數學

我們研究的主題是曲率，且以高中所學的函數為主。雖然大學已有曲率公式，但我們將其表示成高中生較易了解的型式，並且以f(x) 的方式呈現。我們在函數曲線上取不共線三點，構成一個三角形，並求出此三角形的外接圓半徑。再將所取三點逼近，所求之半徑即為特定點的密切圓，也就是曲率半徑。而此曲率半徑的倒數，就是所求的曲率，同時我們將公式帶入高中各常見函數，以導出函數上各點曲率。;Our study is about curvature, especially about the fuctions we learn in senior high school. In university, there is a certain formula for curvature, but we hope to change it into a form that can be easily accepted by senior high school students, and express the formula with f(x), the symbol of functions. We pick three incollinear points from the curve of a function, making the three points into a triangle, and figure out the circumradius of this triangle. Then, we approximate the three points to one of them, and the circumradius will also be the radius of the osculating circle of the point. We define the radius as radius of curvature. The reciprocal of the radius of curvature will be the curvature. Then, we use the formula to figure out the curvature of the functions we learn in senior high school.

In any , we prove that there exist cycles which have any length between 3 and 3n and paths which have any length between their smallest distance and longest Hamiltonian paths in any two different nodes; for any two nodes, there exist varied Hamiltonian cycles, making the two nodes locate on any possible counterpart position(only limited by the distance between the two nodes). In , there are 2n internally-disjoint spanning cycles, and 2n-1 internally-disjoint spanning paths. Besides, we also prove has no more than 2n disjoint spanning paths, and calculate its wide diameter. 本報告證明在環狀網路 中，存在有長度3到 3n 的所有迴圈；任何相異兩點都有各種不同的長度的路徑:從最短的距離到最長的漢米頓路徑；取定任意兩點，存在有各種不同的漢米爾頓迴圈，使得兩點位於所有可能的相對位置上(僅被兩點之間的距離限制)。在 中，也具有2n 個彼此不相交、經過所有點的迴圈，以及2n-1 個彼此不相交、經過所有點的路徑。除此之外，也證明了，在兩相異點間，具有個數不超過2n 且互斥的路徑，且這些路徑經過所有點。我們也估算了它的寬直徑。

本文主要在討論一個益智遊戲─瓶蓋問題，我們延伸本問題，觀察多個正交互影響的變數的變化規律，並了解在兌換中的特性，進而找出計算法則來求取可兌換到的瓶子總數。 我們的研究過程如下： 1. 對原始問題進行定義化並求出增減初始瓶數所得之飲料通解。 2. 改變瓶蓋、瓶身兌換所需數量並觀察總共可喝飲料與殘餘瓶蓋、瓶身數量的變化。 3. 以含有高斯符號的等式描述總體變化關係。 4. 求得上述的第2點，並求出演算法則。 5. 增減可用於兌換的事物數量(瓶蓋、瓶身)，並觀察總共可喝飲料與殘餘瓶蓋、瓶身、‧‧‧(其他可用於兌換的事物)數量的變化。 6. 以含有高斯符號的等式描述總體變化關係。 7. 求得上述的第5點，並求出演算法則。 8. 擴張規則，引入m級的概念。 9. 以數團概念描述兌換過程。 10. 求得上述的第8點，並求出演算法則。 11. 調整研究的角度，納入銷售者與經濟學者觀點進行討論。

Most football managers are not aware of the need for analysis of soccer data, which is one of the dynamic sports. In this study, we developed a statistical model with performance indicators and score indicators of individual soccer player based on various event data including dynamic features such as goal, assist, pass, etc. In this study, the correlation between the dependent variables and the explanatory variables, and each explanatory variable was confirmed through the correlation analysis to solve the problem of multiple communicability from the regression model and to analyze the statistically significant preliminary model. In addition, we analyzed the correlation between individual rating of the players and the data recorded in the soccer games, and found that there has been a problem with the rating of the soccer players evaluated by the reporters and the soccer statistics site. To solve this problem, we developed a model that best fit the performance indicators of individual soccer player using the linear regression model and the beta regression model. The performance index model of the athletes was developed by comparing the R-squared value and the mean absolute percentage error of two models, the linear regression model and the beta regression model, and found out the beta regression model is better model to use. By using the estimated regression coefficients of the regression model we made new PI model. Score Index, which is the attractive point of soccer, was developed by comparing Poisson regression model and negative binomial regression model based on AIC value, and the one using negative binomial regression model was found to be better. Through the model developed by this study, it is possible to collect the event data recorded by individual athletes for each soccer game, and obtain the PI & SI index which are the athlete performance index models. This allows us to evaluate each team's players objectively, analyze the team's deficiencies, and provide tools to find players, who can fill in the missing positions of the teams. This study can also be utilized to grasp the performance of athlete in real time by simulating the resultative model.

本研究從Brianchon定理「圓外切六邊形三條對角線共點」以及Pascal定理「圓內接六邊形三組對邊延長線交點共線」，這兩個對偶定理出發，試圖以雙心六邊形串連兩個定理，讓Pascal (1623–1662)及Brianchon (1785–1864)兩位法國數學家「相遇在21世紀」。本研究除了探討雙心六邊形的共點共線問題外，更進一步研究其共點共線圖形的軌跡。研究有更驚人的發現：雙心六邊形將各邊延長取交點，其所形成的新六邊形同時內接於一條圓錐曲線，外接於另一條圓錐曲線。

給定一個簡單圖G（simple graph ），令V(G)、E(G)分別為G的頂點與邊所形成的集合。對於兩個不同的頂點u, v∈V(G)，若存在一條邊連結頂點u, v，則將此邊記為uv，以uv∈E(G)表示。給定函數f: V(G)→ℕ，若對任意的邊uv∈E(G)，函數f皆滿足f(u)≠f(v)，則稱函數f為圖G的一個著色函數（proper coloring）。對於圖G，任意給定每個頂點v∈V(G)一個可用的顏色清單L(v)⊂ℕ，其中清單內各有四個可用顏色（|L(v)|=4 ，每個頂點的顏色清單可以不相同），若總是存在一個著色函數f，滿足∀v∈V(G)，f(v)∈L(v)，則稱圖G為『可四元列表著色（4-choosable ）』。對於平面圖（ plane graph），針對長度較小的圈（cycle）進行限制，我們設計充分條件，使得滿足條件的平面圖為可四元列表著色。

編號1~mn 的mn 個物件已隨機置入m× n 階的矩陣中，另外有一行m 個空格的暫存區供物件暫存用。我們探討將這mn 個物件移至目標區並按照1,2,…, mn 的次序排列，所需的移動步數；每一步的移動中，只能移動每一行最頂層的物件到其他行(含暫存區)的最頂層或目標區。在這篇報告中，我們給出了一個適用於n ? m ?1時的移動方法，此方法在一般的情形下，所需的移動次數未必是最少；但是在最不利於移動的情形下，我們證明此方法所需的移動步數為最少。There are mn objects, numbered from 1 to mn, put on an m× n matrix randomly, and there is another column with m blank spaces for temporary storage purpose during moving. In each step of moving, we can only move the top object from one column to the top of another column or to the target pile. The total steps needed to move these mn objects to the target pile in increasing order from the bottom to the top is studied in this article. A general method for solving this problem when n ? m ?1 is given, and we prove that it provides an optimal solution in the worst cases. However, it may not always provide the minimal steps in all cases.

本研究探討正凸多邊形正則鑲嵌及阿基米德鑲嵌，在限制每一格相鄰格子中至多(或至少)有 格被塗色的情形下的最大(或最小)塗色格子數問題。研究利用塗色格子位於邊線角落、非角落的邊線、鑲嵌內部的共用邊數差異、及與塗色格子總數間的限制條件，採用賦值法解析塗色格子數的最小上界或最大下界。接著建構具最大(或最小)塗色格子數的塗色方式，以歸納法推導塗色格子數，證明其與賦値法解析結果相同，證得存在該塗色格子數。研究結果可應用至貼磚或印染鑲嵌圖案設計、LED點燈遊戲設計、供給-需求組合配置最佳化、LED廣告面板或色差控制等。