二等獎

在這篇報告中，我們以貝斯曲線的做圖原理建立出一種新的曲線－環狀貝斯曲線，進而得到不少有趣的結果。我們發現有名的古典曲線－蚶線，也是屬於二次環狀貝斯曲線。軌跡方程式為：，此時，係數恰符合二項式定理。之後我們推廣至n次環狀貝斯曲線的軌跡方程式：，也符合二項式定理。 在複數平面上，給定z0、z1、z2三點，我們定義出一個二次變換 ，若，，可映射成蚶線的圖形；若z∈實數，則可映射成拋物線。利用此結果類推我們找到一個複數平面上由 z0、z1、...、zn 所決定的n次變換將以原點為圓心的單位圓，映射成n次環狀Bezier曲線。 In this essay, we use the method of forming a Bezier Curve to establish a new curve, circular Bezier Curve, and find a lot of interesting results. We discover the famous classical curve ＂limacon＂, which belongs to the Quadratic Circular Bezier Curve. The locus of Quadratic Circular Bezier Curve is, where. Its coefficients match the binomial theorem. Then we apply it to the locus of nth-circular Bezier Curve:, and it also matches the binomial theorem.On the complex plane, we define a quadratic transformation corresponding to three points—z0,z1 and z2 as .If , where , a limacon is mapped. If z is a real number, a parabola is mapped. With this result, we will find a nth transformation defined by z0、z1、...、zn on the complex plane. It will form a nth-circular Bezier Curve with unit circle centering on the origin.

設n 為正整數，引人興趣的兄弟樹BT(n)是由高欣欣和徐力行教授不久前在[10]所提出的三正則二分圖。本報告證明在兄弟樹BT(n)中，任兩異色點之間存在三條連結線，彼此不相交且經過所有的點；若除去圖中任一點，則與此點同色之任意兩點之間也存在三條連結線，且彼此不相交並經過所有的點。此外，證明在BT(n)中，任兩異色點之間存在一條路徑並經過圖中所有點；若除去圖中任一點，則與此點異色之任意兩點之間也存在一條路徑並經過圖中所有點。除此之外，還證明兄弟樹中存在一漢米頓圈經過任三條邊。

本研究採用受試者內實驗法進行設計，以2*2*2＝3個因子（糖分*刺激口味*咀嚼時間），加上2次不同時間咀嚼運動與對照組，一共有11次實驗。每次實驗量測短期記憶與短期工作記憶共6種題項，並透過心律變異器、簡單的腦波偵測儀與問卷調查，瞭解咀嚼口香糖對於情緒放鬆的影響。 研究結果發現：咀嚼無糖口香糖對於數字廣度記憶有較佳的效果，反倒是咀嚼有糖薄荷口香糖時，則數字廣度記憶效果最差，甚至比單純咀嚼運動或對照組來得差。其次，在情緒放鬆上，受試者都認為咀嚼有糖薄荷口香糖，其糖分的味覺與薄荷的嗅覺有助於放鬆情緒，但透過心律變異器量測交感神經活性指數與副交感神經活性指數，反倒是增加緊張與焦慮的情緒。其次，從記憶力與心律變異結果得知－有糖分的口香糖會使得情緒較為緊繃與無法鬆懈，因此導致記憶力也跟著有降低的趨勢。但如果經過咀嚼15分鐘後，再進行記憶力量測時，其情緒較為輕鬆而記憶力也有所提升。 咀嚼口香糖對於記憶與情緒並沒有太大改變增加，因此，本研究建議如果真的有咀嚼口香糖習慣的人，可咀嚼15到20分鐘再進行相關工作，如此將有稍顯幫助工作績效。但咀嚼有糖分與刺激口味的口香糖，可能會造成反效果。

本研究以增進日光燈管使用效率為目標，設計出一套整合型的系統以減少不必要的浪費。我們將感知用的硬體設備與自行開發的軟體程式作結合，使系統能夠在感知到日光燈管的各種狀態後，做出相對應的處置。其狀況包含:1.無電時系統會警告，以避免誤判的情形；2.有電時燈管正常運作；3.有電時燈管閃爍或不亮，此時系統會切斷日光燈管的電力，並記錄資訊。以上述系統為基礎，我們搭配紅外線模組來達到有人在才開燈，無人時自動斷電；亦加入偵測環境用的光敏電阻來達到光線暗時才開燈，光線亮時會自動斷電。依據測試，閃爍的燈管會比正常燈管多消耗30%的耗電量，壞掉不亮的燈管亦會消耗正常燈管70%的耗電量，對能源越來越少的今天，我們的系統提供了另一種節能的機制。

本研究是[對於正n 邊形A1A2…An邊上一點P(含頂點)，想像自定點P 朝鄰邊發出一條光線，若依逆(順)時針方向依序與每邊皆碰撞一次，經一圈而可回到P 點，則此路徑稱為「光圈」。過程試著追蹤在正n 邊形內能形成光圈的光線行進路徑及其相關問題。 本研究令，且以逆時針得光圈來討論： 1.根據[光的反射原理]，探討光圈之存在性，發現除定點P 在正2m 邊形或正三角形的頂點外，其餘皆有光圈。 2.將可形成光圈的路徑圖展開成[直線路徑圖]來探討。 3.由[直線路徑圖]，觀察到形成光圈的光線行進路徑，可能存在下列情況： (1)不通過正n 邊形的頂點，且產生路徑循環與不循環問題。 (2)通過正n 邊形的頂點。 4.發現正2m 邊形光圈皆為[完美光圈]。 5.發現正2m+1 邊形光圈之路徑與有理數、無理數之特質有關。即當s 值為有理數時，路徑會循環；當s 值為無理數時，路徑不循環。 The research is about [on Point P (including the angles) on the side of regular polygons A1、A2…An , imagine the light goes from Point P to the closest side, then bumps each side sequentially counterclockwise. After going a circle, it’s back to Point P. The track is called “the circle of light.” I try to trace the light track of the circle of light and other correlative questions.] In this research, we suppose,and we discuss the circle of light according counterclockwise direction:1.According to the light reflective principles, we discuss whether the circle of light exists or not. And then we discover that the circle of light really exists except when Point P is on the angles of regular triangle or regular 2m polygons. 2.Spread out the circle of light’s track to [rectilinear track.] 3.By [the picture of rectilinear track], observing there are two kinds of the circle of light’s track: (1)If the light doesn’t go through the angles of regular polygons, it can be a circulative track or a non-circulative track. (2)When the light goes through the angles, it stops. 4.We discover that all the circles of light in regular 2m polygons are [the perfect circles of light.] 5.We discover the circle of light’s track is correlative with rational numbers and irrantional numbers. When s is a rational number, the track is circulative, if s is a irrantional number, the track is not circulative.

本專題的目的是研究以任意實數 a1 、 a2 、 a3 為起始的M&m Sequences 之穩定性質。我們主要關心的問題是：(1) 是否任給定三數a1 、 a2 、 a3 為起始的M&m 數列皆會穩定？(2) 若上述的M&m 數列穩定，則其穩定的長度與a1 、 a2 、 a3的關係為何？(3) 其穩定的值與a1 、 a2 、 a3的關係為何？我們研究的主要步驟及結果如下︰1. 當1 2 3 a 1) 為起始的M&m 數列。3. 我們證明了下列性質：(1) 若M&m 數列中前n 項所成數列的中位數為n m ，則下式成立： (2) 當存在 k > 4 , k ? N ，使得 ?1 ?2 = k k m m 成立時，則此數列穩定，且穩定長度p 滿足：min{ | 4 } ?1 ?2 = > = k k p k k 且m m ，其中p 必為奇數。(3) { n m }為單調遞增且, 5 1 ? ? ? a m n n n4. 如果x ? 41.625，則{?x,1, x}為起始的M&m 數列，其對應的數列有相同的大小次序且此M&m 數列會穩定，穩定值為41.625，且穩定長度為73。5. 我們觀察發現：如果x 1). 3. We prove the following properties: (1) If the median of the former n numbers of the M&m sequence is n m , we obtain (2) There exist k > 4 , k ? N such that ?1 ?2 = k k m m , then the sequence is stable and the stable length min{ | 4 }?1 ?2 = > = k k p k k and m m , where p must be an odd number. (3) { n m } is monotone increasing and , 5 1 ? ? ? a m n n n . 4. Suppose x ? 41.625, then the all M&m Sequences beginning with –x , 1 , x are the same, and the sequences will be stable, the stable value is 41.625 and the stable length is 73. 5. By the computer experiments, we observe that if x is any positive real number less than 41.625, the M&m Sequence starting with –x, 1, x, will be also stable but does not appear to follow any clearly discernible pattern of behavior. However, the stable lengths are much variant and exist some unknown relation with point format of x. Moreover, we have the following properties: (1)If x is a node, then the stable value is x and the stable length equals to the index of median of the node + 2； (2)Near the branch of 41.625, the stable length is almost a constant except at the edge area，the stable length of (-x,1,x) as x around branch 1 is chaos； (3)If x near the node (K= 3, 5, 7, …, 67, 69), then the stable length is l(K)+K?1 where the positive integral l(K) is determined by Prop1 (see Table 6 and 7).

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.

本研究主要探討水滴在高溫鋸齒面上之爬坡現象。實驗中發現在高達四百度的鋸齒面上能形成懸浮水滴，且水滴可持續一段時間而不會立即蒸發，在水平之鋸齒面沿特定方向移動，特別的是，水滴甚至沿著傾斜之鋸齒面向上運動。首先，水滴因受熱在周圍形成一層蒸氣層，阻隔熱的傳遞，使其不會立即蒸發。實驗中改變鋸齒齒廓之兩斜邊的比例，可以驅動水滴往長斜邊所面對的方向運動。進一步實驗顯示，水滴內部的流動趨勢分別為：水滴前端呈順時針方向流動，而水滴後端則呈逆時針方向流動。此外，爬坡中的水滴，其後半部具有較高之蒸發速率，經由設計一個類比的實驗，我們發現，在高溫金屬平面上，透過其平面上的一個凹槽，可以控制水滴前後部位蒸發速率的差異性，也造成水滴向上爬坡的結果。由實驗結果證明，蒸氣壓力與氣體運動對水滴造成的摩擦力為驅動水滴爬坡的力量來源。

由於在銀河系中的多數恆星是以雙星系統存在，因此雙星的形成機制在恆星演化理論中扮演極重要的角色。但目前其形成的機制未有定論，而爭論的焦點主要為兩個假說：一個假說為當分子雲在裂解為雲核時，同一雲核會形成兩顆恆星互繞；另一假說為不同雲核會分別塌縮為不同的恆星。這份研究中，我們以金斯最小質量與半徑為理論依據，找尋上萬筆的觀測數據，並對其資料來源做觀測限制上的確認，進而分析雙星間距的分布，且以雙星的星團與光譜型為分類作圖。分析後的結果中，我們發現前主序雙星間距分布圖出現了一個小於金斯最小半徑的峰值，得出了分裂說必定存在，然無法排除捕獲說的存在；並且發現不同星團、光譜型對間距的關係有顯著的差別。

Water is very vital in our lives as we cannot live without it. However our world is now facing a serious problem. Owing to the increasing population, water resources are scarce. In recent years, we can see that droughts have been affecting millions of people around the globe. In the meantime, people in developed countries have been wasting huge amount of water for flushing the toilet. According to the US Environmental Protection Agency, 30% of household water goes toward flushing the toilet. Countries like China, US, Canada and UK are still using fresh water in flushing, consuming 190 million L of fresh water every day, not to mention the energy needed in pumping the water. In fact, it is not necessary to use so much water to flush away substances like tissue, hair, urine etc. The water we used is far more than we need. However, as we cannot control how much water is used when we flush, all water in the cistern is flushed away. Realizing the seriousness of water shortage and wastage in flushing, we tried to invent a device to conserve water by controlling the amount of flushing water used. Firstly, we study the principle of normal flushing system so as to understand why flushing cannot be controlled. Then, we tried to think of ways to control flushing. We have tried various methods and materials. After the 6-month testing and modification, we successfully invented Fast Fabulous Flush. It is a device which can be fit into existing cistern to conserve water. With our invention, users can control the amount of water flushed according to needs, so as not to waste unnecessary water. Our invention costs a low price which is no more than 2 US dollar. Also, it can be fit into existing cistern within 3 minutes with simple installation process. Most importantly, flushing water can be conserved effectively. It is estimated that around 200L of water can be saved per household every day.