How to write papers in English?


(Here I collected a few expressions/templates/patterns and pieces of advice that may be useful for novice writers of scientific [mainly mathematical] papers. My comments are in parenthesis; part of these and all the expressions outside parenthesis have been taken from native English speakers [who are to be followed]. Any comments, additions, useful links [like the one by D. Knuth] or references are welcome.) This color will denote the notion we are treating right now, e.g. Title is the topic to be considered first.


(Make the title as informative as possible, but not lengthy.
Note that titles may be used in information retrieval; therefore every word counts.
Include relevant important names into the title.
Do not use formulae in the title.)

Running title

Abstract, Summary

(Avoid using formulae in the abstract.
Do not refer to the reference list of the paper in the abstract, because
the abstract should be self-contained and appropriate to be used in data bases.)

Keywords, Key words

(Both spellings exist.)

AMS subject classifications, Mathematics Subject Classification

(Use as many classes as allowed to be informative.)


(The first paragraph of the introduction should be comprehensible to any mathematician.)
In this paper, we are concerned with the following question:
At the outset, it is important to realize ...
My delightful task here will be to survey the startlingly wide range of applications of ... and to sketch the many varied and interesting proofs of ...

Review of the literature

There have been a number of contributions dealing with different aspects of this problem, see e.g. ...
(Even if you are a mathematician it is not prohibited for you to read before writing.)


(Such a section or subsection is almost obligatory.) An outline/The structure of the paper is as follows.

Notations, Preliminaries

We begin with some definitions.
... in the sequel, we shall work with this formulation.


(The statement of a theorem should precede its proof.)
We shall now state two theorems on ...
These theorems are minor restatements of results obtained in ...
Let the function f satisfy the hypotheses of Theorem 1.
It should be noted that although this result covers a very wide range of problems, there are some standard, simple examples for which uniqueness fails.
If we imposed two conditions, Theorem 1 would imply that we would have a unique solution.
Let f be analytic on D and continuous on the closure of D.


(Strive for proofs that are conceptual rather than computational.)
Part 1 is merely a rephrasing of the definition.
The reader is encouraged to prove this using Cauchy's integral formula.


X is said to be a Hilbert space if ...

Results, conclusions, perspectives, discussion

Problems with features like this arise in the model(l)ing of ... (One 'l': American spelling, two 'l's: British spelling.) (Even if you write a mathematical paper this part is very helpful.)


(Footnotes are generally considered to be avoidable if it is possible at all.)

Equations, formulae

(We suggest that you number only those equations which are referred to elsewhere in the text.)
(It is poor style to use =, ∈, ⇒, →, &, or ∀, ∃ etc. in the text unless they are absolutely essential.)

Useful and harmful expressions

A few useful patterns

That being the case, ...
The set X endowed with this topology ...
Without loss of generality ...
Upon multiplying both sides with ...
Straightforward calculation shows ...


By and large.Taking everything into account.
End up.Reach a state eventually.
In that.In so far as.
It goes without saying.Something so obvious that it needn't be said.
On the other hand.From the other point of view.
On the whole.In general, ignoring minor details.
Over and above.In addition to.
Rule of thumb.A rule based on experience and estimation rather than precise calculation.
Start from scratch.To start from the beginning with no help.

Expressions to avoid

To be replacedBy this
by means ofby
conduct an investigationinvestigate
due to the fact thatsince/because
in a position tocan
in order toto
in the case thatif
in the course ofduring
in the event thatif
in the first placefirst
it is apparent thatapparently
it may happen thatthere may/might
most uniqueunique
take into considerationconsider
that is to sayin other words, that is
to begin withto begin

Vocabulary of


boundary value problem
closed set
fundamental theorem of calculus
initial value problem
inverse function theorem
open set
system of equations


column vector
direct sum
divisor of zero
linear combination
linearly independent
linear operator
quotient space
row vector
selfadjoint operator
similarity transformation

Probability theory and mathematical statistics

absolutely continuous
Borel field
conditional expectation
(Lebesgue) dominated convergence theorem
elementary event
measurable space
probability density function
probability distribution function (previously: cumulative distribution function)
probability measure
probability space
random variable


(Books, conference proceedings: Title is in Italics.
Papers: Journal name is in Italics.)


Higham, N. J.: Handbook of writing for the mathematical sciences, SIAM, Philadelphia, 1993.
Lyle, K. M.: The author's guide to the applied probability journals, Applied Probability Trust, Sheffield, 1979.
Steenrod, N. E. (ed.): A manual for authors of mathematical papers, American Mathematical Society, Providence, RI, 1984.
Steenrod, N. E.; Halmos, P. R.; Schiffer, M. M.; Dieudonné, J. A.: How to write mathematics, American Mathematical Society, Providence, RI, 1981.


(Formal acknowledgements of financial assistance or sponsorship; personal acknowledgements, such as thanks to colleagues, thesis supervisors or referees.) I thank my colleagues X and Y, who contributed directly to the contents and approach of the book by reviewing chapters and suggesting changes.
This manuscript has not been previously published in any language anywhere and it is not under simultaneous consideration by another journal. This paper is in final form and no version of it will be submitted for publication elsewhere.


This work was supported by the National Scientific Research Foundation under grant No. T037491. Mr. András Rácz was the first to share his views on this text. Will you be the second, please?