數學

在三角形的外接圓上取一點，作其對三角形三邊的垂足，此時這三個垂足會共線，稱為西姆松線。本研究主要探討的問題為：當三角形以其外心旋轉 時 (我們稱之為對徑三角形)，將此外接圓上一動點P對兩對徑三角形分別做西姆松線，我們想研究當P點在外接圓上轉動時，兩西姆松線的交點軌跡為何。我們將西姆松線放在複數平面上來分析，這兩條西姆松線會互相垂直，並且它們的交點軌跡為一橢圓。此橢圓會相切於兩對徑三角形的六條邊，因此我們將此橢圓稱作這兩對徑三角形的「六點橢圓」，並探討這個橢圓的性質。

本文先針對直線L和三定點A、B(線外)、P(線上)，探討△ABP的尤拉線平行(AB) ⃡的公式，透過函數凹性判定和對函數最小值N和3的比較，提供是否有解的依據，並找到了漂亮的判斷公式4ab/h^2 。接著在確定尤拉線平行(AP) ⃡、(BP) ⃡的存在性及解的公式後，發現最多有三解。 對於前文的P點，作者利用先前發現的一系列定理，設計了能以尺規作圖達成△ABP的尤拉線平行(AB) ⃡、(AP) ⃡、(BP) ⃡的兩條直線，當在那兩條線上分別取A點和B點之後，可用尺規作圖找到P點，甚是有趣。 針對首段P點，兩平行線，尤拉線和△各邊，交角可作為0°，作者推廣至△ABP的尤拉線與¯AB達成交角為指定角的方法。 在探討藉多邊形各邊與分割點連接的子△中，作者發現任意三角形不存在能使各子△尤拉線平行原多邊形與其共用邊的分割點P；但四邊形、五邊形可能存在符合此條件的分割點P，且能利用前文定理創造出這樣的多邊形。

This project aimed to study the motion which occurred from the end point on the circumference of the outermost circle by moving the center on the circumference of a preceding circle and the center of an innermost circle at origin. According to the study, when angular velocity was changed, it caused the different of loci. Based on the above information, finding the locus of the point on circumference of n-th circle that formed by moving the center of any radius circles on circumference of preceding set of circles was studied to get general equation. A set of circle and locus were created with GSP program. First, set the same radius circles on the X-axis with the first circle at origin, then found the relationship that occurred from the characteristics of locus. The result showed that if the ratios of angular velocity are 1:1:1, 2:2:2, 3:3:3, ..., …, n:n:n or 1:2:3, 2:4:6, 3:6:9, …,nw1:nw2:nw3, the characteristics of locus will be the same, while the others will be different. Finally, the equation of locus was found as follow: (x,y) = { ..........see in abstract...........} when .........see in abstract........... Where ri is the radius of i-th circle, zeta i is an angle between the radius of i-th circle and X-axis, wi is the angular velocity, t is elapsed time and alpha i is a starting angle between the radius of i-th circle and X-axis.

本研究從在方格棋盤中走捷徑的問題出發，推廣至由多個相異正多邊形所組成的半正鑲嵌圖形棋盤，其沿格線走捷徑的方法數與最短路徑。研究中，我們針對所有8種1律半正鑲嵌圖形進行分類探討，包括截半六邊形、截角六邊形、扭稜六邊形、小斜方截半六邊形、大斜方截半六邊形、扭稜正方形、異扭稜正方形、截角正方形圖形。我們將每種棋盤進行「轉正」，使它對應於唯一的矩形棋盤，達到「捷徑同構」，因而原本半正鑲嵌圖中的捷徑問題就等價於方格棋盤的捷徑問題。我們將走捷徑方法數的通解分類，發現有組合數類、以及階差與指數混合兩大類，並分析康威表示法與通解的關係。

本研究透過遞迴式及數學歸納法探討 維空間中正多胞體（單純形、超方形、正軸形）之點、線、面的一般化結果。本研究利用頂點圖及線性變換－行列式的方法探討三維空間正多面體至 維空間中凸正多胞體之保距變換方式並使其一般化。另外，本研究也嘗試透過特徵多項式及隸美弗定理分析正多胞體旋轉之旋轉角度。

本研究的目的在於探討在社群網路發達時代中，資訊的傳播範圍之可能性。我們將的智慧上網裝置視為節點，以圖論方式分析節點到另一個節點的訊息傳遞模式。我們研究在傳遞訊息對象人數不同時，及在不同共同朋友數量的網路圖中找出其傳播範圍的關係式。最後我們找到不同結點數與傳遞次數、發源點之關係式，並進行一般化論證。並提出定理以供探討不同節點訊息傳遞時，其網路傳播範圍之關係，應用於社群網路分析參考。

The project is devoted to the study of the Seymour’s Second Neighborhood conjecture by determining the properties of possible counterexamples to it. This problem has remained unsolved for more than 30 years, although there is some progress in its solution. The vector of the research is aimed at the analysis of possible counterexamples to the conjecture with the subsequent finding of some of their characteristic values. In addition, attention is focused on the generalized Seymour’s conjecture for vertex-weighted graphs. Combinatorial research methods and graph theory methods were used in the project. The author determines the values ​​of densities and diameters of possible counterexamples, considers separately directed graphs of diameter 3. The conditions under which specific graphs cannot be counterexamples to the Seymour’s conjecture with the minimum number or vertices are defined. The relationship between the Seymour’s conjecture and vertex-weighted Seymour’s conjecture is explained. It is proved that if there exists at least one counterexample, then there exist counterexamples with an arbitrary diameter not less than 3. Under the same condition, the existence of counterexamples with a density both close to 0 and close to 1 is also proved. The equivalence of the above two conjectures is substantiated in detail. It can be concluded that if the Seymour’s Second Neighborhood Conjecture is true for a directed graph of diameter 3, then it is true for any digraph, so that problem will be solved. Moreover, if the conjecture is true, then vertex-weighted version of this conjecture is true too. That is why a digraph of diameter 3 needs further research.

在比賽時看到許多選手，雖然本身實力不差，卻因為賽制的編排而無緣晉級決賽，因此本研究透過數學分析單淘汰賽（可以很快的找出勝負）、單循環賽（大部分是使用在人數較少時，但是每位選手都會交手到）、雙淘汰賽（可以讓選手有輸一次的機會，選手就算輸一場還是有機會得到冠軍）、循環賽（主辦單位會融合單淘汰賽、單循環賽、雙淘汰賽來衍生出新的賽制），得到可以選出與實力相當的前三名（準確找出前三名）機率，並且將所有機率加以比較，分析出何種賽制準確找出前三名的機率最高，並且利用比較後的結果，製作一個新的賽程。經過分析得到混和賽的機率與單淘汰賽差不多，但考慮場次的使用並沒有優於單淘汰賽，因此並未符合主辦單位採用此種賽程的依據。但雙淘汰賽卻相反，機率偏高且使用的場次適中，符合主辦單位採用的依據。

本研究主要提出新的、有創意的圓錐曲線製造方法。首先，我們利用兩個全等三角形製造兩個線束，來討論基線夾角與線束中心在平面上的相對位置，用以製造各式的二次曲線。接著，分別在五條等距平行線上取點，固定其中四點為梯形，分類第五個點的位置來對應生成不同的圓錐曲線。然後，簡化為共線的四個點A_1、A_2、A_3、A_4，往線外一點B_0投射，滿足(B_k B_0)┴⃑=k⋅(B_0 A_k)┴⃑得B_k，k=1,2,3,4，依(A_1 A_2)┴⃑:(A_2 A_3)┴⃑:(A_3 A_4)┴⃑的不同比例，來分類{B_k }_(k=0)^4生成的二次曲線。特別的是，由此得到兩個特別的應用：一、得到一個過圓錐曲線中心線段的比例，只要給定A_1、A_2、B_0 三點即可快速尺規作圖得到中心；二、新的拋物線切線的尺規作圖法。最後，我們定義了三個線束特定的對應方式，得到六個對應點共橢圓的性質，以及分類了兩個基圓上點與點的角度與圓心的相對位置，用以製造各式的二次曲線。未來，我們希望能定量化我們的結果，以及探討更多個線束的對應。

The project is devoted to the study of the Seymour’s Second Neighborhood conjecture by determining the properties of possible counterexamples to it. This problem has remained unsolved for more than 30 years, although there is some progress in its solution. The vector of the research is aimed at the analysis of possible counterexamples to the conjecture with the subsequent finding of some of their characteristic values. In addition, attention is focused on the generalized Seymour’s conjecture for vertex-weighted graphs. Combinatorial research methods and graph theory methods were used in the project. The author determines the values ​​of densities and diameters of possible counterexamples, considers separately directed graphs of diameter 3. The conditions under which specific graphs cannot be counterexamples to the Seymour’s conjecture with the minimum number or vertices are defined. The relationship between the Seymour’s conjecture and vertex-weighted Seymour’s conjecture is explained. It is proved that if there exists at least one counterexample, then there exist counterexamples with an arbitrary diameter not less than 3. Under the same condition, the existence of counterexamples with a density both close to 0 and close to 1 is also proved. The equivalence of the above two conjectures is substantiated in detail. It can be concluded that if the Seymour’s Second Neighborhood Conjecture is true for a directed graph of diameter 3, then it is true for any digraph, so that problem will be solved. Moreover, if the conjecture is true, then vertex-weighted version of this conjecture is true too. That is why a digraph of diameter 3 needs further research.