【筆記】學術英語寫作

Imakf發表於2024-07-23
筆記

小學期修了學術英語寫作，老師是我們數分三的老師（啊這）。以下是課堂筆記彙總

analogy 類比

constitute 組成

attenuate 減少

convention 公約

referee 審閱

sanity check

Overview

  1. 完整。把數學符號翻譯成英文後，行文應當符合英文語法。

  2. 簡潔。Don't show.

  3. 邏輯。Motivation。

  4. 向大師學習。

Definition

  • 本質： if and only if.（或者只寫 if）

  • what is what. 可能的陳述方式：把滿足 x 的 y 叫做 z。

下面給出例子。

例子 1：（數學分析）

定義：設 \(B \sub \R^2\)。如果對任意給定的 \(\epsilon > 0\) 存在可數個閉矩形序列 \(\{I_i\}\)（\(i=1,2,\cdots\)）使得 \(B\sub \cup_{i=1}^{\infty} I_i\)，\(\sum_{i=1}^{\infty} \sigma(I_i) <\epsilon\)，則稱 \(B\) 為（二維）零測集。

Definition： Let \(B \sub \R^2\). Assume that for any \(\epsilon >0\), there exists a countable sequence \(\{I_i\}_{i=1,2,\cdots}\) of closed rectangles, such that \(B\sub \cup_{i=1}^{\infty} I_i,\sum_{i=1}^{\infty} \sigma(I_i) <\epsilon\). Then \(B\) is said to be a (2-dimensional) null set.（Here \(\sigma\) denotes ...）

Comment：用 Assume that 祈使句替代 If 從句。善用逗號。引入了新的記號必須在附近進行解釋。

例子 2：（數學分析）

Definition: Let \(\Omega \sub \R^3\) be a domain and \(P_0 \in \Omega\). We say that \(\Omega\) is star-shaped with respect to \(P_0\) if the segment \(\overline{PP_0}\) lies in \(\Omega\) for any \(P\in \Omega\).

例子 3：（表示論）

Let \(G\) be a group. Let \(V\) be a vector space. A representation of \(G\) on \(V\) is a linear action of \(G\) on \(V\). That is, for each \(g\in G\), there is a linear transform \(\rho(g):V\to V\) such that \(\rho(g_1g_2)=\rho(g_1)\rho(g_2)\) for any \(g_1,g_2\in G\).

Comment：也就是說可以用 "That is," 或者 "i.e.," 表示。transformation (US) transform (UK)。重要的公式可以居中。非必要不寫記號，如 \(\forall\) 和 for all。

例子 4：（隨機過程）

Let \(\xi = (\xi_1,\cdots,\xi_d)^T\) be a \(d\)-dimensional random vector such that

\[\begin{cases} \xi_1 = a_{11}\eta_1 + \cdots +a_{1m}\eta_m + \mu_1 ,\\ \cdots\\ \xi_d = a_{d1}\eta_1 + \cdots +a_{dm}\eta_m + \mu_d.\\ \end{cases} \]

Here, \(\eta_1,\cdots,\eta_m\) are i.i.d. Gaussians, and \((a_{ij})_{1 \le i \le d\\1 \le j \le m}\); \((\mu_i)_{1\le i \le d}\) are constants. Then \(\xi\) is said to obey \(d\)-dimensional Gaussian distribution.

Notation. \(\xi = A\eta + \mu\) for \(A=(a_{ij}),\mu = (\mu_i)\) in the matrix form.

Comment：當其他內容太多時，把要定義的東西提前。可以使用領域公認的縮寫（如本例中 Independent and identically distributed 縮寫成 i.i.d.）。

例子 5：（數學分析）

Def: Let \(f:X\to Y\) be a mapping between metric spaces. Let \(x_0 \in X\). Say that \(f\) is continuous at \(x_0\) if for any \(\epsilon > 0\), there is \(\delta > 0\) such that \(f(\mathbb{B}_\delta^X(x_0)) \sub \mathbb{B}_{\epsilon}^{Y}(f(x_0))\).

Comment：避免用數學符號作為一句話的開頭。不要引入不必要的記號。

Theorem

  1. 完整準確羅列條件和限制。
  2. 最核心的結論儘量在突出位置一句話凸顯。
  3. 上下文安排。 *必要時可以分割成若干引理。

例子 1：Let \(f:[-\pi,\pi] \to \R\) be a continuous function such that \(f(\pi) = f(-\pi)\). Suppose that \(f\) is piecewise differentiable on \([-\pi,\pi]\), and that \(f'\) is Riemann-integrable. Then the Fourier series of \(f\) on \([-\pi,\pi]\) converges uniformly to \(f(x)\). In fact,

\[f(x)= \frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n\cos nx+b_n \sin nx) \text{ for each } x\in[-\pi,\pi]. \]

Comment：公式後記得打標點符號。Punctuate the maths formulae!

例子 2：

Thm： Let \(U\sub \R^n\) be open, let \(f:U \to \R\) be of class \(C^2\) and let \(x^0\) be a stationary point of \(f\). Then the following holds:

  1. if ..., then ... ;
  2. if ..., then ... ; and
  3. if ..., then ... .

Comment：冒號和分號後小寫，列舉最後兩項間的 and.

Proof of Theorem

有一些固定證明思路：Contradiction, induction, cases...

例子 1：（反證）

命題：設 \(A\) 為 \(\R^n\) 中子集，則以下等價。

  1. \(A\) 為緊緻集；
  2. \(A\) 為序列緊緻集。
  3. \(A\) 為有界閉集。

Proof (1) to (2)

Assume that there were a sequence \(\{b_n\} \sub A\), such that any of its subsequence does not converge in \(A\). By ...., there is an open ball \(\mathbb{B}_{r(a)}(a)\) for any \(a \in A\) which contains at most finitely many terms of \(\{b_n\}\). By compactness of \(A\) (note that \(A \sub \cup B_{{r(a)}}(a)\)), one can find \(a_1,\cdots,a_k\) such that \(A \sub \cup_{i=1}^k B_{r(a_i)}(a_i)\). In particular, only finitely many \(b_n\) are in \(A\).

Contradiction. (/which contradicts ...)

Comment: Contradiction 中可以使用虛擬語氣（Europe）。用以下來代替 according to：By/ in view of/ by virtue of/ thanks to。可以用括號表示原因。特別地使用 in particular。

歸納法：

We induct on \(k\)/We proceed with induction on \(k\).

The base step/case \(k=0\) has been covered by Lemma A.

Now assume the assertion for \(k-1\) and argue for \(k\).

....

Hence, the proof is complete by induction.

designate

predicate

preliminary

cornerstone

manuscript

compromise 損害

usher

typographical

painstakingly

以下是……

  • The following result is a Tauberian theorem.
  • Let us introduce the Tauberian theorem as follow.
  • Below is a Tauberian theorem.

Theorem (Hardy [5, Chapter 7])

Denote the partial sum of \(\sum f_n(x)\) as/by \(S_n(x)\). / Denote by \(S_n(x)\) the partial sum of \(\sum f_n(x)\). Let \(\sigma_n=\cdots\). If \(\sigma_n(x)\) converges uniformly to \(f(x)\) and if \(\{nf_n(x)\}\) is uniformly bounded, the \(\sum f_n(x)\) converges uniformly to \(f\).

Comment: 可以 Denote sth by 記號. 或者 Denote by 記號 sth。多個條件時可以把後面的條件 if 顯式寫出。

Proof. By assumption (the uniform boundness of \(\{nf_n(x)\}\)), there is \(M>0\) such that \(|nf_n(x)| < M\) for every \(x\) and \(n\). Given any \(0<\epsilon<1\), one can find \(N_0\) such that \(\cdots\) for every \(x\) and \(n > N_0\).

Also/ in addition/ moreoever/ further more/ on the other hand, since \(\cdots\), we obtain that/ one sees that/ it holds that \(\cdots\).

Hence/ Thus, ...

Take/Set/Pick \(N > \cdots\), then ...

This yields/leads to ...

Comment: on the other hand 表遞進。表最終結果用 therefore。

Introduction

The main contribution of this paper is to obtain/establish/derive the \(W^{1,p}\) estimate under weaker assumptions; in particular, we assume that \(A^{-1}(x)\) has small BMO norm. (解釋 weaker 這種不精確的詞語) More precisely, ...

不用 Obviously。It is clear that.

引用/參照

  • (see [14])

  • (cf [14])

  • [14]

  • See [14].

  • See [2,3,5] and the many references cited therein.

  • See [1] by Bourgain-Kenig-Tao.

[編號] 作者名（首字母排序）文章標題，雜誌名縮寫（斜體）卷號（加粗）年份（括號中），頁碼（或者文章號）.

常用短語

sufficiently large

well justified

easy/simple/straightforward

readily

deriavtive

（改變運算優先順序的）括號 parenthesis

代入 substitute sth into sth

neglect/ignore the lower order terms

dominate/majerise

as mentioned above / as aforementioned

verbatim

strictly/roughly speaking

For simplicity/ease of notations.

推廣 generalize/extend

斷言 claim/assert/assertion

answer the question in the affirmative/negative.

說明 illustrate/elaborate

To sum up/conclude.

clutch

seduce

assume 顯露(特徵)

succinctly

fourscore

dictum

crude

rigor

from scratch

culminate

a priori/ a posteriori 先驗後驗

compelling

\(\heartsuit \spadesuit \clubsuit \diamondsuit\)

\(\varpi\)

在 Introduction 結尾，描述該文結構

  • The plan of the paper is as follows.

  • The rest of the paper is organised as follows.

defer

公務郵件

Title：Siran Li request for recommendation letter

Sent to: Prof Tao

cc'ed Hua Li (抄送自己一份留底)

Dear Prof. Tao,

​ My name is Hua Li. I took your course "Mathematical Analysis" in Fall Semester 2022-23. This was my first course on analysis, and I enjoyed it immersely. I got 98/100 in the final exam and ranked the first in your class. (最近沒有頻繁聯絡需要介紹自己與收信人關係)

​ I am now applying for graduate school/Ph D programmes in both China and the US (寫明地點，因為各地推薦信內容不一). I am writing to ask if you would like to write a recommendation letter on my behalf/ in support of my application.

​ I am applyig for 12 schools this time (PKU, SJTU, FDU, SUST, Havard, Princeton, Yale, NYU(master), xx, ....). The deadline for the letter is 5th July, 2024. (給足資訊)

​ Many thanks for your time and your consideration! If there is any more information I should provide, please do not hesitate to contact me at any time.

​ Yours Cordially/Sincererly/With Best Regards,

Hua Li

尊敬的陶教授：

​ 您好。我是上海交大數學大二學生李華，正在參考您的教材《數學分析》進行學習。關於第三章定理三（如下）

。。。。

我有一處不明向您請教：為何 \(K\) 必須為緊？

​ 盼您百忙之中撥冗回覆。非常感激！

祝好，

​ 李華

寫證明：

  • 善用 Claim。
  • data-ink ratio

\(\chi\)

善用 newcommand。

\newcommand\e\epsilon
\allowdisplaybreaks[4]

\newtheorem*{theorem*}{Theorem} % 不參與編號

\bibliography{bib file}

\cite[p. 133]{xxx, yyy, zzz}

\nonumber\\
\tag{$\clubsuit$}

\(\newcommand\e\epsilon\)

\(\e\)

~ 連線號

\[\nabla^{\perp} \S \\ \natural \flat \\ \Subset \]

\(\Bigg( \Huge( \bigg( \Big( \big(^\top\)

\textsc small caps

word：12號字 兩倍行距

空格

\qquad \quad \, \.

\(a\qquad b\)

\(a\quad b\)

\(a\,b\)

\(a\.b\)

\(\P \S \dagger \ddagger \copyright \AA \O \ss \pounds \i \j\)

\(u_{\text{NS}}^3\)

這樣的 \(\ell\) 以避免混淆 \(l1\).

\(\| a\|\)

\(\wp\)

\(\surd\)

\(\top\bot \vdash \dashv\)

\(\bigsqcup \odot \biguplus\)

\(\setminus\) 和 \(\backslash\)

\(\ll\)

\(\sqsubseteq\)

\(\lesssim \simeq \approx \cong \bowtie \asymp\)

\(\mbox{hhhh}\)

\begin{eqnarray*}

\(a\hspace{300pt}3\)

\(\vspace{20mm}\)

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