【笔记】学术英语写作

标签：... sub sum 笔记 cdots Let 写作 pi 英语

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

完整。把数学符号翻译成英文后，行文应当符合英文语法。
简洁。Don't show.
逻辑。Motivation。
向大师学习。

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

完整准确罗列条件和限制。
最核心的结论尽量在突出位置一句话凸显。
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:

if ..., then ... ;
if ..., then ... ; and
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。

\(A\) 为紧致集；

\(A\) 为序列紧致集。

\(A\) 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.

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.

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.

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

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

写证明：

善用 Claim。

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

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

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

\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*}

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

\(\vspace{20mm}\)

标签：...,sub,sum,笔记,cdots,Let,写作,pi,英语
From： https://www.cnblogs.com/imakf/p/18318013