第四章

1. 第四章 數學分析

2. 大 綱 前言 4.1 分析章節的基本結構 4.2 時態 4.3 寫作風格與常用句型

3. 前言 • 數學分析 • 說明對問題的分析 • 包括技術之理論基礎或理論模型

4. 4.1 分析章節的基本結構 • 介紹題目 • 說明相關假設、條件或定義 • 敘述主要問題或基本方程式 • 進行分析並得出結果 • 說明或討論上述結果，或根據結果推出某種結論

5. 範例

8. 範例解析 • 先介紹一個題目(“Consider…) • 再說明一些假設及定義(“Let…”) • 列出一個方程式並進行分析及演算 • 根據整個段落內容得出結論

9. 介紹模型包括的項目 • 模型的背景或理論基礎 • 方法或模型的基本假設或根據 • 基本方程式與分析 • 模型的詳細描述 • 說明模型的重要特性 • 敘述如何應用該模型或以實際例子說明

10. 範例 • 介紹模型章節後，比較從模型的預測、推導和實驗數據的差距，以確證模型的有效性 • 以下介紹兩個不同模型的文章

11. 第一篇 • 說明模型的相關背景 ，並詳述其不同的假設或條件 • 提出作者自己的模型 • 解釋自己的模型在許多特定問題上的應用，並討論模型對這些問題的理論涵義

12. [相關背景]The model and analysis presented here were originally developed in Powell (1983). A closely related discrete time model was developed by Switzer and Salant (1980) and recently published in French (Switzer and Salant 1986).... • [不同的假設或條件]We assume that a social planner controls the rate of production of all energy resources. [提出作者自己的模型] These resources are of two types: depletable energy, which is costless to produce but available in a limited amount, and backstop energy, which has a constant production cost and unlimited (physical) availability. The two resources are perfect substitutes; that is, consuming a unit of each provides the same gross utility to society. ...

13. [解釋自己的模型]The social planner's problem is to choose the optimal rates of production of depletable and backstop energy, as well as the rate of investment in backstop capacity. We assume the existence of a concave social utility function defined on total energy production. Furthermore, we assume the planner's objective is to maximize the present value of social utility net of production and investment costs. The operative constraints are, first, that total production of depletable energy may not exceed the fixed stock; second, that backstop capacity increases in each period by the rate of backstop investment; and third, that backstop production is limited in each period by the current level of backstop capacity. Formally, the planner's problem can be written as follows ....

14. 第二篇 • 一開始介紹一個基本方程式，並說明該方程式及一些重要假設 • 接著利用此方程式及其它資料導出一個模型 • 討論模型的各種特性

15. [介紹基本方程 式及重要假設]

16. [導出模型]

18. [討論模型的 各種特性]

19. 分析章節的基本要點 • 先清楚敘述所有假設及概念之定義，再應用這些假設或概念 • 解決任何問題前，先把問題陳述清楚 • 若有許多子題需討論，應逐步解決目前子題，再討論下個子題，且應清楚表達不同子題間的邏輯關係 • 不應假定讀者瞭解文中所有細節，應一步步清楚說明分析章節中的所有內容

20. 方程式的標點符號 • 問題 • 當數學方程式(或公式)與句子的其餘部分區隔開，而自成一行時。若句子結構需要標點，則數學方程式(或公式)後是否應加標點符號(如逗點或句點) ﹖ • Example1 • If Vcc and RL are known, the load line equation may be written as VCC = IC * RL * VCE˙ • 方程式位於句尾，VCE後可有句點

21. Example2 • Under these assumptions, the load line equation may be written as VCC = IC * RL * VCE， where VCC and RL are known. • 字詞where VCC and RL are known 是修飾方程式的非限定關係子句， “VCE”後面應有逗點

22. 兩派說法 • [說法一]在方程式與公式後使用標點符號 • 因標點符號有助於釐清句子的文法結構，可增進文章的可讀性 • 某些例子需要標點符號，以防止讀者誤讀 • [說法二]省略方程式或公式後的標點符號 • 認為釐清含有方程式或公式的句子意義並不需要標點符號 • 認為在含有大量數學公式的文章中，標點符號的使用會使頁面過於擁擠

23. 方程式的位置 • 當數學方程式和文章其餘部份區隔開時 • 方程式上下應至少空一行，且方程式應縮進排印 • 縮排的距離應至少和新段落首行縮進的距離一樣，最好比新段落首行縮進的距離還多一點(如0.5或1.0英寸) • 有些專業期刊把區隔開的方程式排在頁面中央(置中)

24. 有些在數學方程式右邊空白處為每個方程式加上序號有些在數學方程式右邊空白處為每個方程式加上序號 • 序號應放括號內 • 方程式和序號間應留空白 • Ex： f(x, y) = 3x2-4xy+y2 (5) • 不該用虛線把方程式和序號連起來 • Ex： f(x, y) = 3x2-4xy+y2 …………………………… (5)

25. 4.2 時態 • 分析章節中的資料屬於有關數學或邏輯關係、不受時間影響之普遍事實，所以大多數句子使用現在式 • Consider a unitary matrix B whose eigenvalues all lie on the unit circle. • Then we have the following equation: • If A is chosen such that Al = B1, then the two expressions are equivalent.

26. We assume that a social planner controls the rate of production of all energy resources. • The solution can be obtained by solving Equation (1). • The social planner's problem is to choose the optimal rates of production of depletable and backstop energy.

27. 也可以未來式介紹新的題目或轉接到新的段落或章節也可以未來式介紹新的題目或轉接到新的段落或章節 • We will now discuss the nonlinear case. • In the next subsection, we will show how to solve this equation. • 未來式也可以在條件句中使用，以敘述可能的結果 • If an appropriate value of λ is chosen, then the coefficients of gjwillbe small.

28. 如在描述模型或數學分析時，需提及過去已完成的實驗研究須用過去式如在描述模型或數學分析時，需提及過去已完成的實驗研究須用過去式 • 1For three modes, Equation (5) provides；42 equations. 2If the stiffness matrix is assumed to be symmetric, then there are 105 unknown terms in the matrix. 3Consequently, the coupling between nodes must be arbitrarily reduced. 4In the initial evaluation of the experimental data, the coupling was reduced such that only adjacent nodes were coupled in the stiffness matrix.

29. 前三句 • 使用現在式：討論和數學分析相關、不受時間影響之事實 • 第四句 • 使用過去式：指過去實驗研究時的行為

30. 4.3 寫作風格與常用句型 • 分析章節應採用自然、易讀的寫作方式 • 應用簡單、直接的方式清楚詳述分析過程 • 應用正式英文，不該使用任何俚語或不正式的字詞 • 儘量避免使用it is、there is或there are為句子的開頭 • 如採第一人稱複數代名詞(we)可使句子更自然、通順，則可多利用該代名詞

31. 範例 • (Worse) It is assumed here that the social planner controls the rate of production of all energy resources. • (Better) We assume that the social planner controls the rate of production of all energy resources. • (Better)Let us assume that the social planner controls the rate of production of all energy resources. • (Better)Assume that the social planner controls the rate of production of all energy resources.

32. 避免孤立或誤置的修飾詞(缺少修飾對象或在句子中位置不正確)避免孤立或誤置的修飾詞(缺少修飾對象或在句子中位置不正確) • (Worse) Before describing the adaptive controller in detail, the ideal case in which the dynamic model is exactly known will be considered [位在句首的片語缺少修飾的對象]. • (Better) Before describing the adaptive controller in detail, we will consider the ideal case in which the dynamic model is exactly known.

33. (Better) Before describing the adaptive controller in detail, we consider the ideal case in which the dynamic model is exactly known. • (Better) Before describing the adaptive controller in detail, let us consider the ideal case in which the dynamic model is exactly known.

34. (Worse) The relationship between these two variables is now considered [不知道這行為的產生者為何人何物，而且動詞的時態很不自然]. • (Worse) The relationship between these two variables will now be considered [這行為的產生者為何人何物?].

35. (Better) Now let us consider the relationship between these two variables. • (Better) Consider the relationship between these two variables. • (Better) These two variables are related in the following way [直接描述兩者之間的關係].

36. 常用句型 • Consider the case in whichA is equal to B. • As an example, consider the case in which A=B. • Let us now consider the case in which A=B. • We assume that A=B. • Let A be equal to B.

37. If A=B, then we have the following equation: [列出方程式] • Given that A=B, we obtain.. [列出方程式] • This problem can be written as …[列出方程式] • This problem can be expressedas …[列出方程式] • This problem can be written as follows: [列出方程式]

38. SubstitutingA into Equation 2, we obtain…[列出方程式] • We will now integrate Equation 3 in order to derive the solution. • A is inversely proportional to B, as shown below. • The relationship between A and B is as follows:[列出方程式] • The relationship between A and B can be expressed as …[列出方程式]

39. The solution is …[列出方程式] • We can now derive the solution to Equation(3). • We will now reduce Equation(3) to a simpler form. • Given these assumptions, the system can be modeled as follows: [列出方程式] • Under these conditions, the equation can be rewritten as shown below.