外觀數列
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。
Nonlinear Time Series Analysis of Electroencephalogram Tracings of Children with Autism
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.
別鬧了,辛普森先生
We investigate the machinery producing successive Simpson’s paradoxical reverse. Taking advantage of algebraic and geometric techniques, we obtain the following results. Take playing baseball for example. In our study, we find that Simpson’s paradox only occurs when the hitter’s hits over 3 times in one game. Set n equal to the times I will hit in one game. If my batting average in each game is at least(n ?1)/2 times higher than the others’; then I am sure that my total batting average would not be invert by the others. In order to find how many the lattice points in the triangle, we use Pick’s formula. But sometimes, the Pick’s formula is not appropriate to triangles whose vertex are not all lattice points. So we develop New Pick’s formula to estimate the number of lattice points in such kind of triangles. Besides, we also find an iterative algorithm to produce successive “Simpson reverse” phenomenon by using C++ language, and we can therefore produce as many “Simpson’s set of four sequences” terms as we like(not beyond the computers’ upper limit).Moreover, if both sequences of ratios converge, then they must have the same limit.我們探討了一般人乍看之下顯得頗弔詭的辛普森詭論。我們配合GSP 作圖,用解析幾何、設立直角座標系和C++ 程式的運算,找出在特殊情況下或一般情況下所產生的辛普森數列組和特殊的性質,並且以棒球場上的打擊率為例子來做印證。通常一場棒球賽中,每個人平均上場3 次~4 次,經過我們的討論,發現要發生逆轉的機會只有在上場達到4 次或以上時才會發生。?了求出在直角座標系中可以滿足的格子點個數,我們用了Pick公式,但?了更準確的估計,我們引進了虛點的概念,重新推導出了新Pick 公式。另外,我們還發現,假設兩個人上場比賽,若打了2 場,且每場最多上場打擊K 次,其中的一個人的打擊率只要是另一個人的(k-1)/2倍以上就保證不會被逆轉。我們又找到了連續產生辛普森逆轉的演算法,利用C++ 寫出程式,經由演算法和遞迴式,製造出項數可任意多(只要電腦能夠承受)的辛普森數列組,且我們發現若兩個比值數列接收斂,則極限趨近於同一個數值。
由Brocard Point 發現幾何不等式
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.
立體尺規作圖-PES 作球
In this study, we mainly explore the geometric construction in 3D. By conducting some problems about constructing circles, we define the PLC construction in 2D as constructing a circle, either passing through a given point (P), tangent to a given line (L) or tangent to a given circle (C). Besides, we aim to discuss the properties of the PLC construction and the relations between each other. We discover if we find a plane satisfying certain conditions in space, the properties in the PLC construction can apply to such a plane. Furthermore, we extend the properties in PLC to the PES construction in 3D, defined as constructing a sphere, either passing through a given point (P), tangent to a given plane (E) or tangent to a given sphere (S). Also we discuss the relations among them.這個研究主要在探討3D 的尺規實作。藉由歸納某些有關作圓的題目,我們定義2D 中的PLC作圖─作圓,過已知點(P)、切已知線(L)、切已知圓(C)。並探討PLC 作圖的性質及彼此的關聯性。而我們發現:在空間中只要找到滿足特定條件的平面,則2D 幾何作圖性質在該平面仍能沿用。此外,運用PLC 作圖性質,我們進一步推廣到空間中的PES 作圖─作球,過已知點(P)、切已知面(E)、切已知球(S),並探討各個類型間的關聯性。
調和變換之研討與應用
在此研究中,我們用類似反演變換的方法,以一個定圓創立並證明了一種新的幾何變換,稱為 「調和變換」 · 我們得到點、直線、圓與圓錐曲線經過變換的關係 ·。1 .直線可以映射成原直線或一圓錐曲線 · 2.圓可以映射成一種特殊曲線。 3 .圓錐曲線可以映射成兩條圓錐曲線或一條圓錐曲線和一直線。此外我們還發現調和變換和反演變換的特殊關係 · 最後,由於調和變換可以簡化圓錐曲線的關係,我們將調和變換應用在行星輾些的證明上,並得到了良好的結果。In this research, we use a method similar to the inversion to establish a new geometric transformation, called harmonic transformation, by a fixed circle O, we prove some of its properties. We have gotten the relationship among points. lines, circs, conies and their images: 1 .The image of a line is a conic or a line itself. 2.Thc image of a circle is a special category of curve. 3.The image of a conic with its focus at the center of O is two conies or a line and a conic. Further mote, we also find the special connection between harmonic transformation and inversion. Finally, since the harmonic transformation can simplify the conic, we apply the harmonic transformation to identify the orbit of a planet, and obtain a nice conclusion.
摺紙數列-相關問題探討
1. 遊戲規則:將1~ 2m × 2n的連續正整數,由上而下、由左而右依序填入 2m × 2n的方格內。操作規則允許將2m × 2n做往右或往左或往上或往下的完全對摺,直到操作至所有單位方格均疊成一行,此同時有數字也由上而下形成一數列。2. 本研究即是探討操作完成的數列之數量與數字間的關連性。3. 我們發現:(1) 數列之數量與巴斯卡三角形有關。(2) 形成的數列必符合內文的 [ R(L) 性質]、 [ D(U) 性質]、[ R&D 性質]、[D&R 性質]。
1. Rules of thegame: Fill in order the continuous positive integers 1~ 2m × 2n, from top to bottom and from left to right in the 2m × 2n check. The operational rule allows a complete fold of 2m × 2n either rightward or leftward, or upward or downward, until all the check units pile up in a line. At the same time, all the integers form a series from top to bottom. 2. This study explores the relationship between the number of the series and the integers after the operation. 3. Our findings are: (1) The number of the series is related to Pascal triangles. (2) The series formed meet the properties mentioned in the study: [the property of R(L)], [the property of D(U)], [the property of R & D], and [the property of D & R].
殊途同歸-格子點平面最短路徑和之探討
本研究從理想城鎮(Ideal City)街道開始,討論平面上相異n 點到某一點的最短距離和。經研究後發現:當n 為偶數時,則到相異n 點的最短距離和所形成的區域可能是一個點、一個線段或是一個矩形;當n 為奇數時,則相異n 點的最短距離和所形成的區域將會退化成一個點。此外,本研究將理想城鎮的街道換成正三角形的街道幾何平面,同樣是討論平面上相異n 點到某一點的最短距離和。經研究後發現:當n 為偶數時,則相異n 點的最短距離和所形成的區域可能為一個點、一個線段、一個四邊形、一個五邊形及一個六邊形;當n 為奇數時,相異n 點的最短距離和所形成的區域則可能為點、三角形的情況。假使考量各點重要性的比重,分別加權後再求最小點。研究發現無論在理想城鎮或正三角形幾何平面上,皆可將各點視為多個權數相同之點重疊於此點上,便可利用先前的方式求得最小點區域。透過這次的研究,可以利用n 個相異點到某一點的最短距離和實際應用在貨物運送的問題或是消防設施配置等問題。The present study was intended to start with the Ideal City and proceed to discuss the sum of the shortest distance between a point and n different points on a plane. After the discussion, it was found that if n is even, the formed region could be a point, a line segment, or a rectangle. If n is odd, then the formed region must be a mere point. Further, the current study transformed the Ideal City into the geometric plane of an equilateral triangle. Similar to the previous discussion, if n is even, the formed region could be a point, a line segment, a quadrangle, a pentagon, or a hexagon. On the other hand, if n is odd, then the formed region could be a point, or a triangle. The result of this study, which investigated the sum of the shortest distance of a certain point to n different points can be applied to the real life situation, such as transporting goods or distributing fire control facilities.