數學

有二系統A和B，A中一開始有2k個物體，，B中有0個物體。在一個單位時間內，兩系統可以互相轉移最多一個物體。當B中物體的個數為 i-1，i∈{1,2,...,k+1}，我們稱其為狀態 i，從狀態1﹝初態﹞開始計時，到達狀態 k+1﹝相同態﹞便即刻停止實驗，經過之時間為一隨機變數T，稱之為實驗時間。問當兩個系統的物體數剛好相等時，經過的實驗時間之分佈為何？本文將以上述問題為核心，分別探討不同條件下系統的實驗時間所反映出來的現象，如機率、期望值、變異數等等。 Define two systems, A includes 2k objects, and B has none. They can transfer at most one object from one system to another in a time unit. When the number of objects in B is i-1, i∈{1,2,...,k+1} , we say the system is at state i. As soon as system transfer form state 1 ( initial state ) to state k+1 ( the same state ), the experiment stop. Random variable T, called the experiment time, is the time before stop. What would be the distribution of the experiment time if all systems have the same amount of objects within? This article will focus on the described question and discuss what property the experiment time of the system under various conditions has, such as probability, mean, and variance.

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!

本研究以b04課程中的巴斯卡三角形為研究對象，將原先巴斯卡以「1」為首、「+」為運算符號的規律三角形，改為以「-1」及「ω 」為首、「×」為運算符號，分別就其產生的新三角形作探討，發現其中似乎隱藏著原先三角形所沒有的規律性。為了更瞭解這種規律，藉由電腦軟體繪出其圖形，圖形顯示出如碎形般的複製關係，不論放大或縮小，其中的遞迴關係並未改變，頗令人好奇，因此著手研究。研究過程中對於圖形的規律性採用先臆測、接著歸納、最後給予證明的方式呈現。得到以下的結論：一、分別以數列呈現新三角形圖形的規律性。二、分別將新三角形中每一列中的某數字（如-1、ω 或ω 2 ）的個數予以通式表之。三、分別推算出新三角形第n 列第j 行的數是「1」或「?1」及「1」或「ω 」或「ω 2 」。四、相同的模式，在特定的圖形範圍中，不斷重複出現。許多研究將巴斯卡三角形中的所有數，以某數為模的餘數紀錄下，去探討其餘數在新產生的巴斯卡三角形中的分布情形；而在碎形的研究中，大部份著重如何畫出碎形。本研究著重圖形其規律性的探討，提供上述研究不同角度的詮釋與探討。 This research subject is based on Pascal’ s triangle in senior high school curriculum. The regular triangle begins with「1」and use「+」as operation. Let 「1」 be replaced with「-1」and「ω 」, the operation sign「+」be changed into「×」. I do research on the new triangle and discover the seemingly hidden regularity which doesn’t exist in the original one. To understand more about this regularity, I draw figures through the computer. The figures show the relationship of reproduction as fractal. Whether the figure is enlarged or minimized, it’s surprising curious the recursive relationship doesn’t change, so we begin to work on research. In the process of the research, we make careful observations, assumptions and deductions about the regularity of the figure. Finally, we come to some conclusions by means of giving proofs:（1）Present the regularity of the new triangle figure with progression.（2） Present such numbers as「-1」, 「ω 」, 「ω 2 」 in each row of the new triangle with formulas separately. （3）Figure out the number in the row n and in the column j of the new triangle is「1」or「-1」,and「1」or「ω 」or「ω 2 」. （4）The same model appears again and again in the specific range of figure. Many researches record Pascal’s triangle modulo certain number to explore the distribution of remainders in the new triangle. In the research of fractal, how to draw fractal is mostly focused on. The exploration of this research emphasizes the regularity of figure, offering the interpretation and exploration of researches above from different angles.

Two soldiers walk on a checkerboard. They can only walk one step once a time and two directions, front and left, are decided randomly. The gunshot is the column and row where a soldier is located, and one will die if he enters the gunshot area of the other. To treat the probability of winning, we first study the cases of 1×n, 2×n, 3×n, and 4×n rectangles iterately. Then we establish a general form of the probability of winning in a general n×k rectangle by using recurrence technique and generating function, respectively. Finally, we extend to the general n×m×k cuboid case to obtain the first soldier’s probability of winning.在一個長方形的棋盤中，兩士兵行走，每一次只走一步，而且上和左兩個方向是隨機的，射程範圍是所在的此行和此列，而進入他人射程範圍則死亡。探討其獲勝機率，從1×n 、2×n、3×n、4×n 矩形的情形逐步研究，並分別運用遞迴式的技巧及生成函數，導出 n×k 矩形中先走士兵獲勝機率的一般式。更進一步地，我們也獲得了n×m×k 立體空間先走士兵的獲勝機率。

本研究報告以Brocard Point 為核心，所用到的性質均先證明，以確認其正確性，並推演出一些其他的性質，藉由這些性質導出幾何不等式。內容可概分為四部份：(1)以Brocard Angle 及已知的或推演出的基本性質，導出一些不等式。(2)結合「法格乃諾問題」、「費馬點」、「尤拉公式」導出幾個幾何不等式。尤其是三角形邊長與面積，外接、內切圓半徑與邊長間的不等關係，頗為有趣。(3)以向量為工具，分別計算內、重、垂心與Brocard Point 間的距離，並導出邊長的不等關係。其中由內心及重心所導出的不等式，清楚俐落；垂心所導出的不等式則較為複雜。(4)以Brocard Cirle 與內、重心間的關係，導出一系列的不等式。其中Weitgenberk 不等式的無意發現，令我們印象深刻。The Discovery of Geometry Inequalities by Brocard Point This paper takes Brocard Point as a core. We proved some properties about Brocard geometry to confirm its accuracy, and deduce some other properties, and then derive some geometry inequalities by these properties. The content may divide into four parts: a) Derives geometry inequality by Brocard Angle, Crux Mathematicorum and properties which known or deduced. b) Unifies "Fagnano problem", "Fermat Point", "Euler formula" to derive several geometry inequalities. In particular the inequalities between triangle area and length of side, or circumradius inradius and the length of side, is quite interesting. c) Derives geometry inequalities about length of sides in triangle by the distances between incenter centroid circumcenter and Brocard Point. Especially, these inequalities were elegant which derived by incenter and centroid, but it was complicated derived by orthocenter. d) According to the relation about incenter centroid and Brocard Circle derives a series of inequalities. Discover Weitgenberk inequality makes us excited.

Cultures of bacteria were analyzed using fractal geometry and statistics to provide a method for predicting organism growth, paving the way for a better design of treatment drugs. Images of three cultures of isolated Bacillus subtilis were taken at time intervals of two to three hours for eight days. The images were processed using the IDOLON program and quantitatively described using three statistical formulas: fractal dimension D, Renyi dimension and Hausdorff-Besicovitch dimension. The three variables were integrated to compute the maximum of the distribution and were used as coordinates for a 3-dimensional graph f. A 2-dimensional graph g containing the maximum of a distribution under time analysis was also constructed. Topological properties of the graphs, including slope, direction and area were used to determine the interrelationship of the three fractal values. The two graphs, described as φ - : X -? P1 where X is the smooth algebraic assimilation of the four variables under time analysis, was extended using Java. A computer-aided prediction model of the graphs f and g were made which combined the topology of f and g at infinity. The computed fractal values showed the existence of a fractal pattern in the growth of Bacillus subtilis with fractal dimension ranging from 0.900 to 4.000, indicating a linear iteration. This was supported by the values of the Renyi dimension, which showed a horizontal growth pattern of the bacterial cultures, establishing the growth of the bacteria to be inclined to go towards the North East direction. There was consistency in the computed fractal values, maximum of distribution and topographical computations of all three cultures which also indicated the existence of a pattern of growth which could be extended to tinfinity, thereby allowing prediction of the direction and rate of growth of the bacterial colonies. The fractal patterns in the growth of bacteria, in this case Bacillus subtilis, yielded the direction and rate of growth of the bacteria as shown by the analysis of the fractal patterns and statistical values, showing that the growth of harmful organisms can therefore be predicted, making it possible to improve on the design of drugs for the control of perilous cells. By preventing the growth of insidious cells, the potential effects of virulent organisms may be avoided, and treatment may be made more possible.

本文由‘‘分數7/17是否能表示成兩個相異的埃及分數之和’’這個問題出發，藉由簡單數論的性質以及反證法，得到一個真分數可表示成兩個相異埃及分數之和的定理檢驗法(定理1)。有了這個基礎，我們進ㄧ步推廣定理1 的結果，做出了嶄新的結果(定理2、定理3) 。此定理分別可以用來檢驗真分數表示成三個、四個相異埃及分數之和的存在性; 至於將真分數表示為5 項、6 項….k 項相異埃及分數之和的部分尚在嘗試。利用定理1、2，我們寫了兩個Matlab 軟體工具的電腦程式，使得我們可以檢驗任意真分數是否可以表示成兩項及三項的和，並可把所有的解列出來; 最後我們研究的是一個有關埃及分數的猜想(Erdos-Strauss Conjecture)問題，當分子為4，且分母為4k、4k+2、4k+3 時，猜想皆成立。對於分母為4k+1 而言，當k 為3r+1、3r+2 猜想亦成立，k=3r 且r 為奇數時也是成立的，因此目前需解決的問題只剩分母為24t+1 的情況了。值得一提的是，我們用Matlab 的程式檢驗出當分母為1014 至1014 +240000 之內的正整數時，猜想都是成立的，這已經超越了已知文獻的結果。This paper begins with the question: ‘‘Is 7/17 able to be the sum of two different Egyptian fractions?’’ to discuss the problem of Egyptian fractions. According to the complete division properties and the counter-evidence method, we get a back-check theorem which is about a true fraction can be the sum of two different Egyptian fractions (see theorem 1). Using the same method we obtain a new back-check theorem that is a fraction can be the sum of three or four different Egyptian fractions (thereom2, thereom3). Similarly, we can follow the same procedure to get the rule that a fraction can be the sum of five or six …or even more different Egyptian fractions. By the theorem1 and 2, we propose two programs written vie the Matlab software to examine that any true fraction can be the sum of two items and three items or not. Finally we focus on the Erdos-Straus Conjecture, which related about true fractions can be divided by three different Egyptian fractions. The conjecture is when the denominator is 4k, 4k+2, or 4k+3, the problem mentioned above can be solved. As for the denominator is 4k+1, then the conjecture also can be solved, as k equals to 3r+1 or 3r+2. Also, k being 3r and r is an odd number, the conjecture is satisfied. As for the case of r equals to even number, the problem has not been solved. But it is worth to mention here that we use Matlab software to examine the conjecture is agreeable as the denominator is between 1014to 1014+ 240000. This is beyond the results from the literatures.

本文我們研究下列問題： 1. 何時心律會成穩定的狀態呢？ 2. 是否能建立出心律穩定的數學模型？ 3. 什麼樣的函數會使得心律趨於穩定？我們以xmp+i 表示第p+1 個訊息傳到第i 個細胞之前，第i 個細胞的舒張時間，而且xk 和 xk−m, xk−m+1,..., xk−1的關係為 In this paper we study the problems as follows: When will the rhythm of heart beats approach to a steady state? Can we set up the mathematics model with steady rhythm of heart beats? What kind of function will make the rhythm of the heart beating tend to be stable? The result of our study is as follows:

This study was to explore the nature of two basic constitutes of the regular pentagon,With these two constitutes, the regular pentagon could be multiplied into any times in size. We used four multiplication methodsto show how the regular pentagon enlarge and to verify that the enlarged regular pentagons derived from computer did exist. By integrating these four multiplication rules, we were able to arrange regular pentagon of any length of side, and evidenced the equation was ( If m,n is the number of A,B of a regular pentagon respectively ) When we tried to verify if any regular pentagon could be constituted by other smaller regular pentagons, we found that it was un-dividable only if the length of pentagon side were (the number of A, B were the 2n and 2n-1 item of Lucas Sequence), otherwise, any regular pentagon is able to be constituted by other smaller regular pentagons. The divided forms could be multiple. We also found that any pentagon could be divided by two successive un-dividable pentagons, which is called “standard division rule”. We expected to derive all kinds of division by analysis of two successive un-dividable pentagons in standard division rule. 這個研究起源於一個拼圖玩具：利用兩種黃金三角形排出指定大小的正五邊形。我們的研究動機是：一、 假如無限量供應A 和B，能夠拼出哪些邊長的正五邊形？二、 哪些拼好的正五邊形不能拆成一些較小的正五邊形？我們將研究的主要結果分述如下：