由６面Sicherman骰子來分析ｎ面的Sicherman骰子
Sicherman 已經找出與兩顆六面的正常骰子有相同機率分布的Sicherman 骰子，並進一步獲得與三顆六面的正常骰子有相同機率分布的骰子必為一對Sicherman 骰子與一顆六面的正常骰子之結果，我們試圖由已知的Sicherman 六面骰子的處理方法出發，透過對割圓多項式的分析來累積足夠的相關資料，以處理由兩顆四面骰子至兩顆三十面骰子，處理由三顆四面骰子至三顆三十面骰子的各種Sicherman 骰子的答案，來探索兩顆與三顆的n 面Sicherman骰子存在的充要條件與求法，並進一步將所得之結果分類，得到 ”有相同標準分解式的類型的數n，會具有相同組數的Sicherman 骰子”之猜測結果與特殊情形下的證明。 Sicherman has found out the Sicherman dice which have the same probability distribution as the normal two six-sides dice. Furthermore , he also found out a pair of Sicherman dice and a normal six-sides dice has the same result as 3 normal six-sides dice . We try to begin with the given algorithm of six-sides Sicherman dice , through the analysis of Cyclotomic Polynomials to accumulate sufficient related information then to come up with the solution from discussion of 2 four-sides dice to 2 thirty-sides dice , from 3 four-sides dice to 3 thirty-sides dice to explore the existence of necessary and sufficient condition and solution of 2 n-sides Sicherman dice and 3- sides Sicherman dice , and even to classify the results to come to a conclusion of the guessing results and proofs under special cases about “the numbers n which have the same Canonical Prime Factorization will have the same numbers of n-sides Sicherman dice.”
環境因子影響美洲蜚蠊觸角擺動模式之研究
本研究以攝影紀錄的方式，透過電腦進行影像分析，記錄不同刺激下美洲蜚蠊(Periplaneta americana)的觸角擺動模式，計算出各項觸角運動的參數，以瞭解光線(光刺激或光適應)、震動刺激、喝水與進食對其觸角行為的影響。我們發現在不同因子的刺激下，觸角擺動的模式具有差異，若兩種不同的刺激同時發生，蜚蠊觸角的行為亦具整合性的反應。蜚蠊於不同狀態下(如喝水或進食)，對相同的刺激有不同的反應，證明蜚蠊觸角的行為模式，受環境因子與個體狀態調節。透過掃瞄式電子顯微鏡的觀察，也發現觸角具多種感覺毛，且雌雄的感覺毛的分佈與數量具有差異。綜合以上發現，證明觸角除了為敏感的受器，亦為能反映出生理與環境狀態的動器，同時也適合進行發展檢測器的仿生學應用，用來檢測環境中物理及化學因子。The aim of this study is to investigate the different swing motion modes of antennae of American cockroach (Periplaneta americana) by computer-aided Imaging Analysis. The parameters of each swing movement were calculated in order to analyze how light (including light stimulation or light adaptation), vibration, food and drinking water may affect the antennae behavior of American cockroach. It was found that the antennae swing motion modes were significantly different under different types of stimulus. If two different types of stimulus occurred at the same time, the reactions of antennae motion may become conformable. Under different environmental conditions (such as food or water), same stimulus may result in different reactions. The antennae behavior has shown to be significantly affected by environmental conditions and individual physiological status. Through the observation with scanning electron microscope (SEM), it was found that the antennae has many types of sensilla; and the distribution and quantity of these sensilla are significant different between sexes. In conclusion, not only the antennae are considered as the sensitive receptors, but also they are the important effectors to reflect physiological status and environmental conditions. The current model is suitable for the development of specific detectors in the applications of Bionics to detect the physical and chemical factors in certain environments.
將錯就錯的knuth 河內塔
在這篇報告中，我們探索了「將錯就錯的Knuth 河內塔問題」。傳統河內塔問題在電腦科學上佔有重要的地位，是一個極具內涵的模型。由於這個模型的深厚數學內涵，使其和巴斯卡三角形建立了緊密的連結，且利用這個緊密的數學連結，設計出復原任意起始狀態的良好演算法。Knuth 河內塔起因於數學家Knuth 在論文[3]中，描述傳統的河內塔問題時所發生的一次筆誤。在這個新的規則之下，我們意外發現Knuth 河內塔存在著一個和傳統河內塔平行的模型，此模型在電腦科學及數學上有著完全不同於傳統河內塔的內涵。我們的研究主要如下：(分別為內文中的四大段)(一) 結構分析。移動環所需要的次數，如何移動環並分析每一次動作所動的環，及每個環何時被動到並給出演算法。(二) 正整數的分割。所有的移動步驟將正整數做了一個新的分割(Partition)；此分割模k之後有良好的循環性質。(三) 費波那契真分數的排序。這個正整數的分割形成一張表，這張表恰好就是分子分母皆為費波那契真分數之排序。(四) 隨意亂排的Knuth 河內塔復原演算法。在Knuth 河內塔的規定下將起始狀態改變，找出良好的復原演算法，並分析。 In this project we study the "Knuth Hanoi Tower", which is motivated by a typo in a paper of Knuth. This inadvertently typo leads to a new rule of moving the discs on the Hanoi Tower (see introduction below for definition). Although seemingly similar to the traditional Hanoi-Tower problem, it turns out that under this rule the "Knuth Hanoi Tower" problem consists of amazing properties, and is totally different from the traditional one. Our study focuses on the following directions: (1) Structure analyzing: We analysis the sequences recording the disc moving and offer enumeration results and recurrsive/non-recurrsive algorithms. (2) Partition of N: The moving sequence forms a partition (a table) of N, which has an amazing congruence property. (3) The order of Fibonacci proper fraction: The row/column of the partition table is, even more amazing, exactly the order when sorting the Fibonacci proper fraction with fixed denominator/numerator. (4) The Restoration of an arbitrary initial state: We offer an efficient algorithm for restoring any initial state of discs. We hope that our study on the "Knuth Hanoi Tower" offers a simple, neat, and new example on the theory of Algorithm, Number theory and Combinatorics.
半屏山之簷下姬鬼蛛的研究
The spiders, Neoscona nautica, often appear in groups, but individuals have their own sense of territory.They usually spin webs among branches during 6:00~ 8:00 in the evening. When building webs, they will first start with bridges and then spin Y-shaped spokes. Next, they spin meshe of net, silk frame, spokes, spirals and free-zone in order. After finishing webs, they will wait for prey on the free-zone or meshe of net. If they find something inanimate on the web, they will break the spiral attached with the inanimate object that is later removed. If the meshe of net is broken, they will fix it immediately. For them, the time to take webs back is during 2:30~ 6:00 in the morning. Most time they use the first pair and the second pair of legs to take webs back and swallow the webs. Sometimes, they break the spirals by the last pair of legs. The sequence to take webs back is : lower right section, lower middle section, lower left section, upper left section, and upper right section. At last, one thread of bridge will be left. Every early mornings they take webs back and swallow them. The next evening they rebuild webs. Possibly there are two reasons to explain why spiders eat their webs: (1).They swallow webs to get protein. (2).The web threads are easily polluted by dust and humidity and reduce stickiness. The web may also reduce the probability of capturing prey. The body length of them is not related to effective web dimensions. However, the web sizes depend on the width of web-building location. The study shows linear relation among body length, meshe of net and dimensions of free-zone. The linear relation represents that the meshe of net and free-zone have ecological or survival meaning for them. We expect that this study of Neoscona nautica can be helpful to build spider ecological database in Taiwan.簷下姬鬼蛛常成群出現，但個體卻有很強的領域性；常於下午6：00 至8：00 結網於樹枝間，結網時，先以橋絲為出發，織出一Y 形的縱絲，再由此依序織出中空網眼、絲框、縱絲、橫絲、棲息圈，網結好後，簷下姬鬼蛛則在棲息圈或網眼靜候獵物，若發現網上有非生物之異物，則將黏住異物的橫絲弄斷，再把網上的異物丟棄；若網眼被破壞，則會立即修補。收網時間為凌晨2：30 至凌晨6：00，收網時，大部分由第一、二對步足進行收網，偶爾會用最後一對步足將橫絲弄斷，一邊收網一邊將網吞食，收網的順序為：右下、中下、左下、左上、右上，最後留下一條橋絲。簷下姬鬼蛛每天清晨都會收網，並將網吃掉，翌日傍晚再重新結網，其可能原因有兩點：(1)將網吃掉以補充蛋白質。(2)蛛絲容易受灰塵、水氣之污染而減小黏性，降低獵捕功效。簷下姬鬼蛛體長與有效網面積無關，但網的大小視其結網地點寬敞程度而定。體長與網眼、棲息圈面積呈線性關係，表示網眼和棲息圈對簷下姬鬼蛛具有生態或生存意義。我們對簷下姬鬼蛛生態調查之結果，希望能幫助台灣的蜘蛛生態資料庫之建立。
外觀數列
The Look and Say sequence is produced by describing the appearance of the previous row. For example, start with “1,” which can be described as “one 1,” and therefore the second row is “11,” which is "two 1s," making the third row “21,” the fourth row “1211,”and so on. The main goal of this study is to work out the exact formula for this sequence, which means given the row number n, we can know at once what the n-th row is without having to start from the first row and doing the look-and-say iteration for n-1 times. Some of the methods used include dividing groups, repetition and cracks. The formula we derived speeds up the calculation and gives us a better understanding of the look and say sequence.「外觀數列」為依照外觀產生下一列的數列，第一列為「1」，第二列描述第一列「1 個1」而為「11」，第三列則描述第二列「2 個1」而為「21」，第四列「1211」，依此類推。本研究針對外觀數列的各項數學性質作研究探討，並由此推導出外觀數列的一般式，即給定第n 列就可知道該列的內容。我們運用了分組、重複性以及裂縫的方法分析數列，最後得到了其一般式，此一般式有助於運算速度的加快以及我們對數列性質的了解。
『吸凍！』--再造保麗龍的第二個春天
中文摘要 本實驗先尋求將廢棄保麗龍磺酸化為陽離子交換樹脂(本實驗稱”保麗龍膠”)的方法。將保 麗龍依：丙酮溶解→硬化→打碎→與濃硫酸共煮三小時→浸於50%硫酸溶液中→沖洗→以水 浸泡的流程，即可達再造的目的；我們測得其磺酸化比例為62.5%。再利用「碘滴定法」(浸 泡式)與「相對電壓檢測法」(流動式)，依次尋求保麗龍膠吸附金屬離子的最佳條件。其中「碘 滴定法」可有效測出銅離子濃度，但手續繁瑣；「相對電壓檢測法」最大的好處是知道保麗龍 膠何時吸附達飽和必須再生。 目前我們所知，要保麗龍膠達到吸附陽離子的最佳效能，其條件依次為：使用細粒的保 麗龍膠；低濃度的金屬離子溶液；質量愈大的保麗龍膠；低溫下較慢的金屬廢水流速及pH 值約為4.30 的銅離子廢水；鈉型的保麗龍膠吸附效能優於氫型。保麗龍膠對不同金屬離子亦 有吸附力，單位體積所含離子數愈少，初始的相對電壓會愈高；在相同莫耳濃度下，不同離 子的吸附力依次為Cr3＋＞Fe3＋＞Ni2＋＞Cu2＋＞Co2＋；分次吸附確可將金屬離子完全去除；由 吸附等溫線觀察得知，可能保麗龍膠為多孔物質，導致500ppm 以下的吸附模式無法明確判 斷，1000ppm 以上則為物理吸附模式；保麗龍膠可以再生也可被覆在砂粒上達到不錯的吸附 效能；最後，我們將吸附過金屬離子的保麗龍廢膠與硫酸鈣、紙漿及些許的石灰(質量依序為 13 克、13 克、7 克、0.04 克)混合，可製成類似紙黏土，做成造型磁鐵，廢物利用十分有趣。 Abstract The Experiment will, first of all, explore the ways to sulfonate expandable polystyrene into cation ion exchange resin (called “polystyrene rubber” hereafter in the experiment). The procedures of treating expandable polystyrene are as follows: acetone dissolve→hardening→smashing→ boiling with sulfuric acid for three hours→immersing in 50% sulfuric acid solution→washing→ immersing in water so that we may reach the goal of reconstruction. We calculate the sulfonated rate to be 62.5%. Then we make use of “Iodine Titration”(immersion method) and “Opposite Voltage”(floating method) to seek for the best conditions of adsorption the metallic ion through polystyrene rubber. The former can effectively calculate the concentration of copper ion, but the procedures are quite complex. The greatest advantage of the “Opposite Voltage” method is that we may know when the adsorption of polystyrene rubber is saturated and should be regenerated. As far as we know at present, the conditions of obtaining the best effect that polystyrene may adsorb the cation ion are as follows: fine particles of polystyrene rubber; low concentration metallic solution; polystyrene rubber of which the mass is greater; at lower temperature, slower waste water flow speed and the copper ion waste water with pH 4.30; the adsorption effect of sodium type polystyrene rubber is better than the hydrogen type. Polystyrene rubber also has adsorption effect toward different metallic ion. The less ion per cubic contains, the higher the original opposite voltage. With the same mole concentration, different ion adsorption effects may range as follows: Cr3＋＞Fe3＋＞Ni2＋＞Cu2＋＞Co2＋. The batch adsorption definitely may erase metallic ion completely. By observing the adsorption isotherm, possibly because the polystyrene rubber is a multi-apertured matter, we find that it is impossible to judge exactly the adsorption model of those metallic ion solutions of which the concentrations are below 500ppm. Those which are over 1,000ppm belong to physical adsorption models. Polystyrene may be regenerated and get an adsorption effect by coating sand particals. In the last analysis, we may make paper clay and magnets of different styles by mixing the adsorbed metallic ion polystyrene rubber with calcium sulfate, paper pulp and a little lime(the mass are respectively 13g, 13g, 7g, and 0.04g). The reuse of waste is really very interesting.