時，其係表示壓頭已產生偏斜或旋轉的之情況，需要進一步對該偏斜角度ψ 或旋轉角度ω 進行修正。（6）在壓痕偏斜情況下硬度值的修正方法以投影面積修正為最佳，其次是壓痕體積再其次是壓痕接觸面積方法作修正。（7）在壓痕偏斜情況下硬度值修正方法的硬度誤差敏感度則以投影面積為最佳，其次是接觸面積再其次是壓痕體積修正方法。（8）本研究提出之修正方法經由矽、鋁單晶壓痕結果驗證，足以說明適用於工程學上金屬材料進行奈米壓痕硬度檢測時更精確的數據獲得。

雙春海濱公園五年前規劃?紅樹林栽培區，透過學校向管理單位\r 申請調查。經由兩位學姊的協助下和特有生物保育中心所提供的資\r 料，結合目前調查和測試結果，對紅樹林成長過程中所產生的環境和\r 動物相改變，能有更詳細的探討。\r (一)、四種紅樹林品種以海茄苳成長最快繁殖力也比較強，其次是紅\r 海欖。\r (二)、紅樹林的成長伴隨環境因素的改變：\r 1. 砂土的攔截堆積量明顯增加。\r 2. 土壤的酸性因種植的紅樹林品種而有所改變。\r 3. 水質由於指標生物的出現間接可知有所改善。\r 4. 枯枝落葉等有機質增加提供食物網的基層能量。\r 5. 河道縮減導致漲潮時潮水漫流到低漥處。\r (三)、動物相的改變：\r 1. 以魚類魚苗、螺、貝和蟹類的種類和數量增加最明顯。\r 2. 養殖業中常出現的無脊椎寄生蟲數量增加，間接可以得知宿主\r 的數量也會有相對的增加。\r 3. 許多以往紅樹林調查未曾紀錄過的海鞘和星蟲均有新紀錄。\r Shuang-chun Coastal Park is planned as the planting region of mangrove during\r five years. The investigation is granted through our school’s application to the\r authorities concerned. Through the assistance of two senior alumni and the\r information offered by ESRI, combined with our investigation and the testing result,\r we can conduct a more detailed discussion about the changing phases of the\r environment and animals during the growing process of the mangrove.\r (I) Of the four species of mangroves, Avicennia marina grows the fastest with\r superior reproduction, and second comes Rhizophora mucronata.\r (II) The growth of mangrove accompanying the factor in the change of the\r environment:\r 1. The apparent increase in the sand piling\r 2. The variation of the soil acidity subject to the differences of mangroves\r 3. The quality of water is known to improve indirectly due to the existence of\r target living things.\r 4. Withered twigs and fallen leaves form organic substances to offer the bottom\r energy of the food chain.\r 5. The shrinkage of the river width leads to the overflowing of the low-lying area\r when the tide is on the flow.\r (III) The changes in the animal phase\r 1. The apparent increase in the species and number of fish fry, spiral shells, shells,\r and crabs.\r 2. The increase in the spineless parasites existing in the aquaculture, indirectly\r estimating the proportional increase in the hosts of the parasites.\r 3. A new record of tunicate and Sipunculus sp.，which were not recorded in the\r past many mangrove investigations.\r \r

這篇報告要探討下列的「轉沙發的問題」是否有解？有一個長方形的沙發，如圖一，若要求每次只能以「四個頂點逆時針或順時針連續旋轉90度」的方式轉動，請問當長寬具備何種關係時，沙發經數次轉動後，剛好可以「轉」到相鄰的位置，如圖一，而且沙發坐人的正面方向仍保持不變呢？ 我們把原問題看成「平面座標上長方形旋轉的數學問題」，再利用「平面座標、三角函數、複數、複數的極式表示及向量」等數學工具，導出符合題目要求的方程式，最後證出當長與寬的比值為正實數時，有下列的結果： 1.當長與寬比值為無理數時，此問題無解。 2.當長與寬比值是最簡分數時，若分子為奇數，此問題無解。 3.當長與寬比值是最簡分數時，若分子為偶數，分母為奇數，此問題有解。 4.在有解的情況下，我們可以找出特定轉法的最小值。 5.當長與寬比值是最簡分數時，若分子為偶數，分母為奇數，沙發可轉至A點座標為(αp,0) 的位置，其中 α∈Z，且沙發坐人的正面方向保持不變。 6.當長與寬比值是最簡分數時，若分子為奇數，分母為偶數，沙發可轉至A點座標為(0,βq) 的位置，其中 β∈Z，且沙發坐人的正面方向保持不變。 7當的長與寬比值為正實數時，可將沙發轉至A點的座標為(2αp + 2βq,2γp + 2qω)的位置，其中 α,β,γ,ω∈Z，且沙發坐人的正面方向保持不變。 In this paper we discuss the solution of rotating sofa problem as follows : The condition is : Merely allow to rotate the sofa several times by rotating 90 degrees clockwise or counterclockwise around the vertex. (maybe A, B, C, or D in Fig. 1) The question is : What’s the relationship between the length and the width of the sofa, if we request the sofa translated next to the original position with direction unchanged. (as shown in Fig. 1 with A’B’C’D’). We take this problem as a mathematical one of rotating a rectangle in plane coordinates. Then we derive the desired equations by using the tools of plane coordinates, trigonometric functions, complex number, polar form of complex number, and vector. Finally, we prove that： 1. When the ratio of length and width is irrational, the problem has no solution. 2. When the length of sofa is odd in the ratio of length and width, the problem has no solution. 3. When the ratio of length and width is even, the problem has solutions. 4. When the solutions exist , we can find the minimum of the number of rotations. 5. When the ratio of length and width is an irreducible fraction, which has the even numerator and the odd denominator, the sofa can be rotated to the coordinate (αp,0)(α∈Z)which is the new position of A and keep the original position with direction unchanged. 6. When the ratio of length and width is an irreducible fraction, which has the odd numerator and the even denominator, the sofa can be rotated to the coordinate (0,βq)(β∈Z) which is the new position of A and keep the original position with direction unchanged. 7. When the ratio of length and width is a real positive number, the sofa can be rotated to the coordinate (2αp + 2βq,2γp + 2qω)(α,β,γ,ω∈Z)which is the new position of A and keep the original position with direction unchanged.

在物理馬戲團這本書中提到：「當你泡即溶咖啡或攪拌奶精的時候，用湯匙輕敲杯壁看看，添加奶精後攪拌時，敲擊的聲音與添加前明顯不同，為什麼？」這本書的解答是：「當粉末溶解的時候，藏在粉末裡的空氣就會跑出來。因為空氣裡的音速低於水裡的音速，在空氣與水混合的環境裡，音速也比在水裡低。當水裡不斷有空氣混進去時，這個容器的共振頻率和它裡面的音速有關，所以也會降低。因此你會聽到較低的音調，直到空氣全部跑光。」我們利用指向性麥克風以電腦錄音後以Adobe Audition 軟體分析聲波頻率，覺得這個說法有點問題。例鹽水溶液音速較水高，敲擊時的音調卻較水低。由敲擊一黏於裝水水盆中之空杯，與敲擊杯內裝同一水位之水之杯子，頻率非常接近。告訴我們影響頻率的是靠近杯壁一層有效質量。因鹽水溶液密度較高有效質量較大，所以頻率較低。以密度的觀念檢視裝有溶液之杯子被敲後的頻率是對的。但對杯中有懸浮物就不然，例如流體中含有氣泡，則混合體之密度必定變低，有效質量變小頻率應變高。但實驗發現含有氣泡時頻率是變低的。可見氣泡還有其他的影響力高於密度對音調的影響。 流體的振動應是會壓縮到氣泡，氣泡與流體間之力學交互作用為何會使頻率下降，正是我們要找出的。 The Flying Circus of Physics has a question “As you stir instant cream or instant coffee into a cup of water, tap the side with your spoon. The pitch of the tapping changes radically as the powder is added and during the stirring. Why?” The answer is, “The air trapped in the powder is released as the powder dissolves. Since the speed of sound is lower in air than that in water, the speed of sound in the air-water mixture is lower than that in pure water. During that period while the air escapes the container, the resonant frequencies of the water, which depend directly on the speed of sound, will also be lower. Hence, you hear a lower tone until the air escapes. “We then tap the coffee cup and generate an audible tone. The signal picked up by the microphone . The same signal is also studied using Adobe Audition, a waveform processing and analyzing software. We find the assumption is wrong, the speed of sound is higher in sugar solution than that in water, but we hear a lower tone. An effective layer of fluid adjacent to the glass wall is set into motion when we gently rub the rim of the wineglass. The thickness is about the same whether the fluid is inside or outside the glass. This explains why the frequency drops when the liquid is added to the system. When the density of the sugar solution is higher, the mass of the effective layer is higher. But what the presence of the bubbles and the theoretical explanations must NOT rely on are: Use effective density argument: One should not just use a change in the main density to try to explain why the frequency is lower. I would think that the bubbles are compressed a little bit by the vibrational motion of the glass communicated to them through the fluid. But how do the bubbles interact with the fluid under this setting? This is what we need to work out.

這篇報告要探討下列的「轉沙發的問題」是否有解？有一個長方形的沙發(如圖一)，若要求每次只能以「四個頂點逆時針或順時針連續旋轉90 度」的方式轉動，請問當長寬具備何種關係時，沙發經數次轉動後，剛好可以「轉」到相鄰的位置(如圖一)，而且沙發坐人的正面方向仍保持不變呢？ 我們把原問題看成「平面座標上長方形旋轉的數學問題」，再利用「平面座標、三角函數、複數、複數的極式表示及向量」等數學工具，導出符合題目要求的方程式，最後證出下列的結果： 1.當長與寬比值為無理數時，此問題無解 2.當長與寬比值是最簡分數時；若分子為奇數，此問題無解 3.當長與寬比值為偶數時，此問題有解 In this paper we discuss the solution of rotating sofa problem as follows : The condition is : Merely allow to rotate the sofa several times by rotating 90 degrees clockwise or counterclockwise around the vertex. (maybe A, B, C, or D in Fig. 1) The question is : What’s the relationship between the length and the width of the sofa, if we request the sofa translated next to the original position with direction unchanged. (as shown in Fig. 1 with A’B’C’D’). We take this problem as a mathematical one of rotating a rectangle in plane coordinates. Then we derive the desired equations by using the tools of plane coordinates, trigonometric functions, complex number, polar form of complex number, and vector. Finally, we prove that： 1. When the ratio of length and width is irrational, the problem has no solution. 2. When the length of sofa is odd in the ratio of length and width, the problem has no solution. 3. When the ratio of length and width is even, the problem has solutions.

給定一個有n個頂點的簡單圖 G，將頂點標號為1,2,…n；考慮 任意相鄰的兩頂點標號和中最大值的最小值，稱此極值發生時的標號為圖G的擁擠標號。在這個研究中，我們得出方格表、m×n×l長方體、環狀圖、圓柱圖及樹圖的擁擠標號和其極值的通式，並討論相關的問題。 Given a simplicial graph G of n vertices, label the vertices with 1,2,…n. Consider the minimum of the maximum of the sum of any two close vertices’ labeling, and we call the labeling that has this extremum the “crowded labeling”. In this study, we found out the “crowded labeling” and the common equation of the grid, m×n×l cuboid, cycle, cylinder, and trees. And discuss the correlative questions.

本研究將竹炭與銀兩種不同的材料結合,研發出金屬結合非金屬的複合導電材質;利用銀鏡反應,以竹炭當作載體,製作出竹炭-銀複合物,透過自製竹炭-銀電壓與電流的裝置,發現竹炭-銀錠最佳導電的質量比例為竹炭比銀1:9,利用掃描是電子顯微鏡,分析竹炭-銀複合物,發現銀會有效分布在竹炭表面形成包覆,竹炭銀定可導電,電阻介於純銀與炭之間,其電阻極低,將來可應用在代替石墨作為電池的電極,對提升導電度會有幫助;In this work, using the silver-mirror reaction, porous bamboo charcoal has been successfully adopted as novel supports for immobilization of silver nanoparticles by a chemical reduction method and the metal-nonmetallic composites with conductivity efficacy were investigated. Through the test of homemade voltage with the electric current instrument, we found out that the best ratio of conductivity in the bamboo charcoal-silver ingot is 1:9. Scanning electron microscopy (SEM) of the composites show uniform Ag particles distribution on the BC matrix. The bamboo charcoal-silver ingot has the conductivity. The resistance, between the pure silver and the coal (graphite), is extremely low. Thus, this composite will promote conductivity and apply in the battery of electrode for replacing the graphite in the near future.

我們這個作品是先由在長方形中切割出正方形的研究著手，先研究出在平面中，在一個邊長為任意正整數的長方形中，如何找到在其中切割出正方形，但正方形的邊長為最大，而且正方形的個數為最少的方法和規則。 緊接著，我們更進一步想研究這個問題在長方體中的研究：在長方體三邊長a、b、c（a、b、c均為正整數）中，如何在其中切割出正立方體，每次切割出邊長為最大的正立方體，而且正方體的個數為最少的方法和規則。 This study began with investigation of how to segment squares from a rectangle. We studied from a rectangle, with random positive integer sides, trying to figure out the methods and regulations to segments squares with the longest side length but the fewest number of squares within. Moreover, we took further step to examine a cuboid. We found out the methods and regulations to segment cubes with longest side length but fewest number of cubes from a cuboid with sides a, b, and c(a ,b ,c are positive integers).