數學

在 Fibonacci Sequence 中，我將Cassini’s identity 轉換成圖形時發現：邊長為Fibonacci number 的正方形，分割後重新拼成長寬分別為Fibonacci number 前後兩項的矩形，會得到矩形內有縫隙（或重疊）。接著我將Cassini’s identity 的圖形推廣到Catalan’s identity 的圖形，我發現邊長一樣的正方形，拼成的矩形長會變大，寬會變小，矩形內的縫隙（或重疊）面積會以Fibonacci number 平方增長。接下來我再將圖形推廣，邊長為非Fibonacci number 的正方形分割拼成矩形時，我發現若將整數遞迴數列代入Cassini’s identity，圖形將會有規律的方式呈現，且每一種數列的縫隙（或重疊）面積會有所不同；若遞迴數列代入Catalan’s identity，縫隙（或重疊）面積還會再以Fibonacci number 平方增長。所以最後我得到一個通式：只要是遞迴數列[an]的圖形，都會滿足於： 。

Methods of nonlinear time series analysis were compiled for use in the analysis of Electroencephalogram (EEG) tracings of children aged three to seven with varying degrees of autism in order to provide a quantitative means of diagnosing autism and determining its severity in a child. After determining the EEG leads to be used for analysis, the identified methods were coded and saved as functions on Scilab. To test the compiled program, a minimum of five EEG readings per cluster of children diagnosed with mild, moderate, severe and no autism will be obtained. The project was able to identify the mean, standard deviation, skewness, kurtosis and other higher order moments, the autocorrelation function, and the Fourier Series as the time-resolved statistical methods to be used for time series analysis. The nonlinear analysis methods identified include the use of the correlation integral, time-delay embedding and the Lorenz equations. One-way ANOVA testing will then be used on the numerical data obtained from the analysis to determine if a significant numerical differentiation has been obtained between the different clusters of EEG. This will provide a definitive way to medically diagnose autism, pinpointing children afflicted with the disorder and giving them proper treatment.\r Two copies of the "Abstract of Exhibit" (in English) should be sent to the National Taiwan Science Education Center or email to fung@mail.ntsec.gov.tw or yuonne@mail.ntsec.gov.tw before December 31, 2009.

給定一個p∈(0,1)，令k0=0,p0在(0,1)間隨機分布,定義 k1為能使的最小正整數k，而;相同的，對於給定的kn-1,kn為能使的最小正整數k，。若存在kn使得，則稱p∈In;若對於所有的n與kn，，則稱p∈I∞。如此區間(0,1)可分解成集合I1,I2,…I∞。

We inferred the original Feynman triangle theorem from equilateral triangle into common triangle and from same ratios of dividing points into different ones. The conclusions are below： 由原始費曼三角形原理中的正三角形等比例分點問題，推廣至一般三角形的分點等比例及不相同之比例，得到如下之結論：

最後留下數字會是多少？該問題在台灣的全國中小學科學展覽出現多次。而資訊界演算法大師Donlad E. Knuth 在其著作The Art of Programing，CONCRETE MATHEMATICS （具體數學），針對該數列作詳細的說明；但是，不論是歷屆全國中小學科學展覽或是大師著作，對於該問題，都只是談及殺1 留β或是殺α留1。本研究利用獨創α分類、n 及k 分類、d 函數、b 函數及循環、n 及y 分類、碎形數列和演變關係，將約瑟夫問題探討範圍提升至殺α(個數)留β(個數)，直到剩下最後1 個數時就不能再殺了，遊戲終止，倒數第k 個留下的自然數是多少？同時，本研究在殺α(個數)留β(個數)下，指定自然數y 為酋長，酋長不能被殺，殺到酋長時遊戲停止，求剩下的自然數有幾個？會發生什麼情形？The Josephus problem refers to what will be remaining when arranging n natural numbers in a circle and starting killing one and leaving the next one alive. The problem has been on display for many times in Taiwan National Primary and High School Science Exhibitions (as shown in Table 1). And, the information algorithm master, Donald E. Knuth has elaborated on the array in his works The Art of Programming, CONCRETE MATHEMATICS. However, both the past science exhibitions and the master’s works are limited to discussions on cases of killing 1 leaving β or killing α and leaving 1. This research employs uniquely created α classification, n and k classifications, d function, b function and loop theory to extend the Josephus problem scope to killing α leaving β to find out what the remaining natural number is by No. k counted recursively. Meanwhile, this research designates natural number y as the chieftain, which can never be killed. The game is over when the chieftain is to be killed. The problem is to work out how many natural numbers are remaining. And what happened?

本研究所探討的主題是一道分割棋盤的題目：\r 給定一個黑白相間的mxn棋盤(設m為縱向邊，n為橫向邊)，將其分割成若干個面積大小都不等的矩形區域，且每個矩形的黑格和白格個數相等。設f(m,n)為符合上述條件的最多分割矩形的個數，則f(m,n)是多少？\r 在研究過程中，我們首先藉由對各棋盤進行分類，利用不等式的運算找出其中的關係，求得當m=n以及mn=k(k+1) 時(其中K為正整數)的f(m,n)值，並構造出其分割方法。而k(k+1)

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.”

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。

Mathematics and music are two poles of human culture. Listening to music we get into the magic world of sounds. Solving problems we are immersed in strict space of numbers and we do not reflect that the world of sounds and space of numbers have been adjoining with each other for a long time. Interrelation of mathematics and music is one of the vital topics. It hasn’t been completely opened and investigated up to now. This is the point why it draws attention of a lot of scientists and mathematicians to itself. This is the point why it draws attention of a lot of scientists and mathematicians to itself. Having considered the value of these two sciences, it seems to us that they are completely non-comparable. In fact can there be a similarity between mathematics – the queen of all sciences, a symbol of wisdom and music – the most abstract kind of art? But if you peer deeply into it you can notice that the worlds of sounds and space of numbers have been adjoining with each other for a long time. In the work I will try to establish the connection between mathematics and music and to find their common elements, to analyze pieces of music with the help of laws and concepts of mathematics to find a secret of mastery of musicians using mathematics and also to investigate the connection of music with mathematics with the “research part”. They are my own calculations and researches which are an integral part of the work. The connection of mathematic and music is caused both historically and internally in spite of the fact that mathematics is the most abstract of sciences and music is the most abstract kind of art. V. Shafutinskiy, I. Matvienko, m. Fadeev, K. Miladze, Dominik the Joker – modern composers of the XXI century – have used the golden proportion only in 4% of their pieces of music and more often in romances or children’s songs. I have revealed this fact after investigating their pieces of music of different genres. However there is a question: why does modern music attracts all of us more but the classics is being forgotten? Investigating connection between mathematics and music I had come to the conclusion that the more deeply the piece of music gives in to the mathematical analysis, to research and submits to any mathematical laws, the more harmonious and fine its sounding is, the more it excites human soul. Besides I am convinced that many important, interesting and entertaining things have not been opened in this field. We can safely continue our research of these things. I think that I have managed to lift a veil over mathematics in music, to find something common for apparently incompatible science and art. In due time English mathematician D. Silvestre called music as mathematics of feelings, and mathematics – as music of intellect. He expressed hope that each of them should receive the end from the part of the other one. In the future he expected the occurrence of a person in which Beethoven and Gauss’ greatness would unite. Terms ‘science’ and ‘art’ practically didn’t differ during far times of antiquity. And though roads of mathematics and music have gone away since then music is penetrated with mathematics and mathematics is full of poetry and music!

我們從日常生活中的酒瓶填塞問題，延伸出圓在相守條件下產生的衍生圖面\r 積極值及相守圓排法等問題，研究過程中我們發現以下的結果：\r 一、 我們可以利用較少個相守圓的排法，以繁殖或增加的方式排出較多個相守圓\r 的衍生圖。\r 二、 當相守圓數量為2個、4個及6個時，相守圓以對角線排列，且衍生圖為正方形\r 時，面積會產生最大值。而當相守圓數量為3 個及5 個時，衍生圖面積最大\r 值會出現在相守圓排成波浪形時，但其夾角並非特殊角。\r 三、當相守圓數量為2個至6個時，相守圓以直線排列，衍生圖面積會產生最小值。\r 四、透過不同層數、個數的研究得知：當相守圓的數量為11個或14個以上時，存\r 在正三角形排法小於直線排法的情形 。而相守圓個數在10個以下時，直線排\r 法面積都叫正三角形排法小。\r 五、我們可以用質單元分割的方式討論更多圓數時，相守圓間的排列方式，使得\r 生多元間的排列變為數字間的加法排列，可以大大降低討論情形的複雜度。\r 六、我們可以用密度的概念，搭配質單元分割的方式，得到不同圓數時，以不同\r 衍生多元搭配波浪形法形成之衍生圖密度最小值。\r 七、我們可以算幾及柯西不等式來驗證特定類型的衍生圖面積極值，也可以列出\r 面積函數來分析不同類型的衍生圖面積極值。