輸贏一線間-淘汰賽的相關探討
單淘汰賽是一種失敗一次即遭淘汰的賽制；在此假定每位選手都有一相對應的能力數值，本文主要探討在均高的單淘汰賽程表之下，若賽程安排完全依照種子安排原則(亦即最強的選手對最弱的選手、次強隊次弱….)，則對於能力越強的選手越有保障，直觀上而言能力最強的選手應有最大的奪冠機率，探討此種賽程安排是否滿足能力較強的選手有較大的勝率？因發現在某些特殊的選手能力數值分佈之下會發生次強選手勝率大於最強選手的情況，令A、B代表最強與次強選手，P(A)、P(B)代表A、B奪冠的機率，故擬定P(B)/ P(A)為參考依據，尋求P(B)/ P(A)的最大值發生處作為最極端的狀況。發現四位選手的情況下，P(B) / P(A)最大值 = 1；八位選手的情況下，P(B) / P(A)最大值=(196+98) / 343=1.0938，當選手數為2n時，P(B)/ P(A)最大值隨n的增加而遞增。
Knockout Tournament is a highly competitive system in which any player losing a game can no longer play in the tournament. Here we suppose that every player has a numerical value that corresponds to his ability. We consider a totally-seeded knockout tournament with 2n players where in the first round, the strongest player matches the weakest player, the second strongest player matches the second weakest player, and so on. We examine whether a stronger player always has a greater probability of winning the tournament. The answer is in the affirmative for n = 2. For a tournament with eight players(n = 3), the situation is much more complicated. In certain cases, the second strongest player has the greatest probability of winning the tournament. Specifically, let A and B denote the strongest and second strongest players, P(A) and P(B) their respective probability of winning the tournament. We find that the maximum value of P(B)/P(A)equals (196+98) / 343 = 1.0938. For n > 3, we have not obtained the maximum value of P(B) / P(A) . However, it can be readily seen that the maximum value of P(B) / P(A) is non-decreasing as n increases.
光基因新角色-cop8
COP8 is the second unit of COP9 signalsome. In the comparison of BLAST ,we found that\r LLPC14 cDNA and the protein sequence are quit similar to the COP8 of Arabidopsis thaliana\r Therefore, we are interested in the role that COP8 plays in the photomorphogenesis and try to find\r the length of the sequence. So far, no result is obtained about the 3’-RACE, but we’re surprised to\r find that the sequence found in 5’-RACE is shorter than what we’re know, which suggests the RNA\r length of the cDNA used in the RACE combination has decreased.\r Although the final results of the COP8 5’and 3’-RACE haven’t been achieved, it is sure the\r COP8 is highly-conserved in many species. Since the similarity between LLPC14 and the COP8 of\r Arabidopsis thaliana is as high as 90 percent, we used the COP8 of Arabidopsis thaliana to detect the\r COP8 protein in pollen.在過去種綠豆芽的經驗中，不難發現它本身因為環境的不同，而有不同型態白化\r 的現象，例如：黑暗下萌發的豆苗。但尚未深入去理解白化的原因；而在後來的種植\r 經驗上，也發現有許多令自己覺得困惑的相同處。\r 在高二的生命科學第四章中，有提到光照與黑暗生長的植株，在構造比較上有明\r 顯的不同；其中，提及植物的活性組織存在「光敏素」（phytochrome），此色素蛋白的\r 存在於光照後，會引發一連串反應，至此，也稍微了解植物的萌發反應的差別。\r 故藉由個人在中研院植物所的資優生培訓過程，便以此為前題,進一步探究植物基\r 因組COP 8 與白化現象的相關性
數位攝譜儀及其數位分析方法
Color is not a physical quantity, but it is a characteristic of spectra. Traditionally spectra of light sources are characterized by the wavelengths and intensities of the spectral lines. We propose an alternative way of charactering spectra using colors. Using digital cameras, convex lens, and a 600 Lines/mm grating, we design a “Digital Spectrophotometer” (Pic.1), which uses no light sensors and electrical circuits that are necessary for conventional spectrometers. To analyze a spectrum using the “Digital Spectrophotometer”, we take digital images of the diffracted light through the grating emitted by the light source and then analyze the intensity distribution of the color components of the spectral lines. The structure of the “Digital Spectrophotometer” is simple and is easy to operate. The Digital Spectrophotometer includes a computer software program we have developed called the “Digital Spectrological Method”. After enlarging the digital spectrographs to a mosaic scale and regards each mosaic as a basic color block, the Digital Spectrological Method will transform every color block into a four dimensional “color coordinates” (λ (wavelength), R(red), G(green), B(blue)), where the coordinateλ is translated from the spatial position of the spectral line and the R, G, and B coordinates specifies respectively the corresponding intensity of the red, green, and blue color components. Comparing the “color coordinates” of the unknown light sources to the known, we can easily identify the wavelengths of the lights emitted by the unknown illuminant precisely. We have accomplished the following experiments by using the “Digital Spectrophotometer”: 1. Measure the spectra of various gaseous atoms, and establish the “database of digital spectra in color coordinates” (DDSCC). 2. Compare the characters of color presentation between digital camera images and positive film of the optical camera. 3. Identify the absorption spectrum of the Solar spectrum (Fraunhofer Lines) using the DDSCC. 4. Analyze the Orion αandβ spectrum using the DDSCC. 5. Identify the 589.0 and 589.6 nm wavelength difference between the “Double Lines of sodium spectrum”. 6. Measure the range of wavelength of the colored LED and register the results into the (λ, R, G, B) coordinates. 7. Compare the range of wavelength of He-Ne Laser and commercial Laser pointer. 8. Measure the Zeeman splitting of the hydrogen atom spectrum at 0.5 Tesla.
顏色雖不是物理量，卻是光譜的特性，傳統上對光譜的分析只記錄波長及對應的強度，而非以顏色來區分。我們運用數位相機、凸透鏡及600 條/㎜光柵，設計一個以顏色成分為標準來分析各類光譜的「數位攝譜儀」(Pic.1)。這個新的設計無須使用傳統光譜儀所需之光感應器及電路設計，只需拍攝光源透過光柵的繞射影像即可分析對應之光譜。我們製作的「數位攝譜儀」包含了一個自行設計的電腦軟體程式「數位光譜分析法」；將拍攝到光譜數位影像放大成「馬賽克」，作為光譜的最小「色塊」，該程式可將每個色塊轉換為一組四維的「顏色座標」 (λ (波長)，R(紅)，G(綠)，B(藍))，其中的λ 座標係由光譜線的位置轉換而來，而紅、綠、藍座標則記錄對應的紅、綠、藍色成分強度。與已知光源譜線的「顏色座標」比較，「數位攝譜儀」可精確測量各種未知光源放射出的光波波長且操作方便。利用「數位攝譜儀」的數位分析方法，我們完成以下實驗：1. 測量不同種類的原子光譜，建立「數位光譜資料庫」，包括氫、汞及鈉原子。2. 比較數位相機影像與光學相機正片的色彩顯影。3. 利用「數位光譜資料庫」，鑑定太陽光譜中的吸收光譜(Fraunhofer Lines)。4. 利用「數位光譜資料庫」，分析獵戶座α、β的可見光光譜。5. 鑑別波長589.0、589.6 奈米的鈉雙線。6. 用顏色座標(λ,R,G,B)測量發光二極體的波長範圍。7. 比較He-Ne 雷射與雷射光筆放光的波長範圍，發現市售雷射光筆所放之光並非單頻。
數字波的節點探討
數字波是探討在直線上的起始點、位移速度、總數相互變化的節點關係。在直線上，將全部格子數做為總數(m)，開始彈跳的點為起始點(i)，每次彈跳的格子數為位移速度(s)，被踩到的格子就是節點。節點是由位移速度和起始點決定，起始點本身可視為節點之一，之後的節點是由起始點加n 個位移速度產生。我們分別以三種型式討論：起始點等於位移速度，總數增加：使起始點和位移速度所代表的數字相同的彈跳。節點呈2、s、s+2…起始點固定，位移速度與總數增加：觀察位移速度和總數的關係。兩節點的和=s+2位移速度固定，起始點與總數增加：探討起始點和總數的關係。發現節點隨起始點有規律的變化在上述討論的型式中，我們再進一步將位移速度分為質數和合數，進而依其因數變化，可觀測到很多特殊的節點變化。The number wave is to discuss the relationship of the starting point, the moving speed, and the variations of total amount. In straight lines, let all the trellises be total amount (m), and let the starting jumping point be the starting point (i). The trellis number of each jump is the moving speed(s). The trodden trellises are knots. And knots are decided by the moving speed and the starting point. The starting point itself can be viewed as a knot. The following knots produce with the starting point and n moving speeds. We respectively discuss them in three types: When the starting point equals the moving speed, the total amount increases. The number of the starting point is the same with the jumping moving point; the knots are 2, s, s+2…. When the starting point is fixed, the moving speed and the total number increase. From observing the relationship between the moving speed and the total number, the sum of two knots is s+2. When the moving speed is fixed, the starting point and the total number increase. After our research into the relationship between the starting point and the total amount, we find the knots have regular variations with the starting point. From the types discussed above, we further divide the moving speed into prime numbers and non-prime numbers. Furthermore, according to the factor variations, we can see a lot of specific knot variations.
長方體中切割正立方體之研究
在1940 年代，Bouwkamp 提出一系列有關如何將矩形切割成若干個正方形的研究報告，但是如何找出正方形個數最少的方法仍是長久以來懸而未決的問題。在本研究報告中，首先引進「四角切割」的方法，並結合輾轉相除法的概念，來研究矩形的切割問題。我們的方法能大幅度降低正方形的個數，也適合做為此問題的上界函數。有關如何在長方體中切割出正立方體的組合，我們也將輾轉相除法的概念延伸到三維空間，進而建立所切割出最少個正立體數的一個上界模式。此外，藉由四角切割概念的延伸，我們也發現這個上界亦可再予修正。In 1940’s, Bouwkamp proposed the study of dissecting squares from rectangles. Among the study, the problem of the least number of dissected squares has been open for decades. In this project, we first propose a corner dissection method, associated with the famous Euclidean algorithm. By reducing nearly three fourths of the number dissected by the primitive Euclidian algorithm, our method indeed establish a suitable upper bound of the minimal number of dissected squares from the given rectangles Meanwhile, the Euclidean algorithm has also been considered to dissect the cubes from cuboids. We analyze the fundamental properties of the method and establish a prototype of upper bound function for the minimal number of dissected cubes. Moreover, the method of corner dissection has also been implemented for some cuboids, which also exhibits the acceptable improvement being a suitable upper bound.
棋子跳躍問題
This is a study about the solution to a chess flipping game. The game is based on a 4*4 game grid. First, place some chesses on the grid randomly to start a game. Move any chess by jumping over one or two neighboring chesses in the same row (left or right), same column (up or down), or on the same diagonal. Chesses which get jumped over should be flipped. The ultimate aim is to make all the chesses upside down. In this study, I try to find the rules of the beginning arrangement that ensure solution. Here are the steps I take: First, break the restriction of the 4*4 grid, and set the coordinate system. Second, find out a few “basic illustrations” that can be solved and moved in order to cope with certain complicated problems. Third, with “basic illustrations”, find the rules applying to games on n*n grid. 這是關於翻棋遊戲的可行解之探討。棋盤是一個4*4 的方格，遊戲開始時在棋盤上任意擺上一些棋子，均是正面朝上，利用相鄰棋子的水平、垂直、斜向跳躍，棋子被跳躍過一次則翻面一次，遊戲目的在於使所有的棋子都翻為反面。我要探討的是關於棋局可解不可解的問題，找出棋盤上可解棋局的規則。研究步驟大致如下：一、打破棋盤4*4 的限制，將棋盤座標化。二、找出若干個可解並可移動的「基本圖」。三、利用基本圖，找出n 列棋盤可解的規律。
松鶴土石流災害初步調查分析
The heavy rain fall brought by Typhoon Mindulle in 2004 caused debris flows in the mountains of Taiwan. The most serious debris flows took place in the areas along the East-West Expressway. The area from Mt. Li to Tien Leng, namely, from the upper course to the middle course of River Da Chia. There was plenty of debris flowing to the courses of the rivers from the hot spring area in Ku Kuan to the starting place of East-West Expressway, Tien Leng. This situation caused the sedimentation of the river courses. According to the data issued by The Soil and Water Conservation Bureau of the R.O.C, on July 2nd debris flows erupted in the First and the Second branches of the river in Sung Ho Village and caused 1 death and 2 injuries, besides, the disaster destroyed 8 major roads causing transportation breakdown. On August 24th, the Typhoon Aere caused the heavy flow of the river which destroyed Po I Elementary School and Chun Chin Bridge. The researchers employed research reviews and field investigations as the research methodology with the research scope of Sung HoVillage in middle Taiwan and disaster of debris flow. The First and the Second branches of Sung Ho River belong to the category of high potentiality of danger of debris flows. The Chichi Earthquake had accumulated sufficient sedimentation of soil and stone. 2004 年敏督利颱風豐沛的雨量，引起台灣山區發生土石流，中橫公路沿線尤其嚴重。從 大甲溪上游的梨山到中游的天冷都有災情；谷關溫泉區至新中橫起點的天冷，大量土石，流 入溪中，造成河道淤積。據水土保持局的資料顯示，7 月2 日松鶴一、二溪爆發土石流，傷 亡各1 人，2 人失蹤，對外聯絡道路台8 省道崩塌中斷。8 月24 日艾莉颱風來襲，溪水暴漲， 沖毀博愛國小、長青橋及民房7 戶【1】。 本文以松鶴為試區，土石流災害為對象，使用文獻探討及現場調查的方法，進行研究。 結果顯示，松鶴一、二溪，均屬於土石流高危險潛勢溪流；肇因於九二一地震的崩塌地，提 供充足的土石堆積物。
討論顯微鏡下的化學反應
由於想了解化學反應的微觀形態，我們設計微型化學反應裝置來比較巨觀(傳統型)與微觀(創新型)化學反應間的差異，並探討其實用及環保方面的問題。在顯微鏡底下，我們觀察化學反應的沉澱結晶及電解反應，嘗試以各項變因(溫度、濃度、聲波…等)來觀察其結晶的型態。我們已成功地將實驗藥品用量減少到一滴(約0.04ml)，並以微觀的角度觀察化學反應的過程。在實驗中，發現反應進行時，粒子會不斷流動，經查證後為愛因斯坦所提出的布朗運動，並且測得硫顆粒的直徑大小約4.2 ~ 6.7 微米。不同聲波所造成硫粒子的移動速率不同，而不同溫度的部份，我們發現→每增加十度硫粒子移動速率增加約兩倍。在面積4.392×10-4cm2 範圍內大約有250~300 顆硫沉澱的粒子。本實驗成功地將顯微鏡應用在化學領域上，若將此實驗推廣，可達到污染少、觀察實驗的時間短、用量少的目標。此實驗是邁向化學微觀世界，一種值得嘗試且創新的方法。In order to compare the differences between the chemical reactions of macroscopic reactor and microscopic reactor, we have designed a device of chemical reaction and researched into the problems of their environmental protections and practical aspects. Under the microscope, we observed not only their precipitating crystal compound from the chemical reaction and electrolytic reation but their types of crystal. We have successfully reduced the dose to one drop ( about 0.04ml) and observed the process of their chemical reaction from the angle of microscopic reactor. During performing the experiment, we found the particles would keep flowing while the reaction was working. It was proved as "Brown motion" introduced by Einstein. The diameter of these particles were around 4.2~6.7μm. We find that different sound waves and temperatures,the motion speeds are quite different. And the movement rate increases about two times as the sulfer particles increase 10℃ each time .Within the measure of area of 4.392×10-4cm2,there are 250~300 sulfer particles.The experiment has successfully used a microscope in the field of chemistry. If we popularize the experiment, we can reach the goal of less pollution, fewer the dose and time-saving observation. It’s an innovation to step to the world of chemical microscope world.