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This is a general question about mathematical writing especially for writing research papers and the like.

Question: Do you precede an equation with a comma or colon?

Example A:

  1. The following equation is the Yosida-Hawking-Penrose-Dantzig function \begin{equation} f(x) = \frac{1}{2} D_\alpha(x,y) \end{equation}
  2. The following equation is the Yosida-Hawking-Penrose-Dantzig function, \begin{equation} f(x) = \frac{1}{2} D_\alpha(x,y) \end{equation}
  3. The following equation is the Yosida-Hawking-Penrose-Dantzig function: \begin{equation} f(x) = \frac{1}{2} D_\alpha(x,y) \end{equation}

Example B:

  1. In fact, we can express the earlier function using a much simpler expression \begin{equation} f(x) = \varphi(x,y) \end{equation} where $\varphi(x,y)$ is the Demiane functional
  2. In fact, we can express the earlier function using a much simpler expression, \begin{equation} f(x) = \varphi(x,y) \end{equation} where $\varphi(x,y)$ is the Demiane functional
  3. In fact, we can express the earlier function using a much simpler expression: \begin{equation} f(x) = \varphi(x,y) \end{equation} where $\varphi(x,y)$ is the Demiane functional

Can anyone comment on which one is the best practice?

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  • $\begingroup$ I would go for colons in first and comma in the second, at the very least use full stops after equations. this amongst many others may be helpful, but at the same time, should it be moved to a chat or flagged as duplicate? $\endgroup$ – mdave16 Apr 2 '17 at 3:43
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    $\begingroup$ Tangential comment, but the equation and the function are two different things (they are not even the same type of thing), so I might avoid saying that the equation "is" the function. $\endgroup$ – littleO Apr 2 '17 at 3:47
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    $\begingroup$ This has also been discussed at MathOverflow. $\endgroup$ – pjs36 Apr 2 '17 at 3:55
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    $\begingroup$ You should have "following" instead of "follow". ​ ​ $\endgroup$ – user57159 Apr 2 '17 at 4:57
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    $\begingroup$ I agree with what the others have said, but if you precede the equation with a comma as in the second example and then continue the sentence after the equation, you should put a comma after the equation itself also. This is hinted by Olivier's answer. $\endgroup$ – Darren Ringer Apr 2 '17 at 19:04
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Generally, I would treat the equation as if it were any ordinary noun phrase, and use the usual rules for comma, colon, or no punctuation.

A colon is used if the equation is an elaboration, or an item. So, just as you might write

Lips are characterized by the following properties: fleshy, paired, red.

you would write

An ellipse is characterized by the following equation:

$$ \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1 $$

A comma precedes a non-restrictive clause (one that describes rather than identifies the noun phrase), so by analogy with

The line can be assigned to a simpler character, Polonius.

we might write

A line can be described with a simpler equation,

$$ y = mx+b $$

In comparison, with a restrictive clause, we use no comma, so just as we would write

From this, the oiler obtained the formula CH$_3$C$_6$H$_4$C$_2$H$_5$.

we would also write

From this, Euler obtained the formula

$$ e^{i\pi}+1 = 0 $$

I suspect there aren't any hard and fast rules for this, however. Whatever you choose to do, be consistent and reasonable.


ETA: You'll notice that I have no periods at the ends of these equations. The papers I have generally (though not universally) observe this pattern. However, in other fields, equations may have ending punctuation depending on how they occur within a sentence. It may be useful for a writer to consult the publication's style guide, if applicable, or at least examine previous articles within the same publication or outlet.

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    $\begingroup$ Good answer. But are you not forgetting the dots at the end of the sentences? :^) $\endgroup$ – Olivier Apr 2 '17 at 4:02
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    $\begingroup$ What about periods at the end of the equations if they end the sentence? Like $$e^{i\pi}+1=0.$$ $\endgroup$ – Ruslan Apr 2 '17 at 6:39
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    $\begingroup$ @BrianTung I have never seen periods not used after equations ending a sentence (examples of what I read are given in my answer). Can you provide an example of this from what you consider an exemplary publication? $\endgroup$ – Olivier Apr 2 '17 at 21:06
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    $\begingroup$ @Olivier: I'm in computer science. I just looked over some papers that appeared in IEEE Security and Privacy, and lo and behold, it's a mixed bag. About one paper in five seems to have the periods at the end. That confirms the notion that it's a matter of what papers one reads. I'll edit my answer accordingly. Thanks for the tip! $\endgroup$ – Brian Tung Apr 3 '17 at 0:44
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    $\begingroup$ @YashasSamaga Euler's last name is pronounced exactly like "oiler" (at least in the US, where the uninitiated often pronounce it "yoo-ler"), so the last pair of examples is funny to me (maybe "pun" isn't exactly the right word). $\endgroup$ – pjs36 Apr 4 '17 at 3:27
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Equations should be included as part of the sentence, as in the following.

Consider the Yosida-Hawking-Penrose-Dantzig function $$ f(x) = \frac{1}{2}D_\alpha(x,y). $$

It can also be expressed as $$f(x)=\phi(x,y),$$ where $\phi(x,y)$ is the Demiane functional.

In example B, if you don't want to change the sentence:

In fact, we can express the earlier function using the much simpler expression $$f(x)=\phi(x,y),$$ where $\phi(x,y)$ is the Demiane functional.

I use this style example. Milnor's book, referenced in the comments below, follows the same rules, as do all the references and publications in my bookshelf (Polya's Problems and Theorems in Analysis, Rudin's Real and Complex analysis, etc.)

Standard guides

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    $\begingroup$ I think you are wrong about no having punctuation. This is much like writing "I had dinner with Stephen Hawking, a scientist." We wouldn't remove that comma. The grammar of the colon also allows it here. $\endgroup$ – Thomas Andrews Apr 2 '17 at 3:57
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    $\begingroup$ I'm not sure why you are obsessed with adjectives. My example doesn't use any adjectives, either. It still requires a comma. $\endgroup$ – Thomas Andrews Apr 2 '17 at 4:19
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    $\begingroup$ For what it's worth, your principle is correct (the formula is part of the sentence that surrounds it and punctuation should be applied accordingly) and your last two examples look fine to me. But sentences sometimes contain commas or colons, and sometimes those occur just before a formula. I would put a comma after "function" in your first example. $\endgroup$ – David K Apr 2 '17 at 4:52
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    $\begingroup$ I think your first example is standard, grammatically correct, and clear. I read such statements often in math writing. For example I think a similar statement appears on p. 35 of Milnor's book Topology from the Differentiable Viewpoint: "The sum ... is equal to the Euler number $\chi(M) = \sum_{i=0}^m (-1)^i \, \text{rank} \, H_i(M)$." (The formula for $\chi$ is displayed on its own line, as is the formula for $f$ in your first example.) $\endgroup$ – littleO Apr 2 '17 at 4:57
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    $\begingroup$ @ThomasAndrews rather "Consider the scientist Stephen Hawking. It can also be expressed as something, where whatever". Nevertheless, the OP started with examples conatining "follow[ing]". There, I'd use a colon: "The following person is a scientist: Stephen Hawking." $\endgroup$ – Hagen von Eitzen Apr 2 '17 at 22:00
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No punctuation should be used between the word "function" and an immediately following expression that defines it, unless "the function" has already been defined and its expression in the text is just a convenient reminder of its definition. Exactly the same applies in ordinary Language. Compare, for example the following two sentences:

My sister Laura lives in London.

My sister, Laura, lives in London.

The second sentence implies that "my sister" has already been defined, perhaps as the only sister or the sister who was previously discussed. In contrast, the first sentence makes no such implication: Laura, as far as we can tell from the sentence, might be any one of many sisters; but the sentence does specify which sister is being considered.

In general, the punctuation of mathematical writing should follow that of the corresponding natural language. In particular, a mathematical expression that ends a sentence should be followed by a full stop.

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This is what I remember from my technical writing class of twenty years ago.

The most important thing is that a colon must only be used at the end of a complete sentence.

For example

A line can be excpressed as: $$y=mx + b.$$

is incorrect, and

A line can be expressed as $$y=mx + b$$

and

The following equation is the slope-intercept equation of a line: $$y = mx + b$$

are correct

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For papers, it always depends on what is specified in the style guide for the publication or journal. Having said that, however, the guide is often not so specific as to whether the sentence with the equation should be punctuated with commas, colons or little flags.

For example, from the Monthly Notices of the Royal Astronomical Society

Mathematics

...

Equations should be punctuated as part of the sentence. Displayed equations are ranged left (i.e. no indent). Numbering of equations should follow the convention (1), (2)… throughout the whole paper, or (2.1), (2.2)… by section. Equations in appendices should be numbered (A1), (A2), (B1), etc.

This is indented, but still accepted: image from paper at https://arxiv.org/pdf/1401.2593.pdf

edit: added picture

image from paper at https://arxiv.org/pdf/1401.2593.pdf

Much seems to depend how particular/pedantic a reviewer is in relation to the style guide (which they may not know in such detail for specifics like this), and grammar conventions are not universal.

Check the style guide, read some of the published papers from the journal in question, and be consistent in application of grammar rules when inserting equations.

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I was told to use correct punctuation as defined by grammatical rules, but to always place the equation or formula on a line of its own, even if this meant starting a line with a punctuation mark. Even though the layout may look a little odd there can be no confusion as to the meaning. While it would be very rare for a comma to be misconstrued imagine the confusion an exclamation mark would cause.

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  • $\begingroup$ My experience is that an equation earns its own line in a paper only when it is complicated or important enough to merit the additional space. $\endgroup$ – Austin Mohr Apr 5 '17 at 19:23

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