棋子排列的平均值
本研究由下述問題開始：將n1 個黑色棋子和n2 個白色棋子排成一列，規定第一個棋子必為黑棋；對於每一種排列方法中，同色棋相鄰處記為1，異色棋相鄰處記為-1，所有1 和-1 的總和記為 t (n1,n2 )。對所有可能的排列方法所算出來的t( n1,n2 ) 值求其平均值，記為a (n1,n2 ) 。我們先由觀察各種n1 和n2 值，得到這平均值的可能公式，隨後並嚴格證明其正確性，證明方法也經過多次精鍊到十分簡潔的方式。以此為基礎，我們並做了各方向的推廣，研究涉及下列各點：(一) 利用組合數探討原來的問題。(二) 在第一個棋子不限定為黑棋的假設下，求平均值a( n1,n2 ) 。(三) 將棋子由兩種增加到多種。(四) 改變棋子排列以及相鄰的方式。經由研究，我們發現，每一次愈將問題推廣時，愈能找出清晰的概念涵蓋並印證先前的想法。Our study starts with the following problem. Suppose n1 black chesses and n2 white chesses are arranged in a line under the condition that the first chess is black. For any arrangement of these chesses, an adjacent pair of chesses having the same (respectively, different) colors is associated with a value of 1 (respectively, -1). Let t(n1,n2 ) denote the sum of these values. The purpose of this problem is to calculate the average value a (n1,n2 ) of these t (n1,n2 )which runs over all possible arrangements of the chesses described above. We begin from observing various values of n1 and n2 and find a possible formula for the solution. We then give a rigorous proof for the formula. After some refinements, simple proofs are also established. Based on this, we also make some generalizations. In summary, the research includes the following: 1. Study the problem by using binomial coefficients. 2. Calculate a(n1,n2 ) when t( n1,n2 ) runs over all possible arrangements in which the first chess can be black or white. 3. Increase the types of chesses from two to many. 4. Variant the arrangement method of the chesses from a line to other configurations. During the study, we find that whenever we extend the problem to a more general case, we make the ideas for the original problem clearer.
蝌蚪游泳能力之探討
本研究主要探討蝌蚪之游泳運動特性，及游泳速度(V)與尾鰭長度(SL)、尾鰭高度(SH)、身體質量(M)、尾鰭擺動頻率(TBF)、擺動幅度(AMP)之關係，並分析蝌蚪游泳之體軸變化及流場變化。祈能了解蝌蚪之游泳運動特性，進而探討其適應環境之機制。研究結果顯示：黑眶蟾蜍蝌蚪體重(M)愈重，則鰭長、鰭高亦隨之生長，並呈現高度相關性(R2=0.9381、R2=0.9809)。另外，尾鰭生長時之長度增加較多。蝌蚪體重(M)與鰭長(SL)、鰭高(SH)之迴歸方程式(M=0.027SL＋0.342SH－0.078，R2＝0.9832)。黑眶蟾蜍蝌蚪之游泳速度，會隨著尾鰭擺動頻率之增加而提高。尾鰭長度愈短之蝌蚪，增加游泳速度時尾鰭擺動頻率增加較多。蝌蚪游泳速度(V)與鰭長(SL)、擺動頻率(TBF)之迴歸方程式(V=0.480TBF＋4.804SL－4.381，R2＝0.9110)。不同尾鰭長度蝌蚪之擺幅對體長之比率並無明顯變化，其擺動幅度(AMP)的範圍介於0.45(BL)至0.56(BL)之間。蝌蚪游泳時各部分體軸之擺動幅度自吻端開始(P=0)至P 為0.24 時逐漸遞減，且在P 為0.24 時呈現最小擺幅，但P 超過0.24 之後直至尾鰭部分卻又大幅遞增，其最大值出現在尾鰭末端(P=1)。蝌蚪游泳是以尾鰭快速向中心軸擺動，產生較大的前進動力，過了軸線則慢速擺動，以減少阻力。This investigation is to explore the swimming habits of tadpoles- the relationship between their swimming velocity, length and height of their tails, mass, the frequency at which their tails movement, and the amplitude of the tail’s movement, as well as analysis their body axes, and the flow distribution of the water, in order to understand how the swimming patterns of the tadpoles are affected by the changes in their environment. The results of this investigation have shown that as the mass of the tadpoles increases, both the length and the height of their tails also increase according to the R values of the tail increases according to the R values of 0.9381 and 0.9809. However, it is observed the length of the tail increases at a faster rate than its height during the tadpoles’s growth. The formula which models the regression relationship between the tadpole’s mass, tail length, and tail height are found to be (M=0.027SL+0.342SH-0.078,R=0.9832). It’s also noted that as the length of the tadpole’s tail decreases, the velocity and the frequency of the tail would increases (the length of the tail is inversely proportional to the tadpole’s velocity and tail frequency). The formula which models the regression relationship between the tadpole’s velocity, tail length and tail frequency is (V=0.480TBF+4.804SL-4.381,R=0.9110) The different frequency model by tails of different lengths do not appear to have an apparent relationship with the tail length, given that the amplitude is between 0.45(BL) and 0.56(BL). As the tadpole swims, the angle between its oscillating body axes decrease as the P values increases from 0 to 0.24, their force the angle is at a minimum whom the P is at 0.24.Yet when P exceeds 0.24 the angle would increase dramatically. The maximum value is observed when P=1.The tadpole’s swimming motion mainly relays on the rapid oscillations of the tail about the centre of mass (body axis)-producing a stronger driving force, and slowing down towards the end of each oscillation to minimise the friction forces acting on the tadpole, which in furn, decrease its velocity.
環境標籤---地衣與環境污染的探討
隨著工商業發展，環境污染日益嚴重，對多數生長在這塊土地上的人，無疑造成了非常嚴重的影響。但若想要監控目前環境中的空氣品質，則必須具備專業的訓練，及昂貴的實驗設備，對一般民眾而言，根本就做不到。\r 藉由指標植物對所生長環境的高度敏感性，可以發展出一套純天然且免費的環境污染偵測器，不但方便、免攜帶、無須高級儀器協助、更不需要專業的分析技術。為此，我們以對二氧化硫等空氣污染物極為敏感的地衣作為指標植物，對其進行生態與環境污染關係的一系列觀察，並設計相關的實驗，找出環境污染物對地衣的實際影響，使其能夠實際的應用於日常生活，並可加以推廣，讓人人都可以利用地衣來了解自己所處的環境是否遭受污染，為自己家園的環境優劣把關。\r \r 文摘要 :\r With the development of industry and business , environmental pollutions become more and more serious . Undoubtedly , those pollutions have a great effect on us.\r However , by the means of the indicator plant which is highly sensitive to its environment , we can develop a set of natural and free environmental pollution detectors . In this project , we use lichenes,which are very sensitive to air pollution , to do a series of observations and to find out the influence the pollutants have on lichenes . If we can apply this to our regular lives , everyone can use lichenes to see if their environment is polluted or not .
節省能源之社區供水系統
The water-supporting system of community should use high efficiency pumps to\r support water and set up a unique water tower for each floor. Since there are high and\r low period of water-consuming and the variety of water pressure, we have to set up a\r unique water tower for higher floors firstly.\r It help us through high peak period of water-consuming, or it turns on pressure-aid\r pumps to save energy.\r 社區供水系統應採高效率之幫浦高壓供水,分樓層設置專用水塔,在用水有高峰\r 及離峰時段,水壓有高低變化,優先順序供水給較高樓層的專用水塔,樓層專用水\r 塔容量必須足以渡過用水高峰時段之容量,否則就啟動輔助加壓幫浦,達到節約能\r 源之目的。
模擬聲波干涉
在高中光學裡，介紹了許多有關光波之特性，而聲波與光波皆具有波動性，因此聲波應具有如干涉、反射、聚焦等特性，但在物理課本上並未詳加敘述，所以我們開始了本項的研究，希望可以籍由改變聲源及邊界的各項條件，而探討其發生之現象。在本研究中，我們利用聲波之基本原理在電腦上進行聲場的模擬並加以改變其變因（頻率、相位、聲源數、聲源間距、強度、邊界反射），進而明瞭聲場之各項特性及應用與控制方式。經電腦模擬聲場圖中，我們觀察到，兩聲源干涉所形成之圖形為多組雙曲線所組成，近似於光學之雙狹縫干涉，增加聲音頻率與聲源間距離皆可使腹(節)線數目增加。如同現實世界中所知的，隨著頻率的增加，將會具有指向性的產生並且在聲源數目越多時越明顯，但發現頻率增加至一定值之後，指向性反而會降低而形成冠狀面。在延遲了多點聲源間相位之後，聲場分佈有偏轉之現象，利用相位延遲的方法，在多聲源中，將兩旁之聲音偏向中央將可造成聲音的聚焦。在兩聲源干涉中，調整其中一聲源之強度，將可完全消除兩音源連線間一點之聲音，可適當的應用在工業上消除噪音。聲場分佈在具有邊界的環境下，我們試著找出聲源位置及邊界條件對聲場分佈的影響與關係以模擬室內聲場，但在簡化的數學模式下，即無法有我們所希望之最均勻聲場分佈產生。最後我們將實驗中的結果與文獻上的實驗數據加以比較，以探討其誤差。 The optical course in senior high school , which introduced many characteristics of optical wave. However, both of sound and light have the characters of wave; therefore, sound wave should have the characteristics, such as interference, reflection and focalizing. Nevertheless, there are not many details of sound wave in the section of acoustic on our textbook. So we began this research, and discuss the different phenomena by changing many kinds of variables. In our research, we simulated the sound field on the computer, based on sound wave’s principle, furthermore we change many variables, which like frequency, phase, source number, distance, intensity and reflection. It helps us understand the characteristics of\r sound, how to control sound and how to apply these findings. According to the result of computer simulation, we discovered that the graph of two acoustic source’s interference comprised by many pairs of hyperbola, just like optical double slit interference. As the frequency or the sound source distance increased, acoustic direction became more and more obvious. But when the frequency was high enough to over the extreme, instead increasing, the acoustic direction would lower down like a crown. After we make phase differences on one of the two sound sources, sound field generated\r deviation. So if we use this method in multiple sound source, and delay the middle source, the sound field might be converged. In such two-sound-source interference pattern, when we control the intensity of one, a certainly point on the line of the two sources disappeared When the sound field enclose by borderline, the standing wave appear, and we discovered many funny phenomena. We put large amount of source in a narrow slit, the phenomenon of diffraction appeared. Finally, we discussed the discrepancies between interference pattern previously done by others experiments and the simulated one conducted by us.
在generalized Petersen graph P(n,5)中的hyper Hamiltonian
Generalized Petersen graph P(n,k)，定義為n 為不小於2 的整數以及1≤ k ≤ n−1，有頂點{ u0, u1, . . . , un−1, v0 , v1 , . . . , vn−1 }，及路徑{ uiui+1 , uivi , vivi+k：1≤ i ≤ n−1 }。在 [2] 中，我們可以知道P(n,5) 是Hamiltonian 等價於當n≠11。
在這一篇報告中，我們證明當generalized Petersen graph P(n,5) 是hyper Hamiltonian（一種Hamiltonian graph 再去掉任何一點後，仍然是Hamiltonian graph）的充要條件是n 為不等於11 的奇數且n ≥ 7。
The generalized Petersen graph P(n,k), n ≥ 2 and 1≤ k ≤ n−1, has vertex-set { u0, u1, . . . , un−1, v0 , v1 , . . . , vn−1 } and edge-set { uiui+1 , uivi , vivi+k：1≤ i ≤ n−1 with subscripts reduced modulo n}. And we can know that P(n,5) is Hamiltonian if and only if n≠11 from [2].In this paper it is proved that generalized Petersen graph P(n,5) is Hyper Hamiltonian (A Hamiltonian graph can still be a Hamiltonian graph when any one of the nodes fault) if and only if n is odd and n≠11.
會變色的金屬—神奇的奈米科技
本報告研究內容，是利用電化學氧化還原方法合成金、銀、銅三種奈米粒子，以及探討電流是否會影響電解合成奈米粒子，在前半部成功地利用控制電解的部份條件，如界面活性劑、以及電流值大小，而合成出金、銀、銅三種奈米粒子，利用UV-VIS的光譜分析，鑑定其三種奈米粒子不同的吸收波長，其光譜出現吸收的現象是因為金屬表面特殊的表面電漿共振吸收現象而產生的。但是在本實驗中發現在UV-VIS的光譜中，電壓值的大小對金奈米粒子吸收波長並沒有關係，這些奈米粒子在水溶液中藉由界面活性劑的包覆，而溶解的相當好。 The content of thesis focuses on using electrochemistry oxidation-reduction reaction to synthesis gold, silver, and copper nanoparticles. We confer whether current of the electrolysis is an influence for the synthesis of nanoparticles. We succeed in synthesizing nanoparticle by controlling some terms of the electrolysis, like the micelle concentration, and current value. Using UV-VIS spectrum to analyse wavelength of three kinds of nanoparticles. The special phenomenon of absorption spectra is appeared because the surface plasma resonance on the surface of metal. From the UV-Vis spectra, we didn’t find the exact relationship between the potential value and the absorption spectra of gold nanoparticles. Finally, we also obtained good results in spectra observation, which meant that these nanoparticles encapsulated with surfactants were well solved in the solution.
紫蝶幻影
The main purpose of this experiment is to discuss the characteristics of iridescent colors of Taiwanese Euploea’s wings, inclusive of the relations between the colors of wings and squamas. According to the results from scanning electron microscope, we discovered that the iridescent colors had a close relation to nanostructure and arrangements of squamas. We inferred that both the nanostructure and the arrangements would influence the formation of iridescent colors and the basic colors on wings. In addition, the basic colors on wings are related to different types of scales. To compare with the diverse formations of different sorts of Taiwanese Euploea’s wings, we took SEM pictures of Elymnias hypermnestra as well, discovering that its iridescent colors had similar relation with scales. And there was the regulation that Elymnias hypermnestra had only one type of scales at iridescent area, and two different scales at not-iridescent area as well as Euploea’s. 本實驗目的為探討台灣地區紫斑蝶蝴蝶翅膀幻色的特性，以及翅膀幻色與鱗片的相關性。由結果得知，幻色實驗中利用掃描式電子顯微鏡發現紫斑蝶幻色的形成和其鱗片的細微結構與排列方式有密切相關。我們推論紫斑蝶的鱗片細微結構與排列皆會影響其幻色的形成，而顏色的不同則與不同類型的鱗片相關。除此之外，我們亦對同具幻色的紫蛇目碟進行拍照分析，發現其幻色亦與鱗片有相關性。紫蛇目蝶的幻色區具有單一種鱗片構成的規則性，非幻色區則有兩種鱗片，與紫斑蝶相同。