【笔记】学术英语写作

小学期修了学术英语写作，老师是我们数分三的老师（啊这）。以下是课堂笔记汇总

analogy 类比

constitute 组成

attenuate 减少

convention 公约

referee 审阅

sanity check

Overview

1. 完整。把数学符号翻译成英文后，行文应当符合英文语法。

2. 简洁。Don't show.

3. 逻辑。Motivation。

4. 向大师学习。

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Definition

• 本质： if and only if.（或者只写 if）

• what is what. 可能的陈述方式：把满足 x 的 y 叫做 z。

下面给出例子。

例子 1：（数学分析）

定义：设 $B \sub \R^2$。如果对任意给定的 $\epsilon > 0$ 存在可数个闭矩形序列 $\{I_i\}$（$i=1,2,\cdots$）使得 $B\sub \cup_{i=1}^{\infty} I_i$，$\sum_{i=1}^{\infty} \sigma(I_i) <\epsilon$，则称 $B$ 为（二维）零测集。

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 从句。善用逗号。引入了新的记号必须在附近进行解释。

例子 2：（数学分析）

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：（表示论）

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：也就是说可以用 "That is," 或者 "i.e.," 表示。transformation (US) transform (UK)。重要的公式可以居中。非必要不写记号，如 $\forall$ 和 for all。

例子 4：（随机过程）

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：当其他内容太多时，把要定义的东西提前。可以使用领域公认的缩写（如本例中 Independent and identically distributed 缩写成 i.i.d.）。

例子 5：（数学分析）

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：避免用数学符号作为一句话的开头。不要引入不必要的记号。

Theorem

1. 完整准确罗列条件和限制。
2. 最核心的结论尽量在突出位置一句话凸显。
3. 上下文安排。 *必要时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：公式后记得打标点符号。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：冒号和分号后小写，列举最后两项间的 and.

Proof of Theorem

有一些固定证明思路：Contradiction, induction, cases...

例子 1：（反证）

命题：设 $A$ 为 $\R^n$ 中子集，则以下等价。

1. $A$ 为紧致集；
2. $A$ 为序列紧致集。
3. $A$ 为有界闭集。

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 中可以使用虚拟语气（Europe）。用以下来代替 according to：By/ in view of/ by virtue of/ thanks to。可以用括号表示原因。特别地使用 in particular。

归纳法：

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.

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designate

predicate

preliminary

cornerstone

manuscript

compromise 损害

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 记号. 或者 Denote by 记号 sth。多个条件时可以把后面的条件 if 显式写出。

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 表递进。表最终结果用 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. (解释 weaker 这种不精确的词语) More precisely, ...

不用 Obviously。It is clear that.

引用/参照

• (see [14])

• (cf [14])

• [14]

• See [14].

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

• See [1] by Bourgain-Kenig-Tao.

[编号] 作者名（首字母排序）文章标题，杂志名缩写（斜体）卷号（加粗）年份（括号中），页码（或者文章号）.

常用短语

sufficiently large

well justified

easy/simple/straightforward

readily

deriavtive

（改变运算优先级的）括号 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.

推广 generalize/extend

断言 claim/assert/assertion

answer the question in the affirmative/negative.

说明 illustrate/elaborate

To sum up/conclude.

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clutch

seduce

assume 显露(特征)

succinctly

fourscore

dictum

crude

rigor

from scratch

culminate

a priori/ a posteriori 先验后验

compelling

$\heartsuit \spadesuit \clubsuit \diamondsuit$

$\varpi$

在 Introduction 结尾，描述该文结构

• The plan of the paper is as follows.

• The rest of the paper is organised as follows.

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defer

公务邮件

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. (最近没有频繁联系需要介绍自己与收信人关系)

​ I am now applying for graduate school/Ph D programmes in both China and the US (写明地点，因为各地推荐信内容不一). 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. (给足信息)

​ 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

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尊敬的陶教授：

​ 您好。我是上海交大数学大二学生李华，正在参考您的教材《数学分析》进行学习。关于第三章定理三（如下）

。。。。

我有一处不明向您请教：为何 $K$ 必须为紧？

​ 盼您百忙之中拨冗回复。非常感激！

祝好，

​ 李华

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写证明：

• 善用 Claim。

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• data-ink ratio

$\chi$

善用 newcommand。

\newcommand\e\epsilon
\allowdisplaybreaks[4]

\newtheorem*{theorem*}{Theorem} % 不参与编号

\bibliography{bib file}

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

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

$\newcommand\e\epsilon$

$\e$

~ 连接号

$$\nabla^{\perp} \S \\ \natural \flat \\ \Subset$$

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

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\textsc small caps

word：12号字 两倍行距

空格

\qquad \quad \, \.

$a\qquad b$

$a\quad b$

$a\,b$

$a\.b$

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

$u_{\text{NS}}^3$

这样的 $\ell$ 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*}

$a\hspace{300pt}3$

$\vspace{20mm}$