【筆記】學術英語寫作

學術英語寫作（LaTeX 入門）

小學期修了學術英語寫作，老師是我們數分三的老師（啊這）。以下是課堂筆記匯總

analogy 類比

constitute 組成

attenuate 減少

convention 公約

referee 審閱

sanity check

Overview

完整。把數學符號翻譯成英文后，行文應當符合英文語法。
簡潔。Don't show.
邏輯。Motivation。
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

完整準確羅列條件和限制。
最核心的結論盡量在突出位置一句話凸顯。
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：公式后記得打標點符號。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:

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

Comment：冒號和分號后小寫，列舉最后兩項間的 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。

\(A\) 為緊致集；

\(A\) 為序列緊致集。

\(A\) 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。多個條件時可以把后面的條件 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，雜志名縮寫（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

（改變運算優先級的）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(特征)

succinctly

fourscore

dictum

crude

rigor

from scratch

culminate

a priori/ a posteriori 先驗后驗

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. (最近沒有頻繁聯系需要介紹自己與收信人關系)

? 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。關于第三章定理三（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復。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{$\clubsuit$}

\(\newcommand\e\epsilon\)

\(\e\)

~ 連接號

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

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

\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*}

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

\(\vspace{20mm}\)

posted @ 2024-07-23 11:54 Imakf 閱讀(150) 評論(0) 收藏舉報

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