數學

在一個有障礙的平面上，給三個定點，我們探討連接此三點的最佳網絡。我們討論了諸如直線、射線、線段、圓、網格狀、三角形……等類的障礙，當網絡每穿越障礙一次，就必須付出代價，例如「拖延5 分鐘」。所以，設網絡穿越障礙的次數為y ，則網絡除了原本的總長度之外，還額外加入y 倍某固定數值的損耗。我們以費瑪點的各種性質及三角形不等式等方法為工具，就不同的穿越障礙次數綜合比較，而找出最佳網絡。在某些情況下，最佳網絡不是以費瑪點來連接三點，而是在障礙(如：直線)上找出符合某種與餘弦值相關特殊性質的點，以該點來連接三點，而此網絡可用GSP 軟體相當精確地作出。另外，我們也探討在考慮障礙造成損耗的情況下，兩點間的「實際距離」為何。 最後，我們考慮「混合障礙」問題。在此類問題中，除了前面所討論的障礙，還另加了如同「河流」的兩平行直線間區域之障礙，在這種障礙區域中，網絡的長度要乘以數倍來計算。我們發現，此類問題的最佳網絡也可用特定的正弦條件配合GSP 而相當精確地作出來。;Considering various kinds of obstacles in a plane, such as a line, a segment, a ray, a circle, a triangle or chessboard grids, which function like a red light, we research into the problem of finding the optimal network connecting three given points A, B, C in the plane amidst obstacles described above. Each time when the network crosses an obstacle, it will cause losses, such as five minute’s delay or a loss of one hundred dollars. Taking advantage of Fermat points, some basic inequalities concerning triangles and some special qualities about sine or cosine functions, we obtain the optimal networks in different situations. Besides, we consider what the “real distance” between two points is when there are obstacles in a plane. We also put another obstacle, including a line and a weighted region between two parallel lines, into consideration. In the region, like a river or a muddy ground in real life, the length of the network should be multiplied by a fixed time. Furthermore, we can use GSP to make the networks very accurately.

令Sn 為{1,2,…,n}任意排列所成的集合，π ? Sn 為其中的一個元素，我們記π = (π(1), π(2),…, π(n))。今給定π ? Sn ，若對所有i,1? i ? n，都有π (π (i)) = i 時，我們稱π 為involution。假設π ? Sn ，並給定σ ? Sm (m ? n)，當π 中任取m 項，其大小關係的順序都和σ 不同，我們稱π 避開σ，或稱π 是一個σ-avoiding 排列。在這篇報告中，我們主要分析了2143-avoiding involution，1234-avoiding involution，和3412-avoiding involution 中的一些統計量，給出了十數個結果與幾個猜想。Let Sn be the set of permutations on {1,2,…,n} and π ? Sn be an element in Sn. Denote π as π = (π(1), π(2),…, π(n)). We say that π is an involution if π(π(i)) = i for every i, 1? i ? n. Given π ? Sn and σ ? Sm (m ? n) , we say that π avoids σ (or π is an σ-avoiding permutation) if π does not contain any m-term subsequence in the order of σ. In this paper, we discuss some classic statistics on 2143-avoiding involutions, 1234-avoiding involutions and 3412-avoiding involutions. We get many new results in this field and give some interesting conjectures.

Starting from the problem in AMC competition of Australia, we try to find out the locus and its length when a point in a regular polygon rolls in a circle. The result is that the locus has a wonderful and regular cycle.Next, we discuss the regularity of the cycle when a regular polygon（n sides） rolls in another regular polygon. Furthermore,we discuss the the equation of the locus by changing the radius and the angle of rolling. we find out the argument function of the locus of a point inside when a a regular polygon（n sides）rolls in another regular polygon （m sides）： ， Aj is the summits of the regular polygon（m sides）, Bjcorresponds Aj when a point inside the regular polygon (n sides) rolls, ) And then, we do some moving simulation with some computer math software, such as Cabri Geometry、Mupad, etc. We discuss the regularity of the locus and its equation of a point inside when some special cycloids, like asteroids, cardioids, etc, roll in a certain condition. Moreover, with the result of research 2, we create the “plate＂ and apply for a patent on it. We hope to study math by playing games. 從澳洲AMC 競賽題出發，嘗試探討一正n 邊形中的一點在單位圓內滾動軌跡及其軌跡長度，發現該軌跡均會產生奇妙的循環規律。 接下來，推廣探討正n 邊形在其他正多邊形中滾動時循環的規律，並利用旋轉半徑及角度之間的變化深入探討其滾動軌跡方程式，發現正n 邊形繞正m 邊形滾動時其內部一點軌跡參數式為，其中， Aj 為 正m 邊形之各頂點、Bj 為正n 邊形中內部一點旋轉時對應 Aj 之點，。 進一步想嘗試使用數學電腦軟體如：Cabri Geometry、Mupad 等對以上研究去做一些動態模擬，並再探討一些特殊擺線如：星狀線、心臟線…等，在條件下相切滾動時，圖中某一點的軌跡規律性及其方程式。另外，應用研究二中的結果，創造出寓數學於遊戲的「圖形板」，並申請了新型專利。

We understood the definition and meaning of spider number by reading〝Wonders of Numbers〞. It interested us so much. So, we took further step to study the situation of extreme value when the gap sometimes lie on the line and sometimes on the circle or even on both. That is to say, we explored the relation between spider number and the gap when the spider number is maximum or minimum. New research for the application of spider number involves several directions. First, we design a new game called〝Stepping Land Mine〞with the rule of spider number. Give you a net with several hidden gaps, trying to find the right positions of gaps. Second is the further result for a different type of net about regular n-polygon. Third is a tactic for a net with destroying of the strategy points. In this situation, the gaps amount on the circle and on the line are fixed. At the same time, consider the situation of circles and lines designing the tactic of placing the gaps to attain the maximum of the destructive effect. 在本文中我們定義一個蜘蛛網上的蜘蛛數，若在蜘蛛網中加入缺口後，會影響蜘蛛數的大小。我們探討蜘蛛網上的缺口，該如何分配才能夠得到蜘蛛數的極值(最大值及最小值)。先觀察一直線和圓上缺口如何放置蜘蛛數有極值，再探討許多條直線及圓上的情況，進而推展至許多同心圓及通過圓心的許多條放射線的缺口，該如何放置，蜘蛛數才會有極值發生。

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:

在拋物線上置掛正三角形看似簡單，其實不然。本篇文章研究在二次函數的各種不同情況下，可做正三角形的分佈以及其個數。 1. 在一條拋物線上時，最多只能作正三角形。 4. 在三條對稱軸相等的拋物線和共頂點開口大小不同之拋物線上，本篇文章證明一定能找出正三角形落在它們之上。但由於最多有四個分界點，要解四次方乘組過於繁複，於是本篇文章對分界點作了一些估計，找出了分界點的極限值。 5. 本篇文章證明了對於給定的正n 邊形，存在一1 元n-1 次方程式可以通過它所有頂點。 Building a regular triangle on a parabolic curve looks easy . In fact , it doesn’t . This Article researches regular triangles distributions and its numbers in different conditions. 1. On one parabolic curve can only build regular triangles , squares and other regular polygons can’t be built. 4. For three parabolic curves which has same symmetrical axis or three concurrent parabolic curves, we prove that it can build at least one regular triangle on them .But because it can have at most 4 boundary points, to solve quartic equation is to complicated. So we do some estimation of boundary points, and find out some limits. 5. This Article prove that for given regular polygons , there exists a one dimension n-1 orders equation can pass all its apexes.

給定下面範例：\r 058823529411764705882 +235294117647058823532\r =0588235294117647058823529411764705882353，\r 其等式結果與質數17 的倒數結果(1/17)有某種關聯(卻沒有一個決定性的證據)，意即\r 1/17=0.0588235294117647=\r 0.058823529411764705882352941176470588235294117647...... ( Len(17) =16 )\r \r 曾經在下列網站上發現過幾組數字(挑戰試題)，引起我們極大的興趣。\r http://www.domino.research.ibm.com/Comm/wwwr_pondernsf/challenges/March2000.html\r http://www.math.smsu.edu/~les/POW08_96.html\r \r \r The two examples that I have are 0588 2+23532=05882353 and 058823529411764705882+23529411764705882353 2=0588235294117647058823529411764705882353 These were found by the Canadian professor Alf van der Poorten, and he gave a talk on these identities in December at the west coast number theory conference. He was unspecific as to exactly where these identities were coming from, but they are connected with reciprocals of primes:1/17 = 0.0588235294117647= 0.058823529411764705882352941176470588235294117647 ΛΛ ( Len(17) = 16 ) Though not mentioning how to obtain these equations, Prof. Poorten demonstrated the relationship between the above examples and the reciprocal of the prime numbers 17 (1/17 ) without a definitive proof.

本專題的目的是研究以任意實數 a1 、 a2 、 a3 為起始的M&m Sequences 之穩定性質。我們主要關心的問題是：(1) 是否任給定三數a1 、 a2 、 a3 為起始的M&m 數列皆會穩定？(2) 若上述的M&m 數列穩定，則其穩定的長度與a1 、 a2 、 a3的關係為何？(3) 其穩定的值與a1 、 a2 、 a3的關係為何？我們研究的主要步驟及結果如下︰1. 當1 2 3 a 1) 為起始的M&m 數列。3. 我們證明了下列性質：(1) 若M&m 數列中前n 項所成數列的中位數為n m ，則下式成立： (2) 當存在 k > 4 , k ? N ，使得 ?1 ?2 = k k m m 成立時，則此數列穩定，且穩定長度p 滿足：min{ | 4 } ?1 ?2 = > = k k p k k 且m m ，其中p 必為奇數。(3) { n m }為單調遞增且, 5 1 ? ? ? a m n n n4. 如果x ? 41.625，則{?x,1, x}為起始的M&m 數列，其對應的數列有相同的大小次序且此M&m 數列會穩定，穩定值為41.625，且穩定長度為73。5. 我們觀察發現：如果x 1). 3. We prove the following properties: (1) If the median of the former n numbers of the M&m sequence is n m , we obtain (2) There exist k > 4 , k ? N such that ?1 ?2 = k k m m , then the sequence is stable and the stable length min{ | 4 }?1 ?2 = > = k k p k k and m m , where p must be an odd number. (3) { n m } is monotone increasing and , 5 1 ? ? ? a m n n n . 4. Suppose x ? 41.625, then the all M&m Sequences beginning with –x , 1 , x are the same, and the sequences will be stable, the stable value is 41.625 and the stable length is 73. 5. By the computer experiments, we observe that if x is any positive real number less than 41.625, the M&m Sequence starting with –x, 1, x, will be also stable but does not appear to follow any clearly discernible pattern of behavior. However, the stable lengths are much variant and exist some unknown relation with point format of x. Moreover, we have the following properties: (1)If x is a node, then the stable value is x and the stable length equals to the index of median of the node + 2； (2)Near the branch of 41.625, the stable length is almost a constant except at the edge area，the stable length of (-x,1,x) as x around branch 1 is chaos； (3)If x near the node (K= 3, 5, 7, …, 67, 69), then the stable length is l(K)+K?1 where the positive integral l(K) is determined by Prop1 (see Table 6 and 7).

本次科展我們所探討的，是關於圓的分割問題，討論披薩被切出來的面積和及切痕的長度和等相關問題。 我們提出並且用較初等的數學方法證明了幾個與圓的分割有關的問題，包括披薩定理以及另外七個定理。 最後我們也利用GSP軟體驗證定理二、三、四分給2人時的對偶結果和定理二分給3人以上時的其他結果，並對於一般情形的推廣，作合理的猜測。

完全圖Kn是指一個圖中有n個點，且任意一個點都跟其它的點有邊相連。兩個圖G和H的卡氏乘積G□H的點集V(G□H)＝{(g,h)| g∈V(G),h∈V(H)}，兩個點(g1,h1)和(g2,h2)有邊相連若且為若g1=g2 且h1~h2，或g1~g2且h1=h2。 三個完全圖Ka、Kb、Kc 的立方乘積是指Ka□Kb□Kc。一個圖G中的一點v所連的其它點稱為這個點v的鄰居，也就是N(v)={x | x~v}。一個點集S中的所有點的鄰居的聯集稱為這個點集的鄰居，也就是N(S)=∪v∈S N(v)。如果一個點集S和它的鄰居N(S)包含了一個圖G的所有的點，也就是S∪N(S)＝V(G)稱這個點集S是這個圖G的一個控制集。我們把圖G的所有控制集中點數最少的稱為最小控制集，並定最小控制集的點數為最小控制數γ(G)，也就是γ(G)＝min { | S |, S是G的控制}。 本文的目的在於研究完全圖立方乘積的最小控制，也就是要給γ(Ka□Kb□Kc)一個上界。特別當 a = b = c = n時，γ(Ka□Kb□Kc) = 。 A complete graph Kn is a graph with n vertices, which any vertex is adjacency to every other vertices. The Cartesian product of two graph G and H which is denoted G□H is define as follow: the vertex set V(G□H)＝{(g,h)| g∈V(G),h∈V(H)}，and two vertices (g1,h1) and (g2,h2) is adjacent if and only if g1=g2 and h1~h2 or g1~g2 and h1=h2. The Cartesian product of three complete graph Ka,Kb,Kc is Ka□Kb□Kc,which is the same with (Ka□Kb)□Kc. In a graph G, the neighbor of a vertex v N(v) is the set of the vertices adjacent to the vertex v, that is N(v)={x | x~v}。 The neighbor of a vertex set S is N(S), which is the union of the neighbors of vertex v over S, that is N(S)=∪v∈SN(v). For a graph G, if a vertex set S unions its neighbor N(S) equal to the vertex set of G, that is S∪N(S)＝V(G), we say that S is a dominating set of G. The domination number of a graph G will be denoted as γ(G), which is the minimum size of all dominating set of G.. We give an upper bound to γ(Ka□Kb□Kc). And when a=b=c, γ(Ka□Kb□Kc) ≦