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42

This tendency of mathematicians is so well-known and universal that it has been taken as an axiom. See Andrew Appel's seminal work establishing whether different computer science conferences are mathematics or science.


39

In a paper ``Is the null-graph a pointless concept?" Harary and Read examine reasons for assigning certain properties to the empty graph. They observe that from the enumeration perspective it appears to be convenient to consider the empty graph as a forest, but not a tree.


31

I had always heard that there was a famous counterexample to alphabetization, the Zucker-Cox Theorem (where they flipped the order for obvious reasons), but apparently the non-alphebetization in this case was apocryphal. But indeed, non-alphabetization is very rare.


31

Although just beyond your 50-year scope, this may be of interest. Among the series $\mathsf A_n, \mathsf B_n, \mathsf C_n, \mathsf D_n$ in the Cartan-Killing classification of simple Lie groups, everyone (I believe) always agreed to call $\mathsf A_n$ the special linear group, $\mathbf{SL}(n)$, and $\mathsf B_n$ and $\mathsf D_n$ the special orthogonal ...


24

The subject known for decades as recursion theory, studying the class of recursive functions and the recursively enumerable (r.e.) sets and degrees, is now known almost universally, especially amongst the newer generation, as computability theory, studying the computable functions and the computably enumerable (c.e.) sets and degrees. This change, led by ...


23

This question often arises also in the promotion issues, since faculty in other sciences (esp. biological and medical sciences) and humanities have a very different approach to listing the co-authors. I am not totally sure, but I think mathematics is more of an exception than the rule in this respect for whatever cultural reasons. This "ordering" problem ...


23

The whole discussion seems to devolve on whether the empty graph (or empty space) should be considered "connected". Angelo and I are of the school that it should not, but this should be explained since some of the traditional definitions of "connected" apparently allow the empty space to be connected. A general abstract context is as follows. Let $C$ be a ...


22

In general, I think that alphabetical order is very common. However, sometimes this should be alphabetical order in other language, and in english translation this order becomes different. For example, take several papers by Vershik and Kerov - the Russian alphabetical order is VK, but in english this is not alphabetical.


18

I just write $\displaystyle \prod_{p \text{ prime}}$.


18

I feel the answer is NEVER. You must describe the content of the article, not the length. Some journals publish notes separately from regular papers, and often even encourage their submission by offering speedy refereeing and publication (even the Annals encourages "short", i.e. under 20 pp. papers). Anyhow, if your paper is published as a note it will ...


18

An example of a failure to change notation is the movement by Eilenberg, Jacobson, Herstein and others to replace function notation $f(x)$ with $xf$ and then have the composition $fg$ mean first do $f$ and then $g$. The notation has the advantage that diagrams $X\xrightarrow{f}Y\xrightarrow{g} Z$ don't have to be flipped around. It also has the property ...


16

To answer the actual question, I don't know any standard symbol; I've seen $P$, $\mathbb{P}$ and $\mathtt{PRIMES}$. (The last seems more common in the CS-literature, such as this famous paper.) I would like to use this as an opportunity to make my standard plea for using multi-letter symbols; and to argue in this case for $\mathtt{PRIMES}$. There are more ...


16

It is always interesting when I meet professors in other sciences, particularly biology, to see their reaction when the issue of author names on papers comes up. The last time this happened, I was speaking to a cancer researcher at Harvard medical school. When I told him that author names in math are universally in alphabetical order his eyes got really, ...


16

As far as "a place putting this down formally," see the AMS's 2004 "Information Statement on Joint Research and its Publication" at http://www.ams.org/profession/leaders/culture/culture. The AMS has several of these "culture statements," intended to "highlight the ways in which the traditions in mathematics differ from those in other disciplines." I.e. ...


14

My understanding (as someone who hasn't been in this business very long) is that when pure mathematicians co-author a paper, they form a kind of partnership as equal partners, and all credit for everything in the paper goes to the partnership rather than individuals, regardless of what actually happened behind the scenes. As for why: It is seen to be ...


14

This convention is not universal in mathematics, and it's annoying to some of us. See this corrigendum from Inventiones Mathematicae. "On page 79, line 24 from the bottom and on page 110, lines 21, 13, 7 and 5 from the bottom, replace "first" by "second". Editor's note. In the original manuscript the order of the authors was I. Rivin first, C. ...


14

Something is connected if the number of its connected components is equal to one. That being said, in one of my papers, I have sometimes felt the need to say that an object $X$ was either connected or empty. The language I eventually decided to use was: $$ \text{``Let $X$ be a connected possibly empty ...''} $$


12

I think it's generally bad form to have a corollary dependent on an earlier conjecture. I recommend one of the following: Theorem: Assuming Conjecture A, properties X, Y and Z are true. or Theorem: Conjecture A implies X, Y and Z. Most importantly, it should be crystal clear that the result is dependent on the conjecture.


11

I don't consider the empty graph to be a tree, or a connected graph, because I prefer the following definition of connectedness: A graph $G$ is connected if, whenever it is the disjoint union of a family of graphs, then one of the graphs in that family is $G$ itself. The empty set does not satisfy this, because it is the disjoint union of the empty family. ...


10

One practice which supports our practice of listing authors alphabetically is our practice of setting a fairly high bar for what counts as sufficient contribution to merit coauthorship. I am only exaggerating slightly when I say that in some disciplines people become coauthors merely for sitting in on meetings where the paper was discussed. Certainly, in ...


10

I've read $\mathbb{P}$ many times and also use it.


10

Comments. For my book Classics on Fractals I published translations of various relevant papers. I wrote to the copyright owners (such as learned societies who published the journals) for permission to do this. (This was back in the Olden Days, 20 years ago, when email was not as common as today.) My publisher (Addison-Wesley) gave me the wording to use ...


9

Roger Penrose's abstract index notation for tensors is a relatively modest example, but I think it fits all the criteria of the question. Around 1952, Penrose invented a personal graphical notation for tensors and tensor operations such as contraction and covariant derivatives. It's been described as "fornicating ostriches;" variations on it are referred to ...


9

In Bourbaki's terminology, the empty graph is a tree - cf. LIE.IV.Annex.3.


9

I would write "Proposition Z: If X holds, then Y is true." Even if the deduction of Y from X were trivial, I think labelling this a corollary would be confusing. (After all, what is the statement "X implies Y" a corollary of?) However, I wouldn't have a problem writing something like "as we saw above, Y would be a corollary of X" later on. (The subjunctive ...


9

I'm reminded of the following story that I posted on my personal web journal a couple years ago: At the Topology seminar yesterday, the speaker presented a theorem, which he immediately followed with a refinement: a statement that directly and obviously implies the theorem. He labeled his refinement a "corollary". I turned to Noah Snyder, and said that ...


9

A famous (and rare) counterexample is the Rivest-Shamir-Adleman paper on public-key cryptography, which gave us the name RSA cryptosystem. Maybe someone can tell us the reason for this ordering of authors' names.


8

As in other answers and comments: context usually suffices to explain that $p$ is a prime, whether in the rational integers or whatever. That is, when possible, no notation at all is clearer (and less bulky and visually noisy) than any possible notation. Similarly, as I was slow to learn, objects' notations need not make explicit reference to every ...


8

In most cases that I am familiar with, successful changes to terminology were accompanied by other more basic innovations. E.g. Grothendieck's language became standard in algebraic geometry in the late 1950's, because he successfully rewrote the foundations of the subject. An example of a proposed change for its own sake is the word "contrahomology" for ...


7

Making a new Theorem environment that let you have the bolded part say "Corollary to Conjecture X" seems to me a good compromise of concise and unlikely to confuse anyone.



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