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.
Eye gone V.S.eyeless決定果蠅複眼發育基因之協同作用與未知調控基因之尋覓
In this study, we try to know how ectopic eye genes: eyeless(ey), eye gone(eyg), twin of eyeless(toy), twin of eye gone(toe) act cooperatively, and look for some unknown genes which affect the function of eyg. First, through human trans gene screening, we find two human genes change the phenotype of ey>eyg into dorsal out-growth when they co-express with eyg (ey>eyg+X). It means the two genes may relate to cell proliferation. Second, by sequencing the insert genes of mutant fly which was found by EP screening, the result shows the site of the insertion is the same as effete (eff) gene. eff translated wrong proteins which differ from functional ubiquitin-conjugating enzyme may be the major cause of the mutant eye . 本研究係探討果蠅複眼異位基因eyeless(ey)、eye gone(eyg)、twin of eyeless(toy)、 twin of eye gone(toe)間的協同作用，並尋找與eyg 有交互作用的基因、突變株。藉由人類基因轉殖篩選，找到兩株人類基因轉殖株，當其與eyg 共同表現時(ey>eyg+X)，會改變ey>eyg 的複眼性狀，造成dorsal out-growth，顯示這兩個基因可能與細胞增生有關，此外，藉由EP screening 複眼發生突變的果蠅之UAS 下游基因經比對後，位置與effete（eff）部分契合，推測複眼發生突變的原因是eff 的功能發生異常，致使細胞內蛋白質代謝失常所致。
一后多夫否？黑棘蟻(Polyrhachis dives)聚落生殖策略之研究
A vast amount of research has been conducted on various ant species such as Monomonum pharaonis, Formica rufa, and Lasius niger, demonstrating the coexistence of polygyny and monogyny colonies. In polygyny colony, the high heredity variability of filial generation could be expected, but what could the monogyny colony do to hold the high heredity variability in order to increase the fitness? I hope I can account the sex ratio in the colony of spiny weaver ants, Polyrhachis dives, and compare the different colony with different reproductive strategies. I also want to check by V8 that the colony can form a polygyny colony and virgin queens can mate with many males of the same colony in the artificial nest. The results suggest that in the colony of spiny weaver ants, the higher the number of the dealate queens, the lower the number of the male. In the monogyny colony, workers control the sex ratio. According to the observation, we can see the alate adult mate in the artificial nest, and multimating can be recorded for the first time. The following study was there for undertaken to examine what’s the aftereffect made by the Reproductive strategies of polygyny colony and monogyny colony.在多后聚落內，子代變異度高是可以預期的，但是單后聚落中，又如何維持遺傳的變異度以增加其對環境的適應力呢？我們希望能計算黑棘蟻聚落組成，比較不同生殖策略的聚落之各種性比，並且錄影證實黑棘蟻聚落可以為多后聚落並進行巢內自交。我們發現：黑棘蟻聚落內，蟻后數愈高，則雄蟻比例較低；單后聚落之性比決定於工蟻；而根據錄影觀察，聚落中的有翅生殖族可在人工蟻巢中完成交配，多次交配的現象首次被實地紀錄，並且佔所有交配新生蟻后的92.8%。本實驗可利用黑棘蟻來了解單后與多后聚落的生殖策略對其族群發展的影響。
Tree'ting Wood Better:Sunscreen for Trees
The purpose of this experiment is to determine whether or not wood can be used as an alternative testing method for sunscreens and which species of wood are good for this objective. This will be accomplished by applying varying strengths, or SPFs, of sunscreens onto thin wood veneers. They will then be exposed using a weatherometer, in which real light and sun conditions will be simulated. Using a device called a Pulmac, the samples of wood will be pulled apart from the centre at zero span. Based on the consistency of the strengths, it can be determined how the wood reacts to the sunscreen and whether that species of wood is suitable as a testing medium. Trees are an important commodity which enrich the lives of many people, particularly those who live in British Columbia. They provide the means of living and recreation for numerous people, whether it is employment, business, housing, or camping. Since trees are a renewable resource, they are convenient and realistic to use for testing. There are two possible extensions to this project. If the active ingredients in sunscreens can be determined, then we may be able to use these elements to create a new type of stain for fences and other structures. This could make the necessity for painting less frequent, creating cheaper options for the homeowner. The second possible extension of this project is to use the results to as an initiative to reduce the amount of animal and human testing done by manufacturers. This is important, as animal testing not only causes unnecessary pain, but often the results are not very accurate. Overall, I determined that this method of testing sunscreens would work quite well, especially with the following species, listed in decreasing order. 1) Yellow Cedar 2) Yellow Poplar 3) Maple 4) Hemlock These results were based on how tightly grouped together the means of the strength values were for each species. The closer they were, the more consistent that type of wood and therefore the better they are for testing. Overall, I think this method of testing sunscreens would work quite well, particularly with the yellow cedar, maple, yellow poplar and hemlock, as they resulted in fairly consistent strength values. Generally, the sunscreens with a high SPF, or Sun Protection Factor, resulted in a lower strength loss than the sunscreen with SPF 15. In addition to proving that sunscreen is a good method for preventing UV rays from reaching the skin, this experiment has also confirmed that there are other methods of testing sunscreens, which can be researched further. Along with my main presentation I will be showing various graphs, statistics, and pictures. They will be supplemented by a booklet of “Commonly Asked Questions and Answers” and species descriptions for each type of wood.
酒杯發出之音符
When you draw a wet finger around the edge of a half filled wine glass, a sweet musical sound comes forth. The pitch of this sound is directly correlated to the amount of liquid in the glass- the higher the height of the liquid is, the lower the frequency is. It means that the shorter the air column in the glass is, the lower the frequency is. This phenomenon differs from the variance in pitch in a wind instrument. In a wind instrument such as a flute, the shorter the air column in its chamber is, the higher the resulting pitches are. In order to study the wine glass phenomenon, we used a piezoelectric crystal loudspeaker connected to an oscilloscope. We recorded the resulting data by using a digital video recorder to capture the images of the waveform of sound, and than analyzed the waveform by using the computer. Our conclusions are as follows: 1. The frequency of sound thus produced was the same whether we draw our finger around the rim, or we strike the glass rim. The higher the height of the liquid is, the lower the frequency is. But the frequencies vary when we strike the glass and when we blow on the edge. 2. When we used a glass without liquid in it, the frequency emitted when we drew our finger around the edge, this frequency varied inversely as the cube root of their weights. 3. In a glass with liquid, the emitted frequency did not have any correlation to the weight of the contents. By taking two identically filled glasses and placing in each a solid object of the same size but different weight, we were able to see that there was no change in the frequency emitted between the two glasses as long as the height of the liquid remained constant. 4. According to “The Flying Circus of Physics”, if we tap the side of a glass of beer, because of the air bubbles in the beer, the frequency emitted will be lower than that from a glass of pure water. This is according to the book, because the speed of sound is lower in air than in water, therefore the speed of sound in an air-water mixture would be lower than in pure water. The resonant frequencies of the mixture will also be lower. However, in our experiment, we discovered that\r when the glass contained air bubbles, the frequency emitted higher. Our explanation is that the sound emitted since the rim of the glass oscillated transversely, the frequency depends only on the retard of the rim and that the frequency is independent of the speed of sound. The intention of this research is to clarify the many misconceptions of this interesting phenomenon.以溼的手指在玻璃酒杯邊緣摩擦，會有悅耳的聲音，而且頻率會隨著內裝液體減少（空氣柱變長）而變高，這種變化與管樂器隨空氣柱的變長而音調變低不同，為了研究它的原因，我們利用壓電晶片喇叭連接到示波器上，並且利用數位錄影機錄下示波器上的訊號，再以電腦分析出瞬間的頻率，結果發現：一、摩玻璃杯與敲玻璃杯，杯所發出之頻率相同，都是所裝液體愈多發出之頻率愈低。但敲玻璃管與吹玻璃管所發出之頻率不同。二、不裝液體之高腳杯，摩擦時所發出之頻率與重量之立方根成反比。（與鐘相同）\r 三、裝液體之高腳杯發出之頻率，不再與總重量有關，而是與液體之高度有關，保持液體高度不變，即使在杯子中央加入不同重量之固體，杯子振動頻率還是不變。若改裝不同密度之液體，則密度愈大頻率愈低。四、在“The Flying Circus of Physics”書中提到輕敲裝有啤酒之杯時，會因杯中含有氣泡而聽到較低之音調，書中解釋是”空氣中之音速低於水中之音速，混有空氣之水中音速變低，其共振頻率也會降低。”但我們的實驗結果是有氣泡時頻率反而高。我們的解釋是杯子所發出之聲音是由於杯面之振動也就是杯壁的橫向振盪，振盪頻率與液體對杯壁之阻尼有關，但與液中聲速無關，密度愈大之液體阻尼愈大。有氣泡時接觸杯壁之液體變少，阻尼較少所以頻率高。希望本研究能使大多數人對這有趣之現象不再有誤解。
從有限三角和公式研究偶次調和級數之遞迴公式及其相關等式之推廣與應用
本研究中，我們將提出一些新穎結果，著重討論其在三角中的應用；同時，找出其遞迴關係式，得出三角展開式與其所對應之多項式分解式，進而討論出多種的規律性及所涵蓋的內容及推廣性質，我得到很多高中數學公式無法推導出在【4】和【8】中的漂亮公式及創新的結果，且這些等式都是由我們不太瞭解的無理數所構成的。
主要是討論我們在【7】中所得到的收穫與經驗；複數是三角、幾何、代數互動的橋樑，我是以不同的角度及嶄新的方法來綜合探討在【6】中相關的應用。提出關於正整數平方的倒數和公式更為精簡且基本的證明，將 sin−2 x 表示成級數形式的部分分式，進而應用在(a,b) = 1的機率問題上；並研究相關的等式，直接透過三角與代數來研究關於 2p 次方的倒數之求和問題，得出級數 之和的有用遞迴公式，並與最重要的常數扯上關係。
For one thing, we present diverse methods to evaluate finite trigonometric summation and related sums. Trigonometric summations over the angles equally divided on the upper half plane are investigated systematically. Several related trigonometric identities are also exhibited.
What is more, we use methods of calculus, and make several surprising and unexpected transformations. A useful recursive formula for obtaining the infinite sums of even order harmonic series, infinite sums of a few even order harmonic series, which are calculated using the recursive formulas, are tabulated for easy references. Furthermore, is there any interesting results and applications？
Finally, the purpose of this paper is to develop a new proof of and related identities, but their derivations are more complicated. The following studies are completed under the instruction of the professor.