本篇論文討論的是SIR模型的反應擴散方程 　　　　　　　　　s_t = d_1 s_xx − βsi/(s + i), 　　　　　　　　　i_t = d_2 i_xx + βsi/(s + i) − γi, 　　　　　　　　　r_t = d_3 r_xx + γi, 之行進波的存在性，其中模型描述的是在一個封閉區域裡流行疾病爆發的狀態。這裡的 β 是傳播係數，γ 是治癒或移除(即死亡)速率，s 是未被傳染個體數，i 是傳染源個體數，d_1、d_2、d_3分別為其擴散之係數。 　　我們將使用Schauder不動點定理(Schauder fixed point theorem)、Arzela-Ascoli定理和最大值原理(maximum principle)來證明：該系統存在最小速度為c=c*:=2√(d2( β - γ ))之行進波解。我們的結果回答了[11]裡所提出的開放式問題。 / In this thesis, we study the existence of traveling waves of a reaction-diffusion equation for a diffusive epidemic SIR model 　　　　　　　　　s_t = d_1 s_xx − βsi/(s + i), 　　　　　　　　　i_t = d_2 i_xx + βsi/(s + i) − γi, 　　　　　　　　　r_t = d_3 r_xx + γi, which describes an infectious disease outbreak in a closed population. Here β is the transmission coefficient, γ is the recovery or remove rate, and s, i, and r rep-resent numbers of susceptible individuals, infected individuals, and removed individuals, respectively, and d_1, d_2, and d_3 are their diffusion rates. We use the Schauder fixed point theorem, the Arzela-Ascoli theorem, and the maximum principle to show that this system has a traveling wave solution with minimum speed c=c*:=2√(d2( β - γ )). Our result answers an open problem proposed in [11].

本研究先藉由文獻分析法瞭解擴散性思考、數學問題發現與數學學業成就三者的內涵，繼而依據它們的內涵並佐以學習、問題解決的角度，建立起三者間關係的假設，並透過實證調查研究法來驗證這些假設。本研究之研究對象為台北縣市五所國中的318位國三學生，研究工具為「新編創造思考測驗」、「數學問題發現測驗」、「第一次數學科基本學力測驗」，並以次數統計、集群分析、相關分析、變異數分析、逐步迴歸分析進行資料分析。本研究主要的研究結果如下： 一、學生能夠發現各種思考產物類型與數學類型的問題。其中，關係性問題與發現性問題最多人提出，而單位性、類別性與驗證性問題則較少人提出。 二、學生的數學問題發現型態有個別差異。 三、擴散性思考與數學問題發現間為顯著中低度相關。 四、擴散性思考與數學學業成就多為顯著中低度相關。 五、數學問題發現與數學學業成就間為顯著中低度相關。 六、能問大量且層次高數學問題的學生其數學學業成就比較不會問數學問題的學生為佳。 七、擴散性思考之流暢力、數學學業成就、擴散性思考之變通力可以有效預測數學問題發現之問題數。 八、擴散性思考之流暢力、數學學業成就、擴散性思考之變通力可以有效預測數學問題發現之問題獨特性。 九、數學學業成就與擴散性思考之流暢力可以有效預測數學問題發現之問題品質。 十、數學問題發現之問題品質、數學問題發現之問題數可以有效預測數學學業成就。 本研究最後針對數學教育以及未來研究提出若干具體建議。 / First, this study probed into the contents of divergent thinking, mathematical problem finding, and mathematical achievement by literature review. Then the researcher made hypotheses of the relationships between divergent thinking, mathematical problem finding, and mathematical achievement based on the contents of them and the views of learning and problem solving, and designed survey research to examine these hypotheses. The subjects were 318 9th grade students from five junior high schools in Taipei county and Taipei city. The data- collection instruments included：(a) New Creativity Test; (b) Mathematical Problem Finding Test; (c) Basic Educational Indicator Tests of Mathematics. After utilizing frequency, cluster analysis, correlation analysis, ANOVA, and stepwise regression, the main results of this investigation are：(a) Students can find problems of all kinds of intellectual products and mathematics. Among them, problems of relations and problems to find were found most and problems of units and classes and problems to prove were found least ; (b) There are individual differences between mathematical problem finding styles; (c) The correlations between divergent thinking and mathematical problem finding are significantly positive; (d) Most of the correlations between divergent thinking and mathematical achievement are significantly positive; (e) The correlations between mathematical problem finding and mathematical achievement are significantly positive; (f) Students who can finds many high-level problems have higher mathematical achievement than those who can not; (g) Fluency of divergent thinking, mathematical achievement, and flexibility of divergent thinking can be used to predict the number of problems of mathematical problem finding effectively; (h) Fluency of divergent thinking, mathematical achievement, and flexibility of divergent thinking can be used to predict the rarity of problems of mathematical problem finding effectively; (i) Mathematical achievement and fluency of divergent thinking can be used to predict the quality of problems of mathematical problem finding effectively; (j) The quality of problems and the number of problems can be used to predict mathematical achievement effectively. Finally, the researcher brings up some suggestions on mathematical education and the future research.