數學

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.

將一個集合{1,2,...,n}分成數個非空的集合（組，區塊），稱為此集合的一個分割。如果可以找到1 ≦ a 已知無孤立點無交錯分割以Riordan 數{rn}n≥0 =1,0,1,1,3,6,15,36,... 來計數。在這篇文章中我們研究無孤立點無交錯分割的一些性質。 首先我們考慮無孤立點的無交錯分割按區塊的細分。我們得出：集合{1,2,...,n}恰含k個區塊的無孤立點的無交錯分割的個數為： 其次，我們證明bn,k和多邊形的剖分有令人訝異的關連。令dn,k是用不相交對角線將凸n 邊形分成k 塊的方法。我們用代數方法證出 bn,k = dn+2−k ,k，也給了一個新的組合證明。 最後，透過對應的方法，我們找出了七個嶄新的組合結構，這些結構都是以Riordan 數來計數。 Partition the set {1,2,...,n} into several nonempty sets (blocks) and call it a partition. If there exists 1 ≦ a It is known that the nonsingleton noncrossing partitions are counted by Riordan numbers {rn}n≥0 =1,0,1,1,3,6,15,36,... In this paper we study the properties of them. First we consider the enumeration of nonsingleton noncrossing partitions in respect to the blocks. We prove that the number of nonsingleton noncrossing partitions of {1,2,...,n} with k blocks is Then we give a connection between nonsingleton noncrossing partitions and polygon dissections. Let dn,k be the ways to dissect an n –gon with noncrossing diagonals. We prove that bn,k = dn+2−k ,k We also give a combinatorial proof. Furthermore, by way of the technic of bijection, we find 7 new combinatorial structures counted by Riordan numbers.

一 Motivation and Purpose: In this study, we want to completely know about “The number abc…de, which times m/n, 1≦n≦m≦9?N can get ed… cba?”, and also expect to find out “The good rule within them”. 二 Procedure:Using method of enumeration, induction to collect sample of all and beginning from two digits to get information “good rule”. When get some useful idea, put them into the following research for the step easy go on, the method try and error is a very tiresome works, especially when we deal higher digits. till enough information is obtained, we solve problem and find new one, then likewise again research steps, just the basic science research ways, we are glad have the key of these problem. 三 Result and conclusion :Those number we named “converse No.” There are two groups: S=m+n=10 and 11 S=11, then Q=m/n=9/2,8/3,7/4,6/5=4.5,2.6,1.75,1.2 S=10, then Q=m/n=9/1,8/2,7/3,6/4=9,4,2.3,1.5 Each group have four type. When S=11,Q=7/4=1.75,if converse No.each digit is a multiple of 3, then can cancellation or extension of fraction to get another 3 or 4. Growth up rule: Converse No. = type factor x heritable factor x growth factor=rx hx g S=11,r=2~5,h=9, s=10, r=1~4, h=99 一 研究目的：盼能找出”顛倒一族”的族譜。二 研究過程：確定研究題目為ab…cde×m/n=edc...ba,0≦n≦m≦9?N 求ab… cde？以窮舉法收集觀察資料，歸納演繹尋求規律。1.先觀察兩位數，分析共有顛倒對36對。2.建立乘數Q=m/n一覽表，共有27個3.設計顛倒對大/小及其商一覽表，以利觀察、歸納獲得規律。4.接著觀察三位數，共有360對，綜合二、三位數規律，找出選擇式窮舉法：9之倍數法。5.再接著找出四位數，再綜合而知另有 全調法 重現法 半調法 GCD遺傳基因法等來繁衍高位數顛倒數。6.於是依諸法找得六位數資料，得知GCD遺傳基因法為繁衍通則，完成族譜建立模式。7.研究顛倒數位數與其個數間關係式，完成研究。研究結論：1.顛倒一族有兩大類：S=10與S=11 S=m+n。2.每一大類有四型： S=10中，Q =9/1,8/2,7/3,6/4(9,4,2.3,1.5)S=11中，Q =9/2,8/3,7/4,6/5(4.5,2.6,1.75,1.2)3.每一型均有一個顛倒數，除了S=11中，Q=7/4=1.75者可約、擴分而得3or4個。4.顛倒數原則上均為9之倍數，除了Q=7/4經約、擴分可能得非9倍數者。

Motivated by Napoleon theorem, we study the properties of the triangles obtained by moving the midpoint of each side of a given trianle along the perpendicular bisector of corresponding sides, and extend the results to the case of quadrilaterals. On the other hand ,we consider the method of erecting a regular M-gon to each side of a given N-gon and joint the N centers of these M-gons to form a new N-gon. (abbreviated as CRG method),and get the following results. 1. We characterize some kinds of N-gons that can be transformed to regular N-gons via CRG method. 2. Of M,N are nature numbers with M|N, then it is possible to find a N-gon that can be transformed to a regular N-gon by CRG method. \r 3. If a polygon P is symmetric with respect to a fixed point or a fixed line, then P can be transformed by CRG to a polygon with similar symmetries. 4. If a polygon P is transformed by CRG to ′P,there exists a commonpoint G such that ΣGA=0 andΣGB=0, where A and B runs through vertices of and P′P, respectively. 本研究將拿破崙定理加以延伸。先探討由各邊中點沿中垂線延伸得出之三角形的性質並推廣至四邊形之情形條列式報告成果。另一個推廣是將給定的多邊形的每邊外接一個正多邊形，再以這些外接的正多邊形的中心為頂點造出一個新的多邊形。我們發現此幾何變換具有以下性質：（1） 「哪些多邊形能被變換成正多邊形呢?」，我們觀察出能被變換成正多邊形的多邊形其限制條件隨邊數增加而增多，並進一步區分了哪些多邊形可以被變換成正多邊形。 （2） 在將非正N邊形做變換時，不一定須外接正N邊形才能得到正N邊形，我們區分出可外接哪些正多邊形而得到正多邊形。 （3） 對給定的多邊形作此變換時，若原多邊形有點對稱或線對稱等性質，則新多邊形也將具有相同的性質。 （4） 此變換得到的新多邊形會與原多邊形共重心，亦即新舊兩多邊形內到各自的頂點向量和為0的點會是同一點。

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.

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!

在實驗中學2007 年校內科展，參展作品《三角形中的切圓》的研究中，研究三角形內的切圓時，發現連續切圓的圓心與拋物線的軌跡有關。於是去查資料，在偶然的情況下，翻閱《平面幾何中的小花》時，接觸了「六圓定理」。因為覺得這問題非常有趣，於是便著手證明（見報告內文）。 又發現，當移動六個圓中的起始圓時，總是在某種情況下，六個圓會重合成三個圓。繼續研究其重合的狀況，發現了馬爾法蒂問題（Malfatti's Problem）的一種代數解法。 當我試著推廣六圓定理至多邊形時，發現奇數邊的多邊形似乎也有如六圓定理般圓循環的狀況，於是著手證明，但目前尚未證明成功。而偶數邊的多邊形則無類似的結果。 ;In 2007 National Experimental High School Science Exhibition, one of the exhibit works, "Inscribed Circles in Triangles", shows that the centers of the consecutive inscribed circles has something to do with the parabola's trajectory. To learn more about inscribed circles and parabolas, I referred to literature. By accident, I am faced with the problem on six circles theorem, in the book The Small Flower of Plane Geometry(平面幾何中的小花). Out of my interest in this problem, I tried to prove it. The other results are as follows: With the initial circle of six circles moved, in certain circumstances, the six circles merge into three. Further in studying this coincidence leads to an algebraic method to solve the Malfatti's Problem. Applying six circles theorem to the odd-number-sided polygons exists the same characteristic. It indicates that the inscribed circles will form a cycle. However, it hasn’t been successfully proven. The even-number-sided polygons show no similar results.

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.

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.